Download - Rock slope engeneering (1)

Rock Slope Engineering

Rock Slope Engineering

Civil and mining

4th edition

Duncan C. Wyllie andChristopher W. Mah

First published 1974 by the Institute of Mining and MetallurgySecond edition published 1977Third edition published 1981

© The Institute of Mining and Metallurgy and E. Hoek and J. W. Bray2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN

Simultaneously published in the USA and Canadaby Spon Press270 Madison Avenue, New York, NY 10016

Spon Press is an imprint of the Taylor & Francis Group

© 2004 Duncan C. Wyllie and Christopher W. Mah

All rights reserved. No part of this book may be reprinted orreproduced or utilised in any form or by any electronic,mechanical, or other means, now known or hereafterinvented, including photocopying and recording, or in anyinformation storage or retrieval system, without permission inwriting from the publishers.

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

Library of Congress Cataloging in Publication DataRock slope engineering : civil and mining / Duncan C. Wyllie,

Christopher W. Mah — 4th ed.p. cm.

“Based on Rock Slope Engineering (third edition, 1981) byDr Evert Hoek and Dr John Bray.”

1. Rock slopes. I. Mah, Christopher W. II. Wyllie, Duncan C.,1933 — Rock slope engineering. III. Title.

TA706.W98 2004624.1′51363–dc22 2003014937

ISBN 0–415–28000–1 (hbk)ISBN 0–415–28001–X (pbk)

This edition published in the Taylor & Francis e-Library, 2005.

“To purchase your own copy of this or any of Taylor & Francis or Routledge’scollection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”

ISBN 0-203-49908-5 Master e-book ISBN

ISBN 0-203-57083-9 (Adobe eReader Format)

This book is dedicated to

Dr Evert Hoek and Dr John Bray

in recognition of their pioneering work in the field of rock slope engineering.

Contents

Introduction xviiiForeword xxiNotation xxiiNote xxiv

1 Principles of rock slope design 1

1.1 Introduction 11.1.1 Scope of book 21.1.2 Socioeconomic consequences of slope failures 3

1.2 Principles of rock slope engineering 41.2.1 Civil engineering 41.2.2 Open pit mining slope stability 5

1.3 Slope features and dimensions 81.4 Rock slope design methods 8

1.4.1 Summary of design methods 81.4.2 Limit equilibrium analysis (deterministic) 111.4.3 Sensitivity analysis 141.4.4 Probabilistic design methods 151.4.5 Load and Resistance Factor Design 20

2 Structural geology and data interpretation 22

2.1 Objectives of geological investigations 222.2 Mechanism of joint formation 242.3 Effects of discontinuities on slope stability 252.4 Orientation of discontinuities 252.5 Stereographic analysis of structural geology 27

2.5.1 Stereographic projection 272.5.2 Pole plots and contour plots 292.5.3 Pole density 31

viii Contents

2.5.4 Great circles 322.5.5 Lines of intersection 34

2.6 Identification of modes of slope instability 352.6.1 Kinematic analysis 372.6.2 Plane failure 382.6.3 Wedge failure 382.6.4 Toppling failure 392.6.5 Friction cone 392.6.6 Applications of kinematic analysis 40

2.7 Example Problem 2.1: stereo plots of structural geology data 432.8 Example Problem 2.2: slope stability evaluation related to structural geology 44

3 Site investigation and geological data collection 46

3.1 Planning an investigation program 463.1.1 Geology 473.1.2 Rock strength 483.1.3 Ground water 48

3.2 Site reconnaissance 483.2.1 Aerial and terrestrial photography 493.2.2 Geophysics 49

3.3 Geologic mapping 523.3.1 Line and window mapping 533.3.2 Stereogrammetric mapping of discontinuities 533.3.3 Types of discontinuity 533.3.4 Definition of geological terms 54

3.4 Spacing, persistence and roughness measurements 593.4.1 Spacing of discontinuities 603.4.2 Persistence of discontinuity sets 613.4.3 Roughness of rock surfaces 62

3.5 Probabilistic analysis of structural geology 643.5.1 Discontinuity orientation 643.5.2 Discontinuity length and spacing 65

3.6 Diamond drilling 673.6.1 Diamond drilling equipment 673.6.2 Diamond drilling operations 683.6.3 Core logging 693.6.4 Core orientation 71

4 Rock strength properties and their measurement 74

4.1 Introduction 744.1.1 Scale effects and rock strength 74

Contents ix

4.1.2 Examples of rock masses 754.1.3 Classes of rock strength 77

4.2 Shear strength of discontinuities 794.2.1 Definition of cohesion and friction 794.2.2 Friction angle of rock surfaces 814.2.3 Shearing on an inclined plane 814.2.4 Surface roughness 824.2.5 Discontinuity infilling 854.2.6 Influence of water on shear strength of discontinuities 88

4.3 Laboratory testing of shear strength 884.4 Shear strength of rock masses by back analysis of slope failures 904.5 Hoek–Brown strength criterion for fractured rock masses 92

4.5.1 Generalized Hoek–Brown strength criterion 954.5.2 Modulus of deformation 994.5.3 Mohr–Coulomb criterion 994.5.4 Rock mass strength 1004.5.5 Determination of σ′

3 max 1004.5.6 Estimation of disturbance factor D 101

4.6 Rock durability and compressive strength 1024.6.1 Slake durability 1024.6.2 Compressive strength 104

4.7 Example Problem 4.1: analysis of direct shear strength test results 1064.8 Example Problem 4.2: analysis of point load test results 107

5 Ground water 109

5.1 Introduction 1095.2 The hydrologic cycle 1105.3 Hydraulic conductivity and flow nets 111

5.3.1 Hydraulic conductivity 1115.3.2 Porosity 1135.3.3 Flow nets 113

5.4 Ground water flow in fractured rock 1145.4.1 Flow in clean, smooth discontinuities 1155.4.2 Flow in filled discontinuities 1165.4.3 Heterogeneous rock 1175.4.4 Anisotropic rock 1185.4.5 Ground water in rock slopes 118

5.5 Measurement of water pressure 1205.6 Field measurement of hydraulic conductivity 123

5.6.1 Variable head tests 1245.6.2 Pumping test 126

x Contents

5.7 Example Problem 5.1: Influence of geology and weather conditions onground water levels 127

6 Plane failure 129

6.1 Introduction 1296.2 General conditions for plane failure 1296.3 Plane failure analysis 129

6.3.1 Influence of ground water on stability 1336.3.2 Critical tension crack depth and location 1346.3.3 The tension crack as an indicator of instability 1346.3.4 Critical slide plane inclination 1366.3.5 Analysis of failure on a rough plane 137

6.4 Reinforcement of a slope 1386.4.1 Reinforcement with tensioned anchors 1386.4.2 Reinforcement with fully grouted untensioned dowels 1396.4.3 Reinforcement with buttresses 140

6.5 Seismic analysis of rock slopes 1416.5.1 Performance of rock slopes during earthquakes 1416.5.2 Seismic hazard analysis 1426.5.3 Ground motion characterization 1436.5.4 Pseudo-static stability analysis 1446.5.5 Newmark analysis 145

6.6 Example of probabilistic design 1486.7 Example Problem 6.1: plane failure—analysis and stabilization 150

7 Wedge failure 153

7.1 Introduction 1537.2 Definition of wedge geometry 1547.3 Analysis of wedge failure 1567.4 Wedge analysis including cohesion, friction and water pressure 1577.5 Wedge stability charts for friction only 160

7.5.1 Example of wedge analysis using friction-only charts 1707.6 Comprehensive wedge analysis 171

7.6.1 Data for comprehensive analysis 1717.6.2 Computer programs for comprehensive analysis 1747.6.3 Example of comprehensive wedge analysis 175

8 Circular failure 176

8.1 Introduction 176

Contents xi

8.2 Conditions for circular failure and methods of analysis 1768.2.1 Shape of slide surface 1778.2.2 Stability analysis procedure 178

8.3 Derivation of circular failure charts 1808.3.1 Ground water flow assumptions 1808.3.2 Production of circular failure charts 1818.3.3 Use of the circular failure charts 182

8.4 Location of critical slide surface and tension crack 1848.5 Examples of circular failure analysis 185

8.5.1 Example 1—China clay pit slope 1858.5.2 Example 2—highway slope 186

8.6 Detailed stability analysis of circular failures 1878.6.1 Bishop’s and Janbu’s method of slices 1888.6.2 Use of non-linear failure criterion

in Bishop stability analysis 1938.6.3 Example of Bishop’s and Janbu’s methods of analysis 1938.6.4 Circular failure stability analysis computer programs 1958.6.5 Three-dimensional circular failure analysis 1968.6.6 Numerical slope stability analysis 196

8.7 Example Problem 8.1: circular failure analysis 197

9 Toppling failure 200

9.1 Introduction 2009.2 Types of toppling failure 200

9.2.1 Block toppling 2009.2.2 Flexural toppling 2019.2.3 Block-flexure toppling 2029.2.4 Secondary toppling modes 202

9.3 Kinematics of block toppling failure 2049.3.1 Block shape test 2049.3.2 Inter-layer slip test 2049.3.3 Block alignment test 205

9.4 Limit equilibrium analysis of toppling on a stepped base 2059.4.1 Block geometry 2069.4.2 Block stability 2089.4.3 Calculation procedure for toppling stability

of a system of blocks 2109.4.4 Cable force required to stabilize a slope 2109.4.5 Factor of safety for limiting equilibrium analysis

of toppling failures 2119.4.6 Example of limit equilibrium analysis of toppling 2119.4.7 Application of external forces to toppling slopes 213

xii Contents

9.5 Stability analysis of flexural toppling 2149.6 Example Problem 9.1: toppling failure analysis 216

10 Numerical analysis 218

10.1 Introduction 21810.2 Numerical models 220

10.2.1 Joint material models 22110.2.2 Rock mass material models 221

10.3 Modeling issues 22310.3.1 Two-dimensional analysis versus three-dimensional analysis 22310.3.2 Continuum versus discontinuum models 22510.3.3 Selecting appropriate zone size 22610.3.4 Initial conditions 22610.3.5 Boundary conditions 22810.3.6 Incorporating water pressure 22910.3.7 Excavation sequence 23010.3.8 Interpretation of results 230

10.4 Typical stability analysis 23110.4.1 Rock mass failure 23110.4.2 Plane failure—daylighting and non-daylighting 23310.4.3 Wedge failure—daylighting and non-daylighting 23410.4.4 Toppling failure—block and flexural 23410.4.5 Flexural buckling failure 237

10.5 Special topics 23710.5.1 Reinforcement 23710.5.2 Time-dependent behavior 23910.5.3 Dynamic analysis 244

11 Blasting 245

11.1 Introduction 24511.2 Mechanism of rock fracturing by explosives 24611.3 Production blasting 247

11.3.1 Explosive properties 24811.3.2 Bench height 24911.3.3 Burden 25011.3.4 Blast hole diameter 25111.3.5 Nature of the rock 25111.3.6 Sub-drill depth 25211.3.7 Stemming 25211.3.8 Hole spacing 253

Contents xiii

11.3.9 Hole detonation sequence 25311.3.10 Fragmentation 25511.3.11 Evaluation of a blast 256

11.4 Controlled blasting to improve stability 25711.4.1 Pre-shearing and cushion blasting 25711.4.2 Drilling 25911.4.3 Explosive load 25911.4.4 Stemming 26011.4.5 Spacing and burden 26111.4.6 Hole detonation sequence 261

11.5 Blast damage and its control 26211.5.1 Damage from ground vibration 26211.5.2 Control of flyrock 27011.5.3 Control of air blast and noise 270

11.6 Example Problem 11.1: blast design 27311.7 Example Problem 11.2: controlled blasting design 27411.8 Example Problem 11.3: blast damage control 275

12 Stabilization of rock slopes 276

12.1 Introduction 27612.2 Causes of rock falls 27712.3 Rock slope stabilization programs 279

12.3.1 Planning stabilization programs 27912.3.2 Rock slope inventory systems 27912.3.3 Hazard rating criteria 28012.3.4 Database analysis of slope inventory 28212.3.5 Selection of high priority sites 28212.3.6 Selection of stabilization measures 283

12.4 Stabilization by rock reinforcement 28612.4.1 Shear keys 28712.4.2 Rock anchors 28712.4.3 Reaction wall 30112.4.4 Shotcrete 30112.4.5 Buttresses 30412.4.6 Drainage 30412.4.7 “Shot-in-place” buttress 306

12.5 Stabilization by rock removal 30712.5.1 Resloping and unloading 30812.5.2 Trimming 30812.5.3 Scaling 30812.5.4 Rock removal operations 309

12.6 Protection measures against rock falls 309

xiv Contents

12.6.1 Rock fall modeling 31012.6.2 Ditches 31212.6.3 Barriers 31312.6.4 Rock catch fences and attenuators 31612.6.5 Draped mesh 31712.6.6 Warning fences 31712.6.7 Rock sheds and tunnels 318

13 Movement monitoring 320

13.1 Introduction 32013.2 Types of slope movement 322

13.2.1 Initial response 32213.2.2 Regressive and progressive movement 32213.2.3 Long-term creep 324

13.3 Surface monitoring methods 32413.3.1 Crack width monitors 32513.3.2 Surveying 32613.3.3 Laser imaging 32713.3.4 Tiltmeters 32713.3.5 Global positioning system 32713.3.6 Synthetic aperture radar 327

13.4 Sub-surface monitoring methods 32713.4.1 Borehole probes 32813.4.2 Time–domain reflectometry 32813.4.3 Inclinometers 328

13.5 Data interpretation 32813.5.1 Time–movement and time–velocity plots 32913.5.2 Slope failure mechanisms 332

14 Civil engineering applications 334

14.1 Introduction 33414.2 Case Study I—Hong Kong: choice of remedial measures for plane failure 334

14.2.1 Site description 33414.2.2 Geology 33414.2.3 Rock shear strength 33514.2.4 Ground water 33514.2.5 Stability analysis 33514.2.6 Stabilization options 339

14.3 Case Study II—Cable anchoring of plane failure 34114.3.1 Site description 341

Contents xv

14.3.2 Geology 34214.3.3 Rock shear strength 34214.3.4 Ground water 34314.3.5 Earthquakes 34414.3.6 Stability analysis 34414.3.7 Stabilization method 34514.3.8 Construction issues 347

14.4 Case Study III—Stability of wedge in bridge abutment 34814.4.1 Site description 34814.4.2 Geology 34814.4.3 Rock strength 34914.4.4 Ground water 34914.4.5 Seismicity 35014.4.6 External forces 35014.4.7 Stability analysis 350

14.5 Case Study IV—Circular failure analysis of excavation for rock fall ditch 35214.5.1 Site description 35214.5.2 Geology 35314.5.3 Ground water 35314.5.4 Rock shear strength 35314.5.5 Ditch and slope design 35414.5.6 Construction issues 354

14.6 Case Study V—Stabilization of toppling failure 35414.6.1 Site description 35414.6.2 Geology 35514.6.3 Rock strength 35514.6.4 Ground water 35514.6.5 Stability conditions 35514.6.6 Stabilization method 356

15 Mining applications 357

15.1 Introduction 35715.2 Example 1—porphyry deposits 357

15.2.1 Design issues 35815.2.2 Engineering geology 35815.2.3 Rock strength and competency 35815.2.4 Hydrogeology 35915.2.5 Slope stability analyses and slope design 359

15.3 Example 2—stratigraphically controlled deposits 36115.3.1 Design issues 36115.3.2 Engineering geology 36115.3.3 Rock strength and competency 362

xvi Contents

15.3.4 Hydrogeology 36315.3.5 Structural domains 36315.3.6 Kinematic analyses 36315.3.7 Stability analyses 36415.3.8 Slope design concepts 36515.3.9 Preliminary design 367

15.4 Example 3—deep-seated deformation in a weak rock mass 36815.4.1 Design and operational issues 36815.4.2 Engineering geology 36915.4.3 Rock strength and rock mass competency 37015.4.4 Hydrogeology and slope depressurization measures 37015.4.5 Slope stability analyses 37115.4.6 Slope design and operational management 372

15.5 Example 4—overall slope design in a competent rock mass 37215.5.1 Design aspects and issues 37215.5.2 Engineering geology 37315.5.3 Rock strength and competency 37315.5.4 Hydrogeology 37315.5.5 Slope performance 37415.5.6 Slope stability analyses 37515.5.7 Implementation and ongoing evaluation 375

15.6 Conclusions 376

Appendix I Stereonets for hand plotting of structural geology data 377

I.1 Introduction 377I.2 Plotting poles 377I.3 Contouring pole concentrations 377I.4 Plotting great circles 377I.5 Lines of intersection 378

Appendix II Quantitative description of discontinuities in rock masses 381

II.1 Introduction 381II.2 Rock mass characterization parameters 381

II.2.1 Rock material description 381II.2.2 Discontinuity description 383II.2.3 Infilling description 389II.2.4 Rock mass description 389II.2.5 Ground water 392

II.3 Field mapping sheets 394

Contents xvii

Appendix III Comprehensive solution wedge stability 398

III.1 Introduction 398III.2 Analysis methods 398III.3 Analysis limitations 399III.4 Scope of solution 399III.5 Notation 400III.6 Sequence of calculations 400

Appendix IV Conversion factors 408

References 411Index 425

Introduction

Readers will undoubtedly recognize the similarity of this book to Rock Slope Engineering by Dr EvertHoek and Dr John Bray. We hope the following discussion of the origin and evolution of the currentbook will help to demonstrate the relationship between the two.

Rock Slope Engineering was published in three editions (1974, 1977 and 1981) by the Institute ofMining and Metallurgy in London. The original research for the book at the University of Londonwas sponsored by the mining industry in response to a need to develop design methods for increas-ingly deep open pits. The 1960s and 1970s had seen the development of a new generation of highproduction drills, shovels and trucks that made low grade ore deposits economical to mine, and therewas a consequent significant increase in the size of open pits. The investigation and design techniquesoriginally developed in Rock Slope Engineering for mines were soon adopted in civil engineeringwhere the slopes’ heights are usually less than those in open pits, but there is a need for a high level ofreliability in terms of both rock falls and overall stability. In response to the demand for a book thatclearly presents well-proven methods to design rock slopes, Hoek and Bray’s book has continued tosell steadily around the world, and has been translated into a number of languages.

In 1980, one of the authors of this book (DCW) was awarded a contract by the Federal HighwayAdministration (FHWA) in Washington to prepare a manual on rock slope design and constructionspecifically applicable to highways. At that time, I was working with Dr Hoek and he generously agreedthat his manuscript of Rock Slope Engineering could be adapted for this purpose. The manual closelyfollowed the original book, apart from chapters on slope stabilization and movement monitoring.A second FHWA contract was awarded in 1996 as part of an eleven module series on highwaygeotechnical engineering, and this opportunity was taken to embark on a major updating of themanual. The manuals have been used primarily as teaching material for a series of courses sponsoredby the National Highway Institute for highway engineers in the United States; to date over 40 courseshave been presented.

It was realized that a limitation of the FHWA manuals was their focus on highway engineering, andthat their availability was generally limited to course participants. Therefore, in 2001 it was decidedthat it would be worthwhile to produce another update that would cover the wider field of rock slopeengineering, including civil and mining applications. In order to take this step, it was necessary toobtain the permission and co-operation of a number of organizations and individuals—Mr JerryDiMaggio of the Federal Highway Administration, Dr G. P. Jayaprakash of the TransportationResearch Board, both in Washington, and Dr George Munfakh of Parsons Brinckerhoff Quade andDouglas (PBQD) in New York. We are most grateful for their assistance and encouragement.

Of course, the most important participant in this work has been Dr Evert Hoek who generouslyagreed that we could use Rock Slope Engineering as the basis for the new work. Since Dr Hoeklives in the same neighborhood, it has been possible to have a series of meetings to discuss both the

Introduction xix

overall approach and details of the contents and methods. We express our gratitude for the valuableassistance that we have received from Dr Hoek over the two years, as well as his pioneering workwith Dr Bray in establishing the fundamental procedures for rock slope engineering.

Dr Lorin Lorig and Mr Pedro Varona of Itasca Corporation and Mr Alan Stewart and his colleaguesat Piteau Associates Engineering have also made important contributions. Dr Lorig and Mr Varonawrote Chapter 10 on numerical analysis methods, and the personnel from Piteau wrote Chapter 15 oncase studies on open pit mining. We decided these two chapters were best written by persons workingin these specialist fields, and we appreciate their hard work and dedication.

One of our objectives in writing this book has been to maintain much of the original content of RockSlope Engineering, because the basic slope design methods that were developed for the 1974 editionare still valid to this day. Our approach has been to incorporate, within the original framework,technical advances and experience in rock slope design and construction projects over the past 30years. This has allowed us to maintain the structure of Rock Slope Engineering so that those who arefamiliar with Hoek and Bray can readily find their way around this book. In addition to generallyupdating the book, the following is a list of the major topics that have been added:

• Geological data collection—The International Society of Rock Mechanics nomenclature forstructural geology is included in Chapter 3 and Appendix II.

• Rock mass strength—The 2002 version of the Hoek–Brown rock mass strength criterion is includedin Chapter 4.

• Earthquakes—The effects of earthquakes on slope stability and design methods for slopes in seismicareas are described in Chapter 6.

• Probabilistic, and Load and Resistance Factor design methods—The basic principles of probabil-istic and LRFD design are discussed in Chapter 1; an example of probabilistic design is includedin Chapter 6.

• Numerical analysis—Chapter 10 is a new chapter describing the use of numerical analysis methodsin slope design.

• Production blasting—Updated methods of designing production blasts have been added toChapter 11, which covers in addition, blasting methods for final walls and control of damageoutside the blast area.

• Slope stabilization and rock fall protection—Chapter 12 is a new chapter describing rock slopehazard assessment, slope reinforcement and rock fall protection such as ditches, fences and sheds.

• Slope movement monitoring—Surface and sub-surface movement monitoring methods, and datainterpretation are described in Chapter 13.

• Case studies in civil engineering—Case studies of plane, wedge, circular and toppling failures aredescribed in Chapter 14.

• Case studies in open pit mining—Four case studies demonstrating pit slope design in a variety ofgeologic environments are described in Chapter 15.

It may be noticed that some material is very similar to that in the books Foundations on Rock and Land-slides: Investigation and Stabilization. This duplication is inevitable when discussing fundamentals ofrock mechanics that do not change significantly over time.

We would like to express our gratitude to a number of people, in addition to Dr Hoek who haveassisted us with this work. In particular, Ms Sonia Skermer and Mr George Gorczynski preparedthe illustrations, and Ms Glenda Gurtina prepared the original FHWA manual and has worked onmany of the complex portions of the text. We also appreciate the valuable input on seismic design byDr Randy Jibson and Dr Upul Atukorala, the photographs supplied by Rio Tinto Ltd and acknowledge

xx Introduction

our long associations with the Federal Highway Administration in Washington DC, the CanadianPacific Railway, and the British Columbia Ministry of Transportation.

Finally, we are most grateful to our colleagues, fellow geotechnical engineers and contractors withwhom we have worked over many years on a wide range of projects. This book attempts to distillour collectively acquired experience.

Duncan C. Wylliewww.wnrockeng.com

Christopher W. Mahwww.cmrockeng.com

Vancouver, Canada, 2003

Foreword

My work on rock slope engineering started more than thirty years ago while I was Professor ofRock Mechanics at the Imperial College of Science and Technology in London, England. A four-yearresearch project on this topic was sponsored by 23 mining companies around the world, and wasmotivated by the need to develop design methods for the rock slopes in large open pit mines whichwere becoming increasingly important in the exploitation of low-grade mineral deposits. Rock SlopeEngineering, co-authored with my colleague Dr John Bray, was first published in 1974 and thenrevised in 1977 and again in 1981.

While rock slope engineering remains an important subject, my own interests have shifted towardstunnelling and underground excavations. Consequently, when Duncan Wyllie suggested that he andChristopher Mah were willing to prepare a new book on rock slopes I felt that this would be anexcellent move. They have had a long involvement in practical rock slope engineering, mainly for civilengineering construction projects, and are familiar with recent developments in methods of analysisand stabilization. The evolution of their text, from Rock Slope Engineering to a manual on rock slopedesign for the US Federal Highway Administration, is described in their introduction and need not berepeated here.

The text that follows is a comprehensive reference work on all aspects of rock slope engineeringand, while it embodies all the original concepts of my work, it expands on these and introduces asignificant amount of new material, for both mining and civil engineering and several new case studies.As such, I believe that it will be an important source of fundamental and practical information forboth students and designers for many years to come.

I commend the authors for their efforts in producing this volume and I look forward to havinga copy on my own bookshelf.

Evert HoekVancouver, 2003

Notation

A Area of plane (m2)a Material constant for rock mass strength; ground acceleration (m/s2)B Burden distance for blast holes (m)b Distance of tension crack behind crest of face (m); joint spacing (m)Cd Dispersion coefficientc Cohesion (kPa)D Disturbance factor for rock mass strength; depth (m)d Diameter (mm)Em Deformation modulus for rock masse Joint aperture (mm)F Shape factorFS Factor of safetyG Shear modulus (GPa)GSI Geological Strength Indexg Acceleration due to gravity (m/s2)

H Height of slope, face (m)h Water level head above datum (m)i Roughness angle of asperities (degrees)JRC Joint roughness coefficientK Bulk modulus (GPa); hydraulic conductivity (cm/s−1); corrosion service life constant (µm),

rate of slope movement constantk Seismic coefficient; attenuation constant for blast vibrationsL Length of scan line, face, borehole (m)l Persistence of discontinuity (m); unit vectorm Unit vectormb Material constant for rock mass strengthmi material constant for intact rockN Number of joints, readingsn Unit vector, number of toppling blocksP Probabilityp Pressure (kPa)Q External load (kN)R Resultant vector; hole radius (mm)r radius of piezometer (mm)

Notation xxiii

S Shape factor; spacing distance for blast holes (m)SD Standard deviations Spacing of discontinuity (m); material constant for rock mass strengthT Rock bolt tension force (kN)t Time (s, yr)U Uplift force on sliding plane due to water pressure (kN)V Thrust force in tension crack due to water pressure (kN); velocity of seismic waves (m/s),

velocity of slope movementW Weight of sliding block (kN); explosive weight per delay (kg)X Loss of element thickness, corrosion (µm)z Depth of tension crack (m)zw Depth of water in tension crack (m)α Dip direction of plane (degrees)β Attenuation constant for blast vibrations�x Width of toppling slab, slice in circular failure (m)δ Displacement (mm)ε Coefficient of thermal expansion (per ◦C)φ Friction angle (degrees)γ Unit weight (kN/m3)

µ Poisson’s ratioσ Normal stress (kPa)σcm Compressive strength of rock mass (kPa)σci Compressive strength of intact rock (kPa)σ′

1 Major principal effective stress (kPa)σ′

3 Minor principal effective stress (kPa)τ Shear stress (kPa)ψ Dip of plane (degrees)υ Viscosity of water (m2/s)

Note

The recommendations and procedures contained herein are intended as a general guide, and prior totheir use in connection with any design, report, specification or construction procedure, they shouldbe reviewed with regard to the full circumstances of such use. Accordingly, although every care hasbeen taken in the preparation of this book, no liability for negligence or otherwise can be acceptedby the authors or the publisher.

Chapter 1

Principles of rock slope design

1.1 Introduction

A variety of engineering activities requireexcavation of rock cuts. In civil engineering,projects include transportation systems such ashighways and railways, dams for power produc-tion and water supply, and industrial and urbandevelopment. In mining, open pits account for themajor portion of the world’s mineral production.The dimensions of open pits range from areas ofa few hectares and depths of less than 100 m, forsome high grade mineral deposits and quarries inurban areas, to areas of hundreds of hectares anddepths as great as 800 m, for low grade ore depos-its. The overall slope angles for these pits rangefrom near vertical for shallow pits in good qual-ity rock to flatter than 30◦ for those in very poorquality rock.

Figure 1.1 shows two typical rock slopes.Figure 1.1(a) is a rock cut, with a face angleof about 60◦, supported with tensioned anchorsincorporating reinforced concrete bearing padsabout 1 m2 that distribute the anchor load onthe face. The face is also covered with shotcreteto prevent weathering and loosening between thebolts. Water control measures include drain holesthrough the shotcrete and drainage channels onthe benches and down the face to collect surfacerun-off. The support is designed to both ensurelong-term stability of the overall slope, and min-imize rock falls that could be a hazard to traffic.Figure 1.1(b) shows the Palabora open pit inSouth Africa that is 830 m deep and an overallslope angle of 45–50◦; this is one of the steepestand deepest pits in the world (Stewart et al., 2000).

The upper part of the pit is accessed via a dualramp system, which reduces to a single ramp inthe lower part of the pit.

In addition to these man-made excavations, inmountainous terrain the stability of natural rockslopes may also be of concern. For example, high-ways and railways located in river valleys may belocated below such slopes, or cut into the toe,which may be detrimental to stability. One ofthe factors that may influence the stability of nat-ural rock slopes is the regional tectonic setting.Factors of safety may be only slightly greater thanunity where there is rapid uplift of the land massand corresponding down-cutting of the water-courses, together with earthquakes that loosenand displace the slope. Such conditions exist inseismically active areas such as the Pacific Rim,the Himalayas and central Asia.

The required stability conditions of rock slopeswill vary depending on the type of project andthe consequence of failure. For example, for cutsabove a highway carrying high traffic volumesit will be important that the overall slope bestable, and that there be few if any rock fallsthat reach the traffic lanes. This will often requireboth careful blasting during construction, andthe installation of stabilization measures such asrock anchors. Because the useful life of such sta-bilization measures may only be 10–30 years,depending on the climate and rate of rock degrad-ation, periodic maintenance may be required forlong-term safety. In contrast, slopes for open pitmines are usually designed with factors of safety

2 Principles of rock slope design

(a) (b)

Figure 1.1 Examples of rock slopes: (a) rock slope in Hong Kong supported with tensioned rock anchors andreinforced concrete reaction blocks, and shotcrete (photograph by Gary Fu); and (b) 830 m deep Palaboraopen pit copper mine, South Africa. (Photograph courtesy: Rio Tinto Ltd.)

in the range of 1.2–1.4, and it is accepted thatmovement of the slope and possibly some partialslope failures will occur during the life of themine. In fact, an optimum slope design is one thatfails soon after the end of operations.

In the design of cut slopes, there is usually littleflexibility to adjust the orientation of the slopeto suit the geological conditions encountered inthe excavation. For example, in the design ofa highway, the alignment is primarily governedby such factors as available right-of-way, gradesand vertical and horizontal curvature. Therefore,the slope design must accommodate the partic-ular geological conditions that are encounteredalong the highway. Circumstances where geo-logical conditions may dictate modifications tothe slope design include the need for relocationwhere the alignment intersects a major landslidethat could be activated by construction. Withrespect to open pit slope design, the pit mustobviously be located on the ore body, and thedesign must accommodate the geological con-ditions that exist within the area of the pit.This may require different slope designs aroundthe pit.

The common design requirement for rock cutsis to determine the maximum safe cut face anglecompatible with the planned maximum height.

The design process is a trade-off between stabil-ity and economics. That is, steep cuts are usuallyless expensive to construct than flat cuts becausethere is less volume of excavated rock, less acquis-ition of right-of-way and smaller cut face areas.However, with steep slopes it may be necessaryto install extensive stabilization measures such asrock bolts and shotcrete in order to minimize boththe risk of overall slope instability and rock fallsduring the operational life of the project.

1.1.1 Scope of book

The design of rock cuts involves the collection ofgeotechnical data, the use of appropriate designmethods, and the implementation of excavationmethods and stabilization/protection measuressuitable for the particular site conditions. In orderto address all these issues, the book is dividedinto three distinct sections that cover respect-ively design data, design methods and excav-ation/support procedures. Details of the maintopics covered in each section are as follows:

(a) Design data

• Geological data of which structural geo-logy is usually the most important. Thisinformation includes the orientation of

Principles of rock slope design 3

discontinuities and their characteristicssuch as length, spacing, roughness andinfilling. Chapter 2 discusses interpret-ation of these data, while Chapter 3describes methods of data collection.

• Rock strength with the most importantparameter being the shear strength ofdiscontinuity surfaces or rock masses,and to a lesser extent the compressivestrength of the intact rock (Chapter 4).

• Ground water conditions comprise thelikely ground water level within theslope, and procedures to drain the slope,if necessary (Chapters 5 and 12).

(b) Design methods

• Design methods for rock slopes fall intotwo groups—limit equilibrium analysisand numerical analysis. Limit equilib-rium analyses calculates the factor ofsafety of the slope and different proced-ures are used for plane, wedge, circularand toppling failures; the type of fail-ure is defined by the geology of theslope (Chapters 6–9). Numerical ana-lysis examines the stresses and strainsdeveloped in the slope, and stabilityis assessed by comparing the stressesin the slope with the rock strength(Chapter 10).

(c) Excavation and stabilization

• Blasting issues relevant to slope sta-bility include production blasting, con-trolled blasting on final faces, and inurban areas the control of damage fromground vibrations, flyrock and noise(Chapter 11).

• Stabilization methods include rock rein-forcement with rock anchors and dowels,rock removal involving scaling and trimblasting, and rock fall protection meas-ures comprising ditches, fences and sheds(Chapter 12).

• Monitoring of slope movement is oftenan important part of slope manage-ment in open pit mines. Surface and

sub-surface monitoring methods arediscussed, as well as interpretation ofthe data (Chapter 13).

• Civil and mining applications are dis-cussed in Chapters 14 and 15, respect-ively, which describe examples of slopedesign, including stabilization methodsand movement monitoring programs.The examples illustrate the design pro-cedures discussed in the earlier chapters.

Also included in the book are a series of exampleproblems demonstrating both data analysis anddesign methods.

1.1.2 Socioeconomic consequences of slopefailures

Failures of rock slopes, both man-made andnatural, include rock falls, overall slope instabil-ity and landslides, as well as slope failures in openpit mines. The consequence of such failures canrange from direct costs of removing the failedrock and stabilizing the slope to possibly a widevariety of indirect costs. Examples of indirectcosts include damage to vehicles and injury to pas-sengers on highways and railways, traffic delays,business disruptions, loss of tax revenue due todecreased land values, and flooding and disrup-tion to water supplies where rivers are blocked byslides. In the case of mines, slope failures can res-ult in loss of production together with the cost ofremoval of the failed material, and possible lossof ore reserves if it is not possible to mine the pitto its full depth.

The cost of slope failures is greatest in urban-ized areas with high population densities whereeven small slides may destroy houses and blocktransportation routes (Transportation ResearchBoard, 1996). In contrast, slides in rural areasmay have few indirect costs, except perhaps thecosts due to the loss of agricultural land. Anexample of a landslide that resulted in severe eco-nomic costs is the 1983 Thistle Slide in Utah thatresulted in losses of about $200 million when thelandslide dammed the Spanish Fork River sever-ing railways and highways, and flooding the town


Failure-Visited minesFailure-ReviewFailure-Caving minesStable-Visited minesStable-ReviewAitik Today

Slo

pe h

eigh

t (m

)

100

200

300

400

500

600

700

800

900

10 20 30 50 60 70 80 9040

Overall slope angle (degrees)

0

(a) (b)Failure-River bank slopesFailure-Reservoir slopesFailure-Engineering slopesStable river bank slopesStable reservoir slopesStable engineering slopes

0 10 20 4030 50 60 70 80 90

200

400

600

800

1000

1400

1200

Slope angle (degrees)S

lope

hei

ght (

m)

Figure 1.2 Relationship between slope height and slope angle for open pits, and natural and engineered slopes:(a) pit slopes and caving mines (Sjöberg, 1999); and (b) natural and engineered slopes in China (data fromChen (1995a,b)).

of Thistle (University of Utah, 1985). An exampleof a landslide that resulted in both loss of life andeconomic costs is the Vaiont Slide in Italy in 1963.The slide inundated a reservoir sending a waveover the crest of the dam that destroyed five vil-lages and took about 2000 lives (Kiersch, 1963;Hendron and Patton, 1985).

A country that experiences high costs of rockfalls and landslides is Japan. This country hasboth highly developed infrastructure and steepmountainous terrain, and in addition, there arefrequent triggering events such as high rainfall,freeze–thaw cycles and ground shaking due toearthquakes. Documentation of major landslidesbetween 1938 and 1981 recorded total lossesof 4834 lives and 188,681 homes (Ministry ofConstruction, Japan, 1983).

1.2 Principles of rock slope engineering

This section describes the primary issues that needto be considered in rock slope design for civil pro-jects and open pit mines. The basic differencebetween these two types of project are that in

civil engineering a high degree of reliability isrequired because slope failure, or even rock falls,can rarely be tolerated. In contrast, some move-ment of open pit slopes is accepted if productionis not interrupted, and rock falls are of littleconsequence.

As a frame of reference for rock slope design,Figure 1.2 shows the results of surveys of theslope height and angle and stability conditionsfor natural, engineered and open pit mine slopes(Chen, 1995a,b; Sjöberg, 1999). It is of interest tonote that there is some correspondence betweenthe steepest and highest stable slopes for bothnatural and man-made slopes. The graphs alsoshow that there are many unstable slopes at flat-ter angles and lower heights than the maximumvalues because weak rock or adverse structure canresult in instability of even low slopes.

1.2.1 Civil engineering

The design of rock cuts for civil projects suchas highways and railways is usually concernedwith details of the structural geology. That is,


Figure 1.3 Cut face coincident with continuous, lowfriction bedding planes in shale on Trans CanadaHighway near Lake Louise, Alberta. (Photographby A. J. Morris.)

the orientation and characteristics (such as length,roughness and infilling materials) of the joints,bedding and faults that occur behind the rockface. For example, Figure 1.3 shows a cut slope inshale containing smooth bedding planes that arecontinuous over the full height of the cut and dipat an angle of about 50◦ towards the highway.Since the friction angle of these discontinuities isabout 20–25◦, any attempt to excavate this cutat a steeper angle than the dip of the beds wouldresult in blocks of rock sliding from the face onthe beds; the steepest unsupported cut that canbe made is equal to the dip of the beds. However,as the alignment of the road changes so that thestrike of the beds is at right angles to the cut face(right side of photograph), it is not possible forsliding to occur on the beds, and a steeper facecan be excavated.

For many rock cuts on civil projects, the stressesin the rock are much less than the rock strengthso there is little concern that fracturing of intactrock will occur. Therefore, slope design is pri-marily concerned with the stability of blocks ofrock formed by the discontinuities. Intact rockstrength, which is used indirectly in slope design,relates to the shear strength of discontinuitiesand rock masses, as well as excavation methodsand costs.

Figure 1.4 shows a range of geological condi-tions and their influence on stability, and illus-trates the types of information that are importantto design. Slopes (a) and (b) show typical con-ditions for sedimentary rock, such as sandstoneand limestone containing continuous beds, onwhich sliding can occur if the dip of the bedsis steeper than the friction angle of the discon-tinuity surface. In (a) the beds “daylight” on thesteep cut face and blocks may slide on the bed-ding, while in (b) the face is coincident with thebedding and the face is stable. In (c) the overallface is also stable because the main discontinuityset dips into the face. However, there is some riskof instability of surficial blocks of rock formed bythe conjugate joint set that dips out of the face,particularly if there has been blast damage dur-ing construction. In (d) the main joint set alsodips into the face but at a steep angle to form aseries of thin slabs that can fail by toppling wherethe center of gravity of the block lies outside thebase. Slope (e) shows a typical horizontally bed-ded sandstone–shale sequence in which the shaleweathers considerably faster than the sandstoneto form a series of overhangs that can fail sud-denly along vertical stress relief joints. Slope (f)is cut in weak rock containing closely spaced butlow persistence joints that do not form a continu-ous sliding surface. A steep slope cut in this weakrock mass may fail along a shallow circular sur-face, partially along joints and partially throughintact rock.

1.2.2 Open pit mining slope stability

The three main components of an open pit slopedesign are as follows (Figure 1.5). First, the over-all pit slope angle from crest to toe, incorporatesall ramps and benches. This may be a compos-ite slope with a flatter slope in weaker, surficialmaterials, and a steeper slope in more compet-ent rock at depth. In addition, the slope anglemay vary around the pit to accommodate bothdiffering geology and the layout of the ramp.Second, the inter-ramp angle is the slope, or slo-pes, lying between each ramp that will dependon the number of ramps and their widths. Third,


(a) (b)

(c) (d)

(e) (f)

Figure 1.4 Influence of geological conditions on stability of rock cuts: (a) potentially unstable—discontinuities“daylight” in face; (b) stable slope—face excavated parallel to discontinuities; (c) stable slope—discontinuitiesdip into face; (d) toppling failure of thin beds dipping steeply into face; (e) weathering of shale beds undercutsstrong sandstone beds to form overhangs; (f) potentially shallow circular failure in closely fractured,weak rock.


Pit depth

Bench height

Toe of slope

Overall slope angle

Bench width

Inter-ramp angle

Ramp

Bench faceangle

Pit crest

Figure 1.5 Typical open pit slope geometry showing relationship between overall slope angle, inter-rampangle and bench geometry.

the face angle of individual benches depends onvertical spacing between benches, or combinedmultiple benches, and the width of the benchesrequired to contain minor rock falls.

Some of the factors that may influence slopedesign are the slope height, geology, rockstrength, ground water pressures and damage tothe face by blasting. For example, with each suc-cessive push-back of a slope, the depth of the pitwill increase and there may need to be a corres-ponding decrease in the overall slope angle. Also,for slopes on which the ramp is located, the slopeangle may be flatter to limit the risk of failuresthat take out the ramp, compared to slopes withno ramp where some instability may be tolerated.Where there is significant water pressure in theslope, consideration may be given to installing adrainage system if it can be shown that a reduc-tion in water pressure will allow the slope angleto be increased. For deep pits where an increasein slope angle of one or two degrees will result ina saving of several million cubic meters of rockexcavation, an extensive drainage system may be

justified. Such drainage systems could comprisefans of holes with lengths of hundreds of metersdrilled from the slope face, or a drainage aditwith holes drilled into the rock above the tunnel.

With respect to the bench face angle, this maybe governed by the orientation of a predominantjoint set if there are joints that dip out of the faceat a steep angle. If this situation does not exist,then the bench angle will be related to the overallslope geometry, and whether single benches arecombined into multiple benches. One factor thatmay influence the maximum height of individualbenches is the vertical reach of excavating equip-ment, to limit the risk accidents due to collapseof the face.

In order to provide a guideline on stable pitslope angles, a number of studies have been car-ried out showing the relationship between slopeangle, slope height and geology; the records alsodistinguished whether the slopes were stable orunstable (see Figure 1.2). These studies have beenmade for both open pit mine slopes (Sjöberg,1999), and natural and engineered slopes in


China (Chen, 1995a,b). As would be expected,if the slopes were not selected according to geo-logy, there is little correlation between slopeheight and angle for stable slopes. However,sorting of the data according to rock typeand rock strength shows a reasonable correla-tion between slope height and angle for eachclassification.

1.3 Slope features and dimensions

The International Association of EngineeringGeology has prepared definitions of landslide fea-tures and dimensions as shown in Figures 1.6and 1.7 (IAEG, 1990; TRB, 1996). Although thediagrams depicting the landslides show soil-typeslides with circular sliding surfaces, many of theselandslide features are applicable to both rockslides and slope failures in weak and weatheredrock. The value of the definitions shown in

B 9 8 7 6

19

19

4 3 2 1A

15

18 17

14

16520

A

B 10

1211

Figure 1.6 Definitions of landslide features: upperportion, plan of typical landslide in which dashed lineindicates trace of rupture surface on original groundsurface; lower portion, section in which hatchingindicates undisturbed ground and stippling showsextent of displaced material. Numbers refer todimensions defined in Table 1.1 (IAEG Commissionon Landslides, 1990).

Figures 1.6 and 1.7 is to encourage the useof consistent terminology that can be clearlyunderstood by others in the profession wheninvestigating and reporting on rock slopes andlandslides.

1.4 Rock slope design methods

This section summarizes four different proced-ures for designing rock slopes, and shows thebasic data that is required for analyzing slopestability. The design methods and the designdata are common to both mining and civilengineering.

1.4.1 Summary of design methods

A basic feature of all slope design methods is thatshear takes place along either a discrete slidingsurface, or within a zone, behind the face. Ifthe shear force (displacing force) is greater than

B

B

18

2 A

734

A

5

6

Figure 1.7 Definitions of landslide dimensions: upperportion, plan of typical landslide in which dashed lineis trace of rupture surface on original ground surface;lower portion, section in which hatching indicatesundisturbed ground, stippling shows extent ofdisplaced material, and broken line is original groundsurface. Numbers refer to dimensions defined inTable 1.2 (IAEG Commission on Landslides,1990).


Table 1.1 Definition of landslide features

No. Name Definition

1 Crown Practically undisplaced material adjacent to highest parts of main scarp2 Main scarp Steep surface on undisturbed ground at upper edge of landslide caused by

movement of displaced material (13, stippled area) away from undisturbedground; it is visible part of surface rupture (10)

3 Top Highest point of contact between displaced material (13) and main scarp (2)4 Head Upper parts of landslide along contact between displaced material and main

scarp (2)5 Minor scarp Steep surface on displaced material of landslide produced by differential

movements within displaced material6 Main body Part of displaced material of landslide that overlies surface of rupture between

main scarp (2) and toe of surface of rupture (11)7 Foot Portion of landslide that has moved beyond toe of surface of rupture (11) and

overlies original ground surface (20)8 Tip Point on toe (9) farthest from top (3) of landslide9 Toe Lower, usually curved margin of displaced material of a landslide, most distant

from main scarp (2)10 Surface of rupture Surface that forms (or that has formed) lower boundary of displaced material

(13) below original ground surface (20); mechanical idealization of surface ofrupture is called sliding surface in stability analysis

11 Toe of surface ofrupture

Intersection (usually buried) between lower part of surface of rupture (10) of alandslide and original ground surface (20)

12 Surface of separation Part of original ground surface (20) now overlain by foot (7) of landslide13 Displaced material Material displaced from its original position on slope by movement in landslide;

forms both depleted mass (17) and accumulation (18); it is stippled in Figure 1.614 Zone of depletion Area of landslide within which displaced material (13) lies below original

ground surface (20)15 Zone of

accumulationArea of landslide within which displaced material lies above original groundsurface (20)

16 Depletion Volume bounded by main scarp (2), depleted mass (17), and original groundsurface (20)

17 Depleted mass Volume of displaced material that overlies surface of rupture (10) but underliesoriginal ground surface (20)

18 Accumulation Volume of displaced material (13) that lies above original ground surface (20)19 Flank Undisplaced material adjacent to sides of surface of rupture; compass directions

are preferable in describing flanks, but if left and right are used, they refer toflanks as viewed from crown

20 Original groundsurface

Surface of slope that existed before landslide took place

the shear strength of the rock (resisting force)on this surface, then the slope will be unstable.Instability could take the form of displacementthat may or may not be tolerable, or the slopemay collapse either suddenly or progressively.The definition of instability will depend on theapplication. For example, an open pit slope may

undergo several meters of displacement withouteffecting operations, while a slope supporting abridge abutment would have little tolerance formovement. Also, a single rock fall from a slopeabove a highway may be of little consequence ifthere is an adequate ditch to contain the fall, butfailure of a significant portion of the slope that


Table 1.2 Definitions of landslide dimensions

No. Name Definition

1 Width of displaced mass, Wd Maximum breadth of displaced mass perpendicular to length, Ld2 Width of surface of rupture, Wr Maximum width between flanks of landslide perpendicular to

length, Lr3 Length of displaced mass, Ld Minimum distance from tip to top4 Length of surface of rupture, Lr Minimum distance from toe of surface of rupture to crown5 Depth of displaced mass, Dd Maximum depth of surface of rupture below original ground surface

measured perpendicular to plane containing Wd and Ld6 Depth of surface of rupture, Dr Maximum depth of surface of rupture below original ground surface

measured perpendicular to plane containing Wr and Lr7 Total length, L Minimum distance from tip of landslide to crown8 Length of center line, Lcl Distance from crown to tip of landslide through points on original

ground surface equidistant from lateral margins of surface of ruptureand displaced material

reaches the traveled surface could have seriousconsequences.

Based upon these concepts of slope stability,the stability of a slope can be expressed in one ormore of the following terms:

(a) Factor of safety, FS—Stability quantified bylimit equilibrium of the slope, which is stableif FS > 1.

(b) Strain—Failure defined by onset of strainsgreat enough to prevent safe operation of theslope, or that the rate of movement exceedsthe rate of mining in an open pit.

(c) Probability of failure—Stability quantifiedby probability distribution of differencebetween resisting and displacing forces,which are each expressed as probabilitydistributions.

(d) LRFD (load and resistance factor design)—Stability defined by the factored resistancebeing greater than or equal to the sum of thefactored loads.

At this time (2003), the factor of safety isthe most common method of slope design, andthere is wide experience in its application to alltypes of geological conditions, for both rockand soil. Furthermore, there are generally accep-ted factor of safety values for slopes excavatedfor different purposes, which promotes the pre-paration of reasonably consistent designs. The

Table 1.3 Values of minimum total safetyfactors

Failure Category Safetytype factor

Shearing Earthworks 1.3–1.5Earth retainingstructures, excavations

1.5–2.0

Foundations 2–3

ranges of minimum total factors of safety asproposed by Terzaghi and Peck (1967) and theCanadian Geotechnical Society (1992) are givenin Table 1.3.

In Table 1.3, the upper values of the totalfactors of safety apply to usual loads and ser-vice conditions, while the lower values apply tomaximum loads and the worst expected geolo-gical conditions. For open pit mines the factorof safety generally used is in the range of 1.2–1.4,using either limit equilibrium analysis to calculatedirectly the factor of safety, or numerical analysisto calculate the onset of excessive strains in theslope.

Although probabilistic design methods for rockslopes were first developed in the 1970s (Harr,1977; Canada DEMR, 1978), they are not widelyused (as of 2003). A possible reason for this lackof acceptance is that terms such as “5% prob-ability of failure” and “consequence of failure


expressed as lives lost” are not well understood,and there is limited experience on acceptableprobabilities to use in design (see Section 1.4.4).

The calculation of strain in slopes is the mostrecent advance in slope design. The techniquehas resulted from the development of numer-ical analysis methods, and particularly thosethat can incorporate discontinuities (Starfield andCundall, 1988). It is most widely used in the min-ing field where movement is tolerated, and theslope contains a variety of geological conditions(see Chapter 10).

The load and resistance factor design method(LRFD) has been developed for structural design,and is now being extended to geotechnical sys-tems such as foundations and retaining structures.Further details of this design method are discussedin Section 1.4.5.

The actual factor of safety, probability of fail-ure or allowable strain that is used in designshould be appropriate for each site. The designprocess requires a considerable amount of judg-ment because of the variety of geological andconstruction factors that must be considered.Conditions that would require the use of factorsof safety at the high end of the ranges quoted inTable 1.3 include the following:

• A limited drilling program that does notadequately sample conditions at the site, ordrill core in which there is extensive mechan-ical breakage or core loss.

• Absence of rock outcrops so that mapping ofgeological structure is not possible, and thereis no history of local stability conditions.

• Inability to obtain undisturbed samples forstrength testing, or difficulty in extrapolatinglaboratory test results to in situ conditions.

• Absence of information on ground water con-ditions, and significant seasonal fluctuationsin ground water levels.

• Uncertainty in failure mechanisms of the slopeand the reliability of the analysis method.For example, plane type failures can beanalyzed with considerable confidence, whilethe detailed mechanism of toppling failures isless well understood.

• Concern regarding the quality of construction,including materials, inspection and weatherconditions.

• The consequence of instability, with higherfactors of safety being used for dams andmajor transportation routes, and lower valuesfor temporary structures or industrial roadsfor logging and mining operations.

This book does not cover the use of rock massrating systems (Haines and Terbrugge, 1991;Durn and Douglas, 1999) for slope design. Atthis time (2003), it is considered that the frequentinfluence of discrete discontinuities on stabilityshould, and can be, incorporated directly intostability analyses. In the rock mass rating, thegeological structure is only one component ofthe rating and may not be given an appropriateweight in the rating.

A vital aspect of all rock slope design is thequality of the blasting used in excavation. Designassumes that the rock mass comprises intactblocks, the shape and size of which are defined bynaturally occurring discontinuities. Furthermore,the properties of these discontinuities should bepredictable from observations of surface outcropsand drill core. However, if excessively heavyblasting is used which results in damage to therock behind the face, stability could be dependenton the condition of the fractured rock. Since theproperties of the fractured rock are unpredictable,stability conditions will also be unpredictable.Blasting and the control of blast damage arediscussed in Chapter 11.

1.4.2 Limit equilibrium analysis(deterministic)

The stability of rock slopes for the geological con-ditions shown in Figure 1.4(a) and (f) dependson the shear strength generated along the slidingsurface. For all shear type failures, the rock canbe assumed to be a Mohr–Coulomb material inwhich the shear strength is expressed in terms ofthe cohesion c and friction angle φ. For a slid-ing surface on which there is an effective normalstress σ′ acting, the shear strength τ developed on


FS = �/�s

�s

�

She

ar s

tres

s (�

)

Effective normal stress (��)

�= c +��tan�

c

�

��

(a)

�s

W sin �pW cos �p

�pW

��(b)

Figure 1.8 Method of calculating factor of safety of sliding block: (a) Mohr diagram showing shear strengthdefined by cohesion c and friction angle φ; (b) resolution of force W due to weight of block into componentsparallel and perpendicular to sliding plane (dip ψp).

this surface is given by

τ = c + σ′ tan φ (1.1)

Equation (1.1) is expressed as a straight line ona normal stress—shear stress plot (Figure 1.8(a)),in which the cohesion is defined by the intercepton the shear stress axis, and the friction angle isdefined by the slope of the line. The effective nor-mal stress is the difference between the stress dueto the weight of the rock lying above the slidingplane and the uplift due to any water pressureacting on this surface.

Figure 1.8(b) shows a slope containing a con-tinuous joint dipping out of the face and forminga sliding block. Calculation of the factor of safetyfor the block shown in Figure 1.8(b) involves theresolution of the force acting on the sliding sur-face into components acting perpendicular andparallel to this surface. That is, if the dip of thesliding surface is ψp, its area is A, and the weightof the block lying above the sliding surface is W ,then the normal and shear stresses on the slidingplane are

Normal stress, σ = W cos ψp

Aand

shear stress, τs = W sin ψp

A(1.2)

and equation (1.1) can be expressed as

τ = c + W cos ψp tan φ

A(1.3)

or

τsA = W sin ψp and

τA = cA + W cos ψp tan φ (1.4)

In equations (1.4), the term [W sin ψp] definesthe resultant force acting down the sliding planeand is termed the “driving force” (τsA), whilethe term [cA + W cos ψp tan φ] defines the shearstrength forces acting up the plane that resist slid-ing and are termed the “resisting forces” (τA).The stability of the block in Figure 1.8(b) can bequantified by the ratio of the resisting and driv-ing forces, which is termed the factor of safety,FS. Therefore, the expression for the factor ofsafety is

FS = resisting forcesdriving forces

(1.5)

FS = cA + W cos ψp tan φ

W sin ψp(1.6)

The displacing shear stress τs and the resist-ing shear stress τ defined by equations (1.4) areplotted on Figure 1.8(a). On Figure 1.8(a) it isshown that the resisting stress exceeds the displa-cing stress, so the factor of safety is greater thanone and the slope is stable.

If the sliding surface is clean and contains noinfilling then the cohesion is likely to be zero and


hw V

U

T

�p�T

(a)

Unstable

She

ar s

tres

s (�

)

B

A

C

c

Effective normal stress (��)

Stable

�U

�v

�T

�v �T

�

(b)

Figure 1.9 The effect of ground water and bolt forces on factor of safety of rock slope: (a) ground water andbolting forces acting on sliding surface; (b) Mohr diagram of stresses acting on sliding surface showing stableand unstable stability conditions.

equation (1.6) reduces to

FS = cos ψp · tan φ

sin ψp(1.7)

or

FS = 1 when ψp = φ (1.8)

Equations (1.7) and (1.8) show that for a dry,clean surface with no support installed, the blockof rock will slide when the dip angle of the slidingsurface equals the friction angle of this surface,and that stability is independent of the size of thesliding block. That is, the block is at a conditionof “limiting equilibrium” when the driving forcesare exactly equal to the resisting forces and thefactor of safety is equal to 1.0. Therefore, themethod of slope stability analysis described in thissection is termed limit equilibrium analysis.

Limit equilibrium analysis can be applied toa wide range of conditions and can incorporateforces such as water forces acting on the slid-ing surface, as well as external reinforcing forcessupplied by tensioned rock anchors. Figure 1.9(a)shows a slope containing a sliding surface witharea A and dip ψp, and a vertical tension crack.The slope is partially saturated such that the ten-sion crack is half-filled with water, and the water

table exits where the sliding surface daylights onthe slope face. The water pressures that are gen-erated in the tension crack and on the slidingsurface can be approximated by triangular forcediagrams where the maximum pressure, p at thebase of the tension crack and the upper end of thesliding surface is given by

p = γw hw (1.9)

where γw is the unit weight of water and hw isthe vertical height of water in the tension crack.Based on this assumption, the water forces actingin the tension crack, V , and on the sliding plane,U , are as follows:

V = 12

γw h2w and U = 1

2γw hwA (1.10)

and the factor of safety of the slope is calculatedby modifying equation (1.6) as follows:

FS = cA + (W cos ψp − U − V sin ψp) tan φ

W sin ψp + V cos ψp(1.11)

Similarly, an equation can be developed for areinforced slope in which a tensioned rock anchor


has been installed with the anchor below thesliding plane. If the tension in the anchor is T

and it is installed at an angle ψT below the hori-zontal, then the normal and shear forces actingon the sliding plane due to the anchor tension arerespectively:

NT = T sin(ψT + ψp) and

ST = T cos(ψT + ψp) (1.12)

and the equation defining the factor of safety ofthe anchored, partially saturated slope is

FS = cA + (W cos ψp − U − V sin ψp + T sin(ψT + ψp)) tan φ

W sin ψp + V cos ψp − T cos(ψT + ψp)

(1.13)

Figure 1.9(b) shows on a Mohr diagram themagnitude of the normal and shear stresses onthe sliding surface developed by the water andbolting forces, and their influence on the factorof safety. That is, destabilizing forces (e.g. water)decrease the normal stress and increase the shearstress, and tend to cause the resultant of the forcesto be above the limiting strength line, indicatinginstability (Point B). In contrast, stabilizing forces(bolting and drainage) increase the normal stressand decrease the shear stress, and cause the res-ultant to be below the line, indicating stability(Point C).

The force diagram in Figure 1.9(b) can also beused to show that the optimum dip angle for thebolts, that is, the dip that produces the greatestfactor of safety for a given rock anchor force is

ψT(opt) = (φ − ψp

)or φ = ψp + ψT(opt)

(1.14)

Strict application of equation (1.14) may showthat the anchor should be installed above thehorizontal, that is, ψT is negative. However,in practice, it is usually preferable to installanchors below the horizontal because this facil-itates drilling and grouting, and provides a morereliable installation.

These examples of limit equilibrium analysis tocalculate the stability of rock slopes show that

this is a versatile method that can be applied toa wide range of conditions. One limitation of thelimit equilibrium method is that all the forces areassumed to act through the center of gravity ofthe block, and that no moments are generated.

This analysis described in this section is applic-able to a block sliding on a plane. However,under certain geometric conditions the block maytopple rather than slide, in which case a differentform of limit equilibrium analysis must be used.Figure 1.10 shows the conditions that differenti-ate stable, sliding and toppling blocks in relationto the width �x and height y of the block, the dipψp of the plane on which it lies and the frictionangle φ of this surface. Sliding blocks are analyzedeither as plane or wedge failures (see Chapters 6and 7 respectively), while the analysis of topplingfailures is discussed in Chapter 9. Figure 1.10(b)shows that there are only limited conditions underwhich toppling occurs, and in fact this is a lesscommon type of failure compared with slidingfailures.

1.4.3 Sensitivity analysis

The factor of safety analysis described aboveinvolves selection of a single value for each ofthe parameters that define the driving and resist-ing forces in the slope. In reality, each parameterhas a range of values, and a method of examin-ing the effect of this variability on the factor ofsafety is to carry out sensitivity analyses usingupper and lower bound values for those para-meters considered critical to design. However,to carry out sensitivity analyses for more thanthree parameters is cumbersome, and it is diffi-cult to examine the relationship between each ofthe parameters. Consequently, the usual designprocedure involves a combination of analysis andjudgment in assessing the influence on stabilityof variability in the design parameters, and thenselecting an appropriate factor of safety.

An example of sensitivity analysis is shown inSection 4.4 in Chapter 4 that describes the stabil-ity analysis of a quarry slope in which sensitivityanalyses were carried out for both the friction


y

∆x

W sin �p

W cos �p

�p

W

Stable block�p<�

∆x/y > tan �p

Sliding only�p > �

∆x /y > tan�p

Sliding and toppling�p>�

∆x /y < tan�pToppling only�p<�; ∆x /y < tan�p

0 10 20 30 40 50 60 70 80 90

2

1

3

4

5

Rat

io ∆

x/y

Base plane angle �p (degrees)

(a) (b)

�p=

�Figure 1.10 Identification of sliding and toppling blocks: (a) geometry of block on inclined plane;(b) conditions for sliding and toppling of block on an inclined plane.

angle (range 15–25◦) and the water pressure—fully drained to fully saturated (Figure 4.20). Thisplot shows that water pressures have more influ-ence on stability than the friction angle. That is, afully drained, vertical slope is stable for a frictionangle as low as 15◦, while a fully saturated slopeis unstable at an angle of 60◦, even if the frictionangle is 25◦.

The value of sensitivity analysis is to assesswhich parameters have the greatest influence onstability. This information can then be used inplanning investigation programs to collect datathat will define this parameter(s) more precisely.Alternatively, if there is uncertainty in the valueof an important design parameter, this can beaccounted for in design by using an appropriatefactor of safety.

1.4.4 Probabilistic design methods

Probabilistic design is a systematic procedure forexamining the effect of the variability of each

parameter on slope stability. A probability distri-bution of the factor of safety is calculated, fromwhich the probability of failure (PF) of the slopeis determined.

Probability analysis was first developed in the1940s and is used in the structural and aeronaut-ical engineering fields to examine the reliabilityof complex systems. Among its early uses in geo-technical engineering was in open pit mine slopedesign where a certain risk of failure is accept-able, and this type of analysis could be readilyincorporated into the economic planning of themine (Canada DEMR, 1978; Pentz, 1981; Savely,1987). Examples of its use in civil engineering arein the planning of slope stabilization programsfor transportation systems (Wyllie et al., 1979;McGuffey et al., 1980), landslide hazards (Fell,1994; Cruden, 1997) and in design of storagefacilities for hazardous waste (Roberds, 1984,1986).

There is sometimes reluctance to use prob-abilistic design when there is a limited amount


Marginally accepted

AcceptedMobile drill rigs

Foundations

Mine pit slopes

Fixed drill rigs

Dams

US dams (estimated)Commercial aviation

0.1 1 10 100 1000 10,0000

Ann

ual p

roba

bilit

y of

failu

re

100

10–5

10–4

10–3

10–2

10–1

Consequence of failure (lives lost)

Figure 1.11 Risks for selectedengineering projects (Whitman,1984).

of design data that may not be representativeof the population. In these circumstances, it ispossible to use subjective assessment techniquesthat provide reasonably reliable probability val-ues from small samples (Roberds, 1990). Thebasis of these techniques is the assessment andanalysis of available data, by an expert or groupof experts in the field, in order to arrive at aconsensus on the probability distributions thatrepresent the opinions of these individuals. Thedegree of defensibility of the results tends toincrease with the time and cost that is expen-ded in the analysis. For example, the assessmenttechniques range from, most simply, informalexpert opinion to more reliable and defensibletechniques such as Delphi panels (Rohrbaugh,1979). A Delphi panel comprises a group ofexperts who are each provided with the sameset of data and are required to produce a writ-ten assessment of these data. These documentsare then provided anonymously to each of theother assessors who are encouraged to adjust their

assessments in light of their peer’s results. Afterseveral iterations of this process, it should bepossible to arrive at a consensus that maintainsanonymity and independence of thought.

The use of probability analysis in designrequires that there be generally accepted ranges ofprobability of failure for different types of struc-ture, as there are for factors of safety. To assist inselecting appropriate probability of failure values,Athanasiou-Grivas (1979) provides charts relat-ing factor of safety to the probability of failure.Also, Figure 1.11 gives a relationship betweenlevels of annual probability of failure for a varietyof engineering projects, and the consequence offailure in terms of lives lost. For example, for openpit mine slopes for which slope performance isclosely managed and there is little risk to life in theevent of a failure, the acceptable range of annualprobability of failure can be about 10−1 to 10−2.In comparison, for dams where failure could res-ult in the loss of several hundred lives, annualprobability of failure should not exceed about


–2 –1 0 1 2

SD = 0.25

SD = 0.5

SD = 1

f(x)

x

1

1

0.5

0−2(SD) −1(SD) 0 1(SD) 2(SD)z

Φ(z)

(a) (b)

Figure 1.12 Properties of the normal distribution: (a) density of the normal distribution with mean x = 0 andstandard deviations (SD) of 0.25, 0.5 and 1.0; (b) distribution function �(z) of the normal distribution withmean 0 and standard deviation 1.0 (Kreyszig, 1976).

10−4 to 10−5. Despite the wide range of valuesshown in Figure 1.11, this approach provides auseful benchmark for the ongoing developmentof probabilistic design (Salmon and Hartford,1995).

(a) Distribution functions

In probability analysis, each parameter for whichthere is some uncertainty is assigned a rangeof values that is defined by a probability dens-ity function. Some types of distribution func-tions that are appropriate for geotechnical datainclude normal, beta, negative exponential andtriangular distributions. The most common typeof function is the normal distribution in whichthe mean value is the most frequently occurringvalue (Figure 1.12(a)). The density of the normaldistribution is defined by

f (x) = 1

SD√

2πe− 1

2 [(x−x)/SD]2 (1.15)

where x is the mean value given by

x =∑n

x=1 x

n(1.16)

and SD is the standard deviation given by

SD =[∑n

x=1(x − x)2

n

]1/2

(1.17)

As shown in Figure 1.12(a), the scatter in thedata, as represented by the width of the curve, ismeasured by the standard deviation. Importantproperties of the normal distribution are that thetotal area under the curve is equal to 1.0. Thatis, there is a probability of unity that all values ofthe parameter fall within the bounds of the curve.Also, 68% of the values will lie within a range ofone standard deviation either side of the mean,and 95% will lie within two standard deviationseither side of the mean.

Conversely, it is possible to determine the valueof a parameter defined by a normal distributionby stating the probability of its occurrence. This isshown graphically in Figure 1.12(b) where �(z) isthe distribution function with mean 0 and stand-ard deviation 1.0. For example, a value whichhas a probability of being greater than 50% of allvalues is equal to the mean, and a value whichhas a probability of being greater than 16% ofall values is equal to the mean plus one standarddeviation.

The normal distribution extends to infinity inboth directions, but this is often not a realisticexpression of geotechnical data for which thelikely upper and lower bounds of a parameter canbe defined. For these conditions, it is appropri-ate to use the beta distribution which has finitemaximum and minimum points, and can be uni-form, skewed to the left or right, U-shaped orJ-shaped (Harr, 1977). However, where there islittle information on the distribution of the data,


a simple triangular distribution can be used whichare defined by three values: the most likely, andthe minimum and maximum values. Examples ofprobability distributions are shown in the workedexample in Section 6.6.

(b) Probability of failure

The probability of failure is calculated in a sim-ilar manner to that of the factor of safety inthat the relative magnitude of the displacing andresisting forces in the slope is examined (seeSection 1.4.2). Two common methods of calcu-lating the coefficient of reliability are the marginof safety method, and the Monte Carlo methodas discussed later.

The margin of safety is the difference betweenthe resisting and displacing forces, with the slopebeing unstable if the margin of safety is negative.If the resisting and displacing forces are mathem-atically defined probability distributions—fD(r)

and fD(d) respectively in Figure 1.13(a)—then itis possible to calculate a third probability dis-tribution for the margin of safety. As shown inFigure 1.13(a), there is a probability of failure ifthe lower limit of the resisting force distributionfD(r) is less than the upper limit of the displa-cing force distribution fD(d). This is shown asthe shaded area on Figure 1.13(a), with the prob-ability of failure being proportional to the area ofthe shaded zone. The method of calculating thearea of the shaded zone is to calculate the prob-ability density function of the margin of safety:

fD(d )fD(d )fD(r )

fD(r )

Zone representingprobability of failure

f (d ), f(r)d r (f(r) – f (d ))

(a) (b)

Probability offailure

Coefficient ofreliability

0

fD(r – d)

(r – d)

Figure 1.13 Calculation of probability of failure using normal distributions: (a) probability density functions ofthe resisting force fR and the displacing force fD in a slope; and (b) probability density function of differencebetween resisting and displacing force distributions fD(r−d).

the area of the negative portion of this function isthe probability of failure (Figure 1.13(b)). If theresisting and displacing forces are defined by nor-mal distributions, the margin of safety is also anormal distribution, the mean and standard devi-ation of which are calculated as follows (CanadaDEMR, 1978):

Mean, margin of safety = fr − fd (1.18)

Standard deviation, margin of safety

= (SD2r + SD2

d)1/2 (1.19)

where fr and fd are the mean values, and SDr andSDd are the standard deviations of the distribu-tions of the resisting and displacing forces respect-ively. Note that the definition of the deterministicfactor of safety is given by fr/fd.

Having determined the mean and standarddeviation of the margin of safety, the probabilityof failure can be calculated from the properties ofthe normal distribution. For example, if the meanmargin of safety is 2000 MN and the standarddeviation is 1200 MN, then the margin of safetyis zero at (2000 – 0)/1200, or 1.67 standard devi-ations. From Figure 1.12(b), where the marginof safety distribution is represented by �(z), theprobability of failure is 5%.

Note that the margin of safety concept dis-cussed in this section can only be used where theresisting and displacing forces are independentvariables. This condition would apply where thedisplacing force is the weight of the sliding mass,


INPUT

1. Expressions for total resisting and displacing forces

2. Density functions of the independent random variables:

r = r (x1, x2, x3, . . ., xn)

d = d(y1, y2, y3, . . ., yn)

f(xi), i = 1, 2, . . ., n and f(yi) i = 1, 2, . ., m

Generate (n + m) random numbers between 0 and 1.

Select a random value for each xi, i = 1, 2, . . ., nand yi, i = 1, 2, . . ., m from their cumulative

distributions

Calculate the values of r and d

Yes M times No N – M times

Repeat N times

OUTPUT

Probability of failure

PF = (N – M )/N

Checkr > d

Figure 1.14 Flow chart forMonte Carlo simulation tocalculate probability of failureof a slope (Athanasiou-Grivas,1980).

and the resisting force is the installed reinforce-ment. However, where the resisting force is theshear strength of the rock, then this force and thedisplacing force are both functions of the weightof the slope, and are not independent variables.Under these circumstances, it is necessary to useMonte Carlo analysis as described next.

Monte Carlo analysis is an alternative methodof calculating the probability of failure which ismore versatile than the margin of safety methoddescribed earlier. Monte Carlo analysis avoidsthe integration operations that can become quitecomplex, and in the case of the beta distribu-tion cannot be solved explicitly. The particularvalue of Monte Carlo analysis is the ability towork with any mixture of distribution types, andany number of variables, which may or maynot be independent of each other (Harr, 1977;Athanasiou-Grivas, 1980).

The Monte Carlo technique is an iterativeprocedure comprising the following four steps(Figure 1.14):

1 Estimate probability distributions for each ofthe variable input parameters.

2 Generate random values for each parameter;Figure 1.12(b) illustrates the relationship for anormal distribution between a random num-ber between 0 and 1 and the correspondingvalue of the parameter.

3 Calculate values for the displacing and resist-ing forces and determine if the resisting forceis greater than the displacing force.

4 Repeat the process N times (N > 100) andthen determine probability of failure Pf fromthe ratio:

Pf = N − M

N(1.20)


where M is the number of times the resistingforce exceeded the displacing force (i.e. thefactor of safety is greater than 1.0).

An example of the use of Monte Carlo ana-lysis to calculate the coefficient of reliability ofa slope against sliding is given in Section 6.6in Chapter 6 on “Plane failure.” This exampleshows the relationship between the determin-istic and probabilistic analyses. The factor ofsafety is calculated from the mean or most likelyvalues of the input variables, while the probab-ilistic analysis calculates the distribution of thefactor of safety when selected input variables areexpressed as probability density functions. Forthe unsupported slope, the deterministic factorof safety has a value of 1.4, while the prob-abilistic analysis shows that the factor of safetycan range from a minimum value of 0.69 to amaximum value of 2.52. The proportion of thisdistribution with a value less than 1.0 is 7.2%,which represents the probability of failure of theslope.

1.4.5 Load and Resistance Factor Design

This design method is based on the use of prob-ability theory to develop a rational design basisfor structural design that accounts for variabil-ity in both loads and resistance. The objectiveis to produce a uniform margin of safety forsteel and concrete structures such as bridges,and geotechnical structures such as foundationsunder different loading conditions. The LRFDmethod has been developed in structural engin-eering and is becoming widely used in the designof major structures such as bridges (CSA, 1988;Eurocode, 1995; AASHTO, 1996). In order thatfoundations design is consistent with structuraldesign, the LRFD method has been extended toinclude geotechnical engineering (TransportationResearch Board, 1999).

Some of the early LRFD work in geotechnicalengineering was carried out by Myerhoff (1984)who used the term Limit States Design, anddefined the two limit states as follows. First, thestructure and its components must have, during

the intended service life, an adequate margin ofsafety against collapse under the maximum loadsthat might reasonably occur. Second, the struc-ture and its components must serve the designedfunctions without excessive deformations anddeterioration. These two service levels are the ulti-mate and serviceability limit states respectivelyand are defined as follows:

• Ultimate limit state—Collapse of the structureand slope failure including instability due tosliding, toppling and excessive weathering.

• Serviceability limit state—Onset of excessivedeformation and unacceptable deterioration.

The basis of LRFD design is the multiplicationof loads and resistances by factors that reflect thedegree of uncertainty and variability in the para-meters. The requirement of the design is that thefactored resistance is equal to, or greater than,the factored loads. This is stated in mathematicalterms as follows:

φkRnk ≥∑

ηijγijQij (1.21)

where φk is the resistance factor and Rnk is thenominal strength for the kth failure mode or ser-viceability limit state, ηij is the factor to accountfor the ductility, redundancy and operationalimportance of the element or system, γij is theload factor and Qij the member load effect for theith load type in the jth load combination underconsideration.

In the application of equation (1.21), theload factors are greater than unity unless theload is beneficial to stability, and the resist-ance factors are less than unity. On this basis,the Mohr–Coulomb equation for the shear res-istance of a sliding surface is expressed asfollows:

τ = fcc + (σ − fUU)fφ tan φ (1.22)

The cohesion c, friction coefficient tan φ andwater pressure U are all multiplied by partialfactors with values less than unity, while the nor-mal stress σ on the sliding surface is calculated


using a partial load factor greater than unityapplied to the slope weight and any applied loads.Actual values for the resistance factors will varydepending on such factors as the extent of testingduring construction and ratio of the live load tothe dead load.

LRFD would usually only be used for slopedesign where the slope was a component of abridge foundation, for example. For slopes thatare not part of any structure, the design methodsdescribed in Sections 1.4.2–1.4.4 would be morecommonly used.

Chapter 2

Structural geology and datainterpretation

2.1 Objectives of geologicalinvestigations

The stability of rock slopes is often significantlyinfluenced by the structural geology of the rockin which the slope is excavated. Structural geo-logy refers to naturally occurring breaks in therock such as bedding planes, joints and faults,which are generally termed discontinuities. Theproperties of discontinuities relative to stabilityinclude orientation, persistence, roughness andinfilling. The significance of discontinuities is thatthey are planes of weakness in the much stronger,intact rock so failure tends to occur preferen-tially along these surfaces. The discontinuitiesmay directly influence stability such as the slopesshown in Figure 2.1. In Figure 2.1(a) the face isformed by the bedding planes that are continu-ous over the full height of the cut—this conditionis termed a plane failure and is discussed indetail in Chapter 6. Alternatively, slope failuremay occur on two discontinuities that intersectbehind the face (Figure 2.1(b))—this condition istermed a wedge failure and is discussed in detailin Chapter 7.

Alternatively, discontinuities may only indir-ectly influence stability where their length is muchshorter than the slope dimensions, such as anopen pit mine slope where no single discontinuitycontrols stability. However, the properties of thediscontinuities will affect the strength of the rockmass in which the slope is mined.

Almost all rock slope stability studies shouldaddress the structural geology of the site, andsuch studies involve two steps as follows. First,

determine the properties of the discontinuities,which involves mapping outcrops and existingcuts, if any, and examining diamond drill core,as appropriate for the site conditions. Second,determine the influence of the discontinuities onstability, which involves studying the relationshipbetween the orientation of the discontinuity andthe face. The objective of this study, which istermed kinematic analysis, is to identify possiblemodes of slope failure.

The overall purpose of a geological mappingprogram is to define a set or sets of discontinu-ities, or a single feature such as a fault, whichwill control stability on a particular slope. Forexample, the bedding may dip out of the face andform a plane failure, or a pair of joint sets mayintersect to form a series of wedges. It is commonthat the discontinuities will occur in three ortho-gonal sets (mutually at right angles), with possiblyone additional set. It is suggested that four setsis the maximum that can be incorporated into aslope design, and that any additional sets that mayappear to be present are more likely to representscatter in the orientation of the sets. Discontinu-ities that occur infrequently in the rock mass arenot likely to have a significant influence on stabil-ity of the overall slope and so can be discountedin design. However, it is important to identify asingle feature such as a through-going, adverselyorientated fault that may be a controlling featurefor stability.

There are certain geological conditions wherethe discontinuities may be randomly orientated.For example, basalt that has cooled rapidly while

Structural geology and data interpretation 23

still flowing may have a “crackled” structurein which the joints are not persistent, curvedand have no preferred orientation. Also, somevolcanic rocks exhibit sets that only extend overa few meters on the face, and then different setsoccur in an adjacent area.

In the design of rock slopes on civil projects, it isusually advisable that the full length of each slopebe designed with a uniform slope angle; it is notpractical to excavate a slope with varying slopeangles because this will complicate surveying andthe lay out of blast holes. This will require thatin applying geological data to slope design, thedominant geological structure, such as bedding ororthogonal joint sets, be used for the design. Anexception to this guideline may be where there is asignificant change in rock type within the cut, andit may be appropriate to prepare a separate designfor each. However, even in these circumstances, itmay be more economical in terms of constructioncosts to cut the entire slope at the flatter of thetwo slope designs, or to install support in the lessstable material.

Another issue in planning a geological mappingprogram is to decide how many discontinuitiesshould be mapped in order to define the designsets. It is usually possible, by inspection of a nat-ural face or existing cut, to ascertain whether thestructure occurs in sets or is randomly oriented.For locations where there is good rock expos-ure and the structure is uniform, as few as 20measurements should provide information on theorientation of the sets, with a further 50–100measurements required to define typical prop-erties such as persistence, spacing and infilling.Conditions in which a greater number of dis-continuities should be mapped include faulted orfolded structure, or contacts between differentrock types. In these cases, several hundred fea-tures may need to be mapped in order to define theproperties of each unit. A detailed procedure fordetermining the number of joints to be mapped isdescribed by Stauffer (1966).

Structural geology is specifically addressed intwo chapters in the book. Chapter 2 describesthe properties of discontinuities and how theyare used in kinematic analysis, and Chapter 3

(a)

(b)

Figure 2.1 Rock faces formed by persistentdiscontinuities: (a) plane failure formed by beddingplanes parallel to face with continuous lengths overthe full height of the slope (shale on Route 19 nearRobbinsville, North Carolina); (b) wedge failureformed by two intersecting planes dipping out of theface (sedimentary formation on Route 60 nearPhoenix, Arizona).

24 Structural geology and data interpretation

discusses methods of collecting structural geolo-gical data, including mapping and drilling.

2.2 Mechanism of joint formation

All the rocks observed in outcrops and excav-ations have undergone a long history of modi-fication over a time of hundreds of millions, oreven billions of years. The sequence of modi-fication that a sedimentary rock, for example,typically undergoes is deposition at the surface,gradual burial to depths of up to several kilo-meters with imposition of heat and pressure, andthen uplift to the surface (Figure 2.2). Through-out this sequence, the rock may also be subjectto deformation comprising folding and faulting.These processes usually result in the stresses inthe rock exceeding its strength a number of times,

causing the rock to fracture and form joints andfaults. In sedimentary rock, bedding planes thatare coincident with breaks in the continuity ofsedimentation will also form.

Figure 2.2(a) shows how the stresses in rockincrease with burial, assuming that there are nowater, thermal or tectonic pressures acting in therock (Davis and Reynolds, 1996). The verticalstress, which is the major principal stress σ1, isequal to the weight of the overlying rock and isgiven by

σ1 = γrH (2.1)

where γr is the unit weight of the rock and H isthe depth of burial. The horizontal stress, whichis the minor principal stress σ3, also increaseswith the depth of burial due to the effect of the

Tension Stress in rock (MPa) Compression

Depth ofburial, H (km)

Major principal stress, �1

Minor principal stress, �3

(a)

(b) (c)

–20 40200 60 80 100 120 140

1

2

3

4

5

Rockstrength

Shearstress, �(MPa)

H = 5 km

H = 2 km

Normal stress, � (MPa)

2�

20

40

60

80

+�–��1�3

20 40 60 80 100 120

�3

�1

�1

�3

�

Figure 2.2 Development of jointingdue to burial and uplift of rock:(a) stress changes in rock duringburial; (b) Mohr diagram showingconditions for rock fracture;(c) inclination of joints with respectto stress direction (adapted fromDavis and Reynolds (1996)).


Poisson’s Ratio µ, and any temperature increasethat occurs. In ideal conditions σ3 is related to σ1as follows:

σ3 =(

µ

1 − µσ1

)+(

E

1 − µε�T

)(2.2)

where E is the modulus of deformation of therock, ε is the coefficient of thermal expansion and�T is the temperature rise. The first component ofequation (2.2) shows the value of the horizontalstress due to gravitational loading; if the Poisson’sratio is 0.25, for example, then σ3 = 0.33σ1. Ifthe rock were not free to expand, and a temperat-ure rise of 100◦C occurs in a rock with a modulusvalue of 50 GPa and an ε value of 15 × 10−6/◦C,then a thermal stress of 100 GPa will be gener-ated. In reality, the values of the principal stresseswill be modified by tectonic action resulting indeformation such as folding and faulting.

On Figure 2.2(a), the value of σ1 is defined byequation (2.1), and the value of σ3 varies withdepth as follows. The value of σ3 is tensile atdepths less than 1.5 km where the sediments havenot been consolidated into rock, and below thisdepth, σ3 increases as defined by equation (2.2),assuming no temperature change.

The formation of joints in rock during theburial–uplift process shown in Figure 2.2(a) willdepend on the rock strength in comparison to theapplied stresses. A method of identifying condi-tions that will cause the rock to fracture is to usea Mohr diagram (Figure 2.2(b)). In Figure 2.2(b)the rock strength is shown as a straight line undercompressive stress, and curved under tensile stressbecause micro-fractures within the rock act asstress concentrators that diminish the strengthwhen a tensile stress is applied. On the Mohrdiagram, circles represent the σ1 and σ3 stressesat different depths, and where the circle inter-sects the strength line, failure will occur. Thestress conditions show that fracture will occur ata depth of 2 km (σ1 = 52 MPa; σ3 = 0 MPa)because there is no horizontal confining stress act-ing. However, at a depth of 5 km where the rockis highly confined (σ1 = 130 MPa; σ3 = 25 MPa)the rock will not fracture because the strength

exceeds the stress. For conditions where σ3 is lowor tensile (negative), failure occurs more readilycompared to conditions at greater depth whereboth σ1 and σ3 are compressive (positive).

The Mohr diagram also shows the orientationof the fracture with respect to the stress direc-tion (Figure 2.2(c)). Since principal stresses areoriented mutually at right angles, sets of jointstend to form in orthogonal directions.

2.3 Effects of discontinuities on slopestability

While the orientation of discontinuities is theprime geological factor influencing stability, andis the subject of this chapter, other propertiessuch as persistence and spacing are significantin design. For example, Figure 2.3 shows threeslopes excavated in a rock mass containing twojoint sets: set J1 dips at 45◦ out of the face, andset J2 dips at 60◦ into the face. The stability ofthese slopes differs as follows. In Figure 2.3(a), setJ1, which is widely spaced and has a persistencegreater than the slope height, forms a potentiallyunstable plane failure over the full height of thecut. In Figure 2.3(b) both sets J1 and J2 havelow persistence and are closely spaced so thatwhile small blocks ravel from the face, there isno overall slope failure. In Figure 2.3(c), set J2 ispersistent and closely spaced, and forms a seriesof thin slabs dipping into the face that create atoppling failure.

The significance of Figure 2.3 is that, while ananalysis of the orientation of joint sets J1 and J2would show identical conditions on a stereonet,there are other characteristics of these discon-tinuities that must also be considered in design.These characteristics, which are discussed furtherin Chapter 3, should be described in detail as partof the geological data collection program for rockslope design.

2.4 Orientation of discontinuities

The first step in the investigation of discontinuitiesin a slope is to analyze their orientation and


J1J2

(a)

(b)

(c)

J2

J1

Figure 2.3 Effects of joint properties on slopestability: (a) persistent J1 joints dipping out of faceforms potentially unstable sliding blocks; (b) closelyspaced, low persistence joints cause raveling of smallblocks; (c) persistent J2 joints dipping into face formpotential toppling slabs.

identify sets of discontinuities, or single dis-continuities that could form potentially unstableblocks of rock. Information on discontinuityorientation may be obtained from such sourcesas surface and underground mapping, diamonddrill core and geophysics, and it is necessary to

combine these data using a procedure that isreadily amenable to analysis. This analysis is facil-itated by the use of a simple and unambiguousmethod of expressing the orientation of a dis-continuity. The recommended terminology fororientation is the dip and dip direction which aredefined as follows, and shown schematically inFigure 2.4(a) and (b).

1 Dip is the maximum inclination of a discon-tinuity to the horizontal (angle ψ).

2 Dip direction or dip azimuth is the direc-tion of the horizontal trace of the line of dip,measured clockwise from north (angle α).

As will be demonstrated in Section 2.5, the dip/dipdirection system facilitates field mapping, plottingstereonets, and analysis of discontinuity orienta-tion data. Strike, which is an alternative means ofdefining the orientation of a plane, is the traceof the intersection of an inclined plane with ahorizontal reference plane.

The strike is at right angles to the dip direction,and the relationship between the strike and thedip direction is illustrated in Figure 2.4(b) wherethe plane has a strike of N45E and a dip of 50SE.In terms of dip and dip direction, the orienta-tion of the plane is 50/135, which is consideredto be a simpler nomenclature that also facilitatesthe use of stereographic analysis. By always writ-ing the dip as two digits and the dip direction asthree digits, for example, 090 for 90◦, there canbe no confusion as to which set of figures refersto which measurement. Strike and dip measure-ments can be readily converted into dip and dipdirection measurements, if this mapping system ispreferred.

In defining the orientation of a line, the termsplunge and trend are used (Figure 2.4(c)). Theplunge is the dip of the line, with a positive plungebeing below the horizontal and a negative plungebeing above the horizontal. The trend is the dir-ection of the horizontal projection of the linemeasured clockwise from north, and correspondsto the dip direction of a plane.

When mapping geological structure in the field,it is necessary to distinguish between the true and


N

Strike

Dipdirection

Plane

Dip = 50°

W

N

E

S

StrikeN45° E

(a) (b) (c)

North

Trend

Plunge

Dip direction135°

�

�

Figure 2.4 Terminology defining discontinuity orientation: (a) isometric view of plane (dip and dip direction);(b) plan view of plane; (c) isometric view of line (plunge and trend).

apparent dip of a plane. True dip is the steepestdip of the plane, and is always steeper than theapparent dip. The true dip can be found as fol-lows. If a pebble or a stream of water is run downthe plane, it will always fall in a direction that cor-responds to the dip direction; the dip of this lineis the true dip.

2.5 Stereographic analysis of structuralgeology

Previous sections describe structural geologicalfeatures that influence rock slope stability. Thisdata often occurs in three dimensions with adegree of natural scatter, and in order to be able touse the data in design, it is necessary to have avail-able an analysis technique that can address thesematters. It has been found that the stereographicprojection is an ideal tool for this application.

This section describes methods of analyzingstructural geology data using the stereonet toidentify discontinuity sets, and examine theirinfluence on slope stability.

2.5.1 Stereographic projection

The stereographic projection allows the three-dimensional orientation data to be representedand analyzed in two dimensions. Stereographicpresentations remove one dimension from con-sideration so that lines or points can represent

planes, and points can represent lines. Animportant limitation of stereographic projectionsis that they consider only angular relationshipsbetween lines and planes, and do not representthe position or size of the feature.

The stereographic projection consists of a ref-erence sphere in which its equatorial plane ishorizontal, and its orientation is fixed relative tonorth (Figure 2.5). Planes and lines with a specificplunge and trend are positioned in an imagin-ary sense so that the axis of the feature passesthrough the center of the reference sphere. Theintersection of the feature with the lower half ofthe reference sphere defines a unique line on thesurface of the reference hemisphere. For a plane,this intersection with the reference sphere is a cir-cular arc called a great circle, while for a line, theintersection with the reference sphere is a point.In order to develop a stereographic projection ofa plane or line, the intersection with the referencesphere is rotated down to a horizontal surface atthe base of the sphere (Figure 2.6). The rotatedlines and points are unique locations on the ste-reonet that represent the dip (plunge) and dipdirection (trend) of the feature. In slope stabil-ity analysis using stereonets, planes are used torepresent both discontinuities and slope faces.

An alternative means of representing the ori-entation of a plane is the pole to the plane(Figure 2.6(a)). The pole is the point on the sur-face of the reference sphere that is pierced by aradial line in a direction normal to the plane. The


North

Strike

Dipdirection

Dip

Lower halfreference sphere

Great circlerepresentationof a plane

North

Trend

Plunge

Pointrepresentationof line

Lower halfreferencesphere

(a)

(b)

Figure 2.5 Stereographic representation of plane andline on lower hemisphere of reference sphere:(a) plane projected as great circle; (b) isometric viewof line (plunge and trend).

value of the pole projection is that a single pointcan represent the complete orientation of a plane.As described in Section 2.5.2, the use of polesfacilitates the analysis of a large number of planescompared with the use of great circles.

As an aid to interpreting the information shownon stereonets, it can be seen from Figures 2.5 and2.6 that planes and lines with shallow dips havegreat circles and points that plot near the circum-ference of the stereonet, and those with steep dipsplot near the center. In contrast, the pole of a shal-low dipping plane plots close to the center of thecircle, and the pole of a steep plane plots close tothe perimeter.

The two types of stereographic projections usedin structural geology are the polar and equatorialprojections as shown in Figure 2.7. The polar

ZenithReference sphere

Reference sphere

Great circle

Pole to plane

N

Zenith

N

(a)

(b)

Equalarea net

Figure 2.6 Equal area projections of plane and line:(a) plane projected as great circle and correspondingpole; (b) line projected as pole.

Polar projection

Equatorial projection

Figure 2.7 Polar and equatorial projections of asphere.


net can only be used to plot poles, while the equat-orial net can be used to plot both planes and polesas described later. In the case of the equatorialprojection, the most common type of stereonetprojection is the equal area or Lambert (Schmidt)net. On this net, any area on the surface of thereference sphere is projected as an equal area onthe stereonet. This property of the net is used inthe contouring of pole plots to find concentrationsof poles that represent preferred orientations, orsets of discontinuities. The other type of equat-orial projection is the equal angle or Wulff net;both the Wulff and Lambert nets can be usedto examine angular relationships, but only theLambert net can be used to develop contours ofpole concentrations.

These two nets are included in Appendix Iin a size that is convenient for plotting andanalyzing structural data. For hand plotting ofstructural data, the usual procedure is to placetracing paper on the nets and then draw polesand planes on the tracing paper as shown inFigure 2.8. Since plotting of great circles requiresthat the tracing paper be rotated on the net, asdescribed below, a thumbtack is placed at the

Figure 2.8 Geological data plotted and analyzed ona piece of tracing paper that is located over the centerof the stereonet with a pin to allow the paper to berotated.

center point so that the curves can be plottedwithout distortion.1

Further details on stereographic projections aredescribed by Phillips (1971) who discusses thetheoretical background to this technique, andLeyshon and Lisle (1996) who demonstratedapplications of this technique to geological map-ping. Goodman and Shi (1985) demonstratestereographic techniques for identifying wedgesof rock that can slide from the face, or are“removable”; this technique is termed key blocktheory.

2.5.2 Pole plots and contour plots

The pole to a plane, as shown in Figure 2.6,allows a point to represent the orientation of theplane. Pole plots, in which each plane is repres-ented by a single point, are the most convenientmeans of examining the orientation of a largenumber of discontinuities. The plot provides animmediate visual depiction of concentrations ofpoles representing the orientations of sets of dis-continuities, and the analysis is facilitated by theuse of different symbols for different types ofdiscontinuities.

Poles can be plotted by hand on a polar net asshown in Figure 2.9. On the net, the dip direc-tion scale (0–360◦) around the periphery has thezero mark at the bottom of the vertical axis andthe 180◦ mark is at the top of the net. This is aconvenience for plotting such that poles can beplotted directly without the need for rotating thetracing paper; it can be demonstrated that polesplotted on both the polar and equatorial nets arein identical positions.

Pole plots are commonly generated by stereo-graphic computer programs, an example of whichis shown in Figure 2.10. This is a lower hemi-sphere, equal angle projection of 421 original

1 For occasions in remote hotel rooms when a computer orlight table is not available for preparing stereonets, it hasbeen found that a television screen provides an adequatesubstitute. If the set is tuned to “snow,” the static electricityholds the papers to the screen and allows the tracing sheetto be rotated on the stereonet.


N180°

0°S

270° EW 90°

130°

Pole

Pile orientation50°/130°

Figure 2.9 Plotting poles on apolar net. Plot pole of planeoriented at 50/130—locate dipdirection of 130◦ clockwisearound the circumference of acircle starting at the lower end ofthe vertical axis. At 130◦ radialline, count 50◦ out from thecenter of the net, and plot a pointat the intersection between 130◦radial line and 50◦ circle.

FaultsJointsBedding

Surface type

1 [33]2 [253]9 [135]

N

S

E

Equal AreaLower Hemisphere

421 Poles421 Entries

W

Figure 2.10 Example ofpole plot of 421 planescomprising bedding,joints and faults.


poles mapped over an area of about a squarekilometer, at a site where the rock is a beddedlimestone. The rock contains discontinuity setscomprising the bedding and two sets of joints,together with a number of faults that are gener-ally coincident with the bedding. On Figure 2.10,there is a different symbol for each of the threetypes of discontinuity. While there is considerablescatter in the pole orientations, careful exam-ination of this plot shows that there is someclustering particularly in the southwest quadrant.In order to identify discontinuity sets on pole plotswith considerable scatter, it is necessary to pre-pare contours of the pole density, as described inthe next section.

2.5.3 Pole density

All natural discontinuities have a certain amountof variation in their orientations that results inscatter of the pole plots. If the plot contains polesfrom a number of discontinuity sets, it can be dif-ficult to distinguish between the poles from thedifferent sets, and to find the most likely orient-ation of each set. However, by contouring theplot, the most highly concentrated areas of polescan be more readily identified. The usual method

of generating contours is to use the contouringpackage contained with most stereographic pro-jection computer programs. Contouring can alsobe carried out by hand using a counting net suchas the Kalsbeek net that consists of mutually over-lapping hexagons, each with an area of 1/100 ofthe total area stereonet (Leyshon and Lisle, 1996);a Kalsbeek counting net is shown in Appendix I.Contouring is carried out by overlaying the count-ing net on the pole plot and counting the numberof poles in each square. For example, if there areeight poles out of a total of 421 poles in one square,then the concentration in that square is 2%. Oncethe percent concentration in each square has beendetermined, contours can be drawn.

Figure 2.11 shows a contour plot of the polesplotted in Figure 2.10. The contour plot showsthat the orientation of the bedding has relat-ively little scatter—the maximum concentrationis 5–6%, and that the mean orientation of thebedding has a dip of 74◦ and a dip directionof 050◦. In contrast, the joint orientations showmore scatter, and on the pole plot it is difficult toidentify discontinuity sets. However, on the con-toured plot, it is possible to distinguish clearlytwo sets of orthogonal joints. Set A has a shallowdip of about 26◦, and a dip direction of about219◦, that is in a direction at 180◦ to the bedding.

Bedding

Set B

N

S

W E

Set A

Set B

88/326

Bedding74/050

I1:04/138

I3:24/237

I2:75/049 Fisherconcentrations

% of total per 1.0% area

No bias correctionMax. conc. = 5.1298%

Equal areaLower hemisphere

421 Poles421 Entries

0 ~ 1 %1 ~ 2 %2 ~ 3 %3 ~ 4 %4 ~ 5 %5 ~ 6 %6 ~ 7 %7 ~ 8 %8 ~ 9 %9 ~ 10 %

Set A26/219

3 m 1 m

2 m

3 m1 m

2 m

+

Figure 2.11 Contoured plot ofdata shown in Figure 2.10,with great circlescorresponding to meanorientation of bedding andtwo orthogonal joint sets, andlines of intersection betweenplanes.


Faults

Surface type

1 [33]

N

S

E


33 Poles33 Entries

W I

Figure 2.12 Pole plot of faultsselected from data plotted inFigure 2.10.

Set B has a near vertical dip and a dip directionof 326◦, approximately at right angles to Set A.The poles for set B occur in two concentrations onopposite sides of the contour plot because somedip steeply to the north-west and others steeplyto the south-east.

In Figure 2.11, the different pole concentra-tions are shown by symbols for each 1% contourinterval. The percentage concentration refers tothe number of poles in each 1% area of the surfaceof the lower hemisphere.

A further use of the stereographic projectionprogram in analyzing structural data is to prepareplots of data selected from the total data collected.For example, joints with lengths that are only asmall fraction of the slope dimensions are unlikelyto have a significant influence on stability. How-ever, faults usually have greater persistence andlower friction angle than joints. Therefore, itwould facilitate design to prepare a stereographicplot showing only faults (Figure 2.12). This plotshows that only 33 discontinuities are faults,and that their orientations are similar to thoseof the bedding. Selections can also be made, forexample, of discontinuities that have a certaintype of infilling, or are slickensided, or showevidence of seepage, provided that the mappingidentifies this level of detail for each surface.

Appendix II contains field mapping sheets forrecording details of discontinuity properties bythe use of codes that are input directly into thestereographic analysis program.

The assignment of poles into discontinuity setsis usually achieved by a combination of contour-ing, visual examination of the stereonet, andknowledge of geological conditions at the site,which will frequently show trends in orientationof the sets. It is also possible to identify discon-tinuity sets by rigorous and less subjective ana-lysis of clusters in orientation data. A techniquepresented by Mahtab and Yegulalp (1982) iden-tifies clusters from random distributions of orien-tations using the Poisson’s distribution. However,in applying such techniques, a result that identi-fies more than about four concentrations shouldbe carefully examined before being used in design.

2.5.4 Great circles

Once the orientation of the discontinuity sets, aswell as important single discontinuities such asfaults, have been identified on the pole plots, thenext step in the analysis is to determine if thesediscontinuities form potentially unstable blocksin the slope face. This analysis is carried out byplotting great circles of each of the discontinuity


set orientations, as well as the orientation of theface. In this way the orientation of all the surfacesthat have an influence on stability are representedon a single diagram. Figure 2.11 shows the greatcircles of the three discontinuity sets identified bycontouring the pole plot in Figure 2.10. It is usu-ally only possible to have a maximum of aboutfive or six great circles on a plot, because witha greater number, it is difficult to identify all theintersection points of the circles.

While computer generated great circles are con-venient, hand plotting is of value in developing

an understanding of stereographic projections.Figure 2.13 illustrates the procedure for drawinggreat circles on an equal area net. As shown inFigure 2.8, the procedure involves overlaying thestereonet with tracing paper on which the greatcircles are plotted.

The primary purpose of plotting great circlesof discontinuity sets in a slope is to determ-ine the shape of blocks formed by intersectingdiscontinuities, and the direction in which theymay slide. For example, in Figure 2.1 the slopefailures only occurred for conditions where single

+

N(a)

(b)

(c) N

Pole

Greatcircle

Pole

Greatcircle

N

� = 130°

W E

� = 50° � = 50°

Figure 2.13 Construction of great circles and a pole representing aplane with orientation 50 (dip)/130 (dip direction) on an equal areanet: (a) with the tracing paper located over the stereonet by means ofthe center pin, trace the circumference of the net and mark the northpoint. Measure off the dip direction of 130◦ clockwise from northand mark this position on the circumference of the net; (b) rotatethe net about the center pin until the dip direction mark lies onthe W–E axis of the net, that is, the net is rotated through 40◦counterclockwise. Measure 50◦ from the outer circle of the net andtrace the great circle that corresponds to a plane dipping at this angle.The position of the pole, which has a dip of (90–50), is found bymeasuring 50◦ from the center of the net as shown, or alternatively40◦ from the outside of the net. The pole lies on the projection ofthe dip direction line which, at this stage of the construction, iscoincident with the W–E axis of the net; (c) the tracing is now rotatedback to its original position so that the north mark on the tracingcoincides with the north mark of the net. The final appearance of thegreat circle and the pole representing a plane dipping at 50◦ in a dipdirection of 130◦ is as illustrated.


discontinuities (Figure 2.1(a)), or pairs of inter-secting discontinuities (Figure 2.1(b)), dip out ofthe face. It is, of course, important to identifysuch potential failures before movement and col-lapse occurs. This requires an ability to visu-alize the three-dimensional shape of the wedgefrom the traces of the discontinuities on the faceof the original slope. The stereographic pro-jection is a convenient means of carrying outthe required three-dimensional analysis, keepingin mind that this procedure examines only theorientation of the discontinuities and not theirposition or dimensions. If the stereonet showsthe possible occurrence of a potentially unstable

block, examination of the location of the dis-continuities on the geological map would help todetermine if they intersect the slope.

2.5.5 Lines of intersection

The intersection of two planes defines a line inspace that is characterized by a trend (0–360◦)and plunge (0–90◦). In the stereographic projec-tion, this line of intersection is defined as the pointwhere the two great circles cross (Figure 2.14).The two intersecting planes may form a wedge-shaped block as shown in Figure 2.1(b), andthe direction in which this block may slide is

N

N

N

�i = 20.5°

(a)

(b)

W E

(c)

�i = 200.5°

Figure 2.14 Determination of orientation(plunge and trend) of line intersection betweentwo planes with orientations 50/130 and30/250: (a) the first of these planes has alreadybeen drawn in Figure 2.13. The great circledefining the second plane is obtained bymarking the 250◦ dip direction on thecircumference of the net, rotating the tracinguntil the mark lies on the W–E axis andtracing the great circle corresponding to a dipof 30◦; (b) the tracing is rotated until theintersection of the two great circles lies alongthe W–E axis of the stereonet, and the plungeof the line of intersection is measured as20.5◦; (c) the tracing is now rotated until thenorth mark coincides the north point on thestereonet and the trend of the line ofintersection is found to be 200.5◦.


determined by the trend of the line of intersec-tion. However, the existence of two intersectinggreat circles on the stereonet does not necessar-ily mean that a wedge failure will occur. Thefactors that influence the stability of the wedgeinclude the direction of sliding relative to the slopeface, the dip of the planes relative to the frictionangle, external forces such as ground water, andwhether the planes are located so that they actu-ally intersect behind the face. These factors arediscussed further in Section 2.6.3.

+

+

+

+

140°

N

A

A

64°

B

B

N

240°

(a)

(b)

Figure 2.15 Determination of angle between lineswith orientations 54/240 and 40/140: (a) the points Aand B that define the poles of these two lines aremarked on the stereonet as described in Figure 2.13for locating the pole; (b) the tracing is rotated untilthe two poles lie on the same great circle on thestereonet. The angle between the lines is determinedby counting the small circle divisions between A andB, along the great circle; this angle is found to be 64◦.The great circle on which A and B lie defines the planethat contains these two lines. The dip and directionof this plane are 60◦ and 200◦ respectively.

Figure 2.14 shows the procedure for measuringthe trend and plunge of the line of intersectionof two planes on the equal area stereonet, whileFigure 2.15 shows the procedure for measuringthe angle between two planes. The value of thesetwo measurements is that the orientation of theline of intersection shows the direction of sliding,and the angle between the planes gives an indica-tion of the wedging action where two planes inter-sect. If the angle between the planes is small, anarrow, tight wedge will be formed with a higherfactor of safety compared to a wide, open wedgein which the angle between the planes is large.

For the data shown on Figures 2.10 and 2.11,intersections occur between the bedding and jointset A (I1) and joint set B (I2), and betweenjoint sets A and B (I3). The orientations of thethree lines of intersection are also shown onFigure 2.11. Intersection line I3 has a trend of237◦ and a plunge of 24◦, and joint sets A and Bcould together form a wedge failure that wouldslide in the direction of the trend. Intersection lineI1 is almost horizontal so the wedge formed by thebedding and joint set A is unlikely to slide, whileintersection line I2 is near vertical and would forma thin wedge in the face.

2.6 Identification of modes of slopeinstability

Different types of slope failure are associated withdifferent geological structures and it is importantthat the slope designer be able to recognize poten-tial stability problems during the early stages ofa project. Some of the structural patterns thatshould be identified when examining pole plotsare outlined on the following pages.

Figure 2.16 shows the four types of failure con-sidered in this book, and typical pole plots ofgeological conditions likely to lead to such fail-ures. Note that in assessing stability, the cut faceof the slope must be included in the stereo plotsince sliding can only occur as the result of move-ment towards the free face created by the cut.The importance of distinguishing between thesefour types of slope failure is that there is a spe-cific type of stability analysis for each as shown


N

N

N

�f�s

�f

�i�s

�f�t

Randomly orienteddiscontinuities

Legend

Pole concentrations

Great circle representingface

Great circle representingplane corresponding to centersof pole concentrations

�f dip direction of face�s direction of sliding�t direction of toppling�i dip direction, line of intersection

(a)

(b)

(c)

(d)

N

Figure 2.16 Main types of block failuresin slopes, and structural geologyconditions likely to cause these failures:(a) plane failure in rock containingpersistent joints dipping out of the slopeface, and striking parallel to the face;(b) wedge failure on two intersectingdiscontinuities; (c) toppling failure instrong rock containing discontinuitiesdipping steeply into the face; and(d) circular failure in rock fill, very weakrock or closely fractured rock withrandomly oriented discontinuities.

in Chapters 6–9, and it is essential that the correctanalysis method be used in design.

The diagrams given in Figure 2.16 have beensimplified for the sake of clarity. In an actual rockslope, several types of geological structures maybe present, and this may give rise to additionaltypes of failure. For example, in Figure 2.11, aplane failure could occur on joint set A, while thebedding could form a toppling failure on the sameslope.

In a typical field study in which structuraldata have been plotted on stereonets, a num-ber of significant pole concentrations may bepresent. It is useful to be able to identify thosethat represent potential failure planes, and toeliminate those that represent structures that areunlikely to be involved in slope failures. Tests foridentifying important pole concentrations havebeen developed by Markland (1972) and Hocking(1976). These tests establish the possibility of


a wedge failure in which sliding takes place alongthe line of intersection of two planar discontinu-ities as illustrated in Figure 2.16(b). Plane failureshown in Figure 2.16(a) is also covered by thistest since it is a special case of wedge failure. Fora wedge failure, contact is maintained on bothplanes and sliding occurs along the line of inter-section between the two planes. For either planeor wedge failure to take place, it is fundamentalthat the dip of the sliding plane in the case of planefailure, or the plunge of the line of intersection inthe case of wedge failure, be less than the dip ofthe slope face (i.e. ψi < ψf) (Figure 2.17(a)). Thatis, the sliding surface “daylights” in the slope face.

The test can also differentiate between slidingof a wedge on two planes along the line of inter-section, or along only one of the planes suchthat a plane failure occurs. If the dip directionsof the two planes lie outside the included anglebetween αi (trend of intersection line) and αf (dipdirection of face), the wedge will slide on bothplanes (Figure 2.17(b)). If the dip direction of oneplane (A) lies within the included angle betweenαi and αf , the wedge will slide on only that plane(Figure 2.17(c)).

2.6.1 Kinematic analysis

Once the type of block failure has been identifiedon the stereonet, the same diagram can also beused to examine the direction in which a blockwill slide and give an indication of stability con-ditions. This procedure is known as kinematicanalysis. An application of kinematic analysisis the rock face shown in Figure 2.1(b) wheretwo joint planes form a wedge which has slidout of the face and towards the photographer.If the slope face had been less steep than theline of intersection between the two planes, orhad a strike at 90◦ to the actual strike, thenalthough the two planes form a wedge, it wouldnot have been able to slide from the face. Thisrelationship between the direction in which theblock of rock will slide and the orientation ofthe face is readily apparent on the stereonet.However, while analysis of the stereonet gives agood indication of stability conditions, it does

N

Plane A

Plane B Direction of sliding, �i, �s

Dip direction of face, �f

�f

�i

Face

N

Plane A

Plane B

�A

Face

�B

Plane A

Plane B

FaceN

�i

�f

�i�A

�f

�B

(a)

(b)

(c)

Figure 2.17 Identification of plane and wedge failureson stereonet: (a) sliding along line of intersection ofplanes A and B is possible where the plunge of thisline is less than the dip of the slope face, measured inthe direction of sliding, that is, ψi < ψf ; (b) wedgefailure occurs along line of intersection (dip directionαi) on slope with dip direction αf because dipdirections of planes A and B (αA and αB) lie outsideincluded angle between αi and αf ; (c) plane failureoccurs on plane A (dip direction αA) on slope withdip direction αf because dip direction of planes A liesinside included angle between αi and αf .


not account for external forces such as waterpressures or reinforcement comprising tensionedrock bolts, which can have a significant effect onstability. The usual design procedure is to usekinematic analysis to identify potentially unstableblocks, followed by detailed stability analysis ofthese blocks using the procedures described inChapters 6–9.

An example of kinematic analysis is shown inFigure 2.18 where a rock slope contains three setsof discontinuities. The potential for these discon-tinuities to result in slope failures depends on theirdip and dip direction relative to the face; stabil-ity conditions can be studied on the stereonet asdescribed in the next section.

A(a)

(b)

B

�B > �f: stable

�A < �f: sliding possible

�B�C

Toppling set

B

A

C

C�f

�A

Great circle offace, dip �f

20°10°

20° 10°

PAAPBB

Pf

(90° – �f) �j PCC �f

Legend

Daylight envelope for wedges

Daylight envelope for planar failures

Toppling envelope

N

Figure 2.18 Kinematic analysis of blocks of rock inslope: (a) discontinuity sets in slope; and (b) daylightenvelopes on equal area stereonet.

2.6.2 Plane failure

In Figure 2.18(a), a potentially unstable planarblock is formed by plane AA, which dips at a flat-ter angle than the face (ψA < ψf ) and is saidto “daylight” on the face. However, sliding isnot possible on plane BB which dips steeper thanthe face (ψB > ψf ) and does not daylight. Simil-arly, discontinuity set CC dips into the face andsliding cannot occur on these planes, althoughtoppling is possible. The poles of the slope faceand the discontinuity sets (symbol P) are plottedon the stereonet in Figure 2.18(b), assuming thatall the discontinuities strike parallel to the face.The position of these poles in relation to the slopeface shows that the poles of all planes that day-light and are potentially unstable, lie inside thepole of the slope face. This area is termed the day-light envelope and can be used to identify quicklypotentially unstable blocks.

The dip direction of the discontinuity sets willalso influence stability. Plane sliding is not pos-sible if the dip direction of the discontinuitydiffers from the dip direction of the face by morethan about 20◦. That is, the block will be stableif |αA − αf | > 20◦, because under these condi-tions there will be an increasing thickness of intactrock at one end of the block which will have suf-ficient strength to resist failure. On the stereonetthis restriction on the dip direction of the planesis shown by two lines defining dip directions of(αf + 20◦) and (αf − 20◦). These two lines desig-nate the lateral limits of the daylight envelope onFigure 2.18(b).

2.6.3 Wedge failure

Kinematic analysis of wedge failures (Figure2.16(b)) can be carried out in a similar mannerto that of plane failures. In this case the pole ofthe line of intersection of the two discontinuitiesis plotted on the stereonet and sliding is possibleif the pole daylights on the face, that is (ψi < ψf ).The direction of sliding of kinematically permiss-ible wedges is less restrictive than that of planefailures because there are two planes to formrelease surfaces. A daylighting envelope for the


line of intersection, as shown on Figure 2.18(b),is wider than the envelope for plane failures.The wedge daylight envelope is the locus of allpoles representing lines of intersection whose dipdirections lie in the plane of the slope face.

2.6.4 Toppling failure

For a toppling failure to occur, the dip directionof the discontinuities dipping into the face mustbe within about 10◦ of the dip direction of the faceso that a series of slabs are formed parallel to theface. Also, the dip of the planes must be steepenough for interlayer slip to occur. If the faces ofthe layers have a friction angle φj, then slip willonly occur if the direction of the applied com-pressive stress is at angle greater than φj with thenormal to the layers. The direction of the majorprincipal stress in the cut is parallel to the faceof the cut (dip angle ψf ), so interlayer slip andtoppling failure will occur on planes with dip ψpwhen the following conditions are met (Goodmanand Bray, 1976):

(90◦ − ψf) + φj < ψp (2.3)

These conditions on the dip and dip directionof planes that can develop toppling failures aredefined on Figure 2.18(b). The envelope definingthe orientation of these planes lies at the oppositeside of the stereonet from the sliding envelopes.

2.6.5 Friction cone

Having determined from the daylight envelopeswhether a block in the slope is kinematically per-missible, it is also possible to examine stabilityconditions on the same stereonet. This analysisis carried out assuming that the shear strength ofthe sliding surface comprises only friction and thecohesion is zero. Consider a block at rest on aninclined plane with a friction angle of φ betweenthe block and the plane (Figure 2.19(a)). For anat-rest condition, the force vector normal to theplane must lie within the friction cone. When theonly force acting on the block is gravity, the poleto the plane is in the same direction as the normal

Normal toplane

Weightvector

(a)

(b)

Frictioncone, � = 35°

�f = 30°

N �f = 80° �f = 60°

Frictioncone

10°

10°

LegendEnvelopes of potential instability:

Wedges;

Plane failures;

Toppling failures;

Envelopes for �f = 80°;

Envelopes for �f = 60°.

�

Figure 2.19 Combined kinematics and simple stabilityanalysis using friction cone concept: (a) friction conein relation to block at rest on an inclined plane (i.e.φ > ψp); and (b) stereographic projection of frictioncone superimposed on “daylighting” envelopes.

force, so the block will be stable when the polelies within the friction circle.

The envelopes on Figure 2.19(b) show thepossible positions of poles that may form unstable


blocks. Envelopes have been drawn for slopeface angles of 60◦ and 80◦, which show that therisk of instability increases as the slope becomessteeper as indicated by the larger envelopes forthe steeper slope. Also, the envelopes become lar-ger as the friction angle diminishes. The envelopesalso indicate that, for the simple gravity loadingcondition, instability will only occur in a limitedrange of geometric conditions.

2.6.6 Applications of kinematic analysis

The techniques demonstrated on Figures 2.16–2.19 to identify both potentially unstable blocksof rock on the slope and the type of instabilitycan readily be applied to the preliminary stagesof slope design. This is illustrated in the twoexamples that follow.

Highway: A proposed highway on a north–southalignment passes through a ridge of rock in whicha through-cut is required to keep the highwayon grade (Figure 2.20(a)). Diamond drilling andmapping shows that the geological conditions inthe ridge are consistent so that the same structurewill be exposed in each face. The predominantgeological structure is the bedding that strikesnorth–south, parallel to the highway alignmentand dips to the east at angles of between 70◦ and80◦ (i.e. dip and dip direction of 70–80/090).

The stereonets in Figure 2.20(b) show polesrepresenting the dip and dip direction of thebedding, and great circles representing the orient-ations of the left and right cut faces. Also plottedon the stereonets is a friction cone representinga friction angle of 35◦ on the bedding. Thesestereonets show that on the left (west) face, thebedding dips towards the excavation at a steeperangle than the friction angle so sliding can occuron the bedding. The cut face has been made alongthe bedding to create a stable face.

On the right (east) face the bedding dips steeplyinto the face and there is a possibility that theslabs formed by these joints will fail by toppling.According to equation (2.3), toppling is possibleif (90◦ − ψf) + φj < ψp. If the face is cut at 76◦(0.25V:1H) and the friction angle is 35◦, then the

left side of this equation equals 49◦, which is lessthan the dip of the bedding (70◦–80◦). The poten-tial for toppling is indicated by the poles to thebedding planes lying inside the toppling envelope.

This preliminary analysis shows that the right(east) cut slope has potential stability problemsand that more detailed investigation of struc-tural geology conditions would be required beforefinalizing the design. The first step in this invest-igation would be to examine the spacing of thebedding planes and determine if the center ofgravity of the slabs will lie outside the base, inwhich case toppling is likely. Note that it is rarelypossible to change the alignment sufficiently toovercome a stability problem, so it may be neces-sary to either reduce the slope angle on the rightside, or stabilize the 76◦ face.

Open pit slopes: During the feasibility studies ona proposed open pit mine, an estimate of the safeslope is required for the calculation of ore-to-waste ratios and the preliminary pit layout. Theonly structural information that may be avail-able at this stage is that which has been obtainedfrom diamond drill cores drilled for mineral eval-uation purposes, and from the mapping of surfaceoutcrops. Scanty as this information is, it doesprovide a basis for preliminary slope design.

A contour plan of the proposed open pit mineis presented in Figure 2.21 and contoured ste-reo plots of available structural data are super-imposed on this plan. Two distinct structuralregions, denoted by A and B, have been identifiedand the boundary between these regions has beenmarked on the plan. For the sake of simplicity,major faults have not been shown. However, it isessential that any information on faults should beincluded on large-scale plans of this sort and thatthe potential stability problems associated withthese faults should be evaluated.

Overlaid on the stereonets are great circles rep-resenting the orientations of the east and west pitfaces, assuming an overall slope angle of 45◦. Alsoshown on the stereonets is a friction cone of 30◦,which is assumed to be the average friction anglesof the discontinuity surfaces. The stereonets showthat the western and southern portions of the pit


N

Left side

(a)

(b)

Potential planar sliding zone Potential toppling zone

Right side

Slip limit

Bedding58/090

Face76/270

(90–76°) �= 35°

Face parallelto bedding

Friction cone(�= 35°)

± 20° ± 10°

N

S S

E EW W

Figure 2.20 Relationship between structural geology and stability conditions on slope faces in through-cut:(a) photograph of through-cut showing two failure mechanisms—plane sliding on left side (west), and topplingon right side (east) on Route 60 near Globe AZ; (b) stereographic plots showing kinematics analysis of leftand right cut slopes.

are likely to be stable at the proposed slope of45◦. This suggests that, if the rock is strong andfree of major faults, these slopes could probablybe steepened or, alternately, this portion of the pit

wall could be used as a haul road location withsteep faces above and below the haul road.

On the other hand, the northeastern portionof the pit contains a number of potential slope


A1

A1

A1

A3

Potentialplane failures

Potential wedgefailures

Potentialtoppling

A2

A2

A3

A3

Stable

B2

B1

B2

B1

Structural region B Structural region A

Proposed pitbottom

N

ToppleA2

Wedge(A1, A3)

Plane (A1)

EW

N

S

W

S

N

E

Figure 2.21 Presentation of structural geology on stereonets, and preliminary evaluation of slope stability ofproposed open pit mine.

problems. The northern face is likely to sufferfrom plane sliding on discontinuity set A1, sincethis set will daylight in the face at an angle steeperthan the friction angle. Wedge failures on theintersection of sets A1 and A3 are possible in thenortheastern corner of the pit, and toppling fail-ure on set A2 may occur in the eastern slopes.

Indications of potential instability would suggestthat consideration be given to flattening the slopesin the northeastern part of the proposed pit.

It is interesting to note that three types ofstructurally controlled slope failure can occur inthe same structural region, depending upon theorientation of the slope face.


2.7 Example Problem 2.1: stereo plotsof structural geology data

Statement

A structural geology mapping program for a pro-posed highway produced the following resultsfor the orientation of the discontinuities (format-dip/dip direction).

40/080 45/090 20/160 80/310 83/312 82/30523/175 43/078 37/083 20/150 21/151 39/07470/300 75/305 15/180 80/010 31/081

Required

(a) Plot the orientation of each discontinuity as apole on a stereonet using the equal area polarnet and tracing paper.

(b) Estimate and plot the position of themean pole of each of the three sets ofdiscontinuities.

(c) Determine the angle between the mean poleswith the steepest and shallowest dips.

(d) Plot the great circles of the mean pole of eachset on the equal area equatorial net.

(e) Determine the plunge and trend of the line ofintersection between the joint sets with thesteepest and intermediate dip angles.

Solution

(a) The poles of the 17 planes are plotted onFigure 2.22, which shows that there are threesets of discontinuities.

(b) The mean pole of each discontinuity set is asfollows:

Set 1 : 78/305; Set 2 : 40/081;

Set 3 : 20/163

There is one pole (80/010) that does notbelong to any of the three discontinuity sets.

(c) The angle between the mean poles of jointsets 1 and 3 is determined by rotating the ste-reonet until both mean poles lie on the same

N180

0

90 270

2

3

1

20/163

40/081

78/305

94°

Poles (17)Mean pole of set

Figure 2.22 Example problem 2.1—plot of poles ofstructural mapping data (lower hemisphereprojection).

N0

180

270 90

78/3051

2

320/163

40/081

I1,2= 27/029

�i = 27°

Figure 2.23 Example problem 2.1—great circles ofmean poles plotted using equatorial net.

great circle. The number of divisions on thisgreat circle is 94◦ as shown by the dotted lineon Figure 2.22. The actual included angle is86◦ (180 − 94).

(d) The great circles of the three mean poles areplotted on Figure 2.23.


(e) The plunge and trend of the line of intersec-tion of joint sets 1 and 2 are also shown onFigure 2.23. The values are as follows:

Dip, ψi = 27; dip direction, ψi = 029

2.8 Example Problem 2.2: slope stabilityevaluation related to structuralgeology

Statement

In order to make a 90◦ curve in a highway, a rockcut has been excavated that follows the curve ofthe highway, the face is cut at an angle of 50◦.

The joints in the rock at the site form three setswith the following orientations:

Set 1 : 78/305; Set 2 : 40/081;

Set 3 : 20/163

Figure 2.24 shows the alignment of the highwayand the orientation of the three sets in the slope.The friction angle of the joint surfaces is 25◦.

Required

(a) On a piece of tracing paper draw a greatcircle representing the 50◦ slope face and a25◦ friction circle.

(b) Determine the most likely mode of failure,that is, plane, wedge or toppling, on thefollowing slopes:

• East dipping slope.• North dipping slopes.

(c) State the joint set or sets on which slidingwould occur on each slope.

(d) Determine the steepest possible slope anglefor these two slopes assuming that only theorientation of the discontinuities and the

2 1

3

3

2

1

78°

40°

20°

20°

78°40°

Cut slope

Highway

Geological structure:Strike

Dip

100

90

80 7060 Figure 2.24 Cut slope and

geological structures in cut face.


friction angle of the surfaces have to beconsidered.

Solution

(a) Figure 2.25 shows the great circle of theslope dipping to the east at a face angle of50◦; and a 25◦ friction circle plotted as asmall circle.

(b) The stability evaluation is carried out byplacing first the tracing of the great circles(Figure 2.23) and then the tracing of theslope face and friction circle (Figure 2.25)on the equal area net. The tracing of theslope great circle is rotated to the corres-ponding orientation of the slope face to givethe following results:

East dipping slope: Plane failure possible onjoint set 2 (Figure 2.26). Sliding could be pre-vented by cutting the slope at 40◦ coincidentwith the joint surfaces. If the friction anglehad been greater than 40◦, then sliding maynot occur on these planes.North dipping slope: Wedge failure possibleon joint sets 1 and 2 (Figure 2.27). Sliding

270 90

N0

180

Great circle ofslope face

Frictioncircle

�f = 50°

�= 25°

Figure 2.25 Great circle of slope dipping at 50◦, and25◦ friction circle.

could be prevented by cutting the slope atan angle of 27◦ so that the wedges are notundercut by the slope.

N0

180

270 90

2

50/090

40/081

40°

50°

�2

�f

Figure 2.26 Stability conditions on east dippingslope—plane failure on joint set 2.

N0 �f

�2

�i

�1

180

270 90

40/081

50/360

78/305

Set

1

1

Set2

2

Figure 2.27 Stability conditions on north dippingslope—wedge failure on joint set 1 and joint set 2.

Chapter 3

Site investigation and geologicaldata collection

3.1 Planning an investigation program

The design of rock cuts is often an iterativeprocess that proceeds from initial reconnaissance,through preliminary and final design followed byconstruction. This process involves progressivelycollecting more detailed design data, specific tothe site conditions and needs of the project. Typ-ically, the three stages of a complete investigationare as follows:

• Reconnaissance: examination of publishedgeological maps and reports, study of air pho-tographs, gathering of local experience, fieldvisits to examine, if possible, the performanceof existing slopes in similar geological condi-tions, and geophysics studies if outcrops arelimited.

• Route selection/preliminary pit slope design:if the project involves the evaluation of alter-native routes, limited investigations could becarried out of each route comprising out-crop mapping, geophysics to find overburdenthickness and index tests of rock properties.For an open pit mine, there will usually beconsiderable geological information on theproperty generated during the explorationprogram. This will often include mapping,geophysics and drilling from which geotech-nical data can be obtained. It is beneficial tothe design of the pit slopes if geotechnical datacan be collected as part of the explorationprogram.

• Detailed investigations: final design wouldusually require detailed mapping of outcrops

and existing cuts to study structural geo-logy, test pits to obtain information onoverburden thickness and properties, anddiamond drilling to investigate rock condi-tions at depth. Components of the drillingcould include core orientation to obtain struc-tural geology information, and installation ofpiezometers to measure ground water levels,and possibly measure permeability. Rockstrength testing could comprise laboratorytesting of drill core to determine the frictionangle of discontinuities, uniaxial compres-sion strength tests and slake durability tests.Figure 3.1 shows a diamond drilling programin progress where an inclined hole and tripletube core barrel are being used to investigatetargeted geological structure and collect highquality core in closely fractured rock. Someof the current (2003) requirements for drillingand sampling programs may include collec-tion of all circulation water (“zero dischargedrilling”), and complete restoration of the drillsite and access roads.

Because of the wide variety of both site conditionsand slope designs, it is not considered appro-priate to draw up any rules on the types andquantity of investigation programs. That is, everyinvestigation is unique. The only general rule thatapplies to investigation for rock slope design isthat information is required on geology, rockstrength and ground water. These three sets ofdesign parameters are discussed in the followingsections.

Site investigation and geological data collection 47

Figure 3.1 Photograph of typical diamond drilling equipment for hole depths up to about 300 m.

3.1.1 Geology

A distinguishing feature of many investigationsfor rock slopes is that it is particularly importantto focus on the details of the structural geology.For example, the orientation of one clay-filledfault that dips out of the face can make thedifference between stability and instability. Struc-tural geology data provided by surface mapping,where available, is usually more reliable than thatobtained by diamond drilling because outcropsand cuts show larger scale features and undis-turbed in situ conditions compared to the verysmall volume of a drill core. Furthermore, the ori-entation of discontinuities in core is not knownunless the core is oriented.

It is recommended wherever possible, that themapping be carried out by the same person orengineering group who will carry out the designso that the objectives of the mapping program areclearly identified and the data collected is relev-ant to the design. For example, a large number ofshort, non-persistent joints that have little influ-ence on the rock mass strength or stability should

be given less attention during mapping than alimited number of shears with continuous lengthsequal to that of the slope height. A design engineerwho is analyzing the data and is not familiar withthe site, may be unable to distinguish on a con-toured stereonet the relative importance betweenthe many non-persistent joints and the shears.

Alternative approaches to geological investiga-tions are as follows. First, there may be a numberof existing slopes, either natural or excavated,near the site where the geological conditions aresimilar to those on the project. In this case, strongreliance could be placed on extrapolating theperformance of these slopes to the new design.In these circumstances, it may not be necessary tocollect additional data, except to carefully doc-ument existing slope performance and assessinghow this may be applied to the proposed design.Alternatively, where there is little local experiencein cut slope stability, it may be necessary to con-duct an extensive investigation program involvingmapping, drilling, and laboratory testing. As thisprogram develops, it should be modified to suit

48 Site investigation and geological data collection

particular conditions at the site. For example, thedrilling and mapping may show that although therock is strong and the jointing is favorable to sta-bility, there are a number of faults that couldcontrol stability conditions. The investigationprogram would then concentrate on determin-ing the location and orientation of these faults,and their shear strength properties. This chapterdescribes geological investigation methods.

3.1.2 Rock strength

The rock strength parameters that are used inslope design are primarily the shear strength ofdiscontinuities and the rock mass, the weather-ing characteristics of the rock where applicable,and to a lesser extent the compressive strengthof intact rock. The shear strength of discontinui-ties can be measured in the laboratory on samplesobtained from drill core, or samples cut fromlumps of rock that are intersected by a discontinu-ity. The shear strength of the rock mass can eitherbe determined by back-analysis of slope fail-ures, or calculated by an empirical method thatrequires information on the intact rock strength,the rock type and the degree of fracturing. Thecompressive strength of rock can be measured oncore samples, or from index tests applied to out-crops in the field. The susceptibility of rock toweathering can also be measured in the laborat-ory, or assessed by field index tests. Details ofrock testing methods are described in Chapter 4.

3.1.3 Ground water

The investigation of ground water plays animportant part in any slope design program. Inclimates with high precipitation levels, waterpressures should always be included in the design.The design water pressures should account forlikely peak pressures that may develop duringintense rainfall events or snow melt periods,rather than the pressure due to the average sea-sonal water table. Furthermore, if drainage mea-sures are installed, the design may account for thepossible degradation of these systems due to lackof maintenance.

Similarly to geological investigations, theextent of the ground water investigation will alsodepend on site conditions. In most cases, it issufficient to install piezometers to measure theposition and variation in the water table so thatrealistic values of water pressure can be used indesign. However, if it is planned to install extens-ive drainage measures such as a drainage adit,then measurements of permeability are benefi-cial to assess whether the adit will be successfulin draining the slope, and then to determine theoptimum location and layout of the adit and drainholes. Details of ground water investigations aredescribed in Chapter 5.

3.2 Site reconnaissance

The following is a discussion on some of thereconnaissance techniques that may be used earlyin a project, mainly for the purpose of projectevaluation. It is rare that the information gatheredat this stage of a project would be adequate for usein final design, so these studies would be followedby more detailed investigations such as surfacemapping and drilling once the overall projectlayout has been finalized.

Part of any site reconnaissance is the collectionof all relevant existing data on the site, rangingfrom published data from both government andprivate sources to observations of the perform-ance of existing natural slopes and cut faces.These sources will provide such information asrock types, depth of weathering, likely slope fail-ure modes and the frequency and size of rockfalls.

An important first step in the reconnaissancestage of a project is to define zones, in eachof which the geological properties are uniformwith regards to the requirements of the pro-ject (ISRM, 1981a). Typical boundaries betweenzones include rock type contacts, faults or majorfolds. The zoning of the rock mass should provideinformation on the location, orientation andtype of boundary between zones, as well assome information on the engineering propertiesof the rock mass in each. By defining the bound-aries of each zone, it is possible to determine


the extent to which stability conditions will varyalong the alignment or around the pit, and planmore detailed investigations in those zones inwhich there is potential for instability.

Figure 3.2 shows an example of areconnaissance-level geologic map for a highwayproject. This map shows basic location featuressuch as universal transverse Mercator (UTM) co-ordinates, a river, the existing highway and rail-way, and the proposed new alignment. Note thatground contours have been left off the map forthe sake of clarity. The geologic data includes therock type, a landslide, thrust faults, and the strikeand dip orientation of discontinuities mappedon outcrops. The numbers such as “10-3” arereferences linked to a table that provides moreinformation on the characteristics of each discon-tinuity; the locations of the discontinuities weredetermined using a GPS (geographical position-ing system) unit. Orientations of the bedding andtwo primary orthogonal joint sets are shown onthe stereonets on Figures 2.11–2.13.

3.2.1 Aerial and terrestrial photography

The study of stereographic pairs of vertical aerialphotographs or oblique terrestrial photographsprovides much useful information on the larger-scale geological conditions at a site (Petersonet al., 1982). Often these large features will bedifficult to identify in surface mapping becausethey are obscured by vegetation, rock falls ormore closely spaced discontinuities. Photographsmost commonly used in geotechnical engineer-ing are black and white, vertical photographstaken at heights of between 500 and 3000 m withscales ranging from 1:10,000 to 1:30,000. Onsome projects, it is necessary to have both highand low level photographs, with the high levelphotographs being used to identify landslidesfor example, while the low level photographsprovide more detailed information on geologicalstructure.

One of the most important uses of aerial photo-graphs is the identification of landslides thathave the potential for causing movement, oreven destruction of facilities. Landslide features

that are often readily apparent on vertical aerialphotographs are scarps along the crest of the slide,hummocky terrain in the body of the slide andareas of fresh disturbance in the toe, includingsudden changes in river direction. By compar-ing photographs taken over a number of years,it may be possible to determine the rate of move-ment of a slide, and whether it is growing in size.Figure 3.3 shows stereo pair aerial photographs ofthe Hope Slide in British Columbia that occurredin 1965. The volume of rock that failed was about47 million m3; a 3.2 km length of the highway wasburied to a depth of about 80 m. The slide massfailed on a continuous foliation plane dipping atabout 30◦ out of the slope face.

Other features that may be evident on aerialphotographs are major geological structures suchas faults, bedding planes and continuous jointsets. The photographs may provide informationon the position, length and continuity of thesefeatures (Goodman, 1976).

3.2.2 Geophysics

Geophysical methods are often used in the recon-naissance or preliminary stages of a site invest-igation program to provide such information asthe depth of weathering, the bedrock profile,contacts between rock types of significantly dif-ferent density, the location of major faults, andthe degree of fracturing of the rock (Griffithsand King, 1988). The results obtained from geo-physical measurements are usually not sufficientlyaccurate to be used in final design and shouldpreferably be calibrated by putting down a num-ber of test pits or drill holes to spot check actualproperties and contact elevations. However, geo-physical surveys provide a continuous profile ofsubsurface conditions and this information can beused as a fill-in between drill holes. For rock slopeengineering purposes the most common geophy-sical investigation method is seismic refraction asdescribed next.

Seismic methods. Seismic surveys are used todetermine the approximate location and densityof layers of soil and rock, a well-defined water


11-3

81°

11-30

11-3180°

86°

15°

11-32

11-33

11-34

11-35

10-1

10-1

10-2

10-3

10-4

11-22

11-26

11-27

11-2511-28

10-510-6

10-7

11-21

89°

85° 85°80°

80°

90°

65°

90°

86°

88°

89°

90°

85°

72°

85°

80°78°52°

86°

83°

514500E

514500E

514500E

514300E

5694

100N

5694

200N

5694

000N

5684

100N

71°

RailwayCliffs

River

Thin/thick bedded,grey LIMESTONE

Hig

hway

Park

Brid

ge S

lide

New alignment

Limestone

Shaley limestone

Limestone

Fault

N

Drainage adit

Legend

Discontinuity orientation withreference number

Surface trace of fault withmovement direction

For stereonet of geologicalstructure—See Figure 2.10

Figure 3.2 Typical reconnaissance-level geological map showing structural geology features (mapping byDr C. H. B. Leitch).


Figure 3.3 Stereo pair aerial photographs of Hope Slide, British Columbia; slide had volume of 47 million m3

of rock and buried a 3.2 km length of highway.

table, or the degree of fracturing, porosity andsaturation of the rock. Seismic velocities of a vari-ety of rock types have also been correlated withtheir rippability, which is a useful guideline inthe selection of rock excavation methods (Cater-pillar, 2001). The seismic method is effective todepths in the range of tens of meters to a max-imum of a few hundred meters. Discontinuitieswill not be detected by seismic methods unlessthere is shear displacement and a distinct eleva-tion change of a layer with a particular density asa result of fault movement.

Seismic surveys measure the relative arrivaltimes, and thus the velocity of propagation, ofelastic waves traveling between a shallow energysource and a number of transducers set out in astraight line along the required profile. The energysource may be a hammer blow, an explosion ofa propane–oxygen mixture in a heavy chamber(gas-gun), or a light explosive charge. In elast-ically homogeneous ground subject to a suddenstress near its surface, three elastic pulses travel

outward at different speeds. Two are body wavesthat are propagated as spherical fronts affectedto only a minor extent by the free surface of theground. The third wave is a surface wave (Raleighwave) that is confined to the region near the sur-face; its amplitude falling off rapidly with depth.The two body waves, namely the primary or “P”wave and the secondary or “S” wave, differ inboth their direction of motion and speed. TheP wave is a longitudinal compressive wave inthe direction of propagation, while the S waveinduces shear stresses in the medium. The velocit-ies of the primary (Vp) and secondary (Vs) wavesare related to the elastic constants and density ofthe medium by the equations:

Vp =(

(K + 4G/3)

γr

)1/2

(3.1)

Vs =(

G

γr

)1/2

(3.2)


Air

Water

Ice

Superficialdeposits

Saturated shalesand sandstones

Limestones

Chalk

Salt

Common igneous andmetamorphic rocks

Ultramafic rocks

Vp (m/s × 1000)

0 1 2 3 4 5 6 7 8 9

Figure 3.4 Approximate range of P wave velocities(Vp) for some common geological materials (Griffithsand King, 1988).

where K is the bulk modulus, G is the shear mod-ulus and γr is the rock density. The velocity ofthe S wave in most rocks is about one half that ofthe P wave, and the S wave is not propagated atall in fluids. The ratio Vp/Vs depends only on thePoisson’s ratio of the medium. Figure 3.4 showstypical values of the P wave for a variety of rocktypes.

Some of the rock properties that can be assessedfrom the behavior waves include the strength (orconsolidation), the density and degree of frac-turing. For example, the amplitude of the wavesdecreases with distance from their source becausethe energy wave spreads over the increasing wavefront area. Earth materials are imperfectly elasticleading to energy loss and attenuation of theseismic waves that is greater than would be expec-ted from geometric spreading alone. This reduc-tion in amplitude is more pronounced for lessconsolidated rocks.

The sonic velocity of the elastic wave will begreater in higher density material, and in moremassive rock compared to low density, closelyfractured rock; this information is used to assessthe rippability of rock. Where a layer of lowdensity material overlies a denser layer, suchas soil overlying bedrock, then the elastic wave

velocity will be greater in the bedrock and thecontact between the layers will act as a refract-ing surface. In a specific range of distances fromthe shot point, the times of first arrival at differentdistances from the shot point will represent wavestraveling along this surface. This information canbe used to plot the contact profile between thetwo layers.

3.3 Geologic mapping

Geological mapping of surface outcrops or exist-ing cuts, in similar geological formations to thatin which the excavation will be made, usuallyfurnishes the fundamental information on siteconditions required for slope design. While map-ping is a vital part of the investigation program,it is also an inexact process because a certainamount of judgment is usually required to extra-polate the small amount of information availablefrom surface outcrops to the overall cut slope(McClay, 1987).

In order to produce geological maps anddescriptions of the engineering properties of therock mass that can be used with confidence indesign, it is important to have a well-defined pro-cess that produces comparable results obtainedby different personnel working at several sites.To meet these requirements, standard mappingprocedures have been drawn up which have thefollowing objectives:

• Provide a language that enables observers totransmit their general impression of a rockmass, particularly with regard to its antici-pated mechanical behavior. The language ofthe geological description must be unambigu-ous so that different observers of a given rockmass describe the rock mass in the same way.

• Contain as far as possible quantitative dataof interest to the solution of definite practicalproblems.

• Whenever possible, use simple measurementsrather than visual observations alone.

• Provide a complete specification of the rockmass for engineering purposes.


3.3.1 Line and window mapping

Methods of structural mapping that will system-atically examine all significant geological featuresare “line” and “window” mapping (see alsoAppendix II).

Line mapping comprises stretching a tape alongthe face and mapping every discontinuity thatintersects the line; line lengths are normallybetween 50 and 100 m. If the ends of the lineare surveyed, then the location of all the discon-tinuities can be determined. Window mappingcomprises mapping all discontinuities within arepresentative segment or “window” of fixed size,spaced at regular intervals along the exposure.The intervening areas are examined for similarityof structure. The dimensions of a window wouldnormally be about 10 m. Either of these mappingtechniques may be used in both the reconnaissanceand final design stages of a project, depending onthe extent of the face available for mapping. If theinitial investigations identify a particular featurethat is likely to have a significant effect on stability,then more detailed mapping, such as roughnessand persistence measurements, could be carriedout on these structures.

3.3.2 Stereogrammetric mapping ofdiscontinuities

There are circumstances where it is not possible todirectly access a rock face for mapping because,for example, there is a hazard to the geolo-gists of rock falls or the face is overhanging inplaces. Under these conditions, there are availableindirect methods of geological mapping using ter-restrial photography. The basic principle involvesobtaining the co-ordinates of at least three pointson each surface from which its orientation can becalculated.

One system that provides this facility is SIRO-JOINT (CSIRO, 2001). A digital image of theslope face taken by a camera at a known loc-ation, and the image is converted into three-dimensional spatial data defining the surface ofthe rock face. Each spatial point has a position(x, y, z co-ordinates) in space, and each local set of

three spatial points defines a triangle. From theseco-ordinates, the orientation of the triangles canbe defined in terms of the dip and dip direction,together with the co-ordinates of the centroid.The software allows a mouse to be used to outlinethe surface to be analyzed, and the calculated dipand dip direction can then be imported directly toa stereonet.

3.3.3 Types of discontinuity

Geological investigations usually categorize dis-continuities according to the manner in whichthey were formed. This is useful for geotech-nical engineering because discontinuities withineach category usually have similar properties asregards both dimensions and shear strength prop-erties that can be used in the initial review ofstability conditions of a site. The following arestandard definitions of the most common typesof discontinuities:

(a) Fault—Discontinuity along which there hasbeen an observable amount of displacement.Faults are rarely single planar units; nor-mally they occur as parallel or sub-parallelsets of discontinuities along which move-ment has taken place to a greater or lessextent.

(b) Bedding—Surface parallel to the surface ofdeposition, which may or may not have aphysical expression. Note that the originalattitude of the bedding plane should not beassumed to be horizontal.

(c) Foliation—Parallel orientation of platy min-erals, or mineral banding in metamorphicrocks.

(d) Joint—Discontinuity in which there hasbeen no observable relative movement. Ingeneral, joints intersect primary surfacessuch as bedding, cleavage and schistosity. Aseries of parallel joints is called a joint set;two or more intersecting sets produce a jointsystem; two sets of joints approximately atright angles to one another are said to beorthogonal.


(e) Cleavage—Parallel discontinuities formed inincompetent layers in a series of beds ofvarying degrees of competency are knownas cleavages. In general, the term impliesthat the cleavage planes are not controlledby mineral particles in parallel orientation.

(f) Schistosity—Foliation in schist or othercoarse grained crystalline rock due to theparallel arrangement of mineral grains of theplaty or prismatic type, such as mica.

3.3.4 Definition of geological terms

The following is a summary of information thatmay be collected to provide a complete descrip-tion of the rock mass, and comments on howthese properties influence the performance of therock mass. This information is based primarily onprocedures developed by the International Societyof Rock Mechanics (ISRM, 1981b), with someadditional information from the Geological Soci-ety Engineering Group (1977). More details ofthe mapping data is provided in Appendix II,which includes mapping field sheets and tablesrelating descriptions of rock mass properties toquantitative measurements.

Figure 3.5(a) illustrates the 12 essential featuresof geological structure, each of which is describedin more detail in this section. The diagram andphotograph in Figure 3.5 show that sets of discon-tinuities often occur in orthogonal sets (mutuallyat right angles) in response to the stress field thathas deformed the rock; the photograph showsthree orthogonal joints in massive granite. Ortho-gonal structure is also illustrated in the stereonetin Figure 2.11. The value of recognizing ortho-gonal structure on an outcrop or in a stereonet isthat these features are often the most prevalent ina cut and are likely to control stability.

The following is a list, and a description of theparameters that define the characteristics of therock mass.

A Rock type—The rock type is defined by theorigin of the rock (i.e. sedimentary, meta-morphic or igneous), the mineralogy, thecolor and grain size (Deere and Miller, 1966).

The importance of defining the rock typeis that there is wide experience in the per-formance of different rock types (e.g. graniteis usually stronger and more massive thanshale), and this information provides a usefulguideline on the likely behavior of the rock.

B Discontinuity type—Discontinuity typesrange from clean tension joints of limitedlength to faults containing several metersthickness of clay gouge and lengths of manykilometers; obviously the shear strength ofsuch discontinuities will be very different.Section 3.3.3 provides a definition of the sixmost common types of discontinuity.

C Discontinuity orientation—The orientationof discontinuities is expressed as the dip anddip direction (or strike) of the surface. The dipof the plane is the maximum angle of the planeto the horizontal (angle ψ), while the dip dir-ection is the direction of the horizontal traceof the line of dip, measured clockwise fromnorth, angle α (see Figure 2.4). For the planeshown in Figure 3.5 that dips to the north-east, the orientation of the plane can be com-pletely defined by five digits: 30/045, wherethe dip is 30◦ and the dip direction is 45◦. Thismethod of defining discontinuity orientationfacilitates mapping because the dip and dipdirection can be read from a single compassreading (Figure 3.6). Also, the results can beplotted directly on a stereonet to analyze thestructural geology (see Section 2.5). In usingthe compass shown in Figure 3.6, the dip isread off a graduated scale on the lid hinge,while the dip direction is read off the compassscale that is graduated from 0◦ to 360◦.

Some compasses allow the graduated circleto be rotated to account for the magneticdeclination at the site so that measured dipdirections are relative to true north. If thecompass does not have this feature, then themagnetic readings can be adjusted accord-ingly: for a magnetic declination of 20◦ east,for example, 20◦ is added to the magneticreadings to obtain true north readings.

D Spacing—Discontinuity spacing can bemapped in rock faces and in drill core, with


A:rock type

(a)

(b)

C:orientation�= dip dirn.

�= dip

L:number ofsets B, J1

I:aperture(open)

K:seepage

M:block

size/shapeF:

roughnessB:

discontinuitytype:

bedding, faultetc.

J: fillingtype, width

G:wall strength(Table 3.1)

H:weathering(Table 3.2)

D:spacing

(S)

E:persistence

(l)

S bed

l

BJ1

�

�

i

Figure 3.5 Characteristics ofdiscontinuities in rock masses:(a) parameters describing therock mass; letters (“A” etc.) referto description of parameter intext (Wyllie, 1999); (b)photograph of blocky graniticrock containing three jointsorientated in orthogonaldirections (near Hope, BritishColumbia).

the true spacing being calculated from theapparent spacing for discontinuities inclinedto the face (see also Section 3.4.1); spacingcategories range from extremely wide (>2 m)to very narrow (<6 mm). Measurementof discontinuity spacing of each set of

discontinuities will define the size and shapeof blocks and give an indication of stabil-ity modes such as toppling failure. Also,the rock mass strength is related to spacingbecause in closely fractured rock the indi-vidual discontinuities will more readily join


Figure 3.6 Geological compass (Clar type) used to directly measure dip and dip direction of surfaces. Lid isplaced on surface and body is rotated until it is level, as indicated by the spirit level; the needle is released toindicate dip direction, and the dip is read off scale on the hinge. (The Brunton Co., Riverton, WY.)

to form a continuous zone of weakness. Amethod of calculating the strength of frac-tured rock masses taking into account discon-tinuity spacing is discussed in Section 4.5.

E Persistence—Persistence is the measure ofthe continuous length or area of the disconti-nuity; persistence categories range from veryhigh (>20 m) to very low (<1 m). This para-meter defines the size of blocks and the lengthof potential sliding surfaces, so the map-ping should concentrate on measuring thepersistence of the set of discontinuities thatwill have the greatest influence on stability.Where it is not possible to measure the lengthof discontinuities directly on the face, theprocedure described in Section 3.4.2 can beused to estimate the average length of frac-tures that extend beyond the dimensions ofthe face.

F Roughness—The roughness of a discontinu-ity surface is often an important component

of the shear strength, especially where thediscontinuity is undisplaced and interlocked.Roughness becomes less important where thediscontinuity is infilled, or displaced andinterlock is lost. Roughness should be meas-ured in the field on exposed surfaces withlengths of at least 2 m, if possible, in theanticipated direction of sliding, and can bedescribed in terms of a combination of boththe large- and small-scale features:

Shape: stepped Roughness: roughundulating smoothplanar slickensided

The degree of roughness can be quantified interms of the i◦ value, which is a measure ofthe inclination of the irregularities (or asperit-ies) on the surface (see inset F, Figure 3.5(a)).The total friction angle of a rough surfaceis (φ + i). Values for i can be determined


either by direct measurement of the surface,or by comparing the surface with standardprofiles of irregular joint surfaces. These tech-niques are described in Section 3.4.3. Usualpractice would be to use the standard rough-ness profiles during the preliminary mapping.If there is a critical discontinuity that willcontrol stability, then the values estimatedfrom the profiles could be calibrated with alimited number of detailed measurements ofroughness.

G Wall strength—The strength of the rock form-ing the walls of discontinuities will influ-ence the shear strength of rough surfaces.Where high stresses, compared to the wall

strength, are generated at local contact pointsduring shearing, the asperities will be shearedoff resulting in a reduction of the rough-ness component of the friction angle. In theinitial stages of weathering, there is oftena reduction in rock strength on the discon-tinuity surfaces that may result in a dimin-ished roughness value. It is usually adequateto estimate the compressive strength fromthe simple field tests as shown in Table 3.1(ISRM, 1981b), or if core or lump samplesare available, by carrying out point load tests.The Schmidt hammer test is also a method ofestimating the compressive strength of rock atdiscontinuity surfaces.

Table 3.1 Classification of rock and soil strengths (ISRM, 1981b)

Grade Description Field identification Approx. rangeof uniaxialcompressivestrength (MPa)

R6 Extremely strong rock Specimen can only be chipped with geologicalhammer.

>250

R5 Very strong rock Specimen requires many blows of geologicalhammer to fracture it.

100–250

R4 Strong rock Specimen requires more than one blow ofgeological hammer to fracture it.

50–100

R3 Medium strong rock Cannot be scraped or peeled with a pocketknife, specimen can be fractured with singlefirm blow of geological hammer.

25–50

R2 Weak rock Can be peeled by a pocket knife with difficulty,shallow indentations made by firm blow withpoint of geological hammer.

5.0–25

R1 Very weak rock Crumbles under firm blows with point ofgeological hammer and can be peeled bya pocket knife.

1.0–5.0

R0 Extremely weak rock Indented by thumbnail. 0.25–1.0S6 Hard clay Indented with difficulty by thumbnail. >0.5S5 Very stiff clay Readily indented by thumbnail. 0.25–0.5S4 Stiff clay Readily indented by thumb but penetrated

only with great difficulty.0.1–0.25

S3 Firm clay Can be penetrated several inches by thumbwith moderate effort.

0.05–0.1

S2 Soft clay Easily penetrated several inches by thumb. 0.025–0.05S1 Very soft clay Easily penetrated several inches by fist. <0.025

NotesDiscontinuity wall strength will generally be characterized by grades R0–R6 (rock). Some rounding of strength values has beenmade when converting to SI units (ISRM, 1981b).


The strength of the intact rock is also oneof the parameters used to calculate rock massstrength as discussed in Section 4.5.

H Weathering—Reduction of rock strengthdue to weathering will reduce the shearstrength of discontinuities as described in(G). Weathering will also reduce the shearstrength of the rock mass due to the dimin-ished strength of the intact rock. Weatheringcategories range from fresh rock to residualsoil. Weathering of rock takes the form ofboth disintegration and decomposition. Dis-integration is the result of environmental con-ditions such as wetting and drying, freezingand thawing that break down the exposedsurface layer. Disintegration is most preval-ent in sedimentary rocks such as sandstonesand shales, particularly if they contain swell-ing clays, and in metamorphic rocks with ahigh mica content. Decomposition weather-ing refers to changes in rock produced bychemical agents such as oxidation (e.g. yel-low discoloration in rock containing iron),hydration (e.g. decomposition of feldspar ingranite to kaolinite clay) and carbonation(e.g. solution of limestone). Table 3.2 (ISRM,1981b) lists weathering grades that categorize

the rock mass according to the degree ofdisintegration and decomposition.

I Aperture—Aperture is the perpendiculardistance separating the adjacent rock wallsof an open discontinuity, in which the inter-vening space is air or water filled; categoriesof aperture range from cavernous (>1 m), tovery tight (<0.1 mm). Aperture is therebydistinguished from the “width” of a filleddiscontinuity. It is important in predict-ing the likely behavior of the rock mass,such as hydraulic conductivity and deform-ation under stress changes, to understandthe reason that open discontinuities develop.Possible causes include scouring of infillings,solution of the rock forming the walls of adiscontinuity, shear displacement and dila-tion of rough discontinuities, tension featuresat the head of landslides and relaxation ofsteep valley walls following glacial retreator erosion. Aperture may be measured inoutcrops or tunnels provided that extremecare is taken to discount any blast-inducedopen discontinuities, in drill core if recov-ery is excellent, and in boreholes using aborehole camera if the walls of the holeare clean.

Table 3.2 Weathering grades

Term Description Grade

Fresh No visible sign of rock material weathering; perhaps slightdiscoloration on major discontinuity surfaces.

I

Slightly weathered Discoloration indicates weathering of rock material and discontinuitysurfaces. All the rock material may be discolored by weathering andmay be somewhat weaker externally than in its fresh condition.

II

Moderately weathered Less than half of the rock material is decomposed and/or disintegratedto a soil. Fresh or discolored rock is present either as a continuousframework or as corestones.

III

Highly weathered More than half of the rock material is decomposed and/ordisintegrated to a soil. Fresh or discolored rock is present either as adiscontinuous framework or as corestones.

IV

Completely weathered All rock material is decomposed and/or disintegrated to soil. Theoriginal mass structure is still largely intact.

V

Residual soil All rock material is converted to soil. The mass structure and materialfabric are destroyed. There is a large change in volume, but the soilhas not been significantly transported.

VI


J Infilling/width—Infilling is the term formaterial separating the adjacent walls ofdiscontinuities, such as calcite or fault gouge;the perpendicular distance between the adja-cent rock walls is termed the width ofthe filled discontinuity. A complete descrip-tion of filling material is required to pre-dict the behavior of the discontinuity includethe following: mineralogy, particle size,over-consolidation ratio, water content/conductivity, wall roughness, width andfracturing/crushing of the wall rock. If thefilling is likely to be a potential sliding sur-face in the slope, samples of the material(undisturbed if possible) should be collectedfor shear testing.

K Seepage—The location of seepage from dis-continuities provides information on aperturebecause ground water flow is confined almostentirely in the discontinuities (secondary per-meability); seepage categories range from verytight and dry to continuous flow that canscour infillings. These observations will alsoindicate the position of the water table, orwater tables in the case of rock masses con-taining alternating layers of low and highconductivity rock such as shale and sandstonerespectively. In dry climates, the evaporationrate may exceed the seepage rate and it may bedifficult to observe seepage locations. In coldweather, icicles provide a good indication ofeven very low seepage rates. The flow quantit-ies will also help anticipate conditions duringconstruction such as flooding and pumpingrequirements of excavations.

L Number of sets—The number of sets of dis-continuities that intersect one another willinfluence the extent to which the rock masscan deform without failure of the intact rock.As the number of discontinuity sets increasesand the block size diminishes, the greater theopportunity for blocks to rotate, translate andcrush under applied loads. Mapping shoulddistinguish between systematic discontinuit-ies that are members of a set and randomdiscontinuities, the orientation of which areless predictable.

M Block size/shape—The block size and shapeare determined by the discontinuity spacingand persistence, and the number of sets. Blockshapes include blocky, tabular, shattered andcolumnar, while block size ranges from verylarge (>8 m3) to very small (<0.0002 m3).The block size can be estimated by select-ing several typical blocks and measuring theiraverage dimensions.

Using the terms outlined in this section, a typicaldescription for a rock material would be asfollows:

Moderately weathered, weak, fine grained,dark gray to black carbonaceous shale.

Note that the rock name comes last since this isless important than the engineering properties ofthe rock.

An example of a rock mass description is asfollows:

Interbedded sequence of shale and sandstone;typically the shale units are 200–400 mm thickand the sandstone units are 1000–5000 mmthick. In the shale, the bedding spacing is100–200 mm and many contain a soft clayinfilling with width up to 20 mm; the beddingis planar and often slickensided. The sand-stone is blocky with the spacing of the bedsand two conjugate joint sets in the range of500–1000 mm; the bedding and joint surfacesare smooth and undulating, and contain noinfilling. The upper contacts of the shale bedsare wet with some dripping; the sandstoneis dry.

3.4 Spacing, persistence and roughnessmeasurements

A component of the geological mapping of rockexposures may involve detailed measurements ofsurface roughness, and discontinuity spacing andpersistence. Usually these detailed measurementswould only be carried out on discontinuity sets orspecific features that have been identified to have


a significant influence on stability because, forexample, they are persistent and dip out the face.

3.4.1 Spacing of discontinuities

The spacing of discontinuities will determine thedimensions of blocks in the slope, which willinfluence the size of rock falls and the design ofbolting patterns. A factor to consider in the inter-pretation of spacing measurements from surfacemapping is the relative orientation between theface and the discontinuities. That is, the relat-ive orientation introduces a bias to the numberand spacing of discontinuities in which the actualspacing is less than the apparent spacing, and theactual number of discontinuities is greater thanthe number mapped. The bias arises because alldiscontinuities oriented at right angles to the facewill be visible on the face at their true spacing,while few apparently widely spaced discontinuit-ies, oriented sub-parallel to the face will be visible(Figures 3.5 and 3.7). The bias in spacing can becorrected as follows (Terzaghi, 1965):

S = Sapp sin θ (3.3)

where S is the true spacing between discontinuit-ies of the same set, Sapp is the measured (apparent)spacing, and θ is the angle between face and strikeof discontinuities.

The number of discontinuities in a set canbe adjusted to account for the relative orient-ation between the face and the strike of the

Discontinuityset

Sapp Sapp SappRock face

�

SS

S S = Sapp sin�

Figure 3.7 Relationship between apparent and truespacing for a set of discontinuities.

discontinuity as follows:

N = Napp

sin θ(3.4)

where N is the adjusted number of discontinuitiesand Napp is the measured number of discontinuit-ies. For example, a vertical drill hole will intersectfew steeply dipping discontinuities, and a verticalface will intersect few discontinuities parallel tothis face; the Terzaghi correction will calculatean appropriate increase in the number of thesesurfaces. Some stereonet programs can apply theTerzaghi correction to increase the number ofdiscontinuities to allow for the bias in samplingorientation, and more accurately represent thepopulation of discontinuities.

In rock outcrops, the spacing between discon-tinuities in a set will be variable, and the followingis a discussion on methods of calculating the aver-age spacing of a joint set, and the applicationof the Terzaghi correction. Figure 3.8 shows arock outcrop in which it is planned to make asteep cut; the rock contains a set of joints that dipout of the face at angle ψ . The characteristics of

Scan line

c tl

tc

cc

t

ct

L1

�

L2

s

Figure 3.8 Measurement of average spacing andpersistence of a set of discontinuities in an outcrop.


this joint set can be determined using line map-ping by hanging a tape vertically down the faceand recording the characteristics of each joint thatintersects the tape. The tape can be termed a scanline, which in this case has a length of 15 m.

The average true spacing (s) of joint set inFigure 3.8 is calculated using equation (3.5), forthe conditions where the scan line is vertical andis not at right angles to the joints. If there are N ′joints with dip ψ intersecting a scan line of lengthL1, then the value of s is given by

s = L1 cos ψ/N ′ (3.5)

On Figure 3.8 there are nine joints (N ′ = 9) withan average dip of 35◦ intersecting the scan line,which has a length of 15 m (L1 = 15). The aver-age spacing of these joints is 1.5 m; this averagespacing is plotted to scale on Figure 3.8.

One approach that may be taken to study thespacings of different sets of discontinuities is tomake measurements along scan lines with differ-ent orientations; it is preferable that the scan linesare at right angles to each set (Hudson and Priest,1979, 1983).

The Terzaghi correction can be applied to thejoints measured on the scan line as follows. Thereare nine joints intersecting the scan line (Napp =9), and the average angle between the joints dip-ping at 35◦ and the vertical scan line is 55◦.Therefore, equation (3.4) shows that approxi-mately 11 joints would have been intersected bya 15 m long scan line orientated at right anglesto the joints.

3.4.2 Persistence of discontinuity sets

Persistence of discontinuities is one of the mostimportant rock mass parameters because itdefines, together with spacing, the size of blocksthat can slide from the face. Furthermore, asmall area of intact rock between low persist-ence discontinuities can have a positive influenceon stability because the strength of the rock willoften be much higher than the shear stress actingin the slope. Unfortunately, persistence is one ofthe more difficult parameters to measure because

often only a small part of the discontinuity isvisible in the face. In the case of drill core, noinformation on persistence is available.

A number of procedures have been developedto calculate the approximate average persistenceof a set of discontinuities by measuring theirexposed trace lengths on a specified area of theface (Pahl, 1981; Priest and Hudson, 1981;Kulatilake and Wu, 1984).

The procedure developed by Pahl (1981) com-prises first, defining a mapping area on a facewith dimensions L1 and L2 (Figure 3.8). Thenthe total number of discontinuities (N ′′) of a par-ticular set (with dip ψ) in this area is counted,and the numbers of these discontinuities that areeither contained within (Nc), or transect (Nt)

the mapping area are identified. Contained dis-continuities are short and have both ends visiblewithin the area, while transecting discontinuitiesare relatively long and have neither end visible.The approximate average length (l) of a set ofdiscontinuities is calculated from equations (3.6)to (3.8) that are independent of the assumed formof the statistical distribution of the lengths.

l = H ′ (1 + m)

(1 − m)(3.6)

where

H ′ = L1 · L2

(L1 · cos ψ + L2 · sin ψ)(3.7)

and

m = (Nt − Nc)

(N ′′ + 1)(3.8)

As demonstrated by these equations, the basis ofthis method of estimating the average length ofdiscontinuities is to count discontinuities on a facewith a known area, and does not involve measur-ing the length of individual discontinuities whichcan be a much more time-consuming task.

For the joints depicted on Figure 3.8, the aver-age persistence can be calculated as follows. Thetotal number of joints (ψ ∼ 35◦) within the scan


area is 14 (N ′′ = 14), of which five are containedwithin the scan area (Nc = 5), and four transectthe scan area (Nt = 4). If the dimensions of thescan area are L1 = 15 and L2 = 5, then the valueof m is −0.07 and the value of H ′ is 4.95. Fromequation (3.6), the average length of the joints inthis set is 4.3 m. This average persistence is plottedto scale on Figure 3.8.

If there was a second set of discontinuities inthe scan area that would influence stability, thenthese could be counted seperately using the sameprocedure to determine their average spacing andpersistence.

3.4.3 Roughness of rock surfaces

The friction angle of a rough surface comprisestwo components—the friction of the rock mater-ial (φ), plus interlocking produced by the irreg-ularities (asperities) of the surface (i). Becauseroughness can be a significant component of thetotal friction angle, measurement of roughness isoften an important part of a mapping program.

During the preliminary stages of an investiga-tion it is usually satisfactory to make a visualassessment of the roughness as defined bythe Joint Roughness Coefficient (JRC) (Barton,1973). JRC varies from zero for smooth, planarand particularly slickensided surfaces to as muchas 20 for rough, undulating surfaces. The value of

JRC can be estimated by visually comparing thesurface condition with standard profiles based ona combination of surface irregularities (at a scaleof several centimeters) and waviness (at a scale ofa several meters) as shown in Figure 3.9.

JRC is related to the roughness of the surface,i value by the following equation:

i = JRC log10

(JCSσ′

)(3.9)

where JCS (Joint Compressive Strength) is thecompressive strength of the rock on the discon-tinuity surface (see Table 3.1), and σ′ is theeffective normal stress on the surface due to theweight of the overlying rock less any uplift waterpressure on the surface. Equation (3.9) shows thati diminishes as the asperities are ground off whenthe rock strength is low compared to the appliednormal stress. The application of equation (3.9)to determine the shear strength of rock surfacesis discussed in more detail in Section 4.2.

In the final design stage of a project, a few dis-continuities having a significant effect on stabilitymay be identified, and there are a number of meth-ods of accurately measuring the surface roughnessof these critical surfaces. A method developed byFecker and Rengers (1971) consists of measuringthe orientation of the discontinuity using a geolo-gical compass with a series of plates of different

Examples of roughness profiles

A

B

C

A Rough undulating—tension joints, rough sheeting,rough bedding

B Smooth undulating—smooth sheeting, non-planar foliation,undulating bedding

C Smooth nearly planar—planar shear joints, planar foliation,planar bedding

JRC = 20

JRC = 10

JRC = 5

50 cm 500 cm

Figure 3.9 Standard profiles definingjoint roughness coefficient, JRC(Barton, 1973).


diameters attached to the lid. If the diameter ofthe larger plates is about the same dimensions asthe wavelength of the roughness, then the meas-ured orientation will be approximately equal tothe average orientation of the surface. However,the smaller diameter plates will show a scatterin the orientation measurements as the plates lieon irregularities with shorter wavelengths. If theorientation measurements are plotted on a ste-reonet, the degree of scatter in the poles about themean orientation is a measure of the roughness.

Quantitative methods of profile measurementhave been established by Tse and Cruden (1979)using a mechanical profilometer, and Mearz et al.(1990) have developed a shadow profilometerthat records the shape of the surface with a videocamera and image analyzer.

The method developed by Tse and Cruden isillustrated in Figure 3.10(a). The profile is definedby measuring the distance (yi) of the surface froma fixed reference line at specified equal intervals(�x) over a length of M intervals. From thesemeasurements the coefficient Z2 is defined as

Z2 =[

1M(�x)2

M∑i=1

(yi+1 + yi)2

]1/2

(3.10)

A study has also been carried out to assess theeffect of the size of the sampling interval (�x)

along the profile on the calculated value of JRC(Yu and Vayssade, 1991). This study found thatthe calculated value of JRC was dependent onthe size of �x, and that the most accurate res-ults were obtained with small sampling intervals.The value of JRC can be calculated from the coef-ficient Z2 using one of the following equations forthe appropriate sampling interval:

JRC = 60.32(Z2) − 4.51 for �x = 0.25 mm(3.11a)

JRC = 61.79(Z2) − 3.47 for �x = 0.5 mm(3.11b)

JRC = 64.22(Z2) − 2.31 for �x = 1 mm(3.11c)

yi yi +1

∆xi = M

0 1 2 3 cm

(a)

(b)

(c)

Figure 3.10 Measurement of joint roughness:(a) procedure for measuring roughness withmechanical profilometer (modified from Tse andCruden (1979); reprinted with kind permission fromElsevier Science Ltd. The Boulevard, Langford Lane,Kidlington, UK); (b) photograph of profilometer;(c) profile of joint with JRC of 11.4.

One means of making profile measurements is touse a carpenter’s comb that consists of a series ofmetal rods, positioned in a frame, such that theycan slide relative to each other (Figure 3.10(b)). Ifthe comb is pressed against a rock face, the rodswill slide to conform to the shape of the surface.This profile can then be traced on to a piece ofpaper and the shape of the rock surface quanti-fied by measuring the distance of each rod from areference line.

Figure 3.10(c) shows the results of a profilemeasurement with a carpenter’s comb of a rough,planar joint in granite. The profile of the surface


was digitized, with 145 measurements made ona 1 mm interval (M = 145; �x = 1), fromwhich Z2 was calculated from equation (3.11c)to be 0.214 with a corresponding JRC valueof 11.4.

3.5 Probabilistic analysis of structuralgeology

As discussed in Section 1.4.4, a measure ofthe stability of a slope is the probability offailure. Calculation of the probability of fail-ure involves expressing the design parameters interms of probability distributions that give themost likely value of each parameter (e.g. aver-age), as well as the probability of its occurrencewithin a range of possible values (e.g. standarddeviation).

This section demonstrates techniques to deter-mine the probability distributions of structuralgeology data. The distribution in orientation canbe calculated from the stereonet, while the distri-butions of persistence and spacing are calculatedfrom field measurements.

3.5.1 Discontinuity orientation

The natural variation in orientation of discon-tinuities results in there being scatter of the poleswhen plotted on the stereonet. It can be usefulto incorporate this scatter into the stability ana-lysis of the slope because, for example, a wedgeanalysis using the mean values of pair of discon-tinuity sets may show that the line of intersectionof the wedge does not daylight in the face andthat the slope is stable. However, an analysisusing orientations other than the mean valuesmay show that some unstable wedges can beformed. The risk of occurrence of this conditionwould be quantified by calculating the mean andstandard deviation of the dip and dip direction asdescribed next.

A measure of the dispersion, and from this thestandard deviation, of a discontinuity set can becalculated from the direction cosines as follows(Goodman, 1980). The direction cosines of anyplane with dip ψ and dip direction α are the unit

vectors l, m and n, where

l = sin ψ · cos α

m = sin ψ · sin α

n = cos ψ

⎫⎪⎬⎪⎭ (3.12)

For a number of poles, the direction cosines (lR,mR and nR) of the mean orientation of the discon-tinuity set is the sum of the individual directioncosines, as follows:

lR =∑

li|R| ; mR =

∑mi

|R| ; nR =∑

ni

|R|(3.13)

where |R| is the magnitude of the resultant vectorgiven by

|R| =[(∑

li)2 +

(∑mi

)2 +(∑

ni

)2]1/2

(3.14)

The dip ψR and dip direction αR of the meanorientation are

ψR = cos−1(nR)

αR = + cos−1(lR/ sin ψR) for mR ≥ 0

αR = − cos−1(lR/ sin ψR) for mR < 0

⎫⎪⎪⎬⎪⎪⎭

(3.15)

A measure of the scatter of a set of disconti-nuities comprising N poles can be obtained fromthe dispersion coefficient Cd, which is calculatedas follows:

Cd = N

(N − |R|) (3.16)


If there is little scatter in the orientation of thediscontinuities, the value of Cd is large, and itsvalue diminishes as the scatter increases.

From the dispersion coefficient, it is possible tocalculate from equation (3.17) the probability, P ,that a pole will make an angle θ◦ or less than themean orientation.

θ = cos−1[1 + (1/Cd) ln(1 − P)] (3.17)

For example, the angle from the mean defined byone standard deviation occurs at a probability P

of 0.16 (refer to Figure 1.12). If the dispersionis 20, one standard deviation lies at 7.6◦ fromthe mean.

Equation (3.17) is applicable when the disper-sion in the scatter is approximately uniform aboutthe mean orientation, which is the case in joint setA in Figure 2.11. However, in the case of the bed-ding in Figure 2.11, there is less scatter in the dipthan in the dip direction. The standard deviationsin the two directions can be calculated approx-imately from the stereonet as follows. First, twogreat circles are drawn at right angles corres-ponding to the directions of dip and dip directionrespectively. Then the angles corresponding to the7% and 93% levels, P7, and P93 respectively,are determined by counting the number of polesin the set and removing the poles outside thesepercentiles. The equation for the standard devi-ation along either of the great circles is as follows(Morriss, 1984):

SD = tan−1{0.34[tan(P93) − tan(P7)]}(3.18)

More precise methods of determining the stand-ard deviation are described by McMahon(1982), but the approximate method given byequation (3.18) may be sufficiently accurate con-sidering the difficulty in obtaining a represent-ative sample of the discontinuities in the set. Animportant aspect of accurate geological invest-igations is to account for bias when mapping asingle face or logging a single borehole when few

of the discontinuities aligned parallel to the line ofmapping are measured. This bias in the data canbe corrected by applying the Terzaghi correctionas described in Section 3.4.1.

3.5.2 Discontinuity length and spacing

The length and spacing of discontinuities deter-mines the size of blocks that will be formed inthe slope. Designs are usually concerned withpersistent discontinuities that could form blockswith dimensions great enough to influence overallslope stability. However, discontinuity dimen-sions have a range of values and it is useful tohave an understanding of the distribution of thesevalues in order to predict how the extreme val-ues may be compared to values obtained froma small sample. This section discusses proba-bility distributions for the length and spacing ofdiscontinuities, and discusses the limitations ofmaking accurate predictions over a wide rangeof dimensions.

The primary purpose of making length andspacing measurements of discontinuities is toestimate the dimensions of blocks of rock formedby these surfaces (Priest and Hudson 1976;Cruden, 1977; Kikuchi et al., 1987; Dershowitzand Einstein, 1988; Kulatilake, 1988; Einstein,1993). This information can then be used, ifnecessary, to design appropriate stabilizationmeasures such as rock bolts and rock fall bar-riers. Attempts have also been made to use thisdata to calculate the shear strength of “stepped”surfaces comprising joints separated by portionsof intact rock (Jennings, 1970; Einstein et al.,1983). However, it has since been found thatthe Hoek–Brown method of calculating the shearstrength of rock masses is more reliable (seeSection 4.5).

Probability distributions

Discontinuities are usually mapped along a scanline, such as drill core, slope face or wall of atunnel. Individual measurements are made of theproperties of each fracture, including its visible


length and the spacing between discontinuities ineach set (Appendix II). The properties of discon-tinuities typically vary over a wide range and itis possible to describe the distribution of theseproperties by means of probability distributions.A normal distribution is applicable if a particularproperty has values in which the mean value isthe most commonly occurring. This conditionwould indicate that the property of each discon-tinuity, such as its orientation, is related to theproperty of the adjacent discontinuities reflect-ing that the discontinuities were formed by stressrelief. For properties that are normally distri-buted, the mean and standard deviation are givenby equations (1.16) and (1.17).

A negative exponential distribution is applic-able for properties of discontinuities, such astheir length and spacing, which are randomlydistributed indicating that the discontinuities aremutually independent. A negative exponentialdistribution would show that the most com-monly occurring discontinuities are short andclosely spaced, while persistent, widely spaceddiscontinuities are less common. The generalform of a probability density function f (x) ofa negative exponential distribution is (Priest andHudson, 1981)

f (x) = 1x

(e−x/x) (3.19)

and the associated cumulative probability F(x)

that a given spacing or length value will be lessthan dimension x is given by

F(x) = (1 − e−x/x) (3.20)

where x is a measured value of length or spa-cing and x is the mean value of that parameter. Aproperty of the negative exponential distributionis that the standard deviation is equal to the meanvalue.

From equation (3.20) for a set of discontinuitiesin which the mean spacing is 2 m, the probabilitiesthat the spacing will be less than 1 m and 5 m,

respectively, are

F(x) = (1 − e−1/2) = 40%

and

F(x) = (1 − e−5/2) = 92%

Equation (3.20) could be used to estimate theprobability of occurrence of discontinuities witha specified length. This result could be used, forexample, to determine the likelihood of a planebeing continuous through a slope.

Another distribution that can often be usedto describe the dimensions of discontinuities isthe lognormal distribution which is applicablewhere the variable x = ln y is normally distributed(Baecher et al., 1977). The probability densityfunction for a lognormal distribution is (Harr,1977)

f (x) = 1

y SDx

√2π

exp

[−1

2

(ln y − x

SDx

)2]

(3.21)

where x is the mean value and SD is the standarddeviation.

Figure 3.11 shows the measured lengths of 122joints in a Cambrian sandstone for lengths ofless than 4 m; the mean length l is 1.2 m (Priest

Exponential (r = 0.69)Lognormal (r = 0.89)

N =122=1.2 m

Measured trace length (m)

Freq

uenc

y (%

)

5

10

15

20

25

l

l

00 1.0 2.0 3.0 4.0

Figure 3.11 Histogram of joint trace lengths, withbest fit exponential and lognormal curves (Priest andHudson, 1981).


and Hudson, 1981). To these data have beenfitted both exponential and lognormal curves forwhich the correlation coefficients r are 0.69 and0.89, respectively. While the lognormal curve hasa higher correlation coefficient, the exponentialcurve has a better fit at the longer discontinuitylengths. This demonstrates that for each set ofdata the most appropriate distribution should bedetermined.

3.6 Diamond drilling

On many projects, surface mapping is supplemen-ted by diamond drilling to obtain core samplesof the subsurface rock. The extent of the drillingwill depend on such factors as the soil cover, theavailability of rock outcrops and the confidencewith which surface data can be extrapolated overthe full depth of the cut. For example, if the rockat the surface is weathered or disturbed by blast-ing, then drilling may be required to find rockconditions at depth.

The type of information that can be obtainedby diamond drilling may be somewhat differ-ent from surface mapping information. Whilesurface mapping is the primary means of obtain-ing information on geological structure, in drill

core there is no information on persistence, andthe orientation of discontinuities can only beobtained if the core is oriented (see Section 3.6.4).The types of information that are provided bydrill core are in situ rock strength, fracture fre-quency and the characteristics of shear zones.The core can also be used for laboratory strengthtesting (Chapter 4), and instrumentation suchas piezometers can be installed in the holes(Chapter 5).

3.6.1 Diamond drilling equipment

Figure 3.1 shows a typical diamond drill. Themajor components of the drill comprise a motor,usually gasoline or diesel powered, a head togenerate torque and thrust to the drill rods, amast to support the wire line equipment, anda string of drill rods at the end of which is acore barrel and a diamond impregnated bit. Drillrods are flush coupled, usually in 3 m lengths,and the diameters available for common NorthAmerican equipment are listed on Table 3.3,together with the corresponding hole and coresizes. The rod diameters are designated by fourletters (e.g. BQTT)—the first letter indicates thediameter, the second that it is used with wire line

Table 3.3 Dimensions of diamond drilling equipment (Christensen Boyles Corp.) triple tube core barrels

Drill size AQTT BQTT NQTT HQTT PQTT

Hole diameter, mm (in.) 48.0 60 75.7 96 122.6(1.89) (2-23/64) (2-63/64) (3-25/32) (4-53/64)

Core diameter, mm (in.) 26.9 33.5 45 61.1 83(1.06) (1-5/16) (1-25/32) (2-13/32) (3-9/32)

Hole volume, l/100 m 181 282 451 724 1180(Usgal/100 ft) (14.6) (22.7) (36.3) (58.3) (95.1)

Casing ID, mm (in.) 48.4 60.3 76.2 101.6 127(1-29/32) (2-3/8) (3.0) (4.0) (5.0)

Casing OD, mm (in.) 57.1 73 88.9 114.3 139.7(2-1/4) (2-7/8) (3-1/2) (4-1/2) (5-1/2)

Casing weight, kg/3 m (lb/10 ft) 17 31.3 38.4 50.5 64.3(38) (70) (86) (113) (144)

Drill rod ID, mm (in.) 34.9 46 60.3 77.8 103.2(1-3/8) (1-13/16) (2-3/8) (3-1/16) (4-1/16)

Drill rod OD, mm (in.) 44.5 55.6 69.9 88.9 117.5(1-3/4) (2-3/16) (2-3/4) (3-1/2) (4-5/8)

Drill rod weight, kg/3 m (lb/10 ft) 14 18 23.4 34.4 47.2(31) (40) (52) (77) (106)


equipment and TT indicates a triple tube core bar-rel. Also listed on the table are casing diameters;casing is used in the upper part of holes, in soilor weathered rock, to prevent caving. The cas-ing and rod diameters are arranged such that Nrods fit inside N casing, and that B rods fit inside Bcasing, for example. This allows holes to be “tele-scoped” down to a smaller diameter, if necessary,as the hole is advanced. It is usual in geotechnicaldrilling to use NQTT rods, or HQTT if the rockis highly broken because better core recovery isachieved with the larger diameter.

For geotechnical drilling where one of theobjectives is to recover the weakest portion of therock, recommended practice is to use a triple tubecore barrel. The triple tube barrel comprises anouter tube coupled to the drill rods and bit, whichturns with the rods while drilling. The middle tubelocks into a roller bearing at the top of the corebarrel such that it remains stationary while thedrill rods rotate. The inner tube is split into twopieces longitudinally, and also remains stationaryduring drilling.

The drilling procedure involves rotating therods and bit at high speed (up to about 1000 rpm),while applying a steady thrust to the bit andpumping water down the center of the drill rods.In this way the bit is advanced into the rock whilethe water cools the bit, and removes the cuttingsin the annular space outside the rods. The core fillsthe core barrel as the bit advances, and the drillingis stopped when the core barrel is full—the corebarrel is usually 3 m long. In holes with depthsless than about 20 m, it is possible to recover thecore at the end of each run by removing the rodsfrom the hole. For deeper holes the usual prac-tice is to lower a wireline down the hole with acore catcher on the end which locks into the corebarrel and releases it from the drill rods. The corebarrel is lifted from the hole without moving therods. The core is extracted from the core barrel bypumping the inner split tube out of the barrel andthen removing the upper half of the tube. The corecan then be logged in the core barrel with minimaldisturbance. In contrast, if a double tube barrelis used, the core has to be pumped or hammeredout of the tube, which inevitably results in some

damage and disturbance, especially to the weakerportions of the rock that are the most importantto recover.

In very poor quality rock, it is possible tomodify the triple tube by inserting a clear plasticliner inside the split inner tube. The core iscontained in the plastic tube so that it can belogged and then stored with minimal disturbance.

3.6.2 Diamond drilling operations

The following are some of the factors that mayneed to be considered in a diamond drilling oper-ation. First, the drill rig should be set up on levelground, orona levelplatformif thegroundsurfaceis irregular. Furthermore, the platform should berobust and the rig should be securely attached tothe ground or platform. If the rig is able to moveduring drilling, vibration in the rods will diminishcore quality and may damage the rods.

It is essential that the drill bit be continuouslyflushed with a fluid, usually water, to cool the bitand remove the cuttings. The circulating wateralso lubricates the drill string to reduce the torquerequired to turn the rods, and reduces vibrationof the rods. Water is usually supplied to the siteby pumping from a nearby river, or by a tankertruck. Factors to consider in the supply of waterinclude the pumping head between the supplyand the site, freezing of the pipeline, and availableroad access. It is usual that the return drill wateris collected in a settling tank at the site to removethe cuttings, and is then recirculated down thehole. This reduces the quantity of supplied water,and eliminates environmental contamination bysilt-laden water.

The addition of certain chemicals and solidsto the drilling water can improve the propertiesof the circulating fluid, and can be essential forsuccessfully drilling highly permeable or unstableformations (Australian Drilling Industry, 1996).The most common additive in diamond drillingis organic, long chain polymers that have thixo-tropic properties. That is, they have low viscositywhen they are stirred or pumped, but gel whenallowed to stand. The effect of these propertiesis that the mud can be readily circulated in the


hole to remove the cuttings in the narrow annu-lar space around the rods, but will gel to forma cake to seal and stabilize the walls of the hole.

The correct application of these dual propertiesof mud will greatly enhance diamond drillingoperations. For example, if the hole intersectsa zone of broken and weak rock, the mud canstabilize the walls, and there will be no need toextend the casing down to this level. Similarly,if the circulation fluid is lost in permeable zones,these can be sealed with a mud cake. Ideally, themud cake should be thin and have low permeabil-ity, and should penetrate the formation so that itis not broken up by the rotation of the drill rods.The fluid pressure in the hole then helps to keepthe cake in place. Where the polymer muds are notsufficiently viscous to form a mud cake, possibleadditives to the mud include fine flakes of mica orpaper to help seal fine openings in the rock on thewall of the hole. In the case of artesian flows, thedensity of the mud can be increased by bentonite–barytes mixtures, so that the mud weight balancesthe upward pressure of the water in the artesianformation.

For holes that are to be used for hydrologictesting, such as permeability measurements andthe installation of piezometers, it is necessary thatthere be no mud cake on the walls. For these con-ditions, there are biodegradable muds that breakdown with time to leave the walls of the holeclean.

At the completion of the hole and any associ-ated down-hole testing, there is often a require-ment to fill the hole completely with cement groutto prevent changes to the hydrological conditionsafter the hole is completed.

3.6.3 Core logging

Recording the properties of the recovered drillcore involves making a detailed and complete logof the rock; an example of a diamond drill corelog is shown in Figure 3.12. The log should beprepared using the same properties and descrip-tions of the rock mass discussed in Section 3.3so that there is consistency between the surfaceand sub-surface data. This data will include the

rock description, the properties of the discontinu-ities and their orientation with respect to the coreaxis. Measurements can also be made of the RockQuality Designation (RQD), fracture index andcore recovery, which are indicative of the rockmass quality, as described. Also, the log recordsall testing carried out in the hole such as per-meability measurements, the results of strengthtests on core samples and the position of instru-mentation such as piezometers. Finally, the coreis photographed, complete with a color referenceand dimension scale.

Where rock types such as shales are recoveredthat are highly susceptible to degradation whenexposed to the atmosphere, it may be necessaryto preserve the rock as soon as it has been logged.Preservation of core samples would be necessarywhere they are to be transported to the laboratoryfor strength testing and it is necessary that theybe tested close to their in situ state. Often thelowest strength rock will influence slope design,and these samples should not be allowed to breakdown and loose strength, or bake in the sun andgain strength.

One method of preservation is to wrap the corein plastic film and then dip it in molten wax to cre-ate a barrier to moisture loss. Finally, the sealedcore can be embedded in a rigid foam so that itcan be transported without damage. A furtherprecaution that may be necessary is to preventthe sample from freezing because the formationof ice can fracture weak rock.

The following are measurements of the corethat are routinely made to assess the intact rockstrength and the degree of fracturing.

(a) RQD (rock quality index) is an index relatedto the degree of fracturing of the core. TheRQD is calculated by measuring the totallength of all pieces of core in a drill runwith lengths greater than 100 mm (4 in.),discounting fractures due to drilling. Theselengths are then added together, and the totallength is expressed as a percentage of thelength of the drill run. A low RQD valuewould be indicative of a closely fracturedrock, while an RQD of 100% means that allpieces of core are longer than 100 mm. RQD


is calculated as follows.

RQD=∑

(lengths of core pieces with lengths>100 mm)

Total length of core run

×100% (3.22)

(b) Fracture index is a count of the number ofnatural fractures in the core measured over afixed length of say 0.5 m. This parameter isrelated to the RQD value but is standardizedto a fixed length so is not influenced by the

(a)

WN Rock

(b)

Figure 3.12 Diamond drill records: (a) typical core log with histograms indicating values for recovery andRQD; (b) photograph of drill core with length scale and color reference chart.


length of the core runs.

Fracture index = number of natural discontinuities0.5 m length of core

(3.23)

(c) Core recovery is a measure of how muchrock has been lost during drilling. Core lossmay result from weak zones being washedout by the drilling water, or grinding of thecore during drilling, or the presence of anopen cavity. Drillers can often detect areasof very weak rock or cavities where there isa sudden increase in the advance rate; thesezones should be noted on the log. The corerecovery value is determined by measuringthe length of recovered core compared to thetotal length drilled. If a length of lost core isidentified when logging the core, it is goodpractice to install a wooden spacer at thelocation of the lost core.

Recovery = total length of recovered corelength of drill run

× 100

(3.24)

On Figure 3.12, values for these three para-meters are shown graphically so that areas ofweak or broken rock can be readily identifiedwhen scanning down the log. When making RQDand fracture index measurements it is importantthat drilling breaks in the core are identified andnot included in the reported values.

3.6.4 Core orientation

For conditions where there is insufficient designdata on discontinuity orientation from surfacemapping, it may be necessary to obtain this datafrom drill core. This will require that the core beoriented.

The first step in orienting core is to deter-mine the plunge and trend of the drill hole usinga down-hole survey tool. One such tool comprisesan aluminum (non-magnetic) drill rod that con-tains a dip meter and a compass, both of whichcan be photographed at specified time intervals.The orientation tool is lowered down the hole onthe end of the drill string, and is held stationary at

the times specified for the camera operation. Ateach time interval a photograph is taken of thedip meter and compass, and the depth is recor-ded. When the tool is recovered from the hole,the film is developed to show the hole orientationat the recorded depths. Other hole orientationtools include the Tropari single shot instrument,and gyroscopes for use in magnetic environments(Australian Drilling Industry, 1996).

Most methods of orienting core involve mark-ing a line down the core representing the top of thehole. Since the orientation of this line is knownfrom the hole survey, the orientation of all dis-continuities in the core can be measured relativeto this line, from which their dip and strike canbe calculated (Figure 3.13). Figure 3.13 showsthat a plane intersected by the core has the shapeof an ellipse, and the first step in the calcula-tion process is to mark the down-hole end majoraxis of this ellipse. The dip (δ) of this plane isthen measured relative to the core axis, and areference angle (α) is measured clockwise (look-ing down-hole) around the circumference of thecore from the top-of-core line to the major axisof the ellipse. The dip and dip direction of theplane is calculated from the plunge and trend ofthe hole and the measured angles δ and α. Thetrue dip and dip direction of a discontinuity in thecore can be determined by stereographic meth-ods (Goodman, 1976), or by spherical/analyticalgeometry methods (Lau, 1983).

In a few cases, the core may contain a dis-tinct and consistent marker of known orientation,

Reference angle, �

Discontinuitysurface (ellipse)

Top of core (in situ)

Direction of drilling

Maximumdip vector

Dip,�

Figure 3.13 Measurement of orientation ofdiscontinuity in oriented drill core.


such as bedding, which can be used to orient thecore and measure the orientation of the otherdiscontinuities. However, minor, and unknownvariations in the orientation of the marker bedare likely to lead to errors. Therefore, it is usu-ally preferable to use one of the following threemethods to determine the top-of-the-core.

The clay impression method to orient coreinvolves fabricating a wireline core barrel withone side weighted so that the barrel can rotateand position the weight at the bottom of thehole (Figure 3.14) (Call et al., 1982). A piece

Eccentrically loadedinner core tube

Plate welded across upperend of lifter case

Core lifter with core springremoved and packed withmodelling clay whichprotrudes 10 mm

Diamond drill bit

Clay to take inprintof core stub

Core stub left at the endof the previous drilling run

Figure 3.14 Clay impression core barrel to orient drillcore in an inclined hole (Call et al., 1982).

of clay is placed at the lower end of the barrelsuch that it protrudes past the drill bit when thecore barrel is lowered down the rods and lockedinto place. A light pressure is then applied to therods so that the clay takes an impression of therock surface at the end of the hole. The core bar-rel is then removed and the clay impression, withtop-of-core reference line, is retrieved. The nextdrill run proceeds normally. When this length ofcore is removed, the top-of-the-core is matchedwith the clay impression and the top-of-core lineis transferred from the clay to the core run. Thediscontinuities in the core run are then orientedrelative to the top-of-core line using the methodshown in Figure 3.13.

The advantages of the clay impression core ori-entation method are its simplicity and low equip-ment cost. However, the time required to take animpression on each drill run slows drilling, andthe method can only be used in holes inclinedat angles flatter than about 70◦. Also, the ori-entation line will be lost at any place where thecore is broken and it is not possible to extend thetop-of-the-core line past the break.

A more sophisticated core orientation tool isthe Christiensen–Hugel device that scribes a con-tinuous line down the core during drilling; theorientation of the scribed line is determined bytaking photographs of a compass in the head ofthe core barrel. The advantage of this method

NW

S E

Discontinuity

(a) (b)

Intersection line ofdiscontinuity onborehole wall

Point P on boreholewall

Trace ofintersectionline onunwrappedimage

Point P onunwrappedimage

S W N E S

Figure 3.15 Orientation ofdiscontinuity in borehole using imagefrom 360◦ scanning video camera: (a)core-like image of drill hole wallshowing elliptical intersection betweendiscontinuity and core; (b)“unwrapped” view of borehole wallwith discontinuity displayed as a sinewave (Colog Inc., 1999).


is that a continuous reference line is scribed sothere is no loss of the line in zones of brokencore. However, this is a more expensive andsophisticated piece of equipment than the clayimpression tool.

The most recent advance in core orientationis the use of the scanning borehole camera.A camera developed by Colog Inc. takes acontinuous 360◦ image of the wall of the hole asthe camera is lowered down the hole. The imagecan then be processed to have the appearance ofa piece of core that can be rotated and viewedfrom any direction, or can be “unwrapped”(Figure 3.15). In the core view, planes intersect-ing the core have an elliptical shape, while in theunwrapped view the trace of each discontinuityhas the form of a sine wave. The dip and dipdirection of planes intersecting the core can bedetermined from the plunge and trend of the hole,

and the orientation of the image from the compassincorporated in the camera. The dip direction ofthe plane is found from the position of the sinewave with respect to the compass reading, and thedip of the plane with respect to the core axis canbe determined by the amplitude of the sine wave.The software with the camera system allows ori-entation data indicated by the sine waves to beplotted directly on a stereonet.

The significant advantages of Colog camerasystem are that the camera is run down the holeat the completion of the hole so there is no inter-ruption to drilling. Also, the image provides acontinuous record of rock conditions, includingcavities and zones of broken rock that may belost in the recovered core. The disadvantage ofthe system is the cost and the need for a stablehole with clean walls, that is either dry or is filledwith clean water.

Chapter 4

Rock strength properties andtheir measurement

4.1 Introduction

In analyzing the stability of a rock slope, the mostimportant factor to be considered is the geometryof the rock mass behind the face. As discussed inChapter 3, the relationship between the orient-ation of the discontinuities and of the excavatedface will determine whether parts of the rock massare free to slide or topple. After geology, thenext most important factor governing stability isthe shear strength of the potential sliding surface,which is the subject of this chapter.

4.1.1 Scale effects and rock strength

The sliding surface in a slope may consist of asingle plane continuous over the full area of thesurface, or a complex surface made up of bothdiscontinuities and fractures through intact rock.Determination of reliable shear strength values isa critical part of slope design because, as will beshown in later chapters, small changes in shearstrength can result in significant changes in thesafe height or angle of a slope. The choice ofappropriate shear strength values depends notonly on the availability of test data, but also ona careful interpretation of these data in light ofthe behavior of the rock mass that makes up thefull-scale slope. For example, it may be possibleto use the results obtained from a shear test ona joint in designing a slope in which failure islikely to occur along a single joint similar to theone being tested. However, these shear test resultscould not be used directly in designing a slope inwhich a complex failure process involving several

joints and some failure of intact rock is expected.In this book the term rock mass is used for rockmaterials in which this complex failure processoccurs.

This discussion demonstrates that the selec-tion of an appropriate shear strength of a slopedepends to a great extent on the relative scalebetween the sliding surface and structural geo-logy. For example, in the open pit mine slopeillustrated in Figure 4.1, the dimensions of theoverall slope are much greater than the discon-tinuity length so any failure surface will passthrough the jointed rock mass, and the appropri-ate rock strength to use in design of the pit slope

Intact rock

Single joint set

Two joint sets

Many joints

Jointed rockmass

Figure 4.1 Idealized diagram showing transitionfrom intact rock to jointed rock mass with increasingsample size.

Rock strength properties and their measurement 75

is that of the rock mass. In contrast, the benchheight is about equal to the joint length so sta-bility could be controlled by a single joint, andthe appropriate rock strength to use in design ofthe benches is that of the joints set that dips outof the face. Finally, at a scale of less than thejoint spacing, blocks of intact rock occur and theappropriate rock strength to use in the assessmentof drilling and blasting methods, for example,would be primarily that of the intact rock.

Based on this relationship between sample sizeand rock strength characteristics, this chapterexamines methods of determining the strength ofthe following three classes of rock:

(i) Discontinuities—Single bedding planes,joints or faults. The properties of dis-continuities that influence shear strengthinclude the shape and roughness of the sur-faces, the rock on the surface which may befresh or weathered, and infillings that maybe low strength or cohesive.

(ii) Rock mass—The factors that influence theshear strength of a jointed rock mass includethe compressive strength and frictionangle of the intact rock, and the spacing ofthe discontinuities and the condition of theirsurfaces.

(iii) Intact rock—A factor to consider in meas-uring the strength of the intact rock is that

the strength could diminish over the designlife of the slope due to weathering.

In this book, the subscript “i” is used to designateintact rock, and the subscript “m” to desig-nate the rock mass, for example, the respectivecompressive strengths are σci and σcm.

4.1.2 Examples of rock masses

Figures 4.2–4.5 show four different geologicalconditions that are commonly encountered in thedesign and analysis of rock slopes. These aretypical examples of rock masses in which thestrength of laboratory-size samples may differsignificantly from the shear strength along theoverall sliding surface. In all four cases, instabil-ity occurs because of shear movement along asliding surface that either lies along an existingfracture, or passes partially or entirely throughintact rock. Figures 4.2–4.5 also show that, infractured rock, the shape of the sliding surfaceis influenced by the orientation and length of thediscontinuities.

Figure 4.2 shows a strong, massive limestonecontaining a set of continuous bedding surfacesthat dip out of the slope face. Because the near ver-tical cut face is steeper than the dip of the bedding,the bedding surfaces are exposed, or “daylight”on the face and sliding has occurred with a tension

(a) (b)

Figure 4.2 Plane failure on continuous bedding plane dipping out of the slope (strong, blocky limestone,Crowsnest Pass, Alberta, Canada).

76 Rock strength properties and their measurement

(a) (b)

Figure 4.3 Shallow circular failure in closely jointed rock mass (closely jointed, slightly weathered basalt,Island of Oahu, HI).

(a) (b)

Figure 4.4 Circular failure in residual soil and weathered rock (weathered basalt, Island of Oahu, HI).

crack opening along the sub-vertical, orthogonaljoint set. Under these circumstances, the shearstrength used in stability analysis is that of thebedding surfaces. Section 4.2 of this chapter dis-cusses the strength properties of discontinuitysurfaces, and Section 4.3 describes methods ofmeasuring the friction angle in the laboratory.

Figure 4.3 shows a slope cut in a slightlyweathered, medium weak basalt containing lowpersistence, closely spaced joints that occur in awide variety of orientations. Because the joints arediscontinuous, no single joint controls stability of

the slope. However, if a sliding surface was todevelop in this slope it would follow a steppedpath, which would partly lie along joint surfacesand partly pass through intact rock. The shearstrength of this complex sliding surface cannot bedetermined analytically, so a set of empirical equa-tions has been developed from which the cohesionand friction angle can be calculated with respectto the degree of fracturing and the rock strength.This procedure is described in Section 4.5.

Figure 4.4 shows a slope cut in a weatheredrock in which the degree of weathering varies


(a) (b)

Figure 4.5 Shallow failure in very weak, massive rock containing no discontinuities (volcanic tuff, TransEuropean Highway, near Ankara, Turkey).

from residual soil in the upper part (right) ofthe slope to slightly weathered rock at greaterdepth. For these conditions, the sliding surfacewill lie predominantly in the weaker materials inthe upper part of the slope, and in the stabilityanalysis it is necessary to use different strengthparameters for the upper and lower portions ofthe sliding surface. Because the degree of degrada-tion of weathered rock tends to be highly variable,the strength of the rock mass will also be variableand can be difficult to measure. Consequently, ameans of determining the strength of weatheredrock is to carry out a back analysis of slopesin similar material; this approach is described inSection 4.4.

A fourth geological condition that may beencountered is that of a very weak but intactrock containing essentially no discontinuities.Figure 4.5 shows a cut face in tuff, a rock formedby the consolidation of volcanic ash. A geolo-gical hammer could be embedded in the face witha few blows, indicating the low strength of thisrock. However, because this rock contains no dis-continuities, it has a significant cohesive strengthin addition to a moderate friction angle. There-fore, it was possible to cut a stable, vertical faceto a height of tens of meters in this material,provided water pressures and erosion werecontrolled.

4.1.3 Classes of rock strength

Based on the scale effects and geological condi-tions discussed in the previous sections, it can beseen that sliding surfaces can form either alongdiscontinuity surfaces, or through the rock mass,as illustrated in Figure 4.6. The importance of theclassification shown in Figure 4.6 is that in essen-tially all slope stability analysis it is necessary touse the shear strength properties of either the dis-continuities or of the rock mass, and there aredifferent procedures for determining the strengthproperties as follows:

• Discontinuity shear strength can be measuredin the field and the laboratory as described inSections 4.2 and 4.3.

• Rock mass shear strength is determined byempirical methods involving either back ana-lysis of slopes cut in similar geological con-ditions, or by calculation involving rockstrength indices as described in Sections 4.4and 4.5.

As a further illustration of the effects of geologyon shear strength, relative strength parametersfor three types of discontinuity and two typesof rock mass are shown on the Mohr diagramin Figure 4.7. The slope of these lines repres-ents the friction angle, and the intercept with


Classes ofrock strength

Slidingsurfacealong

discontinuity?

Use discontinuityshear strength

(Sections 4.2, 4.3)

Use rock massshear strength

(Sections 4.4, 4.5)

Yes No

Weak, massive rock

Closely fractured rockJoints parallel to face

Pair of intersecting joints

Figure 4.6 Relationship between geologyand classes of rock strength.

i

Infilledfracture

Smooth, clean fracture

Rough, clean fracture

Fractured, strong rock

Intact,weak rock

Rock masses

Discontinuities

She

ar s

tres

s, �

Effective normal stress, ��Cohesion

�

�

�

�

�

�

�

�

�

�

�

Figure 4.7 Relationshipsbetween shear andnormal stresses on slidingsurface for five differentgeological conditions(Transportation ResearchBoard, 1996).


the shear stress axis represents the cohesion (seeFigure 1.8(a)). A description of these conditionson Figure 4.7 is as follows:

Curve 1 Infilled discontinuity: If the infillingis a weak clay or fault gouge, the infilling fric-tion angle (φinf) is likely to be low, but there maybe some cohesion if the infilling is undisturbed.Alternatively, if the infilling is a strong calcitefor example, which produces a healed surface,then the cohesive strength may be significant (seeSection 4.2.5).

Curve 2 Smooth discontinuity: A smooth,clean discontinuity will have zero cohesion, andthe friction angle will be that of the rock sur-faces (φr). The friction angle of rock is relatedto the grain size, and is generally lower in fine-grained rocks than in coarse-grained rocks (seeSection 4.2.2).

Curve 3 Rough discontinuity: Clean, roughdiscontinuity surfaces will have zero cohesion,and the friction angle will be made up of two com-ponents. First, the rock material friction angle(φr), and second, a component (i) related to theroughness (asperities) of the surface and the ratiobetween the rock strength and the normal stress.As the normal stress increases, the asperities areprogressively sheared off and the total frictionangle diminishes (see Section 4.2.4).

Curve 4 Fractured rock mass: The shearstrength of a fractured rock mass, in whichthe sliding surface lies partially on discontinuitysurfaces and partially passes through intact rock,can be expressed as a curved envelope. At lownormal stresses where there is little confinementof the fractured rock and the individual fragmentsmay move and rotate, the cohesion is low but thefriction angle is high. At higher normal stresses,crushing of the rock fragments begins to takeplace with the result that the friction angle dimin-ishes. The shape of the strength envelope is relatedto the degree of fracturing, and the strength of theintact rock (see Section 4.5).

Curve 5 Weak intact rock: Rock such as thetuff shown in Figure 4.5 is composed of fine-grained material that has a low friction angle.However, because it contains no discontinuities,

the cohesion can be higher than that of a strongintact rock that is closely fractured.

The range of shear strength conditions that maybe encountered in rock slopes as illustrated inFigures 4.2–4.5 clearly demonstrates the import-ance of examining both the characteristics of thediscontinuities and the rock strength during thesite investigation.

4.2 Shear strength of discontinuities

If geological mapping and/or diamond drillingidentify discontinuities on which shear failurecould take place, it will be necessary to determinethe friction angle and cohesion of the sliding sur-face in order to carry out stability analyses. Theinvestigation program should also obtain inform-ation on characteristics of the sliding surfacethat may modify the shear strength parameters.Important discontinuity characteristics includecontinuous length, surface roughness, and thethickness and characteristics of any infilling, aswell as the effect of water on the properties of theinfilling.

The following sections describe the relationshipbetween the shear strength and the properties ofthe discontinuities.

4.2.1 Definition of cohesion and friction

In rock slope design, rock is assumed to be aCoulomb material in which the shear strength ofthe sliding surface is expressed in terms of thecohesion (c) and the friction angle (φ) (Coulomb,1773). The application of these two strengthparameters to rock is discussed in the followingparagraphs.

Assume a number of test samples were cut froma block of rock containing a smooth, planar dis-continuity. Furthermore, the discontinuity con-tains a cemented infilling material such that atensile force would have to be applied to the twohalves of the sample in order to separate them.Each sample is subjected to a force at right anglesto the discontinuity surface (normal stress, σ),and a force is applied in the direction parallel to


Shear displacement, �

Normal stress, �

Shear stress, �

(a)

(c)

(b)

(d)Peak shear strength �= c +� tan�p

Normal stress, �

She

ar s

tres

s, �

Cohesion, c

� tan �p

She

ar s

tres

s, � Peak shear strength

Residual shearstrength

Shear displacement, ��

�

�p

Peak strength �= c +� tan�p

�= � tan �r Residual strengthS

hear

str

ess,

�

Normal stress, �

�r

Figure 4.8 Definition ofshear strength ofdiscontinuity surface;(a) shear test ofdiscontinuity; (b) plotof shear displacement vsshear stress; (c) Mohrplot of peak strength;(d) Mohr plot of peakand residual strength.

the discontinuity (shear stress, τ) while the sheardisplacement (δs) is measured (Figure 4.8(a)).

For a test carried out at a constant normalstress, a typical plot of the shear stress againstthe shear displacement is shown in Figure 4.8(b)).At small displacements, the specimen behaveselastically and the shear stress increases linearlywith displacement. As the force resisting move-ment is overcome, the curve become non-linearand then reaches a maximum that represents thepeak shear strength of the discontinuity. There-after, the stress required to cause displacementdecreases and eventually reaches a constant valuetermed the residual shear strength.

If the peak shear strength values from testscarried out at different normal stress levels areplotted, a relationship shown in Figure 4.8(c)is obtained; this is termed a Mohr diagram(Mohr, 1900). The features of this plot are first,that it is approximately linear and the slopeof the line is equal to the peak friction angleφp of the rock surface. Second, the interceptof the line with the shear stress axis repres-ents the cohesive strength c of the cementingmaterial. This cohesive component of the totalshear strength is independent of the normal

stress, but the frictional component increases withincreasing normal stress. Based on the relation-ship illustrated on Figure 4.8(c), the peak shearstrength is defined by the equation:

τ = c + σ tan φp (4.1)

If the residual shear stress values at each appliednormal stress are plotted on the Mohr diagram,the residual shear strength line is obtained asshown on Figure 4.8(d), and is defined by theequation:

τ = σ tan φr (4.2)

where φr is the residual friction angle. For theresidual strength condition, the cohesion is lostonce displacement has broken the cementingaction; on the Mohr diagram this is representedby the strength line passing through the originof the graph. Also, the residual friction angle isless than the peak friction angle because the sheardisplacement grinds the minor irregularities onthe rock surface and produces a smoother, lowerfriction surface.


Table 4.1 Typical ranges of friction angles for a variety of rock types

Rock class Friction angle range Typical rock types

Low friction 20–27◦ Schists (high mica content), shale, marlMedium friction 27–34◦ Sandstone, siltstone, chalk, gneiss, slateHigh friction 34–40◦ Basalt, granite, limestone, conglomerate

4.2.2 Friction angle of rock surfaces

For a planar, clean (no infilling) discontinuity, thecohesion will be zero and the shear strength willbe defined solely by the friction angle. The fric-tion angle of the rock material is related to thesize and shape of the grains exposed on the frac-ture surface. Thus, a fine-grained rock, and rockwith a high mica content aligned parallel to thesurface, such as a phyllite, will tend to have a lowfriction angle, while a coarse-grained rock such asgranite, will have a high friction angle. Table 4.1shows typical ranges of friction angles for a vari-ety of rock types (Barton, 1973; Jaegar and Cook,1976).

The friction angles listed in Table 4.1 shouldbe used as a guideline only because actual valueswill vary widely with site conditions. Laboratorytesting procedures to determine friction angle aredescribed in Section 4.3.

4.2.3 Shearing on an inclined plane

In the previous section, it was assumed that thediscontinuity surface along which shearing occursis exactly parallel to the direction of the shearstress, τ. Consider now a discontinuity surfaceinclined at an angle i to the shear stress direction(Figure 4.9). In this case, the shear and normalstresses acting on the sliding surface, τi and σi

respectively are given by

τi = τ cos2 i − σ sin i cos i (4.3)

σi = σ cos2 i + τ sin i cos i (4.4)

If it is assumed that the discontinuity surfacehas zero cohesion and that its shear strength is

�i

�i

�

�

�i

�i�

�

i

Figure 4.9 Shear displacement on an inclined plane.

given by

τi = σi tan φ (4.5)

then equations (4.3) and (4.4) can be substitutedin equation (4.5) to give the relationship betweenthe applied shear stress and the normal stress as

τ = σ tan (φ + i) (4.6)

This equation was confirmed in a series of tests onmodels with regular surface projections carriedout by Patton (1966) who must be credited withhaving emphasized the importance of this simplerelationship in the analysis of rock slope stability.

Patton convincingly demonstrated the particu-lar significance of this relationship by measuringthe average value of the angle i from photo-graphs of bedding plane traces in unstable lime-stone slopes. Three of these are reproduced inFigure 4.10 and it will be seen that the rougher thebedding plane trace, the steeper the slope angle.Patton found that the inclination of the beddingplane trace was approximately equal to the sum of


Average dip 56–60°

Average dip43.5°

Average dip31°

1 m

1 m

Figure 4.10 Patton’s observations of bedding planetraces in unstable limestone slopes (Patton, 1966).

the friction angle of the rock φ found from labor-atory tests on planar surfaces, and the averageroughness i angle.

4.2.4 Surface roughness

All natural discontinuity surfaces exhibit somedegree of roughness, varying from polishedand slickensided sheared surfaces with very lowroughness, to rough and irregular tension jointswith considerable roughness. These surface irreg-ularities are given the general term asperities, andbecause they can have a significant effect on thestability of a slope, they should be accountedfor appropriately in design as discussed in thissection.

The discussion in Section 4.2.3 has been sim-plified because Patton found that asperities canbe divided into two classes: first- and second-order asperities as shown in Figure 4.11. Thefirst-order asperities are those that correspondto the major undulations on the bedding sur-faces, while the second-order asperities are smallbumps and ripples on the surface and have higheri values. In order to obtain reasonable agreementbetween field observations of the dip of the

15

4639

2617

10

142537

344132

13

15

0 25 50 cm

Approximate scale

Average i angles for second- order asperities

Average i angles for first-order asperities

Figure 4.11 Measurement of roughness angles i forfirst- and second-order asperities on rough rocksurfaces (Patton, 1966).

unstable bedding planes shown in Figure 4.10 andthe (φ+i) values, it was necessary to measure onlythe first-order asperities.

Later studies by Barton (1973) showed thatPatton’s results were related to the normal stressacting across the bedding planes in the slopesthat he observed. At low normal stresses, thesecond-order projections come into play and Bar-ton quotes (φ + i) values in the range of 69–80◦for tests conducted at low normal stresses rangingfrom 20 to 670 kPa (Goodman, 1970; Paulding,1970; Rengers, 1971). Assuming a friction anglefor the rock of 30◦, these results show that theeffective roughness angle i varies between 40 and50◦ for these low normal stress levels.

The actual shear performance of discontinuitysurfaces in rock slopes depends on the combinedeffects of the surface roughness, the rock strengthat the surface, the applied normal stress and theamount of shear displacement. This is illustratedin Figure 4.12 where the asperities are sheared off,with a consequent reduction in the friction anglewith increasing normal stress. That is, there is atransition from dilation to shearing of the rock.The degree to which the asperities are shearedwill depend on both the magnitude of the normalforce in relation to the compressive strength of therock on the fracture surface, and the displacement

Rock strength properties and their measurement 83S

hear

str

ess,

�

Normal stress, �

�+ i

�

�1

�1

�2

�2

�

�

�

�

i

i = tan–1(�n/�s)

�2�1

Dilation/shearing

Dilation

�s

�n

Figure 4.12 Effect of surfaceroughness and normal stress onfriction angle of discontinuity surface(Transportation Research Board,1996).

distance. A rough surface that is initially undis-turbed and interlocked will have a peak frictionangle of (φ+i). With increasing normal stress anddisplacement, the asperities will be sheared off,and the friction angle will progressively dimin-ish to a minimum value of the basic, or residual,friction angle of the rock. This dilation–shearingcondition is represented on the Mohr diagram asa curved strength envelope with an initial slopeequal to (φ + i), reducing to φr at higher normalstresses.

The shear stress–normal stress relationshipshown in Figure 4.12 can be quantified using atechnique developed by Barton (1973) based onthe shear strength behavior of artificially pro-duced rough, clean “joints.” The study showedthat the shear strength of a rough rock surfacedepends on the relationship between the rough-ness, the rock strength and the normal stress,and can be defined by the following empiricalequation

τ = σ′ tan(

φ + JRC log10

(JCSσ′

))(4.7)

where JRC is the joint roughness coefficient, JCSis the compressive strength of the rock at thefracture surface and σ′ is the effective normalstress. The value of JRC is determined usingthe techniques discussed in Section 3.4.3 thatinvolve either comparing the roughness of thesurface with standard roughness profiles, or mak-ing measurements of the surface. The compress-ive strength of the rock (JCS) can be estimatedusing the simple field observations described inTable 3.1, or with a Schmidt hammer to measurethe rock on the joint surface. The normal stressacting on the surface is the component acting nor-mal to the surface, of the product of the depthof rock above the sliding surface and the unitweight of the rock.

In equation (4.7), the term [JRC log10(JCS/σ′)]is equivalent to the roughness angle i in equa-tion (4.6). At high stress levels, relative to therock strength, when (JCS/σ′) = 1 and the asperit-ies are sheared off, the term [JRC log10(JCS/σ′)]equals zero. At low stress levels the ratio (JCS/σ′)tends to approach infinity and the roughness com-ponent of the shear strength becomes very large.In order that realistic values of the roughness


component are used in design, the term (φ + i)

should not exceed about 50◦ and the usefulrange for the ratio (JCS/σ′) is between about 3and 100.

It is found that both JRC and JCS valuesare influenced by scale effects, that is, as thediscontinuity size increases, there is a corres-ponding decrease in JRC and JCS values. Thereason for this relationship is that small-scaleroughness of a surface becomes less significantcompared to the dimensions of the discontinu-ity, and eventually large-scale undulations havemore significance than the roughness (Barton andBandis, 1983; Bandis, 1993). These effects areconsistent with properties of first- and second-order asperities shown in Figure 4.11. The scaleeffect can be quantified by the following twoequations:

JRCn = JRC0

(Ln

L0

)−0.02JRC0

and

JCSn = JCS0

(Ln

L0

)−0.03JRC0

(4.8)

where L0 is the dimension of the surface used tomeasure JRC such as the wire comb (Figure 3.10),and Ln is the dimension of the sliding surface.

An example of the application of equations (4.7)and (4.8) is as follows. Consider a discontinuitydipping (ψp) at 35◦ and the dimension of this sur-face is 10 m (Ln). If the average depth (H) of thediscontinuity is 20 m below the crest of the slope,and the rock unit weight (γr) is 26 kN/m3, thenthe effective normal stress (σ′) for a dry slope is

σ′ = γrH cos(ψp) = 26 × 20 × cos(35)

= 426 kPa

If the JRC0 value measured with a wire comb(L0 = 0.2 m) is 15, and the rock is strong with aJCS0 value of 50,000 kPa, then the scaled values

are

JRCn = 15(

100.2

)−0.02×15

≈ 5

and

JCSn = 50,000(

100.2

)−0.03×15

≈ 8600 kPa

and the roughness angle is

JRC log10(JCS/σ′) = 5 × log10(8600/426)

≈ 7◦

The concepts of shear strength of rough jointsand scale effects discussed in this section canbe applied to rock slope design as illustrated inFigure 4.13. For example, stress relief, and pos-sibly blast damage during construction, can causeshear movement along discontinuities and dila-tion of the rock at the face. Also, the stress levelson the sliding surface may be high enough to causesome shearing of the asperities (Figure 4.13(a)).For these conditions, the first effect is that anycohesion existing on the undisturbed surface isdiminished. The second effect is loss of interlockon the rough surface so that the second-orderasperities have a diminished effect on the shearstrength. The consequence of this dilation is thatthe roughness angle corresponding to undulatingfirst-order asperities would be used in design.

In contrast to the displaced block shown inFigure 4.13(a), the rock mass can be preventedfrom movement and dilation by the use of care-ful blasting (see Chapter 11), and installation oftensioned rock anchors or passive support suchas dowels and buttresses (see Chapter 12). Underthese conditions, interlock along the sliding sur-face is maintained and the second-order asperitiescontribute to the shear strength of the potentialsliding surface.

The difference in the total friction anglebetween the displaced and undisturbed rockwill have a significant effect on stability and


0.5–2 m

�p

(a)

(b)

i1 = 13°

50–100 mm�p

i2 = 26°

i2 = 28°

Figure 4.13 Effect of asperities on stability of slidingblocks: (a) shear strength of displaced blockcontrolled by first-order asperities (i1); (b) tensionedrock bolts prevent dilation along potential slidingsurface and produce interlock along second-orderasperities (i2).

design of stabilization measures. This demon-strates the value of using construction measuresthat minimize relaxation and dilation of rockmasses.

4.2.5 Discontinuity infilling

The preceding section discussed rough, cleandiscontinuity surfaces with rock-to-rock contactand no infilling, in which the shear strengthis derived solely from the friction angle of therock material. However, if the discontinuity con-tains an infilling, the shear strength properties ofthe fracture are often modified, with both thecohesion and friction angle of the surface beinginfluenced by the thickness and properties of the

infilling. For example, for a clay-filled fault zonein granite, it would be assumed that the shearstrength of the discontinuity would be that of theclay and not the granite. In the case of a healed,calcite-filled fracture, a high cohesion would beused in design, but only if it were certain that thediscontinuity would remain healed after any dis-turbance caused by blasting when excavating theslope.

The presence of infillings along discontinuitysurfaces can have a significant effect on stabil-ity. It is important that infillings be identifiedin the investigation program, and that appropri-ate strength parameters be used in design. Forexample, one of the contributing factors to themassive landslide into the Vaiont Reservoir inItaly that resulted in the death of about 3000people was the presence of low shear strength clayalong the bedding surfaces of the shale (Trollope,1980).

The effect of the infilling on shear strength willdepend on both the thickness and strength prop-erties of the infilling material. With respect to thethickness, if it is more than about 25–50% of theamplitude of the asperities, there will be little orno rock-to-rock contact, and the shear strengthproperties of the fracture will be the properties ofthe infilling (Goodman, 1970).

Figure 4.14 is a plot of the results of directshear tests carried out to determine the peakfriction angle and cohesion of filled discontinu-ities (Barton, 1974). Examination of the testresults shows that the infillings can be dividedapproximately into two groups, as follows:

• Clays: montmorillonite and bentonitic clays,and clays associated with coal measures havefriction angles ranging from about 8 to 20◦and cohesion values ranging from 0 to about200 kPa. Some cohesion values were measuredas high as 380 kPa, which would probably beassociated with very stiff clays.

• Faults, shears and breccias: the materialformed in fault zones and shears in rockssuch as granite, diorite, basalt, and limestonemay contain clay in addition to granular frag-ments. These materials have friction angles


Clay infillingFault gouge,shear zones,low strength rockRange of values

2,111

1157

16 97

8

4

63

4

2621

3

59

1

10 13

14, 20

15

12

25

24

2219

24

18

18 23 17

1. Bentonite shale 2. Bentonite seams in chalk 3. Bentonite; thin layers 4. Bentonite; triaxial tests 5. Clay, over consolidated 6. Limestone, 10–20 mm clay infillings 7. Lignite and underlying clay contact 8. Coal measures; clay mylonite seams 9. Limestone; <1 mm clay infillings 10. Montmorillonite clay 11. Montmorillonite; 80 mm clay seam in chalk 12. Schists/quartzites; stratification, thick clay 13. Schists/quartzites; stratification, thick clay

14. Basalt; clayey, basaltic breccia15. Clay shale; triaxial tests16. Dolomite, altered shale bed17. Diorite/granodiorite; clay gouge18. Granite; clay-filled faults19. Granite; sandy-loam fault fillings20. Granite; shear zone, rock and gouge21. Lignite/marl contact22. Limestone/marl/lignites; lignite layers23. Limestone; marlaceous joints24. Quartz/kaolin/pyrolusite; remolded triaxial25. Slates; finely laminated and altered26. Limestone; 10–20 mm clay infillings

10 20 30 40

Friction angle (degrees)

100

200

300

400

Coh

esio

n (k

Pa)

Figure 4.14 Shear strength of filled discontinuities (modified from Barton, 1970).

ranging from about 25 to 45◦, and cohesionvalues ranging from 0 to about 100 kPa. Faultgouge derived from coarse-grained rocks suchas granites tend to have higher friction anglesthan those from fine-grained rocks such aslimestones.

Some of the tests shown in Figure 4.14 alsodetermined residual shear strength values. It wasfound that the residual friction angle was onlyabout 2–4◦ less than the peak friction angle, whilethe residual cohesion was essentially zero.

Shear strength–displacement behavior is anadditional factor to consider regarding the shearstrength of filled discontinuities. In analyzingthe stability of slopes, this behavior will indic-ate whether there is likely to be a reduction in

shear strength with displacement. In conditionswhere there is a significant decrease in shearstrength with displacement, slope failure canoccur suddenly following a small amount ofmovement.

Filled discontinuities can be divided into twogeneral categories, depending on whether therehas been previous displacement of the discon-tinuity (Barton, 1974). These categories are fur-ther subdivided into either normally consolid-ated (NC) or over-consolidated (OC) materials(Figure 4.15):

• Recently displaced discontinuities—These dis-continuities include faults, shear zones, claymylonites and bedding-surface slips. In faultsand shear zones, the infilling is formed by


Shear strength offilled discontinuities

Faults—often hydrothermal

alteration

Bedding surface

slips

Clay mylonite

Essentially normally consolidated (NC)

� - shear stress� - normal stress� - shear displacement

�, � - peak

peak NC peak OCIIIII

II

IIIIIII, III

residual residual

�, � - residual

Near-surface discontinuities

containing weathering products

Mostly NC clay

Interbedded clay bands,

many hydrothermally altered fillings

Mostly OC clay

Shearzones—often hydrothermal

alteration

Recently displaced Close to residual strength;

therefore, whether normally- or over-consolidated is not of great importance

UndisplacedClose to peak strength;

therefore, whether normally- or over-consolidated is

of considerable importance

�

�

�

�

�

�

�

�

Figure 4.15 Simplified division of filled discontinuities into displaced and undisplaced, and NC and OCcategories (modified from Barton, 1974).

the shearing process that may have occurredmany times and produced considerable dis-placement. The gouge formed in this processmay include both clay-size particles, and brec-cia with the particle orientation and striationsof the breccia aligned parallel to the directionof shearing. In contrast, the mylonites andbedding-surface slips are discontinuities thatwere originally clay bearing, and along whichsliding occurred during folding or sliding.

For these types of discontinuities their shearstrength will be at, or close to, the residualstrength (Figure 4.15, graph I). Any cohesivebonds that existed in the clay due to previousover-consolidation will have been destroyedby shearing and the infilling will be equivalentto the normally consolidated state. In addi-tion, strain softening may occur, with anyincrease in water content resulting in a furtherstrength reduction.

• Undisplaced discontinuities—Infilled discon-tinuities that have undergone no previous dis-placement include igneous and metamorphicrocks that have weathered along the discon-tinuity to form a clay layer. For example,diabase weathers to amphibolite and eventu-ally to clay. Other undisplaced discontinuitiesinclude thin beds of clay and weak shales thatare found with sandstone in interbedded sedi-mentary formations. Hydrothermal alterationis another process that forms infillings that caninclude low strength materials such as mont-morillonite, and high strength materials suchas quartz and calcite.

The infillings of undisplaced discontinuit-ies can be divided into normally-consolidated(NC) and over-consolidated (OC) materi-als that have significant differences in peakstrength values. This strength difference isillustrated on Figure 4.15, graphs II and III.


While the peak shear strength of over-consolidated clay infillings may be high, therecan be a significant loss of strength due tosoftening, swelling and pore pressure changeson unloading. Unloading occurs when rockis excavated for a slope or foundation, forexample. Strength loss also occurs on displace-ment in brittle materials such as calcite.

4.2.6 Influence of water on shear strength ofdiscontinuities

The most important influence of water in a dis-continuity is the diminished shear strength res-ulting from the reduction of the effective normalshear stress acting on the surface. The effect-ive normal stress is the difference between theweight of the overlying rock and the uplift pres-sure produced by the water pressure. The effectof water pressure, u on the shear strength can beincorporated into the shear strength equation asfollows:

τ = c + (σ − u) tan φp − peak strength(4.9a)

or

τ = (σ − u) tan φr − residual strength(4.9b)

These equations assume that the cohesion andfriction angle are not changed by the presenceof water on the surface. In most hard rock andin many sandy soils and gravels, the strengthproperties are not significantly altered by water.However, many clays, shales and mudstones, andsimilar materials will exhibit significant reductionin strength with changes in moisture content. It isimportant, therefore, that the moisture content ofthe test samples be as close as possible to that ofthe in situ rock (see Section 3.6.3 regarding thepreservation of rock samples).

4.3 Laboratory testing of shear strength

The friction angle of a discontinuity surface canbe determined in the laboratory using a directshear box of the type shown in Figure 4.16. Thisis portable equipment that can be used in thefield if required, and is suited to testing sampleswith dimensions up to about 75 mm, such as NQ

b

Vertical displacement gauge

a

Normal load

Pump

Shear load

Lever arm

Jack

Yoke

Hanging weight

Horizontal displacement gauge

Sample

Ball bearing plate

Bottom box

Top box

True vertical displacement = gauge reading × (a /b)

Figure 4.16 Simple equipmentfor performing direct sheartests on rock samples up toabout 75 mm dimensions.


and HQ drill core. The most reliable values areobtained if a sample with a smooth, planar sur-face is used because it is found that, with anirregular surface, the effect of surface roughnesscan make the test results difficult to interpret.

The test procedure consists of using plaster ofparis or molten sulphur to set the two halvesof the sample in a pair of steel boxes (ISRM,1981b). Particular care is taken to ensure that thetwo pieces of core are in their original, matchedposition, and the discontinuity surface is exactlyparallel to the direction of the shear force. A con-stant normal load is then applied using the canti-lever, and the shear load gradually increased untilsliding failure occurs. Measurement of the ver-tical and horizontal displacement of the upperblock relative to the lower block can be mademost simply with dial gauges, while more pre-cise, continuous displacement measurements can

be made with linear variable differential trans-formers (LVDT’s) (Hencher and Richards, 1989).

Each sample is usually tested three or fourtimes, at progressively higher normal loads. Thatis, when the residual shear stress has been estab-lished for a normal load, the sample is reset, thenormal load increased, and another shear testconducted. The test results are expressed as plotsof shear displacement against shear stress fromwhich the peak and residual shear stress valuesare determined. Each test will produce a pair ofshear stress–normal stress values, which are plot-ted to determine the peak and residual frictionangles of the surface.

Figure 4.17 shows a typical result of a dir-ect shear test on a surface with a 4 mm thicksandy silt infilling. The curves on the upper rightare shear stress–shear displacement plots showingan approximate peak shear stress, as well as a

200

150

100

50

41 2 3 8 95 6 750

50

100

100

150

150

200

200

250

250

300350

0.5

0.25

0.05

0.1

�+ i

�r

Residual shear strength

Normal stress, � �2�3 �1 +�n

−�n

�s

Peak shear strength

�=�3

�=�2

�=�1

�1<�2<�3

Shear displacement, �s

kn

1 +

–Normaldisplacement

�s

�n

Shear stress, �

i1�1�2�3

�

�

��

Figure 4.17 Results of direct shear test of filled discontinuity showing measurements of shear strength,roughness (i), and normal stiffness (kn) (modified from Erban and Gill, 1988).


slightly lower, residual shear stress. The samplewas initially undisplaced, so exhibited a differ-ence between peak and residual strengths (seeFigure 4.8). The normal stresses at the peak andresidual shear stress values are calculated fromthe applied normal load and the contact area.When calculating the contact area, an allowanceis made for the decrease in area as shear displace-ment takes place. For diamond drill core in aninclined hole, the fracture surface is in the shapeof an ellipse, and the formula for calculating thecontact area is as follows (Hencher and Richards,1989):

A = πab −[

δsb(4a2 − δ2s )

2a

]1/2

− 2ab sin−1(

δs

2a

)(4.10)

where A is the gross area of contact, 2a is themajor axis of ellipse, 2b is the minor axis of ellipseand δs is the relative shear displacement.

The increase in the normal stress with displace-ment for a constant normal load is shown in theupper left diagram of Figure 4.17 where the nor-mal stress for the residual shear stress is greaterthan that for the peak shear stress.

The measured friction angle is the sum of thefriction angle of the rock (φr), and the roughnessof the surface (i). The roughness of the surfaceis calculated from the plots of shear and normaldisplacement (δs and δn, respectively, on the lowerright side of Figure 4.17) as follows:

i = tan−1(

δn

δs

)(4.11)

This value of i is then subtracted from the frictionangle calculated from the plot of shear and nor-mal stresses at failure to obtain the friction angleof the rock. While the shear test can be conductedon a sawed sample on which there is no roughnesscomponent, the saw may polish the surface result-ing in a low value of the friction angle comparedto a natural surface.

As shown in Figure 4.17, it is usual to testeach sample at a minimum of three normal stresslevels, with the sample being reset to its originalposition between tests. When the tests are runat progressively higher normal stress levels, thetotal friction angle of the surface will diminishwith each test if the asperities are progressivelysheared. This produces a concave upwards nor-mal stress–shear stress plot as shown in the upperleft plot of Figure 4.17. The degree to which theasperities are sheared off will depend on the levelof the normal stress in comparison to the rockstrength, that is, the ratio JCS/σ in equation (4.7).The maximum normal stress that is used in thetest is usually the maximum stress level that islikely to develop in the slope.

It is also possible to measure the normal stiff-ness of the discontinuity infilling during the directshear test as shown in the lower left plot inFigure 4.17. Normal stiffness kn is the ratio ofthe normal stress σ to normal displacement δn, or

kn =(

σ

δn

)(4.12)

The plot of σ against δn is highly non-linear andthe value of kn is the slope of the initial portion ofthe curve. The normal stiffness of a fracture is notusually an issue in rock slope design, and is moreoften used in the estimation of the deformationmodulus of a rock mass (Wyllie, 1999), and innumerical analysis (see Chapter 10).

It can be difficult to measure the cohesion of asurface with the direct shear test because, if thecohesion is very low, it may not be possible toobtain an undisturbed sample. If the cohesion ishigh and the sample is intact, the material hold-ing the sample in the test equipment will have tobe stronger than the infilling if the sample is toshear. Where it is important that the cohesion ofa weak infilling be measured, an in situ test of theundisturbed material may be required.

4.4 Shear strength of rock masses byback analysis of slope failures

For the geological conditions shown in Figure 4.3where a cut has been made in fractured rock,


H = 30.5 m

b = 12.5 m

z = zw = 19.8 m

�p= 20°

�f = 58°

U

V

W

Figure 4.18 Cross-section ofquarry slope failure showinggeometry and water forces.

there is no distinct discontinuity surface on whichsliding can take place. A sliding surface in thisrock mass will comprise both natural discon-tinuities aligned on the sliding surface, togetherwith some shear failure through intact rock. It isgenerally difficult and expensive to sample andtest large samples (∼1 m diameter) of fracturedrock. Consequently, two empirical methods ofdetermining the friction angle and cohesion ofrock masses have been developed—back ana-lysis that is described in this section, and theHoek–Brown strength criterion that is discussedin Section 4.5. In both methods, it is necessaryto categorize the rock mass in terms of both theintact rock strength and the characteristics of thediscontinuities. This may require some judgmentand it is advisable, where possible, to comparethe strength values obtained by both methods toimprove the reliability of values used in design.

Probably the most reliable method of determin-ing the strength of a rock mass is to back analyzea failed, or failing slope. This involves carryingout a stability analysis using available informa-tion on the position of the sliding surface, theground water conditions at the time of failureand any external forces such as foundation loadsand earthquake motion, if applicable. With thefactor of safety set at 1.0, the stability analysisis used to calculate the friction angle and cohe-sion. This section describes the back analysis of afailed slope in a limestone quarry, and the use of

the calculated shear strength values to design anappropriate slope angle for a deeper pit (Robertsand Hoek, 1972).

Figure 4.18 shows the geometry of the slopefailure in which sliding occurred on beddingplanes striking parallel to the face, and dippingout of the face at an angle of 20◦. These condi-tions are applicable to plane failure conditions asshown in Figure 2.16 and described in more detailin Chapter 6.

At the time of failure, there was an open ten-sion crack on the upper, horizontal bench of thequarry and the dimensions of the failed masswere defined by the dimensions H , b, ψf and ψp.From these dimensions and a rock unit weightof 25.1 kN/m3, the weight of the sliding masswas calculated to be 12.3 MN/m. Immediatelyprior to failure, a heavy rainstorm flooded theupper bench so that the tension crack was full ofwater (zw = z). It was assumed that water forcesacting in the tension crack and on the beddingplane could be represented by triangular distri-butions with magnitudes of V = 1.92 MN/m andU = 3.26 MN/m respectively.

In back analysis, both the friction angle andthe cohesion of the sliding surface are unknown,and their values can be estimated by the followingmethod. The likely range of the friction angle canusually be estimated by inspection (see Table 4.1),or by laboratory testing if samples containingbedding planes are available. In this case where


Coh

esio

n, k

Pa

Friction angle, �°

Range of strength values for slope failure

0 10 20 30

50

100

150

Figure 4.19 Shear strength mobilized on beddingplane for slope failure shown in Figure 4.18.

the limestone was fine-grained and the beddingplanes smooth, it was estimated that the frictionangle was in the range of 15–25◦. The next stepwas to carry out a number of stability analyseswith a range of cohesion values and a factor ofsafety of 1.0. The results of this analysis showthat, at a friction angle of 20◦ the correspond-ing cohesion value is about 110 kPa, and thatfor higher friction angles the required cohesionis reduced (Figure 4.19).

The shear strength values calculated in thismanner can be used to design slopes excavatedin this limestone, provided that careful blastingis used to maintain the cohesion on the bed-ding planes. Figure 4.20 shows the relationshipbetween the factor of safety and the face anglefor a 64 m high cut, assuming plane failure on abedding plane dipping at 20◦ out of the face. Ifthe slope is drained, the shear strength is sufficientfor the face to stand vertically, but if the slope issaturated, the steepest stable slope is about 50◦.

In many cases it may not be feasible to carryout a back analysis of a slope in similar geolo-gical conditions to that in which the new slopeis to be excavated. In these circumstances, pub-lished results of rock mass shear strength can beused in design. Figure 4.21 shows the results ofback analyses of slope failures in a variety of geo-logical conditions (as described in Table 4.2), andthe shear strength parameters (φ/c values) calcu-lated at failure. By adding additional points to

Fully drained slope

Saturated slope

Fact

or o

f saf

ety,

FS

Face angle, �f (degrees)

20 30 40 50 60 70 80 90

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Figure 4.20 Relationship between factor of safety andface angle of dry and saturated slope for slope shownin Figure 4.18.

Figure 4.21 for local geological conditions, it ispossible to draw up a readily applicable rock massstrength chart for shear failures. Point 6 is for theslope shown in Figure 4.18.

4.5 Hoek–Brown strength criterion forfractured rock masses

As an alternative to back analysis to determine thestrength of fractured rock masses, an empiricalmethod has been developed by Hoek and Brown(1980a,b) in which the shear strength is repres-ented as a curved Mohr envelope. This strengthcriterion was derived from the Griffith crack the-ory of fracture in brittle rock (Hoek, 1968), aswell as from observations of the behavior of rockmasses in the laboratory and the field (Marsal,1967, 1973; Brown, 1970; Jaeger, 1970).

Hoek and Brown introduced their failure cri-terion to provide input data for the analysesrequired for the design of underground excav-ations in hard rock. The criterion started fromthe properties of intact rock, and then intro-duced factors to reduce these properties based onthe characteristics of joints in a rock mass. The


Residual strength of slickensided surfaces coated with high clay mineral content materials.

Disturbed material with rounded weakly cemented particles and appreciable clay mineral content.

Disturbed soil and unjointed rock masses with relatively low clay mineral content.

Rock masses or fill containing hard clean angular interlocking particles and blocks.

Undisturbed hardrock masses withno major structuralpatterns dippingout of face.

Undisturbed hardrock masses withno through-goingstructures dippingout of face.

Undisturbedjointed soft rockmasses with fewstructures dippingout of face.

Soft rock masses or jointed hard rock disturbed by blasting or excessloading.

Weathered soft rock or discontinuities inhard rock.

Clay or soil or sand.

5

50

1

2

3

4

8

5

9

6

10

7

11

10

10

15

15

20

20

25

25

30

30

35

35

40

40

45

45 50

500

200

100

300

400

Coh

esio

n, c

(p.

s.f.

×10

00)

Coh

esio

n, c

(kP

a)

1512

13 7

6

168 10 20

18

2

1

11

19

3

9

17 14

5

4

Friction angle, � (degrees)

Figure 4.21 Relationship between friction angles and cohesive strength mobilized at failure for slopes analyzedin Table 4.2.

authors sought to link the empirical criterion togeological observations by means of one of theavailable rock mass classification schemes and,for this purpose, they chose the Rock Mass Ratingproposed by Bieniawski (1976).

Because of the lack of suitable alternatives, thecriterion was soon adopted by the rock mechan-ics community and its use quickly spread beyondthe original limits used in deriving the strengthreduction relationships. Consequently, it became

necessary to re-examine these relationships andintroduce new elements from time to time toaccount for the wide range of practical problemsto which the criterion was being applied (Hoeket al., 2002). Typical of these enhancements werethe introduction of the idea of “undisturbed”and “disturbed” rock masses Hoek and Brown(1988), and of a modified criterion to force therock mass tensile strength to zero for very poorquality rock masses (Hoek et al., 1992).


Table 4.2 Sources of shear strength data plotted in Figure 4.21

Pointnumber

Material Location Slope height(m)

Reference

1 Disturbed slates andquartzite’s

Knob Lake, Canada — Coates et al. (1965)

2 Soil Any location — Whitman and Bailey(1967)

3 Jointed porphyry Rio Tinto, Spain 50–110 Hoek (1970)4 Ore body hanging

wall in granite rocksGrangesberg, Sweden 60–240 Hoek (1974)

5 Rock slopes withslope angles of50–60◦

Any location 300 Ross–Brown (1973)

6 Bedding planes inlimestone

Somerset, England 60 Roberts and Hoek(1972)

7 London clay, stiff England — Skempton andHutchinson (1969)

8 Gravelly alluvium Pima, Arizona — Hamel (1970)9 Faulted rhyolite Ruth, Nevada — Hamel (1971a)

10 Sedimentary series Pittsburgh, Pennsylvania — Hamel (1971b)11 Kaolinized granite Cornwall, England 75 Ley (1972)12 Clay shale Fort Peck Dam, Montana — Middlebrook (1942)13 Clay shale Gardiner Dam, Canada — Fleming et al. (1970)14 Chalk Chalk Cliffs, England 15 Hutchinson (1970)15 Bentonite/clay Oahe Dam, South Dakota — Fleming et al. (1970)16 Clay Garrison Dam, North Dakota — Fleming et al. (1970)17 Weathered granites Hong Kong 13–30 Hoek and Richards

(1974)18 Weathered volcanics Hong Kong 30–100 Hoek and Richards

(1974)19 Sandstone, siltstone Alberta, Canada 240 Wyllie and Munn

(1979)20 Argillite Yukon, Canada 100 Wyllie (project files)

One of the early difficulties arose becausemany geotechnical problems, particularly regard-ing slope stability analysis, are more convenientlydealt with in terms of shear and normal stressesrather than the principal stress relationships ofthe original Hoek–Brown criterion, defined by theequation:

σ′1 = σ′

3 + σci

(m

σ′3

σci+ s

)0.5

(4.13)

where σ′1 and σ′

3 are respectively the major andminor effective principal stresses at failure, σci

is the uniaxial compressive strength of the intactrock material and m and s are material constants;s = 1 for intact rock.

An exact relationship between equation (4.13)and the normal and shear stresses at failure wasderived by J. W. Bray (reported by Hoek, 1983)and later by Ucar (1986).

Hoek (1990) discussed the derivation of equi-valent friction angles and cohesive strengths forvarious practical situations. These derivationswere based upon tangents to the Mohr envel-ope derived by Bray. Hoek (1994) suggestedthat the cohesive strength determined by fittinga tangent to the curvilinear Mohr envelope is


an upper bound value and may give optimisticresults in stability calculations. Consequently,an average value, determined by fitting a lin-ear Mohr–Coulomb relationship by least squaresmethods, may be more appropriate. In the 1994paper, Hoek also introduced the concept of theGeneralized Hoek–Brown criterion in which theshape of the principal stress plot or the Mohrenvelope could be adjusted by means of a vari-able coefficient a in place of the 0.5 power termin equation (4.13).

Hoek and Brown (1997) attempted to consol-idate all the previous enhancements into a com-prehensive presentation of the failure criterionand they gave a number of worked examples toillustrate its practical application.

In addition to the changes in the equations,it was also recognized that the Rock Mass Rat-ing of Bieniawski was no longer adequate as avehicle for relating the failure criterion to geo-logical observations in the field, particularly forvery weak rock masses. This resulted in the intro-duction of the Geological Strength Index (GSI)by Hoek et al. (1992), Hoek (1994) and Hoek,Kaiser and Bawden (1995). This index was sub-sequently extended for weak rock masses in aseries of papers by Hoek et al. (1998), Marinosand Hoek (2000, 2001) and Hoek and Marinos(2000).

The GSI provides a system for estimating thereduction in rock mass strength for different geo-logical conditions. Values of GSI are related toboth the degree of fracturing and the condi-tion of fracture surfaces, as shown in Tables 4.3and 4.4 respectively for blocky rock masses andschistose metamorphic rock masses. The strengthof a jointed rock mass depends on the propertiesof the intact rock pieces, as well as the free-dom of the rock pieces to slide and rotate underdifferent stress conditions. This freedom is con-trolled by the geometrical shape of the intact rockpieces and the condition of the surfaces separat-ing the pieces. Angular rock pieces with clean,rough surfaces will result in a much strongerrock mass than one that contains roundedparticles surrounded by weathered and alteredmaterial.

The description of the Hoek–Brown strengthcriterion in this section includes all the data inthis work up to 2002.

4.5.1 Generalized Hoek–Brown strengthcriterion

The generalized Hoek–Brown strength criterionis expressed in terms of the major andminor principal stresses, and is modified fromequation (4.13) as follows (Figure 4.22)

σ′1 = σ′

3 + σci

(mb

σ′3

σci+ s

)a

(4.14)

where mb is a reduced value of the materialconstant mi for intact rock and is given by

mb = mi exp(

GSI − 10028 − 14D

)(4.15)

Table 4.5 gives values of mi for a wide variety ofrock types, and s and a are constants for the rockmass given by

s = exp(

GSI − 1009 − 3D

)(4.16)

a = 12

+ 16

(e−GSI/15 − e−20/3) (4.17)

D is a factor that depends upon the degree ofdisturbance to which the rock mass has been sub-jected by blast damage and stress relaxation. Itvaries from 0 for undisturbed in situ rock massesto 1 for very disturbed rock masses; guidelinesfor the selection of appropriate values for D arediscussed in Section 4.5.6.

The uniaxial compressive strength of the rockmass is obtained by setting σ′

3 = 0 in equa-tion (4.14), giving

σc = σci · sa (4.18)

and, the tensile strength is

σt = − sσci

mb

(4.19)


Table 4.3 GSI values characterizing blocky rock masses on the basis of particle interlockingand discontinuity condition

GEOLOGICAL STRENGTH INDEX FORJOINTED ROCKS (Hoek and Marinos, 2000)From the lithology, structure and surfaceconditions of the discontinuities, estimatethe average value of GSI. Do not try tobe too precise. Quoting a range from 33to 37 is more realistic than stating thatGSI = 35. Note that the table does notapply to structurally controlled failures.Where weak planar structural planes arepresent in an unfavorable orientationwith respect to the excavation face, thesewill dominate the rock mass behavior.The shear strength of surfaces in rocksthat are prone to deterioration as a resultof changes in moisture content will be reduced if water is present. When work-ing with rocks in the fair to very poor categories, a shift to the right may bemade for wet conditions. Water pressureis dealt with by effective stress analysis.

STRUCTURE

BLOCKY/DISTURBED/SEAMY—folded with angular blocksformed by many intersectingdiscontinuity sets. Persistenceof bedding planes or schistosity

LAMINATED/SHEARED—lackof blockiness due to close spacingof weak schistosity or shear planes

DISINTEGRATED—poorly inter-locked, heavily broken rock masswith mixure of angular androunded rock pieces

VERY BLOCKY—interlocked,partially disturbed mass withmulti-faceted angular blocksformed by 4 or more joint sets

BLOCKY—well interlocked un-disturbed rock mass consistingof cubical blocks formed by threeintersecting discontinuity sets

INTACT OR MASSIVE—intactrock specimens or massive in siturock with few widely spaceddiscontinuities

N/A N/A

N/A N/A

DE

CR

EA

SIN

G IN

TE

RLO

CK

ING

OF

RO

CK

PIE

CE

SS

UR

FAC

E C

ON

DIT

ION

S

VE

RY

GO

OD

Ver

y ro

ugh,

fres

h un

wea

ther

ed s

urfa

ces

VE

RY

PO

OR

Slic

kens

ided

, hig

hly

wea

ther

ed s

urfa

ces

with

sof

t cla

yco

atin

gs o

r fil

lings

PO

OR

Slic

kens

ided

, hig

hly

wea

ther

ed s

urfa

ces

with

com

pact

coat

ings

or

fillin

gs o

r an

gula

r fr

agm

ents

FAIR

Sm

ooth

, mod

erat

ely

wea

ther

ed a

nd a

ltere

d su

rfac

es

GO

OD

Rou

gh, s

light

ly w

eath

ered

, iro

n st

aine

d su

rfac

es

10

20

30

40

50

60

70

80

90

DECREASING SURFACE QUALITY

Rock strength properties and their measurement 97T

able

4.4

GSI

valu

esch

arac

teri

zing

schi

stos

em

etam

orph

icro

ckm

asse

son

the

basi

sof

folia

tion

and

disc

onti

nuit

yco

ndit

ion

GS

I FO

R H

ET

ER

OG

EN

EO

US

RO

CK

MA

SS

ES

SU

CH

AS

FLY

SC

H(M

arin

os P

. and

Hoe

k E

., 20

00)

From

a d

escr

iptio

n of

the

litho

logy

, str

uctu

re a

nd s

urfa

ce c

ondi

tions

(pa

rtic

ular

lyof

the

bedd

ing

plan

es),

cho

ose

a bo

x in

the

char

t. Lo

cate

the

posi

tion

in th

e bo

xth

at c

orre

spon

ds to

the

cond

ition

of t

he d

isco

ntin

uitie

s an

d es

timat

e th

e av

erag

eva

lue

of G

SI f

rom

the

cont

ours

. Do

not a

ttem

pt to

be

too

prec

ise.

Quo

ting

a ra

nge

from

33

to 3

7 is

mor

e re

alis

tic th

an g

ivin

g G

SI =

35.

Not

e th

at th

e H

oek-

Bro

wn

crite

rion

does

not

app

ly to

str

uctu

rally

con

trol

led

failu

res.

Whe

re u

nfav

oura

bly

orie

nted

con

tinuo

us w

eak

plan

ar d

isco

ntin

uitie

s ar

e pr

esen

t, th

ese

will

dom

inat

eth

e be

havi

our

of th

e ro

ck m

ass.

The

str

engt

h of

som

e ro

ck m

asse

s is

red

uced

by

the

pres

ence

of g

roun

dwat

er a

nd th

is c

an b

e al

low

ed fo

r by

a s

light

shi

ft to

the

right

in th

e co

lum

ns fo

r fa

ir, p

oor

and

very

poo

r co

nditi

ons.

Wat

er p

ress

ure

does

not c

hang

e th

e va

lue

of G

SI a

nd it

is d

ealt

with

by

usin

g ef

fect

ive

stre

ss a

naly

sis.

CO

MP

OS

ITIO

N A

ND

ST

RU

CT

UR

E

A. T

hick

bed

ded,

ver

y bl

ocky

san

dsto

neT

he e

ffect

of p

eliti

c co

atin

gs o

n th

e be

ddin

gpl

anes

is m

inim

ized

by

the

conf

inem

ent o

fth

e ro

ck m

ass.

In s

hallo

w tu

nnel

s or

slo

pes

thes

e be

ddin

g pl

anes

may

cau

se s

truc

tura

llyco

ntro

lled

inst

abili

ty.

G. U

ndis

turb

ed s

ilty

or c

laye

y sh

ale

with

or w

ithou

t a fe

w v

ery

thin

san

dsto

ne la

yers

F. T

ecto

nica

lly d

efor

med

, int

ensi

vely

fold

ed/fa

ulte

d, s

hear

ed c

laye

y sh

ale

or s

iltst

one

with

bro

ken

and

defo

rmed

sand

ston

e la

yers

form

ing

an a

lmos

tch

aotic

str

uctu

re

H. T

ecto

nica

lly d

efor

med

silt

y or

clay

ey s

hale

form

ing

a ch

aotic

stru

ctur

e w

ith p

ocke

ts o

f cla

y.T

hin

laye

rs o

f san

dsto

ne a

retr

ansf

orm

ed in

to s

mal

l roc

k pi

eces

.

C,D

,E a

nd G

—m

ay b

e m

ore

orle

ss fo

lded

than

Ilus

trat

ed b

utth

is d

oes

not c

hang

e th

e st

reng

th.

Tect

onic

def

orm

atio

n, fa

ultin

g an

dlo

ss o

f con

tinui

ty m

oves

thes

eca

tego

ries

to F

and

H.

: Mea

ns d

efor

mat

ion

afte

r te

cton

ic d

istu

rban

ce

SURFACE CONDITIONS OFDISCONTINUITIES(Predominantly bedding planes)

VERY GOOD—Very rough,fresh unweathered surfaces

FAIR—Smooth, moderatelyweathered and altered surfaces

POOR—Very smooth, occasionallyslickensided surfaces with compactcoatings or fillings with angularfragments

VERY POOR—Very smooth slicken-sided or highly weathered surfaceswith soft clay coatings or fillings

GOOD—Rough, slightlyweathered surfaces

50

40

30

20

10

A

BC

DE F

GH

70

60

B. S

and-

ston

e w

ithth

in in

ter-

laye

rs o

fsi

ltsto

ne

C. S

and-

ston

e an

dsi

ltsto

ne in

sim

ilar

amou

nts

D. S

iltst

one

or s

ilty

shal

ew

ith s

and-

ston

e la

yers

E. W

eak

silts

tone

or c

laye

ysh

ale

with

sand

ston

ela

yers


Table 4.5 Values of constant mi for intact rock by rock Group (values in parenthesis are estimates)

Rocktype

Class

Clastic

Non-Clastic

Group TextureCoarseConglomerates(21 ± 3)

CrystallineLimestone(12 ± 3)

Siltstones7 ± 2

SpariticLimestones

(10 ± 2)

Gypsum8 ± 2

Anhydrite12 ± 2

Chalk7 ± 2

MicriticLimestones

(9 ± 2)

Dolomites(9 ± 3)

Claystones4 ± 2

Shales(6 ± 2)

Marls(7 ± 2)

Greywackes(18 ± 3)

Sandstones17 ± 4

Breccias(19 ± 5)

Fine Very fineMedium

Carbonates

Evaporites

Organic

Non foliated

Slightly foliated

Foliated*

Plutonic

Hypabyssal

Volcanic

Light

Dark

Lava

Pyroclastic

Marble9 ± 3

Migmatite(29 ± 3)

Gneiss28 ± 5

Granite32 ± 3

Gabbro27 ± 3 Dolerite

(16 ± 5)

Porphyries(20 ± 5)

Dacite(25 ± 3)

Obsidian(19 ± 3)

Agglomerate(19 ± 3)

Tuff(13 ± 5)

Breccia(19 ± 5)

Basalt(25 ± 5)

Rhyolite(25 ± 5)

Andesite25 ± 5

Peridotite(25 ± 5)

Diabase(15 ± 5)

Norite20 ± 5

Diorite25 ± 5

Granodiorite(29 ± 3)

Schists12 ± 3

Slates7 ± 4

Phyllites(7 ± 3)

Amphibolites26 ± 6

Hornfels(19 ± 4)

Quartzites20 ± 3

Metasandstone(19 ± 3)

IGN

EO

US

ME

TAM

OR

PH

ICS

ED

IME

NTA

RY

* These values are for intact rock specimens tested normal to bedding or foliation. The value of mi willbe significantly different if failure occurs along a weakness plane.


50

30

40

20

10

–5 0 5 10

Maj

or p

rinci

pal s

tres

s (�

1�)

Minor principal stress (�3�)

�t

��1=��3 +�ci

��3�ci

(mb + s)a

��1= ��3

��3max

+ 2c�cos��1 – sin��

1+ sin��1 – sin��

Figure 4.22 Relationships between major and minorprincipal stresses for Hoek–Brown and equivalentMohr–Coulomb criteria.

Equation (4.19) is obtained by setting σ′1 = σ′

3 =σt in equation (4.14). This represents a conditionof biaxial tension. Hoek (1983) showed that, forbrittle materials, the uniaxial tensile strength isequal to the biaxial tensile strength.

Note that the “switch” at GSI = 25 for thecoefficients s and a (Hoek and Brown, 1997) hasbeen eliminated in equations (4.16) and (4.17)which give smooth continuous transitions for theentire range of GSI values.1

Normal and shear stresses are related to prin-cipal stresses by the equations published by

1 Note that the numerical values of s and a, given by equa-tions (4.16) and (4.17), are very close to those given bythe previous equations in Hoek and Brown (1997) andit is not necessary to revisit and make corrections to oldcalculations.

Balmer2 (1952):

σ′n = σ′

1 + σ′3

2− σ′

1 − σ′3

2

(dσ′

1/dσ′3 − 1

dσ′1/dσ′

3 + 1

)(4.20)

τ = (σ′1 − σ′

3)

√dσ′

1/dσ′3

dσ′1/dσ′

3 + 1(4.21)

where

dσ′1/dσ′

3 = 1 + amb(mbσ′3/σci + s)a−1 (4.22)

4.5.2 Modulus of deformation

The Hoek–Brown failure criterion also allowsthe rock mass modulus of deformation to becalculated as follows:

Em =(

1 − D

2

)√σci

10010((GSI−10)/40)

(units: GPa) (4.23)

Note that the original equation proposed by Hoekand Brown (1997) has been modified, by theinclusion of the factor D, to allow for the effectsof blast damage and stress relaxation.

The principal use of the rock mass modulus ofdeformation is in numerical analysis to calculatestrain in rock slopes (see Chapter 10).

4.5.3 Mohr–Coulomb criterion

The analysis of slope stability involves examina-tion of the shear strength of the rock mass on thesliding surface expressed by the Mohr–Coulombfailure criterion. Therefore, it is necessary todetermine friction angles and cohesive strengthsthat are equivalent between the Hoek–Brownand Mohr–Coulomb criteria. These strengths arerequired for each rock mass and stress range alongthe sliding surface. This is done by fitting an aver-age linear relationship to the curve generated by

2 The original equations derived by Balmer contained errorsthat have been corrected in equations (4.20) and (4.21).


solving equation (4.14) for a range of minor prin-cipal stress values defined by σt < σ3 < σ3 max,as illustrated in Figure 4.22. The fitting processinvolves balancing the areas above and belowthe Mohr–Coulomb plot. This results in the fol-lowing equations for the angle of friction φ′ andcohesive strength c′ (Figure 4.23):

φ′ = sin−1

[6amb(s + mbσ′

3n)a−1

2(1 + a)(2 + a) + 6amb(s + mbσ′3n)a−1

]

(4.24)

c′ = (σci[(1 + 2a)s + (1 − a)mbσ′

3n](s + mbσ′

3n)a−1)/ [(1 + a)(2 + a)

×√

1 + (6amb(s + mbσ′3n)a−1)/((1 + a)(2 + a))

](4.25)

where σ3n = σ′3 max/σci.

Note that the value of σ′3 max, the upper

limit of confining stress over which the rela-tionship between the Hoek–Brown and theMohr–Coulomb criteria is considered, has to bedetermined for each individual case. Guidelinesfor selecting these values for slopes are presentedin Section 4.5.5.

The Mohr–Coulomb shear strength τ, for agiven normal stress σ, is found by substitutionof these values of c′ and φ′ into the followingequation:

τ = c′ + σ tan φ′ (4.26)

The equivalent plot, in terms of the major andminor principal stresses, is defined by

σ′1 = 2c′ cos φ′

1 − sin φ′ + 1 + sin φ′

1 − sin φ′ σ′3 (4.27)

4.5.4 Rock mass strength

The uniaxial compressive strength of the rockmass σc is given by equation (4.18). For under-ground excavations, instability initiates at theboundary of the excavation when the compressivestrength σc is exceeded by the stress induced on

that boundary. The failure propagates from thisinitiation point into the biaxial stress field andit eventually stabilizes when the local strength,defined by equation (4.14), is higher than theinduced stresses σ′

1 and σ′3. Most numerical mod-

els can follow this process of fracture propagationand this level of detailed analysis is very import-ant when considering the stability of excavationsin rock and designing support systems.

However, for slope stability, failure is ini-tiated along a sliding surface within the slopewhere the rock is subject to a biaxial stress fieldand it is useful to consider the overall behaviorof a rock mass rather than the detailed failurepropagation process described earlier. This leadsto the concept of a global “rock mass strength”and Hoek and Brown (1997) proposed that thiscould be estimated from the Mohr–Coulombrelationship:

σ′cm = 2c′ cos φ′

1 − sin φ′ (4.28)

with c′ and φ′ determined for the stress range σt <σ′

3 < σci/4 giving the following value for the rockmass strength σ′

cm:

σ′cm = σci

(mb + 4s − a(mb − 8s))(mb/4 + s)a−1

2(1 + a)(2 + a)

(4.29)

4.5.5 Determination of σ′3 max

The issue of determining the appropriate valueof σ′

3 max for use in equations (4.24) and (4.25)depends upon the specific application. For thecase of slopes, it is necessary that the calculatedfactor of safety and the shape and locationof the failure surface be equivalent. Stabilitystudies of rock slopes using Bishop’s circularfailure analysis for a wide range of slope geo-metries and rock mass properties have beencarried out for both Generalized Hoek–Brownand Mohr–Coulomb criteria to find the valueof σ′

3 max that gives equivalent characteristiccurves. These analyses gave the following rela-tionship between σ′

3 max the rock mass strength


1.4

1.2

1.0

0.8

0.6

0.4

0.2c = 0.19

MPa

0.0 0.2 0.4 0.6 0.8 1.0

Normal stress � (MPa)

Hoek–Brown �= 0.48 MPa �= 0.71 MPa

Mohr–Coulomb �= 0.49 MPa �= 0.69 MPa

She

ar s

tres

s �

(MP

a)

�n

�

� = 45.5°

Figure 4.23 Non-linear Mohr envelope for fracturedrock mass defined by equations (4.24) and (4.25); bestfit line shows cohesion and friction angle forapplicable slope height. Rock mass parameters:σc = 30 MPa, GSI = 50, mi = 10, D = 0.7,H = 20 m, γ = 0.026 MN/m3.

σ′cm and the stress level on the sliding surface, σ0

(Figure 4.24):

σ′3 max

σ′cm

= 0.72(

σ′cm

σ0

)−0.91

(4.30)

The stress level on the sliding surface is related tothe slope height H and the unit weight of the rockγr is given by

σ0 = H · γr (4.31)

4.5.6 Estimation of disturbance factor D

Experience in the design of slopes in very largeopen pit mines has shown that the Hoek–Browncriterion for undisturbed in situ rock masses(D = 0) results in rock mass properties that aretoo optimistic (Pierce et al., 2001; Sjöberg et al.,2001). The effects of heavy blast damage as wellas stress relief due to removal of the overburden

40

20

30

10

0 0.40.2 0.30.1

Rat

io o

f �� 3m

ax/�

� cm

Ratio of rock mass strength to in situ stress��cm/�0

��3max

��cm

��cm

�0= 0.72

–0.91

Figure 4.24 Relationship for the calculation of σ′3 max

for equivalent Mohr–Coulomb and Hoek–Brownparameters for slopes.

result in disturbance of the rock mass (Hoek andBrown, 1988). It is considered that the “dis-turbed” rock mass properties using D = 1 inequations (4.14) and (4.15) are more appropriatefor these rock masses.

A number of other studies to assess the degreeof disturbance of the rock mass have been carriedout by observing the performance of surface andunderground excavations. For example, Lorigand Varona (2001) showed that factors suchas the lateral confinement produced by differentradii of curvature of slopes (in plan) as comparedwith their height also have an influence on thedegree of disturbance. Also, Sonmez and Ulusay(1999) back-analyzed five slope failures in openpit coal mines in Turkey and attempted to assigndisturbance factors to each rock mass based upontheir assessment of the rock mass properties pre-dicted by the Hoek–Brown criterion. Unfortu-nately, one of the slope failures appears to bestructurally controlled while another consists ofa transported waste pile. Hoek considers that theHoek–Brown criterion is not applicable to thesetwo cases. In addition, Cheng and Liu (1990)report the results of very careful back analysis of


deformation measurements, from extensometersplaced before the commencement of excavation,in the Mingtan power cavern in Taiwan. It wasfound that a zone of blast damage extended for adistance of approximately 2 m around all largeexcavations. The back-calculated strength anddeformation properties of the damaged rock massgive an equivalent disturbance factor D = 0.7.

From these references it is clear that a largenumber of factors can influence the degree ofdisturbance in the rock mass surrounding anexcavation, and that it may never be possible toquantify these factors precisely. However, basedon experience and on the analysis of the detailscontained in these papers, Hoek et al. (2002) havedrawn up a set of guidelines for estimating thefactor D (Table 4.6).

The influence of this disturbance factor can belarge. This is illustrated by a typical example inwhich σci = 50 MPa, mi = 10 and GSI = 45.For an undisturbed in situ rock mass surroundinga tunnel at a depth of 100 m, with a disturb-ance factor D = 0, the equivalent friction angleis φ′ = 47.16◦ while the cohesive strength isc′ = 0.58 MPa. A rock mass with the same basicparameters but in highly disturbed slope of 100 mheight, with a disturbance factor of D = 1, hasan equivalent friction angle of φ′ = 27.61◦ and acohesive strength of c′ = 0.35 MPa.

Note that these are guidelines only, and thereader would be well advised to apply the valuesgiven with caution, and to compare the calculatedresults with those obtained by back analysis asshown in Figure 4.21. It is considered that theHoek–Brown calculations can be used to providea realistic starting point for any design and, if theobserved or measured performance of the excav-ation turns out to be better than predicted, thedisturbance factors can be adjusted downwards.

These methods have all been implemented ina Windows program called “RocLab” that canbe downloaded (free) from www.rocscience.com.This program includes tables and charts for estim-ating the uniaxial compressive strength of theintact rock elements (σci), the material constantmi, the Geological Strength Index (GSI) andDisturbance Factor (D).

4.6 Rock durability and compressivestrength

The strength parameters that are usually of mostsignificance to the analysis of slope stability arethe cohesion and friction angle on the slidingsurface, as discussed previously in this chapter.However, both the durability and compress-ive strength of the rock may be of importancedepending on the geological and stress condi-tions at the site, as discussed later. The dur-ability and compressive strength test proceduresare index tests best used to classify and compareone rock with another; if necessary, the indexmeasurements can be calibrated by more preciselaboratory tests.

4.6.1 Slake durability

Widely occurring rock materials are prone todegradation when exposed to weathering pro-cesses such as wetting and drying, and freezingand thawing cycles. Rock types that are partic-ularly susceptible to degradation are shale andmudstone, which usually have a high clay content.The degradation can take the form of swelling,and the time over which weakening and disin-tegration can occur after exposure may rangefrom minutes to years. The effect of degrada-tion on slope stability can range from surficialsloughing and gradual retreat of the face to slopefailures resulting from the loss of strength withtime (Wu et al., 1981). In sedimentary form-ations comprising alternating beds of resistantsandstone and relatively degradable shale, theweathering process can develop overhangs in thesandstone and produce a rock fall hazard due tosudden failure of the sandstone (see Figure 1.4(e)).

A simple index test of the tendency of rockto weather and degrade is the slake durabilitytest (ISRM, 1981a) (Figure 4.25). It is import-ant that undisturbed samples are used that havenot been excessively broken in the sampling pro-cedure, or allowed to freeze. The test procedurecomprises placing the sample in the wire meshdrum, drying it in an oven at 105◦ for 2–6 hours,and then weighing the dry sample. The drum isthen partially submerged in water and rotated


Table 4.6 Guidelines for estimating disturbance factor D

Appearance of rock mass Description of rock mass Suggested value of D

Excellent quality controlled blasting orexcavation by Tunnel Boring Machineresults in minimal disturbance to theconfined rock mass surrounding atunnel.

D = 0

Mechanical or hand excavation in poorquality rock masses (no blasting) res-ults in minimal disturbance to thesurrounding rock mass.

Where squeezing problems result insignificant floor heave, disturbance canbe severe unless a temporary invert, asshown in the photograph, is placed.

D = 0

D = 0.5 No invert

Very poor quality blasting in a hard rocktunnel results in severe local damage,extending 2 or 3 m, in the surroundingrock mass.

D = 0.8

(Table 4.6 continued)


Table 4.6 Continued

Appearance of rock mass Description of rock mass Suggested value of D

Small-scale blasting in civil engineeringslopes results in modest rock mass dam-age, particularly if controlled blasting isused as shown on the left hand side of thephotograph. However, stress relief resultsin some disturbance.

D = 0.7 Good blastingD = 1.0 Poor blasting

Very large open pit mine slopes suffersignificant disturbance due to heavy pro-duction blasting, and also due to stressrelief from overburden removal.

D = 1.0 Productionblasting

In some softer rocks, excavation can becarried out by ripping and dozing, and thedegree of damage to the slopes is less.

D = 0.7 Mechanicalexcavation

at 20 revolutions per minute for a period of 10minutes. The drum is then dried a second timeand the loss of weight recorded. The test cycleis then repeated and the slake durability index iscalculated as the percentage ratio of final to initialdry sample masses. A low slake durability indexwill indicate that the rock is susceptible to degrad-ation when exposed. For highly degradable rocks,it is useful to carry out soil classification testssuch as Atterberg limits, as well as X-ray dif-fraction tests to identify clay mineral types anddetermine if swelling clays such as bentonites andmontmorillonites are present.

4.6.2 Compressive strength

In many slopes in moderately strong to strongrock, the stress level due to gravity loads will

be less than the strength of the rock. There-fore, there will be little tendency for intact rockwithin the slope to fracture, and consequentlycompressive strength is a less important designparameter than shear strength. The compress-ive strength of the rock on the sliding surfaceis only used indirectly in stability analysis whendetermining the roughness of a fracture (JCS termin equation (4.7)), and in the application of theHoek–Brown strength criteria (Section 4.5). Forboth these applications, it is satisfactory to use anestimate of the compressive strength because theresults are not particularly sensitive to the valueof this parameter. The compressive strength isalso used in evaluating excavation methods andcosts. For example, the progress rate for blasthole drilling and excavation by ripping, as well asselection of explosive types and design of blasts


Figure 4.25 Slake durability testing equipment.

are all influenced by the compressive strength ofthe rock.

The point load test is an appropriate method toestimate the compressive strength for rock slopedesign (Figure 4.26(a)). The equipment is port-able, and tests can be carried out quickly andinexpensively in the field on both core and lumpsamples (ISRM, 1985). Because the point load testprovides an index value for the strength, usualpractice is to calibrate the results with a limitednumber of uniaxial compressive tests on preparedcore samples.

The test procedure involves placing the samplebetween the platens and applying a load with thehydraulic jack to break the sample in tension. If P

is the point load breaking strength, then the pointload index, Is is given by

Is = P

D2e

(4.32)

where De is the equivalent core diameter, definedas

D2e = D2 (diametral tests where D is the

core diameter)

or

D2e = 4WD

π(axial, block or lump tests)

where W is the specimen width and D is thedistance between the platens. The term (WD)

is the minimum cross-sectional area of a lumpsample for the plane through the platen contactpoints.

The size-corrected point load strength indexIs(50) of a rock specimen is defined as the value ofIs that would have been measured by a diametraltest with D = 50 mm. For tests conducted onsamples with dimensions different from 50 mm,the results can be standardized to a size-correctedpoint load strength index by applying a correctionfactor kPLT as follows:

Is(50) = IskPLT (4.33)

The value of the size-correction factor kPLT isshown in Figure 4.26(b) and is given by

kPLT =(

De

50

)0.45

(4.34)


1.1

0.9

0.7

1.0

0.8

0.61.0

20 30

AQ NQ HQBQ

40 50 60 (mm)

1.5 2.0 2.5 (in.)

Siz

e co

rrec

tion

fact

or (

k PLT

)

Sample DediameterSeries Qdrill core

(a) (b)

Figure 4.26 Point load testing: (a) point load test equipment; and (b) relationship between sample equivalentcore diameter De, and size correction factor kPLT.

It has been found, on average, that the uni-axial compressive strength is about 20–25 timesthe point load strength index. However, testson many different types of rock show that theratio can vary between 15 and 50, especially foranisotropic rocks. Consequently, the most reli-able results are obtained if uniaxial calibrationtests are carried out.

Point load test results are not acceptable if thefailure plane lies partially along a pre-existingfracture in the rock, or is not coincident with theline between the platens. For tests in weak rockwhere the platens indent the rock, the test resultsshould be adjusted by measuring the amount ofindentation and correcting the distance D.

If no equipment is available to measure thecompressive strength, simple field observationscan be used to estimate the strength with suf-ficient accuracy for most purposes. Table 3.1describes a series of field index tests andobservations of rock behavior, and gives the cor-responding range of approximate compressivestrengths.

4.7 Example Problem 4.1: analysis ofdirect shear strength test results

Statement

The following table of results was obtainedfrom a direct shear box test on a planar dis-continuity in a sample of weathered granite.The average normal pressure on the sample was200 kPa.

Shear stress (kPa) Shear displacement (mm)

159 0.05200 1.19241 3.61228 4.50214 8.51207 9.40200 11.61193 12.60179 17.09179 19.81


100

200

0

0 200 400100 300 500

5

peak

Shear displacement (mm)

Normal stress (kPa)

She

ar s

tres

s (k

Pa)

She

ar s

tres

s (k

Pa)

1510 20

500(b)

(a)

300

400

200

100

residual

�= 45.5°

41°50°

Normal stress = 200 kPa(assumed constant)

�r = 179 kPa

�0 = 240 kPa

Figure 4.27 Analysis of directshear strength tests, ExampleProblem 4.1: (a) plot of shearstress against sheardisplacement; (b) plot of shearstress against normal stress.

Required

(a) Plot a graph of shear stress against sheardisplacement with shear stress on the verticalaxis; from the graph determine the peak andresidual shear strengths of the surface.

(b) Plot a graph of the peak and residual shearstrengths (on the vertical axis) against theaverage normal stress on the surface; fromthe graph determine the peak and residualfriction angles of the surface.

Solution

(a) The graph of shear stress against sheardisplacement is shown on Figure 4.27(a); the

peak strength is 241 kPa and the residualstrength is 179 kPa.

(b) The graph of shear stress against normalstress is shown in Figure 4.27(b); the peakfriction angle is 50.3◦ and the residual fric-tion angle is 41.8◦.

4.8 Example Problem 4.2: analysis ofpoint load test results

Statement

A series of point load tests on pieces of NQ core48 mm in diameter gave an average point loadbreaking strength (P) of 17.76 kN when the corewas loaded diametrically.


Required

Determine the approximate average uniaxialcompressive strength of the samples.

Solution

The point load strength index (Is) is calculatedfrom the point load breaking strength by

Is = P/D2

where D is the core diameter of 48 mm.

Is = 17.76E3/(0.048)2 = 7.71 MPa

Correction factor,

kPLT = (48/50)0.48 = 0.98

The size-corrected point load strength,

Is(50) = IskPLT = 7.71 · 0.98 = 7.56 MPa

The approximate compressive strength of therock samples,

σci ≈ 24 · 7.56 ≈ 180 MPa

Chapter 5

Ground water

5.1 Introduction

The presence of ground water in a rock slope canhave a detrimental effect upon stability for thefollowing reasons:

• Water pressure reduces the stability of theslopes by diminishing the shear strengthof potential failure surfaces as described inChapter 1. Water pressure in tension cracksor similar near vertical fissures reduces sta-bility by increasing the forces that inducesliding.

• Changes in moisture content of some rock,particularly shales, can cause acceleratedweathering and a decrease in shear strength.

• Freezing of ground water can cause wedgingin water-filled fissures due to temperature-dependent volume changes in the ice. Also,freezing of surface water on slopes can blockdrainage paths resulting in a build-up ofwater pressure in the slope with a consequentdecrease in stability.

• Erosion of weathered rock by surface water,and of low strength infillings by ground watercan result in local instability where the toe ofa slope is undermined, or a block of rock isloosened.

• Excavation costs can be increased when work-ing below the water table. For example,wet blast holes require the use of water-resistant explosives that are more expens-ive than non-water-resistant ANFO. Also,flow of ground water into the excavation orpit will require pumping and possibly treat-ment of the discharge water, and equipment

trafficability may be poor on wet haulroads.

By far the most important effect of ground waterin a rock mass is the reduction in stability result-ing from water pressures within the discontinuit-ies. Methods for including these water pressuresin stability calculations and designing drainagesystems are dealt with in later chapters of thisbook. This chapter describes the hydrologic cycle(Section 5.2), methods that are used to analyzethe flow of water through fractured rock, andthe pressures developed by this flow (Sections 5.3and 5.4). Sections 5.5 and 5.6 discuss, respect-ively methods of making hydraulic conductivityand pressure measurements in the field.

In examining rock or soil slopes, it may bea mistake to assume that ground water is notpresent if no seepage appears on the slope face.The seepage rate may be lower than the evap-oration rate, and hence the slope surface mayappear dry and yet there may be water at signi-ficant pressure within the rock mass. It is waterpressure, and not rate of flow, which is respons-ible for instability in slopes and it is essentialthat measurement or calculation of this waterpressure forms part of site investigations for sta-bility studies. Drainage, which is discussed inChapter 12, is one of the most effective andeconomical means available for improving thestability of rock slopes. Rational design of drain-age systems is only possible if the water flowpattern within the rock mass is understood, andmeasurement of hydraulic conductivity and waterpressure provides the key to this understanding.

110 Ground water

A useful means of assessing ground water con-ditions in a slope is to make observations duringperiods of below freezing temperatures. At thesetimes even minor seeps on the face may formicicles that can show both the location of thewater tables, and the set(s) of discontinuities inwhich flow is occurring.

5.2 The hydrologic cycle

A simplified hydrologic cycle illustrated inFigure 5.1 shows some typical sources of groundwater, and emphasizes that ground water cantravel considerable distances through a rockmass. It is important, therefore, to consider theregional geology of an area when starting a rockslope design program. In general, ground waterflows from recharge areas to discharge areas. Arecharge area is one in which the net saturatedflow of ground water is directed away from thewater table, while in a discharge area the net sat-urated flow is directed towards the water table. InFigure 5.1, discharge areas occur at the rock cut,tailings dam and open pit, and there is a rechargearea from the ocean to the pit.

Clearly, precipitation in the catchment areais the most important source of ground water,and Figure 5.2 illustrates the typical relation-ship between the precipitation and ground waterlevels in three climatic regions. In tropical anddesert climates the ground water table is usuallymore predictable and consistent than temperateclimates where the precipitation levels are morevariable. In assessing the relationship between cli-mate and ground water levels in the slope, boththe average precipitation and the peak eventsshould be considered because the peak events arethose that usually cause instability. Examples ofpeak precipitation events that can lead to highinfiltration rates include typhoons, intense rain-storms and rapid snowmelt. If these climaticconditions exist at the site, then it is advis-able to use correspondingly high water pressuresin design, or to design high-capacity drainagesystems.

Sources of ground water in addition toprecipitation may include recharge from adja-cent rivers, tailings dams, reservoirs or theocean as shown in Figure 5.1. There are sev-eral instances of substantial quarries and openpit mines (e.g. Dutra Minerals in California, and

GlacierSnow

Surface flow

Rock cut Rain

Evaporation

Tailings dam

Open pit

Transpiration

Evaporation

Ocean

Figure 5.1 Simplified representation of a hydrologic cycle showing some typical sources of ground water(modified from Davis and de Wiest (1966)).

Ground water 111

Almost constant rain

Tropical

Abundant recharge

Watertable

Frequent rain

Temperate

Variable recharge

Fluctuating water table

Infrequent rain

Semi-desert

Almost no recharge

Water table

Figure 5.2 Relationship between water table level and precipitation (modified from Davis and de Wiest (1966)).

Granisle Copper and Island Copper in Canada)that successfully operated below, and close to,substantial bodies of water. However, in suchoperations significant seepage may develop intothe pit, as well as instability resulting from highwater pressures.

Another important factor influencing groundwater within a slope is distribution of rock types,and details of the structural geology such as faultinfillings, persistence of joint sets and the pres-ence of solution cavities. These features can resultin regions of low and high hydraulic conduct-ivity within the slope that are termed aquitardsand aquifers, respectively. These matters are dis-cussed in more detail in Section 5.4 later in thischapter.

5.3 Hydraulic conductivity and flow nets

Where ground water effects are to be included inslope design, there are two possible approachesto obtaining data on distributions of the waterpressures within a rock mass:

(a) Deduction of the ground water flow patternfrom consideration of the hydraulic conduct-ivity of the rock mass and sources of groundwater.

(b) Direct measurement of water levels in bore-holes or wells, or of water pressure by meansof piezometers installed in boreholes.

Because of the important influence of water pres-sure on slope stability, it is essential that the bestpossible estimates of the likely range of pressuresshould be available before a detailed stability ana-lysis is attempted. There are a large number offactors that control ground water flow in jointedrock masses, and it is only possible in this bookto highlight the general principles that may apply.If detailed studies of ground water conditions arerequired, it is advisable to obtain additional datafrom such sources as Freeze and Cherry (1979)and Cedergren (1989) on ground water flow ana-lysis, and Dunnicliff (1993) on instrumentation.

5.3.1 Hydraulic conductivity

The basic parameter defining the flow of groundwater, and the distribution of water pressure,in geologic media is hydraulic conductivity. Thisparameter relates the flow rate of water throughthe material to the pressure gradient appliedacross it (Scheidegger, 1960; Morgenstern,1971).

Consider a cylindrical sample of soil or rockbeneath the water table in a slope as illustratedin Figure 5.3. The sample has a cross-sectionalarea of A and length l. Water levels in boreholesat either end of this sample are at heights h1 andh2 above a reference datum and the quantity ofwater flowing through the sample in a unit oftime is Q. According to Darcy’s law, the coef-ficient of hydraulic conductivity K of this sample

112 Ground water

is defined as

K = Ql

A(h1 − h2)= Vl

(h1 − h2)(5.1)

where V is the discharge velocity. Substitution ofdimensions for the terms in equation (5.1) showsthat the hydraulic conductivity K has the samedimensions as the discharge velocity V , that islength per unit time. The units most commonlyused in ground water studies is centimeters persecond, and a number of hydraulic conductivityconversion factors are given in Table 5.1.

Equation (5.1) can be rearranged to showthe volume of water, Q flowing through thesample shown in Figure 5.3 under a specified

Table 5.1 Hydraulicconductivity conversion table

To convert cm/s to Multiply by

m/s 1.00 × 10−2

ft/s 3.28 × 10−2

US gal/day/ft2 2.12 × 104

ft/year 1.03 × 106

m/year 3.14 × 105

head, as follows:

Q = KA(h1 − h2)

l(5.2)

In most rock types flow through intact rockis negligible (defined by Kprimary), and essen-tially all flow occurs along the discontinuities(defined by Ksecondary). For example, the primaryhydraulic conductivity for intact granite andbasalt is about 10−10 cm/sec, while for somecoarse grained, poorly indurated sandstones theprimary hydraulic conductivity may be as highas 10−4 cm/sec. The term secondary hydraulicconductivity refers to flow in the rock mass andencompasses flow in both the intact rock andany discontinuities that are present. These condi-tions result in secondary hydraulic conductivitieshaving a wide range of values depending on thepersistence, width and infilling characteristics ofthe discontinuities. For example, granite that hasa very low primary hydraulic conductivity usuallycontains tight, clean, low persistence joints, so thesecondary hydraulic conductivity is also low. Incontrast, sandstone may have some primary con-ductivity, and the presence of persistent beddingplanes may result in high secondary conductiv-ity in the direction parallel to the bedding. Forfurther discussion on flow in fractured rock seeSection 5.4.

Borehole

Borehole

h1

l

h2

Area A

Water table

Referencedatum

z1

z2

�w

P1

�w

P2

Figure 5.3 Illustration of Darcy’sLaw for definition of hydraulicconductivity.

Ground water 113

Karstic limestone

Fractured/jointed basaltFractured igneous and metamorphic rocks

Carbonate rocks (limestone and dolomite)Sandstone

Unfractured igneous and metamorphic rocks

ShaleMarine clay

Glacial tillSiltSilty sand

Clean sandGravel

Rocks

Unconsolidateddeposits

10–1110–1010–9 10–810–7 10–6 10–5 10–4 10–3 10–2 10–1 1 10 102

10–710–8 10–6 10–5 10–4 10–3 10–2 10–1 1 10 102 103 104 105

K (cm/s)

K (m/d)

Figure 5.4 Hydraulic conductivity of various geologic materials (Atkinson, 2000).

Typical ranges of secondary hydraulic con-ductivity for a variety rock types, as well asunconsolidated deposits, are shown in Figure 5.4.The range of hydraulic conductivities for geolo-gical materials covers 13 orders of magnitude,and for any single rock type the range can be fourorders of magnitude. This shows the difficulty inpredicting water inflow quantities and pressureswithin slopes.

Figure 5.3 also shows that the total head h atany point can be expressed in terms of the pres-sure P and the height z above a reference datum.The relationship between these parameters is

h = P

γw+ z (5.3)

where γw is the density of water. The total headh represents the level to which water will rise in aborehole standpipe.

Darcy’s law is applicable to porous media andso can be used to study ground water flow in bothintact rock, and rock masses on a macroscopicscale. However, it is required that the flow be lam-inar, so Darcy’s law is not applicable in the eventof non-linear or turbulent flow in an individualfracture.

5.3.2 Porosity

The total volume VT of a rock or soil is made upof the volume of solid portion Vs and the volumeof the voids Vv. The porosity, n, of a geologicmaterial is defined as the ratio:

n = Vv

VT(5.4)

In general, rocks have lower porosities thansoil. For example, the porosities of sand and clayare in the range of 25–50% respectively. In com-parison, fractured basalt and karstic limestonemay have porosities in the range of 5–50%, whilethe porosity of dense crystalline rock is usually inthe range of 0–5%.

The significance of the porosity to rock slopesis in the design of drainage systems. For example,drains installed in a low porosity granite needonly discharge a small amount of water to lowerthe water pressures, while drains in karstic lime-stone may discharge large volumes of water withlittle effect on the ground water table.

5.3.3 Flow nets

The graphical representation of ground waterflow in a rock or soil mass is known as a flow net

114 Ground water

Standpipes

Water table

Flow lines

Equipotentiallines

Pressure increasewith depth along anequipotential line

Referencedatum

h A

B

ZA

ZB

PA

PB�w

�w

Figure 5.5 Two-dimensional flow net in a slope.

and a typical example is illustrated in Figure 5.5.A flow net comprises two sets of intersecting linesas follows:

(i) Flow lines are paths followed by the water inflowing through the saturated rock or soil.

(ii) Equipotential lines are lines joining points atwhich the total head h is the same. As shownin Figure 5.5, the water level is the same instandpipes that terminate at points A and B

on the same equipotential line. Water pres-sures at points A and B are not the samesince, according to equation (5.3), the totalhead h is given by the sum of the pressurehead P /γw, and the elevation z of the meas-uring point above the reference datum. Thewater pressure increases with depth along anequipotential line.

There are characteristics of flow nets that areapplicable under all conditions and must beused in drawing flow nets. First, equipotentiallines must meet impermeable boundaries at rightangles and be parallel to constant head boundar-ies. Second, there is a uniform head loss betweenadjacent equipotential lines. Third, equipoten-tial and flow lines intersect at right angles toform curve-linear squares in rock with isotropichydraulic conductivity. For a flow net such as that

shown in Figure 5.5, the equipotential lines illus-trate how the ground water pressure varies withinthe slope, and that flow quantity is equal betweenthe adjacent flow lines.

An example of the application of flow nets tostudy the distribution of pressures in rock slopesis illustrated in Figure 5.6. If the pit is located ina recharge area, flow occurs towards the pit, andartesian pressures can be developed below the pitfloor (a, b). In contrast, for pits in discharge areas,flow is away from the pit and there will be lowpressures below the pit floor (c, d).

A complete discussion on the construction orcomputation of flow nets exceeds the scope of thisbook and the interested reader is referred to thecomprehensive texts by Cedergren (1989), Haar(1962) and Freeze and Cherry (1979) for furtherdetails. The use of graphical methods for con-structing flow nets is often an important step inunderstanding how geology and drainage systemsinfluence possible ground water conditions withina slope.

5.4 Ground water flow in fractured rock

As discussed in Section 5.2 on hydraulic conduct-ivity, ground water flow in fractured rock massesoccurs predominately along the discontinuities

Ground water 115

a

b

Equipotential line

c

d

Equipotential line

Flow line

(b)

(a)

Flow line

Figure 5.6 Ground water conditions for pit slopes inregional (a) discharge and (b) recharge areas (Pattonand Deere, 1971).

because of the very low primary hydraulic con-ductivity of most intact rock. Therefore, theconductivity of rock masses will be influenced bythe characteristics of the discontinuities, with anecessary condition required for flow being thatthe persistence of the discontinuities is greaterthan the spacing. Figure 5.7 shows a rock masscontaining two vertical joint sets and one hori-zontal set in which the persistence of the verticaljoints is much greater than the spacing, but thepersistence of the horizontal set is less than thespacing. For these conditions, the hydraulic con-ductivity would be significantly greater in thevertical direction than that in the horizontaldirection.

The analysis of flow in fractured rock can becarried out either assuming that the rock is acontinuum, as has been assumed in the deriva-tion of Darcy’s equation and drawing flow nets,or that the rock is a non-continuum in whichlaminar flow occurs in individual discontinuities.Rock can be assumed a continuum if the dis-continuities spacing is sufficiently close that thefractured rock acts hydraulically as a granular

Figure 5.7 Rock mass with persistent vertical jointsand relatively high vertical hydraulic conductivity(modified from Atkinson (2000)).

porous media so that the flow occurs through anumber of discontinuities.

5.4.1 Flow in clean, smooth discontinuities

The flow of water through fissures in rock hasbeen studied in detail by Huitt (1956), Snow(1968), Louis (1969), Sharp (1970), Maini(1971) andothers. Subsequent to this, extensiveresearch has been carried out on this topic in rela-tion to the design of underground nuclear wastestorage facilities; this work has provided muchadditional information on fluid flow in fracturedrock. However, for the purpose of this discussion,the problem is simplified to that of the determin-ation of the equivalent hydraulic conductivity ofan array of parallel, smooth, clean discontinu-ities (Davis, 1969). The hydraulic conductivityparallel to this array is given by

K ≈ ge3

12υb(5.5)

where g is the gravitational acceleration(9.81 m/s2), e and b are respectively thediscontinuity aperture and spacing, and υ is

116 Ground water

10–7

10–8

10–6

10–5

10–4

10–3

10–2

10–1

1.0

10

0.01 0.05 0.1 0.5 1.0

Joint aperture, e (mm)

Hyd

raul

ic c

ondu

ctiv

ity, K

(cm

/s)

Spacing 0.01m

Spacing 0.1m

Spacing 1.0m

e

b

Figure 5.8 Influence of joint aperture eand spacing b on hydraulic conductivityK in the direction of a set of smoothparallel joints in a rock mass.

the coefficient of kinematic viscosity (1.01 ×10−6 m2/s for pure water at 20◦C).

The equivalent conductivity of a parallel arrayof discontinuities in relation to aperture and spa-cing is shown in Figure 5.8. Since the hydraulicconductivity is proportional to the third powerof the aperture, small changes in the aperturedue, for example, to increasing stress in the rockwill significantly decrease the conductivity. Thiscondition could develop at the toe of a steepslope where high stresses decrease the apertureand result in a build up of water pressure in theslope.

Figures 5.4 and 5.8 demonstrate the applica-tion of equation (5.5) to the actual hydraulicconductivities of rock masses. For example, theconductivity of sandstone is about 10−6 cm/s,while that of fractured and jointed basalt is about10−2 cm/s. This difference in conductivity of fourorders of magnitude can be attributed to the jointspacing decreasing from 1 to 0.1 m, and the aper-ture increasing by a small amount from 0.02to 0.2 mm.

The relationship between discontinuity aper-ture and hydraulic conductivity was studied for

the construction of the ship locks at the ThreeGorges Project in China, which involved mak-ing parallel excavations with depths up to 170 min strong, jointed granite (Zhang et al., 1999).The excavations caused relaxation of the rockin the walls of the locks and the opening of thejoints, which resulted in the hydraulic conductiv-ity increasing by a factor of 18. The applicationof a support pressure of 2 MPa on the verticalwalls of the excavation resulted in the hydraulicconductivity only increasing by a factor of 6 fromthe in situ condition.

5.4.2 Flow in filled discontinuities

Equation (5.5) applies only to laminar flow inplanar, smooth, parallel discontinuities and rep-resents the highest equivalent hydraulic conduct-ivity for fracture systems. The lowest equivalenthydraulic conductivity occurs for infilled discon-tinuities, and is given by

K = eKf

b+ Kr (5.6)

Ground water 117

where Kf is the hydraulic conductivity of thefilling and Kr is that of the intact rock. The termKr is included in equation (5.6) to account for thecondition where there is flow in both the intactrock and along the discontinuities.

While equations (5.5) and (5.6) illustrate theprinciples of water flow along discontinuityplanes, this simple model cannot be used to cal-culate hydraulic conductivity of actual fracturedrock masses. Methods of modeling ground waterflow in rock have been developed using probabil-istic techniques to simulate, in three dimensions,the likely ranges of discontinuity characteristicsthat may occur. One such modeling technique istermed FRACMAN (Dershowitz et al., 1994; Weiet al., 1995).

5.4.3 Heterogeneous rock

Figure 5.9 shows a shallow dipping sequence ofsandstone and shale beds. The shale, which is afine-grained rock with few persistent discontinuit-ies, has low hydraulic conductivity and is termedan aquitard. In contrast, the sandstone, whichis coarse grained, has a relatively high hydraulic

conductivity and is termed an aquifer. Because ofthe significant difference between the hydraulicproperties of the shale and sandstone, this is aheterogeneous rock mass. Flow nets in heterogen-eous rock are modified from the simple net shownin Figure 5.5 because flowlines preferentially usethe high conductivity formations as conduits andtraverse the low conductivity formations by theshortest possible route. The equipotentials tendto loose a greater proportion of the head in thelow conductivity formation than in the higherconductivity formation. This behavior results inrefraction of flow lines at the formation bound-ary, depending on the relative conductivities,according to the following relationship:

Ksandstone

Kshale= tan θ1

tan θ2(5.7)

The refraction angles θ1 and θ2 are definedon Figure 5.9 for a conductivity ratio ofK1/K2 = 10.

Features of the flow conditions shown inFigure 5.9 are as follows. First, flow in theupper, unconfined aquifer tends to flow down-dip

Spring

Piezometer

Water table

Flowline refraction: =K2

K1

tan�2

tan�1

Unconfined aquifer (K1)

Shale

Shale

Sandstone

Confined aquifer (K1)

Aquitard (K2)

Aquitard (K2)

�1

�2

Figure 5.9 Water flow and pressure distribution in aquifers and aquitards formed by dipping sandstone andshale beds (Dr P. Ward, plots by W. Zawadzki).

118 Ground water

in the sandstone and exits the slope at thesandstone/shale contact. This seepage line on thevalley wall would be an indication of the locationof the contact. Second, flow in the lower, confinedaquifer is recharged from a source up-dip from thevalley that develops artesian pressure in the sand-stone. This condition could be demonstrated bycompleting a piezometer in the lower sandstone,in which the water would rise above the groundsurface to the level of the equipotential in whichit is sealed. Third, flow in the confined aquifer isrefracted at the boundary and flows upwards inthe shale to exit in the valley floor.

5.4.4 Anisotropic rock

In formations such as that shown in Figure 5.7, inwhich the conductivity of one set or sets of discon-tinuities is higher than another set, the rock masswill exhibit anisotropic hydraulic conductivity.For the rock shown in Figure 5.7, the verticalhydraulic conductivity will be considerably morethan that in the horizontal direction. On theflow net, anisotropic hydraulic conductivity isdepicted by squares formed by the flow lines andequipotentials being elongated in the direction ofthe higher hydraulic conductivity. In general, theaspect ratio of the flow line/equipotential squaresis equal to (K1/K2)1/2.

Examples of flow nets in isotropic and aniso-tropic rock are shown in Figure 5.10. The sig-nificance of these conditions to slope stability isas follows. First, in rock with high hydraulicconductivity in the horizontal direction, such ashorizontally bedded sandstone, the ground watercan readily drain from the slope (Figure 5.10(b)).For these conditions there will be relatively lowwater pressures on potential sliding surfaces com-pared to the isotropic case. Second, in rock withhigh hydraulic conductivity parallel to the facesuch as a slope cut parallel to bedding, flow tothe face will be inhibited and high water pressureswill develop in the slope. For the slope shownin Figure 5.10(c), the use of horizontal drainsthat connect the high conductivity bedding planesto the face would be effective in lowering waterpressures within the slope.

Isotropic(a)

(b)

(c)

11

110

1

10

Figure 5.10 Flow nets in slopes with isotropic andanisotropic hydraulic conductivity: (a) isotropic rock;(b) Khorizontal = 10 × Kvertical;(c) Kparallel to slope = 10 × Kperpendicular to slope(plots by W. Zawadzki).

5.4.5 Ground water in rock slopes

The discussion on ground water flow in rockmasses shows that details of the geology can havea significant effect on water pressures and seep-age quantities in rock slopes. In addition to theconditions shown in Figures 5.9 and 5.10 thatrelate to heterogeneous and anisotropic rock, avariety of other possible ground water conditions

Ground water 119

(b)

(c)

(a)

Porousrock

Jointedrock

Fault (low conductivity)

Fault (high conductivity)

Figure 5.11 Relationship between geology and ground water in slopes: (a) variation in water pressure in jointsrelated to persistence; (b) comparison of water tables in slopes excavated in porous and jointed rock; (c) faultsas low conductivity ground water barrier, and high conductivity sub-surface drain (Patton and Deere, 1971).

are related to geology as follows:

(a) Low persistence joints that are not connectedto the slope face may develop high tran-sient water pressures, compared to jointswith greater persistence that are connectedto the face and allow water to drain at theface (Figure 5.11(a)). It should be noted thatblast damage is one of the causes of per-sistent joints and fractures close to the face.

However, any improvement in slope stabilitydue to the increase in conductivity is prob-ably out-weighed by the decrease in stabilityresulting from blast damage to the rock.

(b) The porosity of the rock mass will affectthe level of the transient water table inresponse to the same precipitation event(Figure 5.11(b)). In a porous rock mass theinfiltrating water will be contained withinthe rock with the result that there will be

120 Ground water

little increase in the water table. In con-trast, a strong rock with widely spaced jointswill have low porosity so ground water flowwill rapidly fill the joints and increase thewater pressure within the slope. It is com-monly found that rock falls on steep rockfaces occur soon after heavy rainfalls, par-ticularly if the water freezes and expandsbehind the face.

(c) Faults comprising clay and weathered rockmay have low conductivity and act as groundwater barriers behind which high waterpressures could develop. In contrast, faultscomprising crushed and broken rock mayhave high conductivity and act as a drain(Figure 5.11(c)). Measurement of water pres-sures on either side of the fault will indicatethe hydraulic properties of these features.

5.5 Measurement of water pressure

The importance of water pressure to the stabi-lity of slopes has been emphasized in previouschapters. If a reliable estimate of stability is tobe obtained or if the stability of a slope is to becontrolled by drainage, it is essential that waterpressures within the slope be measured. Suchmeasurements are most conveniently carried outby piezometers. Piezometers are devices sealedwithin the ground, generally in boreholes, whichrespond only to ground water pressure in theimmediate vicinity, and not to ground water pres-sures at other locations. Piezometers can also beused to measure the in situ hydraulic conduct-ivity of rock masses using variable head tests asdescribed in Section 5.6.

The following are a number of factors thatmay be considered when planning a piezometerinstallation to measure water pressures in a rockslope:

(a) The drill hole should be oriented such thatit will intersect the discontinuities in whichthe ground water is likely to be flowing. Forexample, in a sedimentary rock containingpersistent beds but low persistence joints, thehole should intersect the beds.

(b) The completion zone of the piezometershould be positioned where the rock masscontains discontinuities. For example, ifdrill core is available, it should be stud-ied to locate zones of fractured or shearedrock where ground water flow is likely tobe concentrated. Positioning the completionzone in massive rock with few discontinuitiesmay provide limited information on groundwater pressures. The length of the com-pletion zone in rock is usually longer thanthat in soil because of the need to intersectdiscontinuities.

(c) Other geological features that may be con-sidered in piezometer installations are faultzones. These may act as conduits for groundwater if they contain crushed rock, or theymay be barriers to ground water flow ifthey contain clay gouge. In the case ofhigh hydraulic conductivity faults, the com-pletion zone may be located in the fault,and for low hydraulic conductivity faults,the completion zones may be located eitherside of the fault to determine any pressuredifferential.

(d) The number of piezometers, or the numberof completion zones in a single piezometer,may be determined by the geology. Forexample, in a sedimentary deposit contain-ing low hydraulic conductivity shale andrelatively high hydraulic conductivity sand-stone, it may be necessary to install comple-tion zones in each rock unit.

(e) The hydrodynamic time lag is the volume ofwater required to register a head fluctuationin a piezometer standpipe. The time lag isdependent primarily on the type and dimen-sions of the piezometer and can be significantin rock with low hydraulic conductivity.Standpipe piezometers have a greater hydro-dynamic time lag than diaphragm piezo-meters because a greater movement of poreor joint water is required to register. Theterm slow response time is used to describea long hydrodynamic time lag.

(f) In rock slopes where the piezometer is beingused to measure joint water pressure in

Ground water 121

which pressure fluctuations are not likelyto be significant, a standpipe piezometer islikely to be suitable. However, if the pur-pose of the piezometer is to measure theresponse of the ground water pressures toa drainage system such as a series of hori-zontal drains, or to detect transient waterpressures in response to precipitation, thena diaphragm piezometer with a much shortertime lag would be more appropriate.

(g) The filter material in the completion zoneshould be suited to the rock type. Installa-tions in clay shales or weathered micaceousrocks should use fine grained filter materialthat will not be clogged by rock weather-ing products washed in from the walls ofthe hole.

(h) Cost and reliability are other factors to con-sider in selecting piezometer types. Stand-pipe piezometers are simple to install andcan be read with inexpensive well sounders,while pneumatic and vibrating wire piezo-meters are expensive and require more costlyreadout units. In situations where the slopeis moving and piezometers may be lost,it would be preferable to install standpipepiezometers for reasons of economy.

The following is a brief description of piezometertypes and the conditions in which they may beused (Dunnicliff, 1993).

• Observation wells—Ground water pressuresmay be monitored in open holes if the per-meability of the rock mass is greater thanabout 10−4 cm/s, such as coarse-grained sand-stones and highly fractured rock. The majorlimitation of observation wells is that theycreate a vertical connection between strataso their only application is in consistentlypermeable rock in which the ground waterpressure increases continuously with depth.Observation wells are rarely utilized in mon-itoring ground water pressures in rock.

• Standpipe piezometers—A standpipe piezo-meter consists of a length of plastic pipe, witha perforated or porous section at the lower

end, which is encased in clean gravel or sandto provide a good hydraulic connection withthe rock (Figure 5.12). This perforated sec-tion of the piezometer, which is the point

50 mm PVC casing

50 mm PVC well screenwith 0.25 mm slot open

Filter pack (clean medium to coarse silica sand)

Silica fine-grained sand (mortar sand)

Granular bentonite seal

2% Bentonite-cement seal

Granular bentonite seal

Concrete pad (min. 0.10 m thick on undisturbed or compacted soil)

Concrete sealFrost sleeve

Drain

Master lockVented cap

Steel (schedule 40)protective casing withhinged cap

0.9 m

Sand

50 mm PVC casing (schedule 40, flush joint,threaded)

1.5 m

1.4 m1.2 m

1.1 m

0.8 m0.6 m

0.3 m

0.6 m

0.3 m

1.5 m

0.6 m

Figure 5.12 Typical standpipe piezometerinstallation.

122 Ground water

where the water pressure is measured, is isol-ated from the rest of the hole with a bentoniteseal and a filter layer to prevent contamina-tion of the clean sand around the perforatedsection. The bentonite is usually placed inthe form of compacted pellets that will fall aconsiderable depth down a water-filled holebefore they expand. In very deep holes, theballs can be first soaked in oil to form aprotective layer that delays their expansion.However, cement is preferred as a seal forholes with depths greater than about 300 m.

The water level in a standpipe piezometercan be measured with a well sounder con-sisting of a graduated electrical cable, withtwo bared ends, connected to an electrical cir-cuit consisting of a battery and an ammeter.When the bared ends contact water, the cir-cuit is closed and a current is registered onthe ammeter. The advantages of this type ofpiezometer are that it is simple and reliable,but the disadvantages are that there must beaccess to the top of the hole, and there can besignificant time lag in low conductivity rock.

• Pneumatic piezometers—A rapid responsetime can be achieved using pneumatic piezo-meters that comprise a valve assembly anda pair of air lines that connect the valve tothe surface. The valve is placed in the sealedsection of the piezometer to measure the waterpressure at that point. The operating prin-ciple is to pump air down the supply line untilthe air pressure equals the water pressure act-ing on a diaphragm in the sealed section andthe valve opens to start air flowing in thereturn tube. The pressure required to openthe valve is recorded on a pressure gauge at thesurface.

Pneumatic piezometers are suitable for lowconductivity rock installations, and are par-ticularly useful where there is no access to thecollar of the hole since readings can be madeat a remote location. The disadvantages of thistype of piezometer are the risk of damage tothe lines either during construction or oper-ation, and the need to maintain a calibratedreadout unit.

• Electronic transducers—Water pressure meas-urements with electrical transducers allowvery rapid response time and the opportun-ity to record and process the results at aconsiderable distance from the slope. Com-mon types of electrical transducers includestrain gauges and vibrating wire gauges thatmeasure pressure with a high degree of accur-acy. It is recommended that all transducersbe thoroughly tested and calibrated beforeinstallation. It should also be kept in mind thatthe long-term reliability of these sensitive elec-trical instruments may not equal the design lifeof the slope and provision should be made fortheir maintenance and possible replacement.

• Multi-completion piezometers—For slopesexcavated in rock types with differinghydraulic conductivities, it is possible thatzones of high ground water pressure existwithin a generally depressurized area. In suchcircumstances, it may be desirable to meas-ure the ground water pressure at a number ofpoints in a drill hole. This can be achieved byinstalling multiple standpipe piezometers in asingle drill hole with bentonite or cement sealsbetween each section of perforated pipe. Themaximum number of such standpipes that canbe installed in an NX borehole is three; withmore pipes, placement of filter and effectiveseals becomes very difficult.

An alternative method of measuring waterpressures at a number of different points in adrill hole is to use a Multi-port (MP) systemwhich also allows measurement of hydraulicconductivity and retrieval of water samples(Black et al., 1986) (Figure 5.13). The MPsystem is a modular multiple-level monitoringdevice employing a single, closed access tubewith valved ports. The valved ports provideaccess to several different levels of a drillhole in a single well casing, and the modu-lar design permits as many monitoring zonesas desired to be established in a drill hole.The system consists of casing componentsthat are permanently installed in the drill hole,and pressure transducers, sampling probesand specialized tools that are lowered down

Ground water 123

Casing

(a)

Pressure probe

Transducer

Location arm

Valve

MP measurementport coupling

Backing shoe

(b)

Figure 5.13 Multiplecompletion piezometerinstallation (MP system,Westbay Instruments) withprobe positioned to makepressure measurements(Black et al., 1986). (a) Probelocated at measurement portcoupling; (b) probemeasuring fluid pressureoutside coupling.

the hole. The casing components includecasing sections of various lengths, two typesof valved port couplings with capabilities toeither measure pressure or take samples. Theport assemblies can be isolated in the drill holeby sealing the annulus between the monitoringzones using either pairs of packers, or by fillingthe annulus with a cement grout or bentoniteseal. The MP system has been used in drillholes up to 1200 m deep.

5.6 Field measurement of hydraulicconductivity

Determination of the hydraulic conductivity of arock mass is necessary if estimates are requiredof ground water discharge from a slope, or in thedesign of a drainage system.

For evaluation of the stability of the slopes,it is the water pressure rather than the volumeof ground water flow in the rock mass that is

important. The water pressure at any point isindependent of the hydraulic conductivity of therock mass at that point, but it does depend uponthe path followed by the ground water in arriv-ing at the point (Figures 5.9 and 5.10). Hence,the heterogeneity and anisotropy of the rock masswith respect to the distribution of hydraulic con-ductivity is of interest in estimating the waterpressure distribution in a slope.

Because ground water flow in fractured rocktakes place predominately in the discontinuities,it is necessary that hydraulic conductivity meas-urements be made in situ; it is not possible tosimulate water flow in a fractured rock mass in thelaboratory. The following is a brief descriptionof the two most common methods of in situ con-ductivity testing, namely variable head tests andpumping tests. Detailed procedures for hydraulicconductivity tests are described in the literature,and the tests themselves are usually conducted byspecialists in the field of hydrogeology.

124 Ground water

5.6.1 Variable head tests

In order to measure the hydraulic conductivity ata “point” in a rock mass, it is necessary to changethe ground water conditions at that point and tomeasure the time taken for the original conditionsto be re-established. These tests are most conveni-ently carried out in a borehole, and the test lengthmay represent the general rock mass propertiesin the slope, or the test may be located in a spe-cific geologic feature such as a fault. An essentialrequirement of the test is that the borehole wall isclean and there is no clogging of the discontinu-ities by drill cuttings or drilling mud. This willrequire flushing of the hole and the use of polymermuds that break down some time after drilling toleave a clean hole (see Section 3.6.2).

Test configurations. A number of boreholetest configurations are possible. A piezometerinstalled in a drill hole will isolate a sectionat the end of the hole (Figure 5.14(a)), or atsome point up the hole. It is also possible toconduct hydraulic conductivity measurements inopen holes (Figure 5.14(b)), although it is neces-sary that the geologic conditions are consistentover the test length.

Test procedure. The procedure for the vari-able head test is first to establish the rest waterlevel, which is the static equilibrium level ofthe water table at the drill hole location (Fig-ure 5.14(a) and (b)). Pumping of circulationwater during drilling will disturb this equilibriumand the permeability results will be in error ifinsufficient time is allowed for equilibrium to bere-established. Once equilibrium has been estab-lished, water can either be removed from (bailtest) or added to (slug test) the standpipe (piezo-meter test) or hole to change the water level. If thetest is conducted above the water table it is neces-sary to perform a slug test, while for a test belowthe water table, a bail test is preferred becausethe flow of water out of the formation minimizesclogging of the fractures.

Water is added or removed from the holeto change the water level by about 1–2 m andthe rate at which the water level recovers to

the equilibrium level is measured. For the testshown in Figure 5.14(a), the hydraulic conduct-ivity K is calculated from the following generalrelationship:

K = A

F(t2 − t1)ln(

h1

h2

)for

L

R> 8 (5.8)

where F is the shape factor. For a drill hole withradius R and a test zone of length L, the shapefactor is given by

F = 2πLln(L/R)

(5.9)

In equation (5.8), A is the cross-section areaof the standpipe (A = πr2) and r is the internalradius of the standpipe. For a bail test, t1 andt2 are the times at which the water levels are atdepths h1 and h2 respectively below the equilib-rium water level. The differential heads h1 andh2, as well as the initial equilibrium head h0, aredefined in Figure 5.14(a), while a typical semilogplot of the rise in water level in the casing withtime is shown in Figure 5.14(c).

The types of test shown in Figure 5.14 aresuitable for measurement of the hydraulic con-ductivity of reasonably uniform rock. Aniso-tropic hydraulic conductivity coefficients cannotbe measured directly in these tests but allowancecan be made for this anisotropy in the calculationsas follows. If an estimate of the ratio of verticaland horizontal hydraulic conductivities Kv andKh respectively is made, the ratio m is given by

m =√

Kh

Kv(5.10)

and the shape factor F given by equation (5.9) ismodified as follows:

F = 2πL

ln(L(m/R))(5.11)

When this value of F is substituted inequation (5.8), then the calculated value of the

Ground water 125

1.0

0.5

0.2

0 2 4 6 8 10 12

Elapsed time, t

ht

h0

Equilibrium water level, t = infinite

h1

h2

t1 t2

2r

h1

L

2R

h2

h0

t = 0 D

h2

h1 h0

2R

t = 0

(a)

(c)

(b)

Figure 5.14 Method of calculating hydraulic conductivity for variable head test: (a) piezometer withcompletion length L; (b) open hole to depth D below water table; (c) typical plot of head increase (recovery)against time.

hydraulic conductivity, K, is the mean hydraulicconductivity given by

√(Kv × Kh).

Packer tests. Where conductivity tests arerequired at specific locations within a drill hole, atest zone can be isolated using packers positionedat any location in the hole, and over any required

length (Figure 5.15). Packer tests can be madeduring diamond drilling using a triple packer sys-tem that is lowered through the rods so that thetest is conducted in a portion of the hole belowthe drill bit. The packer system consists of threeinflatable rubber packers, each 1 m long that is

126 Ground water

Packer

Packer

Packer

Inlet

Bit

Spacer pipe

Open hole

Bit stop

Perforated pipe

Inflation line

Inflation line

Water level indrill rods

Drill rods

Wireline

Testinterval

Figure 5.15 Triple packer arrangement for makingvariable head hydraulic conductivity tests inconjunction with diamond drilling (Wyllie, 1999).

sufficient to minimize the risk of leakage past thepacker. The lower two packers are joined by aperforated steel pipe, the length of which dependson the required test length, while the top andmiddle packers are joined by a solid pipe. Thewhole packer assembly is lowered down the drillhole on the wire line through the drill rods and thelower two packers extend through the bit into theopen hole, while the upper packer is located inthe lower end of the core barrel. The three pack-ers are then inflated with nitrogen through a smalldiameter plastic tube that runs down the hole.The inflated packers seal the packer assembly intothe rods, and isolate a length of drill hole belowthe bit. If water is removed from the drill rods,formation water will flow from the rock, overthe test interval isolated by the two lower pack-ers, and through the perforated pipe to restorethe water level in the rods. This flow of wateris measured by monitoring the change of waterlevel in the drill rods with time. A plot of the res-ults is head recovery plot such as that shown inFigure 5.14(c). Cedergren (1989) provides a com-prehensive discussion of variable head hydraulicconductivity testing.

5.6.2 Pumping test

The main limitations of conductivity tests carriedout in drill holes are that only a small volume ofrock in the vicinity of the hole is tested, and itis not possible to determine the anisotropy of therock mass. Both these limitations are overcome byconducting pump tests, as described briefly in thenext paragraph.

A pump test arrangement consists of a ver-tical well equipped with a pump, and an arrayof piezometers in which the water table elevationcan be measured in the rock mass surrounding thewell. The piezometers can be arranged so that theinfluence of various geologic features on groundwater conditions can be determined. For instance,piezometers could be installed on either side ofa fault, or in directions parallel and perpendi-cular to sets of persistent discontinuities such asbedding planes. Selection of the best location forboth the pumped well and the observation wells

Ground water 127

requires considerable experience and judgment,and should only be carried out after thoroughgeological investigations have been carried out.

The test procedure consists of pumping waterat a steady rate from the well and measuring thedrop in water level in both the pumped well andthe observation wells. The duration of the test canrange from as short as eight hours to as long asseveral weeks, depending on the permeability ofthe rock mass. When the pumping is stopped, thewater levels in all the wells are measured until astatic water level is determined—this is known asthe recovery stage of the test. Plots of draw down(or recovery) against time can be used to calcu-late permeability values using methods describedby Cedergren (1989), Todd (1959), Jacob (1950)and Theis (1935).

Because of the cost and time required for con-ducting a pump test, they are rarely carried outfor rock slope engineering. An example of a situ-ation where a pump test may be justified would beto assess the feasibility of driving a drainage aditfor stabilization of a landslide. Generally, install-ation of piezometers to measure the ground watertable and conduct variable head tests to measurehydraulic conductivity in boreholes provide suffi-cient information on ground water conditions forslope design purposes.

5.7 Example Problem 5.1: Influence ofgeology and weather conditions onground water levels

Statement

Figure 5.16 shows a slope, cut in isotropic, frac-tured rock, under a variety of operational andclimatic conditions; in all cases ground water isinfiltrating the horizontal ground surface behindthe crest of the slope (Terzaghi, 1962). InFigure 5.16(a) the slope has been recently excav-ated, and prior to excavation, the water tablewas horizontal and at a shallow depth belowthe ground surface. In Figure 5.16(b) and (c) theslope has been open for a sufficiently long timefor ground water equilibrium conditions to beestablished, but the climatic conditions vary. The

Infiltration

Original ground surface

Original phreatic line

Infiltration

Lower K

Higher K

Arbitrary boundary

Infiltration

(c)

(b)

(a)

Figure 5.16 Cut slope in rock with water infiltrationon ground surface behind crest for Example 5.1:(a) position of the ground water surface beforeexcavation; (b) slope with variety of surfaceinfiltration and rock conductivity conditions; (c) slopewith higher conductivity rock close to face, andvarious climatic conditions.

objective of the exercise is to sketch in the groundwater table on each slope based on the generalbehavior of ground water in slopes as governedby Darcy’s Law.

Required

A On the cross-section in Figure 5.16(a), drawthe approximate position of the ground watertable after excavation of the slope.

B On the cross-section in Figure 5.16(b), drawthe approximate positions of the ground

128 Ground water

Infiltration

Original ground surface

Original phreatic line

Drawn-down phreatic line

Infiltration

Phreatic line for largeinflow low k value

Phreatic line for smallinflow high k value

Infiltration

Higher K

Lower K

Arbitrary boundary

Wet seasonDry season

Joints plugged with ice

Heavy rainstorm

(c)

(b)

(a)

Figure 5.17 Positions of ground water surface forconditions shown in Figure 5.16 for Example 5.1:(a) position of the ground water surface before andafter excavation; (b) relative positions of groundwater table for variations of inflow and conductivity;(c) hypothetical positions of the ground water surfacein jointed rock for variety of climatic conditions.

water table for the following conditions:

• large infiltration, low hydraulic conduct-ivity; and

• small infiltration, high hydraulicconductivity.

C On the cross-section in Figure 5.16(c) drawthe approximate positions of the groundwater table for the following conditions:

• joints on slope face plugged with ice;• immediately following a heavy rainstorm;• wet season; and• dry season.

Solution

Figure 5.17(a), (b) and (c) show the positionsof the ground water table under the variousconditions shown in Figure 5.16.

In general, the rock near the slope face hasbeen disturbed by blasting and has undergonestress relief so it will have a higher hydraulic con-ductivity than the undisturbed rock. When thehydraulic conductivity is high, the rock drainsreadily and the ground water table has a relativelyflat gradient.

If the face freezes and the water cannot drainfrom the slope, the ground water surface will risebehind the face. The same situation arises whenheavy infiltration exceeds the rate at which therock will drain.

Chapter 6

Plane failure

6.1 Introduction

A plane failure is a comparatively rare sight inrock slopes because it is only occasionally thatall the geometric conditions required to producesuch a failure occur in an actual slope. How-ever, it would not be right to ignore the two-dimensional case because there are many valuablelessons to be learned from a consideration of themechanics of this simple failure mode. Plane fail-ure is particularly useful for demonstrating thesensitivity of the slope to changes in shear strengthand ground water conditions—changes that areless obvious when dealing with the more complexmechanics of a three-dimensional slope failure.This chapter describes the method of analysis forplane failure (Sections 6.2 and 6.3), and demon-strates its application to the design of reinforcedslopes (Section 6.4), analysis methods for slopessubject to seismic ground motion (Section 6.5),and probabilistic design methods (Section 6.6).Two case studies describing the stabilization ofplane failures are described in Chapter 14.

6.2 General conditions for plane failure

Figure 6.1 shows a typical plane failure in a rockslope where a block of rock has slid on a singleplane dipping out of the face. In order for thistype of failure to occur, the following geometricalconditions must be satisfied (Figure 6.2(a)):

(a) The plane on which sliding occurs muststrike parallel or nearly parallel (withinapproximately ±20◦) to the slope face.

(b) The sliding plane must “daylight” in theslope face, which means that the dip of theplane must be less than the dip of the slopeface, that is, ψp < ψf .

(c) The dip of the sliding plane must be greaterthan the angle of friction of this plane, thatis, ψp > φ.

(d) The upper end of the sliding surface eitherintersects the upper slope, or terminates in atension crack.

(e) Release surfaces that provide negligibleresistance to sliding must be present in therock mass to define the lateral boundaries ofthe slide. Alternatively, failure can occur ona sliding plane passing through the convex“nose” of a slope.

6.3 Plane failure analysis

The slope geometries and ground water con-ditions considered in this analysis are definedin Figure 6.3, which shows two geometries asfollows:

(a) slopes having a tension crack in the uppersurface; and

(b) slopes with a tension crack in the face.

When the upper surface is horizontal (ψs = 0),the transition from one condition to anotheroccurs when the tension crack coincides with theslope crest, that is when

z

H= (1 − cot ψf tan ψp) (6.1)

130 Plane failure

Figure 6.1 Plane failure onsmooth, persistent beddingplanes in shale (Interstate 40,near Newport, Tennessee).

Face

Upper slope

Slide plane

Tension crack

�f�p

�

For sliding �f >�p >�

�s

(a)

Slice of unit thickness

(c)(b)

Release surfaces

Slide plane

Figure 6.2 Geometry of slope exhibiting plane failure: (a) cross-section showing planes forming a plane failure;(b) release surfaces at ends of plane failure; (c) unit thickness slide used in stability analysis.

where z is the depth of the tension crack, H is theslope height, ψf is the slope face angle and ψp isthe dip of the sliding plane.

The following assumptions are made in planefailure analysis:

(a) Both sliding surface and tension crack strikeparallel to the slope.

(b) The tension crack is vertical and is filled withwater to a depth zw.

(c) Water enters the sliding surface along thebase of the tension crack and seeps along

the sliding surface, escaping at atmosphericpressure where the sliding surface daylightsin the slope face. The pressure distributionsinduced by the presence of water in the ten-sion crack and along the sliding surface areillustrated in Figure 6.3.

(d) The forces W (the weight of the slidingblock), U (uplift force due to water pressureon the sliding surface) and V (force due towater pressure in the tension crack) all actthrough the centroid of the sliding mass. Inother words, it is assumed that there are no

Plane failure 131

Face

Tension crack in upper surface of slope

Slide plane

H

W

U

V

b

z

zw

�s

�p�f

W�p�f

Face

Tension crack in face

Slide plane

H

U

Vzw

z

(a)

(b)

Figure 6.3 Geometries of plane slope failure: (a) tension crack in the upper slope; (b) tension crack in the face.

moments that would tend to cause rotationof the block, and hence failure is by slid-ing only. While this assumption may notbe strictly true for actual slopes, the errorsintroduced by ignoring moments are smallenough to neglect. However, in steep slopeswith steeply dipping discontinuities, the pos-sibility of toppling failure should be kept inmind (see Chapter 9).

(e) The shear strength τ of the sliding surfaceis defined by cohesion c and friction angleφ that are related by the equation τ =c + σ tan φ, as discussed in Chapter 4. Inthe case of a rough surface or a rock masshaving a curvilinear shear strength envelope,the apparent cohesion and apparent frictionangle are defined by a tangent that takes intoaccount the normal stress acting on the slid-ing surface. The normal stress σ acting on asliding surface can be determined from thecurves given in Figure 6.4.

(f) It is assumed that release surfaces are presentso that there is no resistance to sliding

at the lateral boundaries of the failingrock mass (Figure 6.2(b)).

(g) In analyzing two-dimensional slope prob-lems, it is usual to consider a slice of unitthickness taken at right angles to the slopeface. This means that on a vertical sectionthrough the slope, the area of the sliding sur-face can be represented by the length of thesurface, and the volume of the sliding blockis represented by the cross-section area of theblock (Figure 6.2(c)).

The factor of safety for plane failure is calcu-lated by resolving all forces acting on the slopeinto components parallel and normal to the slid-ing plane. The vector sum of the shear forces,∑

S acting down the plane is termed the drivingforce. The product of the total normal forces,∑

N and the tangent of the friction angle φ, plusthe cohesive force is termed the resisting force(see Section 1.4.2). The factor of safety FS of thesliding block is the ratio of the resisting forces to

132 Plane failure

H

z

10

20

30

4050

60

7080

0 10 20 30 40 50 60 70 80 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Slope face angle �f (degrees)

Dim

ensi

onle

ss n

orm

al s

tres

s ra

tio, �

/�rH

2(1 – z/H)

[(1 – (z/H)2)cot�p– cot�f] sin�p��rH

where z/H = 1 – (cot�f tan�p)1/2, and �s= 0

�p�p�f

�

=

Figure 6.4 Normal stress acting on the slide plane ina rock slope.

the driving forces, and is calculated as follows:

FS = Resisting forceDriving force

(6.2)

= cA +∑N tan φ∑S

(6.3)

where c is the cohesion and A is the area of thesliding plane.

Based on the concept illustrated in equa-tions (6.2) and (6.3), the factor of safety forthe slope configurations shown in Figure 6.3 isgiven by

FS = cA + (W cos ψp − U − V sin ψp) tan φ

W sin ψp + V cos ψp

(6.4)

where A is given by

A = (H + b tan ψs − z) cosec ψp (6.5)

The slope height is H , the tension crack depth isz and it is located a distance b behind the slopecrest. The dip of the slope above the crest is ψs.When the depth of the water in the tension crackis zw, the water forces acting on the sliding planeU and in the tension crack V are given by

U = 12 γwzw(H + b tan ψs − z) cosec ψp

(6.6)

V = 12 γwz2

w (6.7)

where γw is the unit weight of water.The weights of the sliding block W for the two

geometries shown in Figure 6.3 are given by equa-tions (6.8) and (6.9). For the tension crack in theinclined upper slope surface (Figure 6.3(a)),

W = γr

[(1 − cot ψf tan ψp)

(bH + 1

2H2 cot ψf

)+ 1

2b2(tan ψs − tan ψp)]

(6.8)

and, for the tension crack in the slope face(Figure 6.3(b)).

W = 12

γrH2[ (

1 − z

H

)2cot ψp

× (cot ψp tan ψf − 1)

](6.9)

where γr is the unit weight of the rock.Figure 6.3 and equations (6.4)–(6.9) illus-

trate that the geometry of a plane failure andthe ground water conditions can be completelydefined by four dimensions (H , b, z and zw) andby three angles (ψf , ψp and ψs). These simplemodels, together with the ground water, rockbolting and seismic ground motion concepts dis-cussed in the following sections allow stabilitycalculations to be carried out for a wide varietyof conditions.

Plane failure 133

6.3.1 Influence of ground water on stability

In the preceding discussion, it has been assumedthat it is only the water present in the tensioncrack and along the sliding surface that influ-ences the stability of the slope. This is equivalentto assuming that the rest of the rock mass isimpermeable, an assumption that is certainly notalways justified. Therefore, consideration must begiven to water pressure distributions other thanthose presented in this chapter. Under some con-ditions, it may be possible to construct a flow netfrom which the ground water pressure distribu-tion can be determined from the intersection ofthe equipotentials with the sliding surface (seeFigure 5.10). Information that would assist indeveloping flow nets includes the rock mass per-meability (and its anisotropy), the locations ofseepage on the face and recharge above the slope,and any piezometric measurements.

In the absence of actual ground water pressuremeasurements within a slope, the current state ofknowledge in rock engineering does not permit aprecise definition of the ground water flow pat-terns in a rock mass. Consequently, slope designshould assess the sensitivity of the factor of safetyto a range of realistic ground water pressures, andparticularly the effects of transient pressures dueto rapid recharge (see Figure 5.11(b)).

The following are four possible ground waterconditions that may occur in rock slopes, andthe equations that can be used to calculate thewater forces U and V . In these examples, the pres-sure distributions in the tension crack and alongthe sliding plane are idealized and judgment isrequired to determine the most suitable conditionfor any particular slope.

(a) Ground water level is above the base of ten-sion crack so water pressures act both in thetension crack and on the sliding plane. If thewater discharges to the atmosphere wherethe sliding place daylights on the slope face,then it is assumed that the pressure decreaseslinearly from the base of the tension crack tozero at the face. This condition is illustratedin Figure 6.3 and the method of calculating

forces U and V is given by equations (6.6)and (6.7), respectively.

(b) Water pressure may develop in the tensioncrack only, in conditions for example, wherea heavy rainstorm after a long dry spell res-ults in surface water flowing directly into thecrack. If the remainder of the rock mass isrelatively impermeable, or the sliding surfacecontains a low permeability clay filling, thenthe uplift force U could also be zero or nearlyzero. In either case, the factor of safety of theslope for these transient conditions is givenby equation (6.4) with U = 0 and V givenby equation (6.7).

(c) Ground water discharge at the face may beblocked by freezing (Figure 6.5(a)). Wherethe frost penetrates only a few meters behindthe face, water pressures can build up

H

b

U

V zw

zw

z

�p

�p

Uhw

(a)

(b)

Figure 6.5 Possible ground water pressures in planefailures: (a) uniform pressure on slide plane fordrainage blocked at toe; (b) triangular pressure onslide plane for water table below the base of tensioncrack.

134 Plane failure

in the slope and the uplift pressure U canexceed that shown in Figure 6.3. For theidealized rectangular pressure distributionshown in Figure 6.5(a), the uplift force U isgiven by

U = Ap (6.10)

where A is the area of the sliding plane givenby equation (6.5) and p is the pressure in theplane (and at the base of the tension crack)given by

p = γwzw (6.11)

The condition shown in Figure 6.5(a) mayonly occur rarely, but could result in a lowfactor of safety; a system of horizontal drainsmay help to limit the water pressure in theslope.

(d) Ground water level in the slope is below thebase of the tension crack so water pressureacts only on the sliding plane (Figure 6.5(b)).If the water discharges to the atmospherewhere the sliding plane daylights on the face,then the water pressure can be approximatedby a triangular distribution, from which theuplift force U is given by

U = 12

zw

sin ψphwγw (6.12)

where hw is the estimated depth of water atthe mid-point of the saturated portion of thesliding plane.

6.3.2 Critical tension crack depthand location

In the analysis, it has been assumed that theposition of the tension crack is known from itsvisible trace on the upper surface or on the faceof the slope, and that its depth can be establishedby constructing an accurate cross-section of theslope. However, the tension crack position maybe unknown, due for example, to the presence

of soil above the slope crest, or an assumed loc-ation may be required for design. Under thesecircumstances, it becomes necessary to con-sider the most probable position of a tensioncrack.

When the slope is dry or nearly dry (zw/z = 0),equation (6.4) for the factor of safety can bemodified as follows:

FS = c · A

W sin ψp+ cot ψp tan φ (6.13)

The critical tension crack depth zc for a dry slopecan be found by minimizing the right-hand sideof equation (6.13) with respect to z/H . This givesthe critical tension crack depth as

zc

H= 1 −√

cot ψf tan ψp (6.14)

and the corresponding position of the criticaltension crack bc behind the crest is

bc

H= √

cot ψf cot ψp − cot ψf (6.15)

Critical tension crack depths and locations for arange of dimensions for dry slopes are plotted inFigure 6.6(a) and (b). However, if the tensioncrack forms during heavy rain or if it is locatedon a pre-existing geological feature such as a ver-tical joint, equations (6.14) and (6.15) no longerapply.

6.3.3 The tension crack as an indicator ofinstability

Anyone who has examined excavated rock slopescannot have failed to notice the occasional ten-sion cracks behind the crest (Figure 6.7). Someof these cracks have been visible for tens of yearsand, in many cases, do not appear to have had anyadverse influence upon the stability of the slope.It is interesting, therefore, to consider how suchcracks are formed and whether they can give anyindication of slope instability.

In a series of very detailed model studies on thefailure of slopes in jointed rocks, Barton (1971)

Plane failure 135

H

zc

bc

�p

�f

90

89

85

80

70

6050

4030

2010

0 10 20 30 40 50 60 70 80 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0(a)

(b)

�p

�f

z c/H

0

10

20

30

40

50

60

70

80

90

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

10

20

3040

50

607080

Ratio (bc/H )

�f

�p

Figure 6.6 Critical tensioncrack locations for a dry slope:(a) critical tension crack depthrelative to crest of cut;(b) critical tension cracklocation behind crest of cut.

found that the tension crack resulted from smallshear movements within the rock mass. Althoughthese individual movements were very small, theircumulative effect was that there was a significantdisplacement of the slope surfaces—sufficient tocause separation of vertical joints behind the slopecrest and to form “tension” cracks. The fact thatthe tension crack is caused by shear movementsin the slope is important because it suggests that,when a tension crack becomes visible in the sur-face of a slope, it must be assumed that shearfailure has initiated within the rock mass.

It is impossible to quantify the significance oftension cracks since their formation is only the

start of a complex progressive failure processwithin the rock mass, about which little is known.It is quite probable that, in some cases, theimproved drainage resulting from dilation of therock structure, combined with the interlocking ofindividual blocks within the rock mass, could giverise to an increase in stability. However, wherethe failure surface comprises a single discontinu-ity surface such as a bedding plane daylightingin the slope face, initial movement could be fol-lowed by a very rapid decrease in stability becausea small amount of movement could result in areduction in the shear strength from the peak tothe residual value.

136 Plane failure

Figure 6.7 A tension crackbehind a sliding rock mass inwhich significant horizontaldisplacement has occurred(above Kooteney Lake, BritishColumbia).

In summary, the presence of a tension crackshould be taken as an indication of potentialinstability and that, in the case of an importantslope, this should signal the need for a detailedinvestigation of stability.

6.3.4 Critical slide plane inclination

When a persistent discontinuity such as a beddingplane exists in a slope and the inclination of thisdiscontinuity is such that it satisfies the conditionsfor plane failure defined in Figure 6.2, stability ofthe slope will be controlled by this feature. How-ever, where no such feature exists and a sliding

surface, if it were to occur, would follow minorgeological features and, in some places, passthrough intact material, how can the inclinationof such a failure path be determined?

The first assumption that must be made con-cerns the shape of the slide surface. In a weakrock slope or a soil slope with a face angle lessthan about 45◦, the slide surface would have a cir-cular shape. The analysis of such a failure surfaceis discussed in Chapter 8.

In steep rock slopes, the slide surface is approx-imately planar and the inclination of such a planecan be found by partial differentiation of equa-tion (6.4) with respect to ψp and by equating the

Plane failure 137

resulting differential to zero. For dry slopes thisgives the critical slide plane inclination ψpc as

ψpc = 12 (ψf + φ) (6.16)

The presence of water in the tension crack willcause the slide plane inclination to be reducedby as much as 10%, but in view of the uncer-tainties associated with the inclination of thisslide surface, the added complication of includingthe influence of ground water is not consideredto be justified. Consequently, equation (6.16)can be used to obtain an estimate of the criticalslide plane inclination in steep slopes that do notcontain through-going discontinuities.

6.3.5 Analysis of failure on a rough plane

The stability analyses discussed so far in thissection have used shear strength parameters thatare constant throughout the slope. However, asdiscussed in Section 4.2.4 on the shear strength ofrough rock surfaces, the friction angle that will bemobilized in the slope may depend on the normalstress acting on the surface. That is, the frictionangle will decrease with increasing normal stressas the asperities on the surface are ground off, asdefined by equation (4.7). The significance of thisrelationship between friction angle and normalstress is illustrated in this section.

Consider a plane slope failure with the geo-metry as shown in Figure 6.3(a). For a dry slope(U = V = 0), the normal stress σ acting on thesliding surface is given by

σ = W cos ψp

A(6.17)

where W is the weight of the sliding block, ψpis the dip of the sliding surface and A is thearea of this surface. If the sliding plane containsno cohesive infilling so that the shear strengthcomprises only friction, then the factor of safetycan be calculated using equations (1.2)–(1.6) forlimit equilibrium analysis, equation (4.7) to definethe shear strength of the rough surface, andequation (6.17) to define the normal stress on this

surface. For these conditions the factor of safetyis given by

FS = τA

W sin ψp(6.18)

= σ tan(φ + JRC log10( JCS/σ))A

W sin ψp(6.19)

= tan(φ + JRC log10( JCS/σ))

tan ψp(6.20)

= tan(φ + i)

tan ψp(6.21)

The application of these equations and the effectof a rough surface on the factor of safety can beillustrated by the following example. Considera slope with dimensions H = 30 m, z = 15 m,ψp = 30◦ and ψf = 60◦, in which the propertiesof the clean rough joint forming the sliding sur-face are φ = 25◦, JRC = 15 and JCS = 5000 kPa.From Figure 6.4, the normal stress ratio σ/γrH is0.36, and the value of σ is 281 kPa if the rockdensity γr is 26 kN/m3. The value of the σ calcu-lated from Figure 6.4 is the average normal stressacting on the sliding surface. However, the max-imum stress acting on this surface is below thecrest of the slope where the depth of rock is 20 m.The calculated maximum stress is

σmax = 20 · 26 · cos(30)

= 450 kPa

Using equation (4.7) and the roughness prop-erties quoted in the previous paragraph, the shearstrength of the sliding surface and the correspond-ing factors of safety at the average and maximumnormal stresses can be calculated as

σ = 281 kPa τ = 269 kPa; (φ + i) = 44◦

and FS = 1.66

σ = 450 kPa τ = 387 kPa; (φ + i) = 41◦

and FS = 1.49

These results indicate that the effect of increas-ing normal stress level on the sliding surface is to

138 Plane failure

diminish the friction angle (due to the asperitiesbeing ground off) and there is a correspondingdecrease in the factor of safety (10% in this case).

6.4 Reinforcement of a slope

When it has been established that a slope is poten-tially unstable, reinforcement may be an effectivemethod of improving the factor of safety. Meth-ods of reinforcement include the installation oftensioned anchors or fully grouted, untensioneddowels, or the construction of a toe buttress.Factors that will influence the selection of anappropriate system for the site include the sitegeology, the required capacity of the reinforce-ment force, drilling equipment availability andaccess, and time required for construction. Thissection describes the design methods for slopereinforcement, while Section 12.4.2 discussesconstruction aspects of slope reinforcement, andSection 14.3 discusses a case study where a planefailure was reinforced with tensioned cables.

If rock anchors are to be installed, it is neces-sary to decide if they should be anchored at thedistal end and tensioned, or fully grouted anduntensioned. Untensioned dowels are less costlyto install, but they will provide less reinforcementthan tensioned anchors of the same dimensions,and their capacity cannot be tested. One technicalfactor influencing the selection is that if a slopehas relaxed and loss of interlock has occurred onthe sliding plane, then it is advisable to install ten-sioned anchors to apply normal and shear forceson the sliding plane. However, if the reinforce-ment can be installed before the excavation takesplace, then fully grouted dowels are effective inreinforcing the slope by preventing relaxation onpotential sliding surfaces (see Figure 12.5). Unten-sioned dowels can also be used where the rock israndomly jointed and there is a need to reinforcethe overall slope, rather than a particular plane.

6.4.1 Reinforcement with tensioned anchors

A tensioned anchor installation involves drilling ahole extending below the sliding plane, installinga rock bolt or strand cable that is bonded into

T sin (�p+�T)

T cos (�p+�T)

�T

�p

T

Figure 6.8 Reinforcement of a slope with tensionedrock bolt.

the stable portion of the slope, and then tension-ing the anchor against the face (Figure 6.8). Thetension in the anchor T modifies the normal andshear forces acting on the sliding plane, and thefactor of safety of the anchored slope is given by

FS = cA + (W cos ψp − U − V sin ψp + T sin(ψT + ψp)) tan φ

W sin ψp + V cos ψp − T cos(ψT + ψp)

(6.22)

where T is the tension in the anchor inclined at anangle ψT below the horizontal. Equation (6.22)shows that the normal component of the anchortension (T sin(ψp + ψT)) is added to the normalforce acting on the sliding plane, which has theeffect of increasing the shear resistance to sliding.Also, the shear component of the anchor tension(T cos(ψp+ψT)) acting up the sliding plane is sub-tracted from the driving forces, so the combinedeffect of the anchor force is to improve the factorof safety (if (ψp + ψT) < 90◦).

The factor of safety of a slope reinforced withtensioned rock anchors varies with the inclinationof the bolt. It can be shown that the most efficientangle (ψT(opt)) for a tensioned rock is when

φ = (ψT(opt) + ψp) or ψT(opt) = (φ − ψp)

(6.23)

This relationship shows that the optimum install-ation angle for a tensioned bolt is flatter thanthe normal to the sliding plane. In prac-tice, cement grouted anchors are installed atabout 10–15◦ below the horizontal to facilitate

Plane failure 139

grouting, while resin grouted anchors may beinstalled in up-holes. It should be noted that boltsinstalled at an angle steeper than the normal to thesliding plane (i.e. (ψp + ψT) > 90◦) can be detri-mental to stability because the shear componentof the tension, acting down the plane, increasesthe magnitude of the displacing force.

Since the stability analysis of plane failures iscarried out on a 1 m thick slice of the slope, thecalculated value of T for a specified factor ofsafety has the units kN/m. The procedure fordesigning a bolting pattern using the calculatedvalue of T is as follows. For example, if the ten-sion in the each anchor is TB, and a pattern ofbolts is installed so that there are n bolts in eachvertical row, then the total bolting force in eachvertical row is (TB · n). Since the required boltingforce is T , then the horizontal spacing S betweeneach vertical row is given by

S = TB n

T

(kN

kN/m

)(6.24)

This design method is illustrated in the workedexample at the end of this chapter.

6.4.2 Reinforcement with fully grouteduntensioned dowels

Fully grouted, untensioned dowels comprise steelbars installed in holes drilled across the poten-tial sliding plane, which are then encapsulated incement or resin grout. The steel acts as a rigidshear pin across any plane of weakness in therock. A method of calculating the reinforcementprovided by dowels, developed by Spang andEgger (1990), is discussed here.

Figure 6.9 shows the results of a finite elementanalysis of a steel bar grouted into a drill holein a rock containing a joint surface; the angle α

between the axis of the bolt and the normal tothe joint is 30◦. Shear displacement on the jointcauses deformation of the bolt that takes place inthree stages as follows:

(a) Elastic stage—Afterovercomingthecohesionof the joint, the blocks begin to slide relative

to each other. The shear resistance of thedoweled joint comprises the shear strengthdue to friction on the joint, and the elasticresponse of the steel dowel, grout and rock.

(b) Yield stage—At displacements of less thanabout 1 mm in installations where the rockis deformable and the grout thickness is atleast equal to the dowel radius, the steel isdeformed in order to mobilize shear resist-ance. As a result of the deformation, the yieldstrengths of the steel and grout are reachedin bending and compression respectively.

(c) Plastic stage—All the materials in the grouteddowel installation yield at an early stageof shear displacement and at low shearforces. Therefore, the shear resistance of thedoweled joint depends on the shear force–displacement relationship of the plasticizedmaterials. The contribution of the dowel tothe total shear strength of the joint is a func-tion of the friction angle (φ) and roughness(i) of the joint, the dowel inclination (α),the compressive strength of the rock andgrout (σci) and the tensile strength of the steelbar (σt(s)). In general, the shear resistance isenhanced where the joint has a high frictionangle and is rough so there is some dilationduring shearing, the inclination angle α isbetween about 30◦ and 45◦, and the rockis deformable but not so soft that the dowelcuts into the rock.

Based on the tests conducted by Spang andEgger, the shear resistance Rb (kN) of a dowelledjoint is given by

Rb = σt(s)[1.55 + 0.011σ1.07ci sin2(α + i)]

× σ−0.14ci (0.85 + 0.45 tan φ) (6.25)

where the units of σci are MPa and of σt(s) are kN.The corresponding displacement δs of a dowelledjoint is given by

δs = (15.2 − 55.2σ−0.14ci + 56.2σ−0.28

ci )

× (1 − tan α(70/σc)0.125(cos α)−0.5)

(6.26)

140 Plane failure

Steel dowel

Joint plane

Joint plane

Detail

�

Zone of plastic strain in steel

Grout

Rock

Figure 6.9 Strain in fully grouted steel dowel due to shear movement along joint (modified from Spang andEgger (1990)).

In the stability analysis of a plane failure in whichfully grouted dowels have been installed acrossthe sliding plane, equation (6.4) for the factor ofsafety is modified as follows to account for theincreased shear resistance to sliding:

FS = cA + N tan φ + Rb

S(6.27)

6.4.3 Reinforcement with buttresses

The two previous sections discussed reinforce-ment by installing anchors across the potentialsliding surface. An alternate method is to con-struct a buttress at the toe to provide externalsupport to the slope, using the methods shown

on Figure 6.10. In both cases, the factor ofsafety is calculated using equation (6.27) usingthe appropriate value for Rb for the systeminstalled.

Near the crest of the slope where the bed-ding forms a series of slabs, steel bars can begrouted into holes drilled into the rock andthen encased in shotcrete or concrete. The steelprovides shear resistance to movement, while theconcrete provides continuous support betweenthe dowels and keeps small fragments of rockin place. These buttresses are particularly applic-able where the rock along the crest is slightlyweathered, and if rock anchors were installed,the on-going weathering would eventually exposethe bolts. It is likely that the maximum thickness

Plane failure 141

Steel dowel/concrete buttress

Waste rock buttress

Figure 6.10 Reinforcement of slope with buttresses.

of slab that can be supported in this manner isabout 2 m.

Larger scale support can be provided by pla-cing a waste rock buttress at the toe of the slope.The support provided by such a buttress dependson the buttress weight, and the shear resistancegenerated along the base of the buttress that is afunction of the weight of the rock, and the rough-ness and inclination of the base. This method canonly be used, of course, if there is sufficient spaceat the toe to accommodate the required volume ofrock. It is also important that the waste rock befree draining so that water pressures do not buildup behind the buttress.

6.5 Seismic analysis of rock slopes

In seismically active areas of the world, thedesign of rock slopes should take into accountthe effects on stability of earthquake-inducedground motions. This section describes the influ-ence of ground motions on stability, and designprocedures that incorporate seismic acceleration.

6.5.1 Performance of rock slopes duringearthquakes

There are numerous records of rock falls andlandslides being induced by seismic ground mot-ions (Youd, 1978; Van Velsor and Walkinshaw,1992; Harp and Noble, 1993; Ling and Cheng,

1997). For example, in 1980 a magnitude 6.0–6.1earthquake at Mammoth Lakes, California dis-lodged a 21.4 ton boulder that bounced androlled a horizontal distance of 421 m from itssource, and in 1983 in an earthquake in Idaho,a 20.5 ton boulder traveled a distance of about95 m (Kobayashi, 1990). With respect to land-slides, a detailed inventory of landslides inducedby the magnitude 6.7 Northridge earthquakein Los Angeles identified about 11,000 land-slides in an area of approximately 10,000 km2.(Jibson and Harp, 1995). These slides occurredprimarily in the Santa Susana Mountains wherethe slopes comprise Late Miocene through Pleis-tocene clastic sediments that have little or nocementation, and that have been folded and uplif-ted by rapid tectonic deformation (Jibson et al.,1998). For these three events, the maximum epi-central distance to the landslide limit was about70 km, which is about the average in relationto historical worldwide earthquakes of this mag-nitude (Keefer, 1984). Another example of seis-mically induced landslides was the occurrence ofslides in strong rock with volumes of millionsof cubic meters during the Denali, Alaska eventin 2002 (Harp et al., 2003).

Studies of the number and distribution of land-slides and rock falls near earthquakes has shownthat the concentrations of landslides can be ashigh as 50 events per square kilometer. These datahave been used to assess the geological and topo-graphical conditions for which the landslide androck fall hazard is high (Keefer, 1992). It has alsobeen found that the following five slope paramet-ers have the greatest influence on stability duringearthquakes:

• Slope angle—Rock falls and slides rarely occuron slopes with angles less than about 25◦.

• Weathering—Highly weathered rock com-prising core stones in a fine soil matrix, andresidual soil are more likely to fail than freshrock.

• Induration—Poorly indurated rock in whichthe particles are weakly bonded is morelikely to fail than stronger, well-induratedrock.

142 Plane failure

Steeperthan 25°?

Intenselyweathered?

Extremelyhigh

Poorlyindurated?

Fissuresclosely

spaced?

Fissuresopen?

Low

Wet?

High Moderate

High

Veryhigh

Fissuresclosely

spaced?

Low

Yes No

Yes

Yes

Yes

Yes

Yes

Yes

No

No

No

No

No

NoFigure 6.11 Decision tree for susceptibility of rock slopes to earthquake-induced failure (Keefer, 1992).

• Discontinuity characteristics—Rock contain-ing closely spaced, open discontinuities aremore susceptible to failure than massive rockin which the discontinuities are closed andhealed.

• Water—Slopes in which the water table ishigh, or where there has been recent rainfall,are susceptible to failure.

The relationship between these five conditionsand the slope failure hazard is illustrated in thedecision tree in Figure 6.11. It is also of interestthat the hazard is high for pre-existing landslideswith slopes flatter than 25◦. Furthermore, there isa hazard for slopes with local relief greater thanabout 2000 m, probably because seismic shakingis amplified by the topography (Harp and Jibson,2002), and possible freeze–thaw action at highaltitude that loosens the surficial rock.

The decision tree shown in Figure 6.11 can beused, for example, as a screening tool in assessingrock fall and slide hazards along transportationand pipeline corridors. If a more rigorous haz-ard assessment of a specific site is required, it ispossible to use a technique developed by Harpand Wilson (1995) that involves calculating theArias intensity of the ground motion at the loca-tion of interest. The Arias intensity Ia is a measure

of the total energy of the ground motion, and isdefined as

Ia = π

2g

∫ Td

0[a(t)]2 dt (units: m/s) (6.28)

where a(t) is a single component acceleration–time series for a strong-motion record with totalduration Td (seconds), t is the time in seconds,and g is the acceleration of gravity. The occur-rence of rock falls and slides has been correlatedwith the Arias intensity of the recorded strong-motion records for the Whittier and Superstitionearthquakes in California. The study showed thatthe Arias intensity threshold for slides in Mio-cene and Pliocene deposits was in the range of0.08–0.6 m/s, and the threshold for Precambrianand Mesozoic rocks, where the discontinuitieswere open, was in the range of 0.01–0.07 m/s.Note that this method is restricted to sites whereacceleration-time histories are available, or canbe estimated.

6.5.2 Seismic hazard analysis

The design of slopes that may be subjectedto seismic ground motion requires quantitativeinformation on the magnitude of the motion

Plane failure 143

(Abrahamson, 2000). This information may beeither the peak ground acceleration (PGA) orthe acceleration–time history of the motions,depending on the method of stability analysisthat is to be used. The process by whichthe design motion parameters are established istermed “seismic hazard” analysis, which involvesthe following three steps (NHI, 1998; Glass,2000):

(a) Identification of seismic sources capable ofproducing strong ground motions at the site.

(b) Evaluation of seismic potential for eachcapable source.

(c) Evaluation of the intensity of design groundmotions at the site.

Implementation of these three steps involves thefollowing activities.

Seismic sources. Earthquakes are the resultof fault movement, so identification of seismicsources includes establishing the types of faultsand their geographic location, depth, size andorientation. This information is usually avail-able from publications such as geological mapsand reports prepared by government geologicalsurvey groups and universities, and any previ-ous projects that have been undertaken in thearea. Also, the identification of faults can bemade from the study of aerial photographs, geo-logical mapping, geophysical surveys and trench-ing. On aerial photographs, such features asfault scarplets, rifts, fault slide ridges, shutterridges and fault saddles, and off-sets in such fea-tures as fence lines and road curbs (Cluff et al.,1972) may identify active faults. In addition,records of seismic monitoring stations provideinformation on the location and magnitude ofrecent earthquakes that can be correlated to faultactivity.

Seismic potential. Movement of faults withinthe Holocene Epoch (approximately the last11,000 years) is generally regarded as the cri-terion for establishing that the fault is active(USEPA, 1993). Although the occurrence inter-val of some major earthquakes may be greater

than 11,000 years, and not all faults ruptureto the surface, lack of evidence that movementhas occurred in the Holocene is generally suffi-cient evidence to dismiss the potential for groundsurface rupture. Most Holocene fault activity inNorth America has occurred west of the RockyMountains, and may be identified by detailedmapping, followed by trenching, geophysics ordrilling. In regions where there is no surfaceexpression of fault rupture, seismic source char-acterization depends primarily on micro-seismicstudies and the historic record of felt earthquakes.

Ground motion intensity. Once the seismicsources capable of generating strong groundmotions at a site have been identified and char-acterized, the intensity of the ground motionscan be evaluated either from published codesand standards, or from seismic hazard ana-lysis as discussed in Section 6.5.3. The buildingcodes of countries with seismic areas publishmaps in which the country is divided intozones showing, for example, the effectivepeak acceleration levels (as a fraction of grav-ity acceleration) with a 10% probability ofbeing exceeded in a 50-year period (Frankelet al., 1996). The information on these mapsmay also be available on the internet. Forexample, in the United States it is possibleto find acceleration levels for postal zip codes(http:/geohazards.cr.usgs.gov/eq/). These pub-lished accelerations can be used in geotechnicaldesign and have the value of promoting standarddesigns within each zone.

6.5.3 Ground motion characterization

As a complement to the published maps, a seismichazard analysis for a specific site can be conductedby evaluating the magnitude of ground motionsfrom all capable sources with the potential forgenerating strong ground motions at the site. Thevalue of seismic analysis, in contrast to the useof published codes as discussed in the previousparagraph, is the ability to incorporate the latestdevelopments in local seismicity. Furthermore, itis possible to develop site-specific ground motions

144 Plane failure

compared with the regional seismicity on whichthe codes are based.

The three steps in this analysis are first, toestablish the location and style of faulting of allpotential sources, and assign each a representativeearthquake magnitude. Second, an appropriateattenuation relationship is selected as a functionof magnitude, fault mechanism, site-to-sourcedistance and site conditions. Third, the capablesources are screened based on magnitude andground motion intensity at the site to determinethe governing source.

Attenuation. Attenuation equations, discussedin the previous paragraph, define the relationshipbetween the source moment magnitude (Mw) andpeak ground acceleration (PGA) at the site. Theequations are based upon either statistical ana-lysis of values observed in previous earthquakes,or from theoretical models of the propagation ofstrong ground motions, depending on the amountof observed data available. For example, for sub-duction zones and in the eastern United States thePGA on rock at a hypocentral distance R is givenby (Youngs et al., 1988):

ln(PGA) = 19.16 + 1.045Mw − 4.738 ln[R+ 154.7 exp(0.1323Mw)] (6.29)

for 20 < R ≤ 40 km, and Mw > 8. The momentmagnitude is a measure of the kinetic energyreleased by the earthquake, and the hypocenteris the point from which the seismic waves firstemanate.

Time histories. If deformation analyses areto be carried out, it is necessary to use a rep-resentative time history of the ground motions.Time histories can be selected from previouslyrecorded motions, or by simulation techniquesto generate a project-specific synthetic time his-tory. In selecting a representative time historyfrom the catalogue of available records, the relev-ant characteristics of the project and source sitesshould be matched as closely as possible. Some ofthe characteristics that are important in matchingtime histories include magnitude, source mech-anism, focal depth, site-to-source distance, site

geology, PGA, frequency content, duration andenergy content.

6.5.4 Pseudo-static stability analysis

The limit equilibrium method of determining thefactor of safety of a sliding block as described inSection 6.3 can be modified to incorporate theeffect on stability of seismic ground motions. Theanalysis procedure, known as the pseudo-staticmethod, involves simulating the ground motionsas a static horizontal force acting in a directionout of the face. The magnitude of this force isthe product of a seismic coefficient kH (dimen-sionless) and the weight of the sliding block W .The value of kH may be taken as equal to thedesign PGA, which is expressed as a fraction ofthe gravity acceleration (i.e. kH = 0.1 if the PGAis 10% of gravity). However, this is a conservat-ive assumption since the actual transient groundmotion with a duration of a few seconds is beingreplaced by a constant force acting over the entiredesign life of the slope.

In the design of soil slopes and earth dams, it iscommon that kH is fraction of the PGA, providedthat there is no loss of shear strength during cyc-lic loading (Seed, 1979; Pyke, 1999). Study ofslopes using Newmark analysis (see Section 6.5.5)with a yield acceleration ky equal to 50% of thePGA (i.e. ky = 0.5 · amax/g) showed that per-manent seismic displacement would be less than1 m (Hynes and Franklin, 1984). Based on thesestudies, the California Department of Mines andGeology (CDMG, 1997) suggests that it is reas-onable to use a value of kH equal to 50% ofthe design PGA, in combination with a pseudo-static factor of safety of 1.0–1.2. With respectto soil slopes, and rock slopes where the rockmass contains no distinct sliding surface and somemovement can be tolerated, it may be reason-able to use the CDMG procedure to determinea value for kH. However, for rock slopes thereare two conditions for which it may be advisableto use kH values somewhat greater than 0.5 timesthe PGA. First, where the slope contains a dis-tinct sliding surface for which there is likely tobe a significant decrease in shear strength with

Plane failure 145

limited displacement; sliding planes on which thestrength would be sensitive to movement includesmooth, planar joints or bedding planes with noinfilling. Second, where the slope is a topographichigh point and some amplification of the groundmotions may be expected. In critical situations, itmay also be advisable to check the sensitivity ofthe slope to seismic deformations using Newmarkanalysis as discussed in Section 6.5.5.

The factor of safety of a plane failure usingthe pseudo-static method is given by modifyingequation (6.4) as follows (assuming the slope isdrained, U = V = 0):

FS = cA + (W(cos ψp − kH sin ψp)) tan φ

W(sin ψp + kH cos ψp)

(6.30)

The equation demonstrates that the effect of thehorizontal force is to diminish the factor of safetybecause the shear resistance is reduced and thedisplacing force is increased.

Under circumstances where it is considered thatthe vertical component of the ground motion willbe in phase with, and have the same frequency,as the horizontal component, it may be appro-priate to use both horizontal and vertical seismiccoefficients in stability analysis. If the verticalcoefficient is kV and the ratio of the vertical tothe horizontal components is rk (i.e. rk = kV/kH),then the resultant seismic coefficient kT is

kT = kH(1 + r2k)1/2 (6.31)

acting at an angle ψk = atn(kV/kH) above thehorizontal, and factor of safety is given by

FS = cA + (W(cos ψp − kT sin(ψp + ψk))) tan φ

W(sin ψp + kT cos(ψp + ψk))

(6.32)

Study of the effect of the vertical componenton the factor of safety has shown that incor-porating the vertical component will not changethe factor of safety by more than about 10%,

provided that kV < kH (NHI, 1998). Furthermore,equation (6.32) will only apply when the ver-tical and horizontal components are exactly inphase. Based on these results, it may be acceptableto ignore the vertical component of the groundmotion.

6.5.5 Newmark analysis

When a rock slope is subject to seismicshaking, failure does not necessarily occur whenthe dynamic transient stress reaches the shearstrength of the rock. Furthermore, if the factor ofsafety on a potential sliding surface drops below1.0 at some time during the ground motion itdoes not necessarily imply a serious problem.What really matters is the magnitude of perman-ent displacement caused at the times that thefactor of safety is less than 1.0 (Lin and Whitman,1986). The permanent displacement of rock andsoil slopes as the result of earthquake motionscan be calculated using a method developed byNewmark (1965). This is a more realistic methodof analyzing seismic effects on rock slopes thanthe pseudo-static method of analysis.

The principle of Newmark’s method is illus-trated in Figure 6.12 in which it is assumed thatthe potential sliding block is a rigid body on ayielding base. Displacement of a block occurswhen the base is subjected to a uniform horizontalacceleration pulse of magnitude ag and durationt0. The velocity of the block is a function of thetime t and is designated y(t), and its velocity attime t is y. Assuming a frictional contact betweenthe block and the base, the velocity of the blockwill be x, and the relative velocity between theblock and the base will be u where

u = x − y (6.33)

The resistance to motion is accounted for by theinertia of the block. The maximum force thatcan be used to accelerate the block is the shear-ing resistance on the base of the block, whichhas a friction angle of φ◦. This limiting forceis proportional to the weight of the block (W)

146 Plane failure

W tan�

W = Mg

y (t )

Acceleration

t0 Time

t0 tm Time

Velocity

ag

ay= g tan �

Vg = agt

V = agt0

Vb = gt tan �

(a)

(b)

(c)

Figure 6.12 Displacement of rigid block on rigidbase (Newmark, 1965): (a) block on moving base;(b) acceleration plot; (c) velocity plot (Newmark,1965).

and is of magnitude (W tan φ), correspondingto a yield acceleration ay of (g tan φ), as shownby the dashed line on the acceleration plot (Fig-ure 6.12(b)). The shaded area shows that theground acceleration pulse exceeds the accelera-tion of the block, resulting in slippage.

Figure 6.12(c) shows the velocities as a func-tion of time for both the ground and the blockaccelerating forces. The maximum velocity forthe ground accelerating force has a magnitude v

which remains constant after an elapsed time oft0. The magnitude of the ground velocity vg isgiven by

vg = agt0 (6.34)

while the velocity of the block vb is

vb = gt tan φ (6.35)

After time tm, the two velocities are equal andthe block comes to rest with respect to the base,that is, the relative velocity, u = 0. The value oftm is calculated by equating the ground velocityto the velocity of the block to give the followingexpression for the time tm:

tm = vb

g tan φ(6.36)

The displacement δm of the block relative to theground at time tm is obtained by computing thearea of the shaded region on Figure. 6.12(c) asfollows:

δm = 12vtm − 1

2vt0

= v2

2g tan φ− v2

2ag

= v2

2g tan φ

(1 − tan φ

a

)(6.37)

Equation (6.37) gives the displacement of theblock in response to a single acceleration pulse(duration t0, magnitude ag) that exceeds the yieldacceleration (g tan φ), assuming infinite grounddisplacement. The equation shows that the dis-placement is proportional to the square of theground velocity.

While equation (6.37) applies to a block ona horizontal plane, a block on a sloping planewill slip at a lower yield acceleration and showgreater displacement, depending on the directionof the acceleration pulse. For a cohesionless sur-face where the factor of safety of the block FS isequal to (tan φ/ tan ψp) and the applied accelera-tion is horizontal, Newmark shows that the yieldacceleration ay, is given by

ay = (FS − 1)g sin ψp (6.38)

where φ is the friction angle of sliding surface,and ψp is the dip angle of this surface. Note thatfor ψp = 0, ay = g tan φ. Also equation (6.38)shows that for a block on a sloping surface, theyield acceleration is higher when the acceleration

Plane failure 147

pulse is in the down-dip direction compared tothe pulse in the up-dip direction.

The displacement of a block on an inclinedplane can be calculated by combining equa-tions (6.37) and (6.38) as follows:

δm = (agt)2

2gay

(1 − ay

a

)(6.39)

In an actual earthquake, the pulse would befollowed by a number of pulses of varying mag-nitude, some positive and some negative, whichwill produce a series of displacement pulses. Thismethod of displacement analysis can be appliedto the case of a transient sinusoidal accelera-tion (a(t)g) illustrated in Figure 6.13 (Goodmanand Seed, 1966). If during some period of theacceleration pulse the shear stress on the slidingsurface exceeds the shear strength, displacementwill take place. Displacement will take place morereadily in a downslope direction; this is illustratedin Figure 6.13 where the shaded areas are the por-tion of each pulse in which movement takes place.For the conditions illustrated in Figure 6.13, itis assumed that the yield acceleration diminisheswith displacement, that is, ay1 > ay2 > ay3 due toshearing of the asperities in the manner describedin Section 4.2.4.

Integration of the yield portions of the acceler-ation pulses gives the velocity of the block. It willstart to move at time t1 when the yield acceler-ation is exceeded, and the velocity will increaseup to time t2 when the acceleration drops belowthe yield acceleration. The velocity drops to zeroat time t3 as the acceleration direction begins tochange from down slope to up slope. Integrationof the velocity pulses gives the displacement ofthe block, with the duration of each displacementpulse being (t3 − t1).

The simple displacement models shown inFigures 6.12 and 6.13 have since been developedto more accurate model displacement due toactual earthquake motions, with much of thiswork being related to earth dams (Sarma, 1975;Franklin and Chan, 1977). With respect to slopestability, and Jibson (1993) and Jibson et al.(1998) have developed procedures for estimat-ing the probability of landslide occurrence asa function of Newmark displacement based onobservations of landslides caused by the 1994Northridge earthquake in California.

Newmark displacement analysis is useful fordesign if there are guidelines on the relation-ship between slope stability and the calculateddisplacement. While the Newmark method ofanalysis is highly idealized and the calculated

Acceleration

ay1 ay2 ay3

t

t (time)

�p

t

t1 t3

Velocity

Displacement

+ a (t )g

+ a(t )g

– a (t )g

– a(t )g

t1 t2

t3t1

Figure 6.13 Integration ofaccelerograms to determinedownslope movement(Goodman and Seed, 1966).

148 Plane failure

displacements should be considered order-of-magnitude estimates of actual field behavior,CDMG (1997) has developed the followingguidelines on likely slope behavior:

• 0–100 mm displacement—unlikely to corres-pond to serious landslide movement;

• 100–1000 mm—slope deformations may besufficient to cause serious ground cracking orenough strength loss to result in continuingpost-seismic failure; and

• >1000 mm displacement—damaging landslidemovement and slopes should be consideredunstable.

When applying these displacement criteriain rock slope design, consideration should begiven to the amount of displacement that willhave to occur before the residual shear strengthis reached. For example, if the sliding sur-face is a single discontinuity surface containinga weak infilling, a few centimeters of move-ment may be sufficient for the strength to bereduced to the residual value. In contrast, afractured rock mass may undergo several metersof displacement with little reduction in shearstrength.

6.6 Example of probabilistic design

The design procedures discussed so far in thischapter all use, for each design parameter, singlevalues that are assumed to be the average orbest estimate values. In reality, each parameterhas a range of values that may represent naturalvariability, changes over time, and the degree ofuncertainty in measuring their values. Therefore,the factor of safety can be realistically expressedas a probability distribution, rather than a singlevalue. In design, this uncertainty can be accoun-ted for by applying judgment in using a factor ofsafety consistent with the variability/uncertaintyin the data. That is, a high factor of safety wouldbe used where the values of the parameters are notwell known. Alternatively, the uncertainty canbe quantified using probabilistic analysis, such asMonte Carlo analysis, to calculate the probabilityof failure (see Section 1.4.4).

Examples of variability in design parametersare as follows. The orientation of a discontinuitymay vary across the slope due to surface irregular-ities or folding. This variation will be evident fromscatter in the pole locations on the stereonet, andcan be quantified in terms of means and standarddeviations of the dip and dip direction using theprocedure shown in Section 3.5. Also, the shearstrength may vary over the sliding surface becauseof variations in surface roughness and infilling,and can be quantified by testing a number of drillcore or lump samples in the laboratory. The waterpressure is likely to vary with time in response toprecipitation events such as heavy rain storms ormelting snow.

Figure 6.14 shows the results of a probabil-istic stability analysis of the slope described inSection 4.4 (see Figures 4.18 and 4.19). Sec-tion 4.4 describes the calculation of the shearstrength properties of the bedding planes, assum-ing that the factor of safety was 1.0 when thetension crack filled with water and the slopefailed. The purpose of the probabilistic ana-lysis described in this section is to show therange in factor of safety that is likely to exist inpractice due to the variability in the slope para-meters. Figure 6.14(b)–(c) show the probabilitydistributions of the following parameters:

• Dip of the sliding plane, ψp—Normal dis-tribution with a mean value of 20◦ and astandard deviation of 2.4◦.

• Cohesion, c—Skewed triangular distributionwith most likely value of 80 kPa and max-imum and minimum values of 40 and 130 kPa,respectively (4–13.3 ton /m2).

• Friction angle, φ—Normal distribution witha mean value of 20◦ and a standard deviationof 2.7◦.

• Water pressure expressed as percent fillingof tension crack—Triangular distribution ran-ging from dry (0%) to full (100%), with themost likely value being 50%.

The triangular distribution is used where themost likely value and the upper and lower boundscan only be estimated, whereas the normaldistribution is used where there is sufficient data

Plane failure 149

Rel

ativ

e fr

eque

ncy

Rel

ativ

e fr

eque

ncy

Rel

ativ

e fr

eque

ncy

Rel

ativ

e fr

eque

ncy

Rel

ativ

e fr

eque

ncy

Cohesion (×9.81 kPa)

Friction angle (degrees)

Safety factor

Tension crack percent filled zw/z (%)

Sliding plane angle �p (degrees)

�f = 58°H = 30.5 m

(a)

(b)

(c)

U

V

zzw

Probabilityoffailure ~ 7%

�p

0.00

0.22

0.10

4.33 6.10 7.89 9.00 11.400.00

0.15

0.10

0.05

15.3 17.3 19.3 21.3 23.3

0.000.00

0.090.09

0.050.05

0.020

0.010

0.00015.3 17.3 19.3 21.3 23.3 3.62 22.4 43.2 62.9 82.7

0.0

0.5

1.0

1.4

0.561 1 1.44 1.88 2.32

Figure 6.14 Probabilistic analysis of plane failure: (a) slope model showing water pressures U and V ;(b) probability distributions of cohesion, friction angle, sliding plane angle and depth of water in tension crack;(c) probability distribution of factor of safety showing 7% of probability of failure (refer to Figures 4.18–4.20).

150 Plane failure

available to calculate the mean and standard devi-ation.Otherdistributionscanbeusedasapplicable.

Figure 6.14(c) shows the distribution of thefactor of safety generated, using the Monte Carlomethod (see Section 1.4.4(b)), as the result of10,000 iterations with values randomly selec-ted from the input parameter distributions. Thehistogram shows that the mean, maximum andminimum factors of safety are 1.36, 2.52 and0.69 respectively. Also, the factor of safety wasless than 1.0 for 720 iterations, so the probabil-ity of failure is 7.2%. If the mean values of all theinput parameters are used in the stability analysis,the calculated deterministic factor of safety is 1.4.The sensitivity analysis associated with these cal-culations shows that the factor of safety is moststrongly influenced by the dip of the slide plane,and least influence by the depth of water in thetension crack. This analysis was performed usingthe computer program ROCPLANE (Rocscience,2003).

6.7 Example Problem 6.1: planefailure—analysis and stabilization

Statement

A 12-m high rock slope has been excavated at aface angle of 60◦. The rock in which this cut hasbeen made contains persistent bedding planes thatdip at an angle of 35◦ into the excavation. The4.35-m deep tension crack is 4 m behind the crest,and is filled with water to a height of 3 m abovethe sliding surface (Figure 6.15). The strengthparameters of the sliding surface are as follows:

Cohesion, c = 25 kPaFriction angle, φ = 37◦

The unit weight of the rock is 26 kN/m3, and theunit weight of the water is 9.81 kN/m3.

Required

Assuming that a plane slope failure is the mostlikely type of instability, analyze the followingstability conditions.

H=

12m

4 m

z = 4.35 m

zw= 3 m

�f= 60°

�p= 35°T�T

Figure 6.15 Plane failure geometry for ExampleProblem 6.1.

Factor of safety calculations

(a) Calculate the factor of safety of the slope forthe conditions given in Figure 6.15.

(b) Determine the factor of safety if the tensioncrack were completely filled with water dueto run-off collecting on the crest of the slope.

(c) Determine the factor of safety if the slopewere completely drained.

(d) Determine the factor of safety if the cohesionwere to be reduced to zero due to excessivevibrations from nearby blasting operations,assuming that the slope was still completelydrained.

(e) Determine whether the 4.35-m deep tensioncrack is the critical depth (use Figure 6.6).

Slope reinforcement using rock bolts

(a) It is proposed that the drained slope withzero cohesion be reinforced by installing ten-sioned rock bolts anchored into sound rockbeneath the sliding plane. If the rock bolts areinstalled at right angles to the sliding plane,that is, ψT = 55◦, and the total load on theanchors per lineal meter of slope is 400 kN,calculate the factor of safety.

(b) Calculate the factor of safety if the bolts areinstalled at a flatter angle so that the ψT isdecreased from 55◦ to 20◦.

(c) If the working load for each bolt is 250 kN,suggest a bolt layout, that is, the number ofbolts per vertical row, and the horizontal and

Plane failure 151

vertical spacing between bolts to achieve abolt load of 400 kN/m of slope length.

Solution

Factor of safety calculations

(a) The factor of safety is calculated usingequations (6.4)–(6.10).

The weight W of the block is 1241 kN/m(equation 6.8), and the area A of the slid-ing plane is 13.34 m2/m (Equation 6.5).For water in the tension crack to depth,zw = 4.35 m, the values of the water forces U

and V acting on the block are 196.31 kN/mand 44.15 kN/m (equations (6.6) and (6.7)).

FS = {cA + (W cos ψp − U − V sin ψp) tan φ

}/ {

W sin ψp + V cos ψp}

= {25(13.34) + (1241.70 cos 35 − 196.31

−44.15 sin 35) tan 37}/ {1241.70 sin 35 + 44.15 cos 35}= 1.25

(In civil applications, this is usually a marginalfactor of safety for a permanent slope with a highconsequence of failure.)

(b) If the tension crack is completely filled withwater, that is, zw = 4.35 m, and the newfactor of safety is

FS = {333.50 + (1017.14 − 284.57 − 53.23)

× tan 37}/ {712.21 + 76.02}= 1.07

(This indicates that the slope is close to failure.)

(c) If the slope were drained so that there was nowater in the tension crack, that is, zw = 0,then the new factor of safety is

FS = 333.50 + 1017.14 tan 37712.21

= 1.54

0 1 2 3 4 5zw(m)

Fact

or o

f saf

ety

1.6

0.8

1.0

1.2

1.4

Figure 6.16 Plot of factor of safety against depth ofwater in tension crack for example Problem 6.1.

(This is usually an adequate factor of safety). Thefactor of safety values are plotted on Figure 6.16.

(d) If the slope is drained and the cohesion onthe sliding plane is reduced from 25 kPa tozero by blast vibrations, then the new factorof safety is

FS = 0 + 1017.14 tan 37712.21

= 1.08

The loss of cohesion reduces the factor ofsafety from 1.54 to 1.08 which illustrates thesensitivity of the slope to the cohesion on thesliding plane.

(e) Figure 6.6(a) shows that the critical tensioncrack depth is 4.32 m (i.e. z/H = 0.36)which is close to the position of the tensioncrack (i.e. 4.35/12 = 0.36).

Slope reinforcement with rock bolts

(a) The factor of safety of plane slope failurereinforced with rock bolts is calculated usingequation (6.22). In this case, the slope isdrained and the cohesion is zero, that is

c = U = V = 0 (6.40)

Therefore, for a reinforcement force of T of400 kN/m installed at a dip angle ψT of 55◦,

152 Plane failure

the factor of safety is

FS = [W cos ψp + T sin(ψT + ψp)] tan φ

W sin ψp − T cos(ψT + ψp)

= [1241.70 cos 35 + 400 sin(55 + 35)] tan 371241.70 sin 35 − 400 cos(55 + 35)

= 1067.90712.21

= 1.5

(b) If the bolts are installed at a flatter angle,ψT = 20◦, then the factor of safety is

FS = [1241.70 cos 35 + 400 sin(20 + 35)] tan 371241.70 sin 35 − 400 cos(20 + 35)

= 1013.38482.78

= 2.10

This shows the significant improvement thatcan be achieved by installing bolts at an angle

flatter than the normal to the sliding surface.The optimum angle is when

ψTopt = (φ − ψp) (see equation (6.23))

= (37◦ − 35◦)

= 2◦, and factor of safety = 2.41

(c) The rock bolt pattern should be laid out sothat the distribution of bolts on the slope isas even as possible. If four bolts are installedin each vertical row, the horizontal spa-cing S of the vertical rows is calculated asfollows:

S = TBn

T

(kN

kN/m

)(6.24)

= 240.4400

≈ 2.5 m

Chapter 7

Wedge failure

7.1 Introduction

The previous chapter was concerned with slopefailure resulting from sliding on a single planarsurface dipping into the excavation, and strik-ing parallel or nearly parallel to the slope face.It was stated that the plane failure analysis isvalid if the strike of the failure plane is within±20◦ of the strike of the slope face. This chapteris concerned with the failure of slopes contain-ing discontinuities striking obliquely to the slopeface where sliding of a wedge of rock takes placealong the line of intersection of two such planes(Figure 7.1). Wedge failures can occur over amuch wider range of geologic and geometric con-ditions than plane failures, so the study of wedgestability is an important component of rock slopeengineering. The analysis of wedges has beenextensively discussed in geotechnical literature,and the manual draws heavily upon the work ofGoodman (1964), Wittke (1965), Londe (1965),Londe et al. (1969, 1970), John (1970).

In this chapter, the structural geological con-ditions that result in the formation of a wedgeformed by two intersecting planes are defined,and the method of identifying wedges on thestereonet is illustrated. The stereonet defines theshape of the wedge, the orientation of the lineof intersection and the direction of sliding. Thisinformation can be used to assess the potential forthe wedge to slide from the cut face. The proced-ure is termed kinematic analysis, the purpose ofwhich is to identify potentially unstable wedges,although it does not provide precise informationon their factor of safety.

Figure 7.1 Typical wedge failure involving sliding ontwo persistent joints with line of intersection of jointsdaylighting at toe of rock face, and an upper planethat formed a tension crack (strong, volcanic rock onInterstate 5, near Grants Pass, Oregon).

The chapter presents design charts that can beused to find the factor of safety of wedges forwhich friction is the only component of the shearstrength, and there are no external forces such aswater pressures or bolting acting on the wedge.

154 Wedge failure

In addition, equations are presented that can beused to calculate the factor of safety of wedgeswhere the shear strength on the two slide planesis defined by cohesion and friction angle, andeach plane can have different shear strengths. Theanalysis can also incorporate water pressure.

The presence of a tension crack, and the influ-ence of external forces due to water pressures,tensioned anchors, seismic accelerations or bridgefoundations results in a significant increase inthe complexity of the equations. Appendix IIIpresents the complete solution for the wedgeanalysis.

7.2 Definition of wedge geometry

Typical wedge failures illustrated in Figures 7.1and 7.2 show the conditions that are normallyassumed for the analytical treatment of wedges.Figure 7.1 shows a cut slope where a wedge isformed by two continuous, planar discontinu-ities and the line of intersection of these twoplanes daylights just at the toe of the rock face.That is, the trend of the line of intersection andthe dip direction of the face are approximatelyequal. Furthermore, the plunge of the line of

intersection is about 50–55◦, while the frictionangle of these joints is in the range of 35–40◦.That is, the line of intersection dips steeper thanthe friction angle. These conditions meet the kin-ematic requirements for failure of the wedge.Figure 7.1 also illustrates how a slight change inthe site conditions would result in a stable slope.For example, if the line of intersection had beenslightly behind the face, or just one of the jointshad been discontinuous, then no failure wouldhave occurred.

The wedge in Figure 7.2 is formed by beddingon the left and a conjugate joint set on the right.As in Figure 7.1, the line of intersection daylightsin the slope face and failure occurred. However,in this wedge, sliding occurred almost entirely onthe bedding with the joint acting as a release sur-face. Therefore, the shear strength of the joint haslittle effect on stability.

The geometry of the wedge for analyzing thebasic mechanics of sliding is defined in Figure 7.3.Based on this geometry, the general conditions forwedge failure are as follows:

1 Two planes will always intersect in a line(Figure 7.3(a)). On the stereonet, the line of

Figure 7.2 Wedge formed bybedding (left) and a conjugatejoint set (right); slidingoccurred on bedding withjoints acting as a releasesurface (bedded shale, nearHelena, Montana).

Wedge failure 155

Line ofintersection

Face

Wedge

Note: The convention adopted in thisanalysis is that the flatter plane is alwaysreferred to as Plane A.

N

Plane B

Plane B�

Plane B

Face

Face

�i

Plane A

Plane A

�i�

Plane A�

Range of �i for sliding

Line of intersection

�fi �i �

�fi �i �

Direction ofsliding

N

�i

(a) (b)

(c) (d)

Figure 7.3 Geometric conditions for wedge failure: (a) pictorial view of wedge failure; (b) stereoplot showingthe orientation of the line of intersection, and the range of the plunge of the line of intersection ψi where failureis feasible; (c) view of slope at right angles to the line of intersection; (d) stereonet showing the range in thetrend of the line of intersection αi where wedge failure is feasible.

intersection is represented by the point wherethe two great circles of the planes intersect,and the orientation of the line is defined by itstrend (αi) and its plunge (ψi) (Figure 7.3(b)).

2 The plunge of the line of intersection must beflatter than the dip of the face, and steeperthan the average friction angle of the two slideplanes, that is ψfi > ψi > φ (Figure 7.3(b)and (c)). The inclination of the slope face ψfi ismeasured in the view at right angles to the lineof intersection. Note the ψfi would only be thesame as ψf , the true dip of the slope face, if the

dip direction of the line of intersection werethe same as the dip direction of the slope face.

3 The line of intersection must dip in a directionout of the face for sliding to be feasible; thepossible range in the trend of the line of inter-section is between αi and α′

i (Figure 7.3(d)).

In general, sliding may occur if the intersec-tion point between the two great circles of thesliding planes lies within the shaded area onFigure 7.3(b). That is, the stereonet will show ifwedge failure is kinematically feasible. However,

156 Wedge failure

the actual factor of safety of the wedge cannot bedetermined from the stereonet, because it dependson the details of the geometry of the wedge, theshear strength of each plane and water pressure,as described in the following sections.

The trend αi and plunge ψi of the line of inter-section of planes A and B can be determined onthe stereonet, or calculated using equations (7.1)and (7.2) as follows:

αi = tan−1(

tan ψA cos αA − tan ψB cos αB

tan ψB sin αB − tan ψA sin αA

)(7.1)

ψi = tan ψA cos(αA − αi) = tan ψB cos(αB − αi)

(7.2)

where αA and αB are the dip directions, andψA and ψB are the dips of the two planes.Equation (7.1) gives two solutions 180◦ apart; thecorrect value lies between αA and αB.

7.3 Analysis of wedge failure

The factor of safety of the wedge defined inFigure 7.3, assuming that sliding is resisted onlyby friction and that the friction angle φ is the samefor both planes, is given by

FS = (RA + RB) tan φ

W sin ψi(7.3)

where RA and RB are the normal reactionsprovided by planes A and B as illustrated inFigure 7.4, and the component of the weight act-ing down the line of intersection is (W sin ψi).The forces RA and RB are found by resolvingthem into components normal and parallel to thedirection along the line of intersection as follows:

RA sin(β − 1

2 ξ)

= RB sin(β + 1

2 ξ)

(7.4)

RA cos(β − 1

2 ξ)

+ RB cos(β + 1

2 ξ)

= W cos ψi (7.5)

½

½

RBRA

W cos �i

Plane B

Plane A

Face

Direction ofsliding

N

W cos �i

W sin �i

W

�i

(a) (b)

(c)

Figure 7.4 Resolution of forces to calculate factor of safety of wedge: (a) view of wedge looking at faceshowing definition of angles β and ξ, and reactions on sliding planes RA and RB; (b) stereonet showingmeasurement of angles β and ξ; (c) cross-section of wedge showing resolution of wedge weight W .

Wedge failure 157

where the angles ξ and β are defined inFigure 7.4(a). Angles ξ and β are measured onthe great circle containing the pole to the lineof intersection and the poles of the two slideplanes. In order to meet the conditions for equi-librium, the normal components of the reactionsare equal (equation (7.4)), and the sum of theparallel components equals the component ofthe weight acting down the line of intersection(equation (7.5)).

The values of RA and RB are found from equa-tions (7.4) and (7.5) by solving and adding asfollows:

RA + RB = W cos ψi sin β

sin(ξ/2)(7.6)

Hence

FS = sin β

sin(ξ/2)· tan φ

tan ψi(7.7)

In other words,

FSW = K FSP (7.8)

where FSW is the factor of safety of a wedge sup-ported by friction only, and FSP is the factor ofsafety of a plane failure in which the slide plane,with friction angle φ, dips at the same angle as theline of intersection ψi.

K is the wedge factor that, as shown byequation (7.7), depends upon the included angleof the wedge ξ and the angle of tilt β of the wedge.Values for the wedge factor K, for a range ofvalues of ξ and β are plotted in Figure 7.5.

The method of calculating the factor of safetyof wedges as discussed in this section is, of course,simplistic because it does not incorporate differ-ent friction angles and cohesions on the two slideplanes, or ground water pressures. When thesefactors are included in the analysis, the equationsbecome more complex. Rather than develop theseequations in terms of the angles ξ and β, whichcannot be measured directly in the field, the morecomplete analysis is presented in terms of directlymeasurable dips and dip directions. The follow-ing section gives equations for the factor of safety

of a wedge with cohesion and friction acting onthe slide planes and water pressure. The completeset of equations for stability analysis of a wedgeis shown in Appendix III; this analysis includesparameters to define the shape and dimensions ofthe wedge, different shear strengths on each slidesurface, water pressures and two external loads.

This section shows the important influence ofthe wedging action as the included angle of thewedge decreases below 90◦. The increase by afactor of 2 or 3 on the factor of safety determ-ined by plane failure analysis is of great practicalimportance because, as shown in Figure 7.5,the factor of safety of a wedge can be signific-antly greater than that of a plane failure. There-fore, where the structural features that are likelyto control the stability of a rock slope do notstrike parallel to the slope face, the stabilityanalysis should be carried out by means of three-dimensional methods discussed in this chapter.

7.4 Wedge analysis including cohesion,friction and water pressure

Section 7.3 discussed the geometric conditionsthat could result in a wedge failure, but thiskinematic analysis provides limited informationof the factor of safety because the dimensionsof the wedge were not considered. This sectiondescribes a method to calculate the factor of safetyof a wedge that incorporates the slope geometry,different shear strengths of the two slide planesand ground water (Hoek et al., 1973). However,the limitations of this analysis are that there isno tension crack, and no external forces such asbolting can be included.

Figure 7.6(a) shows the geometry and dimen-sions of the wedge that will be considered inthe following analysis. Note that the upper slopesurface in this analysis can be obliquely inclinedwith respect to the slope face, thereby removinga restriction that has been present in the stabil-ity analyses that have been discussed so far in thebook. The total height of the slope H is the differ-ence in vertical elevation between the upper andlower extremities of the line of intersection alongwhich sliding is assumed to occur. The water

158 Wedge failure

View along line ofintersection

Two dimensional plane failurewhen for< ½

10 20 3040

50

60

70

80

90

Angle of tilt

°

Wedge factor K = sin / sin (½)for> ½

Included angle of wedge — degrees

K

0

0.5

2.0

1.0

2.5

1.5

3.0

0 80 16040 12020 100 18060 140

Figure 7.5 Wedge factor K as a function of wedge geometry.

pressure distribution assumed for this analysis isbased upon the hypothesis that the wedge itselfis impermeable and that water enters the top ofthe wedge along lines of intersection 3 and 4 andleaks from the slope face along lines of intersec-tion 1 and 2. The resulting pressure distributionis shown in Figure 7.6(b)—the maximum pres-sure occurring along the line of intersection 5 andthe pressure being zero along lines 1, 2, 3 and 4.This is a triangular pressure distribution with themaximum value occurring at the mid-height ofthe slope, with the estimated maximum pressurebeing equal to (1

2γwH). This water pressure dis-tribution is believed to be representative of theextreme conditions that could occur during veryheavy rain and the slope is saturated.

The two planes on which sliding occurs aredesignated A and B, with plane A having the

shallower dip. The numbering of the five lines ofintersection of the four planes defining the wedgeis as follows:

Line 1 Intersection of plane A with the slopeface

Line 2 Intersection of plane B with the slopeface

Line 3 Intersection of plane A with upperslope surface

Line 4 Intersection of plane B with upperslope surface

Line 5 Intersection of planes A and B

It is assumed that sliding of the wedge alwaystakes place along the line of intersectionnumbered 5, and its factor of safety is given by

Wedge failure 159

Plane A

Upper slope surface,which can be obliquelyinclined with respectto the face

Plane B

Face

H

½ H

Assumed water pressuredistribution

1

53

4

2

(a)

(b)

Figure 7.6 Geometry of wedge used for stabilityanalysis including the influence of friction andcohesion, and of water pressure on the slide surfaces:(a) pictorial view of wedge showing the numbering ofthe intersection lines and planes; (b) view normal tothe line of intersection (5) showing wedge height andwater pressure distribution.

(Hoek et al., 1973):

FS = 3γrH

(cAX + cBY) +(

A − γw

2γrX

)tan φA

+(

B − γw

2γrY

)tan φB (7.9)

where cA and cB are the cohesive strengths, andφA and φB are the angles of friction respectively onplanes A and B, γr is the unit weight of the rock,

γw is the unit weight of the water, H is the totalheight of the wedge. The dimensionless factorsX, Y , A and B depend upon the geometry of thewedge.

The values of parameters X, Y , A and B aregiven in equations (7.10)–(7.13):

X = sin θ24

sin θ45 cos θ2.na(7.10)

Y = sin θ13

sin θ35 cos θ1.nb

(7.11)

A = cos ψa − cos ψb cos θna.nb

sin ψ5 sin2θna.nb

(7.12)

B = cos ψb − cos ψa cos θna.nb

sin ψ5 sin2θna.nb

(7.13)

where ψa and ψb are the dips of planes A andB respectively and ψ5 is the dip of the lineof intersection, line 5. The angles required forthe solution of these equations can be measuredmost conveniently on a stereoplot that defines thegeometry of the wedge and the slope (Figure 7.7).

The application of the equations discussed inthis section is illustrated in the following example,using the parameters shown in Table 7.1.

The total height of the wedge H is 40 m, theunit weight of the rock is 25 kN/m3, and the unitweight of the water 9.81 kN/m3.

The stereoplot of the great circles represent-ing the four planes involved in this example ispresented in Figure 7.7, and all the angles requiredfor the solution of equations (7.10)–(7.13) aremarked in this figure.

Determination of the factor of safety is mostconveniently carried out on a calculation sheetsuch as that presented on Table 7.2. Settingthe calculations out in this manner not onlyenables the user to check all the data, but it alsoshows how each variable contributes to the over-all factor of safety. Hence, if it is required to checkthe influence of the cohesion on both planes fall-ing to zero, this can be done by setting the twogroups containing the cohesion values cA and cBto zero, giving a factor of safety of 0.62. Altern-atively, the effect of drainage can be checked by

160 Wedge failure

Great circleplane B

Great circleplane A

Pole ofplane A

Na

Great circleof face

Great circle ofupper surface

2 1

Pole ofplane B

Nb

�1.nb

�2.nb

�24

�13

�na.nb

N

Direction of sliding

�35

�45

�5

5

3

4

Figure 7.7 Stereoplot of data required for wedge stability analysis.

Table 7.1 Parameters defining properties ofwedge

Plane Dip Dip Propertiesdirection

A 45 105 φA = 20◦, cA = 24 kPaB 70 235 φB = 30◦, cB = 48 kPaSlope face 65 185Uppersurface

12 195

varying the water density to simulate the effectof reducing the water pressure. In this example,the factor of safety is 1.98 when the slope iscompletely drained.

As has been emphasized in previous chapters,this ability to check the sensitivity of the factor ofsafety to changes in material properties or water

pressures is important because the value of theseparameters are difficult to define precisely.

7.5 Wedge stability charts for friction only

A rapid check of the stability of a wedge can bemade if the slope is drained and there is zero cohe-sion on both the slide planes A and B. Under theseconditions, equation (7.9) reduces to

FS = A tan φA + B tan φB (7.14)

The dimensionless factors A and B are found todepend upon the dips and dip directions of thetwo planes. The values of these two factors havebeen computed for a range of wedge geometries,and the results are presented as a series of charts(Figures 7.8–7.15).

Wedge failure 161

Table 7.2 Wedge stability calculation sheet

Input data Function value Calculated values

ψa = 45◦ cos ψa = 0.707A = cos ψa − cos ψb cos θna.nb

sin ψ5 sin2 θna.nb

= 0.707 + 0.342 × 0.1910.518 × 0.964

= 1.548ψb = 70◦ cos ψb = 0.342ψ5 = 31.2◦ sin ψ5 = 0.518ψna.nb = 101◦ cos ψna.nb = −0.191

B = cos ψb − cos ψa cos θna.nb

sin ψ5 sin2 θna.nb

= 0.342 + 0.707 × 0.1910.518 × 0.964

= 0.956sin ψna.nb = 0.982

θ24 = 65◦ sin θ24 = 0.906X = sin θ24

sin θ45 cos θ2.na= 0.906

0.423 × 0.643= 3.336θ45 = 25◦ sin θ45 = 0.423

θ2.na = 50◦ cos θ2.na = 0.643

θ13 = 62◦ sin θ13 = 0.883Y = sin θ13

sin θ35 cos θ1.nb

= 0.8830.515 × 0.5

= 3.429θ35 = 31◦ sin θ35 = 0.515θ1.nb = 60◦ cos θ1.nb = 0.500

φA = 30◦ tan φA = 0.577

FS = 3γrH

(cAX + cBY) +(

A − γw

2γrX

)tan φA +

(B − γw

2γrY

)tan φB

φB = 20◦ tan φB = 0.364γr = 25 kN/m3

γw/2γr = 0.196γw = 9.81 kN/m3 3cA/γH = 0.072

FS = 0.241 + 0.494 + 0.893 − 0.376 + 0.348 − 0.244 = 1.36cA = 24 kPa 3cB/γH = 0.144cB = 48 kPaH = 40 m

Note that the factor of safety calculated fromequation (7.14) is independent of the slope height,the angle of the slope face and the inclination ofthe upper slope surface. This rather surprising res-ult arises because the weight of the wedge occursin both the numerator and denominator of thefactor of safety equation and, for the friction onlycase, this term cancels out, leaving a dimension-less ratio which defines the factor of safety. Asdiscussed in Section 1.4, the factor of safety ofa plane failure is also independent of the slopedimensions if the slope is drained and the cohe-sion is zero. This simplification is very useful inthat it enables the user of these charts to carry outa quick check on the stability of a slope based onthe dips and dip directions of the two discontinu-ities that form the slide planes of the wedge. Anexample of such an analysis is presented later inthis chapter.

Many trial calculations have shown that awedge having a factor of safety in excess of 2.0,as obtained from the friction-only stability charts,

is unlikely to fail under even the most severecombination of conditions to which the slope islikely to be subjected. Consider the example dis-cussed in Section 7.4 in which the factor of safetyfor the worst conditions (zero cohesion and max-imum water pressure) is 0.62. This is 50% of thefactor of safety of 1.24 for the friction only case.Hence, had the factor of safety for the frictiononly case been 2.0, the factor of safety for theworst conditions would have been 1.0, assumingthat the ratio of the factors of safety for the twocases remains constant.

Based on such trial calculations, it is suggestedthat the friction-only stability charts can be usedto define those slopes that are adequately stableand can be ignored in subsequent analyses. Thatis, slopes having a factor of safety in excess of2.0. Slopes with a factor of safety, based uponfriction only, of less than 2.0 must be regarded aspotentially unstable and require further detailedexamination as discussed in Section 7.6 andAppendix III.

162 Wedge failure

Plane A

Note: The flatter of the two planes is always called plane A.

Factor of safety

Plane B

FS = A tan�A+ B tan�B

A/B chart—dip difference 0°

5.0

4.5

4.0

3.5

3.0

Rat

io A

or

B

2.5 Dip of plane 20

3040

5060

7080

2.0

1.5

1.0

0.5

00 20 40 60 80 100 120 140 160

360 340 320 300 280 260

Difference in dip direction—degrees

240 220 200180

Figure 7.8 Wedge stability charts for friction only: A and B charts for a dip difference of 0◦.

Wedge failure 163

20

30

40

50 6070

80

Dip

of p

lane

A

A c

hart

—di

p di

ffere

nce

10°

Ratio A

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0

Diff

eren

ce in

dip

dire

ctio

n—de

gree

s

260

240

220

200

Dip

of p

lane

B

B c

hart

—di

p di

ffere

nce

10°

Ratio B

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0

Diff

eren

ce in

dip

dire

ctio

n—de

gree

s

260

240

220

200

20

20

40

3040 50 60 70

90

80

30

Figu

re7.

9W

edge

stab

ility

char

tsfo

rfr

icti

onon

ly:A

and

Bch

arts

for

adi

pdi

ffer

ence

of10

◦ .

164 Wedge failure

Dip

of p

lane

B

B c

hart

—di

p di

ffere

nce

20°

Ratio B

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

Dip

of p

lane

A

A c

hart

—di

p di

ffere

nce

20°

Ratio A

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

9080

6050

4030

304050

20

3040

5060

70

70

Figu

re7.

10W

edge

stab

ility

char

tsfo

rfr

icti

onon

ly:A

and

Bch

arts

for

adi

pdi

ffer

ence

of20

◦ .

Wedge failure 165

Dip

of p

lane

A

A c

hart

—di

p di

ffere

nce

30°

Ratio A

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

Dip

of p

lane

B

B c

hart

—di

p di

ffere

nce

30°

Ratio B

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

908070605040

6050

40

605040

30

20

Figu

re7.

11W

edge

stab

ility

char

tsfo

rfr

icti

onon

ly:A

and

Bch

arts

for

adi

pdi

ffer

ence

of30

◦ .

166 Wedge failure

Dip

of p

lane

A

A c

hart

—di

p di

ffere

nce

40°

Ratio A

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

Dip

of p

lane

B

B c

hart

—di

p di

ffere

nce

40°

Ratio B

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0

Diff

eren

ce in

dip

dire

ctio

n—de

gree

s

260

240

220

200

5040

30

20

9080706050

6050

Figu

re7.

12W

edge

stab

ility

char

tsfo

rfr

icti

onon

ly:A

and

Bch

arts

for

adi

pdi

ffer

ence

of40

◦ .

Wedge failure 167

Dip

of p

lane

A

A c

hart

—di

p di

ffere

nce

50°

Ratio A

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

Dip

of p

lane

B

B c

hart

—di

p di

ffere

nce

50°

Ratio B

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

40

30

20

6070

8090

Figu

re7.

13W

edge

stab

ility

char

tsfo

rfr

icti

onon

ly:A

and

Bch

arts

for

dip

diff

eren

ceof

50◦ .

168 Wedge failure

Dip

of p

lane

B

B c

hart

—di

p di

ffere

nce

60°

Ratio B

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

Dip

of p

lane

A

A c

hart

—di

p di

ffere

nce

60°

Ratio A

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0

Diff

eren

ce in

dip

dire

ctio

n—de

gree

s

260

240

220

200

7080 90

20

30

Figu

re7.

14W

edge

stab

ility

char

tsfo

rfr

icti

onon

ly:A

and

Bch

arts

for

dip

diff

eren

ceof

60◦ .

Wedge failure 169

Dip

of p

lane

A

A c

hart

—di

p di

ffere

nce

70°

Ratio A

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0D

iffer

ence

in d

ip d

irect

ion—

degr

ees

260

240

220

200

Dip

of p

lane

B

B c

hart

—di

p di

ffere

nce

70°

Ratio B

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5 0

020

4060

8010

012

014

016

018

036

034

032

030

028

0

Diff

eren

ce in

dip

dire

ctio

n—de

gree

s

260

240

220

200

20

80 90

Figu

re7.

15W

edge

stab

ility

char

tsfo

rfr

icti

onon

ly:A

and

Bch

arts

for

dip

diff

eren

ceof

70◦ .

170 Wedge failure

Table 7.3 Wedge stability analysis for friction only

Dip Dip Friction(degrees) direction angle

Plane A 40 165 35Plane B 70 285 20Differences 30 120

In the design of the cut slopes on many projects,it will be found that these friction-only stabilitycharts provide all the information that is requiredfor preliminary design and planning stages of aslope project. This information will help identifypotentially unstable wedges before excavation ofthe slope is started. During construction, thecharts can be used to make a rapid check of sta-bility conditions when, for example, faces arebeing mapped as the excavation is proceeding anddecisions are required on the need for support. Ifthe factor of safety is less than 2.0, the detailedanalysis can be used to design a bolting pattern(see Appendix III).

The following example illustrates the use ofthe friction-only charts, in which Plane A has theflatter dip (Table 7.3).

The first step in the analysis is to calculatethe absolute values of the difference in the dipangles, and the difference in the dip directionangles (third line in Table 7.3). For a dip differ-ence of 30◦, the values of ratios A and B anddetermined from the two charts on Figure 7.11for a difference in dip direction of 120◦. The val-ues of A and B are 1.5 and 0.7 respectively, andsubstitution in equation (7.14) gives the factor ofsafety of 1.30.

The values of A and B give a direct indication ofthe contribution which each of the planes makesto the total factor of safety.

7.5.1 Example of wedge analysis usingfriction-only charts

During the route location study for a pro-posed highway, the layout engineer has requestedguidance on the maximum safe angles that maybe used for the design of the slopes. Extensivegeological mapping of outcrops, together with

Table 7.4 Orientations of discontinuity sets in wedgeanalysis example

Discontinuityset

Dip Dip direction

1 66 ± 2 298 ± 22 68 ± 6 320 ± 153 60 ± 16 360 ± 104 58 ± 6 76 ± 65 54 ± 4 118 ± 2

core logging, identified five sets of discontinuit-ies in the rock mass through which the road willpass. The dips and dip directions of these discon-tinuities are shown in Table 7.4, together with themeasured variation in these measurements.

Note that, because the mapping covers theentire alignment that extends over several kilo-meters, the scatter in the dip and dip directionmeasurements can be taken into account in theanalysis.

Figure 7.16 shows the pole locations for thesefive sets of discontinuities. Also shown on thisfigure are the extent of the scatter in the polemeasurements, and the great circles correspond-ing to the average pole positions. The dashedfigure surrounding the great circle intersectionsis obtained by rotating the stereoplot to find theextent to which the intersection point is influ-enced by the scatter around the pole points. Theintersection of great circles 2 and 5 has beenexcluded from the dashed figure because it definesa line of intersection dipping flatter than 20◦,which is less than the estimated angle of friction.

The factors of safety for each of the discon-tinuity intersections is determined from the wedgecharts, assuming a friction angle of 30◦ (someinterpolation is necessary), and the values aregiven in the circles over the intersection points.Because all of the planes are steep, some of thefactors of safety are dangerously low. Since it islikely that slopes with a factor of safety of lessthan 1.0 will fail as the slope is excavated, theonly practical solution is to cut the slopes at anangle that is coincident with the dip of the line ofintersection. For marginally stable wedges withfactors of safety between 1.0 and 2.0, a detailed

Wedge failure 171

5

4

32

1

.3.4

.3

1.5

.8

.5

1.0

2.4

3

1

45

2

.5

N

> 5

Notes:a. Black triangles mark most likely position of poles of five sets of discontinuities present in rock mass.b. Shaded area surrounding pole position defines extent of scatter in measurements.c. Factors of safety for each combination of discontinuities are shown in circle over corresponding intersection of great circles.d. Dashed line surrounds area of potential instability (FS < 2).

Figure 7.16 Stereoplot of geologicaldata for the preliminary design ofhighway slopes.

analysis can be carried out to determine the bolt-ing force required to increase the factor of safetyto an acceptable level.

The stereoplot in Figure 7.17(a) can be used tofind the maximum safe slope angle for the slopesof any dip direction. This stereographic analysisinvolves positioning the great circle representingthe slope face for a particular dip direction insuch a way that the unstable region (shaded)is avoided. The maximum safe slope angles aremarked around the perimeter of this figure andtheir positions correspond to the orientation ofthe slope face (Figure 7.17(a)). For example, if athrough-cut for a highway is planned in this rock,

the slope on the south side of the cut would be anangle of 30◦ and the slope on the north side wouldbe at 85◦ (Figure 7.17(b)).

7.6 Comprehensive wedge analysis

7.6.1 Data for comprehensive analysis

If the friction-only wedge stability charts showthat the factor of safety is less than 2.0, then acomprehensive stability analysis may be requiredto calculate, for example, the bolting force toachieve a required factor of safety. This analysistakes into account the dimensions and shape of

172 Wedge failure

N

NSouth face

North face

Unstable

30°

30°

32°

30°

36°

44°

65°85°

72°

62°

60°

36°

(a)

(b)

A�

A

8060

40

20

20 80

Plan

Highway

Section A–A�

30° 85°Highway

Figure 7.17 Slope design basedon wedge stability analysis:(a) stereoplot of great circlesrepresenting stable slopes in arock mass containing the fivesets of discontinuities defined inFigure 7.16; (b) stable cut slopeangles based on wedge stabilityanalysis shown in (a).

the wedge, different cohesion and friction angleson each slide plane, water pressure and a numberof external forces. The external forces that mayact on the wedge include earthquake groundmotion, tensioned bolts and possible loads gen-erated by structures that may include a bridgefooting or building located on the wedge.

Figure 7.18 shows the features of the slopeinvolved in the comprehensive wedge analysis,and a complete list of the equations used in theanalysis is included Appendix III. A fundamental

assumption in the analysis is that all the forcesact through the center of gravity of the wedgeso no moments are generated. The following is adescription of the components of the analysis:

• Wedge shape—the shape of the wedge isdefined by five surfaces: the two slide planes(1 and 2) with their line of intersectiondaylighting in the face, the upper slope (3)and face (4), and a tension crack (5)(Figure 7.18(a)). The orientations of these

Wedge failure 173

1 5

2

3

4L

H1


(a) (b)

p

V

U1,2

(c)

Tensioned anchor

�average

�T(opt)

�i

N


�i

�T(opt)

Figure 7.18 Comprehensive wedge analysis: (a) dimensions and surfaces defining size and shape of wedge;(b) water pressures acting in tension crack and along line of intersection; (c) optimum anchor orientation forreinforcement of a wedge.

surfaces are each defined by their dip and dipdirection. The range of orientations that theanalysis can accommodate include an over-hanging face, different dip directions for theupper slope and the face, and a tension crackdipping either towards or away from the face.

• Wedge dimensions—the dimensions of thewedge are defined by the two dimensions H

and L (Figure 7.18(a)). H is the vertical heightbetween the point where the line of intersec-tion daylights on the face and the intersectionof plane 1 with the crest of the slope, L isthe distance measured along plane 1 betweenthe crest of the slope face (4) and the tensioncrack (5).

• Wedge weight—the orientations of the fiveplanes, and the two dimensions can be usedto calculate the volume of the wedge, and theweight is determined from the unit weight ofthe rock (γr).

• Water pressures—if it is assumed that the ten-sion crack (5) is filled with water and thatthe water discharges to the atmosphere whereplanes 1 and 2 intersect the face (plane 4), thentriangular water pressures act on planes 1, 2and 5 (Figure 7.18(b)). The water pressure p

at the base of the tension crack (and the topof the line of intersection) is equal to (h5γw),where h5 is the average vertical depth belowthe top of the tension crack. The water forces

174 Wedge failure

U1, U2 and V are calculated by integrating thepressures over the areas of planes 1, 2 and 5,respectively.

• Shear strengths—the slide planes (1 and 2) canhave different shear strengths defined by thecohesion (c) and friction angle (φ). The shearresistance is calculated by multiplying thecohesion by the area of slide plane, and addingthe product of the effective normal stress andthe friction angle. The normal stresses arefound by resolving the wedge weight in dir-ections normal to the each slide plane (seeequations (7.3)–(7.5)).

• External forces—external forces acting on thewedge are defined by their magnitude andorientation (plunge ψ and trend α). The equa-tions listed in Appendix III can accommodatea total of two external forces; if there arethree or more forces, the vectors are addedas necessary. One external force that maybe included in the analysis is the pseudo-static force used to simulate seismic groundmotion (see Section 6.5.4). The horizontalcomponent of this force would act in thesame direction as the line of intersection ofplanes 1 and 2.

• Bolting forces—if tensioned anchors areinstalled to stabilize the wedge, they are con-sidered to be an external force. The ori ent-ation of the anchors can be optimized tominimize the anchor force required to pro-duce a specified factor of safety. The optimumanchor plunge ψT(opt) and trend αT(opt), withrespect to the line of intersection (ψi/αi), areas follows (Figure 7.18(c)):

ψT(opt) = (φaverage − ψi) (7.15)

and

αT(opt) = (180 + αi) for αT(opt) ≤ 360(7.16)

where φaverage is the average friction angle ofthe two slide planes.

7.6.2 Computer programs for comprehensiveanalysis

Appendix III lists equations that can be used tocarry out a comprehensive stability analysis ofa wedge using the input parameters discussedin Section 7.6.1. These equations were origin-ally developed by Dr John Bray and are includedin the third edition of Rock Slope Engineering(1981). This method of analysis has been used ina number of computer programs that allow rapidand reliable analysis of wedge stability. However,there is a limitation to this analysis that shouldbe noted:

Wedge geometry. The analysis procedure isto calculate the dimensions of a wedge thatextends from the face to the point where planes1, 2 and 3 intersect. The next step is to calculatethe dimensions of a second wedge formed by slid-ing planes 1 and 2, the upper slope (plane 3) andthe tension crack (plane 5). The dimensions of thewedge in front of the tension crack are then foundby subtracting the dimensions of the wedge infront of the tension crack from the overall wedge(see equations (III.54) to (III.57), Appendix III).Prior to performing the subtractions, the programtests to see if a wedge is formed and a ten-sion crack is valid (equations (III.48) to (III.53),Appendix III). The program will terminate if thedip of the upper slope (plane 3) is greater than thedip of the line of intersection of planes 1 and 2,or if the tension crack is located beyond the pointwhere the point where planes 1, 2 and 3 intersect.

While these tests are mathematically valid, theydo not allow for a common geometric conditionthat may exist in steep mountainous terrain. Thatis, if plane 3 is steeper than the line of inter-section of planes 1 and 2, a wedge made upof five planes can still be formed if a tensioncrack is located behind the slope face to createa valid plane 5. Where this physical conditionexists in the field, an alternative method of ana-lysis is to use Key Block Theory in which the shapeand stability condition of removal wedges canbe completely defined (Goodman and Shi, 1985;Kielhorn, 1999; PanTechnica, 2002).

Wedge failure 175

7.6.3 Example of comprehensive wedgeanalysis

The following is an example of the comprehensivestability analysis of a wedge (Rocscience, 2001).Consider the wedge formed by joint sets 3 and5 in Figure 7.16. This has a friction-only factorof safety of 1.0, so water pressures and seismicground motion may result in instability, depend-ing on the cohesions on the slide surfaces. Thisanalysis shows the bolting force required to raisethe factor of safety to 1.5, based on the inputparameters shown in Table 7.5.

The other input parameters are as follows:

Wedge height, H1 = 28 m;Distance of tensioncrack from crestmeasured alongplane 3, L

= 9 m;

Rock density, γr = 26 kN/m3;Water density, γw = 10.0 kN/m3; andSeismic coefficient(horizontal), kH

= 0.1.

The stability analysis of the wedge using theseparameters gave the following results:

(i) Dry, static, c3 = c5 = 0 kPa, FS = 1.05T = 0,

(ii) zw = 50%, static, FS = 0.93c3 = c5 = 0 kPa, T = 0

(iii) zw = 50%, kH = 0.1, FS = 0.76c3 = c5 = 0 kPa, T = 0

(iv) T = 27.2 MN, ψT = −8◦, FS = 1.5αT = 242◦, zw = 50%,kH = 0.1, c3 = c5 = 0 kPa

(v) T = 38.3 MN, ψT = 10◦, FS = 1.5αT = 200◦

(vi) zw = 50%, kH = 0.1, FS = 1.49c3 = c5 = 50 kPa, T = 0

Table 7.5 Orientation of planes forming wedge

Plane Dip Dip Shear strengthdirection

3 60 360 φ3 = 30◦, c3 = 50 kPa5 54 118 φ5 = 30◦, c5 = 50 kPaSlope face 76 060Uppersurface

15 070

Tensioncrack

80 060

Case (i) corresponds to the simplified analysisusing the friction-only charts for a dry, staticslope that gave a factor of safety of 1.1. Withthe tension crack half-filled with water and theseismic coefficient applied, the factor of safetydrops to 0.76 (Case (iii)). For the load condi-tions in Case (iii), it is necessary to install abolting force of 27.2 MN in order to raise thefactor of safety to 1.5 (Case (iv)). In Case (iv),the bolting force is installed in the opposite dir-ection to the line of intersection (αT = 242◦) andat an angle of 8◦ above the horizontal (ψT =−8◦)—this orientation is optimal as defined byequations (7.15) and (7.16). If the bolts areinstalled at 10◦ below the horizontal to facilit-ate grouting (ψT = 10◦) and with a trend of200◦, then the required bolting force to producea factor of safety of 1.5 increases to 38.3 MN(Case (v)). This result illustrates the advantageof installing anchors at, or close to, the optimalorientation.

Case (vi) illustrates that a small amount ofcohesion can be most effective in improvingthe factor of safety because the cohesion actsover large surface areas to produce a significantresisting force.

Chapter 8

Circular failure

8.1 Introduction

Although this book is concerned primarily withthe stability of rock slopes containing well-defined sets of discontinuities, it is also necessaryto design cuts in weak materials such as highlyweathered or closely fractured rock, and rockfills. In such materials, failure occurs along a sur-face that approaches a circular shape (Figure 8.1),and this chapter is devoted to a discussion on thestability analysis of these materials.

In a review of the historical development ofslope stability theories, Golder (1972) traced thesubject back almost 300 years. Much of thedevelopment of circular failure analysis meth-ods was carried out in the 1950s and 1960s,and these techniques have since been used toprepare computer programs that have the ver-satility to accommodate a wide range of geo-logic, geometric, ground water and externalloading conditions. This chapter discusses theprinciples of the theoretical work, and demon-strates their application in design charts and inthe results of computer analyses. During thepast half century, a vast body of literature onthe subject of circular failure has accumulated,and no attempt will be made to summarize thematerial in this chapter. Standard soil mechan-ics text books such as those by Taylor (1937),Terzaghi (1943) and Lambe and Whitman(1969), and papers by Skempton (1948), Bishop(1955), Janbu (1954), Morgenstern and Price(1965), Nonveiller (1965), Peck (1967), Spencer(1967, 1969) and Duncan (1996) all containexcellent discussions on the stability of soil slopes.

The approach adopted in this chapter is topresent a series of slope stability charts for circu-lar failure. These charts enable the user to carryout a rapid check on the factor of safety of aslope, or upon the sensitivity of the factor of safetyto changes in ground water conditions, slopeangle and material strength properties. Thesecharts should only be used for the analysis ofcircular failure in slope materials that are homo-genous and where the conditions apply that wereassumed in deriving the charts (see Section 8.4).More comprehensive methods of analysis arepresented in Section 8.6. These methods can beused, for example, where the material propertiesvary within the slope, or where part of the slidesurface is at a soil/rock interface and the shape ofthe slide surface differs significantly from a simplecircular arc.

This chapter primarily addresses the stabilityof slopes in two dimensions, and assumes thatthe slope can be modeled as a unit slice throughan infinitely long slope, under plane-strain condi-tions. Section 8.6.5 discusses three-dimensionalcircular failure analysis, and Section 10.3.1 dis-cusses the influence of the radius of curvature ofthe slope on stability.

8.2 Conditions for circular failure andmethods of analysis

In the previous chapters, it has been assumedthat the failure of rock slopes is controlled bygeological features such as bedding planes andjoints that divide the rock into a discontinuous

Circular failure 177

Figure 8.1 Circularfailure in highlyweathered, granitic rock(on Highway 1, nearDevil’s Slide, Pacifica,California).

mass. Under these conditions, one or more ofthe discontinuities normally defines the slide sur-face. However, in the case of a closely fracturedor highly weathered rock, a strongly definedstructural pattern no longer exists, and the slidesurface is free to find the line of least resistancethrough the slope. Observations of slope failuresin these materials suggest that this slide surfacegenerally takes the form of a circle, and moststability theories are based upon this observa-tion. Figure 8.1 shows a typical circular failurein a highly weathered rock slope above a high-way. The conditions under which circular failurewill occur arise when the individual particles in asoil or rock mass are very small compared withthe size of the slope. Hence, broken rock in afill will tend to behave as a “soil” and fail ina circular mode when the slope dimensions aresubstantially greater than the dimensions of therock fragments. Similarly, soil consisting of sand,silt and smaller particle sizes will exhibit circularslide surfaces, even in slopes only a few metersin height. Highly altered and weathered rocks, aswell as rock with closely spaced, randomly ori-ented discontinuities such as some rapidly cooled

basalts, will also tend to fail in this manner. It isappropriate to design slopes in these materials onthe assumption that a circular failure process willdevelop.

8.2.1 Shape of slide surface

The actual shape of the “circular” slide surfaceis influenced by the geological conditions in theslope. For example, in a homogenous weak orweathered rock mass, or a rock fill, the failureis likely to form as a shallow, large radius sur-face extending from a tension crack close behindthe crest to the toe of the slope (Figure 8.2(a)).This contrasts with failures in high cohesion, lowfriction materials such as clays where the surfacemay be deeper with a smaller radius that may exitbeyond the toe of the slope. Figure 8.2(b) showsan example of conditions in which the shape ofthe slide surface is modified by the slope geology.Here the circular surface in the upper, weatheredrock is truncated by the shallow dipping, strongerrock near the base. Stability analyses of bothtypes of surface can be carried out using circularfailure methods, although for the latter case it is

178 Circular failure

R

Wi

Ai

(a)

(b)

Forces acting on slice, i

�i�b

�i –1

Eihi

hi –1

Ei –1

�i�Ai

S =(ci +�i tan�i) Ai

FS

Verticalslice

Circular slidingsurface

Non-circular sliding surface

Figure 8.2 The shape oftypical sliding surfaces:(a) large radius circularsurface in homogeneous,weak material, with the detailof forces on slice;(b) non-circular surface inweak, surficial material withstronger rock at base.

necessary to use a procedure that allows the shapeof the surface to be defined.

For each combination of slope parameters therewill be a slide surface for which the factor ofsafety is a minimum—this is usually termed the“critical surface.” The procedure to find thecritical surface is to run a large number of ana-lyses in which the center co-ordinates and theradius of the circle are varied until the surfacewith the lowest factor of safety is found. Thisis an essential part of circular slope stabilityanalysis.

8.2.2 Stability analysis procedure

The stability analysis of circular failure is carriedout using the limit equilibrium procedure similarto that described in earlier chapters for plane andwedge failures. This procedure involves compar-ing the available shear strength along the slidingsurface with the force required to maintain theslope in equilibrium.

The application of this procedure to circu-lar failures involves division of the slope into aseries of slices that are usually vertical, but may


be inclined to coincide with certain geologicalfeatures. The base of each slice is inclined at angleψb and has an area A. In the simplest case, theforces acting on the base of each slice are the shearresistance S due to the shear strength of the rock(cohesion c; friction angle φ), and forces E (dipangle ψ; height h above base) acting on the sidesof the slice (see detail Figure 8.2(a)).

The analysis procedure is to consider equilib-rium conditions slice by slice, and if a conditionof equilibrium is satisfied for each slice, thenit is also satisfied for the entire sliding mass.The number of equations of equilibrium avail-able depends on the number of slices N, andthe number of equilibrium conditions that areused. The number of equations available is 2N

if only force equilibrium is satisfied, and 3N ifboth force and moment equilibrium are satis-fied. If only force equilibrium is satisfied, thenumber of unknowns is (3N − 1), while, ifboth force and moment equilibria are satisfied,the number of unknowns is (5N − 2). Usuallybetween 10 and 40 slices are required to realist-ically model the slope, and therefore, the numberof unknowns exceeds the number of equations.The excess of unknowns over equations is (N −1)for force equilibrium analysis, and (2N − 2) foranalyses that satisfy all conditions of equilib-rium. Thus, the analyses are statically indeterm-inate and assumptions are required to make upthe imbalance between equations and unknowns(Duncan, 1996).

The various limit equilibrium analysis pro-cedures either make assumptions to make upthe balance between known and unknowns, orthey do not satisfy all the conditions of equilib-rium. For example, the Spencer Method assumesthat the inclination of the side forces is thesame for every slice, while the Fellenius andBishop methods do not satisfy all conditions ofequilibrium.

The factor of safety of the circular failure basedon limit equilibrium analysis is defined as

FS = shear strength available to resist sliding (c + σ tan φ)

shear stress required for equilibrium on slip surface(τe)(8.1)

and rearranging this equation, we have

τe = c + σ tan φ

FS(8.2)

The method of solution for the factor of safetyis to use an iterative process in which an initialestimate is made for FS, and this is refined witheach iteration.

The influence of various normal stress distri-butions upon the factor of safety of soil slopeshas been examined by Frohlich (1955) who foundthat a lower bound for all factors of safety thatsatisfy statics is given by the assumption that thenormal stress is concentrated at a single pointon the slide surface. Similarly, the upper boundis obtained by assuming that the normal load isconcentrated at the two ends of the slide surface.

The unreal nature of these stress distributionsis of no consequence since the object of the exer-cise, up to this point, is simply to determinethe extremes between which the actual factor ofsafety of the slope must lie. In an example con-sidered by Lambe and Whitman (1969), the upperand lower bounds for the factor of safety of aparticular slope corresponded to 1.62 and 1.27,respectively. Analysis of the same problem byBishop’s simplified method of slices gives a factorof safety of 1.30, which suggests that the actualfactor of safety may lie reasonably close to thelower bound solution.

Further evidence that the lower bound solutionis also a meaningful practical solution is providedby an examination of the analysis that assumedthe slide surface has the form of a logarithmicspiral (Spencer, 1969). In this case, the factor ofsafety is independent of the normal stress distri-bution, and the upper and lower bounds coincide.Taylor (1937) compared the results from a num-ber of logarithmic spiral analyses with results oflower bound solutions1 and found that the dif-ference is negligible. Based on this comparison,

1 The lower bound solution discussed in this chapter is usu-ally known as the Friction Circle Method and was used byTaylor (1937) for the derivation of his stability charts.


Taylor concluded that the lower bound solutionprovides a value of the factor of safety that issufficiently accurate for most practical problemsinvolving simple circular failure of homogeneousslopes.

The basic principles of these methods of ana-lyses are discussed in Section 8.6.

8.3 Derivation of circular failure charts

This section describes the use of a series of chartsthat can be used to determine rapidly the factor ofsafety of circular failures. These charts have beendeveloped by running many thousands of circularanalyses from which a number of dimensionlessparameters were derived that relate the factor ofsafety to the material unit weight, friction angleand cohesion, and the slope height and face angle.It has been found that these charts give a reliableestimate for the factor of safety, provided that theconditions in the slope meet the assumptions usedin developing the charts. In fact, the accuracy incalculating the factor of safety from the charts isusually greater than the accuracy in determiningthe shear strength of the rock mass.

Use of the stability charts presented in thischapter requires that the conditions in the slopemeet the following assumptions:

(a) The material forming the slope is homogen-eous, with uniform shear strength propertiesalong the slide surface.

(b) The shear strength τ of the material is char-acterized by cohesion: c and a friction angleφ, that are related by the equation τ =c + σ tan φ (see Section 1.4).

(c) Failure occurs on a circular slide surface,which passes through the toe of the slope.2

(d) A vertical tension crack occurs in the uppersurface or in the face of the slope.

2 Terzaghi (1943: 170), shows that the toe failure assumedfor this analysis gives the lowest factor of safety providedthat φ > 5◦. The φ = 0 analysis, involving failure below thetoe of the slope through the base material has been discussedby Skempton (1948) and by Bishop and Bjerrum (1960)and is applicable to failures which occur during or after therapid construction of a slope.

(e) The locations of the tension crack and ofthe slide surface are such that the factor ofsafety of the slope is a minimum for theslope geometry and ground water conditionsconsidered.

(f) Ground water conditions vary from a dryslope to a fully saturated slope under heavyrecharge; these conditions are defined inFigure 8.4.

(g) Circular failure charts are optimized for arock mass density of 18.9 kN/m3. Densitieshigher than this give high factors of safety,densities lower than this give low factorsof safety. Detailed circular analysis maybe required for slopes in which the mate-rial density is significantly different from18.9 kN/m3.

The charts presented in this chapter corres-pond to the lower bound solution for the factorof safety, obtained by assuming that the normalload is concentrated on a single point on the slidesurface. These charts differ from those publishedby Taylor in that they include the influence of acritical tension crack and of ground water.

8.3.1 Ground water flow assumptions

In order to calculate the forces due to water pres-sures acting on the slide surface and in the tensioncrack, it is necessary to assume a set of groundwater flow patterns that coincide as closely as pos-sible with conditions that are believed to exist inthe field.

In the analysis of rock slope failures discussedin Chapters 6, 7 and 9, it is assumed that most ofthe water flow takes place in discontinuities in therock and that the rock itself is practically imper-meable. In the case of slopes in soil or waste rock,the permeability of the mass of material is gener-ally several orders of magnitude higher than thatof intact rock and, hence, a general flow patternwill develop in the material behind the slope.

Figure 5.10(a) shows that, within the rock mass,the equipotentials are approximately perpendi-cular to the phreatic surface. Consequently, theflow lines will be approximately parallel to the


x

Face

H

Face angle

Sliding surface

Sliding surface

Assumed flow lines

Assumed flow lines

Assumed equipotentials

Assumed equipotentials

Phreatic surface

Tensioncrack

Tension crack

(a)

(b)

H

Surface recharge due to heavy rain

Figure 8.3 Definition of groundwater flow patterns used incircular failure analysis ofslopes in weak and closelyfractured rock: (a) ground waterflow pattern under steady statedrawdown conditions where thephreatic surface coincides withthe ground surface at a distancex behind the toe of the slope.The distance x is measured inmultiples of the slope height H ;(b) ground water flow patternin a saturated slope subjected tosurface recharge by heavy rain.

phreatic surface for the condition of steady-statedrawdown. Figure 8.3 shows that this approx-imation has been used for the analysis of thewater pressure distribution in a slope under condi-tions of normal drawdown. Note that the phreaticsurface is assumed to coincide with the groundsurface at a distance x, measured in multiples ofthe slope height, behind the toe of the slope. Thismay correspond to the position of a surface watersource, or be the point where the phreatic surfaceis judged to intersect the ground surface.

The phreatic surface itself has been obtained,for the range of the slope angles and values of x

considered, by solution of the equations proposedby L. Casagrande (1934), and discussed in thetextbook by Taylor (1937). For the case of a sat-urated slope subjected to heavy surface recharge,the equipotentials and the associated flow linesused in the stability analysis are based upon thework of Han (1972). This work involved theuse of an electrical resistance analogue methodto study ground water flow patterns in slopescomprised of isotropic materials.

Figure 8.4 shows five ground water conditionsranging from fully drained to saturated, basedon the models shown in Figure 8.3. For con-ditions 2, 3 and 4, the position of the groundwater table is defined by the ratio x/H . Thesefive ground water conditions are used in conjunc-tion with the circular failure charts discussed inSection 8.4.

8.3.2 Production of circular failure charts

The circular failure charts presented in thischapter were produced by running a searchroutine to find the most critical combination ofslide surface and tension crack for each of a widerange of slope geometries and ground water con-ditions. Provision was made for the tension crackto be located in either the upper surface, or in theface of the slope. Detailed checks were carriedout in the region surrounding the toe of the slopewhere the curvature of the equipotentials resultsin local flow which differs from that illustrated inFigure 8.3(a).


Ground water flow conditions Chart number

1

2

3

4

5

Fully drained slope

Surface water 8x slope heightbehind toe of slope



Saturated slope subjected toheavy surface recharge

Figure 8.4 Ground water flowmodels used with circularfailure analysischarts—Figures 8.6–8.10.

The charts are numbered 1–5 (Figures 8.6–8.10) to correspond with the ground water con-ditions defined in Figure 8.4.

8.3.3 Use of the circular failure charts

In order to use the charts to determine the factorof safety of a slope, the steps outlined here andshown in Figure 8.5 should be followed.

Step 1: Decide upon the ground water con-ditions which are believed to exist in

the slope and choose the chart whichis closest to these conditions, usingFigure 8.4.

Step 2: Select rock strength parameters appli-cable to the material forming the slope.

Step 3: Calculate the value of the dimensionlessratio c/(γ H tan φ) and find this value onthe outer circular scale of the chart.

Step 4: Follow the radial line from the valuefound in step 3 to its intersection withthe curve which corresponds to the slopeangle.


Step 5: Find the corresponding value of tan φ/FSor c/(γ H FS), depending upon whichis more convenient, and calculate thefactor of safety.

Consider the following example:A 15.2-m high cut with a face angle of 40◦ is

to be excavated in overburden soil with a dens-ity γ = 15.7 kN/m3, a cohesion of 38 kPa and afriction angle of 30◦. Find the factor of safety ofthe slope, assuming that there is a surface watersource 61 m behind the toe of the slope.

The ground water conditions indicate the useof chart number 3 (61/15.2 ∼ 4). The value ofc/(γ H tan φ) = 0.28 and the corresponding valueof tan φ/FS, for a 40◦ slope, is 0.32. Hence, thefactor of safety of the slope of 1.80.

12

3

4

4

tan�

FS

c

c�H tan�

�H FS

Figure 8.5 Sequence of steps involved in using circularfailure charts to find the factor of safety of a slope.

0 .012.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

.02 .03 .04 .05.06

.07.08

.09.10

.11.12

.13.14

.15.16

.17.18

.19.20

.25

.30

.35

.40.45.50

.60.70.80.901.0

1.52.0

4.0

80 .02 .04 .06

10°20°

30°40°

50°60°

70°

80°

90°

Slope angle

.08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 .30 .32 .34c

�H FS

c�H tan�

tan�

FS

Figure 8.6 Circular failure chart number 1—fully drained slope.


0 .012.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

.02 .03 .04 .05.06

.07.08

.09.10

.11.12

.13.14

.15.16

.17.18

.19.20

.25

.30

.35

.40.45.50

60.70.80.901.0

1.52.0

4.0

80 .02 .04 .06

10°20°

30°40°

50°60°

70°

80°

90°

Slope angle

.08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 .30 .32 .34c

�H FS

c� H tan�

tan�

FS

Figure 8.7 Circular failure chart number 2—ground water condition 2 (Figure 8.5).

Because of the speed and simplicity of usingthese charts, they are ideal for checking the sens-itivity of the factor of safety of a slope to a widerange of conditions. For example, if the cohesionwere to be halved to 20 kPa and the ground waterpressure increased to that represented by chartnumber 2, the factor of safety drops to 1.28.

8.4 Location of critical slide surfaceand tension crack

During the production of the circular failurecharts presented in this chapter, the locationsof both the critical slide surface and the criticaltension crack for limiting equilibrium (FS = 1)

were determined for each slope analyzed. Theselocations are presented, in the form of charts, inFigures 8.11 and 8.12.

It was found that, once ground water is presentin the slope, the locations of the critical circle andthe tension crack are not particularly sensitive tothe position of the phreatic surface and hence onlyone case, that for chart number 3, has been plot-ted. It will be noted that the location of the criticalcircle center given in Figure 8.12 differs signific-antly from that for the drained slope plotted inFigure 8.11.

These charts are useful for the construction ofdrawings of potential slides and for estimatingthe friction angle when back-analyzing existing


0 .012.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

.02 .03 .04 .05 .06.07

.08.09

.10.11

.12.13

.14.15

.16.17

.18.19

.20

.25

.30

.35

.40.45.50

.60.70.80.901.0

1.52.0

4.0

80 .02 .04 .06

20°30°

40°50°

60°70°

80°

90°

Slope angle

.08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 .30 .32 .34c

�H FS

c� H tan�

tan�

FS


circular slides. They also provide a start inlocating the critical slide surface when carryingout more sophisticated circular failure analysis.

As an example of the application of thesecharts, consider the case of a drained slope havinga face angle of 30◦ in a soil with a friction angle of20◦. Figure 8.11 shows that the critical slide circlecenter is located at X = 0.2H and Y = 1.85H

and that the critical tension crack is at a distanceb = 0.1H behind the crest of the slope. Thesedimensions are shown in Figure 8.13.

8.5 Examples of circular failure analysis

The following two examples illustrate the use ofthe circular failure charts for the study of thestability of slope in highly weathered rock.

8.5.1 Example 1—China clay pit slope

Ley (1972) investigated the stability of a Chinaclay pit slope which was considered to bepotentially unstable, and that a circular failurewas the likely type of instability. The slope pro-file is illustrated in Figure 8.14 and the input dataused for the analysis is included in this figure. Thematerial, a heavily kaolinized granite, was testedin direct shear to determine the friction angle andcohesion.

Two piezometers in the slope and a knownwater source some distance behind the slopeenabled an estimate to be made of the pos-ition of the phreatic surface as shown inFigure 8.14. From Figure 8.14, chart number 2corresponds most closely to these ground waterconditions.


0 .012.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

.02 .03 .04 .05 .06.07

.08.09

.10.11

.12.13

.14.15

.16.17

.18.19

.20

.25

.30

.35

.40.45.50

.60.70.80.901.0

1.52.0

4.0

80 .02 .04 .06

40°50°

60°70°

80°

90°

Slope angle

.08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 .30 .32 .34c

�H FS

c�H tan�

tan�

FS


From the information given in Figure 8.14, thevalue of the ratio c/(γ H tan φ) = 0.0056, andthe corresponding value of tan φ/FS from chartnumber 2, is 0.76. Hence, the factor of safety ofthe slope is 1.01. A number of trial calculationsusing Janbu’s method (Janbu et al., 1956) for thecritical slide circle shown in Figure 8.14, founda factor of safety of 1.03.

These factors of safety indicated that the stabil-ity of the slope was inadequate under the assumedconditions, and steps were taken to deal with theproblem.

8.5.2 Example 2—highway slope

A highway plan called for a cut at an angle of 42◦.The total height of the cut would be 61 m and it

was required to check whether the cut would bestable. The slope was in weathered and alteredmaterial, and failure, if it occurred, would be acircular type. Insufficient time was available forground water levels to be accurately establishedor for shear tests to be carried out. The stabilityanalysis was carried out as follows:

For the condition of limiting equilibrium, FS =1 and tan φ/FS = tan φ. By reversing the pro-cedure outlined in Figure 8.5, a range of frictionangles were used to find the values of the ratioc/(γ H tan φ) for a face angle of 42◦. The value ofthe cohesion c which is mobilized at failure, for agiven friction angle, can then be calculated. Thisanalysis was carried out for dry slopes using chartnumber 1 (line B, Figure 8.15), and for saturatedslopes using chart number 5 (line A, Figure 8.15).


0 .012.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

.02 .03 .04 .05.06

.07.08

.09.10

.11.12

.13.14

.15.16

.17.18

.19.20

.25

.30

.35

.40.45.50

.65.70.80.901.0

1.52.0

4.0

80 .02 .04 .06

10°

20°30°

40°50°

60°70°

80°

Slope angle

.08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 .30 .32 .34c

�H FS

c�H tan�

tan�

FS

Figure 8.10 Circular failure chart number 5—fully saturated slope.

Figure 8.15 shows the range of friction angles andcohesions that would be mobilized at failure.

The shaded circle (D) included in Figure 8.15indicated the range of shear strengths thatwere considered probable for the material underconsideration, based upon the data presented inFigure 4.21. This figure shows that the availableshear strength may not be adequate to maintainstability in this cut, particularly when the cut issaturated. Consequently, the face angle could bereduced, or ground water conditions investigatedto establish actual ground water pressures and thefeasibility of drainage.

The effect of reducing the slope angle can bechecked very quickly by finding the value of the

ratio c/(γ H tan φ) for a flatter slope of 30◦, inthe same way as it was found for the 42◦ slope.The dashed line (C) in Figure 8.15 indicates theshear strength, which is mobilized in a dry slopewith a face angle of 30◦. Since the mobilized shearstrength C is less than the available shear strengthD, the dry slope is likely to be stable.

8.6 Detailed stability analysisof circular failures

The circular failure charts presented earlier in thischapter are based upon the assumption that thematerial forming the slope has uniform proper-ties throughout the slope, and that failure occurs


+X

Location of center ofcritical circle

b

Tensioncrack

Drained slope

Y

H

Failure throughtoe of slope

Friction angle

Distance X

Location of center of critical circle for failure through toe

Dis

tanc

e Y

Rat

io b

/H

�= 20°�= 30°

�= 40°

�= 10°

�= 10°

Slope angle

80°70°

60°

50°40°

30°

20˚10°

�

=50°

�=

40°�

=30°

�=20°

Slope face angle (°)Location of critical tension crack position

0.4

0.3

0.2

0.1

00 10 20 30 40 7050 8060 90

4H

2H

0

3H

H

4H

2H

0

3H

H

H 2H 3H−3H −2H −H 0

H 2H 3H−3H −2H −H 0

Figure 8.11 Location of critical sliding surface and critical tension crack for drained slopes.

along a circular slide path passing through the toeof the slope. When these conditions are not satis-fied, it is necessary to use one of the methods ofslices published by Bishop (1955), Janbu (1954),Nonveiller (1965), Spencer (1967), Morgensternand Price (1965) or Sarma (1979). This sectiondescribes in detail the simplified Bishop and Janbumethods of stability analysis for circular failure.

8.6.1 Bishop’s and Janbu’s method of slices

The slope and slide surface geometries, and theequations for the determination of the factor ofsafety by the Bishop’s simplified method of slices

(1955) and the Janbu’s modified method of slices(1954) are given in Figures 8.16 and 8.17 respect-ively. Bishop’s method assumes a circular slidesurface and that the side forces are horizontal;the analysis satisfies vertical forces and overallmoment equilibrium. The Janbu method allowsa slide surface of any shape, and assumes theside forces are horizontal and equal on all slices;the analysis satisfies vertical force equilibrium.As pointed out by Nonveiller (1965), Janbu’smethod gives reasonable factors of safety whenapplied to shallow slide surfaces (which are typ-ical in rock with an angle of friction in excessof 30◦ and rockfill), but it is seriously in error


+X

Location of center ofcritical circle

b

Tension crackSlope with ground water(chart number 3)

Y

H


Ground watersurface R

atio

b/H

�= 20°

�= 10°

Angle of slope face (°)Location of critical tension crack position

Location of center of critical circle for failure through toe

0.4

0.3

0.2

0.1

00 10 20 30 40 7050 8060 90

�= 30°

�= 40°�= 50°

�= 60°

80°70° 60° 50° 40°

30°

20°

10°

Slope angle

Distance X

Dis

tanc

e Y

4H

2H

0

3H

H

H 2H 3H−3H −2H −H 0

H 2H 3H−3H −2H −H 0

�= 60°�= 50°

�= 40°

�=

30° �=

20°

�=1

0°

Friction angle

Figure 8.12 Location of critical sliding surface and critical tension crack for slopes with ground water present.

b = 0.1 HX = 0.2 H

Y = 1.85 H

Figure 8.13 Location of critical slide surface and critical tension crack for a drained slope at an angleof 30◦ in a material with a friction angle of 20◦.


Measured waterlevel

Critical failurecircle for Janbuanalysis

31°

Input data for analysis:Unit weight �= 21.5 kN/m3

Friction angle �= 37°Cohesion c = 6.9 kPa

76.8 m

Figure 8.14 Slope profile ofChina clay pit slopeconsidered in Example 1.

A

B

C

D

0 10 20 30 40 50

A—Saturated 42° slopeB—Dry 42° slopeC—Dry 30° slopeD—Probable shear strength range for material in which slope is cut (see Figure 4.21).

Friction angle � (°)

Coh

esio

n c

(kP

a)

0

50

100

150

250

200

Figure 8.15 Comparison between shear strengthmobilized and shear strength available for slopeconsidered in Example 2.

and should not be used for deep slide surfaces inmaterials with low friction angles.

The procedures for using Bishop’s and Janbu’smethods of slices are very similar and it is con-venient to discuss them together.

Step 1: Slope and slide surface geometry. Thegeometry of the slope is defined by the actual orthe designed profile as seen in a vertical sectionthrough the slope. In the case of a circular failure,

the charts given in Figures 8.11 and 8.12 can beused to estimate the center of the circle with thelowest factor of safety. In the Janbu analysis, theslide surface may be defined by known structuralfeatures or weak zones within the rock or soilmass, or it may be estimated in the same way asthat for the Bishop analysis. In either case, theslide surface assumed for the first analysis maynot give the lowest factor of safety, and a series ofanalyses are required with variations on this pos-ition to find the surface with the lowest factor ofsafety.

Step 2: Slice parameters. The sliding massassumed in step 1 is divided into a number ofslices. Generally, a minimum of five slices shouldbe used for simple cases. For complex slope pro-files, or where there are different materials in therock or soil mass, a larger number of slices may berequired in order to define adequately the prob-lem. The parameters which have to be defined foreach slice are as given here:

• base angle ψb;• the weight of each slice W is given by the

product of the vertical height h, the unitweight γr of the rock or soil and the widthof the slice �x : W = (h γr�x); and

• uplift water pressure U on the base of eachslice is given by the product of the height hwto the phreatic surface, the unit weight γw ofwater and the width of the slice �x, that is,U = (hw γw �x).


Center of rotation (see Figure 8.12)

R

�

Typical slice

Water force = �whw(∆x/cos �b)

Tension crack

Ground watersurface

Zz/3½ �wz2

Failure through toeof slope

∆x

�b ��

�

Slice weight= �rh∆x

hw

h

Y

X

H

Factor of safety:

FS =∑x /(1 + Y/FS)

∑Z + Q

where

X = [c + (�rh – �whw) tan�] (∆x /cos�b)

Q = ½ �wz 2 (�/R)

Z = �rh ∆x sin�b

Y = tan�b tan�

Note: angle �b is negative when sliding uphill

The following conditions must be satisfied for each slice:

(1)

(2)

��=1 + Y/FS

cos�b (1 + Y/FS) > 0.2

(8.3)

(8.4)

(8.5)

(8.6)

(8.7)

(8.8)

(8.9)

�rh – �whw – c (tan�b/FS)

Figure 8.16 Bishop’s simplifiedmethod of slices for theanalysis of non-circular failurein slopes cut into materials inwhich failure is defined by theMohr–Coulomb failurecriterion.

Step 3: Shear strength parameters. The shearstrength acting on the base of each slice is requiredfor the stability calculation. In the case of a uni-form material in which the failure criterion isassumed to be that of Mohr–Coulomb (equa-tion (1.1) in Section 1.4), the shear strengthparameters c and φ will be the same on thebase of each slice. When the slope is cut in anumber of materials, the shear strength para-meters for each slice must be chosen accordingto the material in which it lies. When the shearstrengths of the materials forming the slope are

defined by non-linear failure criterion as dis-cussed in Section 4.5, it is necessary to determ-ine the cohesion and friction angle for eachslice at the effective normal stress for that slice(see Figure 4.23).

Step 4: Factor of safety iteration. When theslice and shear strength parameters have beendefined, the values of X, Y and Z are calculatedfor each slice. The water force Q is added to

∑Z,

the sum of the components of the weight of eachslice acting parallel to the slide surface. An initialestimate of FS = 1.00 for the factor of safety is


Typical slice

Water force = �whw(∆x/cos�b)

Tension crack

Ground watersurface

zz/3½ �wz2


d

L

∆x

�b ��

Slice weight= �rh∆x

hw

h

H

Factor of safety:

FS =f0∑X/(1 + Y/FS)

∑Z + Qwhere

X = [c + (�rh – �whw) tan� ] (1 + tan2�b) ∆x

Q = ½ �wz2

Z = �rh∆x tan�b

Y = tan�b tan�

(8.11)

(8.10)

(8.12)

(8.13)

(8.14)

(8.15)

Approximate correction factor f0

f0= 1 + K(d /L – 1.4(d/L)2)for c� = 0; K = 0.31

c� > 0, ��> 0; K = 0.50


Figure 8.17 Janbu’smodified method ofslices for the analysis ofnon-circular failure inslopes cut intomaterials in whichfailure is defined by theMohr–Coulomb failurecriterion.

used, and a new factor of safety is calculated fromequations (8.3) and (8.10) given in Figures 8.16and 8.17, respectively. If the difference betweenthe calculated and the assumed factors of safety isgreater than 0.001, the calculated factor of safetyis used as a second estimate of FS for a new factorof safety calculation. This process is repeateduntil the difference between successive factors ofsafety is less than 0.001. For both the Bishop andthe Janbu methods, approximately seven iterationcycles will be required to achieve this result formost slope and slide surface geometries.

Step 5: Conditions and corrections. Figure8.16 lists two conditions (equations (8.8)and (8.9)) that must be satisfied for each slice inthe Bishop analysis. The first condition ensuresthat the effective normal stress on the base

of each slice is always positive. If this con-dition is not met for any slice, the inclusionof a tension crack into the analysis should beconsidered. If it is impossible to satisfy thiscondition by readjustment of the ground waterconditions or the introduction of a tensioncrack, the analysis as presented in Figure 8.16should be abandoned and a more elaborate formof analysis, to be described later, should beadopted.

Condition 2 in Figure 8.16 was suggested byWhitman and Bailey (1967) and it ensures thatthe analysis is not invalidated by conditions whichcan sometimes occur near the toe of a slope inwhich a deep slide surface has been assumed. Ifthis condition is not satisfied by all slices, theslice dimensions should be changed and, if this


fails to resolve the problem, the analysis shouldbe abandoned.

Figure 8.17 gives a correction factor f0, whichis used in calculating the factor of safety bymeans of the Janbu method. This factor allowsfor inter-slice forces resulting from the shape ofthe slide surface assumed in the Janbu analysis.The equation for f0 given in Figure 8.17 has beenderived by Hoek and Bray (1981) from the curvespublished in Janbu (1954).

8.6.2 Use of non-linear failure criterionin Bishop stability analysis

When the material in which the slope is cut obeysthe Hoek–Brown non-linear failure criterion dis-cussed in Section 4.5, the Bishop’s simplifiedmethod of slices as outlined in Figure 8.18 canbe used to calculate the factor of safety. Thefollowing procedure is used, once the slice para-meters have been defined as described earlier forthe Bishop and Janbu analyses:

1 Calculate the effective normal stress σ′ act-ing on the base of each slice by means ofthe Fellenius equation (equation (8.17) onFigure 8.18).

2 Using these values of σ′, calculate tan φ

and c for each slice from equations (4.24)and (4.25).

3 Substitute these values of tan φ and c into thefactor of safety equation in order to obtain thefirst estimate of the factor of safety.

4 Use this estimate of FS to calculate a new valueof σ′ on the base of each slice, using the Bishopequation (equation (8.18) on Figure 8.18).

5 On the basis of these new values of σ′, calcu-late new values for tan φ and c.

6 Check that conditions defined by equa-tions (8.8) and (8.9) on Figure 8.16 are sat-isfied for each slice.

7 Calculate a new factor of safety for the newvalues of tan φ and c.

8 If the difference between the first and secondfactors of safety is greater than 0.001, returnto step 4 and repeat the analysis, usingthe second factor of safety as input. Repeat

this procedure until the difference betweensuccessive factors of safety is less than 0.001.

Generally, about ten iterations will be requiredto achieve the required accuracy in the calculatedfactor of safety.

8.6.3 Example of Bishop’s and Janbu’smethods of analysis

A slope is to be excavated in blocky sandstonewith very closely spaced and persistent discon-tinuities. The slope will consist of three, 15-mhigh benches with two 8-m wide berms, theprimary function of which are to collect surfacerunoff and control erosion (Figure 8.19). Thebench faces will be at 75◦ to the horizontal, andthe slope above the crest of the cut will be atan angle of 45◦. The assumed position of thewater table is shown on the figure. It is requiredto find the factor of safety of the overall slope,assuming that a circular type stability analysis isappropriate for these conditions.

The shear strength of the jointed rock mass isbased on the Hoek–Brown strength criterion, asdiscussed inSection4.5, whichdefines the strengthas a curved envelope. The cohesion and frictionangle for this criterionare calculatedusing thepro-gram ROCLAB 1.004 (RocScience, 2002a), forwhich the input parameters are as follows:

• Very poor quality rock mass, GSI = 20;• Uniaxial compressive strength of intact rock

(from point load testing) ≈ 150 MPa;• Rock material constant, mi = 15;• Unit weight of rock mass, γr = 0.025 MN/m3;• Unit weight of water, γw = 0.00981 MN/m3;• For careful blasting used in excavation, dis-

turbance factor D = 0.7; and• Average slice height = 24 m (this height

together with the rock mass unit weightdefines the average vertical stress on the slidingsurface).

Using these parameters, ROCLAB calculates, atthe appropriate vertical stress level, a best fit lineto the curved strength envelope to define a frictionangle of 43◦ and a cohesion of 0.145 MPa. This is


∆x

�b ��

�

hw

h

Typical slice

Tension crack

Ground watersurface

zz/3

Failure through toeof slope

Center of rotation

R

�

b

Y

X

H


Factor of safety:

FS =∑(c i� +�� tan�i�) (∆x /cos �b)

∑�rh ∆x sin �b + ½ �wz2�/R

where

��=

��=

�rh cos2 �b– �whw

�rh – �whw – (c i� tan �b/FS)and

1+ (tan�i� tan �b/FS)(Bishop solution)

(Fellenius solution)

��= sin–16amb(s + mb��3n)a–1

2(1 + a)(2 + a) + 6 amb(s + mb��3n)a–1

c� =�ci [(1 + 2a)s + ( 1– a)mb��3n] (�+ mb��3n)a–1

(1 + a)(2 + a) – 1+ [6amb (s + mbs�3n)a–1][(1 + a)(2 + a)]

where =�3n=��3 max/�ci

The conditions which must be satisfied for each slice are:

(1) ��> 0, where �� is calculated by Bishop’s method

(2) cos �b [1 + (tan �b tan�i�)/FS] > 0.2

(8.16)

(8.17)

(8.18)

(4.24)

(4.25)

½ �wz2

�rh

Figure 8.18 Bishop’s simplified method of slices for the analysis of circular failure in slope in material inwhich strength is defined by non-linear criterion given in Section 4.5.

the equivalent Mohr–Coulomb shear strength. Inaddition, the program calculates the instantan-eous friction angle and cohesion correspondingto the effective normal stress on the base of eachslice.

Table 8.1 shows the input parameters for theBishop and Janbu stability analyses assuming alinear shear strength, and for the Janbu analysisassuming a non-linear shear strength. The slope isdivided into eight slices, and for each slice the base


angle, weight, pore pressure and width are meas-ured. The program SLIDE (RocScience, 2002b)is then used to calculate the factors of safety forthe linear and non-linear shear strengths. In thecase of the non-linear strength analysis, the val-ues of the effective normal stress on the base of

Critical center for �= 45°

�

1

2

3

45

678

5 m

n/R = 0.401

d/L = 0.117

0 10 20

Scale - m

Figure 8.19 Section of sandstone slope showing watertable, slice boundaries, tension crack and the expectedcircular sliding surface.

the slices, and the corresponding instantaneousfriction angle and cohesion are also calculated, asshown in the last three columns of Table 8.1. Thecalculated factors of safety estimates are

Bishop simplified method of slices forMohr–Coulomb shear strength

= 1.39

Janbu modified method of slices forMohr–Coulomb shear strength

= 1.26

Bishop simplified method of slices fornon-linear shear strength

= 1.39

Note that the stability analysis of this slopeusing numerical analysis methods is shown inSection 10.4.1.

8.6.4 Circular failure stability analysiscomputer programs

The circular failure charts discussed in Section 8.3provide a rapid means of carrying out stabilityanalyses, but are limited to simple conditions asillustrated in the examples. More complex ana-lyses can be carried out using the Bishop andJanbu methods discussed in Section 8.6.3, and thepurpose of providing details on the procedures isto show the principles of the analyses.

Table 8.1 Calculated shear strength values of slices using Mohr–Coulomb and Hoek–Brown failure criteria

Slice parameters for all cases Mohr–Coulombvalues for Bishopand Janbu analyses

Non-linear failure criterionvalues for Bishop analysis

Slicemember

Angleof slicebase, ϕ(degrees)

Sliceheight(MN)

Porepressure,γwhw(MPa)

Slicewidth(m)

Frictionangle,�(degrees)

Cohesion,c (MPa)

Baseeffectivenormalstress,σn(MPa)

Inst.frictionangle,φi(degrees)

Inst.cohesion,ci (MPa)

1 25 1.312 0.017 6.1 43 0.145 0.139 53.8 0.0682 29 1.597 0.047 6.1 43 0.145 0.169 53.8 0.0683 34 2.603 0.071 6.1 43 0.145 0.270 49.8 0.0964 39 2.635 0.087 6.1 43 0.145 0.261 51.3 0.0855 44 3.501 0.095 6.1 43 0.145 0.323 49.0 0.1046 50 3.914 0.087 6.1 43 0.145 0.324 48.6 0.1077 57 3.592 0.047 6.1 43 0.145 0.240 50.3 0.0928 65 2.677 0 6.1 43 0.145 0.109 54.5 0.064


There are, of course, computer programsavailable to carry out stability analyses ofslopes, where the circular failure charts are notapplicable. Important features included in theseprograms, which allows them to be used for awide range of conditions, are as follows:

• Slope face can include benches and a varietyof slope angles;

• Boundaries between the materials can be posi-tioned to define layers of varying thickness andinclination, or inclusions of any shape;

• Shear strength of the materials can be definedin terms of Mohr–Coulomb or Hoek–Browncriterion;

• Ground water pressures can be defined onsingle or multiple water tables, or as specifiedpressure distributions;

• External loads, in any direction within theplane of the slope cross-section, can be posi-tioned at their correct location on the slope.Such loads can include bridge and buildingfoundations and bolting forces;

• Earthquake acceleration which is applied as ahorizontal force in order to carry out pseudo-static stability analysis;

• The shape and position of the slide surfacecan be defined as a circular arc or straight linesegments;

• A search routine finds the slide surface withthe minimum factor of safety;

• Deterministic and probabilistic analysis meth-ods that calculate the factor of safety andprobability of failure, respectively. The prob-abilistic analysis requires that the design para-meters be defined as distributions rather thansingle values;

• Error messages which identify negativestresses along the slide surface; and

• Drawing of slope showing slope geometry,material boundaries, ground water table(s)and slide surface(s).

One program that contains all these functionsis the program XSTABL (Sharma, 1991). Anexample of the output (partial) produced byXSTABL is shown in Figure 8.20 for a benched

slope excavated in sandstone, shale and siltstoneabove a proposed highway. For these conditionsthe shape of the failure surfaces are influencedby the position and thickness of the beds of theweaker material.

8.6.5 Three-dimensional circular failureanalysis

The program XSTABL examines the stabilityof a unit width slice of the slope, which is atwo-dimensional analysis that ignores any shearstresses on the sides of the slice (this is thesame principle that is used in the plane fail-ure analysis described in Chapter 6). Whiletwo-dimensional procedures have found to be areliable method of analysis, there may be cir-cumstances where three-dimensional analysis isrequired to define the slide surface and slopegeometry more precisely. One program thatprovides a three-dimensional analysis is CLARA(Hungr, 1987), which divides the sliding massinto columns, rather than slices as used in the two-dimensional mode. Figure 8.21 shows an exampleof the CLARA analysis for a partially saturatedslope in which the water table is below the bottomof the tension crack.

8.6.6 Numerical slope stability analysis

This chapter has been concerned solely with thelimit equilibrium method of analysis in whichthe factor of safety is defined by the ratio ofthe resisting to the displacing forces on the slidesurface.

An alternative method of analysis is to exam-ine the stresses and strains within the slope asa means of assessing stability conditions. If theslope is close to failure, then a zone of high strainwill develop within the slope with a shape thatwill be approximately coincident with the circu-lar slide surface. If the shear strength propertiesare progressively reduced, there will be a suddenincrease in the movement along the shear zoneindicating that the slope is on the point of failure.The approximate factor of safety of the slope canbe calculated from the ratio of the actual shear


Overburden

Sandstone

Shale

Siltstone andsandstone

Shale

Sandstone

Shale

Siltstone and sandstone

X-axis (m)

Y-a

xis

(m)

1085

995

905

815

725

835

0 90 180 270 360 450 540 630 720

10 most critical surfaces Surface of minimum Janbu

FS = 1.101

Figure 8.20 Two-dimensional stability analysis of a highway cut using XSTABL.

Surcharge oncrest of slope

Retaining wall

Failure surface

05

10 m

Figure 8.21 Three-dimensional stability analysis of aslope incorporating a retaining wall and a surchargeat the crest.

strength of the rock to the shear strength at whichthe sudden movement occurred. This method ofstability analysis is described in more detail inSection 10.3.

8.7 Example Problem 8.1: circularfailure analysis

Statement

A 22-m high rock cut with a face angle of 60◦has been excavated in a massive, very weak vol-canic tuff. A tension crack has opened behind thecrest and it is likely that the slope is on the pointof failure, that is, the factor of safety is approx-imately 1.0. The friction angle of the material isestimated to be 30◦, its density is 25 kN/m3, andthe position of the water table is shown on thesketch of the slope (Figure 8.22). The rock con-tains no continuous joints dipping out of the face,and the most likely type of failure mode is circularfailure.

Required

(a) Carry out a back analysis of the failure todetermine the limiting value of the cohesionwhen the factor of safety is 1.0.


b = 2.9 m

X = –7.7 m

Y = 22 m

Estimated slide surface

Ground water table

60°

22 m

Tension crack(a)

(b)

Figure 8.22 Slope geometry for ExampleProblem 8.1: (a) slope geometry for circular failurewith ground water table corresponding to circularfailure chart number 3; (b) position of critical slidesurface and critical tension crack.

(b) Using the strength parameters calculated in(a), determine the factor of safety for a com-pletely drained slope. Would drainage of theslope be a feasible method of stabilization?

(c) Using the ground water level shown inFigure 8.22 and the strength parameters cal-culated in (a), calculate the reduction inslope height, that is, amount of unloading ofthe slope crest required to increase the factorof safety to 1.3.

(d) For the slope geometry and ground waterlevel shown in Figure 8.22, find the coordin-ates of the center of the critical circle and theposition of the critical tension crack.

Solution

(a) The ground water level shown in Figure 8.22corresponds to ground water condition 3 inTable 8.4 in the manual, so circular failure

chart number 3 (Figure 8.8) is used in theanalysis. When φ = 30◦ and FS = 1.0,

tan φ/FS = 0.58

The intersection of this value for tan φ/FSand the curve for a slope angle of 60◦ gives

c

γ H FS= 0.086

∴ c = 0.086 × 25 × 22 × 1.0

= 47.3 kPa

(b) If the slope were completely drained, circularfailure chart 1 could be used for analysis.

c

γ H tan φ= 47.3

25 × 22 × tan(30)

= 0.15

The intersection of this inclined line with thecurved line for a slope angle of 60◦ gives

tan φ

FS= 0.52

∴ FS = tan 300.52

= 1.11

This factor of safety is less than that usuallyaccepted for a temporary slope, that is, FS =1.2, so draining the slope would not be aneffective means of stabilization.

(c) When FS = 1.3 and φ = 30◦, thentan φ/FS = 0.44.

On circular failure chart 3, the inter-section of this horizontal line with the curvedline for a slope angle of 60◦ gives

c

γ H FS= 0.11

∴ H = 47.325 × 1.3 × 0.11

= 13.2 m

This shows that the slope height must bereduced by 8.8 m to increase the factor of


safety from 1.0 to 1.3. Note that a factor ofsafety of 1.3 would only be achieved if theground water level dropped by an amountequivalent to the unloading.

(d) The critical circle and critical tension crackfor a slope with ground water present arelocated using the graphs in Figure 8.12.

For a slope angle of 60◦ and a frictionangle of 30◦, the coordinates of the centerof the circle are:

X = −0.35 H

= −7.7 m, that is, 7.7 m horizontallybeyond the toe

Y = H

= 22 m, that is, 22 m above the toe

The location of the tension crack behind thecrest is

b/H = 0.13

b = 2.9 m

This critical circle is shown inFigure 8.22(b).

Chapter 9

Toppling failure

9.1 Introduction

The failure modes discussed in the three previouschapters all relate to sliding of a rock or soil massalong an existing or induced sliding surface. Thischapter discusses a different failure mode—thatof toppling, which involves rotation of columnsor blocks of rock about a fixed base. Similar tothe plane and wedge failures, the stability analysisof toppling failures involves, first, carrying outa kinematic analysis of the structural geology toidentify potential toppling conditions, and then,if this condition exists, performing a stabilityanalysis specific to toppling failures.

One of the earliest references to toppling fail-ures is by Muller (1968) who suggested that blockrotation or toppling may have been a contribut-ory factor in the failure of the north face of theVaiont slide (Figure 9.1). Hofmann (1972) car-ried out a number of model studies under Muller’sdirection to investigate block rotation. Similarmodel studies carried out by Ashby (1971), Soto(1974) and Whyte (1973), while Cundall (1971),Byrne (1974) and Hammett (1974) who incor-porated rotational failure modes into computeranalysis of rock mass behavior. Figure 9.2 showsa computer model of a toppling failure in whichthe solid blocks are fixed and the open blocksare free to move. When the fixed blocks at theface are removed, the tallest columns of blockstopple because their center of gravity lies outsidethe base. The model illustrates a typical featureof toppling failures in which the tension cracksare wider at the top than at the base. This con-dition, which can best be observed when looking

along strike, is useful in the field identification oftopples.

Papers concerning field studies of toppling fail-ures include de Freitas and Waters (1973) whodiscuss slopes in Britain, and Wyllie (1980) whodemonstrates stabilization measures for topplingfailures related to railway operations.

Most of the discussion that follows in thischapter is based on a paper by Goodman andBray (1976) in which a formal mathematical solu-tion to a simple toppling problem is shown. Thissolution, which is reproduced here, representsa basis for designing rock slopes in which top-pling is present, and has been further developedinto a more general design tool (Zanbak, 1983;Adhikary et al., 1997; Bobet, 1999; Sagesetaet al., 2001).

9.2 Types of toppling failure

Goodman and Bray (1976) have described anumber of different types of toppling failures thatmay be encountered in the field, and each is dis-cussed briefly on the following pages. The import-ance of distinguishing between types of topplingis that there are two distinct methods of stabilityanalysis for toppling failures as described in thefollowing pages—block and flexural toppling—and it is necessary to use the appropriate analysisin design.

9.2.1 Block toppling

As illustrated in Figure 9.3(a), block topplingoccurs when, in strong rock, individual columns

Toppling failure 201

Figure 9.1 Suggested toppling mechanism of the north face of Vaiont slide (Muller, 1968).

Figure 9.2 Computer generated model of toppling failure; solid blocks are fixed in space while open blocksare free to move (Cundall, 1971).

are formed by a set of discontinuities dippingsteeply into the face, and a second set of widelyspaced orthogonal joints defines the columnheight. The short columns forming the toe of theslope are pushed forward by the loads from thelonger overturning columns behind, and this slid-ing of the toe allows further toppling to develophigher up the slope. The base of the failuregenerally consists of a stepped surface rising fromone cross joint to the next. Typical geological con-ditions in which this type of failure may occur arebedded sandstone and columnar basalt in whichorthogonal jointing is well developed.

9.2.2 Flexural toppling

The process of flexural toppling is illustratedin Figure 9.3(b) that shows continuous columns

of rock, separated by well developed, steeplydipping discontinuities, breaking in flexure asthey bend forward. Typical geological condi-tions in which this type of failure may occur arethinly bedded shale and slate in which ortho-gonal jointing is not well developed. Generally,the basal plane of a flexural topple is not as welldefined as a block topple.

Sliding, excavation or erosion of the toe of theslope allows the toppling process to start andit retrogresses back into the rock mass with theformation of deep tension cracks that becomenarrower with depth. The lower portion of theslope is covered with disordered fallen blocks andit is sometimes difficult to recognize a topplingfailure from the bottom of the slope. Detailedexamination of toppling slopes shows that theoutward movement of each cantilevered column

202 Toppling failure

(b)(a)

(c)

Figure 9.3 Common classes of toppling failures: (a) block toppling of columns of rock containing widelyspaced orthogonal joints; (b) flexural toppling of slabs of rock dipping steeply into face; (c) block flexuretoppling characterized by pseudo-continuous flexure of long columns through accumulated motions alongnumerous cross-joints (Goodman and Bray 1976).

produces an interlayer slip and a portion of theupper surface of each plane is exposed in a seriesof back facing, or obsequent scarps, such as thoseillustrated in Figure 9.3(a).

9.2.3 Block-flexure toppling

As illustrated in Figure 9.3(c), block-flexuretoppling is characterized by pseudo-continuousflexure along long columns that are divided bynumerous cross joints. Instead of the flexuralfailure of continuous columns resulting in flex-ural toppling, toppling of columns in this caseresults from accumulated displacements on thecross-joints. Because of the large number of smallmovements in this type of topple, there are fewertension cracks than in flexural toppling, and feweredge-to-face contacts and voids than in blocktoppling.

9.2.4 Secondary toppling modes

Figure 9.4 illustrates a number of possiblesecondary toppling mechanisms suggested byGoodman and Bray. In general, these failures areinitiated by some undercutting of the toe of theslope, either by natural agencies such as scouror weathering, or by human activities. In allcases, the primary failure mode involves slidingor physical breakdown of the rock, and topplingis induced in the upper part of the slope as a resultof this primary failure (Figure 9.4(a) and (b)).

Figure 9.4(c) illustrates a common occurrenceof toppling failure in horizontally bedded sand-stone and shale formations. The shale is usu-ally significantly weaker and more susceptible toweathering than the sandstone, while the sand-stone often contains vertical stress relief joints. Asthe shale weathers, it undermines support for the


Tension cracks

(b)(a)

(d)(c)

Circular slidingsurfaces

Toppling atpit crest

Talus

ConglomerateSiltstone

Sandstone

Fault1750

1800

1950

1850

2000

1900

2050E

leva

tion

(m)

Figure 9.4 Secondary toppling modes: (a) toppling at head of slide; (b) toppling at toe of slide with shearmovement of upper slope (Goodman and Bray, 1976); (c) toppling of columns in strong upper material due toweathering of underlying weak material; (d) toppling at pit crest resulting in circular failure of upper slope(Wyllie and Munn, 1978).

sandstone and columns of sandstone, with theirdimensions defined by the spacing of the verticaljoints, topple from the face. At some locationsthe overhangs can be as wide as 5 m, and failuresof substantial volumes of rock occur with littlewarning.

The example of the slide base toppling modeshown in Figure 9.4(d) is the failure of a pit slopein a coal mine where the beds at the crest of thepit dipped at 70◦ into the face, and their strikewas parallel to the face. Mining of the pit slope atan angle of 50◦ initiated a toppling failure at thecrest of the pit, which in turn resulted in a circularfailure that extended to a height of 230 m abovethe base of the topple. Detailed monitoring ofthe slope showed that a total movement of about

30 m occurred on the slope above the pit, result-ing in cracks opening in the crest of the mountainthat were several meters wide and up to 9 m deep.Continuous movement monitoring was used toallow mining to proceed under the moving slope,and finally the slope was stabilized by back-fillingthe pit (Wyllie and Munn, 1979).

A further example of the toppling mechan-ism is illustrated in Figure 9.5 (Sjöberg, 2000).In open pit mines where the depth of the slopeprogressively increases, minor toppling move-ment may eventually develop into a substantialfailure. Careful monitoring of the movement, andrecognition of the toppling mechanism, can beused to anticipate when hazardous conditions aredeveloping.


CrestElasticrebound

Joints

New miningstep

Joint slip

Stressredistribution Toe

Joint slip fullydeveloped(exaggerateddisplacements)

Compression andbending of columns

Tensile bending failureat base of rotation

Tensile bending failurepropagated to crest

Movement on slidesurface

Displacementsstarting from toe

I II III

IV V VI

Figure 9.5 Failure stages for large-scale toppling failure in a slope (Sjöberg, 2000).

9.3 Kinematics of block toppling failure

The potential for toppling can be assessed fromtwo kinematic tests described in this section.These tests examine first the shape of the block,and second the relationship between the dip ofthe planes forming the slabs and the face angle. Itis emphasized that these two tests are useful foridentifying potential toppling conditions, but thetests cannot be used alone as a method of stabilityanalysis.

9.3.1 Block shape test

The basic mechanics of the stability of a block ona plane are illustrated in Figure 9.6(a) (see alsoFigure 1.10). This diagram shows the conditionsthat differentiate stable, sliding or toppling blockswith height y and width �x on a plane dippingat an angle ψp. If the friction angle between thebase of the block and the plane is φp, then theblock will be stable against sliding when the dipof the base plane is less than the friction angle,

that is, when

ψp < φp (Stable) (9.1)

but will topple when the center of gravity of theblock lies outside the base, that is, when

�x/y < tan ψp (Topple) (9.2)

For example, for a 3 m wide block on a base planedipping at 10◦, toppling will occur if the heightexceeds 17 m.

9.3.2 Inter-layer slip test

A requirement for toppling to occur in the mech-anisms shown in Figures 9.3 and 9.5 is sheardisplacement on the face-to-face contacts on thetop and bottom faces of the blocks. Sliding onthese faces will occur if the following conditionsare met (Figure 9.6(b)). The state of stress closeto the slope face is uniaxial with the direction ofthe normal stress σ aligned parallel to the slope


∆x

y

W

�p

�f

�f

�d

�d�d

�

�

20°

(90 –�f)

(180 –�f – �d)

�f�d

�d

(a)

(c) (d)

(b)

Figure 9.6 Kinematic conditions for flexural slip preceding toppling: (a) block height/width test for toppling;(b) directions of stress and slip directions in rock slope; (c) condition for interlayer slip; (d) kinematic testdefined on lower hemisphere stereographic projection.

face. When the layers slip past each other, σ mustbe inclined at an angle φd with the normal to thelayers, where φd is the friction angle of the sidesof the blocks. If ψf is the dip of slope face andψd is the dip of the planes forming the sides ofthe blocks, then the condition for interlayer slipis given by (Figure 9.6(c)):

(180 − ψf − ψd) ≥ (90 − φd) (9.3)

or

ψd ≥ (90 − ψf) + φd (9.4)

9.3.3 Block alignment test

The other kinematic condition for toppling isthat the planes forming the blocks should strikeapproximately parallel to the slope face so thateach layer is free to topple with little constraintfrom adjacent layers. Observations of topples in

the field shows that instability is possible wherethe dip direction of the planes forming sides of theblocks, αd is within about 10◦ of the dip directionof the slope face αf , or

|(αf − αd)| < 10◦ (9.5)

The two conditions defining kinematic stabilityof topples given by equations (9.4) and (9.5)can be depicted on the stereonet (Figure 9.6(d)).On the stereonet, toppling is possible for planesfor which the poles lie within the shaded area,provided also that the base friction properties andshape of the blocks meet the conditions given byequations (9.1) and (9.2), respectively.

9.4 Limit equilibrium analysis of topplingon a stepped base

The method of toppling analysis described in thissection utilizes the same principles of limiting


H

∆x

a1

�f

(�f–�p)

b

yn

a2

n

1

2

�d

�s

�b

�p

Stable

Topple

Slide

Figure 9.7 Model forlimiting equilibriumanalysis of toppling on astepped base (Goodmanand Bray, 1976).

equilibrium that have been used throughout thisbook. While this method of analysis is limited toa few simple cases of toppling failure, it providesa basic understanding of the factors that areimportant in toppling, and allows stabilizationoptions to be evaluated. The stability analysisinvolves an iterative process in which the dimen-sions of all the blocks and the forces acting onthem are calculated, and then stability of each isexamined, starting at the uppermost block. Eachblock will either be stable, toppling or sliding,and the overall slope is considered unstable if thelowermost block is either sliding or toppling. Abasic requirement of this analysis is that the fric-tion angle on the base of each block is greaterthan the dip angle of the base so that slidingon the base plane does not occur in the absenceof any external force acting on the block (seeequation (9.1)).

The limit equilibrium method of analysis isideally suited to incorporating external forcesacting on the slope to simulate a wide varietyof actual conditions that may exist in the field.For example, if the lower block or blocks areunstable, then tensioned anchors with a specifiedtensile strength and plunge can be installed inthese blocks to prevent movement. Also, groundmotion due to earthquakes can be simulated by

a pseudo-static force acting on each block (seeSection 6.5.4), water forces can act on the baseand sides of each block, and loads produced bybridge foundations can be added to any specifiedblock (Wyllie, 1999).

As an alternative to the detailed analysisdescribed in this section, Zanbak (1983)developed a series of design charts that can beused to identify unstable toppling slopes, and toestimate the support force required for limitingequilibrium.

9.4.1 Block geometry

The first step in toppling analysis is to calculatethe dimensions of each block. Consider the regu-lar system of blocks shown in Figure 9.7 in whichthe blocks are rectangular with width �x andheight yn. The dip of the base of the blocks isψp and the dip of the orthogonal planes formingthe faces of the blocks is ψd(ψd = 90 − ψp). Theslope height is H , and the face is excavated atangle ψf while the upper slope above the crest isat angle ψs.

Angle of base plane (ψb). The base of the top-pling blocks is a stepped surface with an overalldip of ψb (Figure 9.7). Note that there is no


�

�

�

∆xUpslopeDownslope

Dilatancy movement, �

�p

�= ∆x (1 – cos �)

Figure 9.8 Dilatancy of topplingblocks with base plane coincidentwith normal to dip of blocks(Zanbak, 1983).

explicit means of determining a value of the para-meter ψb. However, it is necessary to use anappropriate value for ψb in the analysis becausethis has a significant effect on the stability of theslope. That is, as the base angle become flatter,the lengths of the blocks increase and there isa greater tendency of the taller blocks to toppleresulting in decreased stability of the slope. Ifthe base angle is coincident with the base of theblocks (i.e. ψb = ψp), then the geometry of top-pling requires dilatancy δ of the blocks along thebase plane and shearing on the faces of the blocks(Figure 9.8). However, if the base is stepped (i.e.ψb > ψp), then each block can topple withoutdilatancy, provided there is displacement on theface-to-face contacts (Figure 9.7). It is expectedthat more energy is required to dilate the rockmass than to develop shear along existing dis-continuities, and so a stepped base is more likelythan a planar base. Examination of base friction,centrifugal and numerical models (Goodman andBray, 1976; Pritchard and Savigny, 1990, 1991;Adhikary et al., 1997) show that base planes tendto be stepped, and the approximate dip angle is

in the range

ψb ≈ (ψp + 10◦) to (ψp + 30◦) (9.6)

It is considered that an appropriate stabilityanalysis procedure for situations where the valueof ψb is unknown is to carry out a sensitivityanalysis within the range given by equation (9.6)and find the value that gives the least stablecondition.

Based on the slope geometry shown inFigure 9.7, the number of blocks n making upthe system is given by

n= H

�x

[cosec(ψb) +

(cot(ψb) − cot(ψf)

sin(ψb − ψf)

)sin(ψs)

](9.7)

The blocks are numbered from the toe of theslope upwards, with the lowest block being 1 andthe upper block being n. In this idealized model,the height yn of the nth block in a position below


the crest of the slope is

yn = n (a1 − b) (9.8)

while above the crest

yn = yn−1 − a2 − b (9.9)

The three constants a1, a2 and b that are definedby the block and slope geometry and are given by

a1 = �x tan(ψf − ψp) (9.10)

a2 = �x tan(ψp − ψs) (9.11)

b = �x tan(ψb − ψp) (9.12)

9.4.2 Block stability

Figure 9.7 shows the stability of a system ofblocks subject to toppling, in which it is possibleto distinguish three separate groups of blocksaccording to their mode of behavior:

(a) A set of stable blocks in the upper part of theslope, where the friction angle of the baseof the blocks is greater than the dip of thisplane (i.e. φp > ψp), and the height is limitedso the center of gravity lies inside the base(y/�x < cot ψp).

(b) An intermediate set of toppling blocks wherethe center of gravity lies outside the base.

(c) A set of blocks in the toe region, whichare pushed by the toppling blocks above.Depending on the slope and block geometries,the toe blocks may be stable, topple or slide.

Figure 9.9 demonstrates the terms used to definethe dimensions of the blocks, and the posi-tion and direction of all the forces acting onthe blocks during both toppling and sliding.Figure 9.9(a) shows a typical block (n) with thenormal and shear forces developed on the base(Rn, Sn), and on the interfaces with adjacentblocks (Pn, Qn, Pn−1, Qn−1). When the block isone of the toppling set, the points of applicationof all forces are known, as shown in Figure 9.9(b).The points of application of the normal forces

Pn are Mn and Ln on the upper and lower facesrespectively of the block, and are given by thefollowing.

If the nth block is below the slope crest, then

Mn = yn (9.13)

Ln = yn − a1 (9.14)

If the nth block is the crest block, then

Mn = yn − a2 (9.15)

Ln = yn − a1 (9.16)

If the nth block is above the slope crest, then

Mn = yn − a2 (9.17)

Ln = yn (9.18)

For an irregular array of blocks, yn, Ln and Mn

can be determined graphically.When sliding and toppling occurs, frictional

forces are generated on the bases and sides ofthe blocks. In many geological environments, thefriction angles on these two surfaces are likely tobe different. For example, in a steeply dippingsedimentary sequence comprising sandstone bedsseparated by thin seams of shale, the shale willform the sides of the blocks, while joints in thesandstone will form the bases of the blocks. Forthese conditions, the friction angle of the sides ofthe blocks (φd) will be lower than friction angleon the bases (φp). These two friction angles can beincorporated into the limit equilibrium analysis asfollows.

For limiting friction on the sides of the block:

Qn = Pn tan φd (9.19)

Qn−1 = Pn−1 tan φd (9.20)

By resolving perpendicular and parallel to thebase of a block with weight Wn, the normal andshear forces acting on the base of block n are,


∆x

Qn – 1

Pn – 1

Pn – 1

Pn – 1 tan�d

Rn tan�b

Pn tan�d

�T

QnPn

Mnyn

Ln

�p

Rn

kn

Sntan�b

Pn

Wn

Sn Rn

T

L1Rn

Wn

Qn

Pn

n

n

Pn – 1 Qn – 1

(b)(a)

(c)

Figure 9.9 Limiting equilibrium conditions for toppling and sliding of nth block: (a) forces acting on nth block;(b) toppling of nth block; (c) sliding of nth block (Goodman and Bray, 1976).

respectively,

Rn = Wn cos ψp + (Pn − Pn−1) tan φd (9.21)

Sn = Wn sin ψp + (Pn − Pn−1) (9.22)

Considering rotational equilibrium, it is foundthat the force Pn−1 that is just sufficient to preventtoppling has the value

Pn−1,t = [Pn(Mn − �x tan φd) + (Wn/2)

(yn sin ψp − �x cos ψp)]/Ln (9.23)

When the block under consideration is one of thesliding set (Figure 9.9(c)),

Sn = Rn tan φp (9.24)

However, the magnitudes of the forces Qn−1, Pn−1and Rn applied to the sides and base of theblock, and their points of application Ln andKn, are unknown. Although the problem isindeterminate, the force Pn−1 required to pre-vent sliding of block n can be determined if it isassumed that Qn−1 = (tan φd · Pn−1). Then theshear force just sufficient to prevent sliding has


the value

Pn−1,s = Pn − Wn(cos ψp tan φp − sin ψp)

(1 − tan φp tan φd)

(9.25)

9.4.3 Calculation procedure for topplingstability of a system of blocks

The calculation procedure for examining topplingstability of a slope comprising a system of blocksdipping steeply into the faces is as follows:

(i) The dimensions of each block and thenumber of blocks are defined usingequations (9.7)–(9.12).

(ii) Values for the friction angles on the sidesand base of the blocks (φd and φp) areassigned based on laboratory testing, orinspection. The friction angle on the baseshould be greater than the dip of the baseto prevent sliding (i.e. φp > ψp).

(iii) Starting with the top block, equation (9.2)is used to identify if toppling will occur,that is, when y/�x > cot ψp. For the uppertoppling block, equations (9.23) and (9.25)are used to calculate the lateral forcesrequired to prevent toppling and sliding,respectively.

(iv) Let n1 be the uppermost block of thetoppling set.

(v) Starting with block n1, determine the lat-eral forces Pn−1,t required to prevent top-pling, and Pn−1,s to prevent sliding. IfPn−1,t > Pn−1,s, the block is on the point oftoppling and Pn−1 is set equal to Pn−1,t , orif Pn−1,s > Pn−1,t , the block is on the pointof sliding and Pn−1 is set equal to Pn−1,s.

In addition, a check is made that there isa normal force R on the base of the block,and that sliding does not occur on the base,that is

Rn > 0 and (|Sn| > Rn tan φp)

(vi) The next lower block (n1 − 1) and all thelower blocks are treated in succession using

the same procedure. It may be found that arelatively short block that does not satisfyequation (9.2) for toppling, may still toppleif the moment applied by the thrust force onthe upper face is great enough to satisfy thecondition stated in (v) above. If the condi-tion Pn−1,t > Pn−1,s is met for all blocks,then toppling extends down to block 1 andsliding does not occur.

(vii) Eventually a block may be reached forwhich Pn−1,s > Pn−1,t . This establishesblock n2, and for this and all lower blocks,the critical state is one of sliding. Thestability of the sliding blocks is checkedusing equation (9.24), with the block beingunstable if (Sn = Rn tan φb). If block 1 isstable against both sliding and toppling (i.e.P0 < 0), then the overall slope is consideredto be stable. If block 1 either topples orslides (i.e. P0 > 0), then the overall slope isconsidered to be unstable.

9.4.4 Cable force required to stabilize a slope

If the calculation process described inSection 9.4.3 shows that block 1 is unstable, thena tensioned cable can be installed through thisblock and anchored in stable rock beneath thezone of toppling to prevent movement. The designparameters for anchoring are the bolt tension, theplunge of the anchor and its position on block 1(Figure 9.9(c)).

Suppose that an anchor is installed at a plungeangle ψT through block 1 at a distance L1 aboveits base. The anchor tension required to preventtoppling of block 1 is

Tt = W1/2(y1 sin ψp − �x cos ψp) + P1(y1 − �x tan φd)

L1 cos(ψp + ψT)

(9.26)

while the required anchor tension to preventsliding of block 1 is

Ts = P1(1 − tan φp tan φd) − W1(tan φp cos ψp − sin ψp)

tan φp sin(ψp + ψT ) + cos(ψp + ψT )

(9.27)


When the force T is applied to block 1, the normaland shear force on the base of the block are,respectively,

R1 = P1 tan φd + T sin(ψp + ψT) + W1 cos ψp(9.28)

S1 = P1 − T cos(ψp + ψT) + W1 sin ψp(9.29)

The stability analysis for a slope with a tensionedanchor in block 1 is identical to that described inSection 9.4.3, apart from the calculations relatingto block 1. The required tension is the greater ofTt and Ts defined by equations (9.26) and (9.27).

9.4.5 Factor of safety for limiting equilibriumanalysis of toppling failures

For both reinforced and unreinforced slopes, thefactor of safety can be calculated by finding thefriction angle for limiting equilibrium. The pro-cedure is first to carry out the limiting equilibriumstability analysis as described in Section 9.4.3using the estimated values for the friction angles.If block 1 is unstable, then one or both of the fric-tion angles are increased by increments until thevalue of P0 is very small. Conversely, if block 1 isstable, then the friction angles are reduced untilP0 is very small. These values of the friction anglesare those required for limiting equilibrium.

The limiting equilibrium friction angles aretermed the required friction angles, while theactual friction angles of the block surfaces aretermed the available friction angles. The factorof safety for toppling can be defined by dividingthe tangent of the friction angle believed to applyto the rock layers (tan φavailable), by the tangentof the friction angle required for equilibrium(tan φrequired).

FS = tan φavailable

tan φrequired(9.30)

The actual factor of safety of a toppling slopedepends on the details of the geometry of the top-pling blocks. Figure 9.7 shows that once a column

overturns by a small amount, there are edge-to-face contacts between the blocks, and the frictionrequired to prevent further rotation increases.Hence, a slope just at limiting equilibrium is meta-stable. However, rotation equal to 2(ψb − ψp)

will convert the edge-to-face contacts along thesides of the columns into continuous face contactsand the friction angle required to prevent furtherrotation will drop sharply, possibly even belowthat required for initial equilibrium. The choiceof factor of safety, therefore, depends on whetheror not some deformation can be tolerated.

The restoration of continuous face-to-facecontact of toppled columns of rock is probablyan important arrest mechanism in large-scale top-pling failures. In many cases in the field, largesurface displacements and tension crack forma-tion can be observed and yet the volumes of rockthat fall from the face are small.

9.4.6 Example of limit equilibriumanalysis of toppling

The following is an example of the applicationof the Goodman and Bray limit equilibrium ana-lysis to calculate the factor of safety and requiredbolting force of the toppling failure illustrated inFigure 9.10(a).

A rock face 92.5 m high (H) is cut at an angleof 56.6◦(ψf) in a layered rock mass dipping at60◦ into the face (ψd = 60◦); the width of eachblock is 10 m (�x). The angle of the slope abovethe crest of the cut is 4◦(ψs), and the base of theblocks is stepped 1 m at every block (atn (1/10) =5.7◦, and ψb = (5.7+ψp) = 35.7◦). Based on thisgeometry, there are 16 blocks formed between thetoe and crest of the slope (equation 9.7); block10 is at the crest. Using equations (9.10)–(9.12),the constants are a1 = 5.0 m, a2 = 5.2 m andb = 1.0 m. These constants are used to calcu-late the height yn of each block, and the heightto width ratio yn/�x as shown on the table inFigure 9.10(b).

The friction angles on the faces and basesof the blocks are equal and have a value of38.15◦(φavailable). The unit weight of the rock is


92.5

m

4° 56.6°5.8°(a)

(b)

(c)

30°

T

Rn

Sn

12

3

4

5

6

7

8

9

1011 12 13 14 15

16

Sn

Rn

For

ce (

MN

)

Block

n yn Mn Ln Pn·t Pn·s Pn Rn Sn Sn /Rn Mode

STABLE________

________

SLIDING

0

4

1

5

3

7

8

2

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

16151413121110987654321

4.010.016.022.028.034.040.036.032.028.024.020.016.012.08.04.0

0.41.01.62.22.83.44.03.63.22.82.42.01.61.20.80.4

yn /∆x

172329353632282420161284

2228343531272319151173–

0000

292.5825.7

1556.02826.73922.14594.84837.04637.53978.12825.61103.1

–1485.1

0000

–2588.7–3003.2–3175.0–3150.8–1409.4

156.81300.12013.02284.12095.41413.5472.2

0000

292.5825.7

1556.02826.73922.14594.84837.04637.53978.12825.61413.5472.2

866216534634533.45643.36787.67662.16933.86399.85872.05352.94848.14369.43707.32471.41237.1

500125020002457.52966.83520.03729.33404.63327.33257.83199.53159.43152.52912.11941.3971.8

0.5770.5770.5770.5420.5260.5190.4870.4910.5200.5550.5980.6520.7220.78550.78550.7855

TOPPLING

Figure 9.10 Limited equilibrium analysis of a toppling slope: (a) slope geometry; (b) table listing blockdimensions, calculated forces and stability mode; (c) distribution of normal (R) and shear (S) forces on base ofblocks (Goodman and Bray, 1976).

25 kN/m3. It is assumed that the slope is dry, andthat there are no external forces acting.

The stability analysis is started by examiningthe toppling/sliding mode of each block, startingat the crest. Since the friction angle on the base

of the blocks is 38.15◦ and the dip of the base is30◦, the upper blocks are stable against sliding.Equation (9.2) is then used to assess the topplingmode. Since cot ψp = 1.73, blocks 16, 15 and14 are stable, because for each the ratio yn/�x is


less than 1.73. That is, these three blocks are shortand their center of gravity lies inside the base.

For block 13, the ratio yn/�x has the value 2.2,which is greater than 1.73 and the block topples.Therefore, P13 is equal to 0 and P12 is calcu-lated as the greater of P12,t and P12,s given byequations (9.23) and (9.25) respectively. Thiscalculation procedure is used to examine thestability of each block in turn progressing downthe slope. As shown in the table of forces inFigure 9.10(b), Pn−1,t is the larger of the twoforces until a value of n = 3, whereupon Pn−1,s islarger. Thus blocks 4 to 13 constitute the poten-tial toppling zone, and blocks 1 to 3 constitutea sliding zone.

The factor of safety of this slope can befound by increasing the friction angles until thebase blocks are just stable. It is found that therequired friction angle for limit equilibrium con-ditions is 39◦, so the factor of safety as givenby equation (9.30) is 0.97 (tan 38.15/ tan 39).The analysis also shows that the required tensionin an anchor installed horizontally in block 1to just stabilize the toe blocks is 500 kN permeter length of slope. This compares with themaximum value of P (in block 5) equal to4837 kN/m.

If tan φ is reduced to 0.650, it will be foundthat blocks 1 to 4 in the toe region will slidewhile blocks 5 to 13 will topple. The tension inan anchor installed horizontally through block1, required to restore equilibrium, is found tobe 2013 kN/m of slope crest. This is not alarge number, demonstrating that support of the“keystone” is remarkably effective in increas-ing the degree of stability. Conversely, remov-ing or weakening the keystone of a topplingslope that is near failure can have seriousconsequences.

When the distribution of P forces has beendefined in the toppling region, the forces Rn

and Sn on the base of the blocks can be calcu-lated using equations (9.21) and (9.22). Assuming[Qn−1 = Pn−1 tan φs], the forces Rn and Sn

can also be calculated for the sliding region.Figure 9.10(c) shows the distribution of theseforces throughout the slope. The conditions

defined by Rn > 0 and |Sn| < Rn tan φp aresatisfied everywhere.

9.4.7 Application of external forcesto toppling slopes

There may be circumstances where there areexternal forces acting on the slope and it isnecessary to investigate their effect on stability.Examples of external forces include water forcesacting on the sides and bases of the blocks, earth-quake ground motion simulated as a horizontalforce acting on each block (see Section 6.5.4),and point loads produced by bridge piers locatedon a specific block(s). Another external force thathas been considered in the worked example inSection 9.4.6 are rock anchors that are secured instable ground beneath the toppling mass and thentensioned against the face.

A feature of limit equilibrium analysis is thatany number of forces can be added to the analysis,provided that their magnitude, direction andpoint of application are known. Figure 9.11shows a portion of a toppling slope in whichthere is a sloping water table. The forces act-ing on block n include the force Q inclined at anangle ψQ below the horizontal, and three waterforces V1, V2 and V3, as well as the forces Pn andPn−1 produced by the blocks above and below.By resolving all these forces normal and parallelto the base of the blocks, it is possible to modifyequations (9.23) and (9.25) as follows. Consider-ing rotational equilibrium, the force Pn−1,t that isjust sufficient to prevent toppling of block n hasthe value:

Pn−1,t = {Pn(Mn − �x tan φd)

+ Wn/2(yn sin ψp − �x cos ψp)

+ V1yw/3 + γw�x2/6

× cos ψp(zw + 2yw) − V3zw/3

+ Q[− sin(ψQ − ψp)�x/2

+ cos(ψQ − ψp)yn]}L−1n (9.31)

Assuming that the blocks are in a state of limitingequilibrium, the force just sufficient to prevent


yn

Mn

yw

V1

Sn

Rn

Wn

V3

Zw

Ln

V2

∆x

�b

Pn

Pn – 1

�Q

Q

n + 1n

n – 1

P n ta

n � d

P n–

1 ta

n � d

Figure 9.11 Toppling block with external forces.

sliding of block n has the value

Pn−1,s = Pn + {−W(cos ψp tan φp − sin ψp)

+ V1 − V2 tan φp − V3

+ Q[− sin(ψQ − ψp) tan φp

+ cos(ψQ − ψp)]}× (1 − tan φp tan φd)−1 (9.32)

where

V1 = 12γw cos ψp · y2

w;

V2 = 12γw cos ψp(yw + zw)�x

V3 = 12γw cos ψpz2

w (9.33)

The limit equilibrium stability analysis thenproceeds as before using the modified versions ofthe equations for Pn−1,t and Pn−1,s.

9.5 Stability analysis of flexural toppling

Figure 9.3(b) shows a typical flexural topplingfailure in which the slabs of rock flex andmaintain fact-to-face contact. The mechanism offlexural toppling is different from the block top-pling mechanism described in Section 9.4. There-fore, it is not appropriate to use limit equilibriumstability analysis for design of toppling slopes.Techniques that have been used to study the sta-bility of flexural toppling include base frictionmodels (Goodman, 1976), centrifuges (Adhikaryet al., 1997) and numerical modeling (Pritchard


50 m

50 m

50 m

(a) (b)

(c)

Figure 9.12 UDEC model of toppling pit slope: (a) pure flexural toppling deformation with grid point velocityvectors; (b) contours of horizontal displacement for toppling slopes; (c) area of failed nodes due to flexure(Pritchard and Savigny, 1990).

and Savigny, 1990, 1991). All these models showthe common features of this failure mechanismincluding interlayer shearing, obsequent scarps atthe crest, opening of tension cracks that decreasein width with depth, and a limiting dip angleψb for the base of toppling. As discussed inSection 9.4.1, dip angle ψb is steeper than theplane ψp (normal to the dip of the slabs) byabout 10–30◦.

The centrifuge modeling by Adhikary et al. hasbeen used to develop a series of design charts thatrelate stability to the slope face angle, the dipof the blocks into the face and the ratio of theslope height to the width of the slabs. Anotherinput parameter is the tensile strength of the rock,because bending of the slabs induces tensile crack-ing in their upper face. The design charts provide

information, for example, on the allowable faceangle for specific geological conditions and slopeheight.

Alternatively, computer simulations of blockmovement provide a means of studying a widerange of geometric and material properties. Oneof the most suitable computer programs forthese simulations is UDEC (Universal DistinctElement Code), which has been developed byItasca Consulting Group in Minnesota (Itasca,2000). Figure 9.12 shows the results of an analysiscarried out on an open pit mine slope (Pritchardand Savigny, 1990). The main features of UDECanalysis for studying toppling slopes are that it

• incorporates a number of materials each withdiffering strength properties;


• recognizes the existence of contacts orinterfaces between discrete bodies, such asslabs of rocks formed by discontinuities dip-ping steeply into the slope face;

• calculates the motion along contacts byassigning a finite normal stiffness along thediscontinuities that separate the columns ofrock. The normal stiffness of a discontinuityis defined as the normal closure that occurson the application of a normal stress andcan be measured from a direct shear test (seeFigure 4.17);

• assumes deformable blocks that undergobending and tensile failure;

• allows finite displacements and rotationsof the toppling blocks, including completedetachment, and recognizes new contactsautomatically as the calculation progresses;

• uses an explicit “time”-marching scheme tosolve the equation of motion directly. Thisallows modeling of progressive failure, or theamount of creep exhibited by a series of top-pling blocks for a chosen slope condition,such as excavation at the toe of the slope.Note that the time step in the analysis is notactual time but a simulation of progressivemovement; and

• allows the user to investigate differentstabilization measures, such as installing rockbolts or installing drain holes, to determinewhich scenario has the most effect on blockmovements.

Because of the large number of input paramet-ers that are used in UDEC and the power of theanalysis, the most reliable results are obtained ifthe model can be calibrated against an existingtoppling failure in similar geological conditionsto those in the design slope. The ideal situationis in mining operations where the developmentof the topple can be simulated by UDEC as thepit is deepened and movement is monitored. Thisallows the model to be progressively updatedwith new data. Chapter 10 discusses numericalmodeling of slopes in more detail.

The application of kinematic stability testsand reinforcement design for a flexural toppling

failure is described by Davies and Smith (1993).The toppling occurred in siltstones in which thebeds were very closely spaced and dipped atbetween 90◦ and 70◦ into the face. Excavationfor a bridge abutment resulted in a series of ten-sion cracks along the crest, and stabilization of theslope required the installation of tensioned rockbolts and excavation to reduce the slope angle.

9.6 Example Problem 9.1: topplingfailure analysis

Statement

Consider a 6 m high slope with an overhangingface at an angle of 75◦ There is a fault, dippingat an angle of 15◦ out of the face, at the toe ofthe slope that is weathering and undercutting theface. A tension crack, which is wider at the topthan at the bottom, has developed 1.8 m behindthe crest of the slope indicating that the face ismarginally stable (Figure 9.13). The friction angleφ of the fault is 20◦ and the cohesion c is 25 kPa.The slope is dry.

Required

(a) Calculate the factor of safety of the blockagainst sliding if the density of the rock is23.5 kN/m3.

(b) Is the block stable against toppling as definedby the relation:

�x/y > tan ψp—stable?

� =15°

Y = 6 m

∆x = 1.8 m

Figure 9.13 Toppling block illustrating ExampleProblem 9.1.


(c) How much more undercutting of the faultmust occur before toppling failure takesplace?

(d) What stabilization measures would beappropriate for this slope?

Solution

(a) The factor of safety against sliding isdetermined by the methods described inChapter 6; the equation for a dry slope is

FS = cA + W cos ψp · tan φp

W sin ψp

= ((25 × 1.8) + (254 × cos 15

× tan 20))/(254 × sin 15)

≈ 2.0

where

A = base area of block

= 1.8 m2/m

W = weight of block/m

= 23.5 × 1.8 × 6

= 254 kN/m

(b) From the dimensions given on Figure 9.13,the following values are obtained to teststability conditions:

�x/y = 0.3

tan 15 = 0.27

The block is stable against toppling because0.3 > 0.27.

(c) If weathering results in the width of the baseof the block being reduced by a further 0.2 m,toppling is likely to occur because: �x/y =(1.8 − 0.2)/6 ≈ 0.27.

(d) Stabilization measures which could be usedon this slope include the following:

• Prevention of ground water infiltrationto limit build-up of water pressure bothin the tension crack and on the fault atthe toe.

• Application of reinforced shotcrete to thefault to prevent further weathering.

• Trim blasting to reduce the slope angleand the dimension y, if this can beachieved without destabilizing the block.

Chapter 10

Numerical analysis

Dr Loren Lorig and Pedro Varona1

10.1 Introduction

The previous four chapters discussed limit equi-librium methods of slope stability analysis forrock bounded by specified slide planes. In con-trast, this chapter discusses numerical analysismethods to calculate the factor of safety withoutpre-defining slide planes. These methods are morerecent developments than limit equilibrium meth-ods and, at present (2003), are used predomin-ately in open pit mining and landslides studies,where interest often focuses on slope displace-ments rather than on the relative magnitude ofresisting and displacing forces.

Numerical models are computer programs thatattempt to represent the mechanical response of arock mass subjected to a set of initial conditionssuch as in situ stresses and water levels, bound-ary conditions and induced changes such as slopeexcavation. The result of a numerical model sim-ulation typically is either equilibrium or collapse.If an equilibrium result is obtained, the resultantstresses and displacements at any point in the rockmass can be compared with measured values. If acollapse result is obtained, the predicted mode offailure is demonstrated.

Numerical models divide the rock mass intozones. Each zone is assigned a material modeland properties. The material models are idealizedstress/strain relations that describe how thematerial behaves. The simplest model is a linearelastic model, which uses the elastic properties

1 Itasca Consulting Group, Inc., Minneapolis, Minnesota55401 USA.

(Young’s modulus and Poisson’s ratio) of thematerial. Elastic–plastic models use strengthparameters to limit the shear stress that a zonemay sustain.2

The zones may be connected together, termed acontinuum model, or separated by discontinuities,termed a discontinuum model. Discontinuummodels allow slip and separation at explicitlylocated surfaces within the model.

Numerical models tend to be general purpose innature—that is, they are capable of solving a widevariety of problems. While it is often desirable tohave a general-purpose tool available, it requiresthat each problem be constructed individually.The zones must be arranged by the user to fit thelimits of the geomechanical units and/or the slopegeometry. Hence, numerical models often requiremore time to set up and run than special-purposetools such as limit equilibrium methods.

There are several reasons why numericalmodels are used for slope stability studies.

• Numerical models can be extrapolated confid-ently outside their databases in comparison toempirical methods in which the failure modeis explicitly defined.

• Numerical analysis can incorporate key geo-logic features such as faults and ground waterproviding more realistic approximations ofbehavior of real slopes than analytic models.

2 In numerical analysis the terms “elements” and “zones” areused interchangeably. However, the term element is usedmore commonly in finite element analysis, and the termzone in finite difference analysis.

Numerical analysis 219

In comparison, non-numerical analysis meth-ods such as analytic, physical or limit equi-librium may be unsuitable for some sites ortend to oversimplify the conditions, possiblyleading to overly conservative solutions.

• Numerical analysis can help to explainobserved physical behavior.

• Numerical analysis can evaluate multiple pos-sibilities of geological models, failure modesand design options.

Many limit equilibrium programs exist todetermine factors of safety for slopes. Theseexecute very rapidly, and in the case of the methodof slices for circular failure, use an approximatescheme in which a number of assumptions aremade, including the location and angle of inter-slice forces (see Section 8.2). Several assumed slidesurfaces are tested, and the one giving the lowestfactor of safety is chosen. Equilibrium is satis-fied only on an idealized set of surfaces. Withnumerical models, a “full” solution of the coupledstress/displacement, equilibrium and constitutiveequations is made. Given a set of properties, thesystem is found either to be stable or unstable.By performing a series of simulations with vari-ous properties, the factor of safety can be foundcorresponding to the point of stability.

The numerical analysis is much slower, butmuch more general. Only since the late 1990s,with the advent of faster computers, has it becomea practical alternative to the limit equilibriummethod. Even so, while limit equilibrium solu-tions may require just a few seconds, numericalsolutions to large complex problems can take anhour or more, particularly when discontinuumbehavior is involved. The third section of thischapter presents typical safety factor analyses forthe most common discontinuum failure modes inrock slopes.

For slopes, the factor of safety often is definedas the ratio of the actual shear strength to the min-imum shear strength required to prevent failure.A logical way to compute the factor of safety witha finite element or finite difference program is toreduce the shear strength until collapse occurs.The factor of safety is the ratio of the rock’s actual

strength to the reduced shear strength at failure.This shear-strength reduction technique was usedfirst with finite elements by Zienkiewicz et al.(1975) to compute the safety factor of a slopecomposed of multiple materials.

To perform slope stability analysis with theshear strength reduction technique, simulationsare run for a series of increasing trial factorsof safety (f ). Actual shear strength properties,cohesion (c) and friction angle (φ), are reducedfor each trial according to the equations

ctrial =(

1f

)c (10.1)

φtrial = arctan(

1f

)tan φ (10.2)

If multiple materials and/or joints are present, thereduction is made simultaneously for all materi-als. The trial factor of safety is increased graduallyuntil the slope fails. At failure, the factor of safetyequals the trial factor of safety (i.e. f = FS).Dawson et al. (1999) show that the shear strengthreduction factors of safety are generally within afew percent of limit analysis solutions when anassociated flow rule, in which the friction angleand dilation angle are equal, is used.

The shear strength reduction technique has twomain advantages over limit equilibrium slope sta-bility analyses. First, the critical slide surface isfound automatically, and it is not necessary tospecify the shape of the slide surface (e.g. circular,log spiral, piecewise linear) in advance. In general,the failure surface geometry for slopes is morecomplex than simple circles or segmented sur-faces. Second, numerical methods automaticallysatisfy translational and rotational equilibrium,whereas not all limit equilibrium methods do sat-isfy equilibrium. Consequently, the shear strengthreduction technique usually will determine asafety factor equal to or slightly less than limitequilibrium methods. Itasca Consulting Group(2002) gives a detailed comparison of four limitequilibrium methods and one numerical methodfor six different slope stability cases.

220 Numerical analysis

Table 10.1 Comparison of numerical and limit equilibrium analysis methods

Analysis result Numerical solution Limit equilibrium

Equilibrium Satisfied everywhere Satisfied only for specific objects,such as slices

Stresses Computed everywhere using fieldequations

Computed approximately on certainsurfaces

Deformation Part of the solution Not consideredFailure Yield condition satisfied everywhere;

slide surfaces develop “automatically”as conditions dictate

Failure allowed only on certainpre-defined surfaces; no check onyield condition elsewhere

Kinematics The “mechanisms” that develop satisfykinematic constraints

A single kinematic condition isspecified according to the particulargeologic conditions

A summary of the differences between anumerical solution and the limit equilibriummethod is shown in the Table 10.1.

10.2 Numerical models

All rock slopes involve discontinuities. Represent-ation of these discontinuities in numerical modelsdiffers depending on the type of model. There aretwo basic types of models: discontinuum modelsand continuum models. Discontinuities in discon-tinuum models are represented explicitly—thatis, the discontinuities have a specific orientationand location. Discontinuities in continuum mod-els are represented implicitly, with the intentionthat the behavior of the continuum model is sub-stantially equivalent to the real jointed rock massbeing represented.

Discontinuum codes start with a methoddesigned specifically to model discontinua andtreat continuum behavior as a special case. Dis-continuum codes generally are referred to asDiscrete Element codes. A Discrete Element codewill typically embody an efficient algorithm fordetecting and classifying contacts, and maintaina data structure and memory allocation schemethat can handle many hundreds or thousandsof discontinuities. The discontinuities divide theproblem domain into blocks that may be eitherrigid or deformable; continuum behavior is

assumed within deformable blocks. The mostwidely used discrete element codes for slopestability studies are UDEC (Universal DistinctElement Code; Itasca Consulting Group, 2000)and 3DEC (3-Dimensional Distinct ElementCode; Itasca Consulting Group, 2003). Of fun-damental importance in discontinuum codes isthe representation of joint or discontinuity beha-vior. Commonly used relations for represent-ing joint behavior are discussed later in thissection.

Continuum codes assume material is con-tinuous throughout the body. Discontinuitiesare treated as special cases by introducinginterfaces between continuum bodies. Finiteelement codes such as PHASE2 (Rocscience,2002c) and its predecessor PHASES (P lasticHybrid Analysis of Stress for Estimation ofSupport) and finite difference codes such as FLAC(Fast Lagrangian Analysis of Continua; ItascaConsulting Group, 2001) cannot handle gen-eral interaction geometry (e.g. many intersectingjoints). Their efficiency may degenerate drastic-ally when connections are broken repeatedly.Typical continuum-based models may have lessthan ten non-intersecting discontinuities. Of fun-damental importance to continuum codes anddeformable blocks in discrete element codesis representation of the rock mass behavior.Continuum relations used to represent rock massbehavior are discussed later in this section.


Finite element programs are probably morefamiliar, but the finite difference method isperhaps the oldest numerical technique used tosolve sets of differential equations. Both finiteelement and finite difference methods producea set of algebraic equations to solve. While themethods used to derive the equations are differ-ent, the resulting equations are the same. Finitedifference programs generally use an “explicit”time-marching scheme to solve the equations,whereas finite element methods usually solvesystems of equations in matrix form.

Although a static solution to a problem is usu-ally of interest, the dynamic equations of motionare typically included in the formulation of finitedifference programs. One reason for doing this isto ensure that the numerical scheme is stable whenthe physical system being modeled is unstable.With non-linear materials, there is always thepossibility of physical instability—for example,the failure of a slope. In real life, some of thestrain energy is converted to kinetic energy. Expli-cit finite-difference programs model this processdirectly, because inertial terms are included. Incontrast, programs that do not include inertialterms must use some numerical procedure totreat physical instabilities. Even if the procedureis successful at preventing numerical instability,the path taken may not be realistic. The con-sequence of including the full law of motion infinite difference programs is that the user musthave some physical feel for what is happening.Explicit finite-difference programs are not blackboxes that will “give the solution.” The behaviorof the numerical system must be interpreted.

FLAC and UDEC are two-dimensional finite-difference programs developed specifically forgeomechanical analysis. These codes can simu-late varying loading and water conditions, andhave several pre-defined material models for rep-resenting rock mass continuum behavior. Bothcodes are unique in their ability to handlehighly non-linear and unstable problems. Thethree-dimensional equivalents of these codes areFLAC3D (Fast Lagrangian Analysis of Continuain 3 Dimensions; Itasca Consulting Group, 2002)and 3DEC (Itasca Consulting Group, 2003).

10.2.1 Joint material models

The material model used most commonly torepresent joints is a linear-elastic–perfectly-plasticmodel. The limiting shear strength is defined bythe usual Mohr–Coulomb parameters of frictionangle and cohesion (see Section 4.2). A peak andresidual shear strength relation can also be spe-cified for the joints. The residual strength is usedafter the joint has failed in shear at the peakstrength. The elastic behavior of the joints is spe-cified by joint normal and shear stiffnesses, whichmay be linear or piece-wise linear.

10.2.2 Rock mass material models

It is impossible to model all discontinuities ina large slope, although it may be possible tomodel the discontinuities for a limited number ofbenches. Therefore, in large slopes much of therock mass must be represented by an equivalentcontinuum in which the effect of the discontinu-ities is to reduce the intact rock elastic propertiesand strength to those of the rock mass. Thisis true whether or not a discontinuum modelis used. As mentioned in the introduction tothis chapter, numerical models divide the rockmass into zones. Each zone is assigned a mater-ial model and material properties. The materialmodels are stress/strain relations that describehow the material behaves. The simplest modelis a linear elastic model that uses only the elasticproperties (Young’s modulus and Poisson’s ratio)of the material. Linear elastic–perfectly plasticstress–strain relations are the most commonlyused rock mass material models. These modelstypically use Mohr–Coulomb strength paramet-ers to limit the shear stress that a zone maysustain. The tensile strength is limited by the spe-cified tensile strength, which in many analysesis taken to be 10% of the rock mass cohe-sion. Using this model, the rock mass behavesin an isotropic manner. Strength anisotropy canbe introduced through a ubiquitous joint model,which limits the shear strength according to aMohr–Coulomb criterion in a specified direction.The direction often corresponds to a predominantjointing orientation.


A more complete equivalent-continuum modelthat includes the effects of joint orientationand spacing is a micropolar (Cosserat) plasticitymodel. The Cosserat theory incorporates a localrotation of material points as an independentparameter, in addition to the translation assumedin the classical continuum, and couple stresses(moments per unit area) in addition to the clas-sical stresses (forces per unit area). This model, asimplemented in FLAC, is described in the contextof slope stability by Dawson and Cundall (1996).The approach has the advantage of using a con-tinuum model while still preserving the abilityto consider realistic joint spacing explicitly. Themodel has not yet (as of 2003) been incorporatedinto any publicly available code.

The most common failure criterion for rockmasses is the Hoek–Brown failure criterion (seeSection 4.5). The Hoek–Brown failure criterionis an empirical relation that characterizes thestress conditions that lead to failure in intact rockand rock masses. It has been used successfullyin design approaches that use limit equilibriumsolutions. It also has been used indirectly innumerical models by finding equivalent Mohr–Coulomb shear strength parameters that providea failure surface tangent to the Hoek–Brown fail-ure criterion for specific confining stresses, orranges of confining stresses. The tangent Mohr–Coulomb parameters are then used in traditionalMohr–Coulomb type constitutive relations andthe parameters may or may not be updated duringanalyses. The procedure is awkward and time-consuming, and consequently there has been littledirect use of the Hoek–Brown failure criterionin numerical solution schemes that require fullconstitutive models. Such models solve for dis-placements, as well as stresses, and can continuethe solution after failure has occurred in some loc-ations. In particular, it is necessary to develop a“flow rule,” which supplies a relation betweenthe components of strain rate at failure. Therehave been several attempts to develop a full con-stitutive model from the Hoek–Brown criterion:for example, Pan and Hudson (1988), Carteret al. (1993) and Shah (1992). These formulationsassume that the flow rule has some fixed relation

to the failure criterion and that the flow rule isisotropic, whereas the Hoek–Brown criterion isnot. Recently, Cundall et al. (2003) has proposeda scheme that does not use a fixed form of the flowrule, but rather one that depends on the stresslevel, and possibly some measure of damage.

Real rock masses often appear to exhibit pro-gressive failure—that is, the failure appears toprogress over time. Progressive failure is a com-plex process that is understood poorly and diffi-cult to model. It may involve one or more of thefollowing component mechanisms:

• Gradual accumulation of strain on principalstructures and/or within the rock mass;

• Increases in pore pressure with time; and• Creep, which is time-dependent deformation

of material under constant load.

Each of these components is discussed briefly laterin the context of slope behavior.

Gradual accumulation of strain on principalstructures within the rock mass usually resultsfrom excavation, and “time” is related to theexcavation sequence. In order to study the pro-gressive failure effects due to excavation, onemust either introduce characteristics of the post-peak or post-failure behavior of the rock massinto a strain-softening model or introduce similarcharacteristics into the explicit discontinuities. Inpractice, there are at least two difficulties asso-ciated with strain-softening rock mass models.The first is estimating the post-peak strength andthe strain over which the strength reduces. Thereappear to be no empirical guidelines for estimat-ing the required parameters. This means that theproperties must be estimated through calibration.The second difficulty is that, for a simulation inwhich the response depends on shear localizationand in which material softening is used, the res-ults will depend on the zone sizes. However, itis quite straightforward to compensate for thisform of mesh-dependence. In order to do this,consider a displacement applied to the boundaryof a body. If the strain localizes inside the body,the applied displacement appears as a jump acrossthe localized band. The thickness of the band


contracts until it is equal to the minimum allowedby the grid, that is a fixed number of zone widths.Thus, the strain in the band is

ε = u/n�z (10.3)

where n is a fixed number, u is the displacementjump, and �z is the zone width.

If the softening slope is linear, the change in aproperty value �p is proportional to strain, thechange in property value with displacement is:

�p

�u= s

n · �z(10.4)

where s is the softening slope.In order to obtain mesh-independent results, a

scaled softening slope s can be input, such that

s = s′�z (10.5)

where s′ is constant.In this case, (�p/�u) is independent of �z. If

the softening slope is defined by the critical strain,εs

crit, then

εscrit ∝ 1

�z(10.6)

For example, if the zone size is doubled, thecritical strain must be halved for comparableresults.

Strain-softening models for discontinuities aremuch more common than similar relations forrock masses. Strain-softening relations for dis-continuities are built into UDEC and 3DEC, andcan be incorporated into interfaces in FLAC andFLAC3D via a built-in programming languagesuch as FISH functions. Strain-softening modelsrequire special attention when computing safetyfactors. If a strain-softening constitutive modelis used, the softening logic should be turned offduring the shear strength reduction process or thefactor of safety will be underestimated. Whenthe slope is excavated, some zones will haveexceeded their peak strength, and some amountof softening will have taken place. During the

strength reduction process, these zones should beconsidered as a new material with lower strength,but no further softening should be allowed dueto the plastic strains associated to the gradualreduction of strength.

Increases in pore pressure with time are notcommon in rock slopes for mines. More com-monly, the pore pressures reduce due to deepen-ing of the pit and/or drainage. However, there arecases in which the pore pressures do increase withtime. In such cases, the slope may appear to failprogressively.

Creep, which is time-dependent deformation ofmaterial under constant load, is not commonlyconsidered in the context of slope stability. Itis much more common in underground excav-ations. Several material models are available tostudy creep behavior in rock slopes. These includeclassical viscoelastic models, power law models,and the Burger-creep viscoplastic model. Applic-ation of a creep model to the study of slope beha-vior at Chuquicamata mine in Chile is discussedlater in this chapter (see Section 10.5.2).

10.3 Modeling issues

Modeling requires that the real problem be ideal-ized, or simplified, in order to fit the constraintsimposed by factors such as available materialmodels and computer capacity. Analysis of rockmass response involves different scales. It isimpossible—and undesirable—to include all fea-tures, and details of rock mass response mechan-isms, into one model. In addition, many of thedetails of rock mass behavior are unknown andunknowable; therefore, the approach to model-ing is not as straightforward as it is, say, in otherbranches of mechanics. This section discusses thebasic issues that must be resolved when setting upa numerical model.

10.3.1 Two-dimensional analysis versusthree-dimensional analysis

The first step in creating a model is todecide whether to perform two-dimensionalor three-dimensional analyses. Prior to 2003,


three-dimensional analyses were uncommon, butadvances in personal computers have permittedthree-dimensional analyses to be performedroutinely. Strictly speaking, three-dimensionalanalyses are recommended/required in thefollowing:

1 The direction of principal geologic structuresdoes not strike within 20–30◦ of the strike ofthe slope.

2 The axis of material anisotropy does not strikewithin 20–30◦ of the slope.

3 The directions of principal stresses are notneither parallel nor perpendicular to theslope.

4 The distribution of geomechanical units variesalong the strike of the slope.

5 The slope geometry in plan cannot be rep-resented by two-dimensional analysis, whichassumes axisymmetric or plain strain.

Despite the forgoing, most design analysis forslopes assumes a two-dimensional geometry com-prising a unit slice through an infinitely longslope, under plane strain conditions. In otherwords, the radii of both the toe and crest areassumed to be infinite. This is not the conditionencountered in practice—particularly in open pitmining where the radii of curvature can have animportant effect on safe slope angles. Concaveslopes are believed to be more stable than plainstrain slopes due to the lateral restraint providedby material on either side of a potential failure ina concave slope.

Despite its potential importance in slope sta-bility, very little has been done to quantify thiseffect. Jenike and Yen (1961) presented the resultsof limit theory analysis of axisymmetric slopes ina rigid, perfectly plastic material. However, Hoekand Brown (1981) concluded that the analysisassumptions were not applicable to rock slopedesign.

Piteau and Jennings (1970) studied the influ-ence of curvature in plan on the stability of slopesin four diamond mines in South Africa. As aresult of caving from below the surface, slopeswere all at incipient failure with a safety factor

of 1. The average slope height was 100 m. Piteauand Jennings (1970) found that the average slopeangle for slopes with radius of curvature of 60 mwas 39.5◦ as compared to 27.3◦ for slopes witha radius of curvature of 300 m.

Hoek and Bray (1981) summarize their experi-ence with the stabilizing effects of slope curvatureas follows. When the radius of curvature ofa concave slope is less than the height of theslope, the slope angle can be 10◦ steeper thanthe angle suggested by conventional stability ana-lysis. As the radius of curvature increases to avalue greater than the slope height, the correc-tion should be decreased. For radii of curvaturein excess of twice the slope height, the slope anglegiven by a conventional stability analysis shouldbe used.

To better quantify the effects of slope curvatureon stability, a series of generic analyses were per-formed. All analyses assumed a 500 m high dryslope with a 45◦ face angle excavated in an iso-tropic homogeneous material with a density of2600 kg/m3. Initial in situ stresses are assumedto be lithostatic, and the excavation was made in40 m decrements beginning from the ground sur-face. For these conditions, pairs of friction angleand cohesion values were selected to produce afactor of safety of 1.3 using circular failure chartnumber 1 (see Section 8.3). A factor of safety of1.3 is a value that is frequently used in the designof slopes for open pit mines. The actual valuesused are shown in Table 10.2.

A series of analyses was performed using FLACfor different radius of curvature for both concaveand convex slopes. For concave slopes, the radiusof curvature is defined as the distance between

Table 10.2 Janbu’s Lambda coefficient for variouscombinations of friction angle and cohesion

Friction angle Cohesion(MPa)

λ = γH tan φ/c

45 0.22 5935 0.66 1425 1.18 515 1.8 2


–10 –2–8 0 2 4 6 8 10–6

Slope height/radius of curvature

�= 15°�= 25°

�= 35°

�= 45°

–4

Convex slopes Concave slopes

FS

curv

ed s

lope

s

FS

plan

e st

rain

Figure 10.1 Results of FLAC axisymmetric analyses showing effect on factor of safety of slope curvature.

the axis of revolution and the toe of the slope.For the convex slopes, the radius of curvature isdefined as the distance between the axis of revolu-tion and the crest of the slope—not the toe. Underboth definitions, cones have a radius of curvatureof zero.

Figure 10.1 shows the results withFS/FSplane strain versus height/radius of curvature(H/Rc), which is positive for concave and negat-ive for convex slopes. The figure shows that thefactor of safety always increases as the radius ofcurvature decreases, but the increase is faster forconcave slopes. One unexpected result is that asthe friction angle increases, the effect of curvaturedecreases. One possible explanation is that asJanbu’s lambda coefficient (λ = γH tan φ/c)

increases, the slide surface is shallower with onlya skin for purely frictional material. This, makesthe slope less sensitive to the confining effect inconcave slopes, and to the ratio of active/passivewedges for the convex ones.

One reason that designers are reluctant totake advantage of the beneficial effects of con-cave slope curvature is that the presence ofdiscontinuities can often negate the effects.However, for massive rock slopes, or slopeswith relatively short joint trace lengths, the

beneficial effects of slope curvature should not beignored—particularly in open pit mines, wherethe economic benefits of steepening slopes can besignificant. The same is true for convex slopes,which are also more stable than plane strainslopes. This goes against observed experiencein rock slopes. If the slide surface is definedin terms of active (top) and passive (bottom)wedges, the ratio of the surface (and weight)of the passive wedge to the active wedge in aconvex slope is greater than the plane straincondition. However, this only applies to a homo-geneous Mohr–Coulomb material that mightbe found, for example, in waste dumps. Thereason why “noses” in rock slopes are usu-ally less stable may be related to the fact thatthey are more exposed to structurally controlledfailures.

10.3.2 Continuum versus discontinuummodels

The next step is to decide whether to use acontinuum code or a discontinuum code. Thisdecision is seldom straightforward. There appearto be no ready-made rules for determining whichtype of analysis to perform. All slope stability


problems involve discontinuities at one scale oranother. However, useful analyses, particularlyof global stability, have been made by assum-ing that the rock mass can be represented as anequivalent continuum. Therefore, many analysesbegin with continuum models. If the slope underconsideration is unstable without structure, thereis no point in going to discontinuum models. If,on the other hand, a continuum model appearsto be reasonably stable, explicit incorporation ofprincipal structures should give a more accurateestimation of slope behavior.

Selection of joint geometry for input to a modelis a crucial step in discontinuum analyses. Typic-ally, only a very small percentage of joints canactually be included in a model in order to createmodels of reasonable size for practical analysis.Thus, the joint geometry data must be filteredto select only those joints that are most crit-ical to the mechanical response. This is done byidentifying those that are most susceptible to slipand/or separation for the prescribed loading con-dition. This may involve determining whether suf-ficient kinematic freedom is provided, especiallyin the case of toppling, and calibrating the ana-lysis by comparing observed behavior to modelresponse.

10.3.3 Selecting appropriate zone size

The next step in the process is to select anappropriate zone size. The finite difference zonesassume that the stresses and strains within eachzone do not differ with position within the zone—in other words, the zones are the lowest-orderelements possible. In order to capture stress andstrain gradients within the slope adequately, it isnecessary to use relatively fine discretizations. Byexperience, the authors have found that at least20 (and preferably 30) zones are required overthe slope height of interest. As discussed later, ifflexural toppling is involved, a minimum of fourzones across the rock column are required. Finiteelement programs using higher-order elementslikely would require less zones than the constantstrain/constant stress elements common in finitedifference codes.

10.3.4 Initial conditions

Initial conditions are those conditions that existedprior to mining. The initial conditions ofimportance at mine sites are the in situ stressfield and the ground water conditions. The roleof stresses has been traditionally ignored in slopeanalyses. There are several possible reasons forthis:

• Limit equilibrium analyses, which are widelyused for stability analyses, cannot include theeffect of stresses in their analyses. Neverthe-less, limit equilibrium analyses are thoughtto provide reasonable estimates of stabilityin many cases, particularly where structure isabsent, such as soil slopes.

• Most stability analyses have traditionally beenperformed for soils, where the range of pos-sible in situ stresses is more limited than forrocks. Furthermore, many soil analyses havebeen performed for constructed embankmentssuch as dams, where in situ stresses do notexist.

• Most slope failures are gravity driven, andthe effects of in situ stress are thought to beminimal.

• In situ stresses in rock masses are not routinelymeasured for slopes, and their effects arelargely unknown.

One particular advantage of stress analysis pro-grams such as numerical models is their ability toinclude pre-mining initial stress states in stabilityanalyses and to evaluate their importance.

In order to evaluate the effects of in situ stressstate on stability, five cases were run using asimple model similar to the model shown inFigure 10.2. For each of the five cases, the slopeangle was 60◦, the slope height was 400 m, thematerial density was 2450 kg/m3, the frictionangle was 32◦, and the cohesion was 0.92 MPa.The results of FLAC analyses are shown inTable 10.3.

In general, it is impossible to say what effectthe initial stress state will have on any partic-ular problem, as behavior depends on factors


Density = 2600 kg/m3

Friction angle = 35°Cohesion = 660 kPa

45°Step –138.9 < x < 2639 –1264 < y < 1514

0 500

1500500

Horizontal axis (m)

1000 2000

Ver

tical

axi

s (m

)

–1000

0

1000

–500

500

Water table

Grid plot

FLAC (Version 3.40)

Legend

Figure 10.2 Problem geometry used to determine the effect of in situ stresses on slope stability.

Table 10.3 Effect of in situ stress on slope stability[x—horizontal in-plane direction; y—vertical in-planedirection; z—out-of-plane direction]

In-planehorizontal stress

Out-of-planehorizontal stress

Factor ofsafety

σxx = σyy σzz = σyy 1.30σxx = 2.0 σyy σzz = 2.0 σyy 1.30σxx = 0.5 σyy σzz = 0.5 σyy 1.28σxx = 2.0 σyy σzz = 0.5 σyy 1.30σxx = 0.5 σyy σzz = 2.0 σyy 1.28

such as orientation of major structures, rock massstrength and water conditions. However, someobservations on the effects of in situ stress onstability can be made:

• The larger the initial horizontal stresses, thelarger the horizontal elastic displacements.This is not much help, as elastic displacements

are not particularly important in slopestudies.

• Initial horizontal stresses in the plane of ana-lysis that are less than the vertical stressestend to slightly decrease stability and reducethe depth of significant shearing with respectto a hydrostatic stress state. This observa-tion may seem counter-intuitive; smaller hori-zontal stresses would be expected to increasestability. The explanation lies in the fact thatthe lower horizontal stresses actually provideslightly decreased normal stress on potentialshearing surfaces and/or joints within theslope. This observation was confirmed in aUDEC analysis of a slope in Peru wherein situ horizontal stresses lower than the ver-tical stress led to deeper levels of joint shear-ing in toppling structures compared to casesinvolving horizontal stresses that were equalto or greater than the vertical stress.


• It is important to note that the regionaltopography may limit the possible stressstates, particularly at elevations aboveregional valley floors. Three-dimensionalmodels have been very useful in the past inaddressing some regional stress issues.

10.3.5 Boundary conditions

Boundaries are either real or artificial. Realboundaries in slope stability problems corres-pond to the natural or excavated ground surfacethat is usually stress free. Artificial boundariesdo not exist in reality. All problems in geo-mechanics, including slope stability problems,require that the infinite extent of a real prob-lem domain be artificially truncated to includeonly the immediate area of interest. Figure 10.3shows typical recommendations for locations ofthe artificial far-field boundaries in slope sta-bility problems. Artificial boundaries can be oftwo types: prescribed displacement, or prescribedstress. Prescribed displacement boundaries inhibitdisplacement in either the vertical direction orhorizontal direction, or both. Prescribed displace-ment boundaries are used to represent the condi-tion at the base of the model and toe of the slope.

Displacement at the base of the model is alwaysfixed in both the vertical and horizontal directionsto inhibit rotation of the model. Two assumptions

W

H

�

>H/2

>W

Figure 10.3 Typical recommendations for locationsof artificial far-field boundaries in slope stabilityanalyses.

can be made regarding the displacement bound-aries near the toe of any slope. One assumption isthat the displacements near the toe are inhibitedonly in the horizontal direction. This is the mech-anically correct condition for a problem that isperfectly symmetric with respect to the plane oraxis representing the toe boundary. Strictly speak-ing, this condition only occurs in slopes of infinitelength, which are modeled in two-dimensionsassuming plane strain, or in slopes that are axi-ally symmetric in which the pit is a perfect cone.In reality, these conditions are rarely satisfied.Therefore, some models are extended laterally toavoid the need to specify any boundary condi-tion at the toe of the slope. It is important tonote that difficulties with the boundary conditionnear the slope toe are usually a result of the two-dimensional assumption. In three-dimensionalmodels, this difficulty generally does notexist.

The far-field boundary location and condi-tion must also be specified in any numericalmodel for slope stability analyses. The generalnotion is to select the far-field location so thatit does not significantly influence the results.If this criterion is met, whether the boundaryis prescribed-displacement or prescribed-stress isnot important. In most slope stability studies, aprescribed-displacement boundary is used. Theauthors have used a prescribed-stress bound-ary in a few cases and found no significantdifferences with respect to the results from aprescribed-displacement boundary. The mag-nitude of the horizontal stress for the prescribed-stress boundary must match the assumptionsregarding the initial stress state in order for themodel to be in equilibrium. However, followingany change in the model, such as an excavationincrement, the prescribed-stress boundary causesthe far-field boundary to displace toward theexcavation while maintaining its original stressvalue. For this reason, a prescribed-stress bound-ary is also referred to as a “following” stress, orconstant stress boundary, because the stress doesnot change and follows the displacement of theboundary. However, following stresses are mostlikely where slopes are cut into areas where the


topography rises behind the slope. Even whereslopes are excavated into an inclined topography,the stresses would flow around the excavationto some extent, depending on the effective widthof the excavation perpendicular to the downhilltopographic direction.

A summary of the effects of boundary condi-tions on analysis results is as follows:

• A fixed boundary causes both stresses anddisplacements to be underestimated, whereasa stress boundary does the opposite.

• The two types of boundary condition“bracket” the true solution, so that it is pos-sible to conduct tests with smaller modelsto obtain a reasonable estimate of the truesolution by averaging the two results.

A final point to be kept in mind is that all open pitslope stability problems are three-dimensional inreality. This means that the stresses acting in andaround the pit are free to flow both beneath andaround the sides of the pit. Therefore, it is likelythat, unless there are very low strength faultsparallel to the analysis plane, a constant stressor following stress boundary will over-predict thestresses acting horizontally.

10.3.6 Incorporating water pressure

The effect of water pressure in reducing effectivestresses and, hence, slope stability is well under-stood. However, the effect of various assump-tions regarding specification of pore pressuredistributions in slopes is not as well understood.Two methods are commonly used to specifypore pressure distributions within slopes. Themost rigorous method is to perform a completeflow analysis, and use the resultant pore pres-sures in the stability analyses. A less rigorous,but more common method is to specify a watertable, and the resulting pore pressures are givenby the product of the vertical depth below thewater table, the water density and gravity. In thissense, the water table approach is equivalent tospecifying a piezometric surface. Both methodsuse similar phreatic surfaces. However, the watertable method under-predicts actual pore pressure

concentrations near the toe of a slope, and slightlyover-predicts the pore pressure behind the toe byignoring the inclination of equipotential lines.

Seepage forces must also be considered in theanalysis. The hydraulic gradient is the differencein water pressure that exists between two pointsat the same elevation, and results from seepageforces (or drag) as water moves through a porousmedium. Flow analysis automatically accountsfor seepage forces.

To evaluate the error resulting from specifyinga water table without doing a flow analysis, twoidentical problems were run. In one case, a flowanalysis was performed to determine the porepressures. In the second case, the pressures weredetermined using only a piezometric surface thatwas assumed to be the phreatic surface taken fromthe flow analysis. The material properties andgeometry for both cases are shown in Figure 10.2.The right-hand boundary was extended to allowthe far-field phreatic surface to coincide with theground surface at a horizontal distance of 2 kmbehind the toe. Hydraulic conductivity within themodel was assumed to be homogeneous and iso-tropic. The error caused by specifying the watertable can be seen in Figure 10.4. The largesterrors, under prediction of up to 45%, are foundjust below the toe, while over prediction errorsin pore pressure values behind the slope are gen-erally less than 5%. The errors near the phreaticsurface are insignificant, as they result from therelatively small pore pressures just below thephreatic surface where small errors in small valuesresult in large relative errors.

For a phreatic surface at the ground surfaceat a distance of 2 km, a factor of safety of 1.1is predicted using circular failure chart number3 (refer to Section 8.3). The factor of safetydetermined by FLAC was approximately 1.15 forboth cases. The FLAC analyses give similar safetyfactors because the distribution of pore pressuresin the area behind the slope where failure occursis very similar for the two cases. The conclusiondrawn here is that there is no significant differ-ence in predicted stability between a completeflow analysis and simply specifying a piezometricsurface. However, it is not clear if this conclusion


–4.5

–5.0

Contour interval = 5.00E-02(zero contour omitted)

–3.5–2.5–1.5

1000

Horizontal axis (m)

Ver

tical

axi

s (m

)

200 600 1400 1800

–800

0

800

–400

400

Pore pressuresunderpredicted bywater table in thisarea

Pore pressuresoverpredicted bywater table in thisarea

FLAC (Version 4.00)

Legend

Step 20000

Pore pressure error

Cons. time 1.3380E + 11

Figure 10.4 Error in pore pressure distribution caused by specifying water table compared to performing a flowanalysis.

can be extrapolated to other cases involving, forexample, anisotropic flow.

10.3.7 Excavation sequence

Simulating excavations in numerical modelsposes no conceptual difficulties. However, theamount of effort required to construct a modeldepends directly on the number of excavationstages simulated. Therefore, most practical ana-lyses seek to reduce the number of excavationstages. The most accurate solution is obtainedusing the largest number of excavation steps,because the real load path for any zone in theslope will be followed closely. In theory, it isimpossible to prove that the final solution is inde-pendent of the load path followed. However, formany slopes, stability seems to depend mostlyon slope conditions, such as geometry and porepressure distribution at the time of analysis, andvery little on the load path taken to get there.

A reasonable approach regarding the numberof excavation stages has evolved over the years.Using this approach, only one, two or threeexcavation stages are modeled. For each stage,two calculation steps are taken. In the first step,the model is run elastically to remove any iner-tial effects caused by sudden removal of a largeamount of material. Second, the model is runallowing plastic behavior to develop. Followingthis approach, reasonable solutions to a largenumber of slope stability problems have beenobtained.

10.3.8 Interpretation of results

As noted in the introduction, finite differenceprograms are not black boxes that “give thesolution.” The behavior of the numerical systemand results from finite difference models mustbe interpreted in much the same way as slopemovement data are interpreted. Finite difference


programs record displacements and velocities atnominated points within the rock mass. Duringthe analysis, the recorded values can be examinedto see if they are increasing, remaining steady, ordecreasing. Increasing displacements and velocit-ies indicate an unstable situation; steady displace-ments and decreasing velocities indicate a stablesituation. In addition, velocity and displacementvectors for every point in the model can be plot-ted. Fields of constant velocity and displacementindicate failure.

The authors have found that velocities below1e−6 indicate stability in FLAC and FLAC3D;conversely, velocities above 1e−5 indicateinstability. Note that no units are given for velocit-ies. This is because the velocities are not real, dueto the damping and mass scaling used to achievestatic solutions. While the displacements are real,the velocities are not, and there is no informationon the “time” the displacement occurs.

It is also possible to examine the failure (plas-ticity) state of points within the model, wherefailure is defined as failure in tension or shear.Care must be used in examining the failure stateindicators. For example, local overstressing atthe base of toppling columns can appear to forma deep-seated slip surface when, in reality, it isjust compressive failure of the columns. There-fore, the failure (plasticity) indicators must bereviewed in the context of overall behavior beforeany definitive conclusions can be drawn.

10.4 Typical stability analysis

In this section, typical stability analyses for a vari-ety of failure modes are discussed. The objectiveof this section is to show how numerical modelscan be used to simulate slope behavior, and com-pute safety factors for typical problems. Modelingissues important in each of the following failuremodes are discussed:

• Rock mass failure;• Plane failure—daylighting and

non-daylighting;• Wedge failure—daylighting and

non-daylighting;

• Toppling failure—block and flexural; and• Flexural buckling failure.

The two-dimensional distinct element code,UDEC, was used for most of the analyses, and thecharacteristics of the slopes that were modeled areas follows:

Slope height 260 mSlope angle 55◦Water pressure none/dryDensity 2660 kg/m3

(26.1 kN/m3)

Rock mass friction angle 43◦Rock mass cohesion 675 kPaRock mass tension 0Rock mass bulk modulus 6.3 GPaRock mass shear modulus 3.6 GPaJoint friction angle 40◦Dilation angle 0◦

A maximum zone size of 15 m was used exceptwhere noted. In all cases, the factor of safetywas estimated by simultaneously reducing therock mass properties and the joint friction untilfailure occurs using the procedure shown in equa-tions (10.1) and (10.2). The safety factor wasassumed to be the reciprocal of the reductionrequired to produce failure. For example, if thestrengths must be reduced by 25% (i.e. 75% oftheir “real” strength) in order to achieve failure,the safety factor is 1.33.

10.4.1 Rock mass failure

Numerical analysis of slopes involving purelyrock mass failure is studied most efficiently usingcontinuum codes such as PHASE2, FLAC orFLAC3D. As mentioned in the previous section,discontinuities are not considered explicitly incontinuum models; rather, they are assumed tobe smeared throughout the rock mass. Assumingthat rock mass shear strength properties can beestimated reasonably, the analysis is straightfor-ward. The process for initially estimating the rockmass properties is often based on empirical rela-tions as described, for example, by Hoek and


–100

200

500

0

300

100

400

200 4001000

Horizontal axis (m)

300 500 600

UDEC (Version 3.20)

Legend

Boundary plotUser defined grid valueDisplacement contour interval = 0.5

Ver

tical

axi

s (m

)

0.51.01.52.02.5

0.0

Figure 10.5 Rock mass failure mode for slope determined with UDEC.

Brown (1997). These initial properties are thenmodified, as necessary, through the calibrationprocess.

Failure modes involve mainly shearing throughthe rock mass. For homogeneous slopes wherethe slide surface is often approximately circular,intersecting the toe of the slope and becom-ing nearly vertical near the ground surface. Thefailure mode for the parameters listed earlier isshown in Figure 10.5. The calculated safety factoris 1.64.

The following is a comparison of a slopestability analysis carried out using limit equilib-rium circular failure analysis (Bishop method)and numerical stability analysis. In Chapter 8, thestability of a benched slope in strong, but closelyfractured, sandstone including a water table andtension crack is described (see Figure 8.19). Therock mass is classified as a Hoek–Brown materialwith strength parameters:

mi = 0.13GSI = 20

σc = 150 MPaDisturbance factor, D = 0.7

The tensile strength is estimated to be 0.012 MPa.For the Bishop’s analysis method, the Mohr–Coulomb strength is estimated by fitting a straightline to the curved Hoek–Brown failure envelopeat the normal stress level estimated from the slopegeometry. Using this procedure, friction angleand cohesion were

φ = 43◦

c = 0.145 MPa

The mass density of the rock mass and waterwere 2550 kg/m3 (25.0 kN/m3) and 1000 kg/m3

(9.81 kN/m3) respectively. The phreatic surfaceis located as shown in Figure 8.19. Based uponthese parameters, the Bishop method produces alocation for the circular slide surface and tensioncrack, as shown in Figure 8.19, and a factor ofsafety of 1.39.


In using FLAC to analyze the stability of theslope in Figure 8.19, the slide surface can evolveduring the calculation in a way that is represent-ative of the natural evolution of the physical slidesurface in the slope. It is not necessary to makean estimate for the location of the circular slidesurface when beginning an analysis, as it is withlimit equilibrium methods. FLAC will find theslide surface and failure mechanism by simulat-ing the material behavior directly. A reasonablyfine grid should be selected to ensure that the slidesurface will be well defined as it develops. It is bestto use the finest grid possible when studying prob-lems involving localized failure. Here, a zone sizeof 2 m was used.

The FLAC analysis showed a factor of safety of1.26, with the slide surface closely resembling thatproduced from the Bishop solution (Figure 10.6).However, the tensile failure extends farther upthe slope in the FLAC solution. It is importantto recognize that the limit equilibrium solutiononly identifies the onset of failure, whereas the

FLAC solution includes the effect of stress redis-tribution and progressive failure after movementhas been initiated. In this problem, tensile failurecontinues up the slope as a result of the tensilesoftening. The resulting factor of safety allowsfor this weakening effect.

10.4.2 Plane failure—daylighting andnon-daylighting

Failure modes that involve rigid blocks sliding onplanar joints that daylight in the slope face aremost efficiently solved using analytical methods.For comparison purposes, a UDEC analysis is per-formed for blocks dipping at 35◦ out of the slope.The joints are assumed to have a cohesion of100 kPa and a friction angle of 40◦. The resultingsafety factor is 1.32, which agrees with the ana-lytic value given by equation (6.4) in Chapter 6,assuming that no tension crack forms. The planefailure mode in the UDEC analysis is shown inFigure 10.7.

– 40

20

80

–20

40

0

60

– 40 –20 20 600 40 80 100

Contour interval = 0.002

Max. shear strain increment

Boundary plot

200

Horizontal axis (m)

FLAC (Version 4.00)

Ver

tical

axi

s (m

)

Legend

0 0.004 0.008 0.012 0.016 0.020 0.024

Figure 10.6 Failure mode and tension crack location determined with FLAC for slope in closely fracturedsandstone slope (refer to Figure 8.19).


Cycle 487840

100 200 400 600300

Horizontal axis (m)

500 700

Ver

tical

axi

s (m

)

–100

200

500

0

300

100

400

0

Block plotVelocity vectors

Maximum = 0.043

0.20

UDEC (Version 3.10)

Legend

Figure 10.7 Plane failure mode with rigid blocks determined with UDEC.

If a tension crack does form, then the factorof safety is slightly reduced. Deformable blockswith elastic–plastic behavior are required to formtension cracks within the UDEC analysis. Whendeformable zones are used, the resultant safetyfactor is 1.27, similar to the value of 1.3 given bythe analytic solution. The difference may be thatthe analytic solution assumes a vertical tensioncrack, whereas the UDEC analysis indicates thatthe tension crack curves where it meets the slidingplane (see Figure 10.8).

Similar analyses can be performed for non-daylighting failure planes. In this case, failureinvolves sliding on discontinuities and shearingthrough the rock mass at the toe of the slope, asshown in Figure 10.9. Here, the cohesionless slid-ing planes dip at 70◦ and are spaced 20 m apart.The resultant safety factor is about 1.5.

10.4.3 Wedge failure—daylighting andnon-daylighting

Analyses involving wedge failures are similar tothose involving plane failures, except that the

analyses must be performed in three dimensions.As with plane failure, sliding analysis of day-lighting rigid blocks is best solved using analyticmethods, as described in Chapter 7. Analysesinvolving formation of tension cracks and/or non-daylighting wedges require numerical analysis.Candidate codes include FLAC3D and 3DEC.The plasticity formulation in FLAC3D uses amixed discretization technique and presentlyprovides a better solution than 3DEC in caseswhere rock mass failure dominates. On the otherhand, setting up problems involving more thanone sliding plane in FLAC3D is more difficult andtime consuming than similar problems in 3DEC.

10.4.4 Toppling failure—block and flexural

Toppling failure modes involve rotation and thususually are difficult to solve using limit equilib-rium methods. As the name implies, block top-pling involves free rotation of individual blocks(Figure 9.3(a)), whereas flexural toppling involvesbending of rock columns or plates (Figure 9.3(b)).


X displacement contoursContour interval = 0.2

Cycle 541651

(zero contour line omitted)

Block plot

100 200 400 600300

Horizontal axis (m)

500 700

Ver

tical

axi

s (m

)

–100

200

500

0

300

100

400

0

UDEC (Version 3.20)

Legend

0.20.40.60.81.01.21.41.61.8

Figure 10.8 Plane failure mode with deformable blocks determined with UDEC.

Y displacement contoursContour interval = 0.2

100 200 400 600300

Horizontal axis (m)

500 700

Ver

tical

axi

s (m

)

–100

200

500

0

300

100

400

0

UDEC (Version 3.20)

Cycle 1246860 Time 1.286E + 03 sec

Legend

–0.6–0.4–0.2 0.2


Block plot

Figure 10.9 Non-daylighting plane failure mode determined with UDEC.



100 200 400 600300

Horizontal axis (m)

500 700

Ver

tical

axi

s (m

)

–100

200

500

0

300

100

400

0

UDEC (Version 3.20)

Cycle 1153501Time 1.451E + 03 sec

Legend

–10–12

–8–6–4–20


Block plot

Figure 10.10 Forward block toppling failure mode determined with UDEC.

Block toppling occurs where narrow slabsare formed by joints dipping steeply into theface, combined with flatter cross-joints (seeSection 9.4). The cross-joints provide releasesurfaces for rotation of the blocks. In the mostcommon form of block toppling, the blocks,driven by self-weight, rotate forward out ofthe slope. However, backward or reverse top-pling can also occur when joints parallel to theslope face and flatter cross-joints are particularlyweak. In cases of both forward and backwardtoppling, stability depends on the location ofthe center of gravity of the blocks relative totheir base.

Figure 10.10 shows the results of an analysisinvolving forward block toppling. The steep jointset dips at 70◦ with a spacing of 20 m. The cross-joints are perpendicular and are spaced at 30 m.The resultant safety factor is 1.13. Figure 10.11shows the result of an analysis involving back-ward block toppling. In this case, the face-paralleljoints are spaced at 10 m, and the horizontal jointsare spaced at 40 m. The factor of safety for thisfailure mode is 1.7.

Flexural toppling occurs when there is onedominant, closely spaced, set of joints dip-ping steeply into the face, with insufficientcross-jointing to permit free rotation of blocks.The columns bend out of the slope like cantileverbeams. Figure 10.12 shows the results of analysiswith joints spaced at 20 m. The factor of safetyis 1.3, with the safety factor being reduced asthe joint spacing decreases. Problems involvingflexural toppling require finer zoning than prob-lems involving block toppling. Because flexuraltoppling involves high stress gradients across anyrock column, it is necessary to provide sufficientzones to represent accurately the stress gradientsdue to bending. In the modeling of centrifugetests reported by Adhikary and Guo (2000),UDEC modeling required four zones across eachcolumn, resulting in a model with nearly 20,000three-noded triangular zones. In contrast, a finiteelement model with Cosserat plasticity elementsrequired only about 1200, eight-noded isopara-metric quadrilateral elements. Both models pro-duced good agreement with the laboratory results(see also Section 9.5).


Figure 10.11 Reverse(backward) block topplingfailure mode determined withUDEC; arrows showmovement vectors.

10.4.5 Flexural buckling failure

Buckling failures (see Figure 10.13) are alsodifficult to reproduce in numerical modelsbecause of the large number of zones requiredto represent the high stress gradients involved inbuckling. One of the most complete studies onthe topic of numerical analysis of buckling inrock slopes is given by Adhikary et al. (2001),who provide design charts based on numericalanalysis using the proprietary finite element pro-gram AFENA (Carter and Balaam, 1995) and aCosserat material model to simulate behavior offoliated rock.

10.5 Special topics

10.5.1 Reinforcement

Reinforcement is often used to stabilize civilslopes, and occasionally critical mine slopes.

Three different types of reinforcement can berepresented in numerical models:

• fully grouted rock bolts (local reinforcement);• cable bolts; and• end-anchored rock bolts.

The basic formulation for each type of reinforce-ment is discussed briefly.

The local reinforcement formulation considersonly the local effect of reinforcement where itpasses through existing discontinuities. This con-dition immediately implies that some form ofdiscontinuous behavior is being modeled in therock mass. The formulation results from obser-vations of laboratory tests of fully grouted unten-sioned reinforcement in good quality rock withone discontinuity, which indicate that strains inthe reinforcement are concentrated across the dis-continuity (Bjurstrom, 1974; Pells, 1974; Spangand Egger, 1990). This behavior can be achieved



100 200 400 600300

Horizontal axis (m)

500 700

–100

200

500

0

300

100

400

0

UDEC (Version 3.20)

Cycle 750880Time 1.083E + 03 sec

Legend

Ver

tical

axi

s (m

)

–10–12

–8–6–4–20


Block plot

Figure 10.12 Flexural toppling failure mode determined with UDEC.

(a) (b)

Figure 10.13 A schematicrepresentation of slopes in a foliatedrock mass: (a) flexural toppling; and(b) flexural buckling (Adhikary et al.,2001).

in the computational model by calculating, foreach zone, the forces generated by deformation ofan “active length” of the element where it crossesa discontinuity (see Figure 6.9). This formula-tion exploits simple force–displacement relationsto describe both the shear and axial behaviors ofreinforcement across discontinuities. Large sheardisplacements are accommodated by consider-ing the simple geometric changes that developlocally in the reinforcement near a discontinu-ity. Although the local reinforcement model canbe used with either rigid blocks or deformable

blocks, the representation is most applicable tocases in which deformation of individual rockblocks may be neglected in comparison withdeformation of the reinforcing system. In suchcases, attention may be focused reasonably onthe effect of reinforcement near discontinuities.The original description of a local reinforcementmodel is given by Lorig (1985).

In assessing the support provided by rockreinforcement, two components of restraintshould be considered. First, the reinforce-ment provides local restraint where it crosses


m

m

m

Excavation

Reinforcing element (steel)

Grout annulus

Shear stiffness of grout

Axial stiffnessof steel

Slider (cohesivestrength of grout)

Reinforcement(nodal point)

Figure 10.14 Conceptual mechanicalrepresentation of fully bondedreinforcement, accounting for shearbehavior of the grout annulus.

discontinuities. Second, there is restraint to intactrock due to inelastic deformation in the failedregion surrounding an excavation. Such situ-ations arise in modeling inelastic deformationsassociated with failed rock and/or reinforcementsystems such as cable bolts, in which the cementor resin grout bonding agent may fail in shearover some length of the reinforcement. Cable ele-ments allow the modeling of a shearing resistancealong their length, as provided by the shear res-istance generated by the bond between the groutand either the cable or the rock. The cable isassumed to be divided into a number of segmentsof length L, with nodal points located at each seg-ment end. The mass of each segment is lumpedat the nodal points, as shown in Figure 10.14.Shearing resistance is represented by spring/sliderconnections between the structural nodes and therock in which the nodes are located.

End-anchored rock bolts are the simplest tomodel. They simply supply axial restraint to theportions of the model in which they are anchored.The axial stiffness K, is given by

K = AE

L(10.7)

where A is the cross-sectional area of the bolt,E, the modulus of the steel and L the distancebetween the anchoring points

10.5.2 Time-dependent behavior

The issue of time-dependent behavior is dis-cussed in reference to behavior of the west wallof Chuquicamata mine, which experiences largeon-going displacements of the order of 2–4 mper year. The slope behavior is affected by thepresence of a pervasive fault and an adjacentzone of sheared rock near the toe of the slope(Figure 10.15). Deformation of these mater-ials is expressed in toppling further upslope.Previous analyses attempted to estimate safetyfactors for the slope using UDEC. However,difficulties in identifying both a clear point offailure and a failure mode suggested that othercriteria should be considered in assessing theacceptability of west-wall slope designs. Slopedisplacement and displacement rates were con-sidered as other criteria. However, these criteriarequired use of material models that could repres-ent time-dependent behavior. Such models werenot available in programs used to study discon-tinuum slope behavior. The modeling described


P3

P5

N-5500

N-5000

N-4500

N-4000

N-3500

N-3000

N-2500 N-2500

E-2

000

E-3

500

E-3

000

E-2

500

N-3000

N-3500

N-4000

N-4500

N-5000

N-5500E

-200

0

E-3

500

E-2

500

E-3

000

Moderateshear zone

West fault

Intense shear zone

Figure 10.15 Plan of Chuquicamataopen pit showing shear zone (modeled asa time-dependent material), and ProfilesP3 and P5.

here demonstrates that the time-dependent beha-vior of the west wall can be reasonably sim-ulated when the sheared zone is representedby a two-component power-law creep modelcombined with a Mohr–Coulomb elasto-plasticmodel. Measured slope movements and changesin displacement rates over a period of six years arecompared to model predictions for two profiles(see Figure 10.16).

Profile P3 is an east–west section located atmine coordinate N3600; it is a good section forcalibration of behavior of the west wall for tworeasons. First, the location of P3 is near the

middle of the west wall and therefore is rep-resentative of conditions in this wall. Second,its orientation is perpendicular to the west walland the geology, allowing representative two-dimensional analysis. Profile P5 is a good sectionfor calibration because mining with steep slopeangles within the shear zone led to a slope fail-ure in February 2002. Both profiles have goodhistorical information about slope movementfrom prism monitoring, as described in the nextsection.

Records of slope movement were availablefrom monitoring records. The prism locations for


Pit advance 1995–2000

S-221-2000

S-223-1996

Prism Start FinishLevel(m)

S-221

S-222

S-223

S-225S-224

S-221

S-222

S-223

S-224

S-225

S-221-2000

S-223-1996

2844.16

2820.18

2672.23

2593.32

2414.34

2852.52

2691.55

28 Mar 98

26 Aug 00

9 Jun 96

14 Feb 01

14 Feb 01

14 Feb 01

14 Feb 01

21 Mar 95

21 Mar 95

21 Mar 95

1 Aug 99

21 Mar 95

12 Mar 00

10 Jul 96

Figure 10.16 Location of movement monitoring prisms in Profile P3 (see Figure 10.15 for location ofProfile P3).

Profile P3 are shown in Figure 10.16, which alsoshows the position of the pit at the end of years1995–2000. The prism records for both profilesshow the following important characteristics:

• horizontal displacements greater than verticaldisplacements, which is characteristic oftoppling behavior;

• decreasing displacement rates from mid-1995to the end of 1999; and

• increasing displacement rates starting near theend of 1999 in Profile P3 and the end of 2000in Profile P5; these periods correspond to thetimes when the shear zone was being mined.

UDEC models for Profiles P3 and P5 are shownin Figures 10.17 and 10.18, respectively, and havethe following features.

• Rock mass behavior for all units except theshear zone is represented by an elasto-plasticmodel, with a bilinear Mohr–Coulomb failuresurface. The bilinear model approximates aHoek–Brown failure surface, and is easier to

use than the non-linear Hoek–Brown failureenvelope.

• Faults and other major discontinuities wereincluded explicitly in the models. Predominatejointing structure was included implicitlythrough a ubiquitous joint model inGranodiorita Fortuna, which is locatedupslope from the sheared zone.

• Water pressures were taken from MINEDW(a three-dimensional finite element developedby Hydrologic Consultants Inc.), and trans-ferred to the UDEC model at yearlyintervals.

• Lithologic units were obtained from Chuquica-mata’s block model and imported into theUDEC model using a recently developedtransfer algorithm.

• Hydrostatic initial in situ stresses wereassumed. Initial in situ stresses with deviatoriccomponents would induce creep under initialpre-mining conditions, a condition that is notbelieved to be correct.

• Small deformation logic was used to avoidproblems with poor zone geometry resultingfrom large deformations within the shear zone.


Intense shear zone

Figure 10.17 UDEC model for Profile P3 with lithology, discontinuities and annual pit geometries.

Moderate shear zone

Figure 10.18 UDEC model of Profile P5 with lithology, discontinuities and annual pit geometries(see Figure 10.15 for location of Profile P5).

Creep behavior was believed to be concentratedwithin the sheared zone located just to thewest of the West Fault. In the model discussedhere, the behavior of the sheared rock wasrepresented using a viscoplastic model thatcombined the behavior of the viscoelastic two-component Norton Power Law model andthe Mohr–Coulomb elasto-plastic model. Thestandard form of the Norton Power Law (Norton,1929) is

εcr = Aσn (10.8)

σ =(

32

)1/2 (σd

ijσdij

)1/2 (10.9)

where εcr is the creep rate, σdij is the deviatoric

part of σij, and A and n are material propertiesthat were found by calibration.

Modeling was performed for conditions repres-entative of the years 1996 through 2002. Initially,the model was brought to equilibrium under con-ditions representative of January 1996. At thispoint, the creep model was turned on and run forone year of simulated time. For each subsequentyear, new water pressures and slope geometrywere introduced, and the model was run foranother year of simulated time.

Calibration was performed by adjusting thepower law parameters until reasonable agree-ment was reached between the prism records and


Prism east

UDEC east

UDEC vertical

Prism vertical

Time–10

–5

0

5

Dis

plac

emen

t (m

)

10

15

20

May-00Aug-99Dec-98Apr-98Aug-97Dec-96 Jan-01 Sep-01Mar-96

Figure 10.19 Comparison of movement record for prism S-223 records with UDEC results for Profile P3;A = 1e–23 and n = 2.5 (equations (10.8) and (10.9)).

Prism

UDEC

Apr-950123456789

Dis

plac

emen

t (m

)

101112

Feb-96 Dec-96 Nov-97 Sep-98

Time

Aug-99 Jun-00 Apr-01 Mar-02

Figure 10.20 Comparison of actual prism movement record with UDEC results for Profile P5; A = 1e–24 andn = 2.5 (equations (10.8) and (10.9)).

the numerical results. Representative results areshown in Figures 10.19 and 10.20 for P3 and P5,respectively. The sharp increase in displacementP5 resulted in slope failure in the upper portionof the slope.

In the analyses reported here, it has beenassumed that creep behavior initiates as soon asdeviatoric stresses are present, and the state ofstress is not hydrostatic. However, it is more

likely that creep behavior starts after a thresholddeviatoric stress is reached. Evidence for this canbe seen by examining the pre-mining in situ stressstate, which has been shown through measure-ments to include deviatoric stresses. There was noevidence of creep behavior in the pre-mining con-dition. Thus, it can be concluded that a thresholddeviatoric stress exists below which no creepbehavior occurs.


10.5.3 Dynamic analysis

Traditional approaches to dynamic analysis arebased on a pseudo-static approach in whichthe effects of an earthquake are representedby constant horizontal and/or vertical accelera-tions. The first explicit application of the pseudo-static approach to the seismic slope stability hasbeen attributed to Terzaghi (1950). The applic-ation of horizontal and/or vertical accelerationscan be made in limit equilibrium methods andnumerical methods alike. The results of pseudo-static analyses depend on the value of the seis-mic coefficient as discussed in Section 6.5.4.Difficulty in assigning appropriate pseudo-staticcoefficients and in interpretation of pseudo-static safety factors, coupled with the advanceof numerical models has provided an alternat-ive to the use of the pseudo-static approachfor seismic slope stability analyses. Numericalmethods, in addition to the Newmark methoddiscussed in Section 6.5.5, allow permanent slope

deformations resulting from seismic excitation tobe computed.

Both finite-element and finite-differenceapproaches can be used to compute permanentdeformations. Typical analyses involve applica-tion of a seismic record to the base of a modeland propagating the wave through the model.Small amounts of damping are sometimes appliedto account for real energy losses that are notrepresented by either the joint behavior or therock mass behavior.

Although there are no documented cases oflarge-scale failures of open pits under seismicloads, there are many instances of failure of nat-ural slopes during earthquakes (see Section 6.5.1).In open pits, smaller-scale failures comprisingrock fall and bench-scale structurally controlledfailures may occur under severe shaking. Wheresuch failures are an operational hazard, mitig-ation can usually be provided by suitable catchbench configurations.

Chapter 11

Blasting

11.1 Introduction

Excavation of rock slopes usually involves blast-ing, and it is appropriate that the subject receivesattention in this book on rock slope engineer-ing. The fragmentation of rock by explosives is amajor subject in its own right and the fundament-als are dealt with in a number of excellent text-books and handbooks (Langefors and Kihlstrom,1973; Hemphill, 1981; CIL, 1984; Du Pont,1984; FHWA, 1985; Atlas Powder Co., 1987;Persson et al., 1993; ISEE, 1998; Oriard, 2002).

The responsibility for the design and imple-mentation of blasting operations varies with thetype of project. On mines, there will usually bea department responsible for all aspects of blast-ing, possibly with some assistance from technicalrepresentatives of the explosive supplier. Thisapproach is suited to the circumstances whereblasting is carried out frequently, and there isgood experience of local conditions. This situ-ation allows the blast designs to be adjusted toaccommodate such factors as variations in geo-logy and optimization of equipment operations.The primary requirements of blasting in open pitmines are to produce a muck pile that is frag-mented to suit operation of the excavating andhauling equipment. Also, times for equipmentmoves before and after the blast are minimizedif flyrock is controlled. A further requirement ofblasting is to control damage to the rock and min-imize slope instability when excavating at finalfaces. Production blasting methods are discussedin Section 11.3.

In contrast to mining operations, on civil pro-jects blasting is usually the responsibility of thecontractor, with the principle duty of the owner’srepresentative being to check that the desired res-ults are produced. That is, the work is carriedout under a performance-type contract in whichthe required results are specified, but the meth-ods of achieving these are left to the contractor.This situation requires that the owner under-stands blasting methods so that the proceduresand results can be reviewed and, if necessary,modifications discussed with the contractor. Theowner should also ensure that accurate recordsare kept of each blast as required by most legisla-tion governing blasting. The records are also use-ful in relating the results obtained to the methodsused, and for cost control purposes.

Rock excavation for civil construction oftenrequires the formation of cut faces that will bestable for many years, and as steep as possibleto minimize excavation volume and land use.While these two requirements are contradictory,the stability of cuts will be enhanced, and themaximum safe slope angle increased, by using ablasting method that does the least possible dam-age to the rock behind the final face. Section 11.4describes methods of minimizing blasting damagethat are included in the general term “controlledblasting.”

When blasting in urban or industrial areas, pre-cautions are required to control damage to resid-ences and other sensitive structures. Section 11.5describes methods of controlling structuraldamage due to blast vibrations, as well asminimizing hazards of flyrock, air blast and noise.

246 Blasting

11.2 Mechanism of rock fracturing byexplosives

The mechanism by which rock is fractured byexplosives is fundamental to the design of blastingpatterns, whether for production or controlledblasting. It also relates to the damage to surround-ing structures and disturbance to people living inthe vicinity. The following is a description of thismechanism (FHWA, 1991; Hagan, 1992; Perssonet al., 1993; Oriard, 2002).

When an explosive is detonated, it is conver-ted within a few thousandths of a second into ahigh temperature and high pressure gas. Whenconfined in a blast hole, this very rapid reactionproduces pressures, that can reach 18 000 atm, tobe exerted against the blast hole wall. This energyis transmitted into the surrounding rock mass inthe form of a compressive strain wave that travelsat a velocity of 2000–6000 m/s.

As the strain wave enters the rock surroundingthe blast hole, the material for a distance of one totwo charge radii (in hard rock, more in soft rock)is crushed by compression (Figure 11.1(a)). As thecompressive wave front expands, the stress levelquickly decays below the dynamic compressivestrength of the rock, and beyond this pulverizedzone the rock is subjected to intense radial com-pression that causes tangential tensile stresses todevelop. Where these stresses exceed the dynamictensile breaking strength of the rock, radial frac-tures form. The extent of these fractures dependson the energy available in the explosive and thestrength properties of the rock, and can equal40–50 times the blast hole diameter. As the com-pressive wave passes through the rock, concentricshells of rock undergo radial expansion result-ing in tangential relief-of-load fractures in theimmediate vicinity of the blast hole.

These concentric fractures follow cylindricalsurfaces, and are subsequently created nearerand nearer to the free face. When the compress-ive wave reaches a free face, it is reflected as atensile strain wave. If the reflected tensile waveis sufficiently strong, “spalling” occurs progress-ively from any effective free face back towardsthe blast hole. This causes unloading of the

Tangential strains(a)

(b)

(c)

Pulverized rock

Radial cracking

Compressive wave positions

Free face

A B C

Tensile wave

Spall

Expanding blast hole

Relief-of-load fractures

High-pressure explosive gases

Expanding blast hole

Figure 11.1 Mechanism of rock breakage byexplosives.

rock mass, producing an extension of previouslyformed radial cracks (Figure 11.1(b)). Rock ismuch weaker in tension than compression, so thereflected strain wave is particularly effective infracturing rock.

The process of fracture formation due to strainwave energy typically occurs throughout the bur-den within 1 or 2 ms after detonation, whereasthe build-up of explosive gases takes in the order

Blasting 247

of 10 ms. As the rock is unloaded due to radialexpansion and reflection of the compressive wave,it is now possible for the expanding gases towedge open the strain wave-generated cracks andbegin to expel the rock mass (Figure 11.1(c)). Thisstage is characterized by the formation of a domearound the blast hole. As wedging action takesplace due to the heaving and pushing effect ofthe expanding gases, considerably more fractur-ing occurs due to shear failure as the rock mass isexpelled in the direction of the free face. In highlyfissured rocks, fragmentation and muck pilelooseness are caused mostly by expanding gases.

The fragmentation achieved by the processdescribed in the previous paragraph is highlydependent upon the confinement of the explosive,the coupling of charges within the blast holes, theburden distance and the sequencing of the blast.That is, if confinement of the charge by stemmingis inadequate, some energy will be lost from theblast holes, and inadequate explosive/rock coup-ling results in poor transmission of strain energyto the rock mass. Also, excessive burdens res-ult in choking and poor movement of the rock,whereas inadequate burden results in waste ofexplosive energy and excessive throw of blastedrock. The best results are produced when effectivedelaying of individual blast holes ensures max-imum development and utilization of free facesby reducing the effective burden. This providesfreedom for the rock to move toward the free faceand reduces damage to the surrounding rock (seeSection 11.3).

To prevent damage to rock behind the face, thezone of crushed rock and radial cracking aroundthe holes in the final row is controlled by reducingthe explosive energy, and decoupling the chargein the holes (see Section 11.4, Controlled blast-ing). As the shock wave travels beyond the limitof rock breakage and into the surrounding rock,it sets up vibrations both within the rock and atthe ground surface. Structures located close to theblast, and through which these vibration wavespass, may be damaged by twisting and rockingmotion induced by the ground motion. Dam-age can be controlled by reducing the explosiveweight detonated per delay (see Section 11.5).

11.3 Production blasting

The basic economics of rock excavation usingexplosives are shown in Figure 11.2 (Harriesand Mercer, 1975). The production of a well-fragmented and loosely packed muck pile thathas not been scattered around the excavationarea, facilitates loading and hauling operations.This condition is at the minimum total cost pointon the graph. However, close to the final face,drilling and blasting costs will increase becausemore closely spaced and carefully loaded holeswill be required to limit damage to the rock. Con-versely, the production of riprap will involve theuse of holes with a spacing greater than the largestblock size required.

In order to achieve the optimum results of blast-ing under all conditions, a thorough understand-ing of the following parameters is required:

1 type, weight, distribution of explosive;2 nature of the rock;3 bench height;4 blast hole diameter;

Total

Loading and hauling

Drilling and blasting

FragmentationFiner Coarser

Cos

t

Figure 11.2 Effect of fragmentation on the cost ofdrilling, blasting, loading and hauling.

248 Blasting

BurdenSpacing

Blast hole

Column load

Sub-drill

Toeload

Stemming

Free face Bench height

Figure 11.3 Definition of bench blastingterms.

5 burden;6 spacing;7 subdrill depth;8 stemming;9 initiation sequence for detonation of

explosives; and10 powder factor.

Factors 3–8 are illustrated in Figure 11.3, andthis section describes procedures for calculatingthe parameters involved in designing produc-tion blasts. This discussion is based on work byDr C. J. Konya (FHWA, 1991). It should be notedthat the equations provided in the following sec-tions defining blasting parameters are guidelinesonly and will need to be modified, as necessary,to suit local site conditions.

11.3.1 Explosive properties

The strength of an explosive is a measure ofthe work done by a certain weight or volumeof explosive. This strength can be expressed in

absolute units, or as a ratio relative to a standardexplosive. Usually the bulk strength of explos-ives is related to the strength of ANFO that isassigned an arbitrary bulk strength of 100. ANFOis the term used for the most widely used explos-ive comprising ammonium nitrate prills (0.5 mmdiameter spheres) and 5.5% fuel oil.

One measure of the strength of an explosive isits velocity of detonation (VOD); the higher thevelocity the greater the shattering effect. How-ever, explosive strength, density and degree ofconfinement are also factors that should be con-sidered in selecting an explosive for a specificpurpose. Table 11.1 lists the velocity of deton-ation, specific gravity and water resistance ofdifferent classes of explosives.

Explosive strength is defined by weight andbulk strengths. Weight strengths are useful whencomparing blast designs in which explosives ofdifferent strengths are used, and when compar-ing the cost of explosives because explosives aresold by weight. The relative bulk (or volume)strength (RBS) is related to the weight strength

Blasting 249

Table 11.1 Typical properties of explosives products

Explosive type Density(g/cc)

VOD (m/s) Relative bulkstrength∗(ANFO = 100)

Waterresistance

Packaged, detonator-sensitiveemulsions

1.12–1.2 4600–5200 115–170 Excellent

Packaged, booster-sensitiveemulsions

1.24–1.26 4300–5050 125–155 Excellent

Watergels 1.20 4785 129 ExcellentDynamites 1.2–1.42 3350–5600 170–130 Good to excellentWall control dynamites 0.75–1.3 1650–2600 76–114 Good to poorBoosters 1.34–1.6 5600–7900 167–280 ExcellentANFO 0.84 4000 100 NoneBulk emulsions 1.25 5200–5500 120–150 Excellent

Note∗ Relative of bulk strength (RBS)—is a comparison of theoretical chemical energy per unit volume of an explosive to ANFO that has beenassigned an RBS of 100.

by the specific gravity, and this value is importantin calculating the volume of blast hole requiredto contain a given amount of explosive energy.A higher bulk strength requires less blast holecapacity to contain a required charge.

The sensitivity of an explosive is a characteristicthat determines the method by which a charge isdetonated, the minimum diameter of the chargeand the safety with which the explosive can behandled. Highly sensitive explosives will deton-ate when used in smaller diameter charges, andcan be detonated with a relatively low strengthdetonator. As the sensitivity of the explosive isdecreased, the diameter of the charge and theenergyof thebooster/detonatormustbe increased.

Table 11.1 provides information on typicalproperties of the main classes of explosives. Eachclass of explosive has a set of characteristics suit-able to specific applications and conditions suchas the size of the blast hole, the presence of waterand the need to control flyrock and noise. Forexample, ANFO and bulk emulsions are com-monly used for large-scale blasts in open pitsand quarries, while watergels and dynamites areused in smaller construction projects. Also, asshown in Figure 11.3, it is necessary when usingANFO as the main explosive, to use a higherstrength explosive in the lower end of the hole(“toe load”). The function of the toe load is to

both ensure complete detonation of the ANFO,and to break the rock in the floor of the benchwhere it is most highly confined.

11.3.2 Bench height

Bench heights are usually determined by the geo-metry of the site, with single benches being usedwhere the excavation depth is up to about 8 m.On larger construction projects and in open pitmines and quarries, multi-benches operations areconducted. For these operations, the selection ofthe optimum bench height to maximize the over-all cost efficiency of drilling and blasting requiresthe correct combination of drilling and loadingequipment. Furthermore, regulations may limitthe bench height, in relation to the maximumreach of the excavating equipment, to minimizethe risk of damage or injury in the event of col-lapse of the face. The following are some of thefactors that should be considered in the selectionof the bench height:

(a) Optimum blast hole diameter increases withbench height. In general, an increase in blasthole diameter results in a decrease in drillingcosts.

(b) For vertical blast holes and sloping benchface, the front row toe burden may become

250 Blasting

excessive as the bench height increases.Where small diameter blast holes are drilledin high benches, blast holes may need to beangled, at least in the front row.

(c) Drilling accuracy becomes more critical inhigher benches. Where precise alignment ofholes on the final face is required, the max-imum bench height is normally limited to8 or 9 m.

11.3.3 Burden

The influence of the effective burden (the distancebetween rows of holes and the nearest free face)on fragmentation is related to the mechanism ofrock fracture described in Section 11.2. The blastis most efficient when the shock wave is reflec-ted in tension from a free face so that the rock isbroken and displaced to form a well-fragmentedmuck pile. This efficiency depends to a largeextent on having the correct burden. Too smalla burden will allow the radial cracks to extendto the free face resulting in venting of the explo-sion gases with consequent loss of efficiency andthe generation of flyrock and air blast problems.Too large a burden, where the shock wave is notreflected from the free face, will choke the blastresulting in poor fragmentation and a general lossof efficiency.

The relationship between the bench height H

and the burden B can be expressed in terms of the“stiffness ratio,” H/B. If this ratio is low suchthat the burden is about equal to the bench height,then the blast will be highly confined resulting insevere backbreak, airblast, flyrock and vibration.In contrast, if the H/B >∼ 4, there is little con-finement and the explosive gases will be vented atthe free face also resulting in airblast and flyrock.It is found that a stiffness ratio of 3–4 producesgood results, or

B = 0.33 × H to 0.25 × H (11.1)

The burden distance calculated from equa-tion (11.1) depends not only upon the blast holepattern, but also upon the sequence of firing. Asillustrated in Figure 11.4(a), a square blast hole

Surface delay

B

S

2 13456

FaceDirection ofrock movement

S

B3

2

1

Face Direction of rock movement

(a)

(b)

(c) Maximum front row burden

B

Figure 11.4 Definition of blast hole spacing (S) andburden (B): (a) burden and spacing for a squareblasting pattern; (b) burden and spacing for anen echelon blasting pattern; (c) effect of face angle onfront row burden.

pattern which is fired row by row from the facegives an effective burden equal to the spacingbetween successive rows parallel to the face. Onthe other hand, an identical pattern of blast holesfired en echelon results in different burdens andspacings, and a spacing to burden ratio greaterthan 1 (Figure 11.4(b)).

Blasting 251

An important factor in designing a blast is thechoice of the front row burden. Once this row hasbeen detonated and effectively broken, new freefaces are created for the next row, until the lastrow is fired. If vertical blast holes are used andthe bench face is inclined, the front row burdenwill increase with depth (Figure 11.4(c)). Allow-ance can be made for this variation by using ahigher energy bottom load in the front row ofholes. Alternatively, the blast hole can be inclinedto give a more uniform burden. When the freeface is uneven, the use of easer holes to reducethe burden to acceptable limits is advisable.

For multiple row blasts, the rows towards theback of the blast are progressively more confined,and consequently the degree of fragmentationmay diminish. This effect can be corrected for bydecreasing the burden of the third and subsequentrows by a factor of 0.9.

11.3.4 Blast hole diameter

Blast hole diameters on construction projects mayrange from about 40 mm for hand-held drills,to 100 mm for “air-trac” drills and 200 mm for“track drills”. All these drills are rotary percuss-ive and are powered with compressed air. In openpit mines, electric powered rotary drills are com-monly used with which holes up to 600 mm aredrilled with roller tri-cone bits (ADI, 1996).

Persson (1975) shows that the cost of drillingand blasting decreases as the hole size increases.This is because the hole volume increases with thesquare of hole diameter so that the same volumeof explosive can be loaded into fewer holes. Thiscost saving is offset by the greater shattering ofthe rock that is produced by the more highlyconcentrated explosive, which can result in lessstable slopes, and larger rock fragments that theexcavating equipment may not be able to handle.

Once the burden B (units: m) has been setto provide an appropriate stiffness ratio (equa-tion (11.1)), the diameter of the explosive dex(units: mm) can be determined as follows:

dex ≈ B × 1000(24γex/γr) + 18

(11.2)

where γex and rock γr are the unit weights of theexplosive and rock, respectively.

Equation (11.2) can be used where there is acorrelation between the energy of the explosiveand its unit weight. However, emulsion slurryexplosives have a range of energies with a nearconstant unit weight, and for these conditions itis appropriate to calculate the burden using therelative bulk strength (RBS) compared to ANFO(RBS = 100). The blast hole diameter taking intoaccount the RBS of the explosive is given by

dex ≈ B × 10008(RBS/γr)0.33

(11.3)

11.3.5 Nature of the rock

The nature and degree of heterogeneity of therock mass is very important in blast design. Thatis, discontinuities such as joints, bedding planes,faults, and soft seams can allow the explosive’senergy to be wastefully dissipated rather thanperform the work intended. In some cases, thediscontinuities can dominate the fracture patternproduced by the explosive, and the influence ofthe structural geology often overshadows that ofthe rock’s mechanical and physical properties.Best fragmentation is usually obtained where theface is parallel to the major discontinuity set.

Heave energy of the explosive is importantin highly fissured rock. The explosion gases jetinto, wedge open, and extend the pre-existingcracks. Therefore, the overall degree of fragment-ation tends to be controlled by the structuralgeology. For example, closely spaced joints andbedding planes result in increased fragmentationbecause few new fracture surfaces need be createdin the blast. For these conditions, longer stem-ming columns and correspondingly lower powderfactors or energy factors can be used, and low-density, low-velocity explosives (e.g. ANFO) arepreferred to higher-velocity explosives that causeexcessive fragmentation immediately around thecharge. Satisfactory results are achieved when thecharges provide sufficient heave energy to dis-place the rock into a loose, readily excavated

252 Blasting

muck pile, without scattering the rock aroundthe site.

The effects of structural geology on blast designcan be quantified by applying two correctionfactors to the calculated burden distance. Thesefactors relate to both the orientation of any per-sistent discontinuities relative to the face andfracture characteristics, as shown in Tables 11.2and 11.3 respectively. The correction factorsare applied to the calculated burden distance asfollows:

B′ = kψ · ks · B (11.4)

11.3.6 Sub-drill depth

Sub-drilling or drilling to a depth below therequired grade is necessary in order to breakthe rock at bench level. Poor fragmentationat this level will form a series of hard “toes”and an irregular bench floor resulting in highoperating costs for loading and hauling equip-ment. However, excessive sub-drilling can resultin unnecessary drilling and explosive costs.

Table 11.2 Correction factors for dip of structure

Orientation of structure Correction factor, kψ

Structure dippingsteeply out of face

1.18

Structure dippingsteeply into face

0.95

Other orientation ofstructure

1.00

Table 11.3 Correction factors for discontinuitycharacteristics

Structural characteristics Correction factor, ks

Closely jointed, closelyspaced weaklycemented seams

1.30

Thin, well-cementedlayers with tight joints

1.10

Massive, intact rock 0.95

Bench floor

Sub-drill depth

15° to 25°

Figure 11.5 Rock breakage at the bottom of a blasthole with the use of sub-drilling.

Breakage of the rock usually projects from thebase of the toe load in the form of an invertedcone with sides inclined at 15–25◦ to the hori-zontal, depending upon the strength and structureof the rock (Figure 11.5). In multi-row blasting,the breakage cones link up to give a reasonablyeven transition from broken to undamaged rock.Experience has shown that a sub-drill depth of0.2–0.5 times the burden is usually adequate foreffective digging to grade. Where benches are tobe formed on the final face, it is advisable toeliminate the sub-drilling in the back rows to helpmaintain stability of the bench crest.

11.3.7 Stemming

The use of stemming to fill the upper portion ofthe hole above the charge contains the explosivegases and directs the explosive effort into the rockmass. Stemming material comprising well graded,angular gravel is more effective than drill cuttingsthat are more easily ejected from the hole. Theoptimum size of the stemming material increaseswith the diameter of the blast hole, and the aver-age size of the stemming particles should be about0.05 times the diameter of the blast hole.

The effects of stemming length on blast resultsare similar to those of burden distance discussedpreviously. That is, a short stemming lengthwill allow the explosive gases to vent, generat-ing flyrock and airblast problems and reducingthe effectiveness of the blast, while too great a

Blasting 253

stemming length will give poor fragmentation ofthe rock above the column load.

The common stemming length is about0.7 times the burden, which is adequate to keepmaterial from ejecting prematurely from the hole.If unacceptably large blocks are obtained from thetop of the bench, even when the minimum stem-ming column consistent with flyrock and airblastproblems is used, fragmentation can be improvedby locating a small “pocket” charge centrallywithin the stemming (Hagan, 1975).

11.3.8 Hole spacing

When cracks are opened parallel to the free faceas a result of the reflected tensile strain wave, gaspressure entering these cracks exerts an outwardforce that fragments the rock and heaves it onto the muck pile. Obviously, the lateral extentto which the gas can penetrate is limited by thesize of the crack and the volume of gas avail-able, and a stage will be reached when the forcegenerated is no longer large enough to fragmentand move the rock. If the effect of a single blasthole is reinforced by holes on either side, the totalforce acting on the strip of burden material willbe evened out and uniform fragmentation of thisrock will result.

Figure 11.6 illustrates drill hole patterns with avariety of burden/spacing ratios. While the squarepattern is the easiest to layout, in some conditionsbetter fragmentation is obtained when the spacingis greater than the burden. For a series of delayedholes, the spacing S can be calculated from thefollowing two equations:

For a stiffness ratio H/B between 1 and 4,

S = (H + 7B)

8(11.5)

and for a stiffness ratio H/B greater than 4,

S = 1.4 × B (11.6)

11.3.9 Hole detonation sequence

A typical blast for a construction project or openpit mine may contain as many as 100 blast holes,

S

B

(a)

(b)

(c)

S

B

E E E

Figure 11.6 Typical blast hole patterns used inproduction blasting: (a) square pattern withburden/spacing ratio 1:1; (b) staggered pattern withburden/spacing ratio 1:1.15; (c) easer holes (E) toassist movement of front row burden.

which in total contain several thousand kilo-grams of explosive. Simultaneous detonation ofthis quantity of explosive would not only pro-duce very poorly fragmented rock, but would alsodamage the rock in the walls of the excavationand create large vibrations in nearby structures.In order to overcome this situation, the blast isbroken down into a number of sequential deton-ations by delays. When the front row is detonatedand moves away from the rock mass to cre-ate a new free face, it is important that timebe allowed for this new face to be establishedbefore the next row is detonated. Figure 11.7show examples of detonation sequences. Row-to-row blasts are parallel to the free face, withthe row closest to the face being detonated first inthe sequence (Figure 11.7(a)). The V cut shown inFigure 11.7(b), where the rows are inclined to theface, is used to open a new free face, and when

254 Blasting

(a)

(c)

(b)

3

2

1

4

1 5433 22

6

175 192 209 226 243 260 277

217 234 251 268 285 302 319

259 276 293 310 327 344 361

301 318 335 352 369 386 403

Free face

17 ms surface delay42 ms surface delay175 ms in-hole delayActual firing time of hole (ms)

403

Free end

Figure 11.7 Typical detonation sequences: (a) square “row-by-row” detonation sequence; (b) square “V”detonation sequence; (c) hole-by-hole detonation using both surface and in-hole non-electric delays(W. Forsyth).

blasting in strongly jointed rock where near ver-tical joints strike across the bench at an angle tothe face.

In many cases, blasting results can be improvedby introducing hole-by-hole firing, where everyblast hole is initiated in sequence at a unique time(Figure 11.7(c)). Where appropriate delays areselected, hole-by-hole initiation exploits the pos-itive benefits of blast hole interaction while avoid-ing most of the negative effects. This can lead tofiner fragmentation, looser muckpiles, less over-break, lower ground vibration, and better controlover the position and profile of the final muckpile.

Hole-by-hole firing can be achieved by using in-hole detonators. However, the available range ofcommercial in-hole delays is limited and, hencehole-by-hole initiation is usually achieved byusing a surface delay system to control blast holesequencing. If long inter-row or inter-hole delaysare required, a combination of surface and in-hole delays will avoid downline cutoffs causedby ground movement during the blast. For thedetonation sequence shown in Figure 11.7(c),

there is an identical in-hole delay of 175 ms ineach hole, together with surface delays of 17 msalong the free face, and 42 ms delays between therows. The diagram shows the actual firing timesof each hole using this delay arrangement.

The required delay interval is related todistance between holes by the following tworelationships.

For row-to-row detonation:

Time delay betweenrows

(units: ms)

≈ (10–13) × (burden)

(units: m)

(11.7a)

For example, for a 5 m burden the delay betweenrows is 50–65 ms.

For hole-to-hole detonation:

Time delaybetween holes

(units: ms)

≈ (delay constant)×(hole spacing)

(units: m) (11.7b)

Blasting 255

Table 11.4 Delay constant–rock type relationship forhole-to-hole delay

Rock type Delay constant (ms/m)

Sand, loam, marl, coal 6–7Soft limestone, shale 5–6Compact limestone andmarble, granite, basalt,quartzite, gneiss,gabbro

4–6

Diabase, diabaseporphyrites, compactgneiss and micaschist,magnetite

3–4

Values for the delay constant depend on the rocktype as shown in Table 11.4. For example, whenblasting granite with a hole-to-hole spacing of6 m, the delay interval would be about 30 ms.

These two relationships for calculating delaytimes take into account both the time required todisplace the broken rock and establish a new freeface, and the natural scatter that occurs in theactual firing times of delays. The error in actualfiring time compared to the rated firing time maybe as high as 15%, and such errors may resultin holes firing out of sequence, especially if thedelay interval is short. The development of elec-tronic delay detonators provides a higher degreeof precision in firing times compared to chemicaldelay elements. A further advantage of electronicdetonators is the ability, with some products, toprogram the delay interval in the field to suitsite conditions (McKinstry et al., 2002; Watson,2002).

11.3.10 Fragmentation

The objective of selecting the appropriate com-bination of design parameters, as discussed onthe preceding pages, is to achieve a desired resultfor the blast. These objectives may include a spe-cified size range of rock fragments, or a final facewith minimal blast damage, or noise and groundvibration levels within specified limits.

The basic design parameter for productionblasting, where the objective is to produce acertain degree of fragmentation, is the “powder

factor”—the weight of explosive required tobreak a unit volume of rock, for example, kg/m3.The powder factor is derived from the proced-ures discussed in this section to calculate the blasthole diameter and depth, the hole pattern and theamount and type of explosive in each hole. Thatis, the weight of explosive and the volume of rockare given by

Weight of explosive per hole, Wex

= (diameter of explosive)

× (unit weight of explosive)

× (bench height − stemming length

+ subdrill depth)

and

Volume of rock per hole, V

= (bench height) × (burden) × (spacing)

Powder factor = Wex

V(11.8)

The effect of the powder factor on fragmentationis shown in Figure 11.8 where the average frag-ment size is related to the powder factor for arange of burdens. These graphs, which are based

0.25

0.501

2

4

8Burden (m)

1.6

1.2

0.8

0.4

0.4 0.5 0.6 0.7 0.8 0.9

Ave

rage

bou

lder

siz

e (m

)

Powder factor (kg/m3)

Figure 11.8 Relationship between average bouldersize L, powder factor q and burden B in benchblasting (Persson et al., 1993).

256 Blasting

on theoretical equations developed by Langeforsand Kihlstrom (1973) and Persson et al. (1993)together with extensive field testing, can be usedto check on the likely results for a given blastdesign and assess the possible effects of designmodifications.

11.3.11 Evaluation of a blast

Once the dust has settled and the fumes have dis-persed after a blast, an inspection of the areashould be carried out. The main features ofa satisfactory blast are as follows (Figure 11.9):

• The front row should have moved out evenlybut not too far—excessive throw is unneces-sary and expensive to clean up. The heights ofmost benches are designed for efficient loaderoperation and low muckpiles, due to excess-ive front row movement, results in low loaderproductivity.

• The main charge should have lifted evenlyand cratering should, at worst, be an occa-sional occurrence. Flat or wrinkled areas areindicative of misfires or poor delaying.

• The back of the blast should be character-ized by a drop, indicating a good forwardmovement of the free face. Tension cracksshould be visible in front of the final excava-tion lines, although excessive cracking behindthe final excavation line represents damage tothe slopes and waste of explosive.

The quality of blast has a significant effect oncomponents of the rock excavation cost suchas secondary drilling and blasting of oversizeboulders, loading rate, the condition of the haulroads, and loader and truck maintenance. Forexample, oversized fragments, hard toes, tightareas and low muck piles (caused by excess-ive throw) have the most significant detrimental

Final wall

Minimum disruption to final wall

Easy loading withno “hard toes”

Even throw

Even lift

No cratering

Good backdropTension cracks

Optimum loading Low productivityclean-up

Low productivity clean-up

Figure 11.9 Features of a satisfactory production blast.

Blasting 257

effect on the excavation rates. Therefore, carefulevaluation of the blast to determine how improve-ments could be made to the design is usuallyworthwhile.

11.4 Controlled blasting to improvestability

Slope instability is often related to blast damage tothe rock behind the face. Blast-induced instabil-ity is usually surficial, extending possibly 5–10 mbehind the face for open pit scale blasts, that canresult in rock falls occurring over time as waterand ice open cracks and loosen blocks. It is alsopossible that the blast damage can cause larger-scale instability where, for example, the slopecontains persistent bedding planes dipping out ofthe face. The explosive gases can travel along,and open, the beds resulting in displacement ofsubstantial blocks of rock.

Control of blast damage to final walls can belimited by implementing one or both of the fol-lowing procedures. First, the production blastshould be designed to limit rock fracturing behindthe final wall, and second, controlled blastingmethods such as line drilling, pre-shearing, andcushion blasting can be used to define final facesprecisely (Hagan and Bulow, 2000). With respectto production blasting, the following precautions

help to avoid excessive backbreak:

(a) Choke blasting into excessive burden orbroken muck piles should be avoided.

(b) The front row charge should be adequatelydesigned to move the front row burden.

(c) Adequate delays and timing intervals shouldbe used for good movement towards freefaces and the creation of new free faces forfollowing rows.

(d) Delays should be used to control the max-imum instantaneous charge.

(e) Back row holes, together with reducedcharge “buffer holes,” should be drilled atan optimum distance from the final face tofacilitate excavation and yet minimize dam-age to the wall. The length of the stemming inthe buffer holes may also need to be adjusteddepending on the degree of fracture along thecrest of the bench.

11.4.1 Pre-shearing and cushion blasting

On permanent slopes for many civil projects, evensmall slope failures are not acceptable, and the useof controlled blasting to limit damage to the finalwall is often required; an example of controlledblasting is shown in Figure 11.10. The prin-ciple behind these methods is that closely spaced,

Figure 11.10 Example ofcontrolled blasting for rock cuton highway project (strong,highly folded gneiss, I-26/US23,near Mars Hill, North Carolina)(courtesy: North CarolinaDepartment of Transportation).

258 Blasting

dh dex

3000

2500

2000

1500

1000

500

01 2 3 4 5

Str

ess

(atm

osph

eres

)

Decoupling ratio (dh /dex)

Figure 11.11 Stress level for decoupled blast holes(FHWA, 1991).

parallel holes drilled on the final face are loadedwith a light explosive charge that has a diametersmaller than that of the hole. The effect of anair-gap around the explosive provides a cushionthat significantly diminishes the shock wave thatis transmitted to the rock. If the decoupling ratio,that is the ratio of hole diameter to the explosivediameter is greater than about 2, then the pres-sure of the shock wave will be about 10–20% ofthat produced by an explosive tightly packed inthe hole (Figure 11.11). This pressure is not suffi-cient to produce crushing of the rock around thehole (see Figure 11.1), but radial fractures formpreferentially between the holes to create a cleanbreak on the plane of the holes. It is found that itis not necessary to fire the holes on the final lineon the same delay, and that the same results areobtained if every hole is fired on a separate delay(Oriard, 2002).

The following is a discussion on various meth-ods of controlled blasting, and their advantagesand disadvantages. The three basic methods ofcontrolled blasting are line drilling, cushion ortrim blasting and pre-shear blasting. Line drillinginvolves drilling holes precisely along the requiredbreak line at a spacing of two to four holesdiameters, and then leaving a number of unloaded

holes between the loaded holes. Line drillingis only used where very precise wall control isneeded, such as corners in excavations. Cushionand pre-shear blasting are the most commonlyused methods, with the main difference betweenthe two being that in cushion blasting the finalrow holes are detonated last in the sequence,while in pre-shearing the final line holes aredetonated first in the sequence. The following aresome of the factors that should be considered inselecting either cushion or pre-shear blasting:

(a) Burden—Pre-shearing should only be usedwhere the burden is adequate to contain theexplosive energy that is concentrated alongthe shear line. For narrow burdens, theenergy may be sufficient to displace the entiremass of rock between the shear line andthe face (Oriard, 2002). As a guideline forthe use of pre-shearing, the burden shouldat least be equal to the bench height, andpreferably greater.

(b) Discontinuities—In closely jointed/shearedrock, the highly confined gases generatedalong the shear line in pre-shearing may ventinto the cracks resulting in damage to therock. In cushion blasting, the rock is some-what less confined, and more stable slopesmay be produced.

(c) Vibration—The shock wave generated byhighly confined pre-shearing may be greaterthan that of cushion blasting. As notedearlier in this section, it is not necessaryto detonate the shear line holes on a singledelay, so employing hole-by-hole delays canlimit ground motion when vibrations mustbe closely controlled.

A general comment on the design of controlledblasts is that it is often necessary to carry out anumber of trail blasts at the start of a project, andwhen the rock conditions change, to determinethe optimum hole layout and explosive charge.This requires flexibility on the part of the con-tractor, and the use of end-product rather thanmethod specifications.

The cost savings achieved by controlled blast-ing cannot be measured directly. However, it is

Blasting 259

generally accepted that, for permanent slopes onmany civil projects, these savings are greater thanthe extra cost of drilling closely spaced, care-fully aligned holes and loading them with specialcharges. The savings are the result of being ableto excavate steeper slopes, and reduce excavationvolume and land take. It is also found that lesstime is spent scaling loose rock from the face afterthe blast, and the resulting face is more stableand requires less maintenance during its opera-tional life. From an aesthetic point of view, steepcuts have smaller exposed areas than flat slopes,although the traces of the final line drill holes onthe face may be considered objectionable.

11.4.2 Drilling

The maximum depth that can be successfullyexcavated by cushion or pre-shear blastingdepends on the accuracy of the hole alignment.Deviations of more than about 150 mm fromthe plane of the holes generally give poor res-ults due to the uneven distribution of explosive onthe shear plane, and the creation of an irregularface. While holes up to 27 m deep have been suc-cessfully cushion blasted, the depths of final lineholes are usually limited to about 8–10 m. Drillhole alignment can be enhanced with the use ofhigh rigidity drill steel, button bits and limitingthe down-hole pressure on the bit.

The penetration rates of the drill should alsobe considered when determining the depth to becushion blasted. If, for example, the penetra-tion beyond a given depth becomes excessivelyslow, it may be more economical to carry outthe excavation in a series of benches in order tokeep penetration rates and drilling costs at accept-able levels. Also, when laying out final line drillholes in a benched excavation, allowance shouldbe made for a minimum 0.3 m offset per benchsince it is not possible to position the drill flush tothe wall of the upper bench. This situation resultsin the overall slope of a multi-bench excavationbeing flatter than the bench face angle, and thisdifference should be allowed for in laying out thefinal line holes (Figure 11.12).

When cushion blasting around curved areas orcorners, closer spacings are required than when

H

H

Drill offset

H = Excavation bench height

�s= Overall slope angle

�f= Bench face angle

�f

�s

Figure 11.12 Alternative explosive placements forcontrolled blasting (adapted from ISEE, 1998).

blasting a straight section. Also, unloaded guideholes can be used to advantage when blastingnon-linear faces (Wyllie, 1999).

In unconsolidated sedimentary formationswhere it is difficult to hold a smooth wall,unloaded guide holes between cushion holes maybe used. Generally, small diameter guide holes areemployed to reduce drilling costs. Where only thetop of the formation is weathered, the guide holesneed be drilled only to that depth and not the fulldepth of the cushion holes.

11.4.3 Explosive load

The required explosive load in the final line holescan be obtained by a combination of steps. First,the charge diameter should be smaller than thehole so that the decoupling ratio is at least 2 (seeFigure 11.11). Second, an explosive with a lowvelocity of detonation should be used that willnot shatter the rock. Third, the charge shouldbe distributed along the hole so that there is

260 Blasting

Collar stemming

Air gap preferable between charge and blast hole wall unless rock is weak or highly fractured in which casestemming should fill entire hole

Heavy core load detonating cord

Charge

Low VOD explosive cartridges taped to detonating cord

Finished wall

2 to 3 times charge/m in bottom to ensure shear at floor

(a) (b)

Figure 11.13 Layout of final line blast holes on multi-bench excavation taking into account drill off-sets.

reasonably uniform explosive load on the face.The required charge distribution can be achievedusing either a continuous column of a explos-ive packaged in a small diameter polyethylenetube (Figure 11.13(a)), or explosive cartridges (orportions of cartridges) are spaced up the hole.Methods of spacing the cartridges include tapingthem to detonating cord (Figure 11.13(b)), usingwooden spacers, or placing them, at the requireddistance apart, in “C” shaped plastic tubes. It ispreferable that the explosive not touch the rockin the walls of the hole, which can be achievedusing sleeves to center the explosive in the hole.To promote shearing at the bottom of the hole, abottom charge two to three times that used in theupper portion of the hole is generally employed.

The approximate explosive load per meter ofblast hole lex that will produce sufficient pres-sure to cause splitting to occur, but not damagethe rock behind the final face, can be approxim-ated by

lex = (dh)2

12,100(kg/m, for dh in mm) (11.9a)

or

lex = (dh)2

28(lb/ft, for dh in inches) (11.9b)

where dh is the hole diameter.

11.4.4 Stemming

Similar to production holes, the upper part of finalline holes is unloaded and stemmed with angu-lar gravel to contain the explosive gases in thehole. For hole diameters up to about 100 mm, thestemming length is in the range of 0.6–1.0 m, andvaries with the formation being shot. In homo-geneous formations, only the top of the hole isstemmed so that air gap around the charge servesas a protective “cushion”. However, when stem-ming is not used around the charges, the explosivegases can find weak zones in the formation andtend to vent before the desired shear betweenholes is obtained. Similarly, the gases may find

Blasting 261

areas of weakness back into the finished walland produce overbreak. Therefore, in weak andhighly fractured, or faulted rock, complete stem-ming between and around individual charges maybe desirable.

11.4.5 Spacing and burden

The spacing between blast holes on the finalline differs slightly between pre-shearing andcushion blasting, as given by the followingrelationships:

Pre-shearing: Spacing

≈ (10–12) × (hole diameter) (11.10a)

Cushion blasting: Spacing

≈ (16) × (hole diameter) (11.10b)

For pre-shearing the burden is effectively infinite,but as discussed in Section 11.4.1, it should be atleast equal the bench height. For cushion blast-ing, the burden should be larger than the spacingso that the fractures preferentially link betweenholes along the final wall and do not extend into

the burden. The burden is given by

Cushion blasting: Burden

≥ (1.3) × (hole spacing) (11.11)

11.4.6 Hole detonation sequence

As discussed in Section 11.4.1, the main differ-ence between pre-shearing and cushion blastingis the detonation sequence. In pre-shearing, thefinal row holes are detonated first in the sequence,and in cushion blasting, the final row holes aredetonated last in the sequence. Furthermore, sat-isfactory results are obtained by detonating eachhole along the final line on separate delays, and itis not necessary to use a single delay for the fulllength of the final wall blast.

Figure 11.14 shows a possible blast hole lay-out and detonation sequence for pre-shearing athrough-cut. This excavation is suitable for pre-shearing because the burden is infinite and there isno possibility for prematurely displacing the bur-den. The shear line holes on either side of the cutare detonated first, followed by the productionblast as a ‘V’ cut. Theoretically, the length of apre-shear shot is unlimited. In practice, however,shooting far in advance of primary excavationcan be troublesome if the rock characteristics

8 86 5 5

7 4 75

6 3 64

5 2 53

4

7

6

5

4

3

7

6

5

4

31

4

3

2

1 42

6

5

4

3

2

Pre-sheared from previous shot

Excavated area

Instantaneouscap

Detonating cord trunkline

Advance pre-shear

Figure 11.14 Blasting procedurefor through-cut showing delaysequence for pre-shearing duringproduction blasting (adaptedfrom ISEE, 1998).

262 Blasting

change and the load causes excessive shatter inthe weaker areas. By carrying the pre-shear onlyone-half shot in advance of the primary blasting,the knowledge gained from the primary blastsregarding the rock can be applied to subsequentpre-shear shots.

11.5 Blast damage and its control

Blasting in urban areas must often be controlledto minimize the risk of damage to structures, anddisturbance to people living and working in thevicinity. Three types of damage that can be causedby blasting include (Figure 11.15):

(a) Ground vibrations—structural or cosmeticdamage resulting from ground vibrationinduced by the shockwave spreading outfrom the blast area.

(b) Flyrock—impact damage by rock ejectedfrom the blast.

(c) Airblast and noise—damage due to over-pressure generated in the atmosphere.

The basic mechanism causing damage is theexcess energy of the shock wave, generated bythe explosive detonation, as it spreads out fromthe immediate blast area. As the shock wavespreads out, its energy diminishes both due tothe energy consumed in breaking and deformingthe rock, and because it occupies a progressivelylarger volume of rock with time. At distanceswhere there is insufficient energy to break the

Ground vibrationNoise

Flyrock

Figure 11.15 Causes of blast damage.

rock, vibrations will be set up that can be largeenough to damage structures. Furthermore, if theexplosive gases are not adequately confined bythe burden, the shock wave released at the facecan generate both flyrock that is ejected from theblast, and considerable noise.

When designing a blast in an urban area, onemust take into consideration not only the poten-tial for damage in the vicinity of the blast, but alsopossible disturbance to people at a considerabledistance from the site. Disturbance to people liv-ing outside the potential damage zone can giverise to complaints and possibly spurious dam-age claims. This section provides information onallowable vibration and air blast levels, and dis-cusses methods of preventing both damage tostructures and disturbance to people.

On most urban construction projects, blastingis only one of the many possible sources of vibra-tion. Figure 11.16 shows how the peak-particlevelocity produced by various types of construc-tion equipment compares with the vibrationsproduced by detonation of 0.45 kg of dynamite.This chart represents approximate values, andactual vibration levels will vary from site to site.Detonation of an explosive charge produces ashort duration, transient vibration compared toother sources, such as heavy machinery whichproduce steady-state (or pseudo-steady) vibra-tions. Generally, the steady-state vibrations aremore disturbing to people than transient vibra-tions, even if the latter have a higher peak particlevelocity. Also, vibrations produced by the steady-state sources can damage structures close to thesource, so in some circumstances the effects ofnon-blasting vibrations should be considered.

11.5.1 Damage from ground vibration

The following is a discussion of the propertiesof ground vibrations, and how these propertiesrelate to damage to structures that are strainedby these motions.

When an explosive charge is detonated near afree surface, the elastic response of the rock tothe shock wave is the generation of two bodywaves and one surface wave. The faster of the

Blasting 263

100

60

40

20

10

6

4

2

1

0.6

0.4

0.2

0.11 2 4 6 8 10 20 40 60 100

Distance (m)

Pea

k pa

rtic

le v

eloc

ity (

mm

/sec

)

0.45 kg Dynam

ite

Diesel pile driver 36,000 ft./lb.

Vibrating pile driver

Plate compactor

(Wacker DVU 3001)

Large bulldozer

Jackhamm

er

Train (in rock tunnel)

Small bulldozer

General road traffic

(On fair road surface) Figure 11.16 Peak particlevelocities produced by constructionmachinery compared to explosivecharge (adapted from Wiss, 1981).

two body waves propagated within the rock is acompressive, or P , wave, while the slower typeis a shear, or S, wave. The surface wave, or R

wave is slower than either the P or S wave; it isnamed after Raleigh who proved its existence. Interms of vibration damage, the R wave is the mostimportant since it propagates along the groundsurface, and because its amplitude decays moreslowly with distance traveled than the P or S

waves.The wide variations in geometric and geolo-

gical conditions on typical blasting sites preclude

the solution of ground vibration magnitudes bymeans of elastodynamic equations. Therefore,the most reliable predictions are given by empir-ical relationships developed from observations ofactual blasts.

Of the three most easily measured propertiesof the stress waves, that is, acceleration, velo-city and displacement, it is generally agreed thatvelocity can be most readily correlated with dam-age to structures. The stress wave has threecomponents—vertical, horizontal and radial, andit is necessary to measure all three components

264 Blasting

and use the greatest, termed the Peak ParticleVelocity (PPV), to assess damage potential. Thedifference between the particle velocity of a stresswave, and the velocity of propagation of the wavecan be explained as follows. As the wave passesthrough the ground, each particle of the rock andsoil undergoes an elliptical motion, and it is thevelocity of this oscillating motion (possibly up to0.5 m/s) that is measured in assessing blast dam-age. In contrast, the velocity of propagation ofthe wave is in the range 300–6000 m/s, and thishas no direct bearing on damage.

Ground motion can be described as a sinus-oidal wave in which the variation of the particlevelocity v with time t is given by (Figure 11.17):

v = A sin(ωt) (11.12)

where A is the amplitude of the wave, and ω is theangular velocity. The magnitude of the angularvelocity given by

ω = 2πf = 2π1T

(11.13)

where f is of the frequency (vibrations persecond, or Hertz) and T is the period (time for onecomplete cycle). The wavelength L of the vibra-tion is the distance from crest to crest of one fullcycle and is related to the period T and the velocityof propagation V by

L = VT (11.14)

CrestPeriod, T

Vel

ocity

Trough

� 2� 3� 4�

Time

A

Figure 11.17 Sinusoidal wave motion for typicalground vibrations.

If the particle velocity v has been measured,then the displacement δ can be found by integ-ration, and the acceleration a by differentiationas follows:

δ = v

ωand a = ωv (m/s2)

or

a = ωv

9.81(g/ms2) (11.15)

The most reliable relationship between blastgeometrics and ground vibration is that relatingparticle velocity to scaled distance. The scaled dis-tance is defined by the function R/

√W , where

R (m) is the radial distance from the point ofdetonation, and W (kg) is the mass of explosivedetonated per delay. Field tests have establishedthat the maximum particle velocity, V (mm/s)is related to the scaled distance by followingattenuation relationship (Oriard, 1971):

V = k

(R√W

)β

(11.16)

where k and β are constants that have to bedetermined by measurements on each particularblasting site. Equation (11.16) plots as a straightline on log–log paper, where the value of k is givenby the PPV intercept at a scaled distance of unity,and the constant β is given by the slope of the line.An example of such a plot is given in Figure 11.18.

In order to obtain data from the preparation ofa plot, such as that in Figure 11.19, it is necessaryto make vibration measurements with a suitablemonitoring instrument. Table 11.5 shows typ-ical specifications for a seismograph suitable formeasuring blast vibrations, as well as vibrationsproduced by a wide range of construction equip-ment such as pile drivers. Of importance formonitoring non-blasting applications is the usefulfrequency range of the equipment; for the Instan-tel DS677, this range is 2–250 Hz, which meansthat vibrations with frequencies outside thisrange will not be detected. Some of the featuresof instruments currently available (2003) aremeasurement of both ground vibration—velocity,

Blasting 265

1000

500400300

200

100

504030

20

10

543

2

1

k = 800

1

= –1.6

1 32 4 5 10 20 304050 100

Pea

k pa

rtic

le v

eloc

ity (

mm

/s)

Scaled distance (m/kg1/2)

SD = 5.65

Typical values of k quoted by Oriard (2002)

k = 1130 (160)—average values= 1700 (242)—typical upper bound (90% bound)= 4270 (605)—high confinement, coupling and

rock strength (99% bound).

units: mm/s, m, kg

units: in./s, ft, lb

Figure 11.18 Hypothetical plot of measured particlevelocity versus scaled distance from blast to determineattenuation constants k and β.

acceleration, displacement and frequency in ver-tical, longitudinal and transverse axes, and airpressure. In addition, the instruments incorpor-ate a trigger that starts recording when a pre-setvibration level or air pressure is detected. Theresults are digitally recorded so the data canbe downloaded to a computer for storage andfurther analysis.

Measurement of ground vibrations involvesinstalling geophones on the ground close to thestructure of interest, or on the structure itself.

It is important that the geophone be properlycoupled to the ground or structure. The Interna-tional Society of Rock Mechanics (ISRM, 1991)suggests that the method of mounting is a func-tion of the particle acceleration of the wave trainbeing monitored. When vertical accelerations areless than 0.2g, the geophone may be placed ona horizontal surface without being anchored inplace. When the maximum particle accelerationsare between 0.2g and 1.0g, the geophones shouldbe completely buried when monitoring in soil,or firmly attached to rock, asphalt or concrete.Appropriate methods of attaching a geophone tothese surfaces include double-sided tape, epoxyor quick setting cement. If these methods areunsatisfactory or if maximum particle accelera-tions exceed 1.0g, then only cement or bolts areappropriate to secure the geophone to a hard sur-face. Note that weighting the geophone with asandbag, for example, is not effective because thebag will move by the same amount as the geo-phone. In all cases, the geophone should be leveland aligned with the longitudinal axis along theradial line to the blast.

Vibration damage thresholds. Much workhas been carried out to estimate threshold vibra-tion levels for damage to many different types ofstructures (Siskind et al., 1976, 1980; Stagg et al.,1984). These peak particle velocity levels, whichare listed in Table 11.6, have been establishedfrom many observations in the field and are nowused with confidence as a basis for design of blastswhere ground vibrations must be controlled.

The damage criterion most frequently used inurban areas is a peak particle velocity of 50 mm/s,which is the level below which the risk of damageto most residential structures is very slight. How-ever, for poorly constructed buildings, and oldbuildings of historic interest, allowable vibrationlevels may be as low as 10 mm/s. The requiredlimit should be determined from the structuralcondition of the building.

Figure 11.19 shows the relationship betweenthe distance of a structure from the blast, theexplosive weight detonated per delay and theexpected peak particle velocity at the structure.

266 Blasting

10,000

5000

1000

500

100

50

10

5

0.5

0.1

1

1 5 10 50 100 500 1000 5000

V =

635

mm

/s

Ons

et o

f roc

k br

eaka

ge

V =

50m

m/s

N

o si

gnifi

cant

stru

ctur

al d

amag

e

V =

8m

m/s

V

ibra

tions

dis

turb

ing

V =

3m

m/s

V

ibra

tions

not

icea

ble

Damage

Typical vibration record

Seconds

Vlongitudinal

Vvertical

Vtransverse

0 0.1 0.2 0.3

Inst

anta

neou

s ch

arge

per

del

ay (

kg)

Distance from blast (m)

Safe

V = Peak particle velocity

Figure 11.19 Typical blastvibration control diagram forresidential structures.

These graphs have been drawn up using the rela-tionship given in equation (11.16) for values ofk = 1600 and β = −1.5. It is intended for useas a guideline in the assessment of blast dam-age for residential structures. If the blast designand this graph indicate that the vibration levelis close to the damage threshold, then it is usu-ally wise to carry out trial blasts and measurethe vibration levels produced. These data can beused to produce a graph similar to that shown inFigure 11.18, from which the values of the con-stants k and β for the site can be determined.The constants can be used to draw up a blastdesign based on the calculated allowable chargeweight per delay. It has been found that the val-ues of k and β will vary considerably from siteto site, so vibration measurements are useful incritical situations unless conservative restrictionsare placed on allowable explosive weights perdelay.

Figure 11.19 shows that 100 kg of explosivedetonated per delay may cause minor cracking of

plaster in houses at distances less than 100 m fromthe blast, while the vibrations will be noticeable atdistances up to about 850 m. Halving of the massof explosive detonated per delay to 50 kg willreduce these distances to 70 m and 600 m respect-ively. Thus, the use of delays to limit the weight ofexplosive detonated per delay is the most effectivemethod of controlling both damage and reactionsof the public to the blasting operations.

Effect of vibration frequency on ground vibra-tions. The frequency of the vibration is alsoof importance in assessing damage potential. Ifthe principle frequency, that is the frequency ofgreatest amplitude pulse, is approximately equalto the natural frequency of the structure, thenthere is a greater risk of damage than if the prin-ciple and natural frequencies are significantly dif-ferent (Dowding, 1985). The natural frequencyof two-storey residential buildings is in the rangeof 5–20 Hz, and the natural frequency diminisheswith increasing height of the structure. The prin-cipal frequency of a blast will vary with such

Blasting 267

Table 11.5 Instantel DS677 technical specifications

Ranges All channels are autoranging. Seismic (three channels): (0–250 mm/s)a

Air pressure (one channel): (up to 140 dB peak (200 kPa)Trigger Seismic: 0.50–50.85 mm/slevels Air pressure: 92–129 dB (0.8–56.4 kPa)

Resolution Seismic: 0.1245 mm/sAir pressure: 1 dB (0.02 Pa)

Frequency All channels 2–250 Hz, ±3 dBresponse (independent of record time)

Recording Standard: 1–7 s in steps of 1 stimes Option: 1–10 s in steps of 1 s

Additional 0.25 s pre-trigger in all recordsAcceleration Computed up to 30 g with a resolution of 0.013 g

(0.01 displayed)Displacement Computed to 38 mm

Metric resolution 0.000127 mm, 0.001 displayed

Notea Measurement units (Imperial or Metric) are user selectable in the field. Air pressure values are rms weighted.

Table 11.6 Peak particle velocity threshold damagelevels

Velocity (mm/s) Effect/damage

3–5 Vibrations perceptible to humans.10 Approximate limit for poorly

constructed and historicbuildings.

33–50 Vibrations objectionable tohumans.

50 Limit below which risk ofdamage to structures is veryslight (less than 5%).

125 Minor damage, cracking ofplaster, serious complaints.

230 Cracks in concrete blocks.

factors as the type of blast, the distance betweenthe blast and the structure, and the materialthrough which the ground vibrations travel. Typ-ical construction blasts produce vibrations withprincipal frequencies in the range of about 50–200 Hz. It is found that large quarry andmineblasts produce vibrations with lower principalfrequencies than construction blasts, and thatprincipal frequencies decrease with increasing dis-tance from the blast due to frequency attenuation.

Figure 11.20 shows the relationship betweenpeak particle velocity, vibration frequency andapproximate damage thresholds for residentialstructures. This diagram demonstrates that withincreasing frequency, the allowable PPV alsoincreases.

Effect of geology on ground vibrations. It hasbeen found that blast vibrations are modifiedby the presence of overburden at the measure-ment location. In general, vibrations measured onoverburden have a lower frequency and higheramplitude than those measured on rock at thesame distance from the blast. A consequence ofthis is that if the particle velocity is approximatelythe same at two locations, the lower frequencyof the vibrations in overburden will make theblast vibrations more readily felt by humans.

Vibrations in uncured concrete. On someconstruction projects there may be a need to carryout blasting operations close to uncured concrete.Under these circumstances, explosive chargeweights per delay should be designed to keepground vibrations within limits that are determ-ined by the age of the concrete, the distance ofthe concrete from the blast and the type of struc-ture (Oriard and Coulson, 1980; Oriard, 2002).Table 11.7 shows an approximate relationship

268 Blasting

Damage

Safe

200

100

10

31 10 100

Par

ticle

vel

ocity

(m

m/s

)

Frequency (Hz)

Plaster 13 mm/s

0.75 mm*

*Allowable displacement (equation (11.15))

Drywall 20 mm/s

0.2 mm*

50 mm/s

Figure 11.20 Allowablevibration levels forresidential structures dueto blasting (Siskind et al.,1980).

Table 11.7 Relationship between age of concrete andallowable peak particle velocity (Oriard, 2002)

Concrete age from Allowablebatching PPV mm/s (in./s)

0–4 h 100 (4) × DF4 h to 1 day 150 (6) × DF1–3 days 225 (9) × DF3–7 days 300 (12) × DF7–10 days 375 (5) × DFMore than 10 days 500 (20) × DF

Distance factor, DF Distance fromblast, m (ft)

1.0 0–15 (0–50)0.8 15–46 (50–150)0.7 46–76 (150–250)0.6 >76 (>250)

between allowable peak particle velocity levelsand the concrete age for mass concrete. At agesless than four hours, the concrete has not yetset and ground vibration levels are permissible,with gradually increasing levels allowed as theconcrete sets.

Table 11.7 also shows that vibration levelsmust be reduced with increasing distance, asindicated by the distance factor (DF). Concretecan withstand higher vibration levels at higherfrequencies, because at low frequencies greaterdeflections will be induced in the structure. Thisis of particular concern for structural walls offreshly poured concrete. Vibration frequenciesdecrease as the distance from the blast increasesbecause there is an attenuation of frequency withdistance. The result of this frequency attenuation

Blasting 269

is that, at equal curing times, higher vibrationlevels are permitted at closer distances.

In critical conditions it is recommended thatvibration measurements and strength tests be con-ducted to confirm the performance of the concreteand the relationships given in Table 11.7.

Electronic equipment and machinery. Sometypes of electronic/electrical equipment are sens-itive to vibrations. Studies have been done oncomputer disk drives, telecommunication equip-ment such as relay stations and fiber optic cables,as well as electrical equipment such as oldermodel power transformers that have mercurycut-out switches. Vibrations from blasting, piledriving, or other construction activities can inter-fere with the operation of this equipment, and inthe absence of guidelines from the manufactureron allowable vibration levels, it may be necessaryto conduct carefully calibrated test blasts.

Human response to blast vibrations. Humansare very sensitive to vibrations and can feel theeffects of a blast well outside the potential dam-age zone. Figure 11.21 shows the relationshipbetween peak particle velocity, frequency andthe possible human response to vibrations. Thisindicates that low frequency vibrations are morereadily felt than high frequency vibrations. Thefrequency of blast vibrations is usually in therange of 50–200 Hz.

Control of vibrations. The magnitude of blastvibrations at a particular location is dependentupon the distance from the blast and the chargeweight per delay. Of the blast design factors dis-cussed in Section 11.4, the most important are theuse of delays and the correct detonation sequenceso that each hole, or row of holes, breaks towarda free face.

To control vibration levels at a particular dis-tance from the blast, it is necessary to limit theexplosive charge detonated per delay accordingto the relationship given in equation (11.16).Calculation of the allowable charge weight perdelay, from either trial blasts or design charts, willdetermine how many holes can be detonated on asingle delay. If the charge weight in a single hole

250

11P

artic

le v

eloc

ity (

mm

/s)

Frequency (cps)

50 mm/s safe structural limit

50

25

5

2.5

0.252 4 6 10060402010

Intolerable

Unpleasant

Perceptible

Figure 11.21 Human response to blast vibrationsrelated to particle velocity and frequency (adaptedfrom Wiss, 1981).

is more than that allowable, then either shorterholes must be drilled, or a decked charge couldbe used. In a decked charge, the explosive loadis separated with stemming material, and eachcharge is detonated on a separate delay. The min-imum delay interval between charges or holes inorder to limit the risk of constructive interferenceof vibrations produced by each charge is about15–20 ms.

Pre-blast surveys. At locations where thereis a potential for damage to structures due toblast vibrations, it is usual to carry pre-blast sur-veys on all structures within the potential damagezone. These surveys should record, with pho-tographs and/or video tape where appropriate,all pre-existing cracks, other structural damageand settlement problems. The Office of SurfaceMining (OSM, 2001) has drawn up a stand-ardized method of recording structural damagewhich ensures that the survey is systematic and

270 Blasting

1

2

3

4Door

Floor plan

Path of inspection

1

2

3

4

5

6

Floor plan

Ceiling

Upperleftquarter

Upperrightquarter

Lowerleftquarter

Lowerrightquarter

Floor

Enter Enter

(a) (b)

(c)

Figure 11.22 Method ofmaking building damagesurveys: (a) wallidentification procedure;(b) multi-wall identificationprocedure; (c) detailed wallidentification procedure(Office of Surface Mining,2001).

thorough (Figure 11.22). In addition to the build-ing survey, a public relations program informingpeople of the blasting operations, and carefullydocumented vibration measurements will usu-ally minimize complaints and spurious damageclaims.

11.5.2 Control of flyrock

When the front row burden is inadequate or whenthe stemming column is too short to contain theexplosive gases, a crater is formed and rock ejec-ted from the crater may be thrown a considerabledistance (Figure 11.23). This figure also showsthat flyrock can be caused by poor drill alignment,and by geologic conditions that allow venting ofthe explosive gases along discontinuities in therock mass.

In practice, the complete control of flyrock isdifficult, even if the blast is designed with the

recommended stemming and burden dimensions.Therefore, in areas where there is a possibilityof damage to structures, blasting mats should beused to control flyrock. Blasting mats consist ofrubber tires or strips of conveyor belting chainedtogether. When there is a possibility that the matcould be displaced by the explosive energy, theyshould be weighted with soil, or anchored to thebedrock.

11.5.3 Control of air blast and noise

These two problems are taken together becausethey both arise from the same cause. Air blast,which occurs close to the blast itself, can causestructural damage such as the breaking of win-dows. Noise, into which the air blast degenerateswith distance from the blast, can cause discom-fort and give rise to complaints from those livingclose to the blasting operations.

Blasting 271

x x x

x

Section Section

Section Section

Section Front elevation of face

(a) (b)

(c) (d)

(e) (f)

Figure 11.23 Commoncauses of flyrock:(a) inadequate front rowburden; (b) holemisalignment resulting inconcentration ofexplosives; (c) weakseams vent gas to rockface; (d) holes loadedclose to bench surface;(e) some holes with nostemming; (f) blockedholes loaded with fixedweight of explosive ornumber of cartridges(after CIL, 1984).

Factors contributing to the development of airblast and noise include overcharged blast holes,poor stemming, uncovered detonating cord, vent-ing of explosive gases along cracks in the rockextending to the face, and the use of inadequateburdens giving rise to cratering. In fact, many ofthe hole layout and charging conditions shown inFigure 11.23 can result in airblast in addition toflyrock.

The propagation of the pressure wave dependsupon atmospheric conditions including temperat-ure, wind and the pressure–altitude relationship.Cloud cover can also cause reflection of the pres-sure wave back to ground level at some distancefrom the blast. Figure 11.24 gives an indication ofhow the propagation of the shockwave is affectedby the variation of temperature with altitude. This

shows that air blast problems can be most severeduring temperature inversions.

Figure 11.25 gives a useful guide to theresponse of structures and humans to sound pres-sure level (Ladegaard-Pedersen and Dally, 1975).The maximum safe air blast levels recommendedby the US Bureau of Mines is 136 dB for pressuremeasurements made using a linear peak weight-ing network. The relationship between decibelsand pressure P (kPa) is given by the followingequation:

dB = 20 log10

(P

P0

)(11.17)

where P0 is the over-pressure of the lowest soundthat can be heard, about 2 × 10−5 kPa.

272 Blasting

TemperatureSoundspeed

(a) (b)A

ltitu

de

Temperature or sound speed

Alti

tude

Totally refractedReflectionCurl up

RaysWavefront

No trouble

No trouble

Wave front

Rays

Quietspot

Intenseregion

Escapes

Inversions—Potential trouble

Potential trouble

Alti

tude

Alti

tude

Alti

tude

Potential trouble

Figure 11.24 Effect of atmosphericconditions on air blast: (a) variationof temperature and sound speedwith altitude; (b) correspondingpattern of sound rays and wavefront (Baker, 1973).

The decrease of sound pressure with distancecan be predicted by means of cube root scaling.The scaling factor with distance KR is given by:

KR = R3√W

(11.18)

where R is the radial distance from the explosionand W is the weight of charge detonated per delay.

Figure 11.26 gives the results in the originalEnglish units of pressure measurements carried

out by the US Bureau of Mines in a number ofquarries. The burden B, was varied and the lengthof stemming was 2.6 ft per inch (0.31 m stem-ming per cm) diameter of borehole. For example,if a 1000 lb (454 kg) charge is detonated with aburden of 10 ft (3.1 m), then the over-pressureat a distance of 500 ft (152 m), is found asfollows:

R3√W

= 5003√1000

= 50

Blasting 273

Maximum safe air blastpressure levels

Most windows breakStructures damaged

Some windows break“No damage” level

Threshold of painThreshold of complaints(dishes and windows rattle)

Riveter

Ordinary conversation

Hospital room

Whisper

Level of hearing

Airblastfrom

explosions

Decibels

180

160

140

120

100

80

60

40

20

0

kPa

20,700

70002000

700210

70

21

7

2

0.7

0.2

0.02

2 × 10–3

2 × 10–4

2 × 10–5

Figure 11.25 Human andstructural response to soundpressure level(Ladegaard-Pedersen andDally, 1975).

and

B3√W

= 103√1000

= 1

From Figure 11.26, over-pressure equals about0.015 psi (0.1 kPa), or 74 db.

These calculations are related to the air blastproduced by the explosive charge itself. However,a significant component of the air blast is pro-duced by detonating cord and a reduction innoise can be achieved by covering the detonatingcord with sand. Alternatively, a detonating sys-tem such as Nonel can be used, which comprisesa fine plastic tube that propagates the shock wavewith little noise.

11.6 Example Problem 11.1: blast design

Statement

A 6 m high rock bench is to be excavated by blast-ing. The rock contains thin, well-cemented layersthat dip steeply out of the face, and the specific

gravity of the rock is 2.6. The available explos-ive has a specific gravity of 1.3. The broken rockwill be excavated by a front-end loader with amaximum vertical reach of 5 m.

Required

(a) Determine a suitable blast hole pattern, thatis, the burden, spacing, and the diameter ofthe explosive.

(b) Determine the depth of subgrade drilling, thelength of the stemming and the explosiveload per hole.

(c) Calculate the powder factor (kg/m3).

Solution

(a) The bench can be blasted in a single liftbecause most percussion drills can drill to adepth of 6 m with good directional controland penetration rate, and it would not bedangerous for the loader to dig a 6 m highmuck pile.

For a stiffness ratio of 0.3 and a benchheight of 6 m, the required burden distance

274 Blasting

Surface

B/3 W = 0

B/3 W = 1 12

1(0.4)

10(4)

100(40)

1000(400)

10 000(4000)

B

ChargeW

B/3

B/3 W = 1

Ove

r-pr

essu

re lb

/in2

(kP

a)

Scaled distance R/3 W—ft/lb1/3 (m/kg)1/3

10(70)

1(7)

0.1(0.7)

0.01(0.07)

0.001(0.007)

0.0001(0.0007)

W = 12

Figure 11.26 Over-pressure as afunction of scaled distance for benchblasting (Ladegaard-Pedersen andDally, 1975).

is 1.8 m (see equation (11.1)). The adjustedburden to account for the rock condition is2.3 m, using equation (11.4) with kψ = 1.18and ks = 1.1.

The explosive diameter, calculated fromequation (11.2), is 78 mm.

The spacing between holes, calculatedfrom equation (11.5), is 2.8 m.

(b) The depth of subgrade drilling is 0.7 m,based on a subgrade depth equal to 0.3 ofthe burden.

The length of the stemming is 1.6 m, basedon a stemming length equal to 0.7 of theburden.

The length of the explosive column is5.1 m, based on a bench height of 6 m, a sub-grade depth of 0.7 m and a stemming lengthof 1.6 m. For this explosive length, a specific

gravity of 1.3 and a diameter of 78 mm, theweight of explosive per hole is 31.7 kg.

(c) The volume of rock to be broken per hole is38.6 m3, for a bench height of 6 m, a burdenof 2.3 m and a spacing of 2.8 m.

The powder factor is 0.8 kg/m3. With ref-erence to Figure 11.8, the average bouldersize of the blasted rock will be about 0.4 m,which can be readily loaded with a front-endloader.

11.7 Example Problem 11.2: controlledblasting design

Statement

A 6 m high slope has been previously excavatedfor a highway by blasting, and it is required that

Blasting 275

the face be trimmed back by 3 m. It is necessarythat controlled blasting be used for this trimmingoperation so that there is minimal overbreak andthe new face is stable.

Required

(a) Determine the most appropriate type of con-trolled blasting for this operation.

(b) If the explosive available has a diameter of25 mm, calculate the blast hole diameter,explosive charge (kg/m of hole) and holespacing.

(c) If the explosive has a specific gravity of 1.1,determine how the explosive can be dis-tributed in the hole to achieve the requiredcharge.

(d) Assess if it is necessary for a second rowof blast holes to be drilled between thepresent face and the final line. If a secondrow is necessary, determine how far shouldthis row be from the final line, and whatdetonation sequence should be used.

Solution

(a) Cushion blasting should be used to removethe 3 m thick slice of rock. In highwayconstruction, it is rarely necessary to useline drilling because it is expensive and isonly used where very high quality slopes arerequired. Pre-shearing would be used onlywhere the burden is at least equal to thebench height.

(b) For an explosive diameter of 25 mm, a50 mm diameter hole would produce adecoupling ratio of 2. The required explosivecharge per meter of blast hole is 0.21 kg/m,from equation (11.9a).

The spacing between holes would be0.8 m, from equation (11.10b).

(c) The weight of explosive for a 25 mm dia-meter cartridge with a specific gravity of1.1 is 0.54 kg/m. To produce an explosivecharge of 0.21 kg/m, a possible distribu-tion of this explosive up the hole would

be to have cartridge length of 200 mm,with 300 mm long spacers between eachcartridge.

(d) From equation (11.11), the required burdenis at least 1 m. Since the total burden on thecushion blast is 3 m, it would be necessaryto drill a row or holes about 1.5 m behindthe face. The detonation sequence would beto fire the front row at least 20 ms before thefinal row, based on equation (11.7a).

11.8 Example Problem 11.3: blastdamage control

Statement

A historic building is located 140 m from the blastsite and a hospital is located at a distance of1 km.

Required

(a) Determine the maximum allowable chargeweight per delay to minimize the risk ofdamage to the historic building.

(b) Determine if the ground vibrations would bedisturbing to patients in the hospital.

Solution

(a) Allowable charge weights per delay aredetermined from damage threshold vibra-tion levels for different structures listed inTable 11.6 and from equation (11.16).

If k = 1600 and β = −1.5 and thethreshold for damage to the historic buildingis 10 mm/s, then at R = 140 m, the allowableinstantaneous charge, W is 23 kg/delay.

(b) Figure 11.21 shows that vibrations are objec-tionable to humans when the peak particlevelocity exceeds about 33–50 mm/s. For acharge weight per delay of 23 kg, equa-tion (11.16) shows that the vibration levelsat the hospital would be about 0.5 mm/s, sothey are unlikely to be even perceptible bythe patients (perceptible threshold is about3–5 mm/s).

Chapter 12

Stabilization of rock slopes

12.1 Introduction

In mountainous terrain, the operation ofhighways and railways, power generation andtransmission facilities, and the safety of residen-tial and commercial developments often requirestable slopes and control of rock falls. This appliesto both excavated and natural slopes. In con-trast, open pit mines tolerate a certain degree ofslope instability unless there is a hazard to theminers or a significant loss of production. Forexample, minor failures of benches usually havelittle effect on operations unless the fall lands ona haul road and results in tire or equipment dam-age. In the event of a large-scale slope failure inan open pit mine, often the only economic andfeasible stabilization measure is drainage, whichmay involve long horizontal drains, pumpedwells or drainage adits (see Section 12.4.6). Amore common means of managing large-scaleslope instability is to monitor the movementso that mining can continue beneath the mov-ing slope. Procedures for monitoring movementand interpreting the results are discussed inChapter 13.

This chapter is concerned mainly with civilslopes because the high cost of failures meansthat stabilization programs are often economic-ally justified. For example, on highways, evenminor falls can cause damage to vehicles, injuryor death to drivers and passengers, and possiblydischarge of toxic substances where transportvehicles are damaged. Also, substantial slopefailures on transportation systems can severelydisrupt traffic, usually resulting in both direct and

indirect economic losses. For railroads and tollhighways, closures result in a direct loss of rev-enue. Figure 12.1 shows a rock slide that occurredfrom a height of about 300 m above the road andclosed both the road and an adjacent railway.

While the cost of a slide, such as that shown inFigure 12.1 is substantial, the cost of even a singlevehicle accident can be significant. For example,costs may be incurred for hospitalization of thedriver and passengers, for repair to the vehicle,and in some cases for legal charges and com-pensation payments. Often there are additionalcosts for stabilization of the slope that will involveboth engineering and contracting charges, usu-ally carried out at premium rates because of theemergency nature of the work.

Many transportation systems were constructedover a century ago in the case of railroads, anddecades ago in the case of many highways. Atthat time, the blasting techniques that were oftenused in construction caused significant damage tothe rock. Furthermore, since the time of construc-tion deterioration of stability conditions is likelydue to weathering of the rock, and loosening ofthe surficial blocks by ice and water, and by thegrowth of tree roots. All these effects can result inon-going instability that may justify remediationprograms.

For urban developments in mountainous ter-rain, hazards that can threaten or even destroystructures include rolling boulders and landslides.The most effective protection against these condi-tions is initial hazard mapping, and then zoning

Stabilization of rock slopes 277

Figure 12.1 Highway closurecaused by rock fall in very strong,blocky granite from a height ofabout 300 m.

unsafe areas to preclude development where mit-igation cannot be carried out.

In order to minimize the cost of falls, rockslope stabilization programs are often a preferredalternative to relocation or abandonment of thefacility. Such programs involve a number of inter-related issues including geotechnical and environ-mental engineering, safety, construction methodsand costs, and contracting procedures. Methodsfor the design of stabilization measures for rockslopes are described in this chapter, and in otherreferences such as Brawner and Wyllie (1975);Fookes and Sweeney (1976); Piteau and Peckover(1978); Wyllie (1991); Schuster (1992); FederalHighway Administration (FHWA) (1993, 1998);Transportation Research Board (TRB) (1996)and Morris and Wood (1999). These proced-ures have been used extensively since the 1970sand can, therefore, be used with confidence fora wide range of geological conditions. However,as described in this chapter, it is essential thatappropriate method(s) be used for the particularconditions at each site.

The first part of this chapter discusses the causeof rock falls, and the planning and management ofstabilization programs. For systems with a largenumber of rock slopes, these programs usuallyinvolve making an inventory of rock fall hazardsand stability conditions, and maintaining these

records in a database and linked GIS map, andpreparing a prioritized stabilization schedule. Theremainder of the chapter discusses stabilizationmeasures, categorized according to rock rein-forcement, rock removal and rock fall protection.

12.2 Causes of rock falls

The State of California made a comprehensivestudy of rock falls that occurred on the state high-way system to assess both the causes of the rockfalls and the effectiveness of the various remedialmeasures that have been implemented (McCauleyet al., 1985). With the diverse topography andclimate within California, their records provide auseful guideline on the stability conditions of rockslopes and the causes of falls. Table 12.1 show theresults of a study of a total of 308 rock falls onCalifornia highways in which 14 different causesof instability were identified.

Of the 14 causes of rock falls, 6 are directlyrelated to water, namely rain, freeze–thaw, snow-melt, channeled runoff, differential erosion, andsprings and seeps. There is also one cause that isindirectly related to rainfall—the growth of treeroots in cracks that can open fractures and loosenblocks of rock on the face. These seven causesof rock falls together account for 68% of the

278 Stabilization of rock slopes

total falls. These statistics are confirmed by theauthors’ experience in the analysis of rock fallrecords over a 25-year period on a major railroadin western Canada in which approximately 70%of the events occurred during the winter. Theweather conditions during the winter includedheavy rainfall, prolonged periods of freezing tem-peratures, and daily freeze–thaw cycles in thefall and spring. The results of a similar study

Table 12.1 Causes of rock falls on highways inCalifornia

Cause of rock fall Percentage of falls

Rain 30Freeze–thaw 21Fractured rock 12Wind 12Snowmelt 8Channeled runoff 7Adverse planar fracture 5Burrowing animals 2Differential erosion 1Tree roots 0.6Springs or seeps 0.6Wild animals 0.3Truck vibrations 0.3Soil decomposition 0.3

carried out by Peckover (1975) are shown inFigure 12.2. They clearly show that the major-ity of rock falls occurred between October andMarch, the wettest and coldest time of the year inwestern Canada. Also in this geographical area,studies have been conducted on the relationshipbetween the frequency and volume of rock slides(Hungr et al., 1999). The study showed that forfalls with volumes less than 1 m3, there were asmany as 50 falls per year, whereas falls withvolumes of 10,000 m3 occurred every 10–50 yearsapproximately.

The other major cause of rock falls in theCalifornia study is the particular geologic condi-tions at each site, namely fractured rock, adverseplanar fractures (fractures dipping out of slopeface), and soil decomposition. These three causesrepresented 17% of the falls, and the totalrock falls caused by water and geologic factorsaccounted for 85% of the falls. These statist-ics demonstrate that water and geology are themost important factors influencing rock slopestability.

It appears that the study in California wascarried out during a time that there were nosignificant earthquakes, because these events fre-quently trigger rock falls, and cause displacement

25 20.0

17.5

15.0

12.5

10.0

7.5

5.0

2.5

20

15

10

5

–5

–15

Jan Feb Mar Apr May JunMonth (1933–70)

Mea

n m

onth

ly te

mpe

ratu

re (

°C)

Mean m

onthly precipitation (cm)

Jul Aug Sep Oct Nov Dec

–10

0

~60

~35 ~30

~10

Mean monthly temperature

Total number of rock falls per monthMean monthly precipitation

Figure 12.2 Correlation of numberof rock falls with temperature andprecipitation on railway lines inFraser Canyon, British Columbia(Peckover 1975).


and failure of landslides (Van Velsor andWalkinshaw, 1991; Jibson and Harp, 1995).Section 6.5 discusses methods to incorpor-ate seismic ground motion in slope stabilityanalyses.

12.3 Rock slope stabilization programs

On transportation systems in mountainous ter-rain, there may be hundreds of rock slopeswith a variety of rock fall hazards resulting ina significant cost to the operator. Under thesecircumstances, a long-term, multi-year stabil-ization program is often justified; this sectiondescribes the steps involved in implementing sucha program.

12.3.1 Planning stabilization programs

When implementing a program to stabilize a largenumber of slopes, the best use of available fundsis often made by setting up a systematic programthat identifies and rates the most hazardous sites.Annual stabilization work can then be scheduled,with the most hazardous sites having the highestpriority. Table 12.2 shows an example of howsuch a program may be structured.

The objective of the program shown inTable 12.2 is to be proactive in identifying andstabilizing slopes before rock falls and accidentsoccur. This requires a careful examination of eachsite to identify the potential hazard, and estim-ate the likely benefit of the stabilization work. In

contrast, a reactive program places the emphasisin areas where rock falls and accidents havealready occurred, and where the hazard may thenbe diminished.

An effective proactive approach to stabiliza-tion requires a consistent, long-term programunder the direction of a team experienced inboth the engineering and construction aspects ofthis work. Another important component of thiswork is to keep accurate records, with photo-graphs, of slope conditions, rock falls and sta-bilization work. This information will documentthe location of hazardous areas and determine thelong-term effectiveness of the program in redu-cing the incidence of rock falls. These records canbe most conveniently handled using database pro-grams that readily allow updating and retrieval ofrecords.

12.3.2 Rock slope inventory systems

The relative rock fall hazard at a site as comparedto other sites can be used in selecting priorities.Early work on this topic by Brawner and Wyllie(1975) and Wyllie (1987), was adapted by Piersonet al. (1990) into a process for the rational man-agement of rock slopes on highways, which hasbeen named the Rock Fall Hazard Rating System(RHRS). The first step in this process is to make aninventory of the stability conditions of each slopeso that they can be ranked according to their rockfall hazard (Steps 1 and 2 in Table 12.2).

The rock fall areas identified in the inventoryare ranked by scoring the categories shown in

Table 12.2 Rock slope stabilization for transportation systems

Step 1

Prepareinventory ofrock slopeswith hazardrating assignedto each slopeand organizerecords indatabase.

Step 2

Rank slopesaccording tohazardrating toidentify thehighest prioritylocations.

Step 3

Select anumber of thehighest prioritylocations forinclusion inannualstabilizationprogram.

Step 4

Determinemostappropriatestabilizationmeasure(s) foreach site, andpreparedesigns andspecifications.

Step 5

Carry outstabilizationwork possibleusing time andmaterialcontract thatcanaccommodatechangingconditions thatdevelop duringthe work.

Step 6

Updatedatabase withrock fall andstabilizationrecords.

Step 7

Reassesshazard ratings,and return toStep 3 to selectsites forsubsequentyearsstabilizationwork.


Table 12.3 Summary sheet of the rock fall hazard rating system (Wyllie, 1987; Pierson et al., 1990)

Category Rating criteria and score

Points 3 Points 9 Points 27 Points 81(a) Slope height (m) 7.5 m 15 m 23 m 30 m(b) Ditch effectiveness Good catchment Moderate

catchmentLimitedcatchment

No catchment

(c) Average vehicle risk(% of time)

25% of the time 50% of the time 75% of the time 100% of the time

(d) Percentage of decisionsight distance (% ofdesign value)

Adequate sightdistance, 100%of design value

Moderate sightdistance, 80% ofdesign value

Limited sightdistance, 60%of design value

Very limited sightdistance 40% ofdesign value

(e) Roadway widthincluding pavedshoulders (m)

13.5 m 11 m 8.5 m 6 m

(f) Geologic characterCase 1

Structural condition Discontinuousjoints, favorableorientation

Discontinuousjoints, randomorientation

Continuousjoints, adverseorientation

Continuous joints,adverseorientation

Rock friction Rough, irregular Undulating Planar Clay infilling, orslickensided

Case 2Structural condition Few differential

erosion featuresOccasionalerosion features

Many erosionfeatures

Major erosionfeatures

Difference in erosionrates

Small difference Moderatedifference

Large difference Extremedifference

(g) Block size 0.3 m 0.6 m 1.0 m 1.2 m— — — — —Quantity of rock fallevent

3 m3 6 m3 9 m3 12 m3

(h) Climate and presence ofwater on slope

Low to moderateprecipitation; nofreezing periods,no water onslope

Moderateprecipitation, orshort freezingperiods, orintermittentwater on slope

High precipitationor long freezingperiods, orcontinual wateron slope

High precipitationand long freezingperiods, orcontinual wateron slope andlong freezingperiods

(i) Rock fall history Few falls Occasional falls Many falls Constant falls

Table 12.3. These categories represent the sig-nificant elements of a rock slope that contributeto the overall hazard. The four columns corres-pond to logical breaks in the hazard representedby each category. The criteria scores increaseexponentially from 3 to 81 points, and repres-ent a continuum of points from 1 to 100. Anexponential system allows for a rapid increasein score which distinguishes the more hazard-ous sites. Using a continuum of points allows

flexibility in evaluating the relative impact ofconditions that are variable by nature. Somecategories require a subjective evaluation whileothers can be directly measured and then scored.

12.3.3 Hazard rating criteria

The following is a brief description of each of thecategories that are used to rate the rock fall hazardalong a highway (refer to Table 12.3).


(a) Slope height—Vertical height of the slopemeasured from the highest point from whichrock fall is expected. If falls originate fromthe natural slope above the cut, the cut heightplus the additional slope height (verticaldistance) is used.

(b) Ditch effectiveness—effectiveness of a ditchis measured by its ability to prevent fall-ing rock from reaching the traveled way.In estimating the ditch effectiveness, fourfactors are to be considered: (1) slope heightand angle; (2) ditch width, depth and shape;(3) anticipated block size and quantity ofrock fall; and (4) effect on rock fall tra-jectories of slope irregularities (launchingfeatures). A launching feature can negate thebenefits expected from a fallout area. Theditch effectiveness can be assessed both bycomparing the dimensions with those recom-mended in the ditch design chart shownin Figure 12.21 (Ritchie, 1963), and frominformation provided by maintenance per-sonnel.

(c) Average Vehicle Risk (AVR)—AVR repres-ents the percentage of time a vehicle willbe present in the rock fall section. Thepercentage is obtained from the average dailytraffic count (vehicles per day, ADT), thelength of the rock fall hazard area (L, km)and the posted speed limit (S, km/h) by thefollowing relationship:

AVR (%) = (ADT/(24 h/d))L

S× 100%

(12.1)

For example, if the length of the slope is0.2 km in an area where the posted speedis 90 kph and the average daily traffic countis 7000 vehicles per day, the average vehiclerisk is 65% and the corresponding hazardscore is 18. A rating of 100% means that onaverage at least one vehicle can be expectedto be under the slope at all times, and thehazard score is 81.

(d) Percentage of decision sight distance(DSD)—The DSD is used to determine the

Table 12.4 Decision site distance to avoidobstacles

Posted speed limit,kph (mph)

Decision site distance,m (ft)

48 (30) 137 (450)64 (40) 183 (600)80 (50) 229 (750)97 (60) 305 (1000)

113 (70) 335 (1100)

length of roadway (in m) needed to makea complex or instantaneous decision. TheDSD is critical when obstacles on the roadare difficult to perceive, or when unex-pected or unusual maneuvers are required.Sight distance is the shortest distance alonga roadway that an object is continuouslyvisible to the driver. Throughout a rockfall section, the sight distance can changeappreciably. Horizontal and vertical high-way curves, together with obstructions suchas rock outcrops and roadside vegetationcan severely limit the available sight dis-tance. The relationship between DSD andthe posted speed limit used in the inventorysystem has been modified from AASHTO’sPolicy on Geometric Design of Highwaysand Streets (1984) as shown in Table 12.4.

The actual site distance is related to the DSDby equation (12.2) as follows:

DSD (%) = Actual site distanceDesision site distance

× 100%

(12.2)

For example, if the actual site distance is restrictedby the road curvature to 120 m in a zone witha posted speed limit of 80 kph, then the DSD is52%. Based on the charts provided in the RHRSmanual, the score for this condition is 42.

(e) Roadway width—It is the dimension thatrepresents the available maneuvering roomto avoid a rock fall, measured perpendicular


to the highway centerline from edge of pave-ment to edge of pavement. The minimumwidth is measured when the roadway widthis not constant.

(f) Geologic character—Geologic conditionsthat cause rock fall generally fit into twocases. Case 1 is for slopes where joints, bed-ding planes or other discontinuities are thedominant structural features of a rock slope.Case 2 is for slopes where differential erosionforming overhangs or over-steepened slopesis the dominant condition that controls rockfall. The case which best fits the slope shouldbe used when doing the evaluation. If bothsituations are present, both are scored butonly the worst case (highest score) is used inthe rating.

Geologic character—Case 1

Structural condition—Adverse discontinuities arethose with orientations that promote plane,wedge or toppling failures.

Rock friction—Friction on a discontinuity isgoverned by the characteristics of the rockmaterial and any infilling, as well as the surfaceroughness (see Chapter 4, Section 4.2.4).

Geologic character—Case 2

Structural condition—Differential erosion orover-steepening is the dominant condition thatleads to rock fall. Erosion features includeover-steepened slopes, unsupported rock units orexposed resistant rocks.

Difference in erosion rates—Different rates oferosion within a slope directly relate to the poten-tial for a future rock fall event. The score shouldreflect how quickly erosion is occurring; the sizeof rocks, blocks or units being exposed; thefrequency of falls; and the amount of materialreleased during a rock fall.

(g) Block size or volume of rock fall per event—Type of event most likely to occur, relatedto the spacings and continuous lengths ofthe discontinuity sets. The score should also

take into account any tendency of the blocksof rock to break up as they fall down theslope.

(h) Climate and presence of water on slope—Water and freeze–thaw cycles contributeto both the weathering and movement ofrock materials. If water is known to flowcontinually or intermittently on the slope,it is rated accordingly. This rating couldbe based on the relative precipitation overthe region in which the ratings are beingmade, and incorporate the influence offreeze–thaw cycles.

(i) Rock fall history—Historical information isan important check on the potential forfuture rock falls. Development of a databaseof rock falls allows more accurate conclu-sions to be made of the rock fall potential.

12.3.4 Database analysis of slope inventory

It is common practice to enter the results of theslope inventory into a computer database. Thedatabase can be used both to analyze the datacontained in the inventory and to facilitate updat-ing the inventory with new information on rockfalls and construction work. The following aresome examples of database analysis:

• Rank slopes in order of increasing point scoreto identify the most hazardous sites.

• Correlate rock fall frequency with such factorsas weather conditions, rock type, and slopelocation.

• Assess severity of rock falls from analysis ofdelay hours or road closures caused by falls.

• Assess effectiveness of stabilization work fromannual number of rock falls.

12.3.5 Selection of high priority sites

It has been found that, when managing on-goingstabilization programs with durations of possiblyseveral decades and involving many hundreds ofrock slopes, there is a need to put in place arational process for selecting high priority sites.


This becomes necessary because slope stabilityconditions deteriorate over time, and it is not pos-sible to re-rate every slope, every year, accordingto the scoring system shown in Table 12.3. How-ever, it is possible to inspect all the higher priorityslopes on an annual basis to assess stability, andfrom this assessment determine whether stabiliza-tion is required and within what time frame. Thisinvolves assigning each slope an “Inspection Rat-ing” and a corresponding “Required Action” (seeTable 12.5). The filled boxes in the table indic-ate allowable Actions for each of the Ratings.For example, for an Urgent slope the permissibleactions are to limit service and work at the sitewithin one month, or to carry out a follow-upinspection to assess stability conditions in moredetail. However, for either an Urgent or Priorityslope it is not permissible to assign “No Action”to the site.

The following are examples of criteria thatcould be used to assign Inspection Ratings, usinga combination of measurement, the rating scoresgiven in Table 12.5, and subjective observationsof stability:

• Urgent—Obvious recent movement or rockfalls, kinematically feasible block withdimensions large enough to be a hazard;weather conditions are detrimental to stabil-ity. Failure possible within next few months.

• Priority—Likely movement since last inspec-tion of block, large enough to be a hazard;failure possible within next two years approx-imately.

• Observe—Possible recent movement, but noimminent instability. Check specific stabilityconditions in next inspection.

• OK—no evidence of slope movement.

It has been found that there are two primary bene-fits in assigning a Required Action for every slope.First, this forces the inspector to make a decisionon the urgency for mitigation, and second, it auto-matically draws up a list of work sites for thecurrent year and the next two years. This listbecomes the basic planning tool for the on-goingprogram.

12.3.6 Selection of stabilization measures

This section provides some guidelines on select-ing the method, or methods, of stabilizationthat are most appropriate for the topograph-ical, geological and operational conditions at thesite. Methods of slope stabilization fall into threecategories:

(a) Reinforcement;(b) Rock removal; and(c) Protection.

Figure 12.3 includes 16 of the more common sta-bilization measures divided into these categories.The following are examples of the factors thatwill influence the selection of appropriate stabil-ization methods. Where the slope is steep andthe toe is close to the highway or railway, therewill be no space to excavate a catch ditch or

Table 12.5 Inspection ratings and corresponding actions

InspectionRating

Required Actions

Limitservice;work within1 month

Work incurrentyear

Follow-upinspection

Work in1–2years

No action

Urgent × ×Priority × × ×Observe ×Okay ×


Rock cutstabilization and

protection

Reinforcement

• Rock bolting• Dowels• Tied-back walls• Shotcrete• Buttresses• Drainage• Shot-in-place buttress

• Resloping• Trimming• Scaling

• Ditches• Mesh• Catch fences• Warning fences• Rock sheds• Tunnels

Rock removal

Stabilizationmeasures

Protectionmeasures

Figure 12.3 Categories or rock slopestabilization measures.

construct a barrier. Therefore, alternative stabil-ization measures may be to remove loose rock,secure it in place with bolts, or to drape meshon the slope. It is generally preferable to removeloose rock and eliminate the hazard, but only ifthis will form a stable face and not undermineother potentially loose rock on the face. If thesource of the rock falls is a zone of boulders inan erodible soil matrix that cannot be stabilizedby bolting of effectively scaled, then a combin-ation ditch–containment structure may be moresuitable. If there is limited space at the toe of theslope for this work, there may be no alternativebut to relocate or realign the facility.

When selecting and designing stabilizationmeasures that are appropriate for a site, geotech-nical, construction and environmental issues mustbe considered. The geotechnical issues—geology,rock strength, ground water, and stabilityanalysis—are discussed in previous chapters.Construction and environmental issues, whichcan affect the costs and schedule of the work,must be addressed during the design phase ofthe project. Issues that are frequently important

are equipment access, available work time duringtraffic closures, and disposal of waste rockand soil.

Another factor to consider in the selectionof stabilization measures is the optimum levelof work. For example, a minor scaling projectwill remove the loosest rock on the slope face,but, if the rock is susceptible to weathering, thiswork may have to be repeated every three tofive years. Alternatively, a more comprehensiveprogram can be carried out using shotcrete andbolting, in addition to scaling. Although the ini-tial costs of this second program would be higher,it would be effective for a longer period, per-haps for 20–30 years. Alternative stabilizationprograms such as these, including the alternativeof doing no work, can be compared using decisionanalysis. Decision analysis is a systematic proced-ure for evaluating alternative courses of actiontaking into account the likely range of construc-tion costs and design life of the stabilization work,as well as the probability and costs of rock fallsoccurring and causing accidents (Wyllie, 1980;Roberds, 1991; Roberds et al., 2002).


The following is a brief discussion of someconstruction issues that may have a significantinfluence on stabilization work.

Blasting. Damage to rock faces by excessivelyheavy blasting is a frequent cause of instability inthe years following excavation of a slope. Meth-ods of controlled blasting, such as pre-shearingand trim blasting as described in Chapter 11,can be used to excavate a slope to a specifiedline with minimal damage to the rock behindthe face.

Topography. If there is a steep slope abovethe crest of a cut, then stabilization work thatinvolves laying back the cut will have the effectof increasing the height of the cut. This increasein the cut height will require a larger catch ditch,and may result in additional stability problems,especially if there is a substantial layer of soil orweathered rock at the surface.

Construction access. Determine the type ofequipment that is likely to be required to carryout the work, and how this equipment will beused at the site. For example, if it is planned toexcavate a substantial volume of rock in order tolay back a slope, then it is likely that airtrac drillsand excavators will have to work on the slope.In steep terrain, it may be found that the con-struction of an access road for this equipment iscostly and causes additional instability. Further-more, a cut width of at least 5 m is required toprovide sufficient working width for this equip-ment. Also, if stabilization work is planned usinglarge diameter rock bolts, then it is essential thatsuitable drilling equipment can access the site. Forexample, on steep faces, holes with a diameterlarger than about 100 mm will have to be drilledwith heavy equipment supported from a crane.Where it is not possible to use such heavy drillingequipment, it would be necessary to drill smal-ler diameter holes with hand-held equipment, anduse a larger number of smaller size bolts.

Construction costs. Cost estimates for stabil-ization work must take into account both thecosts of the work on the slope, and indirectcostssuch as mobilization, traffic control, wastedisposal, and environmental studies as discussed

next. A significant cost issue on active transporta-tion routes is the use of cranes to access the slope.If the work is done from a platform suspendedfrom a crane located on the road, this may blocktwo to three lanes of traffic. In contrast, trafficclosures can be minimized by having the construc-tion crews work off ropes secured behind the crestof the slope.

Waste disposal. The least expensive methodof disposing of waste rock produced by excava-tion and scaling operations in mountainous ter-rain is dumping rock down the slope below thesite. However, disposing of waste rock in thismanner has a number of drawbacks. First, asteeply sloping pile of loose rock may be a visualscar on the hillside which can be difficult to veget-ate. Second, the waste rock may become unstableif not adequately drained or keyed into the exist-ing slope; if it fails, the material may move aconsiderable distance and endanger facilities loc-ated down-slope. Third, where the site is locatedin a river valley, the dumped rock may fall in theriver and have a deleterious effect on fish pop-ulations. In order to minimize these impacts, itis sometimes required that the excavated rock behauled to designated, stable waste sites.

Another problem that may need to beaddressed in the disposal of waste rock is acid-water drainage. In areas of North Carolina andTennessee, for example, some argillite and schistformations contain iron-disulfides; percolation ofwater through fills constructed with this rock pro-duces low pH, acidic runoff. One method that hasbeen used to control this condition is to mix therock with lime to neutralize the acid potential andthen to place the blended material in the center ofthe fill (Byerly and Middleton, 1981). Sometimesit is necessary to encase the rock–lime mixture inan impervious plastic membrane.

Aesthetics. A series of steep, high rock cutsabove a highway may have a significant visualimpact when viewed both by the road user andthe local population. In scenic areas, it may bedesirable to incorporate appropriate landscapingmeasures in the design of the rock cuts in order tominimize their visual impact (Norrish and Lowell,1988). Examples of aesthetic treatments of rock


faces include designing blasts to produce an irreg-ular face with no traces of the blast holes, torecess the heads of rock bolts, and to color andsculpt shotcrete so that it has the appearanceof rock.

Dust, noise, ground vibration. Many rockstabilization operations can produce considerablenoise and dust, and blasting has the additionalrisk of ground vibrations (see Section 11.5.1).Prior to letting the contract, consideration shouldbe given to the acceptable ground vibration levelsfor the site conditions, so that the necessary stepscan be taken to limit their effects.

Biological and botanical effects. On someprojects steps may have to be taken to limitdisturbance to wildlife and vegetation. Typicalprecautions that can be taken are to schedulework outside of specified “windows” of animal

activity, and to relocate protected plants outsidethe work area.

12.4 Stabilization by rock reinforcement

Figure 12.4 shows a number of reinforcementtechniques that may be implemented to securepotentially loose rock on the face of a rock cut.The common feature of all these techniques is thatthey minimize relaxation and loosening of therock mass that may take place as a result of excav-ation. Once relaxation has been allowed to takeplace, there is a loss of interlock between theblocks of rock and a significant decrease in theshear strength. Figure 4.13 illustrates the effectof installing rock bolts to maintain the interlockon high roughness angle, second-order asperit-ies. Once relaxation has taken place, it is not

1

2

3

4

5

6

1 Reinforced concrete shear key to prevent loosening of slab at crest.

4 Shotcrete to prevent raveling of zone of fractured rock.

6 Concrete buttress to support rock above cavity.

5 Drain hole, oriented to intersect water-bearing joints, to reduce water pressure within slope.

3 Tied-back wall to prevent sliding on fault zone.

2 Tensioned rock anchors to secure sliding blocks along crest(Ib—bond length; If—unbonded length).

Figure 12.4 Rock slope reinforcement methods (TRB, 1996).


possible to reverse the process. For this reason,reinforcement of rock slopes is most effective if itis installed prior to excavation—a process knownas pre-reinforcement.

12.4.1 Shear keys

Reinforced shear keys provide support for blocksof rock up to about a meter thick, as well as zonesof loose and weathered rock at the crest of theslope (Figure 12.4, Item 1). Shear keys are usedwhere the support required is limited by the size ofthe blocks, and to prevent raveling and looseningof closely fractured, weak rock. If rock bolts wereto be installed in this rock, the raveling wouldsoon expose the head of the bolt resulting in lossof support.

Shear keys comprise lengths of reinforcing steelabout 25–32 mm diameter and about 1000 mmlong fully grouted into holes about 500–750 mmdeep drilled into stable rock. The holes are locatedclose to the toe of the rock to be supported, andare spaced about 500–1000 mm apart, dependingon the support required. Lengths of reinforcingbars, about 6–8 mm diameter, are then placedhorizontally and secured to the vertical bars.Finally, the reinforcing steel is fully encapsu-lated in shotcrete, or concrete poured in intimatecontact with the rock.

The support provided by the shear key is equalto the shear strength of the vertical steel bars, andpossibly the cohesion of the rock-concrete sur-face. The shear key acts as a resisting force inthe limit equilibrium equations (see Section 6.3),and if the magnitude of this shear force isRk, then the factor of safety for a block withweight W is

FS = W cos ψp tan φ + Rk

W sin ψp(12.3)

where ψp is the dip of the base of the block and φ

is the friction angle on the base of the rock block(assuming a dry slope). The factor of safety calcu-lated by equation (12.3) could be for a unit lengthof the slope, or a specified length, depending onhow forces W and Rk are defined.

Shear keys on a much larger scale have beenused for the stabilization of dam foundations andabutments (Moore and Imrie, 1982). A tunnelwas driven along a distinctly defined shear zone,with the excavation extending into sound rock oneither side of the tunnel. The tunnel was then filledwith concrete to create a high strength inclusionalong the sliding plane.

12.4.2 Rock anchors

Typical applications of rock anchors, as shownin Figure 12.4, items 2 and 3, are to prevent slid-ing of blocks or wedges of rock on discontinuitiesdipping out of the face. It is important to note thatthe primary function of rock anchors is to modifythe normal and shear forces acting on the slidingplanes, rather than to rely on the shear strength ofsteel where the anchor crosses this plane. In thischapter, the term “rock anchor” refers to bothrigid bars and flexible cables that can be used inbundles; the design principles and constructionmethods are similar for both materials.

Rock anchors may be fully grouted and unten-sioned, or anchored at the distal end andtensioned. The different applications of unten-sioned, pre-reinforcement bolts and tensionedanchors are shown in Figure 12.5 (see alsoFigure 4.13). Pre-reinforcement of an excavationmay be achieved by installing fully grouted butuntensioned bolts (dowels) at the crest of the cutprior to excavation. The fully bonded dowels pre-vent loss of interlock of the rock mass becausethe grouted bolts are sufficiently stiff to pre-vent movement on the discontinuities (Moore andImrie, 1982; Spang and Egger, 1990). However,where blocks have moved and relaxed, it is gener-ally necessary to install tensioned anchors to pre-vent further displacement and loss of interlock.The advantages of untensioned bolts are theirlower cost and quicker installation compared totensioned anchors.

Tensioned rock anchors are installed acrosspotential slide surfaces and bonded in sound rockbeyond the surface. The application of a tensileforce in the anchor, which is transmitted into the


(b) Pre-reinforcement of cut face with fully grouted untensioned dowels

(a) Stabilization of displaced block with tensioned rock bolts

Figure 12.5 Reinforcement of a rockslope: (a) tensioned rock bolts in adisplaced block; (b) fully grouted,untensioned dowels installed prior toexcavation to pre-reinforce the rock(TRB, 1996).

rock by a reaction plate at the rock surface, pro-duces compression in the rock mass, and modifiesthe normal and shear stresses on the slide sur-face. Chapters 6, 7 and 9 each contain designinformation on the procedures for calculatingboth the anchor force and anchor orientationrequired to produce a specified factor of safety;as discussed in Section 6.4, there is an optimumanchor orientation that minimizes the requiredforce.

Once the anchor force and hole orientationrequirements have been determined, the follow-ing nine steps are involved in an anchor installa-tion (Littlejohn and Bruce, 1977; FHWA, 1982;BSI, 1989; Xanthakos, 1991; PTI, 1996; Wyllie,1999).

Step 1: Drilling—Determine drill hole dia-meter and length that can be drilled at the sitebased on available equipment and the accessfor this equipment.Step 2: Bolt materials and dimensions—Select anchor materials and dimensions thatare compatible with hole diameter and therequired anchoring force.Step 3: Corrosion—Assess the corrosivity ofthe site, and apply an appropriate level ofcorrosion protection to the anchors.Step 4: Bond type—Select either cement orresin grout or a mechanical anchor to secure

the distal end of the anchor in the hole.Factors influencing this decision include thehole diameter, tensile load, anchor length,rock strength and speed of installation.Step 5: Bond length—Based on the bond-ing type, hole diameter, anchor tension androck strength, calculate the required bondlength.Step 6: Total anchor length—Calculate thetotal anchor length, which is the total of thebond length and free stressing length. The freestressing length should extend from the rocksurface to the top of the bond zone, with thetop of the bond zone being below the potentialsliding plane.Step 7: Anchor pattern—Layout anchor pat-tern so that they are approximately evenlyspaced on the face and produce the requiredoverall anchor force.Step 8: Waterproofing drill holes—Check thatthere are no discontinuities in the bond zoneinto which grout could leak, and seal the holeif necessary by grouting and redrilling.Step 9: Testing—Set up a testing procedurethat will verify that the bonded length cansustain the design load, and that the fulllength of the free stressing length is beingtensioned.

Each of these nine steps is discussed in more detailas follows.


Step 1: Drilling

The diameter of the drill hole is partially determ-ined by the available drilling equipment, but mustalso meet certain design requirements. The holediameter should be large enough to allow theanchor to be inserted in the hole without driv-ing or hammering, and allow full embedmentin a continuous column of grout. A hole dia-meter significantly larger than the anchor will notmaterially improve the design and will result inunnecessary drilling costs and possibly excessivegrout shrinkage. As a guideline, the diameter ofthe drill hole should be about 1.5–2 times the dia-meter of the full anchor assembly, with corrosionprotection, if any.

Percussion drilling. Holes for rock anchorsare usually drilled with percussion equipment thatutilizes a combination of impact and rotation ofa tungsten carbide drill bit to crush the rock andadvance the hole. The cuttings are removed bycompressed air that is pumped down a hole in thecenter of the rods and is exhausted up the annulusbetween the rods and the wall of the hole. Per-cussion drills are either pneumatic or hydraulicpowered, and the hammer is either at the surfaceor down the hole (DTH drill). The advantagesof percussion drills are their high penetrationrates, good availability and the slightly roughwall that is produced. Precautions that must betaken include minimizing hole deviation by con-trolling the down pressure on the rods, and avoidloosing the drill string in zones of weathered andbroken rock.

Where holes are to be drilled through an upperlayer of soil, or intermediate zones of weatheredrock in which collapse of the hole is possible,equipment is available that installs casing as thehole is advanced. Equipment manufactured byTubex1 uses a bit that expands, when torqueis applied, to ream out the hole to a diameterslightly larger than the casing (Figure 12.6). At

1 Manufacturer’s name are given as example only, and arenot intended as endorsements of their products.

1

2

3

4

5

Figure 12.6 Tubex drill bit for advancing casingthrough soil and weathered rock (courtesy: SandvikDrilling) 1, Shoulder; 2, Bit tube; 3, Guide;4, Reamer; 5, Pilot bit.

the completion of drilling, the drill rods andcontracted bit can be withdrawn inside the cas-ing. The maximum hole diameter for Tubex drillsis 356 mm. Drills manufactured by Klemm andBarber advance casing during drilling by apply-ing thrust and torque to the casing, which isindependent of the thrust and torque on thedrill rods.

For holes drilled with hand-held percussiondrills, the limits for efficient operation are amaximum hole diameter of about 60 mm and amaximum length is about 6 m. For track-mountedpercussion drills the hole diameters rangebetween about 75 and 150 mm, and the max-imum hole length for top hammer drills is about60 m, with the main limitation being excessivehole deviation. For DTH drills the maximum holelength is several hundred meters. For hole diamet-ers larger than about 150 mm in hard rock, thereis a substantial increase in the size of the drillingequipment, and this equipment is usually used invertical holes rather than near-horizontal hole.

Rotary drilling. For drilling in weak rocksuch as chalk and some shales it is possible to userotary drilling methods that include augers, dragbits and tri-cone bits; drill hole diameters range


from 150 to 600 mm. These methods generallyrequire that the hole be self-supporting, althoughwhen using hollow-stem augers, bentonite orcement grout can be circulated in the hole tostabilize the walls.

Step 2: Anchor materials and dimensions

Anchors are available as either deformed steelbars, or 7-wire strand cables. Figures 12.7 and12.8 show typical features of these two types ofanchor, both of which incorporate an optionalcorrosion protection system.

Bar anchors. Most bars are manufacturedwith a continuous, coarse thread that is resist-ant to damage, and allows the bars to be cut toany length to suit site conditions. Common steelgrades used for thread bars are 517/690 MPa and835/1030 MPa: yield stress/ultimate stress, with amodulus of elasticity of 204.5 GPa. Bar diametersrange from 19 mm to 57 mm and lengths of barcan be joined by threaded couplers. It is usual thatonly a single bar is installed in each hole, ratherthan groups of bars. If a tension force is appliedto the bar, the head of the bar is secured with areaction plate, washer and threaded nut.

Another type of bar anchor is a self-drillingproduct that comprises a hollow core drill steelwith a continuous coil thread, and a dispos-able bit. The anchor is used most commonlywhen drilling through broken rock and soil seamswhere the hole tends to collapse as soon as thedrill steel is removed. When using this type ofanchor, the drill steel is left in the hole whendrilling is complete, following which cementgrout is pumped down the center hole to fillthe annual space and encapsulate the anchor.It is also possible to use an adaptor with thedrill that allows grout to be circulated down thehole while drilling; this approach may be usedwhere compressed air circulation is not effectivein removing the cuttings. Where the applicationof the anchor is reinforcement of fractured rockmasses, rather than anchoring of defined blocks,then it is appropriate to install fully grouted,untensioned anchors. However, it is possible to

Grout tube

Threadbar

Smooth PE sheathing

Coupling with doublecorrosion protection(if required)

Corrugated PVCsheathing (over fulllength of anchor)

Centeringsleeve1200 mm o/c

Cement grout

Grout tube

Drill hole

Ove

rdril

lB

ond

leng

th (

I b)

Unb

onde

d le

ngth

(I f)

Figure 12.7 Typical threadbar rock anchor withdouble corrosion protection system comprisinggrouted corrugated plastic sleeve over full length ofanchor, and smooth sheath on unbonded length(Class I corrosion protection) (courtesy: DSI AnchorSystems).


A

B

A

BUnbonded length

Bonded length

Bearing plate Trumpet

Seal

Anchored cap

Corrosion inhibitoror grout filled

Corrosion inhibitoror grout filled

Detail 1Anchorage

Encapsulation

External centralizer

End cap

Internal spacer/centralizer

Detail 1

Longitudinal sectionComplete anchor

Corrugated sheathcentralizer

Grout tube(optional)

Grout

Extrusion coatedstrand

Bare strand

Internal spacer/centralizer

Corrugatedsheathing

Section A-AUnbonded length

Section B-BBonded length

Figure 12.8 Typical multi-strand cable anchor with corrosion protection system comprising grouted corrugatedplastic sleeve on bond length, and smooth greased sheath on unbonded length (courtesy: Lang Tendons Inc.).

install tensioned anchors by driving a smoothcasing over the grouted bar to the depth of thetop of the bond zone. The grout above the bondzone is washed out as the casing is driven to cre-ate an unbonded length. Some trade names ofself-drilling anchors are MAI and IBO, and they

are available in diameters ranging from 25 mmto 51 mm, with ultimate strengths of 200 kN to800 kN respectively.

Strand anchors. Wire strand is manufacturedby twisting together seven, 5 mm diameter steelwires to form a strand with a diameter of 12 mm.


Each strand has an ultimate tensile strength of260 kN, and anchors with higher capacities canbe produced by assembling individual strandsinto bundles; bundles as large as 94 strands havebeen used in improving the seismic stability ofconcrete dams. For slope stabilization requiringshallow dip holes, the largest bundle may beabout 12 strands. The strands are flexible whichfacilitates handling in the field, but they cannotbe coupled. When a tension is applied to strand,the end exposed at the surface is secured with apair of tapered wedges that grip the strand andfit tightly into tapered holes in the reaction plate(Figure 12.8, detail 1).

Step 3: Corrosion protection

Corrosion protection for steel bar and strandanchors should be considered for all projects,including temporary installations if the site con-ditions are corrosive (King, 1977; Baxter, 1997).Even if anchors are not subject to corrosion atthe time of installation, conditions may change inthe future that must be accounted for in design.The following list describes conditions that willusually create a corrosive environment for steelanchors (Hanna, 1982; PTI, 1996):

• soils and rocks that contain chlorides;• seasonal changes in the ground water table;• marine environments where they are exposed

to sea water that contains chlorides andsulfates;

• fully saturated clays with high sulfate content;• anchorages that pass through different ground

types which possess different chemical charac-teristics;

• stray direct electrical current that developsgalvanic action between the steel and thesurrounding rock;

• peat bogs; and• cinder, ash or slag fills; organic fills containing

humic acid; acid mine or industrial waste.

Corrosion potential is also related to the soil res-istivity by the magnitude of the current that canflow between the soil and the steel. In general,the corrosion potential decreases with increasing

Table 12.6 Parameter limits for corrosiveness ofground water and soil (PTI, 1996; TRB, 2002)

Non- Aggressiveaggressive

Properties of groundwaterpH 6.5–5.5 <5.5Lime-dissolving 15–30 >30(CO2), mg/l

Ammonium (NH+4 ), mg/l 15–30 >30

Magnesium (Mg2+), mg/l 100–300 >300Sulphate (SO2−

4 ), mg/l 200–600 >600

Properties of SoilResistivity (�), ohm/cm 2000–5000 <2000pH 5–10 <5

resistivity of the soil, as follows:

organic soil > clay > silt > sand > gravel

Table 12.6 lists the properties of ground waterand soil with respect to the site conditions beingnon-aggressive or aggressive for corrosion of steelanchors.

Where aggressive conditions exist, a corrosionprotection system is usually used, which shouldmeet the following requirements for long-termreliability:

• There will be no break down, cracking or dis-solution of the protection system during theservice life of the anchor.

• The fabrication of the protection system canbe carried out either in the plant or on site insuch a manner that the quality of the systemcan be verified.

• The installation and stressing of the anchorcan be carried out without damage to theprotection system.

• The materials used in the protection systemare inert with respect to both the steel anchorand the surrounding environment.

The Post Tensioning Institute (PTI, 1996) cat-egorizes corrosion protection systems as Class Iand Class II. Class I protection is used for per-manent anchors in aggressive environments, or


in non-aggressive environments where the con-sequences of failure are significant. Both the bondand unbonded lengths of the tendon or bar areprotected with either cement grout-filled encap-sulation or an epoxy coating; the head of theanchor is also protected. Class II protection isused for temporary anchors in non-aggressiveenvironments; protection is limited to grout onthe bond length, a sheath on the unbondedlength, and protection of the head if exposed.Figures 12.7 and 12.8 show typical Class I cor-rosion protection systems for bar and strandanchors respectively.

Based on these categories of corrosion, adecision tree has been developed to assess the vul-nerability of rock anchors to corrosion and lossof anchorage capacity (Figure 12.9) (TRB, 2002).High strength steel (ultimate tensile strengthσult > 1000 MPa) is vulnerable to attack fromhydrogen embrittlement and corrosion stresscracking. Generally, high strength steel is usedto manufacture wire strand elements (σult ≈1700–1900 MPa), and strand anchors are morevulnerable than bar anchors because of the largersurface area of steel.

It is also possible to estimate the service life ofanchors based on the rate of corrosion by calcu-lating the loss of element thickness over time. Theservice life t in years is given by

ln(t) = ln(X) − ln(K)

n(12.4)

where X is the loss of thickness or radius (µm),and K and n are constants (Table 12.7). Theloss of thickness X is computed from the originalradius ro , and the critical radius rcrit that is theradius at which the yield stress is reached, atconstant load, due to the loss of cross-sectioncompared to the original cross-section Ao, i.e.

rcrit =√

0.6Ao

π(12.5)

and

X = (ro − rcrit) (12.6)

Approximate values for K are 35 for normal con-ditions, 50 for aggressive conditions, and 340 forvery aggressive conditions, as defined in Table12.6. Equation (12.5) assumes that the workingstress equals 0.6 times the yield stress.

Methods of protecting steel against corrosioninclude galvanizing, applying an epoxy coating,or encapsulating the steel in cement. Cement iscommonly used for corrosion protection, primar-ily because it creates a high pH environment thatprotects the steel by forming a surface layer ofhydrous ferrous oxide. In addition, cement groutis inexpensive, simple to install, has sufficientstrength for most applications, and a long servicelife. Because of the brittle nature of grout and itstendency to crack, particularly when loaded intension or bending, it is usual that the protectionsystem comprise a combination of grout and aplastic (high density polyethylene, HDPE) sleeve.In this way, the grout produces the high pH envir-onment around the steel, while the plastic sleeveprovides protection against cracking. In orderto minimize the formation of shrinkage cracksthat reduce corrosion resistance of the grout, itis usual to use non-shrink grouts for all compon-ents of the installation. Figures 12.7 and 12.8show examples of three-layer corrosion protec-tion systems in which the steel is encapsulated ina grout-filled HDPE sheath, and the outer annu-lus, between the sheath and the rock, formed bythe centering sleeves, is filled with a second groutlayer.

For anchors with unbonded lengths, it is partic-ularly important that the head be protected fromboth corrosion and damage. This is because lossof the nut or wedges, or fracturing of the rockunder the reaction plate, will result in loss of ten-sion in the anchor even if the remainder of theanchor is entirely intact.

Step 4: Bond type

Tensioned anchors comprise two portions—abond length and an unbonded length (Figures 12.7and 12.8). In the bond length, the bar or strandis bonded by one of a variety of means to the sur-rounding rock. In the unbonded length, the bar or


Element type

Installation date

Olderinstallation

PTI corrosion protection level

II NoneI

Anchorage details

Corrosionprotection

Mechanicalanchorage

Highvulnerablility

Highstrength steel

Fpu> 1000 MPa

Lowvulnerablility

Moderatevulnerablility

No

No

No

No

No

Yes

Yes Yes

Yes

Yes

Highstrength steel

Fpu> 1000 MPa

Figure 12.9DecisionTree forassessingvulnerabilityof elementsto corrosionand loss ofanchoragecapacity.

Table 12.7 Values of constants K and n for corrosion rate calculations

Parameter Normal� = 2000–5000;a pH = 5–10

Aggressive� = 700–2000; pH = 5–10

Very aggressive� = <700; pH = <5

K(µm) 35 50 340n 1.0 1.0 1.0

Notea � is soil resistivity (ohm/cm).


strand is unbonded and is free to strain as tensionis applied. Figure 12.4 shows that the bond zoneis located in stable rock below the potential slid-ing plane so that when the anchor is tensioned,stresses are applied to this plane to increase thefactor of safety (see Section 6.4).

Methods of securing the distal end of ananchor in the drill hole include resin, mechan-ical, and cement grout anchors. The selectionof the appropriate anchor will depend on suchfactors as the required capacity of the anchor,speed of installation, strength of the rock inthe anchor zone, access to the site for drillingand tensioning equipment, and the level of cor-rosion protection required. The following isa brief discussion of each of these anchoragemethods.

Resin anchors. These comprise a plastic cart-ridge about 25 mm in diameter and 200 mm longthat contains a liquid resin and a hardener thatset when mixed together (Figure 12.10). Settingtimes vary from about 1 minute to as much as90 minutes, depending on the reagents used. Thesetting time is also dependent on the temperat-ure, with fast-setting resin hardening in about4 minutes at a temperature of −5◦C, and in about25 s at 35◦C.

Figure 12.10 Resin cartridges for anchoring rockbolts (TRB, 1996).

The installation method consists of inserting asufficient number of cartridges into the drill holeto fill the annular space around the bar. It isimportant that the hole diameter, in relation tothe bar size, be within specified tolerances so thatcomplete mixing of the resin is achieved whenthe bar is spun. This usually precludes the useof coupled anchors because the hole diameter toaccommodate the coupling will be too large forcomplete resin mixing. The bar is spun as it isdriven through the cartridges to mix the resin andform a rigid solid anchorage. The required speedof rotation is about 60 revolutions per minute,and spinning is continued for about 30 s after thebar has reached the end of the hole. It is prefer-able that threaded bar be rotated in the directionthat augers the resin into the hole, particularly inup-holes.

The maximum bolt length is limited to about12 m because most drills cannot rotate longer barsat sufficient speed to mix the resin. It is pos-sible to install a tensioned, resin-grouted bolt byusing a fast-setting (about 2 minutes) resin for theanchor, and a slower setting (30 minutes) resinfor the remainder of the bar. The bolt is ten-sioned between the times that the fast and slowresins set.

The primary advantage of resin anchorage isthe simplicity and speed of installation, with sup-port of the slope being provided within minutes ofspinning the bolt. The disadvantages are the lim-ited length and tension capacity (∼ 400 kN) ofthe bolt, and the fact that only rigid bars can beused. Furthermore, the resin is not as effective ascement grout for corrosion protection of the steel.Unlike cement grout, resin does not provide thehigh pH protective layer against corrosion, and itcannot be verified that the cartridges completelyencapsulate the steel.

Mechanical anchors. These comprise a pair ofsteel platens that are pressed against the walls ofthe drill hole. The anchor is expanded by drivingor torquing a steel wedge between the platens.

The advantage of mechanical anchors is thatinstallation is rapid, although not as rapid as resinanchors, and tensioning can be carried out assoon as the anchor has been set. Grouting can


then be carried out using a grout tube attachedto the bar, or through the center hole in thecase of the bolt manufactured by the WilliamsForm Hardware and Rockbolt Co. The disad-vantages of mechanical anchors are that theycan be used only in medium to strong rock inwhich the anchor will grip, and the maximumworking tensile load is about 200 kN. Mech-anical anchors for permanent installations mustalways be fully grouted because the wedge willcreep and corrode in time, resulting in loss ofsupport.

Cement grout. It is the most common methodof anchoring long-service-life rock anchorsbecause the materials are inexpensive, it providescorrosion protection, and installation is simple.Cement grout anchorage can be used in a widerange of rock and soil conditions, and the cementprovides corrosion protection. Figures 12.7and 12.8 show typical installations with cementanchorage and centering sleeves to ensure com-plete encapsulation of the steel. The grout mixusually comprises non-shrink, unsanded cementand water at a water:cement ratio in the rangeof 0.4–0.45. This ratio will produce a grout thatcan be pumped down a small diameter grouttube, yet produce a high strength, continuousgrout column with minimal bleed of water fromthe mix. Admixtures are sometimes added to thegrout to reduce bleeding, and increase the viscos-ity of the grout. In down holes, grout is alwaysplaced with a grout tube extending to the distalend of the hole in order to displace air and waterin the hole.

Step 5: Bond length

For cement and resin grout anchored bolts, thestress distribution along the bond length is highlynon-uniform; the highest stress is concentratedin the proximal end of the bond zone, andideally the distal end of the bond is unstressed(Farmer, 1975; Aydan, 1989). However, it isfound, as a simplification, that the required lengthof the bond zone can be calculated assuming thatthe shear stress at the rock–grout interface is uni-formly distributed along the anchor. Based on this

assumption, the allowable shear stress τa is givenby equation (12.7):

τa = T

πdh1b(12.7)

or, the design bond length lb is

lb = T

πdhτa(12.8)

where T is the design tension force and dh is thehole diameter. Values of τa can be estimated fromthe uniaxial compressive strength (σi) of the rockin the anchor zone according to the followingrelationship (Littlejohn and Bruce, 1977):

τa = σi

30(12.9)

Approximate ranges of allowable bond stress (τa)related to rock strength and rock type are presen-ted in Table 12.8. It is important to note that theallowable shear stress values listed in Table 12.8incorporate a factor of safety of about 3, so thesevalues can be used with confidence in design.

Step 6: Total anchor length

The total length of a tensioned anchor is the sumof the bond length as determined in Steps 4 and 5,and the unbonded length (see Figure 12.4). Theunbonded length extends from the proximal endof the bond zone to the head of the bolt, and thereare three components to this length, as follows.First, the distal end of the unbonded length mustbe beyond the potential sliding surface so that thetensile force applied to the anchor is transferredinto the rock at the sliding surface. If the locationof this surface is precisely known, then the distalend of the unbonded length could be 1–2 m belowthe sliding surface, while if the sliding surface is azone rather than a plane, then this distance shouldbe increased accordingly. The second componentof the length extends from the sliding surface tothe rock surface, and this length will depend onthe slope geometry. The third length component


Table 12.8 Allowable rock–grout bond stresses in cement groutanchorages (PTI, 1996; Wyllie, 1999)

Rockdescription

Compressive strength range(MPa)

Allowable bond stress(MPa)

Strong rock >100 1.05–1.40Medium rock 50–100 0.7–1.05Weak rock 20–50 0.35–0.7Rock typeGranite, basalt 0.55–1.0Dolomitic limestone 0.45–0.70Soft limestone 0.35–0.50Slates, strong shales 0.30–0.45Weak shales 0.05–0.30Sandstone 0.30–0.60Concrete 0.45–0.90

is the distance from the rock surface to the headof the anchor where the bearing plate and nutare located. For strong rock the bearing platecan bear directly on the rock (Figure 12.4, item2), whereas in conditions where the stress underthe bearing plate could crush the rock, a rein-forced concrete or shotcrete reaction pad wouldbe required (Figure 12.4, item 3).

Step 7: Anchor pattern

The layout of the anchors on the face should besuch that there is a reasonably uniform stressapplied to the sliding plane. This will requirethat the horizontal and vertical spacing be aboutequal. Also, the anchors should not be too closeto the toe where the thickness of rock above theslide plane is limited, or close to the crest wherethe anchor may pass through a tension crack. Fora plane failure where the support force T is cal-culated per unit length of slope, then the requiredvertical spacing Sv for an installation comprisingn horizontal rows is given by

Sv = B · n

T(12.10)

where B is the design tension force in each bolt.

Step 8: Waterproofing drill hole

If the drill hole intersects open discontinuities inthe bond length into which there could be signi-ficant leakage of grout, it will be necessary to sealthese discontinuities before installing the anchor.The potential for grout leakage into the rock canbe checked by filling the hole with water andapplying an excess pressure of 35 kPa. If, afterallowing time for some saturation of the rockmass around the hole, the water leakage overa 10 minute period exceeds 9.5 l, then there ispotential that grout in the bond zone will flowinto the rock before it has time to set PTI (1996).Under these conditions, the hole is sealed with alow fluid or sanded grout, and then is re-drilledafter a setting time of about 15–24 h; if the grout isallowed to set fully, it is possible that the drill willwander from the original alignment and inter-sect the ungrouted discontinuity. A second waterinflow test is then conducted. The procedure fortesting, grouting and re-drilling is repeated untilthe hole is sealed.

Step 9: Testing

Where tensioned rock anchors are installed, aprocedure is required to check that the anchorcan sustain the full design load at the required


depth, and that there will be no loss of load withtime. A suitable testing procedure has been drawnup by the Post Tensioning Institute (1996) thatcomprises the following four types of tests:

(a) Performance test;(b) Proof test;(c) Creep test; and(d) Lift-off test.

The performance and proof tests consist of a cyc-lic testing sequence, in which the deflection of thehead of the anchor is measured as the anchor istensioned (Figure 12.11). The design load shouldnot exceed 60% of the ultimate strength of thesteel, and the maximum test load is usually 133%of the design load, which should not exceed80% of the ultimate strength of the steel. Asa guideline, performance tests are usually car-ried out on the first two to three anchors andon 2% of the remaining anchors, while prooftests are carried out on the remainder of theanchors. The testing sequences are as follows,where AL is an alignment load to take slackout of the anchor assembly and P is the designload (Figure 12.12(a)):

Performance test:

AL, 0.25P

AL, 0.25P , 0.5P

AL, 0.25P , 0.5P , 0.75P

AL, 0.25P , 0.5P , 0.75P , 1.0P

AL, 0.25P , 0.5P , 0.75P , 1.0P , 1.2P ,

AL, 0.25P , 0.5P , 0.75P , 1.0P , 1.2P ,

1.33P—hold for creep test∗

AL, P—lock off anchor, carry out lift-off test.

Proof test:

AL, 0.25P , 0.5P , 0.75P , 1.0P , 1.2P ,

1.33P—hold for creep test∗

P—lock off anchor, carry out lift-off test.

∗Creep test—elongation measurements aremade at 1, 2, 3, 4, 5, 6 and 10 minutes. Ifthe total creep exceeds 1 mm between 1 and10 minutes, the load is maintained for an addi-tional 50 minutes with elongation measurementsmade at 20, 30, 40, 50 and 60 minutes.

The usual method of tensioning rock bolts is touse a hollow-core hydraulic jack that allows theapplied load to be precisely measured, as well ascycling the load and holding it constant for thecreep test. It is important that the hydraulic jackbe calibrated before each project to ensure thatthe indicated load is accurate. The deflection of

Figure 12.11 Test set-up for atensioned multi-strand cableanchor comprising hydraulic jackwith pressure gauge to measureload, and dial gauge onindependent mount to measureanchor elongation. (Photographby W. Capaul.)


1

0 0.25

P

0.50

P

0.75

P

1.00

P

1.20

P

1.33

P

AL

2

3

4

5

610 min.

Movement

�t

�e

�r

�t

�e

�t

Load

Residualmovement

Elasticmovement

Load

Acceptancecriteria

6

6

5

5

4

4

3

3

2

2

1

1

0 AL

0.50

P

0.75

P0.25

P

1.00

P

1.20

P

1.33

P

Line B: unbonded length + 50% bond length

Line A: 80% free length

(a)

(b)

Figure 12.12 Results of performance test for tensioned anchor: (a) cyclic load/movement measurements;(b) load/elastic movement plot (PTI, 1996).

the anchor head is usually measured with a dialgauge, to an accuracy of about 0.05 mm, with thedial gauge mounted on a stable reference pointthat is independent of movement of the anchor.Figure 12.11 shows a typical test arrangement fortensioning a cable anchor comprising a hydraulicjack, and the dial gauge set up on tripod.

The purpose of the performance and creep testsis to ensure that the anchor can sustain a con-stant load greater than the design load, and that

the load in the anchor is transmitted into the rockat the location of the potential slide surface. Thecreep test is carried out by holding the maximumtest load constant for a period up to 10 minutes,and checks that there is no significant loss of loadwith time. The creep test also removes some ofthe initial creep in the anchor. The lift-off testchecks that the tension applied during the test-ing sequence has been permanently transferred tothe anchor. The Post Tensioning Institute (PTI)


0.36 mm

1–10 minutes

036.0

36.1

36.2

36.3

36.4

36.5

36.6

36.7

36.8

36.9

37.0

37.1

1 2 3 4Log time (minutes)

5 6 10

Acceptancecriterion(1 mm)

Mov

emen

t (m

m)

Figure 12.13 Results of creep test showing measuredelongation over 10 minutes test period comparedwith acceptance criteria of 1 mm elongation.

provides acceptance criteria for each of the fourtests, and it is necessary that each anchor meetsall the acceptance criteria.

The results of a performance test shown inFigure 12.12(a) are used to calculate the elasticelongation δe of the head of the anchor. Thetotal elongation of the anchor during each load-ing cycle comprises elastic elongation of the steeland residual δr (or permanent) elongation dueto minor cracking of the grout and slippage inthe bond zone. Figure 12.12(a) shows how theelastic and residual deformations are calculatedfor each load cycle. Values for δe and δr at eachtest load, together with the PTI load–elongationacceptance criteria, are then plotted on a separategraph (Figure 12.12(b)). For both performanceand proof tests, the four acceptance criteria fortensioned anchors are as follows:

First, the total elastic elongation is greaterthan 80% of the theoretical elongation of theunbonded length—this ensures that the loadapplied at the head is being transmitted to thebond length.

Second, the total elastic elongation is lessthan the theoretical elongation of the unbon-ded length plus 50% of the bond length—thisensures that load in the bond length is con-centrated in the upper part of the bond andthere is no significant shedding of load to thedistal end.Third, for the creep test, the total elonga-tion of the anchor head during the periodof 1–10 minutes is not greater than 1 mm(Figure 12.13), or if this is not met, isless than 2 mm during the period of 6–60minutes. If necessary, the duration of thecreep test can be extended until the movementis less than 2 mm for one logarithmic cycleof time.Fourth, the lift-off load is within 5% of thedesigned lock-off load—this checks that therehas been no loss of load during the operationof setting the nut or wedges, and releasing thepressure on the tensioning jack.

The working shear strength at the steel–groutinterface of a grouted deformed bar is usually


greater than the working strength at the rock–grout interface. For this reason, the requiredanchor length is typically determined from thestress level developed at the rock–grout interface.

12.4.3 Reaction wall

Figure 12.4, item 3 shows an example where thereis potential for a sliding type failure in closelyfractured rock. If tensioned rock bolts are usedto support this portion of the slope, the frac-tured rock may degrade and ravel from underthe reaction plates of the anchors, and eventu-ally the tension in the bolts will be lost. In thesecircumstances, a reinforced concrete wall can beconstructed to cover the area of fractured rock,and then the holes for the rock anchors can bedrilled through sleeves in the wall. Finally, theanchors are installed and tensioned against theface of the wall. The wall acts as both a pro-tection against raveling of the rock, and a largereaction plate for the rock anchors. Where neces-sary, reinforced shotcrete can be substituted forconcrete.

Since the purpose of the wall is to distributethe anchor loads into rock, the reinforcing forthe wall should be designed such that there is nocracking of the concrete under the concentratedloads of the anchor heads. It is also importantthat there are drain holes through the concreteto prevent build-up of water pressure behindthe wall.

12.4.4 Shotcrete

Shotcrete is a pneumatically applied, fine-aggregate mortar that is usually placed in a50–100 mm layer, and is often reinforced forimproved tensile and shear strength (AmericanConcrete Institute, 1995). Zones and beds ofclosely fractured or degradable rock may be pro-tected by applying a layer of shotcrete to therock face (Figure 12.4, item 4). The shotcretewill control both the fall of small blocks of rock,and progressive raveling that could eventuallyproduce unstable overhangs. However, shotcreteprovides little support against sliding for the

overall slope; its primary function is surface pro-tection. Another component of a shotcrete install-ation is the provision of drain holes to preventbuild-up of water pressures behind the face.

Reinforcement. For permanent applications,shotcrete should be reinforced to reduce the riskof cracking and spalling. The two common meth-ods of reinforcing are welded-wire mesh, or steelor polypropylene fibers. Welded-wire mesh is fab-ricated from light gauge (∼3.5 mm diameter) wireon 100 mm centers, and is attached to the rockface on about 1–2 m centers with steel pins, com-plete with washers and nuts, grouted into therock face. The mesh must be close to the rocksurface, and fully encased in shotcrete, takingcare that there are no voids behind the mesh. Onirregular surfaces it can be difficult to attach themesh closely to the rock. In these circumstances,the mesh can be installed between two layers ofshotcrete, with the first layer creating a smoothersurface to which the mesh can be more readilyattached.

An alternative to mesh reinforcement is to usesteel or polypropylene fibers that are a compon-ent of the shotcrete mix and form a reinforce-ment mat throughout the shotcrete layer (Morganet al., 1989, 1999). The steel fibers are manu-factured from high strength carbon steel with alength of 30–38 mm and diameter of 0.5 mm. Toresist pullout, the fibers have deformed ends orare crimped. The proportion of steel fibers in theshotcrete mix is about 60 kg/m3, while compar-able strengths are obtained for mixes containing6 kg of polypropylene fibers per cubic meter ofshotcrete. The principal function of fibers is tosignificantly increase the shear, tensile and post-crack strengths of the shotcrete compared tonon-reinforced shotcrete; shotcrete will tend tobe loaded in shear and tension when blocks offractured rock loosen behind the face.

The disadvantages of steel fibers are their tend-ency to rust at cracks in the shotcrete, and thehazard of the “pin cushion” effect where per-sons come in contact with the face; polypropylenefibers overcome both these disadvantages.


Mix design. Shotcrete mixes comprise cementand aggregate (10–2.5 mm aggregate and sand),together with admixtures (superplasticizers) toprovide high early strengths. The properties ofshotcrete are enhanced by the use of micro-silicathat is added to the mix as a partial replacementfor cement (USBM, 1984). Silica fume is an ultrafine powder with a particle size approximatelyequal to that of smoke. When added to shotcrete,silica fume reduces rebound, allows thicknesses ofup to 500 mm to be applied in a single pass, andcovers surfaces on which there is running water.There is also an increase in the long-term strengthin most cases.

Shotcrete can be applied as either a wet-mixor a dry-mix. For wet-mix shotcrete the compon-ents, including water, are mixed at a ready-mixconcrete plant and the shotcrete is delivered tothe site by ready-mix truck. This approach issuitable for sites with good road access and theneed for large quantities. For dry-mix shotcretethe dry components are mixed at the plant andthen placed in 1 m3 bags that have a valve in thebottom (Figure 12.14). At the site, the bags aredischarged into the hopper on the pump and

a pre-moisturizer adds 4% water to the mix. Themix is then pumped to the face where additionalwater is added through a ring valve at the nozzle.The advantages of the dry-mix process are its usein locations with difficult access, and where smallquantities are being applied at a time. It is alsouseful to be able to adjust the quantity of waterin areas where there is varying amounts of seepageon the face.

Typical mixes for dry-mix and wet-mix silicafume, steel fiber reinforced shotcrete are shownin Table 12.9 (Morgan et al., 1989).

Shotcrete strength. The strength of shotcreteis defined by three parameters that correspondto the types of loading conditions to whichshotcrete may be subjected when applied to aslope. Typical values for these parameters are asfollows:

(a) Compressive strength of 20 MPa at 3 daysand 30 MPa at 7 days;

(b) First crack flexural strength of 4.5 MPa at 7days; and

(c) Toughness indices of I5 = 4 and I10 = 6.

Figure 12.14 Dry-mix shotcrete processusing bagged mix feeding a pump andpre-moisturizer.


Table 12.9 Typical shotcrete mixes

Material Dry-mix Shotcrete Wet-mix Shotcrete(kg/m3) (% dry materials) (kg/m3) (% dry materials)

Cement, Type I 400 18.3 420 18.3Silica fume 50 2.3 40 1.710 mm coarse aggregate 500 22.9 480 20.9Sand 1170 53.7 1120 48.7Steel fibers 60 2.8 60 2.6Water reduced — — 2l 0.09Superplasticizer — — 6l 0.04Air entraining admixture — — if required if requiredWater 170a — 180 7.8Total wet mass 2350 100 2300 100

Notea Total water from pre-moisturizer and added at nozzle (based on saturated surface dryaggregate concept).

Toughness indices

Load A

C

E

FD

Deformation

B0

�3�

5.5�

1 2 3 4

l 5 =AOAB

AOACD

l 10 =AOABAOAEF

Figure 12.15 Load–deformationcharacteristics of steel-fiber reinforcedshotcrete. 1, without fibers; 2, 1% vol.fibers; 3, 2% vol. fibers. 4, 3% vol. fibers(American Concrete Institute, 1995).

The flexural strength and toughness indices aredetermined by cutting a beam with dimensionsof 100 mm square in section and 350 mm longfrom a panel shot in the field, and testing thebeam in bending. The test measures the deforma-tion beyond the peak strength, and the method ofcalculating the I5 and I10 toughness indices fromthese measurements is shown in Figure 12.15.

Surface preparation. The effectiveness ofshotcrete is influenced by the condition of therock surface to which it is applied—the sur-face should be free of loose and broken rock,soil, vegetation and ice. The surface should also

be damp to improve the adhesion between therock and the shotcrete, and the air temperatureshould be above 5◦C for the first seven days whenthe shotcrete is setting. Drain holes should bedrilled through the shotcrete to prevent build-up of water pressure behind the face; the drainholes are usually about 0.5 m deep, and locatedon 1–2 m centers. In massive rock the drain holesshould be drilled before the shotcrete is applied,and located to intersect discontinuities thatcarry water. The holes are temporarily pluggedwith wooden pegs or rags while applying theshotcrete.


Aesthetics. A requirement on some civil pro-jects is that shotcreted faces should have a naturalappearance. That is, the shotcrete should becolored to match the natural rock color, andthe face sculpted to show a pattern of “discon-tinuities.” This work is obviously costly, but thefinal appearance can be a very realistic replica ofa rock face.

12.4.5 Buttresses

Where a rock fall or weathering has formed acavity in the slope face, it may be necessary to con-struct a concrete buttress in the cavity to preventfurther falls (Figure 12.4, item 6). The buttressfulfills two functions: first, to retain and protectareas of weak rock, and second, to support theoverhang. Buttresses should be designed so thatthe direction of thrust from the rock supportsthe buttress in compression. In this way, bendingmoments and overturning forces are eliminatedand there is no need for heavy reinforcement ofthe concrete, or tiebacks anchored in the rock.

If the buttress is to prevent relaxation of therock, it should be founded on a clean, sound rocksurface. If this surface is not at right angles tothe direction of thrust, then the buttress shouldbe anchored to the base using steel pins to pre-vent sliding. Also, the top of the buttress shouldbe poured so that it is in contact with the under-side of the overhang. In order to meet this secondrequirement, it may be necessary to place the lastpour through a hole drilled downward into thecavity from the rock face, and to use a non-shrinkagent in the mix.

12.4.6 Drainage

As shown in Table 12.1, ground water in rockslopes is often a primary or contributory causeof instability, and a reduction in water pressuresusually improves stability. This improvement canbe quantified using the design procedures dis-cussed in Chapters 6–10. Methods of controllingwater pressure include limiting surface infiltra-tion, and drilling horizontal drain holes or drivingadits at the toe of the slope to create outlets for

the water (Figure 12.16). The selection of themost appropriate method for the site will dependon such factors as the intensity of the rainfall orsnow melt, the permeability of the rock and thedimensions of the slope.

Surface infiltration. In climates that experi-ence intense rainfall that can rapidly saturate theslope and cause surface erosion, it is beneficialfor stability to construct drains both behind thecrest and on benches on the face to interceptthe water (Government of Hong Kong, 2000).These drains are lined with masonry or concreteto prevent the collected water from infiltrating theslope, and are dimensioned to carry the expectedpeak design flows (see Figure 1.1(a)). The drainsare also interconnected so that the water is dis-charged to the storm drain system or nearby watercourses. Where the drains are on steep gradients,it is sometimes necessary to incorporate energydissipation protrusions in the base of the drain tolimit flow velocities. In climates with high rain-fall there is usually rapid vegetation growth, andperiodic maintenance will be required to keep thedrains clear.

Horizontal drain holes. An effective means ofreducing the water pressure in many rock slopes isto drill a series of drain holes (inclined upwards atabout 5◦) into the face. Since most of the groundwater is contained in discontinuities, the holesshould be aligned so that they intersect the dis-continuities that are carrying the water. For theconditions shown in Figure 12.4, the drain holesare drilled at a shallow angle to intersect the morepersistent discontinuities that dip out of the face.If the holes were drilled at a steeper angle, parallelto these discontinuities, then the drainage wouldbe less effective.

There are no widely used formulae from whichto calculate the required spacing of drill holes,but as a guideline, holes are usually drilled on aspacing of about 3–10 m, to a depth of about one-half to one-third of the slope height. The holesare often lined with perforated casing, with theperforations sized to minimize infiltration of finesthat are washed from fracture infillings. Anotheraspect of the design of drain holes is the disposal


Lined collectordrain

Slope immediately behind crest gradedto prevent pools of surface water fromgathering during heavy rain

Lined surface drain to collectrun-off before it can enter topof tension crack

Vertical pumpeddrainage well

Horizontal hole totap base of tensioncrack

Potentialtension crack

Potential slidesurface

Sub-surfacedrainage gallery

Collector drain

Horizontal hole to drainpotential slide surface

Fan of drill holes to increasedrainage efficiency of sub-surface gallery

Figure 12.16 Slope drainage methods.

of the seepage water. If this water is allowed toinfiltrate the toe of the slope, it may result indegradation of low-strength materials, or pro-duce additional stability problems downstream ofthe drains. Depending on site conditions, it maybe necessary to collect all the seepage water in amanifold and dispose of it at some distance fromthe slope.

Drain holes can be drilled to depths of sev-eral hundred meters, sometimes using drilling

equipment that installs the perforated casing asthe drill advances to prevent caving. Also, it iscommon to drill a fan of holes from a single setup to minimize drill moves (Cedergren, 1989).

Drainage adits. For large slides, it may not bepossible to reduce significantly the water pressurein the slope with relatively small drain holes. Inthese circumstances, a drainage tunnel may bedriven into the toe of the slide from which a seriesof drain holes are drilled up into the saturated


rock. For example, the Downie Slide in BritishColumbia has an area of about 7 km2 and a thick-ness of about 250 m. Stability of the slope was ofconcern when the toe was flooded by the con-struction of a dam. A series of drainage tunnelswith a total length of 2.5 km were driven at anelevation just above the high water level of thereservoir. From these tunnels, a total of 13,500 mof drain holes was drilled to reduce the groundwater pressures within the slope. These drain-age measures have been effective in reducing thewater level in the slide by as much as 120 m, andreducing the rate of movement from 10 mm/yearto about 2 mm/year (Forster, 1986). In a miningapplication, ground water control measures forthe Chuquicamata pit in Chile include a 1200 mlong drainage adit in the south wall, and a num-ber of pumped wells (Flores and Karzulovic,2000).

Methods of estimating the influence ofa drainage tunnel on ground water in aslope include empirical procedures (Heuer,1995), theoretical models of ground waterflow in homogeneous rock (Goodman et al.,1965), and three-dimensional numerical mod-eling (McDonald and Harbaugh, 1988). In allcases, the flow and drawdown values will beestimates because of the complex and uncer-tain relationship between ground water flow andstructural geology, and the difficulty of obtainingrepresentative permeability values.

Empirical procedures for calculating inflowquantities are based on actual flow rates measuredin tunnels. Based on these data, a relationship hasbeen developed between the normalized steady-state inflow intensity (l/min/m tunnel length/mhead) and the rock mass conductivity determinedfrom packer tests (Heuer, 1995). The flow quant-ities can be calculated for both vertical rechargewhere the tunnel passes under an aquifer, andradial flow for a tunnel in an infinite rock mass.This empirical relationship has been developedbecause it has been found the actual flows canbe one-eighth of the calculated theoretical valuesbased on measured conductivities.

Approximate inflow quantities can also beestimated by modeling the drainage adit as an

infinitely long tunnel in a homogeneous, isotropicporous medium, with the pressure head on thesurface of the tunnel assumed to be atmospheric.If flow occurs under steady-state conditions suchthat there is no drainage of the slope and the headabove the tunnel H0 is constant with time, theapproximate rate of ground water flow Q0 perunit length of tunnel is given

Q0 = 2πKH0

2.3 log(2H0/r)(12.11)

where r is the radius of the tunnel driven in homo-geneous material with hydraulic conductivity K.For rock formations with low porosity and lowspecific storage it is likely that transient condi-tions will develop where the head diminishes withtime as the slope drains.

An important aspect of slope drainage is toinstall piezometers to monitor the effect of drain-age measures on the water pressure in the slope.For example, one drain hole with a high flowmay only be draining a small, permeable zone inthe slope and monitoring may show that moreholes would be required to lower the watertable throughout the slope. Conversely, in lowpermeability rock, monitoring may show thata small seepage quantity that evaporates as itreaches the surface is sufficient to reduce thewater pressure and significantly improve stabilityconditions.

12.4.7 “Shot-in-place” buttress

On landslides where the slide surface is a well-defined geological feature such as a continuousbedding surface, stabilization may be achieved byblasting this surface to produce a “shot-in-place”buttress (Aycock, 1981; Moore, 1986). The effectof the blasting is to disturb the rock surface andeffectively increase its roughness, which increasesthe total friction angle. If the total friction angleis greater than the dip of the slide surface, thensliding may be halted. Fracturing and dilation ofthe rock may also help reduce water pressures onthe slide surface.


The method of blasting involves drilling a pat-tern of holes through the slide surface and placingan explosive charge at this level that is just suffi-cient to break the rock. This technique requiresthat the drilling begins while it is still safe forthe drills to access the slope, and before the rockbecomes too broken for the drills to operate.Obviously, this stabilization technique shouldbe used with a great deal of caution becauseof the potential for exacerbating stability con-ditions, and probably should only be used inemergency situation when there are no suitablealternatives.

12.5 Stabilization by rock removal

Stabilization of rock slopes can be accomplishedby the removal of potentially unstable rock;

Figure 12.17 illustrates typical removal methodsincluding

• resloping zones of unstable rock;• trim blasting of overhangs;• scaling of individual blocks of rock.

This section describes these methods, and the cir-cumstances where removal should and should notbe used. In general, rock removal is a preferredmethod of stabilization because the work willeliminate the hazard, and no future maintenancewill be required. However, removal should onlybe used where it is certain that the new face willbe stable, and there is no risk of undermining theupper part of the slope. Area 4 on Figure 12.17is an example of where rock removal should becarried out with care. It would be safe to remove

Weatheredrock

Fresh rock

1.

2.

3.

4.

Resloping of unstable weathered materialin upper part of slope

Access bench at top of cut

Removal of rock overhang bytrim blasting

Removal of trees with rootsgrowing in cracks

Hand scaling of loose blocksin shattered rock

Figure 12.17 Rock removal methods for slope stabilization (TRB, 1996).


the outermost loose rock, provided that the frac-turing was caused by blasting and only extendedto a shallow depth. However, if the rock massis deeply fractured, continued scaling will soondevelop a cavity that will undermine the upperpart of the slope.

Removal of loose rock on the face of a slope isnot effective where the rock is highly degradable,such as shale. In these circumstances, exposure ofa new face will just start a new cycle of weatheringand instability. For this condition, more appro-priate stabilization methods would be protectionof the face with shotcrete and rock bolts, or atied-back wall.

12.5.1 Resloping and unloading

Where overburden or weathered rock occurs inthe upper portion of a cut, it is often necessary tocut this material at an angle flatter than the morecompetent rock below (Figure 12.17, item 1).The design procedure for resloping and unload-ing starts with back analysis of the unstable slope.By setting the factor of safety of the unstableslope to 1.0, it is possible to calculate the rockmass strength parameters (see Section 4.4). Thisinformation can then be used to calculate therequired reduced slope angle and/or height thatwill produce the required factor of safety.

Another condition that should be takenaccount of during design is weathering of therock some years after construction, at which timeresloping may be difficult to carry out. A benchcan be left at the toe of the soil or weathered rockto provide a catchment area for minor slope fail-ures and provide equipment access. Where a slidehas developed, it may be necessary to unload thecrest of the cut to reduce its height and diminishthe driving force.

Resloping and unloading is usually carried outby excavating equipment such as excavators andbulldozers. Consequently, the cut width mustbe designed to accommodate suitable excavatingequipment on the slope with no danger of col-lapse of the weak material while equipment isworking; this width would usually be at least5 m. Safety for equipment access precludes the

excavation of “sliver” cuts in which the toe ofthe new cut coincides with that of the old cut.

12.5.2 Trimming

Failure or weathering of a rock slope may forman overhang on the face (Figure 12.17, item 2),which could be a hazard if it were to fail. Inthese circumstances, removal of the overhang bytrim blasting may be the most appropriate sta-bilization measure. Section 11.4 discusses meth-ods of controlled blasting that are applicable tosituations where it is required to trim blast smallvolumes of rock with minimal damage to the rockbehind the trim line.

Where the burden on a trim blast is limited,flyrock may be thrown a considerable distancebecause there is little rock to contain the explosiveenergy. In these circumstances appropriate pre-cautions such as the use of blasting mats wouldbe required to protect any nearby structures andpower lines. Blasting mats are fabricated fromrubber tires or conveyor belts chained or wiredtogether.

12.5.3 Scaling

Scaling describes the removal of loose rock, soiland vegetation on the face of a slope using handtools such as scaling bars, shovels and chainsaws. On steep slopes workers are usually sup-ported by ropes, anchored at the crest of the slope(Figure 12.18). A suitable type of rope for theseconditions is a steel-core, hemp rope that is highlyresistant to cuts and abrasion. The scalers worktheir way down the face to ensure there is no looserock above them.

A staging suspended from a crane is an altern-ative to using ropes for the scalers to access theface. The crane is located at the toe of the slopeif there is no access to the crest of the slope. Thedisadvantages of using a crane, rather than ropes,are the expense of the crane, and on highway pro-jects, the extended outriggers can occupy severallanes of the highway with consequent disruptionto traffic. Also, scaling from a staging suspen-ded from crane can be less safe than using ropes


Figure 12.18 High scalersuspended on rope andbelt while removing looserock on steep rock slope(Thompson River Canyon,British Columbia,Canada) (TRB, 1996).

because the scalers are not able to direct the craneoperator to move quickly in the event of a rockfall from the face above them.

An important component of a scaling opera-tion in wet climates is the removal of trees andvegetation growing on the face, and to a dis-tance of several meters behind the crest of theslope. Tree roots growing in fractures on the rockface can force open the fractures and eventuallycause rock falls. Also, movement of the trees bythe wind produces leverage by the roots on looseblocks. The general loosening of the rock on theface by tree roots also permits increased infiltra-tion of water which, in temperate climates, willfreeze and expand and cause further opening ofthe cracks. As shown in Table 12.1, approxim-ately 0.6% of the rock falls on the Californiahighway system can be attributed to root growth.

12.5.4 Rock removal operations

Where rock removal operations are carried outabove active highways or railroads, or in urbanareas, particular care must be taken to preventinjury or damage from falling rock. This will usu-ally require that all traffic be stopped while rock

removal is in progress, and until the slope hasbeen made safe and the road has been cleared ofdebris. Where there are pipelines or cables bur-ied at the toe of the slope, it may be necessaryto protect them, as well as pavement surfacesor rail track, from the impact of falling rock.Adequate protection can usually be provided byplacing a cover of sand and gravel to a depth ofabout 1.5–2 m. For particularly sensitive struc-tures, additional protection can be provided byrubber blast mats.

12.6 Protection measures against rock falls

An effective method of minimizing the hazardof rock falls is to let the falls occur and controlthe distance and direction in which they travel.Methods of rock fall control and protection offacilities at the toe of the slope include catch-ment ditches and barriers, wire mesh fences, meshhung on the face of the slope and rock sheds.A common feature of all these protection struc-tures is their energy-absorbing characteristics inwhich the rock fall is either stopped over some dis-tance, or is deflected away from the facility that isbeing protected. As described in this section, it is


possible by the use of appropriate techniques, tocontrol rocks with dimensions up to about 2 m,falling from heights of several hundred meters,and impacting with energies as high as 1 MJ.Rigid structures, such as reinforced concrete wallsor fences with stiff attachments to fixed sup-ports, are rarely appropriate for stopping rockfalls.

12.6.1 Rock fall modeling

Selection and design of effective protection meas-ures require the ability to predict rock fallbehavior. An early study of rock falls was madeby Ritchie (1963) who drew up empirical ditchdesign charts related to the slope dimensions (seeSection 10.6.2). Since the 1980s, the predic-tion of rock fall behavior was enhanced by thedevelopment of a number of computer programsthat simulate the behavior of rock falls as theyroll and bounce down slope faces (Piteau, 1980;Wu, 1984; Descoeudres and Zimmerman, 1987;Spang, 1987; Hungr and Evans, 1988; Pfeifferand Bowen, 1989; Pfeiffer et al., 1990; Azzoniand de Freitas, 1995).

Figure 12.19 shows an example of the outputfrom the rock fall simulation program RocFall(Rocscience, 2004). The cross-section shows thetrajectories of 20 rock falls, one of which rollsout of the ditch. Figures 12.19(b) and (c) showrespectively the maximum bounce heights andtotal kinetic energy at intervals down the slope.The input for the program comprises the slopeand ditch geometry, the irregularity (roughness)of the face, the restitution coefficients of the slopematerials, the mass and shape of the block, andthe start location and velocity. The degree ofvariation in the shape of the ground surface ismodeled by randomly varying the surface rough-ness for each of a large number of runs, which inturn produces a range of trajectories.

The results of analyses such as those shown inFigure 12.19, together with geological data onblock sizes and shapes, can be used to estimate thedimensions of a ditch, or the optimum position,required height and energy capacity of a fence orbarrier. In some cases, it may also be necessary to

verify the design by constructing a test structure.Sections 12.6.2–12.6.4 describe types of ditches,fences and barriers, and the conditions in whichthey can be used.

Benched Slopes. The excavation of interme-diate benches on rock cuts usually increases therock fall hazard, and is therefore not recommen-ded for most conditions. Benches can be a hazardwhere the crests of the benches fail due to blastdamage, and the failed benches leave irregularprotrusions on the face. Rock falls striking theseprotrusions tend to bounce away from the faceand land a considerable distance from the base.Where the narrow benches fill with debris, theywill not be effective in catching rock falls. It israrely possible to remove this debris because ofthe hazard to equipment working on narrow,discontinuous benches.

There are, however two situations wherebenched slopes are a benefit to stability.First, in horizontally bedded sandstone/shale/coalsequences the locations and vertical spacing ofthe benches is often determined by the lithology.Benches are placed at the top of the least resistantbeds, such as coal or clay shale, which weatherquicker (Wright, 1997). With this configuration,the more resistant lithology is not undermined asthe shale weathers (Figure 12.20). The width ofintermediate benches may vary from 6 to 8 m,and the face angle depends on the durability ofthe rock. For example, shales with a slake dur-ability index of 50–79 are cut at angles of 43◦(1.33H:1V) and heights up to 9 m, while massivesandstone and limestone may be excavated at aface angles as steep as 87◦ (1/20H:1V) and heightsup to 15 m. Figure 12.20 also shows a bench atthe toe of the overburden slope to contain minorsloughing and provide access for cleaning.

A second application for benched slopes is intropical areas with deeply weathered rock andintense periods of rain. In these conditions, lineddrainage ditches on each bench and down theslope face are essential to collect runoff andprevent scour and erosion of the weak rock(Government of Hong Kong, 2000).


24 42 6030 48 6636 54 72 78

925

932

939

946

953

Off-set (m)

Ele

vatio

n (m

)

Off-set (m)

40300

2

4

6

50 60 70

Hei

ght a

bove

slo

pe (

m)

Off-set (m)4030 50 60 70

0

400

800

1200

1600

Tota

l kin

etic

ene

rgy

(J)

(a)

(b)

(c)

Figure 12.19 Example of analysis of rock fall behavior: (a) trajectories of 20 rock falls; (b) variation in verticalbounce heights along the slope; (c) variation in total kinetic energy along the slope.


Overburden

Limestone

4.5 m

6.0 m

6.0 m

4.3 m

1:2

2:1

20:1

20:1

Coal

Sandstone

Sandstone

Shale

9 m max.

15 m

Roadside ditch

Intermediate bench on shale

Intermediate bencheson weaker rock

Figure 12.20Configuration ofbenched cut inhorizontally beddedshale and sandstone,with weaker coal andshale formations locatedat toe of cut faces.

12.6.2 Ditches

Catch ditches at the toe of slopes are often a cost-effective means of stopping rock falls, providedthere is adequate space at the toe of the slope(Wyllie and Wood, 1981). The required dimen-sions of the ditch, as defined by the depth andwidth, are related to the height and face angle ofthe slope; a ditch design chart developed fromfield tests is shown in Figure 12.21 (Ritchie,1963). The figure shows the effect of slope angleon the path that rock falls tend to follow, and howthis influences ditch design. For slopes steeperthan about 75◦, the rocks tend to stay close to theface and land near the toe of the slope. For slopeangles between about 55◦ and 75◦, falling rockstend to bounce and spin with the result that theycan land a considerable distance from the base;consequently, a wide ditch is required. For slope

angles between about 40◦ and 55◦, rocks will tendto roll down the face and into the ditch.

To up-date the work carried out by Ritchie,a comprehensive study of rock fall behavior andthe capacity of catchment areas has been carriedout by the Oregon Department of Transporta-tion (2001). This study examined rock fall fromheights of 12, 18 and 24 m on slopes with fiveincrements of face angles ranging from vertical to45◦ (1V:1H). The catchment areas at the toe ofthe slope were planar surfaces with inclinationsof 76◦ (1/4H:1V), 80◦ (1/6H:1V) and horizontalto simulate unobstructed highway shoulders. Thetests observed both the impact distance fromthe toe and the roll-out distance. The reportincludes design charts that show, for all combina-tions of slope geometry, the relationship betweenthe percent rock retained and the width of thecatchment area.


60°

45°

30° slope angle, �f

Fence

Roll

Roll

Bounce

FallWidth (W)

Depth (D)

Slo

pe h

eigh

tS

lope

hei

ghtH

(fe

et)

H

Overall slope angle �f (degrees)

Depth Width

Slo

pe h

eigh

tH(m

eter

s)H

120

40

0

20

10

30

160

100

140

180

20

090

0.1Slopegradient

Motion of a falling rock

Free fall Bounce Roll

0.25:1 1.25:11:10.75:10.5:1

4060 507080

0.3:1

7.62 m2.44 m

2.13 m

1.83 m

6.1 m

1.52 m

4.57 m

0.91 m3.05 m

1.22 m

Figure 12.21 Ditch design chart for rock fallcatchment (Ritchie, 1963).

12.6.3 Barriers

A variety of barriers can be constructed either toenhance the performance of excavated ditches,or to form catchment zones at the toe of slopes(Andrew, 1992a). The required type of barrierand its dimensions depend on the energy of the

falling blocks, the slope dimensions and the avail-ability of construction materials. A requirementof all barriers is flexibility upon impact. Barriersabsorb impact energy by deforming, and sys-tems with high impact energy capacity are bothflexible, and are constructed with materials thatcan withstand the impact of sharp rocks withoutsignificant damage. The following is a briefdescription of some commonly used barriers.

Gabions and concrete blocks. Gabions orconcrete blocks are effective protection barriersfor falling rock with diameters up to about0.75 m. Figure 12.22(a) shows an example of aditch with two layers of gabions along the outeredge forming a 1.5 m high barrier.

The function of a barrier is to form a “ditch”with a vertical facing the slope face that trapsrolling rock. Barriers are particularly useful at thetoe of flatter slopes where falling rock rolls andspins down the face but does not bounce signific-antly. Such rocks may land in a ditch at the toe ofthe slope but can roll up the sloping outer side; avertical barrier will help to trap such falls.

Gabions are rock-filled, wire mesh baskets,typically measuring 0.91 m by 0.91 m in cross-section, that are often constructed on-site withlocal waste rock. Advantages of gabions are theease of construction on steep hillsides and wherethe foundation is irregular, and their capacityto sustain considerable impact from falling rock.However, gabions are not immune to damageby impacts of rock and maintenance equipment,and repair costs can become significant. Barriersconstructed with pre-cast concrete with similardimensions as gabions are also used on transport-ation systems for rock fall containment. Althoughconcrete blocks are somewhat less resilient thangabions, they have the advantages of wide avail-ability and rapid installation. In order for con-crete blocks to be effective, flexibility must beprovided by allowing movement at the jointsbetween the blocks. In contrast, mass concretewalls are much less flexible and tend to shatteron impact.

Geofabric-soil barriers. Various barriers havebeen constructed using geofabric and soil layers,


(b)

(a)

Figure 12.22 Rock fallcontainment structures: (a) rockcatch ditch with 1.5 m highgabion along outer edge (FraserRiver Canyon, British Columbia);(b) barrier constructed with MSEwall and wire rope fence on topof wall (Interstate 40 nearAsheville, North Carolina)(Courtesy: North CarolinaDepartment of Transportation).

each about 0.6 m thick, built up to form a bar-rier, which may be as high as 4 m (Threadgoldand McNichol, 1985) (Figure 12.22(b)). By wrap-ping the fabric around each layer it is possible toconstruct a barrier with vertical front and backfaces; the face subject to impact can be protec-ted from damage with such materials as railwayties, gabions and rubber tires (Figures 12.23(a)and (b)). The capacity of a barrier of this type to

stop rock falls depends on its mass in relation tothe impact energy, the shear resistance at the baseand the capacity to deform without failing. Thedeformation may be both elastic deformation ofthe barrier components, and shear displacementat the fabric layers or on the base. A disadvantageof barriers such as those shown in Figure 12.23 isthat a considerable space is required for both thebarrier and the catchment area behind it.


3.3 m(a)

(b)

MSE-wall

Impact catchment bags

Sandy soil

5.3 m

Geo grid

Impact transmission bags

Wall facing unit

4.0

m

1:0.

31:0.2

Geosynthetic

Slope

Rock trap

On-site soil backfill

Facade of choice

Concrete barrier

Roadway

2.5

m

Figure 12.23 Rock fall barriers constructed with soil and geofabric, and a variety of facings: (a) a 4 m highWall with impact energy capacity of 5000 kJ (Protec, 2002); (b) a 2.5 m high wall with energy capacity of950 kJ (Barrett and White, 1991).

Extensive testing of prototype barriers by theColorado Department of Transportation hasshown that limited shear displacement occurson the fabric layers, and that they can with-stand high energy impacts without significantdamage (Barrett and White, 1991). Also, a4 m high geofabric and soil barriersuccessfully

withstood impact from boulders with volumesof up to 13 m3 and impact energies of 5000 kJon Niijima Island in Japan (Protec Engineer-ing, 2002), and a similar 1.8 m wide geofabricbarrier stopped rock impacts delivering 950 kJ ofenergy.


12.6.4 Rock catch fences and attenuators

During the 1980s, various fences and nets suitablefor installation on steep rock faces, in ditches andon talus run-out zones were developed and thor-oughly tested (Smith and Duffy, 1990; Barrettand White, 1991; Duffy and Haller, 1993). Netsare also being used in open pit mines for rock fallcontrol (Brawner and Kalejta, 2002). A designsuitable for a particular site depends on the topo-graphy, anticipated impact loads, bounce heightand local availability of materials. A commonfeature of all these designs is their ability towithstand impact energy from rock falls due totheir construction without any rigid components.When a rock impacts a net, there is deformationof the mesh which then engages energy absorbingcomponents over an extended time of collision.This deformation significantly increases the capa-city of these components to stop rolling rockand allows the use of light, low cost elements inconstruction.

Wire-rope nets. Nets with energy absorptioncapacity ranging from 40 to 2000 kJ have beendeveloped as proprietary systems by a number ofmanufacturers (Geobrugg Corporation and IsoferIndustries). The components of these nets are aseries of steel I-beam posts on about 6 m cen-ters, anchored to the foundation with groutedbolts, and guy cables anchored on the slope.Additional flexibility is provided by incorporat-ing friction brakes on the cable supporting thenets and the guy cables. Friction brakes are loopsof wire in a steel pipe that are activated duringhigh energy events to help dissipate the impactforces (Figure 12.24). It has also been foundthat nets are effective in containing debris flowsbecause the water rapidly drains from the debrismaterial and its mobility is diminished (CaliforniaPolytechnical State University, 1996).

The mesh is a two-layer system comprising a50 mm chain link mesh, and either woven steelwire-rope mesh or interlocking steel rings. The

Grout

3m

3.7m (±0.15m)

Wire rope anchor

16 mm (min.) wire ropewith braking element

W 8 × 48steel post

Friction brake

Concrete0.75 m

0.75 m (min.)

0.06 m (max.)

3 m

Chain link mesh

Ring netDrill hole 100 mm dia.

Figure 12.24 Geobrugg rock fall fence (TRB, 1996).


woven wire-rope net is constructed typically with8 mm diameter wire rope in a diagonal patternon 100–200 mm centers. The wire rope and netdimensions will vary with expected impact ener-gies and block sizes. An important feature of thewire-rope net is the method of fixing the intersec-tion points of the wire rope with high strengthcrimped fasteners. The mesh is attached to theposts by lacing it on a continuous perimeter wirerope that is attached to brackets at the top andbottom of each post.

As an alternative to the woven wire mesh,ring nets are fabricated with 0.31 m diameterrings, each of which is interlocked with four adja-cent rings. The rings are fabricated from 3 mmdiameter high tensile steel wire, and the num-ber of wires in each ring varies between 5 and19 depending on the design energy capacity ofthe net.

Rock fall attenuators. Where rocks fall downa narrow gully or chute bounded by stable rockwalls, it is possible to install a variety of fencesthat slow down and absorb the energy of rockfalls (Andrew, 1992b). The general method ofconstruction is to install an anchor in each rockface to support a wire rope from which the fence,spanning the gully, is suspended. For rock fallswith dimensions up to about 200 mm it is possibleto use chain link mesh draped down the chutefrom the support rope; wire-rope mesh or ringnets can be used for larger blocks. Falling rocksare gradually brought to a halt as they bounceand roll under the mesh.

Maintenance requirements and worker safetyof fence systems should be considered in design.A properly designed system should not need fre-quent repairs if the impacts are within the toler-ances of design energies. However, cleaning ofaccumulated rock is necessary for any system.Typically, fixed barriers such as geofabric wallsrequire room behind them for cleaning opera-tions. In contrast, woven wire rope and ring netsdo not have this requirement because of theirmodular design, allowing the nets to be cleanedfrom the front by removing or lifting each panelin turn.

12.6.5 Draped mesh

Wire mesh hung on the face of a rock slope can bean effective method of containing rock falls closeto the face and preventing them from bouncingon to the road (Ciarla, 1986). Because the meshabsorbs some of the energy of the falling rock, therequired dimensions of the ditch at the base of thisslope are considerably reduced from those shownin Figure 12.21. Chain link mesh is a suitablemethod for controlling rock falls with dimensionsless than about 0.6–1 m, and woven wire rope orring nets are suitable for rock with dimensionsup to about 1.3 m. For installations covering ahigh slope where the weight of the lightweightmesh may exceed its strength, the mesh can bereinforced with lengths of wire rope. In all cases,the upper edge of the mesh or net should beplaced close to the source of the rock fall so theblocks have little momentum when they impactthe mesh.

The mesh is not anchored at the bottom ofthe slope or at intermediate points. The freelyhanging mesh permits rocks to work their waydown to the ditch, rather than accumulatingbehind the mesh; the weight of such accumula-tions can fail the mesh.

12.6.6 Warning fences

Fences and warning signals that are triggered byfalling rock are often used to protect railroads,and occasionally highways. The warning fenceconsists of a series of posts and cantilever cross-arms, which support rows of wires spaced about0.5 m apart. The wires are connected to a sig-nal system that shows a red light if a wire isbroken. The signal light is located far enoughfrom the rock slope that the traffic has time tostop, and then to proceed with caution beforeit reaches the rock fall location. Warning sys-tems can also be incorporated into rock fall fences(Section 12.6.4), so that a second level of protec-tion is provided in the event of a large fall thatexceeds the energy capacity of the fence.

Warning fences are most applicable on trans-portation systems where traffic is light enough


to accommodate occasional closures of the line.However, the use of warning fences as a protec-tion measure has a number of disadvantages. Thesignal lights must be located a considerable dis-tance from the slope, and falls may occur afterthe traffic has passed the light. Also, false alarmscan be caused by minor falls of rock or ice, andmaintenance costs can be significant.

12.6.7 Rock sheds and tunnels

In areas of extreme rock fall hazard where sta-bilization of the slope would be very costly,construction of a rock shed or even relocationof the highway into tunnels may be justified.Figure 12.25 shows two alternate configurationsfor sheds depending on the path of the fallingrock. Where the rock falls have a steep tra-jectory, the shed has a flat roof covered witha layer of energy absorbing material such asgravel (Figure 12.25(a)). Sheds are constructedwith reinforced concrete or steel, designed to

withstand the worst case impact loading at theedge of the roof. The design should also considerthe stability under impact loading of the founda-tions for the outer columns that are often locatedat the crest of steep slopes. Figure 12.25(b) showsthree sheds with sloping roofs that are designedto deflect rolling rock over the railway. Becausethese sheds do not sustain direct impact theyare of much lighter construction than that inFigure 12.25(a), and there is no protective layeron the roof.

Extensive research on the design of rock shedshas been carried out in Japan where tens of kilo-meters of sheds have been constructed to protectboth railways and highways (Yoshida et al.,1991; Ishikawa, 1999). Much of the researchhas involved full-scale testing in which bouldershave been dropped on prototype sheds that arefully instrumented to measure deceleration in theboulder, and the induced stresses and strains inthe major structural components. Objectives ofthe tests are to determine the effectiveness of

(a) (b)

Figure 12.25 Typical rock shed construction: (a) reinforced concrete structures with horizontal roof coveredwith layer of gravel (Photograph courtesy: Dr H. Yoshida, Kanazawa, Japan); (b) sheds constructed withtimber and reinforced concrete with sloping roofs that deflect rock falls over the railway (White Canyon,Thompson River, British Columbia) (Courtesy: Canadian National Railway).


For

ce

Deformation

Styrofoam

Rubber tires

Sand

Rock shed roof

Cushion material

Weight impact force

Rock mass

Transmitted force (integration of the transmitted pressure on distributed area)

(a)

(b)

Figure 12.26 Characteristics of cushioning materials:(a) definition of weight impact force and transmittedimpact force due to rock fall impact on cushion;(b) relationship between force and deformation forimpact loading of gravel, styrofoam and rubber tires(Yoshida et al., 1991).

various cushioning materials in absorbing anddispersing the impact energy, and to assess theinfluence of the flexibility of the structure on themaximum impact that can be sustained withoutdamage.

A critical feature of shed design is the weightand energy absorption characteristics of the cush-ioning material. Ideally the cushion should both

absorb energy by compression, and disperse thepoint impact energy so that the energy trans-mitted into the structure occurs over a largearea. Furthermore, the cushion should remainintact after impact so it does not need to bereplaced. The effectiveness of the material can beexpressed as the difference between the “weightimpact force” induced by boulder impact, andthe “transmitted force” that is absorbed by thestructure (Figure 12.26(a)). Gravel is the mostcommonly used cushioning material because it isinexpensive and widely available. However, thedisadvantage of gravel is its weight, and thereis a point where the gravel layer is so thickthat its weight exceeds the rock fall impact load-ing. Rubber tires have also been used, but itis found that they are highly compressible withlittle energy absorption. A viable alternative togravel is reinforced Styrofoam that is an effectiveenergy absorbing material with low unit weight,which allows for some saving in the dimen-sions of the structure (Mamaghani et al., 1999).The disadvantage of Styrofoam is its cost com-pared to gravel, so the cost benefit of its useshould be carefully evaluated. Figure 12.26(b)shows typical force–deformation characteristicsof gravel, Styrofoam and rubber tires (Yoshida,2000).

In locations at which it is impractical to con-struct a rock shed or stabilize the slope by othermeans, it may be necessary to drive a tunnel tobypass the hazard zone. For example, a railwayin British Columbia drove a 1200 m long tunnelto avoid a section of track located on a nar-row bench between a steep, unstable rock cliffabove a 400 m deep lake. Major rock falls were ahazard to train operations and had caused trackclosures lasting as long as two weeks (Leighton,1990).

Chapter 13

Movement monitoring

13.1 Introduction

Many rock slopes move to varying degrees duringthe course of their operational lives. Such move-ment indicates that the slope is in a quasi-stablestate, but this condition may continue for manyyears, or even centuries, without failure occur-ring. However, in other cases, initial minorslope movement may be a precursor for accel-erating movement followed by collapse of theslope. Because of the unpredictability of slopebehavior, movement monitoring programs canbe of value in managing slope hazards, and theyprovide information that is useful for the designof remedial work.

Slope movement is most common in openpit mines, and many mines continue to operatesafely for years with moving slopes that are care-fully monitored to warn of deteriorating stabilityconditions. Other slopes that undergo long-termmovement are landslides that may creep for hun-dreds of years resulting in accumulative move-ment of tens of meters. Such movement maycomprise an approximately uniform creep rate,on which may be superimposed short periods ofmore rapid movement resulting from such eventsas earthquakes, unusually high precipitation peri-ods and human activities. Human activities thatcan be detrimental to slope stability includeexcavations of the base, and changing the groundwater conditions by dam filling or irrigation.

This chapter describes common methods ofmonitoring movement of rock slopes, and inter-pretation of the results. It is considered that

monitoring programs are most appropriate foractively mined slopes such as open pit minesand quarries which have a limited operationallife and where a carefully managed, on-goingsurvey operation can be set up. The survey willbe able to identify accelerating movement of theslope and take measures to minimize the risk bymoving operations away from the active slide.Figure 13.1 shows an example of an open pit slopewhere careful monitoring identified the increas-ing rate of movement, which allowed the actualcollapse to be photographed. There are severalwell-documented cases of slope monitoring atopen pits where mining continued for severalmonths below the moving slope. Eventually therate of movement increased rapidly indicatingthat stability conditions were deteriorating andoperations were halted shortly before the slopefailed (Kennedy and Neimeyer, 1970; Brawneret al., 1975; Wyllie and Munn, 1979; Broadbentand Zavodni, 1982).

Monitoring may also be suitable for large land-slides that threaten facilities such as reservoirs,transportation systems and residential areas. Theweaknesses of such programs are that they mayhave to be maintained for long periods and mayinvolve sophisticated monitoring and telemetry,which will be costly. Also, it may be difficult toidentify deteriorating stability conditions that willclearly show there is a need to evacuate the site. Itis considered that where there is a significant riskto lives and property, remediation is preferred tolong-term monitoring.

Movement monitoring 321

(b)

(c)

(a)

Figure 13.1 Photograph of slopefailure in an open pit mine; failurewas predicted as the result of slopemonitoring: (a) horizontalmovement of 22.9 m prior tocollapse; (b) and (c) toppling blocka few seconds after photograph (a).(Photograph by P. F. Stacey;Brawner et al. (1975).)

322 Movement monitoring

13.2 Types of slope movement

In setting up a movement monitoring programit is useful to have an understanding of the typeof movement that is occurring. This informationcan be used to select appropriate instrumentationfor the site, and assist in interpretation of theresults. For example, if the slope were undergo-ing a toppling failure, then crack width monitorsat the crest would provide direct measurementof horizontal movement. In comparison, if aninclinometer were to be installed, it may not becertain that it extended to a depth below the zoneof movement, which would result in erroneousreadings. Furthermore, the type of movement isrelated to the failure mechanism and this inform-ation can be used to ensure that an appropriatetype of stability analysis is used. That is, out-ward and downward movement at the crest andbulging at the toe would indicate a plane or circu-lar failure, whereas horizontal movement at thecrest only would be more indicative of a topplingfailure.

The following is a discussion on common typesof slope movement, and their implications forslope stability.

13.2.1 Initial response

When a slope is first excavated or exposed, thereis a period of initial response as a result of elasticrebound, relaxation and/or dilation of the rockmass due to changes in stress induced by theexcavation (Zavodni, 2000). This initial responsewill occur most commonly in open pit mineswhere the excavation rate is relatively rapid. Incomparison, the expose of slopes by the retreatof glaciation and the gradual steepening of slopesdue to river erosion at the toe will occur overtime periods that may be orders of magnitudelonger. However, the cumulative strain of suchslopes can be considerable. Elastic rebound straintakes place without the development of a definitesliding surface, and is likely the result of dilationand shear of existing discontinuities.

Martin (1993) reports on the initial responsemeasurements of three open pit mines, which

showed that total displacement varied from150 mm in a strong massive rock mass at Palaborain South Africa to more than 500 mm in highlyfractured and altered rock at the Goldstrike Minein Nevada. The rates of movement during ini-tial response periods decreased with time andeventually showed no movement. Based on themonitoring carried out at Palabora, the followingrelationship has been established between the rateof movement V (mm/day) and the time t (days):

V = A e−bt (13.1)

where A and b are constants that are a functionof the rock mass properties, the slope height andangle, the mining rate, external influences and theultimate failure mechanism. The reported valuesof A range from 0.113 to 2.449, while values forb range from 0.0004 to 0.00294.

The critical property of the relationship shownin equation (13.1) is that the rate of movementdecreases with time, indicating that the slope isnot at risk from failure.

Another characteristic of initial response typeof movement is that it can occur within a largevolume of rock. For example, during the steep-ening of the 150 m deep Berkeley Pit from aslope angle of 45◦ to an angle of 60◦, movementmeasurements in two adits showed that reboundoccurred at a distance of up to 120 m behindthe face at the toe of the slope (Zavodni, 2000).This rebound and relaxation mechanism has beenmodeled using the FLAC and UDEC codes (ItascaGroup, MN) with the objective of predicting suchbehavior on similar pits.

13.2.2 Regressive and progressive movement

Following a period of initial response and thenpossible stability, slope “failure” would be indic-ated by the presence of tension cracks at, or nearthe crest of the slope. The development of suchcracks is evidence that the movement of the slopehas exceeded the elastic limit of the rock mass.However, it is possible that mining can safelycontinue under these conditions with the imple-mentation of a monitoring system. Eventually, an


Ultimate stability

Ultimate collapse

Regressive system

(Type I)

Progressive system

(Type II)

“Onset of failure”

Regressive phase Progressive phase

Time

Tota

l dis

plac

emen

t Curve A

Curve B

Curve C

1

Transitional system

Type I

Regressive

Type II

Progressive

Type III

Regressive/Progressive

�p Mean structure dip

�f Slope angle

�p

�p

�p

�f

�f

�f

�p< �f, �

�p>�f, �

�<�p<�f

� Angle of friction

3

2

(a) (b)

Figure 13.2 Types of slope movement: (a) typical repressive and progressive displacement curves; (b) structuralgeological conditions corresponding to types of slope movement (Broadbent and Zavodni, 1982).

“operational slope failure” may develop, whichcan be described as a condition where the rate ofdisplacement exceeds the rate at which the slidematerial can be safety mined (Call, 1982).

A means of identifying either plastic strain ofthe rock mass or operational failure is to distin-guish between regressive and progressive time–displacement curves (Figure 13.2). A regressivefailure (curve A) is one that shows short-termdecelerating displacement cycles if disturbingevents external to the slope, such as blasting orwater pressure, are removed. Conversely, a pro-gressive failure (curve B) is one that displacesat an increasing rate, with the increase in rateoften being algebraic to the point of collapse,unless stabilization measures are implemented.Correct interpretation of the curves is valuable inunderstanding the slope failure mechanism, andin predicting the future performance of the slope.

Figure 13.2 also shows geological conditionsthat are commonly associated with these types oftime–displacement curves. Where the slope con-tains discontinuities that dip out of the face, butat a shallow angle that is flatter than the fric-tion angle of these surfaces (Type I), then it isusual that some external stimuli such as blast-ing or water pressures will be required to initiatemovement. The onset of movement indicates thatthe factor of safety of the slope has dropped justbelow 1.0, but with a reduction of the externalstimuli, the factor of safety will increase and therate of movement will begin to reduce. In the caseof water pressures causing movement, the open-ing of tension cracks and dilation of the rock massmay temporarily result in the water pressuresdiminishing, but as pressures gradually build up,another cycle of movement may start. Anothercondition associated with regressive movement


is stick-slip behavior, which is related to thedifference between the static and dynamic coef-ficients of friction on rock surfaces (Jaeger andCook, 1976).

Operations can be continued below slopesexperiencing regressive movement, but it is neces-sary that the mining be conducted for shortperiods with frequent pullbacks, with care beingtaken to identify the transition to a progressivefailure (Zavodni, 2000). As shown in Figure 13.2,geological conditions that may be associated withprogressive failure are discontinuities that dip outof the face at a steeper angle than the friction angle(Type II). Also, a slide surface on which the shearstrength gradually diminishes with displacementmay experience progressive failure. The durationof the progressive stage of a failure has variedfrom 4 days to 45 days, with no obvious cor-relation between the time and the site conditions(Zavodni and Broadbent, 1980). However, morerapid failure would be expected where there is awell-defined slide surface.

As shown by curve C in Figure 13.2, a regress-ive failure may transition into a progressive fail-ure and rapidly lead to collapse. Causes of thischange in behavior can include where mining day-lights a sliding surface, break up of the rock at thetoe of the slope, an increase in water pressure,or continued mining causing the slope to acceler-ate beyond recovery. It is obviously important torecognize the onset of progressive failure, whichwill require a diligent monitoring program andcareful analysis of the results.

13.2.3 Long-term creep

In contrast to the rapid excavation, and the con-sequent large scale, relatively fast movementsthat take place in open pit mines, mountainslopes may creep over periods of hundreds ofyears. Long-term creep may occur where thereis no defined failure surface, such as a top-pling failure (Type III, Figure 13.2), or wherethe change in slope geometry is very slow, forexample, due to stress relief following glacialretreat or erosion at the toe by a river. Othercauses of such long-term movement are historical

earthquakes that each cause displacement, andclimatic changes that result in periods of highprecipitation and increased water pressures inthe slope. The Downie and Dutchman’s RidgeSlides in British Columbia, which experiencedtens of meters of ancient, downslope creep priorto reservoir filling at the base, are both examplesof long-term creep (Moore and Imrie, 1993;Moore et al., 1997).

The authors have examined several dozen land-slides in western North America where a seriesof tension cracks at the crest indicate that tensof meters of movement has occurred. In most ofthese cases, there is no evidence of recent move-ment because the rock surfaces are weathered andthere is undisturbed soil and vegetation fillingthe cracks. It is possible that very slow creep isoccurring, but no long-term monitoring programwas available to determine if this was occurring.In one case in Alaska, comparison of historicphotographs in the local museum showed no sub-stantive change in the appearance of the slopeover a period of 120 years. From these observa-tions, it has been concluded that the presence oftension cracks does not necessarily indicate thatthere is risk of imminent collapse. However, thehazard may be significant if there is evidence ofrecent movement such as disturbance to the soiland movement of blocks of rock, or there is aproposed change to the forces acting on the slope,because of excavation at the toe, for example.

13.3 Surface monitoring methods

This section describes common procedures formaking surface measurements of slope move-ment. In general, monitoring of the surface ofa slide is likely to be less costly to set up andmaintain than sub-surface measurements thatwill require drilling holes to install the instru-ments. However, surface measurements can onlybe used where the surface movement accuratelyrepresents the overall movement of the slope.For example, it would not be appropriate tomake surface measurements where loose blocksof rock on the surface were toppling and rotatingindependently of the main slide movement. Other


factors to consider in the selection of a monitor-ing system include the time available to set up theinstruments, the rate of movement and safe accessto the site.

Options for monitoring equipment includeautomatically collecting measurements at pre-setintervals on data loggers, and using telemetryto transmit these results to another location foranalysis and plotting. These systems can alsoincorporate alarms that are triggered if pre-setmovement thresholds are exceeded (Baker, 1991).An important aspect of such automated systems isthe cost of installation, and particularly mainten-ance. These costs usually limit their use to highhazard locations, and for temporary situationswhile longer-term stabilization is implemented.

13.3.1 Crack width monitors

Since tension cracks are an almost universal fea-ture of slope movement, crack width measure-ments are often a reliable and inexpensive meansof monitoring movement. Figure 13.3 shows twomethods of measuring crack widths. The simplestprocedure is to install a pair of pins on either sideof the crack and measure the distance betweenthem with a steel tape (Figure 13.3(a)). If there aretwo pins on either side of the crack, then the diag-onal distance can also be measured to check thetransverse displacement. The maximum practicaldistance between the pins is probably 2 m.

Figure 13.3(b) shows a wire extensometer thatcan be used to measure the total movement acrossa series of cracks over a distance of as muchas 20 m. The measurement station is located onstable ground beyond the cracks, and the cableextends to a pin located on the crest of the slope.The cable is tensioned by the weight, and move-ment is measured by the position of the steel blockthreaded on the cable. If the movement exceedsthe length of the steel rule, the cable can be exten-ded by moving the counterweight and resettingthe steel block to the left end of the rule. The wireextensometer can also incorporate a warning sys-tem comprising a second steel block threaded onthe cable that is set at a selected distance from atrip switch. If the movement exceeds this pre-set

Steel rule

Steel pin set in concrete

Tension crack

Measurement station on

stable ground Tensioned wire Anchor on

crest of slope Alarm

(a)

(b)

Steel rule Trip switch

Tensioningweight

Measurement block

Trip block

Detail of measurement station

Wire guide

Tensioned wire

Figure 13.3 Measurement of tension crack width:(a) measurement of distance between steel pins;(b) wire extensometer with trip switch to warn ofexcessive movement (Wyllie and Munn, 1979).

limit, the trip switch is triggered and an alarm isactivated. As movement occurs it is necessary toreset the position of the front block, with the dis-tance from the trip switch being determined by therate of movement. The selection of an appropriatedistance from the trip switch is important in orderthat the alarm provide a warning of deteriorat-ing stability conditions, while not triggering falsealarms that result in operators loosing confidencein the value of the monitoring.


The main limitations of crack width monitoringare that the upslope pin or reference point mustbe on stable ground, and that it is necessarythat people access the crest of the slide to makethe measurements. This work could be hazard-ous where the slope is moving rapidly. Theselimitations can be overcome to some degree byautomating the system using vibrating wire straingauges and data loggers to automatically read thedistance and record the measurements.

13.3.2 Surveying

On large slides where access to the slope is haz-ardous and/or there is a need to make frequentand precise measurements and rapidly analyzethe results, surveying using EDM (electronic dis-tance measurement) equipment is the most suit-able monitoring method (Vamosi and Berube,1987; ACG, 1998). There are usually three com-ponents of a survey system (Figure 13.4). First,one or several reference points are required onstable ground, but that can be viewed from theinstrument stations closer to the slide. Second,a number of instrument stations are set up onreasonably stable ground at locations from whichthe slide is visible. If the co-ordinate positionsof the movement stations are to be measured, thenthe instrument stations should be arranged suchthat they form an approximately equilateral tri-angle. Third, a series of stations are set up on,and possibly just outside the slide area, which arethen located relative to the instrument stations.It is preferable that the measurement directionbe in the likely direction of movement so thatthe distance readings approximate the actual slidemovement. For example, in Figure 13.4(a), it ispreferable to measure stations on the north por-tion of the slide from instrument station 1, andthose on the south side from instrument station 2.The stations on the slide can be reflectors used onheavy equipment, or survey prisms, depending onthe sight distance and the accuracy required.

The survey arrangement shown in Figure 13.4can be used to measure at any desired frequencyor level of accuracy. For example, for slowmoving slides the readings may be made every

N

500

100200300400

PitBottom

400

Reference station

Movement stationson slide

Limit of movement

Haul roadInstrument stations

Tension crack

Reflector prism

Verticalangle

EDM

Slope failure

Slope distance

(a)

(b)

Figure 13.4 Survey system to remotely measure slopemovement: (a) typical arrangement of reference,instrument and monitoring stations; (b) measurementof vertical angle and distance to determine verticaldisplacement (Wyllie and Munn, 1979).

few weeks or months, while for a rapidly movingslide above an active mining operation, an auto-mated system can be set up that takes a seriesof readings at pre-set intervals, and records andplots the results. Also, quick checks of stabil-ity can be made by making distance measure-ments only, and these can be followed up withtriangulation measurements to determine the co-ordinates of each station at less frequent intervals.Figure 13.4(b) shows that measurement of thevertical angle and the slope distance allows thevertical displacement to be measured, which is ofvalue in determining the mechanism of failure (seeSection 13.5).


13.3.3 Laser imaging

An advance on monitoring individual stationson a slide is laser imaging that involves mak-ing a precise three-dimensional map of the entireface (Spacial Data Services, 2002). The systeminvolves directing the laser at the slope, select-ing the area to be scanned and the density ofthe scan, and then the laser rapidly and auto-matically makes a large number of closely spacedscans to cover the area. The result is a dense,accurate three-dimensional point cloud that canbe processed to produce contour map. By makinga series of such maps over time from the samereference point, the position of the face can becompared between each scan, and the locationand magnitude of the movement measured.

13.3.4 Tiltmeters

It is possible to measure the tilt of a feature to aresolution of about 10 arc seconds using a tilt-meter. These measurements involve bolting orgluing a base plate to the rock face on whichthe tiltmeter is precisely mounted. The instrumentcan either be permanently mounted on the face sothat readings can be made at any time, or can belocated on the mounting plate just when readingsare being made.

The advantage of tiltmeters is that rapid andprecise measurements can be made of the tilt,from which an assumed movement can be calcu-lated. The disadvantages are that the instrumentis costly, and it may be difficult to find a smallportion of the rock face, the movement of whichis representative of the slope movement. It is con-sidered that the primary application of tiltmetersis on structures such as dams and retaining wallsrather than rock slopes.

13.3.5 Global positioning system

The global positioning system (GPS) may be asuitable method of monitoring slope movementwhere the slide covers a large area and extremeprecision is not required. Stations can be set upon the slide and their co-ordinates measured atany desired frequency with the GPS unit. Greater

accuracy can be achieved by setting up a basestation on stable ground outside the slide area,and accurately determining its position. The GPSreadings on the slide are then referenced to theco-ordinates of the base station (differential GPS).

The advantages of GPS monitoring are the lowcost and ease of set up, but the disadvantages arethat the accuracy is limited by the built-in errorfor civilian users, and in steep terrain there maybe an insufficient number of satellites visible toobtain readings.

13.3.6 Synthetic aperture radar

A technique for precise monitoring of movementover large areas is to use radar satellite remotesensing techniques. This technique is known asInterferometric Synthetic Aperture Radar (SAR)and involves capturing a radar image of theground surface, which is then compared withimages taken at a different time to obtain the rel-ative ground movement. Significant features ofthis technique are that the image can cover an areaa large as 2500 km2, relative movements can bemeasured in the range of 5–25 mm, and the meas-urements are independent of the weather, cloudcover and daylight (www.terrainsar.com, 2002).

These attributes mean that SAR is ideally suitedto precise movement monitoring of large areasover long time periods, with no need to set upreference points on the ground. However, somelimitations of the technique are that the frequencyof the measurements are governed by the intervalbetween satellite orbits over the site, whichpresently (2003) is about once every 24–35 days,and the processing of the data can take another35–40 days. Also, movement is most accuratelymonitored in the vertical direction.

13.4 Sub-surface monitoring methods

Sub-surface measurement of slope movement isoften a useful component of a monitoring pro-gram in order to provide a more complete pictureof the slope behavior. In cases where surface mon-itoring is not feasible, then sub-surface measure-ments will be the only measurements available.


The main purpose of these measurements is tolocate the slide surface or surfaces, and monitorthe rate of movement; in some cases the holes areused for monitoring both movement and waterpressures.

13.4.1 Borehole probes

One of the simplest sub-surface monitoringmethods is the borehole probe comprising alength of reinforcing steel about 2 m long that islowered down the drill hole on a length of rope. Ifthe hole intersects a moving slide plane, the holewill be displaced at this level and it will no longerbe possible to pull the bar past this point. Simil-arly, a probe can be lowered down the hole, andin this way both the top and bottom of the slideplane can be located.

The advantages of the probe are the lowcost and simplicity, but it will provide littleinformation on the rate of movement.

13.4.2 Time–domain reflectometry

Time–domain reflectometry is another means oflocating a sliding surface, which can also mon-itor the rate of movement (Kane and Beck, 1996).This method involves grouting into a borehole aco-axial cable comprising inner and outer metallicconductors separated by an insulating material.When a voltage pulse waveform is sent down thecable, it will be reflected at any point where thereis a change in the distance between the conduct-ors. The reflection occurs because the change indistance alters the characteristic impedance of thecable. Movement of a sliding plane that causesa crimp or kink in the cable will be sufficient tochange the impedance, and the instrumentationcan detect the location of the movement.

The primary advantages of time–domain reflec-tometry are that the cable is inexpensive so that itcan be sacrificed in a rapidly moving slide. Also,the readings can be made in a few minutes froma remote location either by extending the cableto a safe location off the slide, or by telemetry.The ability to make remote readings can achievesignificant savings compared to inclinometers (see

Section 13.4.3) because of the reduced travel time,and the readout box directly shows the move-ment without the need to download and plot theresults.

13.4.3 Inclinometers

Inclinometers are instruments ideally suited tolong-term, precise monitoring of the position ofa borehole over its entire length. By making aseries of readings over time, it is also possibleto monitor the rate of movement. The compon-ents of the inclinometer are a plastic casing withfour longitudinal grooves cut in the inside wall,and a probe that is lowered down the casing onan electrical cable with graduated depth mark-ings (Figure 13.5(a)). The probe contains twoaccelerometers, aligned so that they measure thetilt of the probe in two mutually perpendiculardirections. The probe is also equipped with a pairof wheels that run in the grooves in the casing andmaintain the rotational stability of the probe.

The first requirement of accurate monitoring isto extend the borehole below the depth of move-ment so that readings made from the end of thehole are referenced to a stable base. Precautionsare also needed during installation of the casing tomaintain the vertical alignment of the grooves andprevent spiraling. Readings are made by loweringa probe to the end of the hole and then raising itin increments equal to the length of the wheelbaseL of the probe. At each depth increment the tilt ψ

is measured. Figure 13.5(b) shows the procedurefor calculating the displacement (L sin ψ) for eachincrement, and the total displacement at the top ofthe hole �(L sin ψ). A check of the results is usu-ally made by rotating the probe by 180◦ and takinga second set of readings. Another precaution is toallow time during the readings for the probe toreach temperature equilibrium in the hole.

13.5 Data interpretation

Interpretation of the movement data is anessential part of monitoring operation in orderto identify quickly acceleration or decelera-tion of the slope that indicates deteriorating or


Readout unit

Graduated electrical cable

Borehole

Probe containing gravity- sensing transducer

Coupling

Guide casing

Backfill

Probe

Guide wheels

Guide casing

Actual alignment of guide casing (exaggerated)

Distance betweensuccessive readings, L

True vertical

∑L sin�

L sin�

�

(a) (b)

Figure 13.5 Inclinometer for measuring borehole deflection: (a) arrangement of grooved casing andinclinometer probe; (b) principle of calculating deflection from tilt measurement (Dunnicliff, 1993).

improving stability conditions respectively. Thisallows appropriate action to be taken with respectto the safety and economics of the operation.The importance of updating the plots as thereadings are made is that failures have occurredwithin days of cracks first being observed (Stacey,1996). Unfortunately, there are instances of care-ful monitoring measurements being recorded, butbecause they were not plotted, acceleration of theslope that was a clear precursor to failure was notrecognized.

This section describes a number of proced-ures for interpreting monitoring results to provideinformation on both stability conditions and themechanism of failure. With respect to the mech-anism of failure, this information is useful indesigning stabilization measures, which requirethat the appropriate method of analysis be used,such as plane or toppling.

13.5.1 Time–movement and time–velocityplots

The monitoring program, whether carried outby surveying, crack width gauges, GPS, satellitescans or inclinometers will provide readings ofmovement against time. Plots of this data arefundamental to understanding the mechanism ofthe slope movement, and possibly predicting thetime of failure. The following is a discussion ofa monitoring program in an open pit coal minewhere a movement monitoring program was usedto safely mine below a moving slope for most ofthe life of the pit (Wyllie and Munn, 1979).

Figure 13.6(a) shows a cross-section of the finalpit and the slope above the pit, as well as the litho-logy and geological structure of this slope. Soonafter mining commenced at the 1870 m elevationin March 1974, a toppling failure was initiated


Tension cracks

Slide surfaces

Failure on pit crest

TalusSandstone

Siltstone

Fault

2050

2000

1950

1900

1850

1800

1750

Ele

vatio

n (m

)

0 100 m

1.2

1.0

0.8

0.6

0.4

26 5 10 15 20

Shovel operating

at toe

Slo

pe m

ovem

ent (

m)

April March 1975Date

30

6

4

2

0J J A S O J

1975 1976

Time (months)

Failure

Acceleration

Cum

ulat

ive

slop

e m

ovem

ent (

m)

0.50.40.3

0.2

0.1

0.05

0.02

0.01

10 20 30 40 50 60Time (days)

Slo

pe v

eloc

ity (

m/d

)

N D J F M A M

Failure

Detail

(a)

(c)

(d)

(b)

Figure 13.6 Movement monitoring at open pit coal mine: (a) cross-section of pit and hillside showing geologyand extent of slope failure; (b) regressive slope movement when mining at 1840 m level; (c) slope movementover 13 months leading to slope failure; (d) slope velocity over two months prior to failure (Wyllie andMunn, 1979).


in the overturned siltstone beds at the crest of thepit. When mining on the 1840 m bench, a series ofcracks formed on the 1860 m bench and a move-ment monitoring program was set up in February1975 using a combination of wire extensometers(Figure 13.3(b)) and surveying of prisms. Thismonitoring system was used to control the min-ing operation with the objective of mining backto the final wall so that the coal could eventuallybe mined to the bottom of the pit. Figure 13.6(b)shows the sensitivity of the slope movement tomining at the toe of the toppling slope, and typ-ical regressive behavior as soon as the shovel waspulled back. This experience was used to establishthe criterion, based on hourly movement read-ings, that mining would be halted as soon asthe rate of movement reached 25 mm per hour.When this rate reduced to 15 mm per day over aperiod of about 10 days, mining recommenced.Using this control procedure, mining contin-ued towards the final depth of about 1700 mwith the slope moving at an average rate of6 mm per day.

In April 1976, the slope started to accelerate,and over the next two months total movementof about 30 m occurred on the hillside abovethe pit and the maximum velocity reached nearlya meter per day (Figures 13.6(c) and (d)). Theacceleration on the slope movement plots gavean adequate warning of deterioration stabilityconditions and mining was abandoned. The areawith the greatest movement was on the topplingbeds along the crest of the pit, and in early June1976 two separate slope failures occurred witha total volume of 570,000 m3. After the failures,the monitoring system was re-established whichshowed that the rate of movement was gradu-ally decreasing, and after a month it was decidedto restart mining at the bottom of the pit. Thisdecision was also based on borehole probe meas-urements, which showed that the circular slidesurfaces, associated with the toppling at the pitcrest, daylighted in the upper part of the pit slope(Figure 13.6(a)). Therefore, mining at the base ofthe pit would have little effect on stability.

The type of movement monitoring datashown in Figures 13.6(c) and (d) has been

analyzed to help predict the time of failureonce the progressive stage of slope movementhas developed (Zavodni and Broadbent, 1980).Figure 13.7 shows the semi-log time–velocity plotin feet per day preceding the slope failure at theLiberty Pit. On this plot it is possible to identifythe velocities at the start V0 and the mid-point Vmpof progressive stage of movement. A constant K

is defined as

K = Vmp

V0(13.2)

Study of six carefully documented slope fail-ures shows that the average value of K is −7.21,with a standard deviation of 2.11. For example,Figure 13.6(d) shows a K value of about −7(−0.07/0.01).

The general equation for a semi-log straight linegraph has the form

V = C eSt (13.3)

where V is the velocity, C is the intercept of theline on the time axis, e is the base of the nat-ural logarithm, S is the slope of the line and t isthe time. Therefore, the velocity at any time isgiven by

V = V0 eSt (13.4)

Combining equations (13.2) and (13.3) givesthe following relationship for the velocity atcollapse Vcol:

Vcol = K2 V0 (13.5)

The use of equation (13.4) in conjunction witha time–velocity plot allows an estimation to bemade of the time of collapse. For example,from Figure 13.6(d) where K = −7 and V0 =0.01 m/day, the value of Vcol is 0.49 m/day. Extra-polation of the velocity–time line shows that thisrate will occur at about 61 days, which is veryclose to the actual day of collapse.


Predicted collapse dateusing K = 7.21

Predicted (Vcol)using K = 7.21

5.0

2.0

1.0

0.5

0.2

0.1

0.05

0.02

0.01

0.005

0.002

0.00110 20 30 40 50 60 70 80 90 100 110 120

Days prior to collapse

Dis

plac

emen

t rat

e (f

oot/d

ay)

Velocity mid- point progressive stage (Vmp)

Onset of failure point (V0)

Progressive stage Regressive stageFigure 13.7 Liberty Pittransitional systemdisplacement rate curveand failure prediction(Zavodni and Broadbent,1980).

It is likely that the rate at which collapse occursdepends to some degree on the geological condi-tions on the sliding plane. For example, failure islikely to occur faster if sliding is occurring on awell-defined structural feature such as a fault thatdaylights on the face, compared to that shown inFigure 13.6(a) where the slide plane developed byfracture through intact rock. The reason for thisis that in the case of a fault, the shear strengthwill diminish from peak to residual with relat-ively little movement compared to that of failurethrough the rock mass.

13.5.2 Slope failure mechanisms

Figure 13.8 shows a number of methods of ana-lyzing movement monitoring results that mayhelp to identify the mechanism of slope failure.This information can be useful in applying theappropriate method of stability analysis and inthe design of stabilization measures.

Figure 13.8(a) shows a combined displacementand velocity plot which shows that the accelera-tion of the slope stopped after day 5. This changein behavior is clearly evident on the velocity plot

where the velocity is constant after day 5. Incomparison, on the movement plot the change ingradient is not so obvious. This slope movementwould be typical of regressive type instability.

Figure 13.8(b) shows the magnitude and dips ofmovement vectors for survey stations on the crest,mid-height and toe of the slide. The dip anglesapproximately equal the dip of the underlyingfailure surface, and in this instance indicate that acircular failure is taking place in which the slidingsurface is steep near the crest and near-horizontalnear the base. This information would also showthe location of the toe of the slide, which may notbe toe of the slope, as was the case for the slideshown in Figure 13.6(a). Figure 13.8(c) showsa movement vector for a typical toppling failurein which the stations located on the overturningbeds at the crest may move upwards by smallamount, while there is little movement belowthe crest.

Figure 13.8(d) shows contours of slope velocityplotted on a plane of the pit. These plots showboth the extent of the slide, and the area(s) ofmost rapid movement. Such plots, when regularlyupdated, can be useful in identifying an increase


8

6

4

2

0 1 2 3 4 5 6 7 8 9

2.0

1.0

Time (days)

AccelerationConstant velocity

Cum

ulat

ive

slop

e m

ovem

ent (

m)

Slo

pe v

eloc

ity

(m/day)

N

500

100200300400

Pit bottom

400

Haul road

Contours of slopemovement (interval0.01 m/day)

Movement vector Vertical

angle

Instrument station

Failure surface

��

�

�

0

Toppling Movement vector

(a) (b)

(d)(c)

Figure 13.8 Interpretation of movement monitoring data: (a) displacement and velocity plots show onset ofregressive movement; (b) movement vectors showing circular failure mechanism; (c) movement vectors showingtoppling failure mechanism; (d) slope velocity contours show extent of slope movement (Wyllie andMunn, 1979).

in the dimensions of the slide, and/or a change inthe most rapidly moving area. This informationwill assist in the planning of mining operations,for example, to complete mining in the south-east

corner of the pit before failure occurs. Also, if thehaul road is on a relatively slowly moving partof the slide, this may allow time to develop newaccess to the pit.

Chapter 14

Civil engineering applications

14.1 Introduction

When a slope above an important civil engineer-ing structure is found to be unstable, anurgent decision is commonly required on effectiveand economical remedial measures. Evidence ofpotential instability includes open tension cracksbehind the crest, movement at the toe, failuresof limited extent in part of the slope, or failureof an adjacent slope in similar geology. Whateverthe cause, once doubt has been cast upon the sta-bility of an important slope, it is essential thatits overall stability should be investigated and, ifnecessary, appropriate remedial measures imple-mented. This chapter describes five rock slopes oncivil engineering projects in a variety of geologicaland climatic conditions, and the slope stabiliza-tion measures that were implemented. For eachexample, information is provided on the geo-logy, rock strength and ground water conditions,as well as the stability analysis, design of theremedial work and construction issues.

The purpose of these case studies is to describethe application of the investigation and designtechniques described in previous chapters ofthe book.

14.2 Case Study I—Hong Kong: choice ofremedial measures for plane failure

14.2.1 Site description

A 60 m high cut had an overall face angle of50◦, made up of three, 20 m high benches withface angles of 70◦ (Figure 14.1). A small slide

Hc= 60 m

Z = 18.8 m

�b= 70°

Hb= 20 m

�p = 35°�f = 50°

Figure 14.1 Geometry assumed for two-dimensionalplane failure analysis of the slope in Case Study I.

in a nearby slope has caused attention to befocused on this particular cut and concern hasbeen expressed that a major slide could occurresulting in serious damage to an important civilengineering structure at the foot of the cut. Anassessment was required of the short- and thelong-term stability of the cut, and recommend-ations for appropriate remedial measures, shouldthese prove necessary. No previous geological orengineering studies had been carried out on thiscut, and no boreholes were known to exist inthe area. The site was in an area of high rainfallintensity and low seismicity. A horizontal seis-mic coefficient, aH of 0.08g had been suggestedas the maximum to which this cut was likely tobe subjected.

14.2.2 Geology

The cut was in slightly weathered granitecontaining several sets of steeply dipping joints,

Civil engineering applications 335

Table 14.1 Orientation of slope and joint sets shownin Figures 14.1 and 14.2

Feature Dip (◦) Dip direction (◦)

Overall slope face 50 200Individual benches 70 200Sheet joint 35 190Joint set J1 80 233Joint set J2 80 040Joint set J3 70 325

as well as sheet jointing that dipped at 35◦ andformed the natural slopes in the area. Faced withthis problem and having no geological or engin-eering data from which to work, the first task wasto obtain a representative sample of structuralgeology to establish the most likely failure mode.Time would not allow a drilling program to bemounted. Consequently, the collection of struc-tural data had to be based upon surface mapping,which was reasonable because of the extens-ive rock exposure in the cut face and naturalslopes.

The structural mapping identified the geomet-rical and structural geology features listed inTable 14.1.

14.2.3 Rock shear strength

Because no information was available on theshear strength of the sheet joints forming thepotential sliding surface, the strength valuesused in design were estimated from previousexperience of the stability of slopes in granite.Figure 4.21 is a summary of shear strength valuesdeveloped primarily from back analysis of slopefailures; point ‘11’ most closely represented thestrength of the rock at the site. Based on thisexperience, it was considered that even heavilykaolinized granites exhibit friction values in therange of 35–45◦ because of the angular nature ofthe mineral grains. The cohesion of these surfaceswas likely to be variable depending on the degreeof weathering of the surface and the persistenceof the joints; a cohesion range of 50–200 kPa wasselected.

14.2.4 Ground water

There were no boreholes in the slope so the sub-surface ground water conditions were unknown.However, since the site was in an area that exper-ienced periods of intense rainfall, it was expectedthat significant transient ground water pressureswould develop in the slope following these events.

14.2.5 Stability analysis

The stereoplot of the data in Table 14.1 is shownin Figure 14.2, including a friction circle of 35◦.Note that, although the three joint sets provideda number of steep release surfaces, which wouldallow blocks to separate from the rock mass, noneof their lines of intersection, which are circled inFigure 14.2, fall within the zone designated aspotentially unstable (refer to Figure 7.3(b)). Onthe other hand, the great circle representing thesheet joints passes through the zone of potentialinstability. Furthermore, the dip direction of thesheet joints is close to that of the cut face, sothe most likely failure mode was a plane slideon the sheet joints in the direction indicated inFigure 14.2.

N

J3

J2

J1

Direction of potential slide

Sheet joint

Friction circle �= 35°

Overall slope face

Unstable slope

Unstable benches

Benches

Figure 14.2 Stereoplot of geometric and geologicaldata for slope shown in Figure 14.1.

336 Civil engineering applications

�p

�p

�f

H

H

bTension crack

Assumed waterpressure distribution

Assumed groundwater table

Slide planeU

U

V

W

W

kH W

kH W

z Model I

Model II

zw

HW

F = cA +

F =

where

where U = �w Hw2 cosec�p

Hw

[W (cos�p– kH sin�p) – U – V sin�p] tan�

W (sin�p+ kH cos�p) + V cos�p

cA + [W (cos�p–kH sin�p) – U ] tan�

W (sin�p+ kH cos�p)

Z = H [1 – (cot�p tan�p)1/2] A = (H – z) cosec�p

W = � H 2 [(1 – (z/H)2 cot�p – cot�f] U = �w zw A V = �w zw

2

1 2

1 21 2

1 2

1 4

Figure 14.3 Theoretical models of plane slope failures for Case Study I.

The stability check carried out in Figure 14.2suggested that both the overall cut and the indi-vidual benches were potentially unstable, andit was therefore clearly necessary to carry outfurther analysis of both.

The two steeply dipping joint sets J1 andJ2 were oriented approximately parallel to theslope face, and there was a strong possibility

of a tension crack forming on these discontinu-ities behind the crest of the cut. One possiblefailure mode was that illustrated as Model I inFigure 14.3; this theoretical model assumed thata tension crack occurred in the dry state in themost critical position (refer to Figure 6.6), andthat this crack filled to depth zw with waterduring a period of exceptionally heavy rain.


A simultaneous earthquake subjected the slope toground motion that was simulated with a hori-zontal seismic coefficient kH of 0.08, generatinga force of kHW , where W was the weight of thesliding block. The factor of safety of this slopewith the inclusion of a pseudo-static horizontalearthquake loading is given by the equations inFigure 14.3 (refer to Section 6.5.4, pseudo-staticstability analysis).

To allow for the possible presence of sub-stantial sub-surface water, an alternative theor-etical model was proposed. This is illustrated asModel II in Figure 14.3 and, again this modelincludes the pseudo-static earthquake loading.

Having decided upon the most likely failuremode and having proposed one or more the-oretical models to represent this failure mode,a range of possible slope parameter values wassubstituted into the factor of safety equations todetermine the sensitivity of the slope to the dif-ferent conditions to which it was likely to besubjected. Table 14.2 summarizes the input data.The factors of safety of the slopes were calculatedby substituting these values into the equations onFigure 14.3 as follows:

Overall Cut Model I

FS = 80.2c + (18,143 − 393zw − 2.81zw2) tan φ

14,995 + 4.02zw2

Table 14.2 Input data for Case Study I plane stabilityanalysis

Parameter Parameter value

Cut height Hc = 60 mOverall slope angle ψf = 50◦Bench face angle ψb = 70◦Bench height Hb = 20 mFailure plane angle ψp = 35◦Distance to tension crack(slope)

bs = 15.4 m

Distance to tension crack(bench)

bb = 2.8 m

Rock density γr = 25.5 kN/m3

Water density γw = 9.81 kN/m3

Seismic coefficient kH = 0.08

Overall Cut Model II

FS = 104.6c + (20,907 − 4.28Hw2) tan φ

17,279

Individual benches Model I

FS = 17.6c + (2815 − 86.3zw − 2.81zw2) tan φ

2327 + 4.02zw2

Individual benches Model II

FS = 34.9c + (4197 − 4.28Hw2) tan φ

3469

One of the most useful studies of the factor ofsafety equations was to find the shear strengthwhich would have to be mobilized for failure(i.e. FS = 1.0). These analyses examined the over-all cut and the individual benches for a range ofwater pressures. Figure 14.4 gives the results ofthe study and the numbered curves on this plotrepresent the following conditions:

Curve 1 Overall Cut, Model I, dry, zW = 0.Curve 2 Overall Cut, Model I, saturated,

zw = z = 14 m.Curve 3 Overall Cut, Model II, dry, Hw = 0.Curve 4 Overall Cut, Model II, saturated,

Hw = 60 m.Curve 5 Individual bench, Model I, dry, zw = 0.Curve 6 Individual bench, Model II, saturated,

zw = z = 9.9 m.Curve 7 Individual bench, Model II, dry,

Hw = 0.Curve 8 Individual bench, Model II, saturated,

Hw = H = 20 m.

The reader may feel that a consideration of allthese possibilities is unnecessary, but it is onlycoincidental that, because of the geometry of thisparticular cut, the shear strength values foundhappen to fall reasonably close together. In othercases, one of the conditions may be very muchmore critical than the others, and it would takeconsiderable experience to detect this condition

338 Civil engineering applicationsC

ohes

ion

c (k

Pa)

Friction angle � (degrees)

Range of shear strengthsconsidered reasonable forpartially weathered granite(see Figure 4.21)

87

3

4

1

2

6

5

0

50

100

150

200

250

0 5020 30 4010Figure 14.4 Shear strength mobilized forfailure of slope considered in CaseStudy I (Hoek and Bray, 1977).

without going through the calculations requiredto produce Figure 14.4.

The elliptical area in Figure 14.4 surrounds therange of shear strengths considered reasonablefor partially weathered granite. As discussed inSection 14.2.3, these values are based on the plotgiven in Figure 4.21. Figure 14.4 shows that whenthe cut is fully saturated and subject to earthquakeloading (lines 2, 4 and 6), the likely availableshear strength along the sliding surfaces wouldbe exceeded by the driving forces acting on thesliding surface, and failure would be possible.Considering the rate of weathering of granite intropical environments over the operational life ofthe slope, with a consequent reduction in avail-able cohesive strength, these results indicated thatthe cut was unsafe and that the steps should betaken to increase its stability.

Four basic methods for improving the stabilityof the cut were considered:

(a) Reduction of cut height;(b) Reduction of cut face angle;

(c) Drainage; and(d) Reinforcement with tensioned anchors.

In order to compare the effectiveness of these dif-ferent methods, it was assumed that the sheet jointsurface had a cohesive strength of 100 kPa and afriction angle of 35◦. The increase in factor ofsafety for a reduction in slope height, slope angleand water level was found by altering one of thevariables at a time in the equations on Figure 14.3.The influence of reinforcing the cut was obtainedby modifying these equations to include a boltingforce as shown in equation (6.22).

Model I: FS

= cA + (W(cos ψp − kH sin ψp) − U − V sin ψp + T sin(ψT + ψP)) tan φ

W(sin ψp + kH cos ψp) + V cos ψp − T cos(ψT + ψP)

(14.1)Model II: FS

= cA + (W(cos ψp − kH sin ψp) − U + T sin(ψT + ψP)) tan φ

W(sin ψp + kH cos ψp) − T cos(ψT + ψP)

(14.2)

where T is total reinforcing force (kN/m) appliedby the anchors, and ψT is the plunge, or


Tensionedanchor

T

�P= 35°

Model II

Model IRei

nfor

cing

forc

e T

(kN

/m)

Angle (�T + �P ) (degrees)

�T

15,000

0

5000

10,000

90 50 30 1070

(a)

(b)

Figure 14.5 Slope stabilization with tensioned rockanchors: (a) orientation of tensioned anchors; (b) totalreinforcing force required for a factor of safety of 1.5.

inclination of this force below the horizontal(Figure 14.5(a)).

Figure 14.6 compares the different methodsthat were considered for increasing the stabilityof the overall cut. In each case, the change isexpressed as a percentage of the total range ofeach variable: H = 60 m, ψf = 50◦, zw/z = 1,Hw = 60 m. However, the variation of the rein-forcing force is expressed as a percentage of theweight of the wedge of rock being supported. Incalculating the effect of the reinforcement, it wasassumed that the anchors are installed horizont-ally, that is ψT = 0◦. The influence of the anchor

inclination ψT on the reinforcing load requiredto produce a factor of safety of 1.5 is shownin Figure 14.5(b). This shows that the requiredbolting force can be approximately halved byinstalling the bolts horizontally (ψT = 0◦, orψT + ψp = 35◦), rather than normal to theplane (ψT = 55◦, or ψT + ψp = 90◦). As dis-cussed in Section 6.4.1, the generally optimumangle for tensioned rock anchors is given byequation (6.23). In practice, cement groutedanchors are installed at about 10–15◦ below thehorizontal to facilitate grouting.

14.2.6 Stabilization options

The following is a discussion on the stabilizationoptions analyzed in Figure 14.6.

Reduce height. Curves 1 and 2 show thatreduction in cut height is not an effective solutionto the problem. In order to achieve the requiredfactor of safety of 1.5, the slope height would haveto be reduced by 50%. If this solution were to beadopted, it would be more practical to excavatethe entire slope since most of the volume of rockto be excavated is contained in the upper half ofthe slope.

Reduce face angle. Reducing the angle of thecut face would be very effective stabilizationmeasure as shown by line 3. The required factorof safety of 1.5 is achieved for a reduction of lessthan 25% of the slope angle. That is, the slopeangle should be reduced from 50◦ to 37.5◦. Thisfinding is generally true and a reduction in the faceangle is often an effective remedial measure. In thecase of slopes under construction, using a flatterslope is always a prime choice for improving sta-bility. However, a practical consideration is thedifficulty of excavating a “sliver” cut because ofthe limited access for equipment on the narrow,lower part of the cut.

Curve 4 (reduction of face angle for slopewithout tension crack) is an anomaly and demon-strates that calculations can sometimes produceunrealistic results. The reduction in factor ofsafety shown by this curve is a result of thereduction in the weight of the sliding block as the

340 Civil engineering applicationsFa

ctor

of s

afet

y

Percentage reduction in slope height, slope angle andwater level, and percentage of weight of wedge applied asreinforcing force.

Critical slopes

Required factorof safety

873

4

1

2 6

5

0

0.5

1.0

1.5

2.0

2.5

3.0

0 10040 60 8020

1—reduction in slope height, model I2—reduction in slope height, model II3—reduction in slope angle, model I4—reduction in slope angle, model II5—reduction in water level, model I6—reduction in water level, model II7—reinforcement of slope, model I8—reinforcement of slope, model II

Figure 14.6 Comparison betweenalternative methods of increasingstability of overall slope consideredin Case Study I.

face angle is reduced. Since the water pressure onthe sliding surface remains constant, the effect-ive stress acting on the sliding surface decreasesand hence the frictional component of the resist-ing force decreases. When a very thin sliver ofrock remains, the water pressure will “float” itoff the slope. The problem with this analysis liesin the assumption that the block is completelyimpermeable and that the water remains trappedbeneath the sliding surface. In fact, the blockwould break up long before it floated and hencethe water pressure acting on the sliding surfacewould be dissipated.

Drainage. Curves 5 and 6 show that drainagewould not be an effective stabilization option for

both slope models. In neither case is a factor ofsafety of 1.5 achieved. This is something of a sur-prise since drainage is usually one of the mostcost-effective drainage measures. The reasons forthe poor performance of drainage in this case aredue to the combination of the slope geometry andthe shear strength of the failure surface.

Anchoring. Curves 7 and 8 show that reinfor-cing the cut by means of tensioned anchors with aforce equal to 5000 kN per meter of slope lengthwould achieve a factor of safety of 1.5, assumingthe anchors are installed just below the hori-zontal. In other words, reinforcement of a 100 mlength of slope would require the installation of500 anchors, each with a capacity of 1 MN.


The two most attractive options for long-termremediation were reinforcement using tensionedcables or bar anchors, or reduction of the slopeface angle. Reinforcement was rejected becauseof the high cost, and the uncertainty of long-term corrosion resistance of the steel anchors. Theoption finally selected was to reduce the face angleto 35◦ by excavating the entire block down tothe sheet joints forming the sliding surface. Thiseffectively removed the problem. Since good qual-ity aggregate is always needed in Hong Kong, itwas decided to work the slope as a quarry. It tookseveral years to organize this activity and duringthis time, the water levels in the slope were mon-itored with piezometers. Although the road wasclosed twice during this period, no major prob-lems occurred and the slope was finally excavatedback to the sliding plane.

14.3 Case Study II—Cable anchoring ofplane failure


A potential rock fall hazard had developed on a38 m high, near-vertical rock face located abovea two-lane highway (Figure 14.7). A stabilizationprogram was required that could be implemen-ted with minimal interruption to traffic. Therock falls could vary from substantial failureswith volumes of several hundred cubic metersformed by widely spaced, persistent joints, tofalls of crushed and fractured rock with dimen-sions of tens of centimeters. The rock falls werea hazard to traffic because of the curved align-ment and limited sight distance, together with the2 m wide ditch between the toe of the rock cut

Highway

38 m

Tension crack

Tension crack

Sheet joint J1

Joint J3

Tension crack

J2

Figure 14.7 Cross-section of the slope inCase Study II showing movement alongsheet joints and location of tensioncracks.


and the highway shoulder with limited rock fallcontainment capacity.

The evidence of instability was a series of ten-sion cracks that had opened as the result ofdown-slope movement on the sheet joints. Themovement had created a layer of crushed rock oneach of the sheet joints.

Cross-sections of the slope were developedusing a reflectorless EDM (electronic distancemeasuring), and surface mapping involving rap-pelling down the face was used to collect struc-tural geology data. The positions of the criticaljoints were marked with reference points paintedon the face, the co-ordinates of which were pickedup by the survey. This information was used todetermine accurately the shape and dimensions ofthe major blocks, and the positions of major slid-ing planes. It was decided that diamond drillingwould not be required to obtain geological databecause there was excellent exposure of the rockin the face.

14.3.2 Geology

The cut had been excavated in a very strong,coarse-grained, fresh, blocky granite containingfour sets of joints that were generally orthogon-ally oriented. The predominant structural geologyfeatures were two sets of joints dipping out of theface—sheet joints dipping at an angle of about20–30◦, and a second set (J2) dipping at 45–65◦.A third set (J3) dipped steeply into the face, anda fourth set (J4) also steeply dipping, was ori-ented at right angles to the face (Figure 14.8).Figures 14.8(a) and (b) show respectively the con-toured and great circle stereo plots of the surfacemapping data. The sheet joints had continuouslengths of several tens of meters and the spacingsof all the joints were in the range of 5–8 m.

The cross-section of the slope (Figure 14.7)shows the shape and dimensions of the blocksformed by the joint sets. The blocks were slid-ing on the sheet joints with joint set J3 forminga series of tension cracks behind the face, whileset J4 formed release surfaces at the sides of theblocks. The mapping located tension cracks thatwere up to 150 mm wide, as well as sheared rock

on the sheeting joints at the base of the each majorblock. The substantial volumes of the blocks werethe result of the persistence and wide spacing ofthe discontinuities. Since the dip directions of theslope face and the sheet joints were within 20◦ ofeach other, this geometry formed a plane failureas illustrated in Figures 2.16 and 2.18.


The blocks shown on Figure 14.7 were slidingon the continuous sheet joints because the rockwas very strong and there is no possibility of frac-ture taking place through intact rock. Therefore,stability of the blocks was partially dependenton the shear strength of the sheet joints, whichwere smooth, planar or slightly undulating, andthere was generally no infilling apart from slightweathering of the surfaces. For these conditionswhere the joints had no infilling, the cohesionwas zero and the shear strength comprised onlyfriction.

The total friction angle, comprising the fric-tion (φr) of the granite and surface roughness(i) of the sheet joints, was determined as fol-lows. Block samples of rock containing open, butundisturbed, sheet joints were collected at thesite and 100 mm square test pieces were cut fromthe block. Direct shear tests were then conduc-ted to determine the friction angle of the granite;each sample was tested four times using normalstresses σ of 150, 300, 400 and 600 kPa (refer-ence Figure 4.16). These normal stress test valueswere based on the likely stress acting on the lowersheet joint, calculated from

σ = γrH cos ψp (14.3)

For a rock unit weight (γr) of 26 kN/m3, a rockthickness (H) of 25 m on a plane dipping (ψp) at25◦, the normal stress equals 600 kPa.

The direct shear tests showed that the peakfriction angle (φpeak) for the initial test at σ =150 kPa was 47◦, but at higher stresses thisdiminished as the result of shearing on the slid-ing surface to a residual angle (φresidual) of 36◦(Figure 14.9). Because of the displacement that


N

S

EW

Orientations

J4

J3

J2

Face

�= 36°

Sheetjoint

N

S

EW

Fisherconcentrations

% of total per 1.0% area


123 poles124 entries


No bias correctionMax. conc. = 16.6262%

0–2%2–4%4–6%6–8%8–10%10–12%12–14%14–16%16–18%18–20%

J2

J4J3

ID

Sheet jt.

Face

56/28425/29490/18473/09580/270

Dip/Direction

(a)

(b)

Figure 14.8 Stereonets ofstructural geology of slope inCase Study II: (a) contoured plotshowing pole concentrations;(b) great circles of joint sets, slopeface and friction circle.

had already taken place along the in situ sheetjoints, it was considered appropriate to use theresidual friction angle in design.

Careful examination of the rock along thesheared sheet joints showed that the rock wascrushed and there was little intact rock contact.Therefore it was decided that it would not beappropriate to include a roughness component tofriction angle, and a friction angle of 36◦ used indesign.

14.3.4 Ground water

The site was located in an area with high precipit-ation in the form of both rain and snow. Althoughthere were no piezometers to measure water pres-sure in the slope, the level of the water table wasassessed from observations of seepage from sheetjoint exposed at the base of the lowest block. Itwas expected that the water table would generallybe low because of the dilated nature of the rock


Normal stress, � (kPa)

She

ar s

tres

s, �

(kP

a)

� residual= 36°

�peak = 47° Peak strengthResidual strength

100

200

300

400

500

100 200 300 400 500 600 700 Figure 14.9 Results of direct shear testson sheet joints in the granite for CaseStudy II.

mass would promote drainage. However, duringheavy precipitation events, it was likely that high,transient water pressures would develop and thiswas accounted for in design.

It was assumed for design that water wouldaccumulate in the tension crack to depth zw, andthat water forces would be generated both in thetension crack (V ) and along the sliding plane (U)

(Figure 14.10).

14.3.5 Earthquakes

The site was located in seismically active area, andit was assumed that the actual ground motionswould be made up of both horizontal and ver-tical components that could be in phase. Theseground motions were incorporated in the designby using both horizontal (kH) and vertical (kV)

seismic coefficients as follows:

kH = 0.15; and kV = 0.67 × kH = 0.1

The seismic ground motions were incorporatedinto the slope design assuming that the accelera-tion would act as two pseudo-static forces.


The nominal, static factor of safety of individualblocks sliding on the sheet joints dipping at 25◦

85°25°

70°

25°

14.8

m10.0 m

U

V

T

W

kH · WkV· W

Figure 14.10 Cross-section of block used in design tomodel the assemblage of rock blocks in the slope forCase Study II.

was about 1.5 (tan φ/ tan ψp = tan 36/ tan 25 =1.5). However, the shear movement along thesheet joints and the corresponding pattern of ten-sion cracks behind the face shown in Figure 14.7indicated that, under certain conditions, thefactor of safety diminished to approximately 1.0.


It was considered that the cause of the movementwas a combination of water pressures and icejacking on the joints, seismic ground motions overgeologic time and blast damage during construc-tion. Also, failure could have been progressivein which movement of one block would dragthe adjacent block(s), and as movement occurredcrushing of rock asperities along the slidingsurfaces reduced the friction angle.

The stability of the sliding blocks was stud-ied using a plane stability model in which it wasassumed that the cross-section was uniform atright angles to the slope face, and that sliding tookplace on a single plane dipping out of the face. Inorder to apply this model to the actual slope, asimplifying assumption was made in which thethree blocks were replaced by a single equivalentblock that had the same weight as the total of thethree blocks and the same stability characteristics.

The shape and dimensions of the equivalentsingle block were defined by the following para-meters (Figure 14.10):

Sliding plane, dip ψp = 25◦; tension crack,dip ψt = 85◦; slope face, dip ψf = 70◦;upper slope, dip ψs = 25◦; height of face,H = 18 m; distance of tension crack behindcrest, b = 10 m.

Stability analysis of this block showed that thefactor of safety was approximately 1.0 when thewater in the tension crack was about 1 m deep,and a pseudo-static seismic coefficient of 0.15g

was applied. The static factor of safety for theseconditions was 1.53, and reduced to 1.15 whenthe water level in the tension crack was 50% ofthe crack depth (zw = 7.8 m).

14.3.7 Stabilization method

Two alternative stabilization methods were con-sidered for the slope. Either, to remove theunstable rock by blasting and then, if necessarybolt the new face, or reinforce the existing slopeby installing tensioned anchors. The factors con-sidered in the selection were the need to maintain

traffic on the highway during construction, andthe long-term reliability of the stabilized slope.

The prime advantage of the blasting operationwas that this would have been a long-term solu-tion. In comparison, the service life of the rockanchors would be limited to decades due to cor-rosion of the steel and degradation of the rockunder the head. However, the disadvantage of theblasting operation was that removal of the rockin small blasts required for the maintenance oftraffic on the highway might have destabilized thelower blocks resulting in a large-scale slope fail-ure. Alternatively, removal of all the loose rockin a single blast would have required several daysof work to clear the road of broken rock, andto scale and bolt the new face. Bolting of the newface would probably have been necessary becausethe sheet joints would still daylight in the face andform a new series of potentially unstable blocks.

It was decided that the preferred stabilizationoption was to reinforce the slope by installinga series of tensioned rock anchors extendingthrough the sheet joints into sound rock. Theadvantages of this alternative were that the workcould proceed with minimal disruption to traffic,and there would be little uncertainty as to thecondition of the reinforced slope.

The rock anchoring system was designed usingthe slope model shown in Figure 14.10. For staticconditions and the tension crack half-filled withwater (zw = 7.8 m), it was calculated that ananchoring force of 550 kN per meter length ofslope was necessary to increase the static factor ofsafety to 1.5. With the application of the pseudo-static seismic coefficients, the factor of safety wasapproximately 1.0, which was considered sat-isfactory taking into account the conservatismof this method of analysis. The anchors wereinstalled at an angle of 15◦ below the horizontal,which was required for efficient drilling and grout-ing of the anchors. The factor of safety of 1.5was selected to account for some uncertainty inthe mechanism of instability, and the possibilitythat there may have been additional loose blocksbehind those that could be observed at the face.

The arrangement of anchors on the face wasdictated by the requirements to reinforce each


Trim blasting

Cable anchor

Drain hole

Shotcrete

Borehole wallGrout

Corrugated sheatingGrout tube (inner annulus)

Steel cables (2)Grout tube (outer annulus)

Anchor end detailHighway

Figure 14.11 Cross-section of stabilized slope for Case Study II showing layout of cable anchors, and the trimblast, shotcrete and drain holes; detail shows lower end of cable anchors with arrangement of grout tubes.

of the three blocks, to intersect the sheet jointsand to locate the bond zone for the anchors insound rock (Figure 14.11). Because of the lim-ited area on the face in which anchors could beinstalled, it was necessary to minimize the numberof anchors. This was achieved using steel strandcables, because of their higher tensile strengthcompared to rigid bars. A further advantage ofthe cables was that they could be installed ina hole drilled with a light rig that would beset up on the slope without the support of aheavy crane that would block traffic. Also, theinstallation would be facilitated because cablebundles were lighter than bars, and could beinstalled as a single length without the use ofcouplings.

Details of the anchor design that met thesedesign and construction requirements were as

follows:

Working tensile load of 2-strand, 15 mmdiameter, 7-wire strand anchor at 50% ofultimate tensile strength = 248 kN;

For three rows of anchors arranged asshown on Figure 14.11, the total supportforce = 744 kN (3 × 248 = 744). Thereforethe required horizontal spacing between thevertical rows:

Spacing

= supplied anchor force by three rows of anchorsrequired anchor force for factor of safety of 1.5

= 744 kN550 kN/m

∼ 1.5 m


The bond length (lb) for the anchors wascalculated assuming that the shear stressdeveloped by the tension in the anchor (T )

was uniformly distributed at the rock–groutperipheral surface of the drill hole (diameter,dh = 80 mm). For the strong granitic rockin the bond zone the allowable shear strength(τa) of the rock–grout bond was estimated tobe 1000 kPa (PTI, 1996). The bond length wascalculated as follows:

Bond length = T

π × dhτa

= 248πx0.080 × 1000

∼ 1 m

The actual bond length used for the anchorswas 2 m to allow for loss of grout in frac-ture zones in the rock where the bond zoneswere located, and to ensure that the steel–grout bond strength was not exceeded (Wyllie,1999).

In addition to the cable anchors, which wererequired to prevent large-scale instability, the fol-lowing stabilization measures were implementedto minimize the risk of surficial rock falls thatcould be a hazard to traffic (Figure 14.11):

• Trim blasting was used to remove the over-hang on the face of the upper block. This rockwas fractured and marginally stable, and itwould not have been safe to set up the drillon this face and then drill the anchor holesthrough it.

• The seams of fractured rock along each of thesheet joints were first scaled by hand to removethe loose, surficial rock, and then steel fiberreinforced shotcrete was applied to preventfurther loosening of the blocks of rock.

• Drain holes, 4 m long on 3 m centers weredrilled through the shotcrete to intersect thesheet joints and prevent build up of waterpressure in the slope.

14.3.8 Construction issues

The following is a brief description of a numberof issues that were addressed during construc-tion to accommodate the site conditions actuallyencountered.

• Drilling was carried out with a down-the-holehammer drill, without the use of casing. Par-ticular care had to be taken to keep the holeopen and avoid the loss of the hammer whendrilling through the broken rock on the sheetjoints.

• The thrust and rotation components for thedrill were mounted on a frame that wasbolted to the rock face, with a crane onlybeing used to move the equipment betweenholes. This arrangement allowed drilling toproceed with minimal disruption to highwaytraffic.

• Grouting of the anchor holes to the surfacewas generally not possible because the groutoften flowed into open fractures behind theface. In order to ensure that the 2 m long bondzones were fully grouted, the lower portionof each hole was filled with water and a wellsounder was used to monitor the water level.Where seepage into fractures occurred, theholes were sealed with cement grout and thenredrilled, following which a further water testwas carried out.

• Corrosion protection of the anchors wasprovided with a corrugated plastic sheath thatencased the steel cables, with cement groutfilling the annular spaces inside and outsidethe sheath. In order to facilitate handlingof the cable assemblies on the steep rock face,the grouting was only carried out once theanchors had been installed in the hole. Thisinvolved two grout tubes and a two-stagegrouting process as follows. First, grout waspumped down the tube contained within theplastic sheath to fill the sheath and encapsulatethe cables. Second, grout was pumped downthe tube sealed into the end cap of the sheathto fill the annular space between the sheathand the borehole wall.


• Testing of the anchors to check the load capa-city of the bond zone was carried out using theprocedures discussed in Section 12.4.2 (PTI,1996).

14.4 Case Study III—Stability of wedgein bridge abutment


This case study describes the stability analysis of abridge abutment in which the geological structureformed a wedge in the steep rock face on whichthe abutment was founded (Figure 14.12). Theanalysis involved defining the shape and dimen-sions of the wedge, the shear strength of the twosliding planes, and the magnitude and orienta-tion of a number of external forces. The stabilityof the wedge was examined under a combinationof load conditions, and the anchoring force wascalculated to produce a factor of safety againstsliding of at least 1.5.

The site was located in an area subject to bothhigh precipitation and seismic ground motion.The bridge was a tensioned cable structure with

the cables attached to a concrete reaction blocklocated on a bench cut into the rock face. Thecables exerted an outward force on the abut-ment (15◦ below the horizontal) along the axisof the bridge. The structural geology of the sitecomprised bedding and two sets of faults thattogether formed wedge-shaped blocks in the slopebelow the abutment. The stability of the slopewas examined using the wedge stability ana-lysis method to determine the static and dynamicfactors of safety, with and without rock anchors.Figure 14.12 is a sketch of the abutment showingthe shape of the wedge and the orientations of thebridge force (Q). The anchors were installed inthe upper surface of the abutment, inclined at anangle of 45◦ below the horizontal, and orientedat 180◦ from the direction of the line of inter-section. On Figure 14.12, the five planes formingthe wedge are numbered according to the systemshown on Figure 7.18(a).

14.4.2 Geology

The rock was slightly weathered, strong, massivesandstone with the bedding dipping at an angle

Fault F1 (2)

Fault F2 (5)Bench (3)

Bedding (1)

Face (4)

Line ofintersection

Tensionedbridge cables (Q)

Abutment

Figure 14.12 View ofwedge in bridge abutmentshowing fire planes formingthe wedge in Case Study III.


of 22◦ to the west (orientation 22/270). Thesite investigation identified a persistent beddingplane at a depth of 16 m below the bench levelthat contained a weak shale interbed. This planeformed the flatter of the two sliding planes form-ing the wedge block. There were also two setsof faults in the slope with orientations 80/150(F1) and 85/055 (F2). The faults were planarand contained crushed rock and fault gouge, andwere likely to have continuous lengths of tensof meters. Fault F1 formed the second slidingplane, on the left side of the wedge (Figure 14.12).Fault F2 formed the tension crack at the backof the wedge, and was located at a distance of12 m behind the slope crest, measured along theoutcrop of fault F1.

Figure 14.13 is a stereonet showing the orienta-tions of the great circles of the three discontinuitysets, and the slope face (orientation 78/220), andupper bench (orientation 02/230).

14.4.3 Rock strength

The stability analysis required shear strength val-ues for both the F1 fault and the bedding. Thefault was likely to be a continuous plane over the

length of the wedge, for which the shear strengthof the crushed rock and gouge would comprisepredominately friction with no significant cohe-sion. The shear strength of the bedding planewas that of the shale interbed. The shear strengthof both materials was determined by laborat-ory testing using a direct shear test machine (seeFigure 4.16).

The direct shear tests carried out on faultinfilling showed friction angles averaging 25◦with zero cohesion, and for the shale the fric-tion angle was 20◦ and the cohesion was 50 kPa.Although both the fault and the bedding wereundulating, it was considered that the effectiveroughness of these surfaces would not be incor-porated in the friction angle because shearing waslikely to take place entirely within the weakerinfilling, and not on the rock surfaces.

14.4.4 Ground water

This area was subject to periods of intense rainthat was likely to flood the bench at the crest of theslope. Basedontheseconditions itwasassumedforthe analysis that maximum water pressures wouldbe developed on the planes forming the wedge.

Bench:02/230

I1,2= 19/237

N

S

EW

Face:78/220

Tension crack (F2):85/055

F1: 80/150

Bedding:22/270

Figure 14.13 Stereonet of five planes formingwedge in bridge abutment shown in Figure 14.12.


14.4.5 Seismicity

The seismic coefficient for the site was 0.1. Thestability analysis used the pseudo-static method inwhich the product of the seismic coefficient, thegravity acceleration and the weight of the wedgewas assumed to produce a horizontal force actingout of the slope along the line of intersection ofthe wedge.

14.4.6 External forces

The external forces acting on the wedge com-prised water forces on planes 1, 2 and 5, the seis-mic force, the bridge load and the rock anchors.Figure 14.14 shows the external forces in planand section views.

The water forces were the product of the areasof planes 1 and 2 and the water pressure distri-bution. The seismic force was the product of thehorizontal seismic coefficient and the weight ofthe wedge. The analysis procedure was to run thestability analysis to determine the weight of thewedge (volume multiplied by rock unit weight),from which the seismic force was calculated.

For the bridge, the structural load on the abut-ment due to the tensioned cables had a magnitudeof 30 MN, and trend and plunge values of 210◦and15◦, respectively. The trendcoincidedwith thebridge axis that was not at right angles to the rock

face, and the plunge coincided with the sag angleof the catenary created by the sag in the cables.

The rock anchors were installed in the uppersurface of the bench and extended through thebedding plane into stable rock to apply normaland shear (up-dip) forces to the bedding plane.


The stability of the abutment was analyzedusing the comprehensive wedge analysis proced-ure described in Appendix III, and the computerprogram SWEDGE version 4.01 by Rocscience(2001). The input data required for this ana-lysis comprised the shape and dimensions of thewedge, the rock properties and the external forcesacting on the wedge. Values of these input para-meters, and the calculated results, are listed onthe next page.

(i) Wedge shape and dimensionsThe shape of the wedge was defined by

five surfaces with orientations as shown inFigure 14.13.

(a) Plane 1 (bedding): 22◦/270◦(b) Plane 2 (fault F1): 80◦/150◦(c) Plane 3

(upper slope): 02◦/230◦

Q

Q

U1

U1

N(a) (b)

U2

T

T

W

LegendkhW —horizontal seismic force = 14.1 MN

Q —tension in bridge cables = 30.0 MN

U2 —water force on plane 2 = 6.5 MN

T —tension force in anchor = 10.5 MN

U1 —water force on plane 1 = 19.4 MN

W —weight of wedge = 140.6 MN

khW

khW

Figure 14.14 Sketch showing magnitudeand orientation of external forces on wedge:(a) section view along line of intersection;(b) plan view.


(d) Plane 4 (face): 78◦/220◦(e) Plane 5

(tension crack,fault F2): 85◦/055◦

The orientation of the line of intersectionbetween planes 1 and 2 was calculated to be

(a) Line of intersection: 18.6◦/237◦

The dimensions of the wedge weredefined by two length parameters:

• Height, H1 (vertical height from line ofintersection to crest): 16 m;

• Length, L (length along plane 1 fromcrest to tension crack): 25 m.

(ii) Rock propertiesThe rock properties comprised the shear

strengths of planes 1 and 2, and the rockunit weight:

• Bedding with shale interbed: c1 =50 kPa, φ1 = 20◦;

• Fault F1: c2 = 0 kPa, φ2 = 35◦;• Unit weight of rock, γr = 0.026 MN/m3;

and• Unit weight of water, γw = 0.01 MN/m3.

(iii) External forcesThe magnitude and orientation of the

external forces were as follows.

• Water forces acted normal to each planeand were calculated to have the follow-ing values, for fully saturated condi-tions:

U1 = 19.73 MN;

U2 = 6.44 MN; and

U5 = 1.55 MN.

• The wedge weight acted vertically andwas calculated (from the wedge volumeand the rock unit weight) to havemagnitude:

W = 143.35 MN

• The horizontal component of the seismicforce acted in the direction along the lineof intersection and had magnitude

kHW = 0.1W

= 14.1 MN oriented at 0◦/237◦

• The bridge force, Q acted along the cen-ter line of the bridge at an angle of 15◦below the horizontal:

Q = 30 MN oriented at 15◦/210◦

• The factor of safety of the abutment withno reinforcement provided by tensionedanchors was as follows:

(a) FS = 2.58—dry, static, Q = 0(b) FS = 2.25—saturated, static, Q = 0(c) FS = 1.73—saturated, kH = 0.1,

Q = 0(d) FS = 1.32—saturated, static, Q =

30 MN(e) FS = 1.10—saturated, kH = 0.1,

Q = 30 MN

• It was considered that the factors ofsafety for load conditions (d) and (e)were inadequate for a structure criticalto the operation of the facility, andthat the minimum required static andseismic factors of safety should be 1.5and 1.25, respectively. These factors ofsafety were achieved, with the bridgeload applied, by the installation of ten-sioned anchors (tension load T ), whichgave the following results:

(a) FS = 1.54—saturated, static, T =10.5 MN, ψT = 15◦, αT = 056◦(parallel to the line of intersec-tion); and

(b) FS = 1.26—saturated, kH = 0.1,T = 10.5 MN, ψT = 15◦, αT =056◦.

• It was found that the factor of safety forthe reinforced wedge could be optim-ized by varying the orientation of the


anchors. If the trend of the anchors wasbetween the trends of the line of intersec-tion and the bridge load (i.e. αT = 035◦),it was possible to reduce the anchor forcerequired to achieve the required factor ofsafety to 8.75 MN.

• It is noted that the discussion in this casestudy only addressed the stability of thewedge, and did not discuss the methodof attaching the tensioned bridge cablesto the rock wedge. Also, it is assumedthat all the external forces acted throughthe center of gravity of the wedge so thatno moments were generated.

14.5 Case Study IV—Circular failureanalysis of excavation for rock fallditch


As the result of a series of rock falls from a rockface above a railway, a program was undertakento improve stability conditions (Figure 14.15).

The initial stabilization work involved selectivescaling and bolting of the face, but it was foundthat this only provided an improvement for oneor two years before new rock falls occurred as therock weathered and blocks loosened on joint sur-faces. Rock falls were a potential hazard becausethe curved alignment and stopping distance ofas much as 2 km meant that trains could not bebrought to a halt if a rock fall was observed.In order to provide long-term protection againstrock falls, it was decided to excavate the face tocreate a ditch that was wide enough to containsubstantial falls from the new face. This workinvolved a drilling and blasting operation to cutback the face to a face angle of 75◦, and con-structing a gabion wall along the outer edge of theditch that acted as an energy absorbing barrier tocontain rock falls (Wyllie and Wood, 1981).

The railway and highway were located onbenches cut into a rock slope above a river, andthere were steep rock faces above and belowthe upper bench on which the railway was loc-ated; a 30 m length of the track was supportedby a masonry retaining wall (Figure 14.15). The

Excavated face

Tension crack

Original slope

Ground watersurface

Center of rotation

Gabion

Railroad

Retaining wall

Highway

River

Ditch width

Potentialsliding surface

Figure 14.15 Geometry of slope above railway in Case Study IV. Sketch shows dimensions of ditch afterexcavation of slope, and shape of potential circular sliding surface through rock mass.


original cut above the railway was about 30 mhigh at a face angle of 60◦, and the 2 m wide ditchat the toe of the slope was not adequate to con-tain rock falls. Blasting had been used to excavatethe slope, and there was moderate blast damageto the rock in the face.

The site was in a climate with moderate precip-itation that experienced long periods of freezingtemperatures during the winter. Formation of icein fractures in the rock behind the face couldloosen blocks of rock resulting in the occurrenceof rock falls with little warning; rock falls tendedto occur in the spring when the ice started to melt.

14.5.2 Geology

The cut was in medium strong, slightly to mod-erately weathered volcanic tuff containing jointsspaced at about 0.5–2 m, and lengths up to 3 m.There was one consistent set of joints that had anear vertical dip and a strike at about 45◦ to thestrike of the cut face. However, the orientationsof the other joints were variable over short dis-tances. Many of the joints had calcite infillingsthat had a low cohesive strength.

Because of the variable orientations and lim-ited persistence of the joints throughout the lengthof the cut, there was little structurally controlledinstability on the overall rock face.

14.5.3 Ground water

Because of the low precipitation in the area, it wasassumed that the ground water level in the slopewould have little influence on stability.


An important design issue for the project was thestability of the overall cut face above the railway,and whether it could be cut back safely to create arock fall ditch. The rock strength relevant to thisdesign was that of the rock mass because poten-tial failure surfaces would pass partially throughintact rock, and partially along any low persist-ence joints oriented approximately parallel to thissurface. It was not possible to test samples withdiameters of several meters that would be rep-resentative of the rock mass, or to determine

the proportions of intact rock and joint planethat would form the sliding surface in the slope.Therefore, two empirical methods as describedin the next paragraph were used to estimate thecohesion and friction angle of the rock mass.

The first method of estimating the rock massstrength was to carry out a back analysis of theexisting 30 m high cut above the railway, whichinvolved the following steps. First, there was noevidence of instability of the overall slope, whichhad been standing for over 100 years, or naturalslopes in the same rock type. These slopes hadprobably been subject in the past to earthquakesand occasional periods of high water pressure.Therefore, a factor of safety in the range of 1.5–2.0 was assumed for the existing slope. Second,since there was no geological structure that wouldform a sliding surface, it was likely that instabilitywould take the form of a shallow circular failure,as described in Chapter 8. Third, as discussedin Section 14.3.3, the water table was in thelower part of the slope and it was appropriateto use Chart Number 2 (Figure 8.7) to performstability analyses. Fourth, for blocky rock withno significant clay on the joint surfaces, a fric-tion angle of 35◦ was estimated; the rock unitweight was 26 kN/m3. Using these data, for the30 m high slope at a face angle of 60◦, it waspossible to use the circular failure design chartto calculate the rock mass cohesion as approxim-ately 150 kPa (for FS = 1.75; tan φ/FS = 0.40;c/γ H FS = 0.11). Figure 4.21 was used as anadditional guideline in selecting shear strengthvalues.

As a comparison with the back analysis methodof determining rock mass strength, the Hoek–Brown strength criterion (see Section 4.5), wasused to calculate a friction angle of 38◦ and acohesion of about 180 kPa (input parameters:σci = 40 MPa; GSI = 45; mi = 10; D = 0.9)based on the program ROCLAB version 1.007(Rocscience, 2002a).

The two sets of strength values are reasonablyclose, but the difference illustrates the uncertaintyin determining rock mass strengths, and the needto carry out sensitivity analyses to evaluate thepossible influence on this range in strengths onstability.


14.5.5 Ditch and slope design

The two principle design issues for the projectwere the dimensions of the ditch to contain rockfalls, and the stability of the slope excavated tocreate the ditch.

Ditch. The required depth and width of theditch to contain rock falls is related to boththe height and slope angle of the cut face asillustrated in Figure 12.21 (Ritchie, 1963). Thesedesign recommendations show that the requiredditch dimensions are reduced for a proposedface angle of 75◦, compared to the existing60◦ face. Another factor in the ditch design wasthe face angle of the outside face of the ditch.If this face is steep and constructed with energyabsorbing material, then rocks that land in thebase of the ditch are likely to be contained. How-ever, if the outer face has a gentle slope, they mayroll out of the ditch.

For a 30 m high rock face at an angle of 75◦,the required ditch dimensions were a depth of 2 mand a base width of 7 m. In order to reduce theexcavation volume, the ditch was excavated toa depth of 1 m, and a 1 m high gabion wall wasplaced along the outer side of the excavation tocreate a vertical, energy absorbing barrier.

Slope stability. The stability of the excavatedslope was examined using Circular Chart No. 2.The proposed excavation would increase the faceangle from 60◦ to 75◦ without increasing theheight of 30 m significantly, and the rock massstrength and the ground water conditions in thenew slope would be identical to those in theexisting slope. Chart number No. 2 showed thatthe factor of safety of the new slope was about1.3 (c/(γH tan φ) = 0.275; tan φ/FS ≈ 0.2).Figure 14.15 shows the approximate locationof the potential tension crack, and sliding sur-face with the minimum factor of safety, determ-ined using Figure 8.11 (X = −0.9H ; Y = H ;b/H = 0.15).

14.5.6 Construction issues

The excavation was by drill and blast methodsbecause the rock was too strong to be broken by

rippers. The following are some of the issues thatwere addressed during construction:

The blasting was carried out in 4.6 m liftsusing vertical holes. The “step-out” requiredat the start of each bench to allow clearancefor the head of the drill was 1.2 m, so theoverall slope angle was 75◦. The productionholes were 63 mm diameter on a 1.5 m squarepattern and the powder factor was 0.3 kg/m3.

Controlled blasting was used on the finalface to minimize the blast damage to therock behind the face. The final line holeswere spaced at 0.6 m and charged with de-coupled, low velocity explosive at a loadfactor of 0.3 kg/m of hole length. The finalline holes were detonated last in the sequence(cushion blasting) because the limited burdenprecluded pre-shear blasting.

The detonation sequence of the rows in theblast was at right angles to the face in orderto limit the throw of blasted rock on to therailway and highway, and minimize closuretimes.

The track was protected from the impactof falling rock by placing a 1 m thick layerof gravel on the track before each blast. Thiscould be quickly removed to allow operationsof the train.

Near the bottom of the cut it was necessaryto protect from blast damage the masonryretaining wall supporting the track. This wasachieved by controlling the explosive weightper delay so that the peak particle velocityof the vibrations in the wall did not exceed100 mm/s.

14.6 Case Study V—Stabilization oftoppling failure


A rock slope above a railway was about 25 mhigh, and the rock forming the slope was a blockygranite in which a toppling failure was occurring(Wyllie, 1980). Movement of the upper topplingblock was crushing the rock at the base and


Tensioncrack

Top of slabremoved

~2.5 m

6.0 m

70°

W

Rock anchor

J1

J2

Figure 14.16 Idealized configuration of toppling slabsin Case Study V showing excavation and bolting.

causing rock falls that were a hazard to railwayoperations (Figure 14.16). The site was in a highprecipitation climate, with a moderate risk of seis-mic ground motions. Stabilization measures wereundertaken to limit the rock fall hazard and toprevent additional toppling motion.

14.6.2 Geology

The granite at the site was fresh and very strong,and contained three well-defined sets of jointswith orthogonal orientations. The most promin-ent set (J1) dipped at about 70◦, with the strike atright angles to the railway alignment. The secondset (J2) had the same strike but dipped at about20◦, while the third set (J3) was near vertical withthe strike parallel to the track. The spacing ofthe joints was between 2 and 3 m, and the per-sistence of the J1 joint set was in the range of10–40 m. The joints were planar but rough, andcontained no infilling. Figures 14.16 and 14.17show a sketch of the slope and the dimensions ofthe blocks formed by the jointing.

14.6.3 Rock strength

The compressive strength of the granite was inthe range of 50–100 MPa, and it was estimatedthat the friction angle of the joints was between40◦ and 45◦ with no cohesion. These valueswere determined by inspection because of the lim-ited time available to assess the site and plan astabilization program.

14.6.4 Ground water

The site experienced periods of heavy rainfalland rapid snow melt, so it was expected thattransient high water pressures would develop inthe lower part of the slope. In the upper part ofthe slope, water pressures were unlikely becausewater would not collect in the tension cracksexposed in the face.

14.6.5 Stability conditions

The uniform spacing and orientation of the J1joints formed a series of slabs in the slope thatwere approximately 2.5 m wide and had verticalheights of as much as 20 m. The slabs dipped atabout 70◦ so the center of gravity of the slab layoutside the base when the height exceeded about6 m; this was a necessary condition for toppling(see Figure 1.10). As shown in Figure 14.17, theupper slab had an exposed face about 7 m highand toppling of this slab had opened a tensioncrack about 200 mm wide along the J1 joint set.As the upper block toppled, it generated thrustforces on the lower slabs. The short length ofthese lower slabs meant that their centers of grav-ity were well inside their bases so toppling did notoccur. However, the thrust was great enough tocause the lower blocks to slide on the J2 jointset. This set dipped at 20◦ and had a frictionangle of about 40◦; limit equilibrium analysis ofthe sliding blocks showed that the thrust forcerequired to cause sliding was equal to about 50%of the weight of the block. This shear displace-ment caused some fracturing and crushing of therock that was the source of the rock falls.

The mechanism of instability at the site wasessentially identical to the theoretical topplingmechanism discussed in Chapter 9 and shown in


Figure 14.17 Toppling failure in CaseStudy V. Sketch showing extent of uppertoppling block removed by blasting, andlocation of rock bolts in lower slope.

Figure 9.7. That is, the tall, upper slabs toppledand caused the lower, shorter slabs to slide. Pos-sible stabilization options for these conditionsincluded reducing the height of the toppling slabsso that the center of gravity lay inside the base,or installing a support force in the sliding slabsat the base. These two measures were adopted,with the combined effect of reducing the tend-ency for the upper slabs to topple, and preventingmovement of the lower slabs.

14.6.6 Stabilization method

The following three stabilization measures wereundertaken to reduce the rock fall hazard and toimprove the long-term stability of the slope:

• Scaling was carried out on the face abovethe railway to remove loose rock. This work

included the removal of all trees growing inopen cracks in the rock because these had con-tributed to the loosening of the blocks of rockon the face.

• A row of bolts was installed through one ofthe lower slabs. This work was done prior toexcavation at the crest in order to prevent anyfurther movement due to blasting vibrations.

• Blasting was used to remove the upper 6 m ofthe top slab. The blasting was carried out instages in order to limit blast vibrations in thelower slope and allow additional bolts to beinstalled if further movement occurred. Theblasting pattern comprised 6 m long holes onabout 0.6 m centers, with three rows beingdetonated on each blast. A light explosivecharge of 0.4 kg/m3 was used, with spacersbetween the sticks of explosive in the blastholes.

Chapter 15

Mining applications

Alan F. Stewart, P. Mark Hawley, Nick D. Roseand Brent W. Gilmore∗

15.1 Introduction

Rock slope engineering of open pit mines requirescareful application and adaptation of the fullrange of tools that have been presented in earlierchapters of this book. Each ore body and hostrock mass is unique, and comprises distinct-ive mineralogical assemblages and rock types.In many instances, stratigraphy may be com-plexly deformed by geologic forces. Geologicand geomechanical characteristics, such as litho-logy, mineralogy, alteration, rock strength, in situstress, geologic structure and fabric, and groundwater conditions may vary widely between differ-ent deposits, and even within a given deposit. Thechallenge for the slope designer is first to deter-mine which of these characteristics are importantin terms of stability. The next step is to planand execute focused investigations to obtain theinformation required to define the key stabilityparameters. Stability analyses are then conducted,and results are used in conjunction with experi-ence and judgment to develop slope design criteriafor use by mine planners and operators.

In open pit mining, the optimum slope designis usually one that maximizes overall slope anglesand minimizes the amount of waste stripping.At the same time, it must effectively manage therisk of overall slope instability, and provide forsafe and efficient movement of personnel, equip-ment and materials during mining operations.The general methodology for designing open pit

∗ Piteau Associates Engineering Ltd, North Vancouver, BC,Canada.

mine slopes is described in this chapter by way offour hypothetical examples. These examples rep-resent a range of mine design and rock mechanicsissues in a variety of geologic environments.

Most open pit mines are developed usingbenches that are designed to contain and controlrock falls and small failures. The geometry of thepit and slopes is defined by the shape of the orebody, the height and width of the benches, and thelocations of haul roads and stepouts; Figure 1.5illustrates a typical pit slope geometry. As dis-cussed in the following examples, inter-rampslopes are defined as slope sections comprised ofmultiple benches between haul roads or stepouts.Haul roads are necessary to provide access to theore and waste, and stepouts may be required forreasons of stability or to accommodate the shapeof the ore body. Overall slopes incorporate inter-ramp slopes as well as haul roads and stepouts,and extend from the crest to the toe of the pit wall.

15.2 Example 1—porphyry deposits

This example describes a preliminary slope designinvestigation conducted as part of a feasibilitystudy for a new porphyry copper deposit. Preli-minary mine plans indicated a maximum open pitdepth of 250 m. No mining activity had occurredin the deposit, and no previous design stud-ies had been conducted, other than explorationdrilling, mapping and sampling related to orereserve definition.

A geotechnical investigation program wasconducted that incorporated site reconnais-sance, structural mapping of available outcrops,

358 Mining applications

geomechanical logging of drill cores, and a test-ing program involving point load index testing ofcore, and direct shear testing of selected discon-tinuities. In addition, six geotechnical coreholeswere drilled to obtain oriented core. Piezometerswere targeted for various holes throughout theproperty to monitor ground water levels andobtain an indication of potential pit dewateringrequirements.

15.2.1 Design issues

The proposed pit would have a modest overalldepth of 250 m, and would be excavated in a com-petent rock mass with a consistent, pervasive setof joints and faults related to the genesis of thedeposit. Open pit slope design was expected to becontrolled by the stability of individual benches,and the need to optimize bench geometry to min-imize waste stripping. Due to the combination ofmoderate overall slope height and a competentrock mass, inter-ramp and overall slope stabilitywere not significant concerns.

15.2.2 Engineering geology

The porphyritic intrusion was dacitic in compo-sition, hosted by tertiary andesites and andesitebreccias, and was hydrothermally altered witha distinctive alteration zonation ranging frompotassic to phyllic to propylitic. In terms ofrock mass competency, the potassic altera-tion increased the overall competency of therock, whereas the phyllic alteration significantlyweakened the rock and reduced discontinuityshear strength. Propylitic alteration appearedto have had little influence on overall rockcompetency.

Results of the structural mapping and coreorientation indicated a pattern of radial andtangential jointing and faulting that appearedto be centered around the intrusive core. Theradial joint set (Set 1) dipped sub-vertically andwith a strike approximately radial to the cen-ter of the intrusive complex. These structureswere probably related to the original intrusionand facilitated development of the hydrothermal

NW trending fault

II

IIII

VI

IV

V Structuraldomainboundary

Approximateoutline of intrusivecomplex

Figure 15.1 Distribution of structural domains.

system that deposited the ore. The strike of thetangential set (Set 2) was approximately normalto Set 1 and dipped at 45–60◦ towards the cen-ter. Set 2 was probably formed during collapseof the hydrothermal system. Peak orientations ofthese two principal sets varied depending on theirposition in relation to the intrusive center.

Based on the distribution of discontinuity ori-entations, the deposit was divided into six struc-tural domains distributed radially around thedeposit, as illustrated in Figure 15.1. Withineach structural domain the geologic structuralfabric was expected to be reasonably consistent.Figure 15.2 is a stereonet that shows the distribu-tion of discontinuities in Structural Domain I.

Regionally, northwest trending sub-verticalfaults were present throughout the area. In partic-ular, a large fault zone with a width of about 10 mwas interpreted to intersect the northeast cornerof the proposed pit.

15.2.3 Rock strength and competency

Field estimates of hardness (ISRM, 1981b)obtained during geomechanical logging of thedrill core were correlated with point load indexresults. Both of these measures of rock strengthindicated a moderately hard rock mass, withunconfined compressive strengths (UCS) rangingfrom about 40 to 100 MPa. Local zones ofphyllic alteration had an average UCS as low asabout 5 MPa.

Mining applications 359

1b1a

N

S

EW 2

1a2

1c1b

1c

Set 1

Set 2

+

++

+

Figure 15.2 Stereonet of discontinuities in StructuralDomain I.

Laboratory direct shear testing of selectedjoints collected from the drill core indicatedfriction angles of between about 30◦ and 42◦,depending on the type and intensity of altera-tion present. Results also indicated little or nocohesion. For faults and fault gouge, the aver-age friction angle was about 20◦ with negligiblecohesion.

Geomechanical core logging data, includingRQD, joint spacing, joint condition and hardness,were compiled, and average Rock Mass Ratings(RMR) were determined according to Bieniawski(1976). For purposes of rock mass characteriza-tion, ground water conditions were assumed tobe dry. The average RMR was 65 (good qualityrock mass) for all core, and ranged from approx-imately 35 (poor quality rock mass) for phyllicallyaltered rocks to about 85 (very good quality rockmass) for potassically altered rocks.

15.2.4 Hydrogeology

Initial monitoring of several piezometers installedin exploration drill holes indicated low piezomet-ric pressures in most areas of the proposed pit.However, water levels appeared slightly elevated

in the northeast, probably in response to thelarge regional fault zone described above that mayhave been acting as an aquitard to ground waterflow. Localized horizontal drain holes, targetingareas such as this fault zone, and in-pit sumpswould probably be sufficient to manage expectedground water volumes. Additional hydrogeolo-gical assessments would be required as the pitdeveloped.

15.2.5 Slope stability analyses andslope design

It is usually impractical and uneconomic to designopen pit slopes such that no failures occur. There-fore, a more pragmatic approach is to design thepit with benches, and excavate the slopes undercontrolled conditions such that any failures thatdo occur are caught and effectively controlled onberms.

Initially, slope stability analysis involvedassessment of possible failure modes relating tostructural discontinuities (i.e. joints and faults)that could result in shallow failure of individualbenches, or large-scale failure involving multiplebenches or overall slopes. Subsequent analyseswere conducted to assess the potential for deep-seated rotational rock mass failure of the ultimatepit slopes, based on preliminary inter-ramp slopeangles developed from the bench designs.

As noted earlier, the rock mass was divided intosix structural domains arranged in pie-shapedsegments about the center of the intrusive com-plex (Figure 15.1). Based on the preliminary mineplan, the rock mass was further subdivided intodesign sectors, or zones with consistent geologicstructure as well as uniform pit slope orientation.Within each design sector, kinematic assessmentswere conducted to determine possible failuremodes that could occur (see Figure 2.21). Twobasic failure modes were considered: wedge fail-ures and plane failures. Figure 15.3 is a stereonetthat shows kinematically possible failure modesidentified in a typical design sector in StructuralDomain I. Limit equilibrium stability analyses,utilizing discontinuity shear strengths determinedfrom laboratory direct shear testing, were then


1

4

N

S

EW

105

1c1b

1a2

2

1a1b

1c

Kinematicallypossible wedgefailure

Kinematicallypossible planefailure

Average slope orientationin Design Sector 1

Figure 15.3 Stereonet showingkinematically possible failure modesin Design Sector 1.

conducted for each failure mode to determinewhich failure modes were critical to design. Crit-ical failure modes were defined as kinematicallypossible failures with factors of safety less than orequal to 1.2. In addition, the dip direction of crit-ical plane failures was less than about 30◦ obliqueto the slope, and the trend of the line of intersec-tion of critical wedge failures was less than about45◦ oblique to the slope.

Surface mapping and general reconnaissanceshowed that joints were likely to persist through-out the rock mass, and to have an average con-tinuity of about 10–15 m. Consequently, theywere expected to have a significant impact onbreakback of individual benches, but to have lim-ited importance in terms of overall slope stability.Faults, although not as prevalent, were muchmore continuous and could impact inter-rampand overall slopes as well as individual benches.Based on an assessment of the various criticalfailure modes, associated factors of safety anddegree of development of the joint and fault setsinvolved, the apparent dip or plunge consideredto control bench stability was determined for eachdesign sector. It was expected that the blasted and

excavated bench face angles would range from57◦ to 62◦ (“breakback angle”).

Bench height is usually determined by the sizeof drilling and excavation equipment, and othermine planning considerations. In this example,a bench height increment of 15 m was chosenfor feasibility assessments. In the more com-petent rocks, 30 m high double benches wereconsidered appropriate. Double benches typic-ally allow steeper inter-ramp and overall slopesto be developed, although the size of poten-tial failures increases and wider catchment bermsare generally required. In the less competentrocks, single benches were considered appro-priate to control raveling and rock falls, aswell as bench-scale wedges and plane failures.Bench heights, minimum berm widths, and theapparent dip or plunge considered to controlbench stability determined earlier were usedto determine maximum inter-ramp slope anglesfor each design sector. Minimum berm widthsof 8 and 10 m were recommended for singleand double benches, respectively. Recommen-ded inter-ramp slope design criteria ranged from38◦ to 42◦ for single benches developed in zones


of intense phyllic alteration, to 45◦ to 49◦ fordouble benches developed in competent potassicand propylitic altered rocks. Results of deep-seated limit equilibrium stability assessments ofthe overall slopes indicated adequate stabilityfor the proposed maximum slope heights andrecommended inter-ramp slope angles.

15.3 Example 2—stratigraphicallycontrolled deposits

The process for designing slopes in structurallycomplex, stratigraphically controlled deposits isdemonstrated in the following example using ahypothetical open pit coal mine developed inintensely folded and thrust-faulted sedimentarystrata. While this example was developed basedon the authors’ experience at several mines inthe Canadian Rocky Mountains and Foothills inBritish Columbia and Alberta, the concepts canbe applied to other sedimentary, strataform orstratabound deposits.

15.3.1 Design issues

Layered ore deposits may be tilted, folded and/orfaulted, such as the coal measures of WesternCanada, the iron ore deposits of Brazil, and a vari-ety of other deposits hosted in bedded sediment-ary, foliated metamorphic or layered volcanicrocks. These deposits often present special issuesfor pit slope design. For example, the orientationof bedding or foliation frequently controls wallstability and slope design, and ore horizons maybe narrow, resulting in project economics thatmay be very sensitive to stripping. Also, the struc-tural geology is often complex and can vary sig-nificantly over short distances. Stratigraphy mayalso be complicated by thrust and normal faultingthat can both follow and cross-cut strata, result-ing in apparent stratigraphic thickening, thinningor truncating. It is often difficult or impractical tounderstand fully the geologic complexity of thesetypes of deposits in advance of mining. As a result,material changes in the interpretation may occurduring mining as strata are exposed and mapped.Consequently, design criteria need to be flexible

and readily adaptable to both subtle and dramaticchanges in geologic interpretation.


In this example of an open pit coal mine, thebottom of the stratigraphic sequence was charac-terized by a thick sequence of interbedded Jurassicmarine shales and siltstones (Domain 1). Thesewere overlain by Cretaceous terrestrial siltstones,sandstones and minor mudstones (Domain 2),which in turn underlayed Cretaceous coal meas-ure rocks comprising interbedded coal, car-bonaceous mudstones, siltstones and sandstones(Domain 3). The footwall of the lowest coalseam comprised a relatively massive, thick sand-stone unit. Figure 15.4 shows a typical geologiccross-section through the deposit.

The strata had been deformed into a foldsequence comprising a synclinal core flankedby overturned anticlines. Thrust faults haddeveloped approximately parallel to the axialplanes of the folds, and had thickened the coalsequence in the core of the syncline.

The stereographic projections in Figure 15.5illustrate the main discontinuity sets, as deter-mined by outcrop mapping. The most prominentdiscontinuity set was bedding joints (Set A), andalthough the dip of this set varied widely, thestrike was relatively constant. Peak orientationsgenerally fell on a great circle, which was con-sistent with cylindrical folding (note the inferredfold axis shown on Figure 15.5(a)). In addition tobedding joints, two other discontinuity sets wereapparent:

• Set B: strike approximately perpendicular tobedding, and with sub-vertical dip; and

• Set C: strike approximately parallel to bed-ding, and dip about normal to bedding.

Collectively these three discontinuity sets formedan approximately orthogonal system, which istypical of folded sedimentary rocks. The primaryorientation of regional thrust faulting is alsoindicated on Figure 15.5(b).


W E

Symbols

Domain boundary

Domain(H = highwall, F = footwall)

Trace of axial plane(overturned anticline, syncline)

Trace of bedding plane

Thrust fault

1H

2H

3H

2F

3F

2F

3F

3F

3F

1F

1H

Legend

Coal measures, Cretaceous (Domain 3)

Sandstone

Jurassic shale/mudstone (Domain 1)

Undifferentiatedsediments

Cretaceous(Domain 2)

Figure 15.4 Example 2—typical geologic cross-section of coal formation (modified after Hawley and Stewart(1986)).

A

(a) (b)

B

B

C

Pole to axialplane Pole to region

of thrusting

Figure 15.5Example 2—stereographicprojections of poles todiscontinuities: (a) bedding andbedding joints; (b) cross-joints andfaults.


Estimates of intact rock strength, discontinuityshear strength and general rock mass compet-ency were developed from geomechanical corelogging, point load testing, and laboratory

unconfined compressive strength and direct sheartesting.

The marine shales and siltstones that form thebase of the sedimentary sequence were thinly bed-ded, fissile, fair quality rocks. They had lowdurability and tended to slake and degrade when


exposed. These rocks were highly anisotropicwith UCS ranging from about 35 MPa along bed-ding to about 80 MPa across bedding. Beddingjoints were closely spaced (<0.3 m), and RMRvalues typically ranged from 40 to 50 (i.e. fairquality rock mass).

The Cretaceous siltstones and sandstones thatformed the footwall of the coal measures weremore massive and competent than the under-lying Jurassic rocks. Bedding joint spacing wasabout 1 m, UCS was typically greater than about150 MPa and RMR was greater than about 60(i.e. good quality rock mass).

The competency of the coal measure rocks wasextremely variable. At the low end were shearedcoal seams with UCS of about 14 MPa or less, andRMR of about 30 (i.e. poor quality rock mass).Carbonaceous shales and mudstones were slightlymore competent with UCS of about 25 MPa andRMR of about 40. Interseam siltstones and sand-stones were the most competent, with strengthssimilar to the sedimentary rocks in the immediatefootwall of the coal measures.

The shear strength of the discontinuities alsovaried widely, depending on the discontinuitytype, lithology and infilling materials. Faults,shears and bedding joints in coal had a nominalfriction angle of about 23◦ and negligible cohe-sion. Carbonaceous bedding and cross-joints hada nominal shear strength of about φ = 25◦, c =15 kPa, while non-carbonaceous bedding andcross-joints had a nominal shear strength of aboutφ = 36◦, c = 60 kPa.

15.3.4 Hydrogeology

The ground water flow system was anisotropic,with high hydraulic conductivity in the plane ofbedding compared to that across bedding. Coalseams and fractured sandstone and siltstone unitstended to act as aquifers, and shale/mudstoneunits tended to act as aquitards. The principaldirection of ground water flow was parallel tothe plunging axes of the folds. Because topo-graphy tended to mimic the gross fold struc-ture, artesian conditions could exist in the toesof excavated footwall slopes. Horizontal drain

holes were required to control potentially adversepiezometric pressures in footwall slopes. High-wall slopes were typically moderately to welldrained, and enhanced depressurization was notnormally required for these walls.

15.3.5 Structural domains

Based on stratigraphic and competency consider-ations, the rock mass was first subdivided intothree structural domains: the Jurassic shales andsiltstones (Domain 1), the footwall siltstones andsandstones (Domain 2), and the coal measures(Domain 3). Each domain was further subdividedbased on the orientation of bedding with respectto proposed slope orientations. Footwall domains(F) were defined as domains where bedding strikesparallel to the proposed slope and dips in the samedirection as the slope. Highwall domains (H) weredefined as domains where bedding strikes paral-lel to the proposed slope and dips into the slope.Domains are shown on Figure 15.4.

15.3.6 Kinematic analyses

Kinematic assessments were conducted for eachdomain using stereographic projection techniquesdescribed in Chapter 2 to determine possiblefailure modes. These analyses confirmed that pos-sible failure modes were highly dependent on theorientation (both strike and dip) of the slope withrespect to bedding. Some examples of kinematic-ally possible failure modes that were consideredare illustrated schematically in Figure 15.6.

For footwall domains, the key failure modesthat controlled stability all involved sliding alongbedding discontinuities. Simple plane failure mayhave occurred where the slope undercut bedding(Figure 15.6(a)), or bedding was offset by fault-ing (Figure 15.6(b)). More complex failure modesmay also have occurred, such as bilinear failureinvolving shearing through the toe of the slope(Figure 15.6(c)), ploughing failure where a driv-ing slab forces a key block to rotate out of the toeof the slope (Figure 15.6(d)), or bucking failure(Figure 15.6(e)).


Bedding joint Failure slab

Fault

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 15.6 Example 2—kinematically possible failure modes.

For highwall domains, the key failure modesthat controlled stability included toppling on bed-ding (Figure 15.6(f)), stepped-path plane failureinvolving sliding along cross-joints with releaseon bedding joints (Figure 15.6(g)), and raveling(i.e. rock falls involving individual detached rockblocks) (Figure 15.6(h)).

15.3.7 Stability analyses

Stability analyses were conducted for each of theprimary modes of failure in each domain usinglimit equilibrium techniques. Failure models weredeveloped to assess the sensitivity of stability tovariations in the geometry of the slope, bedding

orientation, bedding joint spacing, rock masscompetency, discontinuity shear strength andground water conditions. The analyses techniquesused for simple plane, wedge and toppling failurewere similar to those presented in Chapters 6, 7and 9. More complex failure modes, such as bilin-ear slab, ploughing and buckling failure, wereanalyzed using limit equilibrium methods similarto those described by Hawley et al. (1986).

Analyses results for footwall domains werepresented in the form of stability curves thatrelated the dip of bedding to slope or benchheight for a given factor of safety. As illustratedschematically in Figure 15.7, multiple curves weredeveloped for each mode to assess sensitivity to


Slo

pe h

eigh

t, H

Slo

pe h

eigh

t, H

Bed

ding

dip

, �p

Ver

tical

dow

el s

paci

ng, V

Incr

easin

g t

Increasin

g � 2

Incr

easi

ng �

p

H

t

Bedding dip, �p Bedding dip, �p

H

Dowels V

Horizontal dowel spacing, Sd

Sb

H

Increasing �f

Spacing of joints, Sb

�p

�f

�p

�p

�p

�2

(a)

(c)

(b)

(d)

Figure 15.7 Schematic illustration of stability analysis results: (a) plane failure; (b) ploughing failure;(c) toppling failure; (d) slab failure with artificial support (modified after Hawley and Stewart (1986)).

variations in key parameters, such as the spacingof bedding joints, or the dip of cross-joints, andto assess the cost/benefit of artificial support.

For highwall domains, analysis results forpotential plane, wedge and stepped path failureswere presented in terms of expected breakbackangles using a similar approach as described inExample 1. Analyses results for potential topplingfailure were presented in the form of stabilitycurves that relate bedding joint dip and spacingto stable bench face angle (see Figure 15.7(c)).

15.3.8 Slope design concepts

To provide the mine planners with flexible designcriteria that could be easily adapted to changinggeologic conditions, a series of slope design con-cepts were developed. Each concept consisted ofa basic slope type, and specific slope design cri-teria. Each concept was applicable within a givendomain over a specified range of geologic condi-tions. Table 15.1 summarizes the various slope

design concepts, associated basic slope types,their range of applicability, and critical failuremodes that control slope design and pertinentcomments.

In developing the slope design concepts, somebasic slope parameters first had to be definedin consultation with the mine planners. Theseincluded fixed criteria, such as bench height incre-ment and minimum catch berm width, whichwere based on the size of the mining equipmentand regulatory requirements, and more subjectiveconsiderations, such as the overall design factorof safety and acceptable level of risk.

In some cases, more than one slope designconcept was applicable. For example, artificialsupport was an alternative that provided a steeperslope design than a conventional approach.Alternative slope design concepts provided themine planners with additional flexibility. Thedecision as to which alternative to adopt wasbased on specific cost/benefit analyses, opera-tional convenience or other criteria.

Table 15.1 Slope design concepts

Slopedesignconcept

Basicslopetype

Beddingorientation

Illustration Criticalfailuremodes

Applicability Design criteria

F-I Benchedfootwallslope; beddingundercut.

Bedding dipsshallowly outof the slope.

Stepped planarfailure onbedding.

Domains wherebedding joints arediscontinuous orbedding dip isflatter than thefriction angle.

Excavate benched slope. Benchesdesigned to limit the size ofpotential stepped failures andprovide catchment for smallfailures and raveling debris.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -F-ll Unbenched

footwallslope; beddingnot undercut.

Bedding dipsshallowly tomoderatelyout of theslope.

Planar failureon bedding.

Domains wherebedding joints arecontinuous orbedding dip issteeper than thefriction angle, butnot steep enoughto initiatebuckling,ploughing,bilinear or otherslab-type failures.

Excavate slope parallel to bedding.Do not undercut bedding.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -F-III Benched

footwallslope; beddingnot undercut.

Buckling,ploughing,bilinear orotherslab-typefailures.

Domains wherebedding joints arecontinuous andbedding dip issignificantlysteeper than thefriction angle.

Excavate bench faces parallel tobedding. Do not undercutbedding. Bench height designed tolimit potential for development ofslab-type failures. Bench widthdesigned to provide catchment forsmall failures and raveling debris.

Bedding dipsmoderately to

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -F-IV Unbenched,

supportedfootwallslope; beddingnot undercut.

steeply out ofthe slope.

Buckling,ploughing,bilinear orotherslab-typefailures.

Domains wherebedding joints arecontinuous andbedding dip issignificantlysteeper than thefriction angle.

Excavate slope parallel to bedding.Apply artificial support to preventdevelopment of major slab-typefailures.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -H-I Benched,

unsupportedhighwallslope.

Toppling;raveling.

Domains wherebedding joints arecontinuous andclosely spaced.

Excavate slope using singlebenches. Flat bench face angledesigned to limit potential fortoppling. Minimal bench widthdesigned to provide catchment forraveling debris.

Beddingdips

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -H-II Benched,

supportedhighwallslope.

steeplyinto slope.

Toppling;raveling.

Domains wherebedding joints arecontinuous andclosely spaced.

Excavate benched slope. Artificialsupport designed to limit potentialfor toppling, maximize benchheight and/or bench face angleand/or increase available benchwidth to contain small slab-typefailures and raveling debris.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -H-III Benched,

unsupportedhighwallslope.

Bedding dipsshallowly tomoderatelyinto slope.

Planer, steppedplanar,wedges orsteppedwedges oncross-joints;raveling.

Domains notsubject to otherkinematicallypossible failuremodes.

Excavate benched slope. Benchesdesigned to limit the size ofpotential planar, wedge andstepped failures and providecatchment for small failures andraveling debris.

Source: Adapted after Hawley and Stewart (1986).


Bedding dip (degrees)

Ben

ch h

eigh

t (m

)

Recommended benchheight design criteria

Range of analysisresults for bilinearfailure

Range of analysis resultsfor ploughing failure

20

60

100

0

40

80

20 6040 8030 7050 90

Figure 15.8 Example 2—typicalfootwall bench height designcriteria (modified after Hawleyand Stewart (1986)).

Specific design criteria were developed for theslope design concept in each domain based onthe results of the stability and sensitivity ana-lyses. Critical failure modes were determined foreach basic slope type, and analyses results wereused to define ranges of acceptable slope andbench geometries. For footwall domains, theresults of plane, bilinear and ploughing failureanalyses were used to define allowable unbenchedslope heights based on the dip of bedding (e.g.Figure 15.8). For highwall domains, toppling fail-ure analyses were used to define the ranges ofbedding dip where toppling failure controlled sta-bility, and to assess appropriate slope geometriesto prevent toppling. For highwall domains notsubject to toppling, design criteria were based onpredicted breakback and minimum catch benchwidths required to control small wedge and planefailures, and raveling.

15.3.9 Preliminary design

A preliminary slope design was prepared basedon the slope design concepts described before.Detailed geotechnical sections were constructedat regular intervals normal to the proposed slopes.

On each section, a provisional pit bottom andslope toe were determined in consultation withmine planners. Based on the domain and ori-entation of bedding in the toe of the slope, anappropriate slope design concept was selected andapplied. This slope design concept was projec-ted upwards for as far as conditions remainedappropriate. When a point was reached where theconditions were no longer applicable, a new slopedesign concept was chosen. This process wasrepeated until the full wall height was developedand the pit crest was reached on each geotech-nical section. Figure 15.9 is an example of howthis iterative process was applied to the typicalcross-section given in Figure 15.4.

To achieve a practical overall slope design,it was necessary to blend the slope geometrybetween sections. Bench heights and berm widthshad to be modified locally to deal with localrolls in the bedding, the occurrence of faults andother geologic complexities. Once the prelimin-ary design was completed, an assessment of thepotential for overall or deep-seated slope instabil-ity was conducted, and slope geometries weremodified as required to meet minimum overallslope stability objectives.


W E

Symbols

Slope design concept

Design sector boundaryLegend

Coal measures, Cretaceous (Domain 3)

Sandstone

Jurassic shale/mudstone (Domain 1)

Undifferentiatedsediments

1H

2F

2F

3F

3FH-III

H-III

H-II

H-I

F-III

F-II

F-IV

F-I

F-II

Pit

Cretaceous(Domain 2)

Figure 15.9 Example 2—slope design concepts applied to typical geologic cross-section (modified after Hawleyand Stewart (1986)).

15.4 Example 3—deep-seated deformationin a weak rock mass

Weak rock mass conditions can occur in manytypes of ore deposits, especially where ore depos-ition is associated with alteration or complexstructural zones. The lithologies associated withthese conditions can be considered to represent awide range of geological environments including:(i) highly fractured plutonic rocks (e.g. copperporphyry deposits); (ii) metasedimentary or meta-volcanic rocks (e.g. shear hosted gold deposits);or (iii) mafic volcanic rocks (e.g. asbestos depo-sits). This example is a generalized case historyof combined experience at four operating mineswith open pit slopes ranging from 300 to 500 mhigh. The main similarity between these projectsis the presence of a weak rock mass associated

with a near-vertical regional fault or shear zone,exposed in the lower portion of one of the pitwalls, that defines a boundary to the ore body.

Overall, these rock masses are variablyaltered, structurally complex and exhibit highground water pressures. Slope depressurization isdifficult due to structural compartmentalizationof ground water and low rock mass conductivity.

15.4.1 Design and operational issues

Large-scale pit slopes may be prone to deep-seated deformation due to stress concentrationsin the highly deformable weak rock mass in thetoe of the slope, or complex modes of failuresuch as squeezing or toppling extending to aconsiderable depth behind the face. These con-ditions often require a number of analytical and


empirical approaches to be applied in design.However, limitations may exist in being able topredict, adequately, initial slope performance.Consequently, early mining phases may requireconservative slope angles to provide experiencefrom which future slope performance can beassessed. Once experience is gained, determ-inistic, probabilistic, empirical and numericalmodeling assessments can be calibrated againstslope performance, allowing ultimate pit slopedesigns to be optimized.

Deep-seated deformation can result in slopemovements of tens of meters per year. There-fore, a comprehensive slope movement monit-oring system is required to provide advancedwarning of increased slope movement rates. Thistypically includes a network of reflective surveyprisms, wire extensometers and borehole inclino-meters in critical areas. Contingency planning isalso required to build flexibility into the minedesigns, such that mitigation measures can beimplemented to remediate areas of instability, ifnecessary.

Depending on the critical nature of thepit slopes, the location of important mine

infrastructure and the flexibility of the mine plan(i.e. single versus dual, or multiple, ramp access),deformation analysis using a discontinuum mod-eling approach such as UDEC (see Chapter 10)may be carried out. Deformation analysis requiresa high level of understanding of the slope deform-ation controls, as well as detailed slope mon-itoring and ground water information. Withadequate calibration, these models can providepredictions of possible slope behavior, allow-ing movement threshold criteria and contingencyplans to be developed.


The engineering geology of the slope is shownon the cross-section in Figure 15.10. The geo-logy was based on detailed pit mapping andcore logging, and the structural geology inter-pretation was based on oriented core drillingand projections of structural mapping from theexisting pit.

A 50 m wide regional fault zone dipped atapproximately 80◦ into the lower pit slope, witha strike approximately parallel to the slope face.

Interim pitwater table

Ultimate pitwater table

Majorfault

Interim pit

Ultimate pit

Piezometer Structuralfabric

Pumpingwell

Sub-horizontaldrain hole

Major faultzone

HangingWall

ShearZone

Footwall

Figure 15.10 Example 4—engineering geology cross-section of open pit slope prone to deep-seateddeformation.


The rock units shown on Figure 15.10 weretermed Hanging Wall, Footwall and Shear Zonerocks relative to the fault. Sympathetic faults andjoints related to the major fault zone were alignedparallel to the slope azimuth and dipped between65◦ and 80◦ into the wall. The combined influ-ence of these features created the potential fordeep-seated toppling. A second structural set con-sisted of faults and joints that dipped towardthe open pit at 45–65◦ (see detail of structuralfabric). Pervasive joints of this set controlledthe geometry of the bench faces, but were non-daylighting on the flatter inter-ramp and overallslopes.

15.4.3 Rock strength and rock masscompetency

In general, rocks were weakest in the faults andShear Zone, and strongest in the Hanging Walland Footwall. Shear strengths of discontinuitysurfaces and fault gouge were based on directshear testing, as well as back analysis of existingfailures. Faults and joints had friction angles of23◦ and 30◦, respectively, and cohesion rangedbetween 0 and 50 kPa for both discontinuitytypes.

Typical ranges of intact rock strength andHoek–Brown “disturbed” rock mass strengthsare shown in Table 15.2, with average parametersin brackets. Equivalent Mohr–Coulomb strengthparameters were estimated using a normal stress

of 1 MPa, which was considered to representaverage stress conditions at the inter-ramp scale.

15.4.4 Hydrogeology and slopedepressurization measures

The hydrogeology of the rock mass was structur-ally compartmentalized by steeply dipping faultsthat acted as structural barriers to ground waterflow. Overall, the conductivity of the rock masswas low due to alteration associated with the orebody and main structural zones. Even if some nat-ural depressurization did occur as the slope wasmined, perched ground water conditions couldoccur, leading to significant residual piezometricpressures in the slope.

Pit slope depressurization measures typicallyconsisted of sub-horizontal (slightly upwardsinclined) drain holes drilled to depths of 200–300 m (Figure 15.10). In rock masses withsteeply dipping, low conductivity structures, sub-horizontal drain holes provided a cost-effectivemeans of slope depressurization. Advanceddepressurization with drain holes that were longenough to penetrate behind the next miningphase provide considerable improvements in sta-bility, especially when consolidation of low shearstrength materials was achieved. Other prac-tical methods of slope depressurization includedvertical pumping wells and vertical drains.Underground drainage galleries, which may bejustified in cases where water pressures have

Table 15.2 Summary of rock mass strength parameters

Rock type Rock MassRating(RMR)

Unconfinedcompressivestrength (MPa)

Intact rockconstant, mi

Frictionangle, φ(◦)

Cohesion,c (kPa)

Hanging Wall 35–45 25–40 15–25 32 300(40) (35) (20)

Shear Zone 25–30 15–25 15 17 190(25) (25) 23a 50a

Footwall 40–60 25–75 15–25 40 400(50) (50) (20)

Notea Strength from back analysis and laboratory direct shear testing.


a very significant influence on stability, wereconsidered too costly for this application.

15.4.5 Slope stability analyses

Limit equilibrium rock mass analyses of kinemat-ically possible failure modes were used to evaluateinter-ramp and overall slope stability. Back ana-lyses of previous instabilities and as-built slopeswere carried out to establish appropriate mater-ial strength parameters for forward analysesof proposed slope designs. These assessmentswere complemented by development of empir-ical slope height–angle relationships to comparestable versus unstable slope geometries. Sensiti-vity analyses were carried out to identify criticalinter-ramp and overall slope heights at variousslope angles. Parametric assessments were alsocarried out to assess the sensitivity of the factorof safety to variations in slope conditions such asground water pressures. Once preliminary slopedesign parameters and initial mine plans weredeveloped, stability analyses were carried out onspecific geotechnical cross-sections to verify theinter-ramp and overall slope designs.

Figure 15.11 shows a typical stability analysismodel that incorporates the major geotechnicalunits and ground water projections for the ulti-mate pit. In the Footwall rocks, Hoek–Brownnon-linear strengths were used in the model.In the Hanging Wall, strength anisotropy wasaccounted for by using joint strengths for dipsof 45–65◦ toward the pit, and Mohr–Coulombstrengths for the remaining Hanging Wall rockmass (Table 15.2). For the Fault and ShearZone materials, Mohr–Coulomb discontinuityshear strengths were used due to the soil-likenature of these materials. The inter-ramp slopegeometries were represented by straight-line seg-ments connecting the inter-ramp slopes at themid-bench level. This simplified the model geo-metry and prevented the breakthrough of shearsurfaces on the bench faces, reducing the poten-tial for computational problems. A minimumfactor of safety of 1.2 was determined for a non-circular analysis of the overall slope. This min-imum factor of safety would not be encountereduntil the final stages of mining, and thereforewas considered acceptable for the overall slopedesign.

Critical shear surfacefactor of safety = 1.2

Figure 15.11 Example 4—non-circular limit equilibrium analysis results for the ultimate pit using SLIDE™.


15.4.6 Slope design and operationalmanagement

The geometry of the 15 m high single benches wascontrolled by plane and wedge failures involvingpervasive jointing dipping at 45–65◦. The averagebreakback angle of the bench faces was 54◦. Fora minimum design catch berm width of 7.6 m,this defined an upper bound allowable inter-ramp slope angle of 39◦. Double benching wasnot considered advisable in this case due to thehazards associated with 30 m high, bench-scalefailures.

Based on the results of deep-seated stabilityanalyses, a 38◦ inter-ramp slope angle was adop-ted with 35 m wide stepouts or ramps placed atmaximum inter-ramp slope heights of 120 m.The inter-ramp slope angle corresponding to the“angle of repose” (38◦) was also chosen to con-trol raveling and rock falls associated with deep-seated squeezing and toppling slope deformation.The ramps and stepouts both provided opera-tional flexibility if instability was encountered,and the de-coupled or re-distributed stresses inthe step-outs had the effect of reducing overallslope displacements. The resulting overall slopeangle was 34◦ at a height of 450 m.

Based on the possible mode and magnitudes ofslope deformation, contingency plans were incor-porated to maximize flexibility in the mine designand manage slope deformations. These included,but were not limited to

• adjustments to the mining rate;• dozing and backfilling of ramp crests;• waste rock buttressing at the toe of unstable

areas;• temporary splitting of mining phases, thus

allowing slope depressurization measures (e.g.drain holes) to be installed;

• offloading of the pit crest; and• provision of secondary (dual) access.

Ongoing slope management included detailedassessment of slope monitoring trends topredict periods of peak velocity or failure.Remedial measures were implemented when

slope movement exceeded threshold criteria, orexhibited potential for progressive failure.

15.5 Example 4—overall slope designin a competent rock mass

This example is based on experience with vari-ous competent rock masses, and is illustrated bya hypothetical case history of a 500 m deep openpit mine that had been operating for a number ofyears with few slope stability or operational pro-blems of any consequence. The geometries of the30 m high double benches and inter-ramp slopeswere designed and excavated to control plane andwedge failures involving steeply dipping bench-scale, and larger discontinuities. The existing pithad inter-ramp and overall slope angles of 55◦and 50◦, respectively. It was planned to increasethe pit depth to 700 m, and there was concernregarding the stability of the proposed slopes withrespect to deep-seated slope deformation.

15.5.1 Design aspects and issues

The following issues were considered duringassessment of the proposed pit slopes. First, studyof the engineering geology showed that the major-ity of the rock mass was strong, with favorablyoriented discontinuities. Limit equilibrium sta-bility analyses for the initial pit slope design,completed many years earlier for a 500 m deepopen pit, determined that the inter-ramp andoverall slope angles were controlled by bench sta-bility and geometry, and not by multi-bench oroverall stability. There were no major geologicstructures or combinations of major structuresthat appeared to control stability. In addition,the strength of the rock mass was sufficiently highthat deep-seated slope deformation was not a con-cern for the 500 m slope height. However, a studywas required of the consistency of the engineeringgeology data to the final proposed pit depth.

The second issue was the actual performanceof the pit walls with respect to the original slopedesign criteria. In this regard, the existing pitslopes were documented by measuring the benchface angles, berm widths and amount of crest


breakback. This information provided valuabledata on both the various geological structures andtheir influence on bench stability for the rangeof slope orientations, and the suitability of theoriginal slope design parameters.

A third issue was the in situ stress environment,and its influence on the potential for deep-seateddeformation that could influence the stability ofthe overall slopes.

Fortunately, due to the extended length of timethat the mine had already been operating, andthe ongoing data collection program, many ofthe parameters that were required for the studywere reasonably well understood. For those para-meters that were not well defined, sensitivityanalyses were conducted to help evaluate a rangeof possible conditions.


The bedrock comprised a series of near-verticalmafic intrusions of gabbro, peridotite and ser-pentinite within a granitic host rock. Occasionalnarrow (<10 m wide), sub-vertical northwest/southeast trending diabase dykes cut throughthe entire assemblage. The majority of the rockwas essentially fresh and unaltered, except foran approximately 50 m surficial layer of moreweathered rock. An east–west striking, sub-vertical to steeply south dipping regional faultzone intersected the floor of the pit and wasprojected to be encountered near the toe of theultimate north pit wall. The fault zone ranged upto about 15 m thick and was characterized as a“zone of breakage,” with little or no soft faultgouge. A cross-section through the north pit wallis illustrated in Figure 15.12.

Information on the structural geology wasobtained both in the pit, and below the pit bottomto the final proposed depth. This investigationinvolved mapping the pit walls to characterize theorientation and distribution of the discontinui-ties, and drilling a number of diamond drill holeswith oriented core. The mapping, supported bythe core orientation measurements, showed thatthe geologic structure was dominated by two per-sistent fault and joint sets, with an additional

four moderately to weakly developed joint setsof limited persistence. All of these structural setsappeared to rotate with the pit wall, such thattheir relationship with the orientation of the pitwalls was approximately constant. One of thepersistent fault and joint sets dipped at 35–40◦into the wall with a strike approximately parallelto the slope, while the second set had an aver-age dip of 65–80◦ out of the slope. The averagespacing of the persistent structures was estimatedto be 25–45 m. Three of the remaining disconti-nuity sets had a strike oblique into the slope at anaverage dip of at least 70◦ and a maximum per-sistence of about 30 m. The fourth joint set hada dip of less than 10◦ and an average persistenceof only 15 m.


Both laboratory testing of intact samples andestimates of strength conducted during mappingof the pit walls indicated that the various rocktypes were all strong, with uniaxial compressivestrengths ranging from 95 to 180 MPa. Labora-tory direct shear testing of natural discontinuitiesshowed that joints in all rock types had negligiblecohesion, and peak and residual friction anglesthat ranged from 30◦ to 35◦ and 26◦ to 29◦,respectively. Faults had peak and residual fric-tion angles of 24◦ and 17◦, respectively, and nocohesion.

Characterization of the various portions ofthe rock mass was conducted using establishedempirical classification systems. All of the rocksin the mine area were considered to be of goodto very good quality, having RMR values in therange of 75–85. In addition, limited triaxial test-ing of the primary rock types was conducted.Rock quality and strength data were used toestimate peak and residual rock mass strengthsusing the Hoek–Brown strength criterion (seeSection 5.4.2).

15.5.4 Hydrogeology

The hydrogeologic regime was based on theinterpreted bedrock geology, piezometer testing


Inclinometer

Survey prism

Small-scale structure

Large-scale structure

Piezometer

Sub-horizontaldrain hole

Proposed pit slope

Regionalfault zone

55° IRA

GabbroPeridotiteSerpentenite

Ramp

Figure 15.12 Example 4—engineering geology cross-section of pit slope illustrating proposed pit deepening.

and monitoring, and seepage observations withinthe open pit. Most of the rock mass in the minearea was interpreted to have a low hydraulicconductivity, typically less than about 10−9 m/s.However, conductivity along the diabase dykecontacts and in the vicinity of the continuousfaults appeared to be one to two orders of mag-nitude higher. Weathered bedrock within theupper 50 m of the ground surface was estimatedto have a hydraulic conductivity that was sim-ilar to, or slightly greater than, that of the dykesand faults. While it was evident that most of theseepage exited the pit walls within about 60 m ofthe pit bottom, it also appeared that the seepagewas compartmentalized in some areas, with localpockets of seepage being evident across the pitwalls.

Two hundred meter long sub-horizontal drainholes had proven to be effective in depressurizingthe pit walls as the pit was deepened. The drainholes had typically targeted the faults and dykecontacts.

15.5.5 Slope performance

Evaluating the performance of the existing pitslopes provided important information in theassessment of the proposed deepening of thepit. It was determined from slope documenta-tion that, with few exceptions, the slopes hadperformed as expected. The safety berms hadgenerally provided sufficient catchment room forrock fall and raveling material, and the occasionalslope failure. Prism monitoring, which had beencarried out for many years, indicated that theslopes were not experiencing undue movement.All slope movement appeared to reflect normalinitial response or slope relaxation. In addition,final wall controlled blasting had caused minimaldamage to the integrity of the rock mass.

A review of the history of the pit slopes withrespect to the nature and size of any incidentsof instability was also carried out. While it wasnoted that instability had been experienced fromtime to time, all of the slope failures were less


than 30 m high, except for one double bench fail-ure (i.e. 60 m high). Localized measures, whichincluded the use of waste rock buttresses, scalingand a minor amount of doweling and anchoring,were implemented to remediate the various areasof instability.

15.5.6 Slope stability analyses

Analysis of the geologic structure was first carriedout to investigate the variation of structural ori-entations with depth for the fault and joint sets ineach structural domain. These analyses showedthat the orientation, distribution and nature ofgeologic structural populations throughout thepit had not changed appreciably since the originalslope design study. In addition, little variationin peak orientations seemed to be occurring withdepth. Based on the updated structural geology,which included information to a depth of approx-imately 700 m, kinematic and stability analyses ofpossible failure modes were conducted. In theseanalyses, emphasis was placed on the data fromthe lower portion of the pit wall to the proposedpit bottom. Not surprisingly, considering the con-sistency of the structural populations with depthand the positive performance of the pit slopesto that point, these assessments yielded favor-able results. Based solely on limit equilibriumanalyses of potential planar, wedge and topplingfailures, it was shown that the previously designedbench, inter-ramp and overall slope designs couldbe continued.

To develop a more comprehensive understand-ing of slope deformation and to evaluate thepotential for deep-seated instability, numericalmodeling of the critical pit slopes was con-ducted. Preliminary assessments were conductedusing finite element analysis of stress conditionswith the computer program Phase2 (Rocscience,2002c). More detailed assessments were conduc-ted using UDEC (see Chapter 10). The modelincorporated the two continuous fault sets notedbefore (see Figure 15.12 for the structural modelinput into UDEC), along with the rock massstrength parameters. Sensitivity analyses wereconducted with respect to the length and spacing

of the two fault sets, ground water and in situstress. Calibration of the model was conductedby comparing the monitored slope displacementswith the model response. After a number of itera-tions, a good correlation between the two wasobtained for both horizontal and vertical dis-placements. Predictive modeling of the proposeddeepened pit was then carried out. To investi-gate the positive influence of confinement of therock mass in the area of the proposed pit bottom,where the pit walls had a relatively limited radiusof curvature, three-dimensional effects were alsochecked numerically. For the range of parametersencompassed by the sensitivity analyses, mod-eling results indicated that the rock mass wassufficiently competent that deepening of the openpit to 700 m should not adversely affect the over-all stability of the pit walls. However, the resultsalso indicated that a certain level of depressuri-zation of the pit walls would be required as thepit was deepened.

15.5.7 Implementation and ongoingevaluation

Notwithstanding the positive results of the sta-bility analyses, it is most important in any slopedesign study to monitor the ongoing behavior ofthe pit walls. This is particularly critical in situ-ations where pit slopes are expanding the boundsof previous experience in a given rock mass. Inthis case, continued monitoring of the prisms andborehole inclinometers was planned so the resultscould be compared with the movements pre-dicted by the modeling. Should monitoring resultsindicate that slope movement was beginning tooutpace predicted movements, further analysesand possibly remedial measures may have beenwarranted.

Since the analyses indicated that the pit wallsshould be depressurized as mining progressed todepth, an aggressive deep drain hole program wasrecommended. Piezometer installation was alsorecommended at a number of locations to mon-itor the depressurization over time. To maintainthe rock mass in as competent state as possible,


and to minimize the potential for rock falls andother modes of instability, the continued use ofcontrolled blasting practices on the final walls wasstrongly recommended.

15.6 Conclusions

The approaches to open pit slope design illus-trated in the first two examples described inthis chapter have been successfully implementedon numerous projects worldwide over the last25 years. They rely predominantly on limit equi-librium stability analysis techniques that are wellestablished and widely used. Early design criteriathat were based largely on analytical studies havebeen confirmed, refined and updated based onobserved slope performance. Refinements in theunderlying analytical tools continue to improve

our understanding of the failure modes and sens-itivity of stability to the key parameters. Oneof the main benefits of these approaches is theirinherent flexibility and adaptability to chan-ging pit geometry, geology and ground waterconditions.

The latter two examples introduced additionaldegrees of complexity in which potential failuremodes could not be adequately described or ana-lyzed using simple limit equilibrium techniquesalone. These examples describe the applicationof sophisticated stability analysis models thatrequire extensive calibration and mature engin-eering judgment. Regardless of the approachemployed, the reliability of the underlying geo-logic structural and engineering geology modelsof the rock mass are critical to the successfulimplementation of any slope design.

Appendix I

Stereonets for hand plotting ofstructural geology data

I.1 Introduction

Analysis of the orientation of structural geologydata involves first, plotting poles representing thedip and dip direction of each discontinuity. Thisplot will help to identify discontinuity sets, forwhich both the average orientation and the scat-ter (dispersion) can be calculated. The second stepin the analysis is to plot great circles representingthe average orientation of each set, major dis-continuities such as faults, and the dip and dipdirection of the cut face. Hand plotting of struc-tural data can be carried out on the stereonetsprovided in this appendix. Further details of theplotting procedures are provided in Chapter 2,Section 2.5.

I.2 Plotting poles

Poles can be plotted on the polar stereonet(Figure I.1) on which the dip direction is indic-ated on the periphery of the circle, and the dip ismeasured along radial lines with zero degrees atthe center. It should be noted that the stereonetshown on Figure I.1 is the lower hemisphere plotin which the dip direction scale starts at the bot-tom of the circle and increases in a clockwisedirection, with the north arrow correspondingto the dip direction of 180◦. The reason for set-ting up the scale in this manner is that if thefield readings, as measured with a structural com-pass, are plotted directly on the stereonet, thepoles are correctly plotted on the lower hemi-sphere plot.

The procedure for plotting poles is to lay a sheetof tracing paper on the printed polar net and markthe north direction and each quadrant positionaround the edge of the outer circle. A mark isthen made to show the pole that represents theorientation of each discontinuity as defined by itsdip and dip direction. Poles for shallow dippingdiscontinuities lie close to the center of the circle,and poles of steeply dipping discontinuities lieclose to the periphery of the circle.

I.3 Contouring pole concentrations

Concentrations of pole orientations can be iden-tified using a counting net such as that shown inFigure I.2. The Kalsbeek net is made up of mutu-ally overlapping hexagons, each with an area of1/100 of the full area of the stereonet. Contour-ing is performed by overlaying the counting neton the pole plot and counting the number of polesin each hexagon; this number is marked on thenet. These numbers of poles are converted intopercentages by dividing each by the total numberof poles and multiplying by 100. Once a percent-age is written in each hexagon, contours can bedeveloped by interpolation.

I.4 Plotting great circles

Great circles are plotted on the equatorial net(Figure I.3), but they cannot be plotted directlyon this net because the true dip can only be scaledoff the horizontal axis. The plotting procedure for

378 Appendix I

20 40 60 8090

80

70

60

50

40

30

20

10 018

0

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340350

100

110

120

130

140

150

160

170

Poles

Figure I.1 Equal-area polar net for plotting poles.

great circles consists of the following steps:

1 Lay a piece of tracing paper on the net witha thumbtack through the center point so thatthe tracing paper can be rotated on the net.

2 Mark the north direction of the net on thetracing paper.

3 Locate the dip direction of the plane on thescale around the circumference of the net andmark this point on the tracing paper. Notethat the dip direction scale on the equatorialnet for plotting great circles starts at the northpoint at the top of the circle and increases ina clockwise direction.

4 Rotate the tracing paper until the dip directionmark coincides with one of the horizontal axesof the net, that is, the 90◦ or 180◦ points ofthe dip direction scale.

5 Locate the arc on the net corresponding tothe dip of the plane and trace this arc on

to the paper. Note that a horizontal planehas a great circle at the circumference of thenet, and a vertical plane is represented bya straight line passing through the center ofthe net.

6 Rotate the tracing paper so that the two northpoints coincide and the great circle is orientedcorrectly.

I.5 Lines of intersection

The intersection of two planes is a straight line,which defines the direction of sliding of a wedgeformed by these two planes. The procedurefor determining the orientation of the line ofintersection between two planes is:

1 Locate the line of intersection between the twoplanes, which is represented by the point atwhich the two great circles intersect.

Figure I.2 Kalsbeek counting net for contouring pole concentrations.

0 1020

30

40

50

60

70

80

90

100

110

120

130

140

150

160170180190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340350

Figure I.3 Equal-area equatorial net for plotting poles and great circles.

380 Appendix I

2 Draw a line from the center of the net throughthe point of intersection and extend it to thecircumference of the net.

3 The trend of the line of intersection is given bythe position where the line drawn in step 2intersects the scale on the circumference ofthe net.

4 Rotate the tracing paper until the linedrawn in step 2 lies over one of the hori-

zontal axes of the net (dip directions 90◦or 180◦). The plunge of the line of inter-section is read off the scale on the hori-zontal axis, with a horizontal plunge havinga point of intersection at the circumfer-ence and a vertical plunge at the center ofthe net.

Appendix II

Quantitative description ofdiscontinuities in rock masses

II.1 Introduction

This appendix provides details of the parametersused in geological mapping and diamond drillingfor the quantitative description of rock masses.The information provided is based entirely onthe procedures drawn up by the InternationalSociety of Rock Mechanics (ISRM, 1981a), andwhich are discussed in more detail in Sections 4.2and 4.3 of this book. The objectives of usingthe ISRM procedures for geological mapping anddrill core logging are as follows. First, these pro-cedures are quantitative so that each parameter ismeasured and the results can be used either dir-ectly, or interpreted, in design. Second, the use ofstandardized procedures allows different person-nel to work to the same standards, and to producecomparable information.

The following is a description of the parametersthat describe the rock mass, together with tableslisting values used to quantify these parameters.Also provided are mapping forms that can be usedto record both geological mapping and orientedcore logging. Further information on geologicalcharacterization and methods of data collectionare discussed in Chapter 4.

II.2 Rock mass characterizationparameters

Figure II.1 illustrates the parameters that char-acterize the rock mass, and Figure II.2 showshow they can be divided into six classes relatedto the rock material and its strength, the dis-continuity characteristics, infilling properties, the

dimensions and shape of the blocks of rock, andground water conditions. Each of the parametersis discussed in this appendix.

II.2.1 Rock material description

A Rock type

The value of including the rock type in describinga rock mass is that this defines the process bywhich the rock was formed. For example, sedi-mentary rocks such as sandstone usually containwell-ordered sets of discontinuities because theyare laid down in layers, and are medium to lowstrength because they have usually only been sub-ject to moderate heating and compression. Also,the rock type gives an indication of the prop-erties of the rock mass from general experienceof their engineering performance. For example,granite tends to be strong and massive andresistant to weathering in comparison to shalewhich is often weak and fissile, and can weatherrapidly when exposed to wetting and dryingcycles.

Table II.1 shows a procedure for definingthe rock type. This procedure involves identi-fying three primary characteristics of rock asfollows:

1 Color, as well as whether light or darkminerals predominate;

2 Texture or fabric ranging from crystalline,granular or glassy; and

3 Grain size that can range from clay particlesto gravel (Table II.2).

382 Appendix II

B: wallstrength

J:Persistence

(I )

K: numberof sets

B, J1, J2

E:orientation�-dip dirn.

�-dip

A: rocktype

I: spacingS1= Sapp sin�

Sbedding

H: fillingtype, width

D:discontinuity

type:bedding, fault

etc.

F:roughness

(i )

L:block

size/shape

M: seepage

G: aperture(open)

Sapp

�

��

N

J1J2

S1

B

I

Sbedding

Figure II.1Diagramillustrating rockmass properties(Wyllie, 1999).

B Rock strength

The compressive strength of the rock comprisingthe discontinuity surfaces is an important com-ponent of shear strength and deformability, espe-cially if the surfaces are in direct rock-to-rockcontact as in the case of unfilled joints. Slightshear displacement of individual joints causedby shear stresses within the rock mass, oftenresults in small asperity contact areas. Wherethe concentrated stresses locally approach orexceed the compression strength of the rock wall

materials, asperity damage occurs. The surfacestrength is quantified in the determination ofshear strength as the Joint Compressive Strength(JCS) as discussed in Section 3.4.2(b). Table II.3defines ranges of rock material strength with acorresponding grade (R6 etc.) related to simplefield identification procedures.

C Weathering

• Rock masses are frequently weathered nearground level, and are sometimes altered byhydrothermal processes. The weathering (andalteration) is generally more pronounced on

Discontinuities in rock masses 383

5. Groundwater M. Seepage

4. Rock mass description I. Spacing J. Persistence K. Number of sets L. Block size and shape

3. Infilling H. Infilling type/Width

2. Discontinuity description D. Type E. Orientation F. Roughness G. Aperture

QUANTITATIVE DESCRIPTIONOF DISCONTINUITIES IN

ROCK MASSES

1. Rock material description A. Rock type B. Wall strength C. Weathering

Figure II.2 Parameters describing rock masscharacteristics.

the rock exposed on the discontinuity surfacesthan in the interior of rock blocks becausewater flow occurs in the discontinuities. Thisresults in the rock strength on the discontinu-ity surfaces being less than that of the fresherrock found in the interior of the rock blocks.A description of the state of weathering oralteration both for the rock material and forthe rock mass is therefore, an essential part ofthe description of the rock mass.

• There are two main results of weathering:one dominated by mechanical disintegration,the other by chemical decomposition includ-ing solution. Generally, both mechanical andchemical effects act together, but, dependingon climatic regime, one or other may be dom-inant. Mechanical weathering results in open-ing of discontinuities, the formation of newdiscontinuities by rock fracture, the openingof grain boundaries, and the fracture or cleav-age of individual mineral grains. Chemicalweathering results in discoloration of the rockand leads to the eventual decomposition ofsilicate minerals to clay minerals; some min-erals, notably quartz, resist this action andmay survive unchanged. Solution is an aspectof chemical weathering, which is particularlyimportant in the case of carbonate and salineminerals.

• The relatively thin “skin” of surface rockthat affects shear strength and deformabilitycan be tested by means of simple index tests.The apparent uniaxial compression strengthcan be estimated both from Schmidt hammertests and from scratch and geological hammertests, since the latter have been approximatelycalibrated against a large body of test dataas shown in Table II.3.

• Mineral coatings will affect the shear strengthof discontinuities to a marked degree, if thesurfaces are planar and smooth. The type ofmineral coatings should be described wherepossible. Samples should be taken when indoubt.

Table II.4 defines grades of rock weathering.

II.2.2 Discontinuity description

D Discontinuity type

• The discontinuity type is useful in the descrip-tion of the rock mass because each typehas properties that influence the behaviorof the rock mass. For example, faults canhave lengths of several kilometers and containlow strength gouge, while joints lengths

Tab

leII

.1R

ock

type

clas

sific

atio

n

Pyro

cla

sti

cC

hem

ical O

rgan

ic

BE

DD

ED

FO

LIA

TE

DM

AS

SIV

E

Da

rk m

ine

rals

Gra

insiz

e

(mm

)

Gra

ins o

f ro

ck, quart

z, fe

ldspar

and

min

era

lsA

t le

ast

50

% o

f g

rain

sa

re o

f fin

e-g

rain

ed

vo

lca

nic

ro

ck

Qu

art

z,

feld

sp

ars

, m

ica

s,

acic

ula

r d

ark

m

ine

rals

Acid

rocks

Inte

rme

dia

te

rocks

Ba

sic

rocks

Ultra

-ba

sic

ro

cks

Very

coars

egra

ined

60

Gra

ins a

re o

f ro

ck f

rag

me

nts

Ro

un

de

d g

rain

s:

CO

NG

LO

ME

RA

TE

MIG

MA

TIT

E

Coars

e

gra

ined

2

An

gu

lar

gra

ins:

BR

EC

CIA

GR

AN

ITE

DIO

RIT

EG

AB

BR

O

MIC

RO

-

GR

AN

ITE

MIC

RO

-

DIO

RIT

E

DO

LE

RIT

E

Very

fin

e

gra

ined

CA

LC

ILU

TIT

EV

ery

fin

e-g

rain

ed

TU

FF

GLA

SS

YT

AC

HY

LY

TE

Re

fere

nce

: G

eo

log

ica

l S

ocie

ty E

ng

ine

eri

ng

Gro

up

Wo

rkin

g P

art

y (

19

77

).

OB

SID

EIA

N a

nd P

ITC

HS

TO

NE

Me

diu

m

gra

ined

ARENACEOUS

Fin

e

gra

ined

AREGILLACEOUS or LUTACEOUS

0.0

6

0.0

02

SA

ND

ST

ON

E:

Gra

ins a

re m

ain

ly m

ine

ral

fra

gm

en

ts Q

UA

RT

Z

SA

ND

ST

ON

E:

95

% q

ua

rts,

vo

ids e

mp

ty

or

ce

me

nte

d

AR

KO

SE

: 7

5%

qu

art

z,

up

to

25

%

feld

sp

ar:

vo

ids e

mp

ty o

f ce

me

nte

d

AR

GIL

LA

RC

EO

US

SA

ND

ST

ON

E: 75%

qu

art

z,

15

% +

fin

e d

etr

ita

l m

ate

ria

l

MU

DS

TO

NE

SH

AL

E:

fissile

mu

dsto

ne

SIL

TS

ON

E 5

0%

fin

e-g

rain

ed

pa

rtic

les

CLA

YS

TO

NE

50%

very

fin

e-g

rain

ed

part

icle

s

CA

LC

AR

EO

US

MU

DS

TO

NE

CA

LC

AR

EN

ITE

PY

RO

XE

NIT

E a

nd

PE

RID

OT

ITE

SE

RP

EN

TIN

E

RH

YO

LIT

EA

ND

ES

ITE

BA

SA

LT

Ign

eo

us

MA

SS

IVE

Lig

ht

co

lou

red

min

era

ls a

re

qu

art

z,

feld

sp

ar,

mic

a a

nd

feld

sp

ar-

like

min

era

ls

CA

LC

ISIL

TIT

EF

ine-g

rain

ed

TU

FF

GN

EIS

S A

lte

rna

te

layers

of gra

nula

r

an

d f

lake

y m

ine

rals

SC

HIS

T

PH

YLLIT

E

SLA

TE

MY

LO

NIT

E

HO

RN

FE

LS

MA

RB

LE

GR

AN

ULIT

E

QU

AR

TZ

ITE

AM

PH

IBO

LIT

ET

UF

F

PE

GM

AT

ITE

Meta

mo

rph

ic

Ro

un

de

d g

rain

s

AG

GLO

ME

RA

TE

An

gu

lar

ga

ins

VO

LC

AN

IC B

RE

CC

IA

SA

LIN

ER

OC

KS

Ha

lite

Anhydrite

Gypsum

CH

ER

T

FLIN

T

CO

AL

OT

HE

RS

Gen

eti

c G

rou

p

Usu

al S

tru

ctu

re

CO

MP

OS

ITIO

N

No

te:

Nu

mb

ers

ca

n b

e u

se

d t

o id

en

tify

ro

ck t

yp

es o

n d

ata

sh

ee

ts (

se

e A

pp

en

dix

III

).

RUDACEOUS

Detr

ital S

ed

imen

tary

BE

DD

ED

At

lea

st

50

% o

f g

rain

s a

re o

f

ca

rbo

na

te

LIMESTONE (undifferentiated)

CA

LC

IRU

DIT

E


Table II.2 Grain size scale

Description Grain size

Boulders 200–600 mm (7.9–23.6 in)Cobbles 60–200 mm (2.4–7.9 in)Coarse gravel 20–60 mm (0.8–0.24 in)Medium gravel 6–20 mm (0.2–0.8 in)Fine gravel 2–6 mm (0.1–0.2 in)Coarse sand 0.6–2 mm (0.02–0.1 in)Medium sand 0.2–0.6 mm (0.008–0.02 in)Fine sand 0.06–0.2 mm (0.002–0.008 in)Silt, clay <0.06 mm (<0.002 in)

Table II.3 Classification of rock material strengths

Grade Description Field identification Approximatecompressive(MPa)

Range ofstrength(psi)

R6 Extremely strong rock Specimen can only be chipped withgeological hammer.

>250 >36,000

R5 Very strong rock Specimen requires many blows ofgeological hammer to fracture it.

100–250 15,000–36,000

R4 Strong rock Specimen requires more than oneblow with a geological hammer tofracture it.

50–100 7000–15,000

R3 Medium weak rock Cannot be scraped or peeled with apocket knife; specimen can befractured with single firm blow ofgeological hammer.

25–50 3500–7000

R2 Weak rock Can be peeled with a pocket knife;shallow indentations made by firmblow with point of geologicalhammer.

5–25 725–3500

R1 Very weak rock Crumbles under firm blows withpoint of geological hammer; can bepeeled by a pocket knife.

1–5 150–725

R0 Extremely weak rock Indented by thumbnail. 0.25–1 35–150S6 Hard clay Indented with difficulty by

thumbnail.>0.5 >70

S5 Very stiff clay Readily indented by thumbnail. 0.25–0.5 35–70S4 Stiff clay Readily indented by thumb but

penetrated only with greatdifficulty.

0.1–0.25 15–35

S3 Firm clay Can be penetrated several inches bythumb with moderate effort.

0.05–0.1 7–15

S2 Soft clay Easily penetrated several inches bythumb.

0.025–0.05 4–7

S1 Very soft clay Easily penetrated several inches byfist.

<0.025 <4

386 Appendix II

Table II.4 Weathering and alteration grades

Grade Term Description

I Fresh No visible sign of rock material weathering;perhaps slight discoloration on majordiscontinuity surfaces.

II Slightly weathered Discoloration indicates weathering of rockmaterial and discontinuity surfaces. All the rockmaterial may be discolored by weatheringand may be somewhat weaker externally than inits fresh condition.

III Moderately weathered Less than half of the rock material is decomposedand/or disintegrated to a soil. Fresh ordiscolored rock is present either as a continuousframework or as corestones.

IV Highly weathered More than half of the rock material isdecomposed and/or disintegrated to a soil. Freshor discolored rock is present either as adiscontinuous framework or as corestones.

V Completely weathered All rock material is decomposed and/ordisintegrated to soil. The original mass structureis still largely intact.

VI Residual soil All rock material is converted to soil. The massstructure and material fabric are destroyed.There is a large change in volume, but the soilhas not been significantly transported.

usually do not exceed a few meters andthey often contain no infilling. Section 3.3.3describes the characteristics of the most com-mon types of discontinuities, which includefaults, bedding, foliation, joints, cleavage andschistosity.

E Discontinuity orientation

• The orientation of a discontinuity in space isdescribed by the dip of the line of steepestdeclination measured from horizontal, andby the dip direction measured clockwise fromtrue north. Example: dip (ψ)/dip direction (α)(45◦/025◦).

• The orientation of discontinuities relative toan engineering structure largely controls thepossibility of unstable conditions or excessivedeformations developing. The importance oforientation increases when other conditionsfor deformation are present, such as lowshear strength and a sufficient number of

discontinuities or joint sets for sliding tooccur.

• The mutual orientation of discontinuities willdetermine the shape of the individual blockscomprising the rock mass.

F Roughness

• The roughness of a discontinuity surface is apotentially important component of its shearstrength, especially in the case of undisplacedand interlocked features (e.g. unfilled joints).The importance of surface roughness declinesas aperture, or infilling thickness, or the degreeof previous displacement increases.

• The roughness can be characterized bywaviness, and the unevenness or asperities.Waviness describes large-scale undulations,which, if interlocked and in contact, causedilation during shear displacement since theyare too large to be sheared off. Unevennessor asperities describe small-scale roughness.


Asperities will tend to be damaged duringshear displacement if the ratio of the rockstrength on the discontinuity surface to thenormal stress level is low, in which casethere will be little dilation on these small-scalefeatures.

• In practice, both waviness and asperities canbe measured in the field, while asperitiescan be measured as a component of a directshear test.

• Roughness can be sampled by linear profilestaken parallel to the direction of sliding, thatis, parallel to the dip (dip vector). In cases

where sliding is controlled by two intersectingdiscontinuity planes, the direction of potentialsliding is parallel to the line of intersection ofthe planes.

• The purpose of all roughness sampling isfor estimating or calculating shear strengthand dilation. Presently available methods ofinterpreting roughness profiles and estimat-ing shear strength include measuring the i

value (or inclination) of the irregularities, orthe Joint Roughness Coefficient (JRC) of thesurface (Figure II.3). The contribution of thesurface roughness to the total friction angle

Typical roughness profiles for JRC range:

0 – 2

2– 4

4 – 6

6 – 8

8 – 10

10 – 12

12 – 14

14 – 16

16 – 18

18 – 20

1

2

3

4

5

6

7

8

9

10

0 5 10cm Scale

Figure II.3 Roughness profiles andcorresponding range of JRC (jointroughness coefficient) values (ISRM,1981a).

388 Appendix II

of a surface is discussed in more detail inSection 3.4.3.

Descriptive terms that can be used to defineroughness are a combination of small-scale fea-tures (several centimeters dimensions): rough,smooth, slickensided, and larger-scale features(several meters dimensions): stepped, undulat-ing, planar. These terms can be combined todescribe decreasing levels of roughness shown inTable II.5.

G Aperture

• Aperture is the perpendicular distance separ-ating the adjacent rock surfaces of an opendiscontinuity, in which the intervening spaceis air or water filled. Aperture is therebydistinguished from the width of a filled discon-tinuity. Discontinuities that have been filled(e.g. with clay) also come under this categoryif the filling material has been washed outlocally.

Table II.5 Descriptive terms for roughness

I Rough, steppedII Smooth, stepped

III Slickensided, steppedIV Rough, undulatingV Smooth, undulating

VI Slickensided, undulatingVII Rough, planar

VIII Smooth, planarIX Slickensided, planar

• Large aperture can result from sheardisplacement of discontinuities having appre-ciable roughness and waviness, from tensileopening, from outwash of infillings, and fromsolution. Steeply dipping discontinuities thathave opened in tension as a result of valleyerosion or glacial retreat may have very largeapertures.

• In most sub-surface rock masses, apertureswill probably be less than half a millimeter,compared to the tens, hundreds or even thou-sands of millimeters width of some of theoutwash or extension varieties. Unless discon-tinuities are exceptionally smooth and planar,it will not be of great significance to the shearstrength that a “closed” feature is 0.1 mmwide or 1.0 mm wide. However, hydraulicconductivity is influenced by minor changesin aperture.

• Unfortunately, visual observation of smallapertures is inherently unreliable since, withthe possible exceptions of drilled holes andbored tunnels, visible apertures are bound tobe disturbed by blasting or surface weatheringeffects. Aperture can be measured indirectlyby hydraulic conductivity testing.

• Apertures are recorded from the point of viewof both their loosening and conducting capa-city. Joint water pressure, inflow of water andoutflow of storage products (both liquid andgas) will all be affected by aperture.

Apertures can be described by the terms listed inTable II.6.

Table II.6 Aperture dimensions

Aperture Description

<0.1 mm Very tight0.1–0.25 mm Tight “Closed” features0.25–0.5 mm Partly open0.5–2.5 mm Open2.5–10 mm Moderately wide “Gapped” features>10 mm Wide1–10 cm Very wide10–100 cm Extremely wide “Open” features>1 m Cavernous


II.2.3 Infilling description

H Infilling type and width

• Infilling is the term for material separatingthe adjacent rock surfaces of discontinuit-ies, e.g. calcite, chlorite, clay, silt, faultgouge, breccia, etc. The perpendicular dis-tance between the adjacent rock surfaces istermed the width of the filled discontinuity, asopposed to the aperture of a gapped or openfeature.

• Due to the variety of occurrences, filled dis-continuities display a wide range of phys-ical behavior, in particular as regards theirshear strength deformability and conductiv-ity. Short-term and long-term behavior may bequite different such that it is easy to be misledby favorable short-term conditions.

• The wide range of physical behavior dependson many factors of which the following areprobably the most important:

• Mineralogy of filling material (Table II.1);• Grading or particle size (Table II.2);• Over-consolidation ratio;• Water content and conductivity

(Table II.11);• Previous shear displacement;• Surface roughness (Figure II.3 and

Table II.5);• Width (Table II.6); and• Fracturing or crushing of surface rock.

• Every attempt should be made to record thesefactors, using quantitative descriptions wherepossible, together with sketches and/or colorphotographs of the most important occur-rences. Certain index tests are suggested fora closer investigation of major discontinuitiesconsidered to be a threat to stability. In spe-cial cases, the results of these field descriptionsmay warrant the recommendation for large-scale in situ testing, possibly in the case ofslopes containing a critical structural featureabove an important facility.

II.2.4 Rock mass description

I Spacing

• The spacing of adjacent discontinuities largelycontrols the size of individual blocks of intactrock. Several closely spaced sets tend togive conditions of low rock mass cohesion,whereas those that are widely spaced are muchmore likely to yield interlocking conditions.These effects depend upon the persistence ofthe individual discontinuities.

• In exceptional cases, a close spacing maychange the mode of failure of a rock massfrom plane or wedge to circular or even to flow(e.g. a “sugar cube” shear zone in quartzite).With exceptionally close spacing the orient-ation is of little consequence as failure mayoccur through rotation or rolling of the smallrock pieces.

• As in the case of orientation, the import-ance of spacing increases when other con-ditions for deformation are present, that is,low shear strength and a sufficient numberof discontinuities or joint sets for sliding tooccur.

• The spacing of individual discontinuities andassociated sets has a strong influence on themass conductivity and seepage characteristics.In general, the hydraulic conductivity of anygiven set will be inversely proportional tothe spacing, if individual joint apertures arecomparable.

Spacing can be described by the terms listed inTable II.7.

Table II.7 Spacing dimensions

Description Spacing (mm)

Extremely close spacing <20Very close spacing 20–60Close spacing 60–200Moderate spacing 200–600Wide spacing 600–2000Very wide spacing 2000–6000Extremely wide spacing >6000

390 Appendix II

J Persistence

• Persistence implies the areal extent or size of adiscontinuity within a plane. It can be crudelyquantified by observing the discontinuity tracelengths on the surface of exposures. It is oneof the most important rock mass parameters,but one of the most difficult to quantify.

• The discontinuities of one particular set willoften be more continuous than those of theother sets. The minor sets will therefore tendto terminate against the primary features, orthey may terminate in solid rock.

• In the case of rock slopes, it is of the greatestimportance to attempt to assess the degreeof persistence of those discontinuities thatare unfavorably orientated for stability. Thedegree to which discontinuities persist beneathadjacent rock blocks without terminating insolid rock or terminating against other discon-tinuities determines the degree to which failureof intact rock would be involved in eventualfailure. Perhaps more likely, it determines thedegree to which “down-stepping” would haveto occur between adjacent discontinuities fora slip surface to develop. Persistence is alsoof the greatest importance to tension crackdevelopment behind the crest of a slope.

• Frequently, rock exposures are small com-pared to the area or length of persistent dis-continuities, and the real persistence can onlybe guessed. Less frequently, it may be possibleto record the dip length and the strike length ofexposed discontinuities and thereby estimatetheir persistence along a given plane throughthe rock mass using probability theory. How-ever, the difficulties and uncertainties involvedin the field measurements will be considerablefor most rock exposures.

Persistence can be described by the terms listed inTable II.8.

K Number of sets

• The mechanical behavior of a rock mass andits appearance will be influenced by the num-ber of sets of discontinuities that intersect one

Table II.8 Persistencedimensions

Very low persistence <1 mLow persistence 1–3 mMedium persistence 3–10 mHigh persistence 10–20 mVery high persistence >20 m

another. The mechanical behavior is especiallyaffected since the number of sets determinesthe extent to which the rock mass can deformwithout involving failure of the intact rock.The number of sets also affects the appearanceof the rock mass due to the loosening anddisplacement of blocks in both natural andexcavated faces (Figure II.4).

• The number of sets of discontinuities may bean important feature of rock slope stability,in addition to the orientation of discon-tinuities relative to the face. A rock masscontaining a number of closely spaced jointsets may change the potential mode of slopefailure from translational or toppling torotational/circular.

• In the case of tunnel stability, three ormore sets will generally constitute a three-dimensional block structure having a con-siderably more “degrees of freedom” fordeformation than a rock mass with less thanthree sets. For example, a strongly foliatedphyllite with just one closely spaced joint setmay give equally good tunneling conditionsas a massive granite with three widely spacedjoint sets. The amount of overbreak in a tun-nel will usually be strongly dependent on thenumber of sets.

The number of joint sets occurring locally (e.g.along the length of a tunnel) can be describedaccording to the following scheme:

I massive, occasional random joints;II one joint set;III one joint set plus random;IV two joint sets;V two joint sets plus random;VI three joint sets;


1

One joint set Three joint sets plus random (R)

2

3

R

1

Figure II.4 Examples illustrating the effect of the number of joint sets on the mechanical behavior andappearance of rock masses (ISRM, 1981a).

VII three joint sets plus random;VIII four or more joint sets; andIX crushed rock, earth-like.

Major individual discontinuities should berecorded on an individual basis.

L Block size and shape

• Block size is an important indicator of rockmass behavior. Block dimensions are determ-ined by discontinuity spacing, by the numberof sets, and by the persistence of the discon-tinuities delineating potential blocks.

• The number of sets and the orientationdetermine the shape of the resulting blocks,which can take the approximate form ofcubes, rhombohedra, tetrahedrons, sheets,etc. However, regular geometric shapes arethe exception rather than the rule since thejoints in any one set are seldom consistentlyparallel. Jointing in sedimentary rocks usuallyproduces the most regular block shapes.

• The combined properties of block size andinterblock shear strength determine the mech-anical behavior of the rock mass under givenstress conditions. Rock masses composedof large blocks tend to be less deformable,and in the case of underground construction,

develop favorable arching and interlocking.In the case of slopes, a small block sizemay cause the potential mode of failure toresemble that of soil, (i.e. circular/rotational)instead of the translational or toppling modesof failure usually associated with discon-tinuous rock masses. In exceptional cases,“block” size may be so small that flowoccurs, as with a “sugar-cube” shear zones inquartzite.

• Rock quarrying and blasting efficiency arerelated to the in situ block size. It may behelpful to think in terms of a block size dis-tribution for the rock mass, in much the sameway that soils are categorized by a distributionof particle sizes.

• Block size can be described either by means ofthe average dimension of typical blocks (blocksize index Ib), or by the total number of jointsintersecting a unit volume of the rock mass(volumetric joint count Jv).

Table II.9 lists descriptive terms give animpression of the corresponding block size.

Values of Jv > 60 would represent crushedrock, typical of a clay-free crushed zone.

Rock masses. Rock masses can be described bythe following adjectives to give an impression ofblock size and shape (Figure II.5).

392 Appendix II

(i) massive—few joints or very wide spacing(ii) blocky—approximately equidimensional(iii) tabular—one dimension considerably smaller

than the other two

Table II.9 Block dimensions

Description Jv (joints/m3)

Very large blocks <1.0Large blocks 1–3Medium-sized blocks 3–10Small blocks 10–30Very small blocks >30

(iv) columnar—one dimension considerablylarger than the other two

(v) irregular—wide variations of block size andshape

(vi) crushed—heavily jointed to “sugar cube”

II.2.5 Ground water

M Seepage

• Water seepage through rock masses resultsmainly from flow through water conduct-ing discontinuities (“secondary” hydraulicconductivity). In the case of certain sedimentary

(a) (b)

(c) (d)

Figure II.5 Sketches of rock masses illustrating block shape: (a) blocky; (b) irregular; (c) tabular; and(d) columnar (ISRM, 1981a).


rocks, such as poorly indurated sandstone,the “primary” hydraulic conductivity of therock material may be significant such thata proportion of the total seepage occursthrough the pores. The rate of seepage isproportional to the local hydraulic gradientand to the relevant directional conductiv-ity, proportionality being dependent on lam-inar flow. High velocity flow through opendiscontinuities may result in increased headlosses due to turbulence.

• The prediction of ground water levels, likelyseepage paths, and approximate water pres-sures may often give advance warning ofstability or construction difficulties. Thefield description of rock masses must inev-itably precede any recommendation for fieldconductivity tests, so these factors shouldbe carefully assessed at early stages of theinvestigation.

• Irregular ground water levels and perchedwater tables may be encountered in rockmasses that are partitioned by persistentimpermeable features such as dykes, clay-filleddiscontinuities or low conductivity beds. Theprediction of these potential flow barriers andassociated irregular water tables is of con-siderable importance, especially for projectswhere such barriers might be penetrated atdepth by tunneling, resulting in high pressureinflows.

• Water seepage caused by drainage intoan excavation may have far-reaching con-sequences in cases where a sinking groundwater level would cause settlement of nearbystructures founded on overlying clay deposits.

• The approximate description of the localhydrogeology should be supplemented withdetailed observations of seepage from indi-vidual discontinuities or particular sets,according to their relative importance to sta-bility. A short comment concerning recent pre-cipitation in the area, if known, will be helpfulin the interpretation of these observations.Additional data concerning ground watertrends, and rainfall and temperature recordswill be useful supplementary information.

• In the case of rock slopes, the preliminarydesign estimates will be based on assumedvalues of effective normal stress. If, as a resultof field observations, one has to conclude thatpessimistic assumptions of water pressure arejustified, such as a tension crack full of waterand a rock mass that does not drain readily,then this will clearly influence the slope design.So also will the field observation of rock slopeswhere high water pressures can develop dueto seasonal freezing of the face that blocksdrainage paths.

Seepage from individual unfilled and filled dis-continuities or from specific sets exposed in atunnel or in a surface exposure, can be assessedaccording to the descriptive terms in Tables II.10and II.11.

In the case of an excavation that acts as a drainfor the rock mass, such as a tunnel, it is helpful ifthe flow into individual sections of the structureare described. This should ideally be performedimmediately after excavation since ground waterlevels, or the rock mass storage, may be depleted

Table II.10 Seepage quantities in unfilleddiscontinuities

Seepagerating

Description

I The discontinuity is very tight anddry, water flow along it does notappear possible.

II The discontinuity is dry with noevidence of water flow.

III The discontinuity flow is dry butshows evidence of water flow, thatis, rust staining.

IV The discontinuity is damp but nofree water is present.

V The discontinuity shows seepage,occasional drops of water, but nocontinuous flow.

VI The discontinuity shows acontinuous flow of water—estimatel/ min and describe pressure, that is,low, medium, high.

394 Appendix II

Table II.11 Seepage quantities in filled discontinuities

Seepagerating

Description

I The filling materials are heavily consolidated and dry,significant flow appears unlikely due to very lowpermeability.

II The filling materials are damp, but no free water ispresent.

III The filling materials are wet, occasional drops of water.IV The filling materials show signs of outwash, continuous

flow of water—estimate l/ min.V The filling materials are washed out locally,

considerable water flow along out-washchannels—estimate l/ min and describe pressure that islow, medium, high.

VI The filling materials are washed out completely, veryhigh water pressures experienced, especially on firstexposure—estimate l/ min and describe pressure.

Table II.12 Seepage quantities in tunnels

Rock mass (e.g. tunnel wall)

Seepage rating Description

I Dry walls and roof, no detectable seepage.II Minor seepage, specify dripping discontinuities.III Medium inflow, specify discontinuities with continuous flow

(estimate l/ min /10 m length of excavation).IV Major inflow, specify discontinuities with strong flows

(estimate l/ min /10 m length of excavation).V Exceptionally high inflow, specify source of exceptional flows

(estimate l/ min /10 m length of excavation).

rapidly. Descriptions of seepage quantities aregiven in Table II.12.

• A field assessment of the likely effectiveness ofsurface drains, inclined drill holes, or drainagegalleries should be made in the case of majorrock slopes. This assessment will depend onthe orientation, spacing and apertures of therelevant discontinuities.

• The potential influence of frost and ice on theseepage paths through the rock mass shouldbe assessed. Observations of seepage fromthe surface trace of discontinuities may bemisleading in freezing temperatures. The pos-sibility of ice-blocked drainage paths should

be assessed from the points of view of sur-face deterioration of a rock excavation, andof overall stability.

II.3 Field mapping sheets

The two mapping sheets included with thisappendix provide a means of recording thequalitative geological data described in thisappendix.

Sheet 1—Rock mass description sheet describesthe rock material in terms of its color, grainsize and strength, the rock mass in terms of theblock shape, size, weathering and the number ofdiscontinuity sets and their spacing.

396 Appendix II

Sheet 2—Discontinuity survey data sheetdescribes the characteristics of each discontinuityin terms of its type, orientation, persistence,aperture/width, filling, surface roughness and

water flow. This sheet can be used for recordingboth outcrop (or tunnel) mapping data, andoriented core data (excluding persistence andsurface shape).

Appendix III

Comprehensive solutionwedge stability

III.1 Introduction

This appendix presents the equations and proce-dure to calculate the factor of safety for a wedgefailure as discussed in Chapter 7. This compre-hensive solution includes the wedge geometrydefined by five surfaces, including a sloped uppersurface and a tension crack, water pressures, dif-ferent shear strengths on each slide plane, andup to two external forces (Figure III.1). Externalforces that may act on a wedge include tensionedanchor support, foundation loads and earthquakemotion. The forces are vectors defined by theirmagnitude, and their plunge and trend. If neces-sary, several force vectors can be combined tomeet the two force limit. It is assumed that allforces act through the center of gravity of thewedge so no moments are generated, and thereis no rotational slip or toppling.

III.2 Analysis methods

The equations presented in this appendix areidentical to those in appendix 2 of Rock SlopeEngineering, third edition (Hoek and Bray,1981). These equations have been found to beversatile and capable of calculating the stabi-lity of a wide range of geometric and geotech-nical conditions. The equations form the basis ofthe wedge stability analysis programs SWEDGE(Rocscience, 2001) and ROCKPACK III (Watts,2001). However, two limitations to the analysisare discussed in Section III.3.

As an alternative to the comprehensive ana-lysis presented in this appendix, there are two

1 5

2

3

4L

H1


Figure III.1 Dimensions and surfaces defining sizeand shape of wedge.

shorter analyses that can be used for a more lim-ited set of input parameters. In Section 7.3, acalculation procedure is presented for a wedgeformed by planes 1, 2, 3 and 4 shown in Fig-ure III.1, but with no tension crack. The shearstrength is defined by different cohesions and fric-tion angles on planes 1 and 2, and the waterpressure condition assumed is that the slope issaturated. However, no external forces can beincorporated in the analysis.

A second rapid calculation method is presen-ted in the first part of appendix 2 in Rock SlopeEngineering, third edition. This analysis alsodoes not incorporate a tension crack or externalforces, but does include two sets of shear strengthparameters and water pressure.

Comprehensive solution wedge stability 399

III.3 Analysis limitations

For the comprehensive stability analysis presen-ted in this appendix there is one geometriclimitation related to the relative inclinations ofplane 3 and the line of intersection, and a specificprocedure for modifying water pressures. Thefollowing is a discussion of these two limitations.

Wedge geometry. For wedges with steepupper slopes (plane 3), and a line of intersec-tion that has a shallower dip than the upper slope(i.e. ψ3 > ψi), there is no intersection betweenthe plane and the line; the program will ter-minate with the error message “Tension crackinvalid” (see equations (III.50) to (III.53)). Thereason for this error message is that the calcula-tion procedure is to first calculate the dimensionsof the overall wedge from the slope face to theapex (intersection of the line of intersection withplane 3). Then the dimensions of a wedge betweenthe tension crack and the apex are calculated.Finally, the dimensions of the wedge between theface and the tension crack are found by subtract-ing the overall wedge from the upper wedge (seeequations (III.54) to (III.57).

However, for the wedge geometry where (ψ3 >

ψi), a wedge can still be formed if a tension crack(plane 5) is present, and it is possible to cal-culate a factor of safety using a different set ofequations. Programs that can investigate the sta-bility wedges with this geometry include YAWC(Kielhorn, 1998) and (PanTechnica, 2002).

Water pressure. The analysis incorporates theaverage values of the water pressure on the slid-ing planes (u1 and u2), and on the tension crack(u5). These values are calculated assuming thatthe wedge is fully saturated. That is, the watertable is coincident with the upper surface of theslope (plane 3), and that the pressure drops tozero where planes 1 and 2 intersect the slope face(plane 4). These pressure distributions are simu-lated as follows. Where no tension crack exists,the water pressures on planes 1 and 2 are givenby u1 = u2 = γwHw/6, where Hw is the ver-tical height of the wedge defined by the two endsof the line of intersection. The second method

allows for the presence of a tension crack andgives u1 = u2 = u5 = γwH5w/3, where H5wis the depth of the bottom vertex of the ten-sion crack below the upper ground surface. Thewater forces are then calculated as the productof these pressures and the areas of the respectiveplanes.

To calculate stability of a partially saturatedwedge, the reduced pressures are simulated byreducing the unit weight of the water, γw. Thatis, if it is estimated that the tension crack is one-third filled with water, then a unit weight of γw/3is used as the input parameter. It is consideredthat this approach is adequate for most purposesbecause water levels in slopes are variable anddifficult to determine precisely.

III.4 Scope of solution

This solution is for computation of the factor ofsafety for translational slip of a tetrahedral wedgeformed in a rock slope by two intersecting dis-continuities (planes 1 and 2), the upper groundsurface (plane 3), the slope face (plane 4), and atension crack (plane 5 (Figure III.1)). The solu-tion allows for water pressures on the two slideplanes and in the tension crack, and for differ-ent strength parameters on the two slide planes.Plane 3 may have a different dip direction to thatof plane 4. The influence of an external load E

and a cable tension T are included in the ana-lysis, and supplementary sections are provided forthe examination of the minimum factor of safetyfor a given external load, and for minimizing theanchoring force required for a given factor ofsafety.

The solution allows for the followingconditions:

(a) interchange of planes 1 and 2;(b) the possibility of one of the planes overlying

the other;(c) the situation where the crest overhangs the

toe of the slope (in which case η = −1); and(d) the possibility of contact being lost on either

plane.

400 Appendix III

III.5 Notation

The wedge geometry is illustrated in Figure III.1;the following input data are required:

ψ, α = dip and dip direction of plane, or plungeand trend of force

H1 = slope height referred to plane 1L = distance of tension crack from crest,

measured along the trace of plane 1u = average water pressure on planes 1

and 2c = cohesion of each slide planeφ = angle of friction of each slide planeγ = unit weight of rock

γw = unit weight of waterT = anchor tensionE = external loadη = −1 if face is overhanging, and +1 if face

does not overhang

Other terms used in the solution are as follows:

FS = factor of safety against sliding alongthe line of intersection, or on plane 1or plane 2

A = area of sliding plane or tension crackW = weight of wedgeV = water thrust on tension crack (plane 5)

Na = total normalforce of plane 1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

when contact isSa = shear force onplane 1 maintained on

Qa = shear resistanceon plane 1

plane 1 only

FS1 = factor of safety

Nb = total normalforce on plane 2

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

when contact isSb = shear force onplane 2 maintained on

Qb = shear resistanceon plane 2

plane 2 only


N1, N2 = effective normalreactions

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

when contact isS = total shear force onplanes 1 and 2 maintained on

Q = total shearresistance onplanes 1 and 2

both planes 1and 2


N′1, N

′2, S

′, etc. = values of N1, N2, S etc.

when T = 0N

′′1, N

′′2S

′′, etc. = values of N1, N2, S etc.

when E = 0a = unit normal vector for plane 1b = unit normal vector for plane 2d = unit normal vector for plane 3f = unit normal vector for plane 4f5 = unit normal vector for plane 5g = vector in the direction of intersection

line of 1, 4g5 = vector in the direction of intersection

line of 1, 5i = vector in the direction of intersection

line of 1, 2j = vector in the direction of intersection

line of 3, 4j5 = vector in the direction of intersection

line of 3, 5k = vector in plane 2 normal to il = vector in plane 1 normal to iR = magnitude of vector iG = square of magnitude of vector g

G5 = square of magnitude of vector g5

Note: The computed value of V is negative whenthe tension crack dips away from the toe of theslope, but this does not indicate a tensile force.

III.6 Sequence of calculations

1 Calculation of factor of safety when the forcesT and E are either zero or completely specifiedin magnitude and direction.

(a) Components of unit vectors in directionsof normals to planes 1–5, and of forcesT and E.


(ax, ay, az)

= (sin ψ1 sin α1, sin ψ1 cos α1, cos ψ1)

(III.1)

(bx, by, bz)


(III.2)

(dx, dy, dz)


(III.3)

(fx, fy, fz)


(III.4)

(f5x, f5y, f5z)


(III.5)

(tx, ty, tz)

= (cos ψt sin αt , cos ψt cos αt , − sin ψt)

(III.6)

(ex, ey, ez)

= (cos ψe sin αe, cos ψe cos αe, − sin ψe)

(III.7)

(b) Components of vectors in the directionof the lines of intersection of variousplanes.

(gx, gy, gz)

= (fyaz − fzay), (fzax − fxaz),

(fxay − fyax) (III.8)

(g5x, g5y, g5z)

= (f5yaz − f5zay), (f5zax − f5xaz),

(f5xay − f5yax) (III.9)

(ix, iy, iz)

= (byaz − bzay), (bzax − bxaz),

(bxay − byax) (III.10)

(jx, jy, jz)

= (fydz − fzdy), (fzdx − fxdz),

(fxdy − fydx) (III.11)

(j5x, j5y, j5z)

= (f5ydz − f5zdy), (f5zdx − f5xdz),

(f5xdy − f5ydx) (III.12)

(kx, ky, kz)

= (iybz − izby), (izbx − ixbz),

(ixby − iybx) (III.13)

(lx, ly, lz)

= (ayiz − aziy), (azix − axiz),

(axiy − ayix) (III.14)

(c) Numbers proportional to cosines ofvarious angles.

m = gxdx + gydy + gzdz (III.15)

m5 = g5xdx + g5ydy + g5zdz (III.16)

n = bxjx + byjy + bzjz (III.17)

n5 = bxj5x + byj5y + bzj5z (III.18)

p = ixdx + iydy + izdz (III.19)

q = bxgx + bygy + bzgz (III.20)

g5 = bxg5x + byg5y + bzg5z (III.21)

r = axbx + ayby + azbz (III.22)

s = axtx + ayty + aztz (III.23)

v = bxtx + byty + bztz (III.24)

w = ixtx + iyty + iztz (III.25)

se = axex + ayey + azez (III.26)

ve = bxex + byey + bzez (III.27)

we = ixex + iyey + izez (III.28)

s5 = axf5x + ayf5y + azf5z (III.29)

v5 = bxf5x + byf5y + bzf5z (III.30)

w5 = ixf5x + iyf5y + izf5z (III.31)

λ = ixgx + iygy + izgz (III.32)

λ5 = ixg5x + iyg5y + izg5z (III.33)

ε = fxf5x + fyf5y + fzf5z (III.34)

402 Appendix III

(d) Miscellaneous factors.

R =√

1 − r2 (III.35)

= 1R2

· nq

|nq| (III.36)

µ = 1R2

· mq

|mq| (III.37)

υ = 1R

· p

|p| (III.38)

G = g2x + g2

y + g2z (III.39)

G5 = g25x + g2

5y + g25z (III.40)

M = (Gp2 − 2mpλ + m2R2)1/2

(III.41)

M5 = (G5p2 − 2m5pλ5 + m25R2)1/2

(III.42)

h = H1|gz| (III.43)

h5 = Mh − |p|LM5

(III.44)

B = [tan2 φ1 + tan2 φ2 − 2 (µ r/ρ)

× tan φ1 tan φ2]/R2 (III.45)

(e) Plunge and trend of line respectively ofline of intersection of planes 1 and 2:

ψi = arcsin(νiz) (III.46)

αi = arctan(−νix

−νiy

)(III.47)

The term −ν should not be cancelled outin equation (III.47) since this is requiredto determine the correct quadrant whencalculating values for dip direction, αi.

(f) Check on wedge geometry.

No wedge ⎧⎨⎩

if p iz < 0, or (III.48)

if n q iz < 0 (III.49)

is formed,terminatecomputation

Tension

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

if ε η q5iz < 0, or (III.50)

if h5 < 0, or (III.51)

if[|m5 h5

m h|]

> 1, or (III.52)

if[| n q5 m5 h5

n5 q m h|]

> 1 (III.53)

crackinvalid,terminatecomputation

(g) Areas of faces and weight of wedge.

A1 = |mq|h2| − |m5q5|h25

2|p| (III.54)

A2 =(|q|m2h2/|n| − |q5|m2

5h25/|n5|)

|2p| (III.55)

A5 = |m5q5|h25

2|n5| (III.56)

W =γ(q2m2h3/|n| − q2

5m25h3

5/|n5|)

6|p| (III.57)

(h) Water pressure.

(i) With no tension crack

u1 = u2 = γwh|m5|6|p| (III.58)

(ii) With tension crack

u1 = u2 = u5 = γwh5|m5|3dz

(III.59)

V = u5 A5 η

(ε

|ε|)

(III.60)


(i) Effective normal reactions on planes 1and 2 assuming contact on both planes.

N1 = ρ{W kz + T (r v − s)

+ E(r ve − se) + V(r v5 − s5)}− u1 A1 (III.61)

N2 = µ{W lz + T (r s − v)

+ E(r se − ve) + V(r s5 − v5)}− u2 A2 (III.62)

(j) Factor of safety when N1 < 0 andN2 < 0 (contact is lost on both planes).

FS = 0 (III.63)

(k) If N1 > 0 and N2 < 0, contact is main-tained on plane 1 only and the factor ofsafety is calculated as follows:

Na = Waz − Ts − Ese − Vs5 − u1A1r

(III.64)

Sx = (Ttx + Eex + Naax + Vf5x + u1A1bx)

(III.65)

Sy = (Tty + Eey + Naay + Vf5y + u1A1by)

(III.66)

Sz = (Ttz + Eez + Naaz

+ Vf5z + u1A1bz) + W (III.67)

Sa = (S2x + S2

y + S2z )1/2 (III.68)

Qa = (Na − u1A1) tan φ1 + c1A1 (III.69)

FS1 =(

Qa

Sa

)(III.70)

(l) If N1 < 0 and N2 > 0, contact is main-tained on plane 2 only and the factor ofsafety is calculated as follows:

Nb = (Wbz − Tv − Eve − Vv5 − u2A2r)

(III.71)

Sx = (Ttx + Eex + Nbbx + Vf5x + u2A2ax)

(III.72)

Sy = (Tty + Eey + Nbby + Vf5y + u2A2ay)

(III.73)

Sz = (Ttz + Eez + Nbbz + Vf5z

+ u2A2az) + W (III.74)

Sb = (S2x + S2

y + S2z )1/2 (III.75)

Qb = (Nb − u2A2) tan φ2 + c2A2 (III.76)

FS2 =(

Qb

Sb

)(III.77)

(m) If N1 > 0 and N2 > 0, contact is main-tained on both planes and the factor ofsafety is calculated as follows:

S = ν(Wiz − Tw − Ewe − Vw5)

(III.78)

Q = N1 tan φ1 + N2 tan φ2

+ c1A1 + c2A2 (III.79)

FS3 =(

Q

S

)(III.80)

2 Minimum factor of safety produced when loadE of given magnitude is applied in the worstdirection.

(a) Evaluate N′′1, N

′′2, S

′′, Q

′′, FS

′′3 by use of

equations (III.61), (III.62), (III.78),(III.79) and (III.80) with E = 0.

(b) If N′′1 < 0 and N

′′2 < 0, even before

E is applied. Then FS = 0, terminatecomputation.

(c) D = {(N ′′1)2 + (N

′′2)2 + 2

(mn

|mn|)

N′′1N

′′2r}1/2

(III.81)

ψe = arcsin{(

− 1G

(m

|m|)

· N′′1az

+ n

|n| · N′′2bz

)}(III.82)

αe = arctan

{m|m| .N

′′1ax + n

|n| .N′′2bx

m|m| .N

′′1ay + n

|n| .N′′2by

}(III.83)

404 Appendix III

If E > D, and E is applied in the directionψe, αe, or within a certain range encom-passing this direction, then contact is loston both planes and FS = 0. Terminatecalculation.

(d) If N′′1 > 0 and N

′′2 < 0, assume contact

on plane 1 only after application of E.Determine S

′′x , S

′′y , S

′′z , S

′′a , Q

′′a, FS

′′1 from

equations (III.65) to (III.70) with E = 0.If FS

′′1 < 1, terminate computation.

If FS′′1 > 1 :

FS1 = S′′aQ

′′a − E{(Q′′

a)2 + ((S

′′a )2 − E2) tan2 φ1}1/2

(S′′a )2 − E2

(III.84)

ψe1 = arcsin

(S

′′z

S′′a

)− arctan

(tan φ1

(FS1)

)(III.85)

αe1 = arctan

(S

′′x

S′′a

)+ 180◦ (III.86)

(e) If N′′1 < 0 and N

′′2 > 0, assume contact

on plane 2 only after application of E.Determine S

′′x , S

′′y , S

′′z , S

′′b , Q

′′b, FS

′′2 from

equations (III.72) to (III.77) with E = 0.If FS


If FS′′2 > 1:

FS2 = S′′bQ

′′b − E{(Q′′

b)2 + ((S

′′b )2 − E2) tan2 φ2}1/2

(S′′b )2 − E2

(III.87)

ψe2 = arcsin

(S

′′z

S′′b

)− arctan

(tan φ2

(FS2)

)(III.88)

αe2 = arctan

(S

′′x

S′′y

)+ 180◦ (III.89)

(f) If N′′1 > 0 and N

′′2 > 0, assume contact

on both planes after application of E.If FS


If FS′′3 > 1:

FS3 = S′′Q

′′ − E{(Q′′)2 + B((S

′′)2 − E2)}1/2

(S′′)2 − E2

(III.90)

χ =√

B + (FS3)2 (III.91)

ex = − ((FS3)νix − ρkx tan φ1 − µlx tan φ2)

χ

(III.92)

ey = − ((FS3)νiy − ρky tan φ1 − µly tan φ2)

χ

(III.93)

ez = − ((FS3)νiz − ρkz tan φ1 − µlz tan φ2)

χ

(III.94)

ψe3 = arcsin(−ez) (III.95)

αe3 = arctan(

ex

ey

)(III.96)

Compute se and ve using equations(III.26) and (III.27)

N1 = N′′1 + Eρ(r ve − se) (III.97)

N2 = N′′2 + Eµ(r se − ve) (III.98)

Check that N1 � 0 and N2 � 0

3 Minimum cable or bolt tension Tmin requiredto raise the factor of safety to some spe-cified value FS.

(a) Evaluate N′1, N

′2, S

′, Q

′by means of equa-

tions (III.61), (III.62), (III.78), (III.79)with T = 0.

(b) If N′2 < 0, contact is lost on plane 2

when T = 0. Assume contact on plane 1only, after application on T . Evalu-ate S

′x, S

′y, S

′z, S

′a and Q

′a using equations

(III.65) to (III.69) with T = 0.

T1 = ((FS)S′a − Q

′a)√

(FS)2 + tan2 φ1(III.99)


ψt1 = arctan(

tan φ1

(FS)

)− arcsin

(S′

z

S′a

)

(III.100)

αt1 = arctan

(S′

x

S′y

)(III.101)

(a) If N ′1 < 0, contact is lost on plane 1

when T = 0. Assume contact on plane 2only, after application of T . Evalu-ate S′

x, S′y, S′

z, S′b and Q′

b using equations(III.72) to (III.76) with T = 0.

T2 = ((FS)S′b − Q′

b)√(FS)2 + tan2 φ2

(III.102)

ψt2 = arctan(

tan φ2

(FS)

)− arcsin

(S′

z

S′b

)(III.103)

αt2 = arctan

(S′

x

S′y

)(III.104)

(a) All cases. No restrictions on values of N ′1

and N ′2. Assume contact on both planes

after application of T .

χ =√

((FS)2 + B) (III.105)

T3 = ((FS)S′ − Q′)χ

(III.106)

tx = ((FS)υix − ρkx tan φ1 − µlx tan φ2)

χ

(III.107)

ty = ((FS)υiy − ρky tan φ1 − µly tan φ2)

χ

(III.108)

tz = ((FS)υiz − ρkz tan φ1 − µlz tan φ2)

χ

(III.109)

ψt3 = arcsin(−tz) (III.110)

αt3 = arctan(

tx

ty

)(III.111)

Compute s and v using equations (III.23)and (III.24).

N1 = N ′1 + T3ρ(rv − s) (III.112)

N2 = N ′2 + T3µ(rs − v) (III.113)

If N1 < 0 or N2 < 0, ignore the resultsof this section.If N ′

1 > 0 and N ′2 > 0, Tmin = T3

If N ′1 > 0 and N ′

2 < 0, Tmin = smallest ofT1, T3If N ′

1 < 0 and N ′2 > 0, Tmin = smallest of

T2, T3If N ′

1 < 0 and N ′2 < 0, Tmin = smallest of

T1, T2, T3

Example Calculate the factor of safety for thefollowing wedge:

Plane 1 2 3 4 5

ψ 45 70 12 65 70α 105 235 195 185 165

η = +1H1 = 100 ft, L = 40 ft, c1 = 500 lb/ft2,

c2 = 1000 lb/ft2

φ1 = 20◦, φ2 = 30◦, γ = 160 lb/ft3.

(1a) T = 0, E = 0, u1 = u2 = u5

u5 calculated from equation (III.59).

(ax, ay, az) = (0.68301, −0.18301, 0.70711)

(bx, by, bz) = (−0.76975, −0.53899, 0.34202)

(dx, dy, dz) = (−0.05381, −0.20083, 0.97815)

(fx, fy, fz) = (−0.07899, −0.90286, 0.42262)

(f5x, f5y, f5z) = (0.24321, −0.90767, 0.34202)

(gx, gy, gz) = (−0.56107, 0.34451, 0.63112)

(g5x, g5y, g5z) = (−0.57923, 0.061627, 0.57544)

(ix, iy, iz) = (−0.31853, 0.77790, 0.50901)

(jx, jy, jz) = (−0.79826, 0.05452, −0.03272)

(j5x, j5y, j5z) = (−0.81915, −0.25630, −0.09769)

(kx, ky, kz) = (0.54041, −0.28287, 0.77047)

(lx, ly, lz) = (−0.64321, −0.57289, 0.47302)

406 Appendix III

m = 0.57833

m5 = 0.58166

n = 0.57388

n5 = 0.73527

p = 0.35880

q = 0.46206

q5 = 0.60945

r = −0.18526

s5 = 0.57407

v5 = 0.41899

w5 = −0.60945

λ = 0.76796

λ5 = 0.52535

ε = 0.94483

R = 0.98269

ρ = 1.03554

µ = 1.03554

ν = 1.01762

G = 0.83180

G5 = 0.67044

M = 0.33371

M5 = 0.44017

h = 158.45

h5 = 87.521

B = 0.56299

ψi = 31.20◦

αi = 157.73◦

piz > 0

nqiz > 0

⎫⎬⎭Wedge is formed

εηq5iz > 0

h5 > 0|m5h5||mh| = 0.5554 < 1

|nq5m5h5||n5qmh| = 0.57191 < 1

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

Tension crack valid

A1 = 5565.01ft2

A2 = 6428.1ft2

A5 = 1846.6ft2

W = 2.8272 × 107 lb

u1 = u2 = u5 = 1084.3 lb/ft2;

V = 2.0023 × 106 lb

N1 = 1.5171 × 107 lb

N2 = 5.7892 × 106 lb

⎫⎬⎭Both positive therefore

contact on planes 1 and 2.

S = 1.5886 × 107 lb

Q = 1.8075 × 107 lb

FS = 1.1378—Factor of Safety

(1b) T = 0, E = 0, dry slope, u1 = u2 = u5 = 0.

As in (1a) except as follows:

V = 0

N1 = 2.2565 × 107lb

N2 = 1.3853 × 107 lb

⎫⎪⎪⎬⎪⎪⎭

Both positive, thereforecontact on both planes 1and 2.

S = 1.4644 × 107 lb

Q = 2.5422 × 107 lb

FS3 = 1.7360—Factor of Safety

(2) As in (1b), except E = 8 × 106 lb. Find the value ofFSmin.

Values of N ′′1 , N ′′

2 , S′′, Q′′, FS′′3 as given in (1b).

N ′′1 > 0, N ′′

2 > 0, FS′′3 > 1, continue calculation.

B = 0.56299

FS3 = 1.04—FSmin (minimum factor of safety)

χ = 1.2798

ex = 0.12128

ey = −0.99226

ez = 0.028243

ψe3 = −1.62◦—plunge of force (upwards)

αe3 = 173.03◦—trend of force

N1 = 1.9517 × 107 lb

N2 = 9.6793 × 106 lb

⎫⎪⎬⎪⎭

Both positive thereforecontact maintained onboth planes.

(3) As in (1a) except that the minimum cable tensionTmin required to increase the factor of safety to 1.5is to be determined

N ′1, N ′

2, S′ and Q′—as given in (1a)

χ = 1.6772T3 = 3.4307 × 106 lb—Tmin (minimum cabletension)

tx = −0.18205

ty = 0.97574


Tensionedanchor

�T(opt)

�i�average

Figure III.2 Optimum anchor orientation forreinforcement of a wedge.

tz = 0.12148

ψt3 = −6.98◦—plunge of cable (upwards)

αt3 = 349.43◦—trend of cable

Note that the optimum plunge and trend of theanchor are approximately:

ψt3 = 12 (φ1 + φ2) − ψi

≈ 25 − 31.2

≈ −6.2◦ (upwards)

αt3 ≈ αi ± 180◦

≈ 157.73 + 180

≈ 337.73◦

That is, the best direction in which to install ananchor to reinforce a wedge is

The anchor should be aligned with the line ofintersection of the two planes, viewed from thebottom of the slope, and it should be inclinedat the average friction angle to the line ofintersection (Figure III.2).

Appendix IV

Conversion factors

Imperial unit SI unit SI unit symbol Conversion factor(imperial to SI)

Conversion factor(SI to imperial)

LengthMile kilometer km 1 mile = 1.609 km 1 km = 0.6214 mileFoot meter m 1 ft = 0.3048 m 1 m = 3.2808 ft

millimeter mm 1 ft = 304.80 mm 1 mm = 0.003 281 ftInch millimeter mm 1 in = 25.40 mm 1 mm = 0.039 37 in

AreaSquare mile square kilometer km2 1 mile2 = 2.590 km2 1 km2 = 0.3861 mile2

Acre hectare ha 1 mile2 = 259.0 ha 1 ha = 0.003 861 mile2

hectare ha 1 acre = 0.4047 ha 1 ha = 2.4710 acresquare meter m2 1 acre = 4047 m2 1 m2 = 0.000 247 1 acre

Square foot square meter m2 1 ft2 = 0.092 90 m2 1 m2 = 10.7643 ft2Square inch square millimeter mm2 1 in2 = 645.2 mm2 1 mm2 = 0.001 550 in2

VolumeCubic yard cubic meter m3 1 yd3 = 0.7646 m3 1 m3 = 1.3080 yd3

Cubic foot cubic meter m3 1 ft3 = 0.028 32 m3 1 m3 = 35.3150 ft3liter l 1 ft3 = 28.32 l 1 liter = 0.035 31 ft3

Cubic inch cubic millimeter mm3 1 in3 = 16 387 mm3 1 mm3 = 61.024 × 10−6 in3

cubic centimeter cm3 1 in3 = 16.387 cm3 1 cm3 = 0.061 02 in3

liter l 1 in3 = 0.016 39 l 1 liter = 61.02 in3

Imperial gallon cubic meter m3 1 gal = 0.004 56 m3 1 m3 = 220.0 galliter l 1 gal = 4.546 l 1 liter = 0.220 gal

Pint liter l 1 pt = 0.568 l 1 liter = 1.7606 ptUS gallon cubic meter m3 1 US gal = 0.0038 m3 1 m3 = 263.2 US gal

liter l 1 US gal = 3.8 l 1 liter = 0.264 US gal

MassTon tonne t 1 ton = 0.9072 tonne 1 tonne = 1.1023 tonton (2000 lb) (US) kilogram kg 1 ton = 907.19 kg 1 kg = 0.001 102 tonton (2240 lb) (UK) 1 ton = 1016.1 kg 1 kg = 0.000 984 tonKip kilogram kg 1 kip = 453.59 kg 1 kg = 0.002 204 6 kipPound kilogram kg 1 lb = 0.4536 kg 1 kg = 2.204 6 lb

(continued)

Conversion factors 409

Continued



Mass densityton per cubic

yard(2000 lb) (US)

kilogram percubic meter

kg/m3 1 ton/yd3 = 1186.49 kg/m3 1 kg/m3 = 0.000 842 8 ton/yd3

(US)

tonne per cubicmeter

t/m3 1 ton/yd3 = 1.1865 t/m3 1 t/m3 = 0.8428 ton/yd3 (US)

ton per cubicyard (2240 lb)(UK)

kilogram percubic meter

kg/cm3 1 ton/yd3 = 1328.9 kg/m3 1 kg/cm3 = 0.000 75 ton/yd3

(UK)

pound percubic foot

1 lb/ft3 = 16.02 kg/m3 1 kg/cm3 = 0.062 42 lb/ft3


t/m3 1 lb/ft3 = 0.01602 t/m3 1 t/m3 = 62.42 lb/ft3

pound percubic inch

gram per cubiccentimeter

g/cm3 1 lb/in3 = 27.68 g/cm3 1 g/cm3 = 0.036 13 lb/in3


t/m3 1 lb/in3 = 27.68 t/m3 1 t/m3 = 0.036 13 lb/in3

Forceton force kilonewton kN 1 tonf = 8.896 kN 1 kN = 0.1124 tonf (US)(2000 lb) (US)

ton force(2240 lb)(UK)

1 tonf = 9.964 KN 1 kN = 0.1004 tonf (UK)

kip force kilonewton kN 1 kipf = 4.448 kN 1 kN = 0.2248 kipfpound force newton N 1lbf = 4.448 N 1 N = 0.2248 lbftonf/ft(2000 lb) (US)

kilonewton permeter

kN/m 1 tonf/ft = 29.186 kN/m 1 kN/m = 0.034 26 tonf/ft (US)

tonf/ft(2240 lb)(UK)

kilonewton permeter

1 tonf/ft = 32.68 kN/m 1 kN/m = 0.0306 tonf/ft (UK)

pound forceper foot

newton permeter

N/m 1 lbf/ft = 14.59 N/m 1 N/m = 0.068 53 lbf/ft

Hydraulicconductivity

centimeter persecond

meter persecond

m/s 1 cm/s = 0.01 m/s 1 m/s = 100 cm/s

foot per year meter persecond

m/s 1 ft/yr = 0.9665 × 10−8 m/s 1 m/s = 1.0346 × 108 ft/yr

foot per second meter persecond

m/s 1 ft/s = 0.3048 m/s 1 m/s = 3.2808 ft/s

Flow ratecubic foot perminute

cubic meter persecond

m3/s 1 ft3/min = 0.000 471 9 m3/s 1 m3/s = 2119.093 ft3/min

liter per second l/s l ft3/min = 0.4719 l/s 1 l/s = 2.1191 ft3/min

(continued)

410 Appendix IV

Continued



cubic foot persecond

cubic meter persecond

m3/s 1 ft3/s = 0.028 32 m3/s 1 m3/s = 35.315 ft3/s

liter per second l/s 1 ft3/s = 28.32 l/s 1 l/s = 0.035 31 ft3/sgallon perminute

liter per second l/s 1 gal/min = 0.075 77 l/s 1 l/s = 13.2 gal/min

Pressure, stresston force persquare foot(2000 lb) (US)

kilopascal kPa 1 tonf/ft2 = 95.76 kPa 1 kPa = 0.01044 ton f/ft2 (US)

ton force persquare foot(2240 lb)(UK)

1 tonf/ft2 = 107.3 kPa 1 kPa = 0.00932 ton/ft2 (UK)

pound forceper squarefoot

pascal Pa 1 lbf/ft2 = 47.88 Pa 1 Pa = 0.020 89 lbf/ft2

kilopascal kPa 1 lbft/ft2 = 0.047 88 kPa 1 kPa = 20.89 lbf/ft2pound forceper squareinch

pascal Pa 1 lbf/in2 = 6895 Pa 1 Pa = 0.000 1450 lbf/in2

kilopascal kPa 1 lbf/in2 = 6.895 kPa 1 kPa = 0.1450 lbf/in2

Weightdensitya

pound forceper cubic foot

kilonewton percubic meter

kN/m3 1 lbf/ft3 = 0.157 kN/m3 1 kN/m3 = 6.37 lbf/ft3

EnergyFoot lbf joules J 1 ft lbf = 1.355 J 1 J = 0.7376 ft lbf

Notea Assuming a gravitational acceleration of 9.807 m/s2.

References

Abrahamson, N. A. (2000) State of the practice ofseismic hazard evaluation. Proc. Geoeng 2000,Melbourne, Australia, 659–85.

Adhikary, D. P. and Guo, H. (2000) An orthotropicCosserat elasto-plastic model for layered rocks.J. Rock Mech. Rock Engng., 35(3), 161–170.

Adhikary, D. P., Dyskin, A. V., Jewell, R. J. andStewart, D. P. (1997) A study of the mechanismof flexural toppling failures of rock slopes. J. RockMech. Rock Engng, 30(2), 75–93.

Adhikary, D. P., Mühlhaus, H.-B. and Dyskin, A. V.(2001) A numerical study of flexural buckling of foli-ated rock slopes. Int. J. Num. Methods Geomech.,25, 871–84.

American Association of State Highway and Trans-portation Officials (AASHTO) (1984) A Policyon Geometric Design of Highways and Streets.American Association of State of Highway andTransportation Officials, Washington, DC.

American Association of State Highway and Trans-portation Officials (AASHTO) (1996) StandardSpecifications for Highway Bridges, 16th edn,AASHTO, Washington, DC.

American Concrete Institute (ACI) (1995) Specifica-tions for Materials, Proportioning and Applicationof Shotcrete. ACI Report 506.2-95, Revised 1995.

Andrew, R. D. (1992a) Restricting rock falls. ASCE,Civil Engineering, Washington, DC, October,pp. 66–7.

Andrew, R. D. (1992b) Selection of rock fall mitigationtechniques based on Colorado Rock fall simulationprogram. Transportation Research Record, 1343,TRB, National Research Council, Washington, DC,pp. 20–22.

Ashby, J. (1971) Sliding and toppling modes of fail-ure in models and jointed rock slopes. MSc. Thesis.London University, Imperial College.

Athanasiou-Grivas, D. (1979) Probabilistic evaluationof safety of soil structures. ASCE, J. Geotech. Engng,105 (GT9), 109–15.

Athanasiou-Grivas, D. (1980) A reliability approachto the design of geotechnical systems. RensselaerPolytechnic Institute Research Paper, Transporta-tion Research Board Conference, Washington, DC.

Atkinson, L. C. (2000) The role and mitigation ofgroundwater in slope stability. Slope Stability inSurface Mines, ch. 9. SME, Littleton, CO, pp. 89–96.

Atlas Powder Company (1987) Explosives and RockBlasting. Atlas Powder Company, Dallas, TX.

Australian Center for Geomechanics (ACG) (1998)Integrated Monitoring Systems for Open Pit WallDeformation. MERIWA Project M236, ReportACG: 1005–98.

Australian Drilling Industry (ADI) (1996) Drilling:Manual of Methods, Applications and Manage-ment, 4th edn. Australian Drilling Industry Train-ing Committee, CRC Press LLC, Boca Raton, FL,615 pp.

Aycock, J. H. (1981) Construction problems involvingshale in a geologically complex environment—StateRoute 32, Grainger County, Tennessee. Proc. 32ndHighway Geology Symp., Gatlinburg, TN.

Aydan, O. (1989) The Stabilization of Rock Engin-eering Structures by Rock Bolts. Doctorate ofEngng Thesis, Dept of Geotechnical Engng, NagoyaUniversity, Japan.

Azzoni, A. and de Freitas, M. H. (1995) Experiment-ally gained parameters decisive for rock fall analysis.Rock Mech. Rock Engng, 28(2), 111–24.

Baecher, G. B., Lanney, N. A. and Einstein, H.(1977) Statistical description of rock properties andsampling. Proc. 18th US Symp. Rock Mech. JohnsonPublishing Co., Keystone, CO.

Baker, D. G. (1991) Wahleach power tunnel monitor-ing. Field Measurements in Geotechnics. Balkema,Rotterdam, pp. 467–79.

Baker, W. E. (1973) Explosives in Air. University ofTexas Press, Austin, TX.

Balmer, G. (1952) A general analytical solution forMohr’s envelope. Am. Soc. Test. Mat., 52, 1260–71.

412 References

Bandis, S. C. (1993) Engineering properties and charac-terization of rock discontinuities. In: ComprehensiveRock Engineering: Principles, Practice and Projects(ed. J. A. Hudson) Vol. 1, Pergamon Press, Oxford,pp. 155–83.

Barrett, R. K. and White, J. L. (1991) Rock fall predic-tion and control. Proc. National Symp. on Highwayand Railway Slope Maintenance, Assoc. of Eng.Geol., Chicago, pp. 23–40.

Barton, N. R. (1971) A model study of the behavior ofexcavated slopes. PhD Thesis, University of London,Imperial College, 520 pp.

Barton, N. R. (1973) Review of a new shear strengthcriterion for rock joints. Engng Geol., Elsevier, 7,287–322.

Barton, N. R. (1974) A Review of the Shear Strengthof Filled Discontinuities in Rock. Norwegian Geo-technical Institute, Pub. No. 105.

Barton, N. R. and Bandis, S. (1983) Effects of blocksize on the shear behaviour of jointed rock. Issuesin Rock Mechanics—Proc. 23rd US Symp. on RockMechanics, Berkeley, CA, Soc. Mining Eng. ofAIME, 739–60.

Baxter, D. A. (1997) Rockbolt corrosion under scru-tiny. Tunnels and Tunneling Int., July, pp. 35–8.

Bieniawski, Z. T. (1976) Rock mass classification inrock engineering. In: Exploration for Rock Engin-eering, Proc. Symp. (ed. Z. T. Bieniawski) Vol. 1,Cape Town, Balkema, pp. 97–106.

Bishop, A. W. (1955) The use of the slip circle inthe stability analysis of earth slopes. Geotechnique,5, 7–17.

Bishop, A. W. and Bjerrum, L. (1960) The relevance ofthe triaxial test to the solution of stability problems.Proc. ASCE Conf Shear Strength of Cohesive Soils,Boulder, CO, pp. 437–501.

Bjurstrom, S. (1974) Shear strength on hard rockjoints reinforced by grouted untensioned bolts. In:Proc. 3rd International Congress on Rock Mechan-ics. Vol. II: Part B, National Academy of Sciences,Washington, DC, pp. 1194–9.

Black, W. H., Smith, H. R. and Patton, F. D. (1986)Multi-level ground water monitoring with the MPSystem. Proc. NWWA-AGU Conf. on Surface andBorehole Geophysical Methods and GroundwaterInstrumentation, Denver, CO, pp. 41–61.

Bobet, A. (1999) Analytical solutions for toppling fail-ure (Technical Note). Int. J. Rock Mech. Min. Sci.,36, 971–80.

Brawner, C. O., Stacey, P. F. and Stark, R. (1975)A successful application of mining with pitwall

movement. Proc. Canadian Institute of Mining,Annual Western Meeting, Edmonton, October,20 pp.

Brawner, C. O. and Kalejta, J. (2002) Rock fallcontrol in surface mining with cable ring net fences.Soc. Mining Eng., Proc. Annual Meeting, Phoenix,AZ, February, 10 pp.

Brawner, C. O. and Wyllie, D. C. (1975) Rock slopestability on railway projects. Proc. Am. RailwayEngng Assoc., Regional Meeting, Vancouver, BC.

British Standards Institute (BSI) (1989) British Stand-ard Code of Practice for Ground Anchorages, BS8081: 1989. BSI, 2 Park Street, London, W1A 2BS,176 pp.

Broadbent, C. D. and Zavodni, Z. M. (1982) Influenceof rock structure on stability. Stability in SurfaceMining, Vol. 3, Ch. 2. Soc. of Mining Engineers,Denver, CO.

Brown, E. T. (1970) Strength of models of rock withintermittent joints. J. Soil Mech. Fdns Div., ASCE96, SM6, 1935–49.

Byerly, D. W. and Middleton, L. M. (1981) Evaluationof the acid drainage potential of certain Precambrianrocks in the Blue Ridge Province. 32nd Annual High-way Geology Symp., Gatlinburg, TN, pp. 174–85.

Byrne, R. J. (1974) Physical and numerical modelsin rock and soil slope stability. PhD Thesis. JamesCook University of North Queensland, Australia.

CIL (1983) Blaster’s Handbook. Canadian IndustriesLimited, Montréal, Québec.

CIL (1984) Technical Literature, Canadian IndustriesLimited, Toronto, Canada.

California Polytechnical State University (1996)Response of the Geobrugg cable net system to debrisflow loading. Dept Civil and Environmental Engng,San Luis Obispo, CA, 68 pp.

Call, R. D. (1982) Monitoring pit slope behaviour.Stability in Surface Mining, Vol. 3, Ch. 9. Soc. ofMining Engineers, Denver, CO.

Call, R. D., Saverly, J. P. and Pakalnis, R. (1982)A simple core orientation device. In: Stability inSurface Mining (ed. C. O. Brawner), SME, AIME,New York, pp. 465–81.

Canada Department of Energy, Mines and Resources(1978) Pit Slope Manual. DEMR, Ottawa, Canada.

Canadian Geotechnical Society (1992) CanadianFoundation Engineering Manual. BiTech PublishersLtd, Vancouver, Canada.

Canadian Standards Association (1988) Design ofHighway Bridges. CAN/CSA-56-88, Rexdale,Ontario.

References 413

Carter, J. P. and Balaam, N. G. (1995) AFENA Users’Manual, Version 5. University of Sydney, Centre forGeotechnical Research.

Carter, T. G., Carvalho, J. L. and Swan, G. (1993)Towards practical application of ground reactioncurves. In: Innovative Mine Design for the 21stCentury (eds W. F. Bawden and J. F. Archibald),Rotterdam, Balkema, pp. 151–71.

Casagrande, L. (1934) Näherungsverfahren zurErmittlung der Sickerung in geschutteten Däm-men auf underchläassiger Sohle. Die Bautechnik,Heft 15.

Caterpillar Tractor Co. (2001) Caterpillar Perform-ance Handbook, edition 32. Caterpillar Tractor Co.,Peoria, IL.

CDMG (1997) Guidelines for Evaluating and Mitigat-ing Seismic Hazards in California. California Div.of Mines and Geology, Special Publication 117,www/consrv.ca.gov/dmg/pubs/sp/117/

Cedergren, H. R. (1989) Seepage, Drainage and FlowNets, 3rd edn. John Wiley & Sons, New York, USA,465 pp.

Chen, Z. (1995a) Keynote Lecture: Recent develop-ments in slope stability analysis. Proc. 8th Int.Congress on Rock Mechanics, Vol. 3, Tokyo, Japan,pp. 1041–8.

Chen, Z. editor in chief (1995b) Transactions of theStability Analysis and Software for Steep Slopesin China. Vol. 3.1: Rock Classification, Statisticsof Database of Failed and Natural Slopes. ChinaInst. of Water Resources and Hydroelectric PowerResearch (in Chinese).

Cheng, Y. and Liu, S. (1990) Power caverns ofthe Mingtan Pumped Storage Project, Taiwan.In Comprehensive Rock Engineering (ed. J. A.Hudson), Pergamon Press, Oxford, Vol. 5,pp. 111–32.

Christensen Boyles Corp. (2000) Diamond DrillProducts—Field Specifications. CBC, Salt LakeCity, UT.

Ciarla, M. (1986) Wire netting for rock fall protec-tion. Proc. 37th Ann. Highway Geology Symp.,Helena, Montana, Montana Department of High-ways, pp. 10–118.

Cluff, L. S., Hansen, W. R., Taylor, C. L., Weaver,K. D., Brogan, G. E., Idress, I. M., McClure, F. E.and Bayley, J. A. (1972) Site evaluation in seismicallyactive regions—an interdisciplinary team approach.Proc. Int. Conf. Microzonation for Safety, Construc-tion, Research and Application, Seattle, WA, Vol. 2,pp. 9-57–9-87.

Coates, D. F., Gyenge, M. and Stubbins, J. B. (1965)Slope stability studies at Knob Lake. Proc. RockMechanics Symp., Toronto, pp. 35–46.

Colog Inc. (1995) Borehole Image Processing System(BIP), Golden, CO, and Raax Co. Ltd., Australia.

Coulomb, C. A. (1773) Sur une application des règlesde Maximis et Minimis a quelques problèmes destatique relatifs à l’Architechture. Acad. Roy. desSciences Memoires de math. et de physique par diverssavans, 7, 343–82.

Cruden, D. M. (1977) Describing the size of discon-tinuities. Int. J. Rock Mech. Min. Sci. & Geomech.Abstr., 14, 133–7.

Cruden, D. M. (1997) Estimating the risks from land-slides using historical data. Proc. Int. Workshopon Landslide Risk Assessment (eds D. M. Crudenand R. Fell), Honolulu, HI, Balkema, Rotterdam,177–84.

CSIRO (2001) SIROJOINT Geological Mapping Soft-ware. CSIRO Mining and Exploration, Pullenvale,Queensland, Australia.

Cundall, P.(1971) A computer model for simulatingprogressive, large scale movements in blocky rocksystems. Proc. Int. Symp. on Rock Fracture. Nancy,France, Paper 11–8.

Cundall, P. A., Carranza-Torres, C. and Hart, R. D.(2003) A new constitutive model based on the Hoek–Brown criterion. Presented at FLAC and Numer-ical Modeling in Geomechanics, Sudbury, Canada,October.

Davies, J. N. and Smith, P. L. P. (1993) Flexural top-pling of siltstones during a temporary excavation fora bridge foundation in North Devon. ComprehensiveRock Engineering, Pergamon Press, Oxford, UK, 5,Ch. 31, 759–75.

Davis, G. H. and Reynolds, S. J. (1996) Structural Geo-logy of Rocks and Regions, 2nd edn. John Wiley &Sons, New York, 776 pp.

Davis, S. N. (1969) Porosity and permeabilityof natural materials. In: Flow through PorousMedia (ed. R. de Weist) Academic Press, London,pp. 54–89.

Davis, S. N. and De Wiest, R. J. M. (1966) Hydrogeo-logy, John Wiley & Sons, New York and London.

Dawson, E. M. and Cundall, P. A. (1996) Slopestability using micropolar plasticity. In: Rock Mech-anics Tools and Techniques (ed. M. Aubertin et al.)Rotterdam, Balkema, pp. 551–8.

Dawson, E. M., Roth, W. H. and Drescher, A.(1999) Slope stability analysis by strength reduction,Géotechnique, 49(6), 835–840.

414 References

Deere, D. U. and Miller, R. P. (1966) EngineeringClassification and Index Properties of Intact Rock.Technical Report No. AFWL-TR-65-116, Air ForceWeapons Laboratory, Kirkland Air Force Base, NewMexico.

Dershowitz, W. S. and Einstein, H. H. (1988) Char-acterizing rock joint geometry with joint systemmodels. Rock Mech. Rock Engng, 20(1), 21–51.

Dershowitz, W. S. and Schrauf, T. S. (1987) Discretefracture flow modeling with the JINX package. Proc.28th US Symp. Rock Mechanics, Tucson, pp. 433–9.

Dershowitz, W. S, Lee, G., Geier, J., Hitchcock, S.and LaPointe, P. (1994) FRACMAN Interactive Dis-crete Feature Data Analysis, Geometric Modeling,and Exploration Simulation. User documentation V.2.4, Golder Associates Inc., Redmond, Washington.

Descoeudres, F. and Zimmerman, T. (1987) Three-dimensional calculation of rock falls. Proc. Int.Conf. Rock Mech., Montreal, Canada.

Dowding, C. H. (1985) Blast Vibration Monitoringand Control. Prentice-Hall, Englewood Cliffs, NJ.

Du Pont De Memours & Co. Inc. (1984) Blaster’sHandbook. Du Pont, Wilmington, DE.

Duffy, J. D. and Haller, B. (1993) Field tests offlexible rock fall barriers. Proc. Int. Conf. onTransportation Facilities through Difficult Terrain,Aspen, Colorado, Balkema, Rotterdam, Nether-lands, pp. 465–73.

Duncan, J. M. (1996) Landslides: Investigation andMitigation, Transportation Research Board, SpecialReport 247, Washington, DC, Ch. 13, Soil SlopeStability Analysis, pp. 337–71.

Dunnicliff, J. (1993) Geotechnical Instrumentation forMonitoring Field Performance, 2nd edn, John Wiley& Sons, New York, 577 pp.

Durn, J. and Douglas, L. H. (1999) Do rock slopesdesigned with empirical rock mass strength stand up?9th Cong. Int. Soc. Rock Mech., Paris, 87–90.

Einstein, H. H. (1993) Modern developments in dis-continuity analysis. Comprehensive Rock Engineer-ing, Pergamon Press, Oxford, 3, Ch. 9, 193–213.

Einstein, H. H., Veneziano, D., Baecher, G. B. andO’Reilly, K. J. (1983) The effect of discontinuity per-sistence on slope stability. Int. J. Rock Mech. Min.Sci. & Geomech. Abstr., 20, 227–36.

Erban, P. J. and Gill, K. (1988) Consideration of theinteraction between dam and bedrock in a coupledmachanic-hydraulic FE-program. Rock Mech. RockEngng, 21(2), 99–118.

European Committee for Standardization (1995) Euro-code 1: Basis of Design and Actions on Structures—

Part 1, Basis of Design. Central Secretariat,Brussels.

Farmer, I. W. (1975) Stress distribution along a resingrouted anchor. Int. J. Rock Mech & Geomech.Abstracts., 12, 347–51.

Fecker, E. and Rengers, N. (1971) Measurement oflarge scale roughness of rock planes by means ofprofilometer and geological compass. Proc. Symp.on Rock Fracture, Nancy, France, Paper 1–18.

Federal Highway Administration (FHWA) (1982)Tiebacks. Publication No. FHWA/RD-82/047,Washington, DC.

Federal Highway Administration (FHWA) (1991)Rock Blasting and Overbreak Control. FHWA, USDepartment of Transportation Contract No. DTFW61-90-R-00058.

Federal Highway Administration (FHWA) (1993)Rockfall Hazard Mitigation Methods. PublicationNo. FHWA SA-95-085, Washington, DC.

Federal Highway Administration (FHWA) (1998)Rock Slopes. Publication No. FHWA-HI-99-007,FHWA, Washington, DC.

Fell, R. (1994) Landslide risk assessment and accept-able risk. Can. Geotech. J., 31, 261–72.

Fleming, R. W., Spencer, G. S. and Banks, D. C.(1970) Empirical study of the behaviour of clay shaleslopes. US Army Nuclear Cratering Group TechnicalReport, No. 15.

Flores, G. and Karzulovic, A. (2000) The role of thegeotechnical group in an open pit: ChuquicamataMine, Chile. Slope Stability in Surface Mining, SME,Littleton, CO, 141–52.

Fookes, P. G. and Sweeney, M. (1976) Stabilizationand control of local rock falls and degrading rockslopes. Quart. J. Engng Geol., 9, 37–55.

Forster, J. W. (1986) Geological problems overcomeat Revelstoke, part two. Water Power and DamConstruction, August.

Frankel, F., Mueller, C., Bernard, T., Perkins, D.,Leyendecker, E. V., Dickman, N., Hanson, S.and Hopper, M. (1996) Interim National HazardsMap: Documentation. Draft Report, US GeologicalSurvey, Denver, CO.

Franklin, A. G. and Chan, F. K. (1977) Earthquakeresistance of earth and rock-fill dams. Report 5:Permanent displacement of earth embankments byNewmark sliding block analysis. US Army EngineerWaterways Experiment Station, Vicksburg, MS,Miscellaneous Paper S-71-17, 59 pp.

Freeze, R. A. and Cherry, J. A. (1979) Groundwater.Prentice Hall, New Jersey, USA, 604 pp.

References 415

Freitas, M. H. and Watters, R. J. (1973) Some fieldexamples of toppling failure. Géotechnique, 23(4),495–514.

Frohlich, O. K. (1955) General theory of the stabilityof slopes. Géotechnique, 5, 37–47.

Geological Society Engineering Group Working Party(1977) The logging of rock cores for engineeringpurposes. Quart. J. Engng Geol., 3, 1–24.

Glass, C. E. (2000) The influence of seismic events onslope stability. Slope Stability in Surface Mining, Soc.Mining, Metallurgy and Exploration, Littleton, CO,97–105.

Golder, H. Q. (1972) The stability of natural andman-made slopes in soil and rock. GeotechnicalPractice for Stability in Open Pit Mining. (edsC. O. Brawner and V. Milligan), AIME, New York,79–85.

Goodman, R. E. (1964) The resolution of stresses inrock using stereographic projection. Int. J. RockMech. Mining Sci., 1, 93–103.

Goodman, R. E. (1970) The deformability of joints.In Determination of the in situ Modulus ofDeformation of Rock. Am. Soc. Testing and Mater-ials, Spec. Tech. Publication, No. 477, 174–96.

Goodman, R. E. (1976) Methods of Geological Engin-eering in Discontinuous Rocks. West Publishing Co.,St Paul, MN, 472 pp.

Goodman, R. E. (1980) Introduction to Rock Mech-anics. John Wiley and Sons, New York.

Goodman, R. E. and Bray, J. (1976) Toppling ofrock slopes. ASCE, Proc. Specialty Conf. on RockEng. for Foundations and Slopes, Boulder, CO, 2,201–34.

Goodman, R. E. and Seed, H. B. (1966) Earthquakeinduced displacements of sand embankments. ASCEVol. 92, SM2, pp. 125–46.

Goodman, R. E. and Shi, G. (1985) Block Theory andits Application to Rock Engineering. Prentice Hall,Englewood Cliffs, NJ.

Goodman, R. E., Moye, D. G., van Schalkwyk, A.and Javandel, I. (1965) Ground water inflows duringtunnel driving. Engng Geol., 70, 396–8.

Griffiths, D. H. and King, R. F. (1988) Applied Geo-physics for Geologists and Engineers, 2nd edn.Pergamon Press, Oxford, UK.

Haar, M. E. (1962) Ground Water and Seepage.McGraw Hill Co., New York.

Hagan, T. N. (1975) Blasting physics—what theoperator can use in 1975. Proc. Aust. Inst. Min-ing and Metall., Annual Conf., Adelaide, Part B,369–86.

Hagan, T. N. (1992) Personal communication, BlastingLtd., Queensland, Australia.

Hagan, T. N. and Bulow, B. (2000) Blast designs toprotect pit walls. Slope Stability in Surface Mining,Soc. Min. Metallurgy and Exploration, Denver, CO,pp. 125–30.

Haines, A. and Terbrugge, P. J. (1991) Preliminaryestimate of rock slope stability using rock massclassification. 7th Cong. Int. Soc. of Rock Mech.,Aachen, Germany, pp. 887–92.

Hamel, J. V. (1970) The Pima Mine slide, PimaCounty, Arizona. Geol. Soc. America, Abstractswith Programs, 2(5), p. 335.

Hamel, J. V. (1971a) The slide at Brilliant cut.Proc. 13th Symp. Rock Mechanics, Urbana, IL,pp. 487–510.

Hamel, J. V. (1971b) Kimberley pit slope failure.Proc. 4th Pan-American Conf. on Soil Mechan-ics and Foundation Engineering, Puerto Rico, 2,117–27.

Hammett, R. D. (1974) A study of the behaviour ofdiscontinuous rock masses. PhD Thesis. James CookUniversity of North Queensland, Australia.

Han, C. (1972) The technique for obtaining equi-potential lines of ground water flow in slopes usingelectrically conducting paper. MSc. Thesis, LondonUniversity (Imperial College).

Hanna, T. H. (1982) Foundations in Tension—GroundAnchors. Trans. Tech. Publications/McGraw HillBook Co., Clausthal-Zellerfeld, West Germany.

Harp, E. L. and Noble, M. A. (1993) An engineer-ing rock classification to evaluate seismic rock fallsusceptibility and its application to the WasatchFront. Bull. Assoc. Eng. Geologists., XXX(3),293–319.

Harp, E. L. and Wilson, R. C. (1995) Shaking intensitythresholds for rock falls and slides: evidence from the1987 Whittier Narrows and Superstition Hills earth-quake strong motion records. Bull. SeismologicalSoc. Am., 85(6), 1739–57.

Harp, E. L. and Jibson, R. W. (2002) Anomalousconcentrations of seismically triggered rock falls inPacoima Canyon: are they caused by highly suscept-ible slopes or local amplification of seismic shaking.Bull. Seismological Soc. Am., 92(8), 3180–9.

Harp, E. L., Jibson, R. W., Kayen, R. E., Keffer,D. S., Sherrod, B. L., Collins, B. D., Moss, R. E. S.and Sitar, N. (2003) Landslides and liquefactiontriggered by the M 7.9 Denali Fault earthquake of3 November, 2002. GSA Today (Geological Societyof America), August, 4–10.

416 References

Harr, M. E. (1977) Mechanics of ParticulateMatter—A Probabilistic Approach. McGraw-Hill,New York, 543 pp.

Harries, G. and Mercer, J. K. (1975) The science ofblasting and its use to minimize costs. Proc. Austr.Inst. Mining and Metall. Annual Conf. Adelaide,Part b, 387–99.

Hawley, P. M. and Stewart, A. F. (1986) Design ofopen pit coal mine slopes: an integrated approach.Proc. Int. Symp. on Geotechnical Stability in SurfaceMining. Calgary, pp. 69–77. © Swets & Zeitlinger.

Hawley, P. M., Martin, D. C. and Acott, C. P.(1986) Failure mechanics and design considerationsfor footwall slopes. CIM Bulletin, Vol. 79, No. 896,47–53.

Hemphill, G. B. (1981) Blasting Operations. McGrawHill Inc., New York.

Hencher, S. R. and Richards, L. R. (1989) Laboratorydirect shear testing of rock discontinuities. GroundEngineering, March, 24–31.

Hendron, A. J. and Patton, F. D. (1985) The Vaiontslide: a geotechnical analysis based on new geolo-gical observations of the failure surface. WaterwaysExperiment Station Technical Report, US ArmyCorps of Engineers, Vicksburg, MS, Vol. 1, 104 pp.

Heuer, R. E. (1995) Estimating ground water flowin tunnels. Proc. Rapid Excavation and TunnelingConf., San Francisco, pp. 41–80.

Hocking, G. (1976) A method for distinguish-ing between single and double plane sliding oftetrahedral wedges. Int. J. Rock Mech. Mining Sci.,13, 225–6.

Hoek, E. (1968) Brittle failure of rock. In: Rock Mech-anics in Engineering Practice (eds K. G. Stagg andO. C. Zienkiewicz), John Wiley & Sons, London,pp. 99–124.

Hoek, E. (1970) Estimating the stability of excavatedslopes in opencast mines. Trans. Inst. Mining andMetallurgy, London, 79, A109–32.

Hoek, E. (1974) Progressive caving induced by min-ing an inclined ore body. Trans. Inst. Mining andMetallurgy, London, 83, A133–39.

Hoek, E. (1983) Strength of jointed rock masses, 23rd.Rankine Lecture. Géotechnique 33(3), 187–223.

Hoek, E. (1990) Estimating Mohr–Coulomb frictionand cohesion values from the Hoek–Brown fail-ure criterion. Int. J. Rock Mech. Mining Sci. &Geomech. Abstr. 12(3), 227–29.

Hoek, E. (1994) Strength of rock and rock masses,ISRM News Journal, 2(2), 4–16.

Hoek, E. and Bray, J. (1977) Rock Slope Engineering,1st edn, IMM, London.

Hoek, E. and Bray, J. (1981) Rock Slope Engineering,3rd edn, Inst. Mining and Metallurgy, London, UK.

Hoek, E. and Brown, E. T. (1980a) Empirical strengthcriterion for rock masses. J. Geotech. Engng Div.,ASCE 106 (GT9), 1013–35.

Hoek, E. and Brown, E. T. (1980b) UndergroundExcavations in Rock, London, Inst. Mining Metal-lurgy, London, UK.

Hoek, E. and Brown, E. T. (1988) The Hoek–Brownfailure criterion—a 1988 update. Proc. 15th Cana-dian Rock Mech. Symp. (ed. J.C. Curran), pp.31–8. Toronto, Dept Civil Engineering, Universityof Toronto.

Hoek, E. and Brown, E. T. (1997) Practical estimatesor rock mass strength. Int. J. Rock Mech. Min. Sci.& Geomech. Abstr., 34(8), 1165–86.

Hoek, E. and Marinos, P. (2000) Predicting tun-nel squeezing. Tunnels and Tunnelling Int. Part1—November 2000, Part 2—December 2000.

Hoek, E. and Richards, L. R. (1974) Rock Slope DesignReview. Golder Associates Report to the PrincipalGovt. Highway Engineer, Hong Kong, 150 pp.

Hoek, E., Bray, J. and Boyd, J. (1973) The stabilityof a rock slope containing a wedge resting on twointersecting discontinuities. Quart. J. Engng Geol.,6(1) 22–35.

Hoek, E., Wood D. and Shah S. (1992) A modifiedHoek–Brown criterion for jointed rock masses. Proc.Rock Characterization, Symp. Int. Soc. Rock Mech.:Eurock ’92, (ed. J. A. Hudson), Brit. Geotech. Soc.,London, pp. 209–14.

Hoek, E., Kaiser P. K. and Bawden W. F. (1995)Support of Underground Excavations in HardRock. Rotterdam, Balkema.

Hoek, E., Marinos, P. and Benissi, M. (1998) Applic-ability of the Geological Strength Index (GSI) classi-fication for very weak and sheared rock masses. Thecase of the Athens Schist Formation. Bull. EngngGeol. Env., 57(2), 151–60.

Hoek, E., Carranza-Torres, C. and Corkum, B. (2002)Hoek–Brown failure criterion—2002 edition. Proc.North Am. Rock Mech. Soc. Meeting, Toronto,Canada, July, 267–73.

Hofmann, H. (1972) “Kinematische Modellstudienzum Boschungs-problem in regalmassig geklüftetenMedien.” Veröffentlichungen des Institutes fürBodenmechanik und Felsmechanik. Kalsruhe,Hetf 54.

Hong Kong Geotechnical Engineering Office (2000)Highway Slope Manual. Civil Engineering Depart-ment, Govt. Hong Kong, Special AdministrativeRegion, 114 pp.

References 417

Hudson, J. A. and Priest, S. D. (1979) Discontinuitiesand rock mass geometry. Int. J. Rock Mech. Min.Sci. & Geomech. Abstr., 16, 336–62.

Hudson, J. A. and Priest, S. D. (1983) Discontinuityfrequency in rock masses. Int. J. Rock Mech. Min.Sci. & Geomech. Abstr., 20, 73–89.

Huitt, J. L. (1956) Fluid flow in a simulated fracture.J. Am. Inst. Chem. Eng., 2, 259–64.

Hungr, O. (1987) An extension of Bishop’s simpli-fied method of slope stability to three dimensions.Géotechnique, 37, 113–17.

Hungr, O. and Evans, S. G. (1988) Engineering evalu-ation of fragmental rock fall hazards. Proc. 5th Int.Symp. on Landslides, Lausanne, July, pp. 685–90.

Hungr, O., Evans, S. G. and Hazzard, J. (1999) Mag-nitude and frequency of rock falls and rock slidesalong the main transportation corridors of south-western British Columbia. Canadian Geotech. J., 36,224–38.

Hutchinson, J. N. (1970) Field and laboratory stud-ies of a fall in upper chalk cliffs at Joss Bay, Isle ofThanet. Proc. Roscoe Memorial Symp., Cambridge.

Hynes, M. E. and Franklin, A. G. (1984) Rational-izing the seismic coefficient method. MiscellaneousPaper GL-84-13, US Army Engineer WaterwaysExperiment Station, Vicksburg, Mississippi, p. 34.

IAEG Commission on Landslides (1990) Suggestednomenclature for landslides. Bull. Int. Assoc. Eng.Geol., 41, 13–16.

International Society of Explosive Engineers (ISEE)(1998) Blasters’ Handbook, 17th edn Int. Soc. ofExplosive Eng., Cleveland, OH, 742 pp.

International Society for Rock Mechanics (ISRM)(1981a) Suggested Methods for the QuantitativeDescription of Discontinuities in Rock Masses (ed.E. T. Brown). Pergamon Press, Oxford, UK, 211 pp.

International Society of Rock Mechanics (ISRM)(1981b) Rock Characterization, Testing and Mon-itoring; ISRM Suggested Method. Pergamon Press,Oxford, UK.

International Society of Rock Mechanics (ISRM)(1981c) Basic geological description of rock masses.Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.,18, 85–110.

International Society of Rock Mechanics (1985) Sug-gested methods for determining point load strength.Int. J. Rock Mech., 22 (2), 53–60.

International Society of Rock Mechanics (1991) Sug-gested methods of blast vibration monitoring.Dowding, C. H., coordinator. Int. J. Rock Mech.Min. Sci. & Geomech. Abstr., Pergamon Press, 129(2), 143–56.

Ishikawa, N. (1999) Recent progress on rock shed stud-ies in Japan. Proc. Joint Japan–Swiss Sci. Seminar onImpact Loading by Rock Falls and Design of Protec-tion Measures, Japan Soc. Civil Engng, Kanazawa,Japan, pp. 1–6.

Itasca Consulting Group Inc. (2000) UDEC (Uni-versal Distinct Element Code), Version 3.1Minneapolis, MN.

Itasca Consulting Group Inc. (2001) FLAC (FastLagrangian Analysis of Continua), Version 4.0,Minneapolis, MN.

Itasca Consulting Group Inc. (2002) FLAC3D (FastLagrangian Analysis of Continua in 3 Dimensions),Version 2.1, Minneapolis, MN.

Itasca Consulting Group Inc. (2003) 3DEC (Three-Dimensional Distinct Element Code), Version 3.0,Minneapolis, MN.

Jacob, C. E. (1950). Flow of ground water. In: Engin-eering Hydraulics (ed. H. Rouse). John Wiley andSons, New York, pp. 321–86.

Jaegar, J. C. (1970) The behaviour of closely jointedrock. Proc. 11th Symp. Rock Mech., Berkeley, CA,pp. 57–68.

Jaeger, J. C. and Cook, N. G. W. (1976) Fundamentalsof Rock Mechanics, 2nd edn. Chapman and Hall,London, UK, 585 pp.

Janbu, N. (1954) Application of composite slidecircles for stability analysis. Proc. European Con-ference on Stability of Earth Slopes. Stockholm, 3,pp. 43–9.

Janbu, N., Bjerrum, L. and Kjaernsli, B. (1956) SoilMechanics Applied to Some Engineering Problems(in Norwegian with English summary). NorwegianGeotech. Inst., Publication 16.

Jenike, A. W. and Yen, B. C. (1961) Slope stabilityin axial symmetry. In: Proc. 5th Rock MechanicsSymposium (University of Minnesota, 1961), pp.689–711 Pergamon, London.

Jennings, J. E. (1970) A mathematical theory for thecalculation of the stability of open cast mines. Proc.Symp. on the Theoretical Background to the Plan-ning of Open Pit Mines, Johannesburg, pp. 87–102.

Jibson, R. W. (1993) Predicting earthquake-inducedlandslide displacements using Newark’s slidingblock analysis. Transportation Research Record1411, Transportation Research Board, Washington,DC, pp. 9–17.

Jibson, R. W. (2003) Personal communication.Jibson, R. W. and Harp, E. L., (1995) Inventory

of Landslides Triggered by the 1994 Northridge,California Earthquake. Dept of the Interior, USGS,Open–File 95-213, 17 pp.

418 References

Jibson, R. W., Harp, E. L., and Michael, J. A. (1998)A method for producing digital probabilistic seis-mic landslide hazard map: an Example from theLos Angeles, California area. Dept. Interior, USGS,Open–File Report 98-113, 17 pp.

John, K. W. (1970) Engineering analysis of three-dimensional stability problems utilizing the referencehemisphere. Proc. 2nd Int. Cong. Int. Soc. RockMech., Belgrade, 2, 314–21.

Kane, W. F. and Beck, T. J. (1996) Rapid slope mon-itoring. Civil Engineering, ASCE, Washington, DC,June, 56–8.

Keefer, D. L. (1984) Landslides caused by earthquakes.Geol. Soc. of America Bull., 95, 406.21.

Keefer, D. L. (1992) The susceptibility of rock slopesto earthquake-induced failure. Proc. 35th AnnualMeeting of the Assoc. of Eng. Geologists (ed. MartinL. Stout), Long Beach, CA, pp. 529–38.

Kennedy, B. A. and Neimeyer, K. E. (1970)Slope monitoring systems used in the predic-tion of a major slope failure at the Chuquica-mata Mine, Chile. Proc. of Symp. On PlanningOpen Pit Mines, Johannesburg, South Africa,pp. 215–25.

Kielhorn, W. (1999) Another wedge calculator.Computer software, [email protected]

Kiersch, G. A. (1963). Vaiont Reservoir Disaster. CivilEngng, 34(3), 32–9.

Kikuchi, K., Kuroda, H. and Mito, Y. (1987)Stochastic estimation and modeling of rock jointdistribution based on statistical sampling. 6th Int.Congress on Rock Mech., Montreal, Balkema,pp. 425–8.

King, R. A. (1977) A review of soil corrosiveness withparticular reference to reinforced earth. Transportand Road Research Laboratory, Crowthorne, UK,Supplementary Report No. 316.

Kobayashi, Y., Harp, E. L. and. Hagawa, T. (1990)Simulation of rock falls triggered by earthquakes.Rock Mech. Rock Engng, 23(1), 1–20.

Kreyszig. E. (1976) Advanced Engineering Mathemat-ics. John Wiley and Sons, New York, pp. 770–6.

Kulatilake, P. H. S. (1988) State of the art in stochasticjoint geometry modeling. Proc. 29th US Symp.Rock Mech. (eds P. A. Cundall, J. Sterling andA. M. Starfield), Balkema, Rotterdam, pp. 215–29.

Kulatilake, P. H. S. and Wu, T. H. (1984) Estimationof the mean length of discontinuities. Rock Mech.Rock Engng, 17(4), 215–32.

Ladegaard-Pedersen, A. and Dally, J. W. (1975) AReview of Factors Affecting Damage in Blasting.

Report to the National Science Foundation. Mech.Engng Dept, Univ. of Maryland, 170 pp.

Lambe, W. T. and Whitman, R. V. (1969). SoilMechanics. Wiley, New York.

Langefors, U. and Kihlstrom, B. (1973). The ModernTechnique of Rock Blasting. John Wiley and Sons,New York, 2nd edn.

Lau, J. S. O. (1983) The determination of true orient-ations of fractures in rock cores. Can. Geotech. J.,20, 221–7.

Leighton, J. C. (1990) New tunnel at Shalath Bluff onBC Rail’s Squamish Subdivision. Proc. 8th AnnualMeeting of the Tunnelling Association of Canada,Vancouver, BC, October, Bitech Publishers Ltd.,pp. 255–66.

Ley, G. M. M. (1972) The Properties of Hydrotherm-ally Altered Granite and their Application to SlopeStability in Open Cast Mining. MSc. Thesis, LondonUniversity, Imperial College.

Leyshon, P. R. and Lisle, R. J. (1996) Stereo-graphic Projection Techniques in Structural Geo-logy. Butterworth and Heinemann, Oxford, UK,104 pp.

Lin, J.-S. and Whitman, R. V. (1986) Earthquakeinduced displacement of sliding blocks. J. Geotech.Engng Div., ASCE, Vol. 112, No.1 Jan. 1986.

Ling, H. I. and Cheng, A. H.-D. (1997) Rock slidinginduced by seismic forces. Int. J. Rock Mech. MiningSci. 34(6), 1021–9.

Littlejohn, G. S. and Bruce, D. A. (1977) RockAnchors—State of the Art. Foundation PublicationsLtd., Brentwood, Essex, UK.

Londe, P. (1965) Une method d’analyse a trois dimen-sions de la stabilite d’une rive rocheuse. Annales desPonts et Chaussees, Paris, 37.60.

Londe, P., Vigier, G. and Vormeringer, R. (1969)Stability of rock slopes—a three-dimensional study.J. Soil Mech. Fndn Engng Div. ASCE, 95 (SM1),235–62.

Londe, P., Vigier, G. and Vormeringer, R. (1970)Stability of rock slopes—graphical methods. J. SoilMech. Fndn Engng Div. ASCE, 96 (SM4), 1411–34.

Lorig, L. J. (1985) A simple numerical representationof fully bonded passive rock reinforcement for hardrocks, Computers & Geotechnics, 1, 79–97.

Lorig, L. and Varona, P. (2001) Practical slope-stabilityanalysis using finite-difference codes. In: Slope Sta-bility in Surface Mining (eds W. A. Hustrulid,M. J. McCarter and D. J. A. Van Zyl). Society forMining, Metallurgy and Exploration, Inc., Littleton,pp. 115–24.

References 419

Louis, C. (1969) A study of ground water flow in join-ted rock and its influence on the stability of rockmasses. Doctorate thesis, Univ. of Karlsruhe (inGerman). English translation: Imperial College RockMech. Research Report No. 10, 50 pp.

McCauley, M. L., Works, B. W. and Naramore, S. A.(1985) Rockfall Mitigation. Report FHWA/CA/TL-85/12. FHWA, US Department of Transportation.

McClay, K. R. (1987) The Mapping of GeologicalStructures, John Wiley & Sons, Sussex, UK.

McDonald, M. and Harbaugh, A. (1988) A modu-lar three-dimensional finite-difference ground waterflow model. US Geological Survey Tech. WaterResources Inv., Bk. 6, Ch. A1.

McGuffey, V., Athanasiou-Grivas, D., Iori, J.and Kyfor Z. (1980) Probabilistic EmbankmentDesign—A Case Study. Transportation ResearchBoard, Washington, DC.

McKinstry, R., Floyd, J. and Bartley, D. (2002) Elec-tronic detonator performance evaluation at BarrickGoldstrike Mines Inc. J. Explosives Engng, Int.Soc. Explosives Eng., Cincinnati, OH, May/June,pp. 12–21.

McMahon, B. K. (1982) Probabilistic Design inGeotechnical Engineering. Australian MineralFoundation, AMF Course 187/82, Sydney.

Mahtab, M. A. and Yegulalp, T. M. (1982) A rejec-tion criterion for definition of clusters in orient-ation data. Proc. 22nd Symp. on Rock Mech.,Berkeley, CA, Soc. Min. Eng., American Inst.Mining, Metallurgical and Petroleum Engineers.pp. 116–24.

Maini, Y. N. (1971) In situ parameters in jointedrock—their measurement and interpretation. PhDThesis, University of London.

Mamaghani, I. H. P., Yoshida, H. and Obata, Y.(1999) Reinforced expanded polystyrene Styrofoamcovering rock sheds under impact of falling rock.Proc. Joint Japan–Swiss Sci. Seminar on ImpactLoading by Rock Falls and Design of ProtectionMeasures, Japan Soc. Civil Eng., Kanazawa, Japan,pp. 79–89.

Marinos, P. and Hoek, E. (2000). GSI—a geologic-ally friendly tool for rock mass strength estimation.Proc. GeoEng 2000 Conference, Melbourne.

Marinos, P., and Hoek, E. (2001) Estimating the geo-technical properties of heterogeneous rock massessuch as flysch. Accepted for publication in the Bull.Int. Assoc. of Engng Geologists.

Markland, J. T. (1972) A useful technique for estimat-ing the stability of rock slopes when the rigid wedge

sliding type of failure is expected. Imperial CollegeRock Mechanics Research Report No. 19, 10 pp.

Marsal, R. J. (1967) Large scale testing of rock fillmaterials. J. Soil Mech. Fdns. Div., ASCE, 93, SM2,27–44.

Marsal, R. J. (1973) Mechanical properties of rockfill. In: Embankment Dam Engineering, CasagrandeVolume, Wiley and Sons, New York, pp. 109–200.

Martin, D. C. (1993) Time-dependent deformationof rock slopes. PhD Thesis, University of London,August, 1993.

Mearz, N. H., Franklin, J. A. and Bennett, C. P. (1990)Joint roughness measurements using shadow profilo-metry. Int. J. Rock Mech. Min. Sci. & Geomech.Abstr., 27(5), 329–43.

Meyerhof, G. G. (1984) Safety factors and limit statesanalysis in geotechnical engineering. Can. Geotech.J., 21, 1–7.

Middlebrook, T. A. (1942) Fort Peck slide. Proc.ASCE, Vol. 107, Paper 2144, p. 723.

Ministry of Construction, Japan (1983) ReferenceManual on Erosion Control Works (in Japanese),Erosion Control Department, Japan, 386 pp.

Mohr, O. (1900) Welche Umstände bedingen die Elast-izitätsgrenze und den Bruch eines Materials? Z. Ver.dt. Ing., 44, 1524–30; 1572–77.

Moore, H. (1986) Construction of a shot-in-placerock buttress for landslide stabilization. Proc. 37thHighway Geology Symp., Helena, Montana, 21 pp.

Moore, D. P. and Imrie, A. S. (1982) Rock slope sta-bilization at Revelstoke Dam site. Trans. 14th Int.Cong. on Large Dams, ICOLD, Paris, 2, pp. 365–85.

Moore, D. P. and Imrie, A. S. (1993) Stabilizationat Dutchman’s Ridge. Proc. 6th Int. Symp. on Land-slides, Christchurch, NZ, Balkema, Vol. 3, 6 pp.

Moore, D. P., Imrie, A. S. and Enegren, E. G. (1997)Evaluation and management of Revelstoke reservoirslopes. Int. Commission on Large Dams, Proc. 19thCongress, Florence, Q74, R1, 1–23.

Morgan, D. R., McAskill, N., Richardson, B. W. andZellers, R. C. (1989) A comparative study of plain,polypropylene fiber, steel fiber, and wire mesh rein-forced shotcretes. Transportation Research Board,Annual Meeting, Washington, DC, 32 pp. (plusappendices).

Morgan, D. R., Heere, R., McAskill, N. and Chan, C.(1999) Comparative evaluation of system ductilityof mesh and fibre reinforced shotcretes. Proc. Conf.on Shotcrete for Underground Support VIII, Engin-eering Foundation (New York) Campos do Jordao,Brazil, 23 pp.

420 References

Morgenstern, N. R. (1971) The influence of groundwater on stability. Proc. 1st. Symp. on Stabilityin Open Pit Mining, Vancouver, Canada, AIME.New York, 65.82.

Morgenstern, N. R. and Price, V. E. (1965) Theanalysis of the stability of general slide surfaces.Geotechnique. 15, 79–93.

Morris, A. J. and Wood, D. F. (1999) Rock slope engin-eering and management process on the Canadianpacific Railway. Proc. 50th Highway GeologySymposium, Roanoke, VA.

Morriss, P. (1984) Notes on the Probabilistic Design ofRock Slopes. Australian Mineral Foundation, notesfor course on Rock Slope Engineering, Adelaide,April.

Muller, L. (1968) New considerations of theVaiont slide. Felsmechanik und engenieurgeologie,6 (1), 1–91.

National Highway Institute (NHI) (1998) Geotech-nical Earthquake Engineering. Publication No.FHWA HI-99-012, Washington, DC, 265 pp.

Newmark, N. M. (1965) Effects of earthquakeson dams and embankments. Geotechnique, 15(2),139–60.

Nonveiller, E. (1965) The stability analysis of slopeswith a slide surface of general shape. Proc. 6th Int.Conf. Soil Mech. Foundation Engng. Montreal. 2,pp. 522.

Norrish, N. I. and Lowell, S. M. (1988) Aesthetic andsafety issues for highway rock slope design. Proc.39th Annual Highway Geology Symp., Park City,Utah.

Norton, F. H. (1929) Creep of Steel at HighTemperatures. McGraw-Hill, New York.

Office of Surface Mining (OSM) (2001) Use of explos-ives: preblasting surveys. US Dept of the Interior,Surface Mining Law Regulations, Subchapter K, 30CRF, Section 816.62, Washington, DC.

Oregon Department of Transportation (2001) RockFall Catchment Area Design Guide. ODOTResearch Group Report SPR-3(032), Salem, OR,77 pp. with appendices.

Oriard, L. L. (1971) Blasting effects and their control inopen pit mining. Proc. Second Int. Conf. on Stabilityin Open Pit Mining, Vancouver, AIME, New York,pp. 197–222.

Oriard, L. L. (2002) Explosives Engineering, Con-struction Vibrations and Geotechnology. Int. Soc.Explosives Engineers, Cleveland, OH, 680 pp.

Oriard, L. L. and Coulson, J. H. (1980) Blast VibrationCriteria for Mass Concrete. Minimizing Detrimental

Construction Vibrations. ASCE, Preprint 80–175,pp. 103–23.

Pahl, P. J. (1981) Estimating the mean length of dis-continuity traces. Int. J. Rock Mech. Min. Sci. &Geomech. Abstr., 18, 221–8.

Pan, X. D. and Hudson, J. A. (1988) A simplified threedimensional Hoek–Brown yield criterion. In: RockMechanics and Power Plants, Rotterdam, Balkema,pp. 95–103.

PanTechnica Corp. (2002) KbSlope Slope Sta-bility Program for KeyBlock Analysis. Pan-Technica Corporation, Caska, Minnesota,www.pantechnica.com/

Patton, F. D. (1966) Multiple modes of shear failurein rock. Proc. 1st. Int. Congress on Rock Mech.,Lisbon, 1, 509–13.

Patton, F. D. and Deere, D. U. (1971) Geologicfactors controlling slope stability in open pit mines.Proc. 1st. Symp. on Stability in Open Pit Mining,Vancouver, Canada, AIME. New York, 23–48.

Paulding, B. W. Jr. (1970) Coefficient of friction ofnatural rock surfaces. Proc. ASCE J. Soil Mech.Foundation Div., Vol. 96 (SM2), 385–94.

Peck, R. B. (1967) Stability of natural slopes. Proc.ASCE, 93 (SM 4), pp. 403–17.

Peckover, F. L. (1975) Treatment of rock fallson railway lines. American Railway Engineer-ing Association, Bulletin 653, Washington, DC,pp. 471–503.

Pells, P. J. N. (1974) The behaviour of fully bondedrock bolts. In: Advances in Rock Mechanics. Vol 2:Part B, pp. 1212–17 National Academy of Sciences,Washington, DC.

Pentz, D. L. (1981) Slope stability analysis techniquesincorporating uncertainty in the critical parameters.Third Int. Conf. on Stability in Open Pit Mining,Vancouver, Canada.

Persson, P. A. (1975) Bench drilling—an important firststep in the rock fragmentation process. Atlas CopcoBench Drilling Symposium, Stockholm.

Persson, P.A., Holmburg, R. and Lee, J. (1993) RockBlasting and Explosive Engineering. CRC Press,Boca Raton FL.

Peterson, J. E., Sullivan, J. T. and Tater, G. A. (1982)The use of computer enhanced satellite imagery forgeologic reconnaissance of dam sites. ICOLD, 14thCong. on Large Dams, Rio de Janeiro, Q53, R26,Vol. II, 449–71.

Pfeiffer, T. J. and Bowen, T. D. (1989) Computer sim-ulation of rock falls. Bull. Assoc. Eng. Geol., XXVI(1), 135–46.

References 421

Pfeiffer, T. J., Higgins, J. D. and Turner, A. K.(1990) Computer aided rock fall hazard analysis.Proc. Sixth Int. Congress, Int. Assoc. of Engin-eering Geology, Amsterdam. Balkema, Rotterdam,Netherlands, pp. 93–103.

Phillips, F. C. (1971) The Use of Stereographic Pro-jections in Structural Geology. Edward Arnold,London, UK, 90 pp.

Pierce, M., Brandshaug, T. and Ward, M. (2001)Slope stability assessment at the Main CressonMine. In: Slope Stability in Surface Mining (edsW. A. Hustralid, M. J. McCarter and D. J. A.Van Zyl). Littleton: Soc. Mining, Metallurgy andExploration, Inc., 239–50.

Pierson, L., Davis, S. A. and Van Vickle, R. (1990) TheRock Fall Hazard Rating System, ImplementationManual. Technical Report #FHWA-OR-EG-90-01,Washington, DC.

Piteau, D. R. (1980) Slope stability analysis for rockfall problems: the computer rock fall model forsimulating rock fall distributions. Rock Slope Engin-eering, Part D. Federal Highway AdministrationReference Manual FHWA-TS-79-208: Departmentof Transportation, Washington, DC.

Piteau, D. R. and Jennings, J. E. (1970) The effectsof plan geometry on the stability of natural slopesin rock in the Kimberley area of South Africa. In:Proceedings of the 2nd Congress of the Interna-tional Society of Rock Mechanics (Belgrade). Vol 3:Paper 7–4.

Piteau, D. R. and Peckover, F. L. (1978) Engineer-ing of rock slopes. Landslides analysis and control,Special Report 176, Chapter 9, National Academyof Sciences, Washington, DC, pp. 192–234.

Post Tensioning Institute (PTI) (1996) Recommenda-tions for Prestressed Rock and Soil Anchors, 2ndedn. Phoenix, Az, 70 pp.

Priest, S. D. and Hudson, J. A. (1976) Discontinuityspacings in rock. Int. J. Rock Mech. Min. Sci. &Geomech. Abstr., 13, 135–48.

Priest, S. D. and Hudson, J. A. (1981) Estimation ofdiscontinuity spacing and trace length using scanlinesurveys. Int. J. Rock Mech. Min. Sci. & Geomech.Abstr., 18, 183–97.

Pritchard, M. A. and Savigny, K. W. (1990) Numer-ical modelling of toppling. Can. Geotech. J., 27,823–34.

Pritchard, M. A. and Savigny, K. W. (1991) TheHeather Hill landslide: an example of a large scaletoppling failure in a natural slope. Can. Geotech. J.,28, 410–22.

Protec Engineering (2002) Rock fall caused by earth-quake and construction of GeoRock Wall in NiijimaIsland, Japan. http://www.proteng.co.jp

Pyke, R. (1999) Selection of seismic coefficients foruse in pseudo-static slope stability analysis. http://www.tagasoft.com/opinion/article2.hmtl, 3 pp.

Rengers, N. (1971) Roughness and friction propertiesof separation planes in rock. Thesis, Tech. Hoch-schule Fredericiana, Karlsruhe, Inst. Bodenmech.Felsmech. Veröff, 47, 129 pp.

Ritchie, A. M. (1963) Evaluation of rock fall andits control. Highway Research Record 17, High-way Research Board, NRC, Washington, DC,pp. 13–28.

Roberds, W. J. (1984) Risk-based decision mak-ing in geotechnical engineering: overview of casestudies. Engineering Foundation Conf. on Risk-based Decision Making in Water Resources. SantaBarbara, CA.

Roberds, W. J. (1986) Applications of decision theoryto hazardous waste disposal. ASCE Specialty Conf.GEOTECH IV, Boston, MA.

Roberds, W. J. (1990) Methods of developingdefensible subjective probability assessments. Trans-portation Research Board, Annual Meeting,Washington, DC.

Roberds, W. J. (1991) Methodology for optim-izing rock slope preventative maintenance pro-grams. ASCE, Geotechnical Engineering Congress,Boulder, CO, Geotechnical Special Publication 27,pp. 634–45.

Roberds, W. J., Ho, K. K. S. and Leroi, E. (2002)Quantitative risk assessment of landslides. Trans-portation Research Record 1786, Paper No.02-3900, pp. 69–75, Transportation ResearchBoard, Washington, DC.

Roberts, D. and Hoek, E. (1972) A study of the stabil-ity of a disused limestone quarry face in the MendipHills, England. Proc. 1st. Int. Conf. on Stability inOpen Pit Mining, Vancouver, Canada. Published byAIME, New York, pp. 239–56.

Rocscience (2001) SWEDGE—Probabilistic Ana-lysis of the Geometry and Stability of Sur-face Wedges. Rocscience Ltd., Toronto, Canada,www.rocscience.com/

Rocscience Ltd. (2002a) ROCLAB Software for Calcu-lating Hoek–Brown Rock Mass Strength. Toronto,Ontario, www.rocscience.com/

Rocscience Ltd. (2002b) SLIDE—2D Slope StabilityAnalysis for Rock and Soil Slopes. Toronto, Ontario,www.rocscience.com/

422 References

Rocscience, I. (2002c) PHASE2 Version 5 (Toronto,Rocscience).

Rocscience (2003) ROC PLANE—Planar Sliding Sta-bility Analysis for Rock Slopes. Toronto, Ontario,www.rocscience.com/

Rocscience (2004) Roc Fall—Statistical Analysis ofRock falls. Toronto, Ontario, www.rocscience.com/

Rohrbaugh, J. (1979) Improving the quality of groupjudgment: social judgment analysis and the Delphitechnique. Organizational Behaviour and HumanPerformance, 24, 73–92.

Ross-Brown, D. R. (1973) Slope Design in Open CastMines. PhD Thesis, London University, 250 pp.

Sageseta, C., Sánchez, J. M. and Cañizal, J. (2001) Ageneral solution for the required anchor force in rockslopes with toppling failure. Int. J. Rock Mech. Min.Sci., 38, 421–35.

Salmon, G. M. and Hartford, N. D. (1995) Risk ana-lysis for dam safety. Int. Water Power and DamConstruction, 21, 38–9.

Sarma, S. K. (1975) Seismic stability of earth dams andembankments. Geotechnique 25 (4), 743–61.

Sarma, S. K. (1979) Stability analysis of embankmentsand slopes. J. Geotech. Engng Div., ASCE, 105(GT12) 1511–24.

Savely, J. P. (1987) Probabilistic analysis of intenselyfractured rock masses. Sixth International Congresson Rock Mechanics, Montreal, 509–14.

Scheidegger, A. E. (1960) The Physics of Flow ThroughPorous Media. Macmillan, New York.

Schuster, R. L. (1992) Recent advances in slope sta-bilization. Keynote Paper, Session G3, Proc. 6thInt. Symp. on Landslides, Auckland, Balkema,Rotterdam, Netherlands.

Seed, H. B. (1979) Considerations in the earthquake-resistant design of earth and rockfill dams.Geotechnique, 29(3), 215–63.

Shah, S. (1992) A Study of the Behaviour of JointedRock Masses, PhD Thesis, University of Toronto.

Sharma, S. (1991) XSTABL, an integrated slope stabil-ity analysis method for personal computers, Version4.00. Interactive Software Designs Inc., Moscow,Idaho, USA.

Sharp, J. C. (1970) Fluid flow through fractured media.PhD Thesis, University of London.

Siskind, D. D., Stachura, V. J. and Raddiffe, K. S.(1976) Noise and Vibrations in Residential Struc-tures from Quarry Production Blasting. US Bureauof Mines, Report of Investigations 8168.

Siskind, D. E., Staff, M. S., Kopp, J. W. andDowding, C. H. (1980). Structure Response and

Damage Produced by Ground Vibrations from Sur-face Blasting. US Bureau of Mines, Report ofInvestigations 8507.

Sjöberg, J. (1999) Analysis of Large Scale RockSlopes. Doctoral thesis 1999:01, Division of Rockmechanics, Luleå University of Technology.

Sjöberg, J. (2000) Failure mechanism for high slopes inhard rock. Slope Stability in Surface Mining, Societyof Mining, Metallurgy and Exploration, Littleton,CO, pp. 71–80.

Sjöberg, J., Sharp, J. C. and Malorey, D. J. (2001) Slopestability at Aznalcóllar. In: Slope Stability in Sur-face Mining (eds W. A. Hustralid, M. J. McCarterand D. J. A. Van Zyl), Society for Mining,Metallurgy and Exploration, Inc., Littleton, CO,183–202.

Skempton, A. W. (1948) The φ = 0 analysis for sta-bility and its theoretical basis. Proc. 2nd Int. Conf.Soil Mech. Foundation Engng, Rotterdam, Vol. 1,pp. 72.

Skempton, A. W. and Hutchinson, J. N. (1969) Stabil-ity of natural slopes and embankment foundations.State of the art report. Proc. 7th Int. Conf. on SoilMechanics, Mexico, Vol. 1, pp. 291–340.

Smith, D. D. and Duffy J. D. (1990) Field Tests andEvaluation of Rock Fall Restraining Nets. Divisionof Transportation Materials and Research, Engin-eering Geology Branch, California Department ofTransportation, Sacramento, CA.

Snow, D. T. (1968) Rock fracture spacings, openingsand porosities. J. Soil Mech. Fndn. Dir., Proc. ASCE,94, 73–91.

Sonmez, H. and Ulusay, R. (1999) Modifications to thegeological strength index (GSI) and their applicabil-ity to the stability of slopes. Int. J. Rock Mech. Min.Sci., 36(6), 743–60.

Soto, C. (1974) A Comparative Study of Slope Model-ling Techniques for Fractured Ground. MSc. Thesis.London University, Imperial College.

Spacial Data Services Inc. (2002) Cyrax Laser Scan-ning. Canada. //www.sds3dscan.com/

Spang, K. and Egger, P. (1990) Action of fully-groutedbolts in joined rock for fractured ground. J. RockMech. Rock Engng., 23, 210–99.

Spang, R. M. (1987) Protection against rock fall—stepchild in the design of rock slopes. Proc. Int.Conf. Rock Mech., Montreal, Canada, ISRM,Lisbon, Portugal, pp. 551–7.

Spencer, E. (1967) A method of analysis of the stabilityof embankments assuming parallel inter-slice forces.Geotechnique, 17, 11–26.

References 423

Spencer, E. (1969) Circular and logarithmic spiralslide surfaces. J. Soil Mech. Foundation Div. ASCE.95(SM 1), 227–234.

Stacey, P. F. (1996) Second workshop on large scaleslope stability. Las Vegas, Sept. 13 (no proceedings).

Stagg, M. S., Siskind, D. E., Stevens, M. G. andDowding, C. H. (1984) Effects of Repeated Blast-ing on a Wood Frame House. US Bureau of Mines,Report of Investigations 8896.

Starfield, A. M. and Cundall, P. A. (1988) Towards amethodology for rock slope modelling. Int. J. RockMechanics, Mining Sci. & Geomech. Abstr., 25(3),99–106.

Stauffer, M. R. (1966) An empirical-statistical studyof three-dimensional fabric diagrams as usedin structural analysis. Can. J. Earth Sci., 3,473–98.

Stewart, A. F., Wessels, F. and Bird, S. (2000)Design, implementation and assessment of open-pitslopes at Palabora over the last 29 years. In: SlopeStability in Surface Mining (eds W. A. Hustralid,M. J. McCarter and D. J. A. Van Zyl), Society forMining, Metallurgy and Exploration, Inc., Littleton,CO, pp. 177–81.

Taylor, D. W. (1937) Stability of Earth Slopes.J. Boston Soc. Civil Engineers, 24, 197 pages.

Terzaghi, K. (1943) Theoretical Soil Mechanics. JohnWiley, New York.

Terzaghi, K. (1950) Mechanisms of Landslides.Engineering Geology (Berkley) Volume (GeologicalSociety of America).

Terzaghi, K. (1962) Stability of steep slopes on hardunweathered rock. Geotechnique, 1–20.

Terzaghi, K. and Peck R. (1967). Soil Mechanics inEngineering Practice. John Wiley and Sons, NewYork, N. Y.

Terzaghi, R. (1965) Sources of errors in joint surveys.Geotechnique, 15, 287–304.

Theis, C. V. (1935) The relation of the lowering of thepiezometric surface, and the rate and duration of dis-charge of a well using ground water storage. Trans.Amer. Geophysical Union, 16, 519–24.

Threadgold, L. and McNichol, D. P. (1985) Designand construction of polymer grid boulder barriers toprotect a large public housing site for Hong KongHousing Authority. Polymer Grid Reinforcement,Thomas Telford, London, UK.

Todd, D. K. (1959) Ground Water Hydrology. JohnWiley and Sons, New York, pp. 47–9 and 78–114.

Transportation Research Board (TRB) (1996) Land-slides, Investigation and Mitigation. National

Research Council, Special Report 247, Ch. 1,Washington DC, 673 pp.

Transportation Research Board (TRB) (1999) Geo-technical Related Development and Implementationof Load and Resistance Factor Design (LRFD) Meth-ods. NCHRP Synthesis No. 276, Washington,DC, 69 pp.

Transportation Research Board (TRB) (2002) Eval-uation of Metal Tensioned Systems in Geotech-nical Applications. NCHRP Project No. 24–13, Washington, DC, 102 pp. plus figures andappendices.

Trollope, D. H. (1980) The Vaiont slope failure. RockMechanics, 13(2), 71–88.

Tse, R. and Cruden, D. M. (1979) Estimating jointroughness coefficients. Int. J. Rock Mech. Min. Sci.& Geomech. Abstr, 16, 303–7.

Ucar, R. (1986) Determination of shear failure envel-ope in rock masses. J. Geotech. Eng. Div. ASCE,112(3), 303–15.

United States Bureau of Mines (USBM) (1984) SelectedPneumatic Gunites for Use in Underground Mines:A Comparative Engineering Analysis. USBM, Dept.of the Interior, Information Circular 8984.

University of Utah, (1984) Flooding and landslides inUtah—an economic impact analysis. University ofUtah Bureau of Economic and Business Research,Utah Dept. of Community and Economic Develop-ment, and Utah Office of Planning and Budget, SaltLake City, 123 pp.

USEPA (1993) Technical manual: solid waste dis-posal facility criteria. United States Environ-mental Protection Agency, EPA/530/R-93/017,Washington, DC.

Vamosi, S. and Berube, M. (1987) Is ParliamentHill moving? Canadian Dept. Energy Mines andResources, Geos Magazine, 4, 18–22.

Van Velsor, J. E. and Walkinshaw, J. L. (1992) Acceler-ated movement of a large coastal landslide followingthe October 17, 1989 Loma Prieta earthquake inCalifornia. Transportation Research Record 1343,Transportation Research Board, Washington, DC,pp. 63–71.

Watson, J. (2002) ISEE (2002) Electronic blastinitiation—a practical users guide. J. ExplosivesEng., Int. Soc. Explosives Eng., Cincinnati, OH,May/June, pp. 6–10.

Watts, C. F. Associates (2001) ROCKPACK III for theAnalysis of Rock Slope Stability. [email protected]

Wei, Z. Q., Egger, P. and Descoeudres, F. (1995)Permeability predictions for jointed rock masses. Int.

424 References

J. Rock Mech. and Min. Sciences, Pergamon Press,32 (3), 251–61.

Whitman, R. V. (1984) Evaluating calculated risk ingeotechnical engineering. J. Geotech. Engng, ASCE,110, 145–88.

Whitman, R. V. and Bailey, W. A. (1967) The use ofcomputers in slope stability analysis. J. Soil Mech.Fndn. Eng., ASCE, 93, SM4, 475–98.

Whyte, R. J. (1973) A Study of Progressive hangingwall caving at Chambishi copper mine in Zambiausing the base friction model concept. MSc. Thesis.London University, Imperial College.

Wiss, J. F. (1981) Construction vibrations: state-of-the-art. Proc. Am. Soc. Civil Engng, Geotech. Eng.Div., Vol. 107. No. GT2, February, pp. 167–80.

Wittke, W. W. (1965) Method to analyze the stabilityof rock slopes with and without additional loading(in German). Felsmechanick und Ingenieurgeolgie.Supp. II. Vol. 30, 52–79. English translation inImperial College Rock Mechanics Research ReportNo. 6, July, 1971.

Wright, E. M. (1997) State of the art of rock cut designin eastern Kentucky. Proc. Highway Geology Symp.,Knoxville, TN, pp. 167–73.

Wu, S.-S. (1984) Rock fall evaluation by com-puter simulation. Transportation Research Board,Washington, Transportation Research Record,No. 1031.

Wu, T. H., Ali, E. M. and Pinnaduwa, H. S. W. (1981)Stability of slopes in shale and colluvium. Ohio Dept.of Transportation and Federal Highway Administra-tion, Prom. No. EES 576. Research Report preparedby Ohio State University, Dept. Civil Engng.

www.terrainsar.com (2002) Webpage describing fea-tures of Synthetic Aperture Radar.

Wyllie, D. C. (1980) Toppling rock slope failures,examples of analysis and stabilization. Rock Mech.,13, 89–98.

Wyllie, D. C. (1987) Rock slope inventory system. Proc.Federal Highway Administration Rock Fall Mitiga-tion Seminar, FHWA, Region 10, Portland, Oregon.

Wyllie, D. C. (1991) Rock slope stabilization and pro-tection measures. Assoc. Eng. Geologists, NationalSymp. on Highway and Railway Slope Stability,Chicago.

Wyllie, D. C. (1999) Foundations on Rock, 2nd edn,Taylor and Francis, London, UK, 401 pp.

Wyllie, D. C. and Munn, F. J. (1979) Use of movementmonitoring to minimize production losses due to pitslope failure. Proc. 1st Int. Symp. on Stability in

Coal Mining, Vancouver, Canada, Miller FreemanPublications, pp. 75–94.

Wyllie, D. C. and Wood, D. F. (1981) Preventat-ive rock blasting protects track. Railway Track andStructures, Chicago, IL, Sept., 34–40.

Wyllie, D. C., McCammon, N. R. and Brumund, W. F.(1979) Use of risk analysis in planning slope stabil-ization programs on transportation routes. Trans-portation Research Board, Research Record 749,Washington DC.

Xanthakos, P. P. (1991) Ground Anchorages andAnchored Structures. John Wiley & Sons Inc., NewYork, 686 pp.

Yoshida, H. (2000) Personal communication.Yoshida, H., Ushiro, T., Masuya, H. and Fujii, T.

(1991) An evaluation of impulsive design load ofrock sheds taking into account slope properties (inJapanese). J. Struct. Eng., 37A (March), 1603–16.

Youd, T. L. (1978) Major cause of earthquake dam-age is ground movement. Civil Engng, ASCE, April,47–51.

Youngs, R. R., Day, S. M. and Stevens, J. L. (1988)Near field ground motions on rock for largesubduction earthquakes. Earthquake Engineeringand Soil Dynamics II—Recent Advances in Ground-Motion Evaluation, ASCE Geotechnical SpecialPublication No. 20, pp. 445–62.

Yu, X. and Vayassde, B. (1991) Joint profiles andtheir roughness parameters. Technical note, Int. J.Rock Mech. Min. Sci. & Geomech. Abstr., 16(4),333–6.

Zanbak, C. (1983) Design charts for rock slope sus-ceptible to toppling. ASCE, J. Geotech. Engng, 109(8), 1039–62.

Zavodni, Z. M. (2000) Time-dependent movements ofopen pit slopes. Slope Stability in Surface Mining.Soc. Mining, Metallurgy and Exploration, LittletonCO, Ch. 8, pp. 81–7.

Zavodni, Z. M. and Broadbent, C. D. (1980) Slopefailure kinematics. Canadian Institute of MiningBulletin, 73 (816).

Zhang, X., Powrie, W., Harness, R. and Wang, S.(1999) Estimation of permeability for the rock massaround the ship locks at the Three Gorges Pro-ject, China. Int. J. Rock Mech. Min. Sci., 36,381–97.

Zienkiewicz, O. C., Humpheson, C. and Lewis, R. W.(1975) Associated and non-associated visco-plasticityand plasticity in soil mechanics, Géotechnique,25(4), 671–89.

Index

3DEC 220, 221, 223; wedge234

aerial photograph 49aesthetics: shotcrete 304;

treatment of rock faces 285air blast, noise 271air pressure: atmospheric 271;

blasting 265Alberta 76, 329allowable bond stress 296anchor see rock anchorandesite: porphyry copper open

pit case study 358anisotropic rock (hydraulic

conductivity) 118, 124aperture 58; ISRM procedure

388aquifer 117aquitard 117Arias intensity 142artesian pressure 114, 118, 363asbestos: slope stability 368asperities: first-/second-order 82atmospheric condition (air blast)

271attenuation: blast vibrations 264;

earthquake ground motion144; seismic waves 52

average vehicle risk 281

back analysis 91; altered rockopen pit case study 371; shearstrength values 92

bail test 124basalt 22, 76base friction model 207bedding: coal mine case study

361; definition 53; groundwater flow 118; kinematic

analysis 40; limestone 75;limestone quarry 91; planefailure 22; probabilisticanalysis 148; roughness 81;shale 5; toppling 36; wedgecase study 348; wedge failure154

bench: drainage ditch 310; drilloff-sets 259; face angle 7;height, blasting 249; open pit7; rock fall hazard 310;stability 75; stability analysis336; stiffness ratio 250; toe ofoverburden 308; wall controlblasting 261

bench stability 310; coal minecase study 364; competentrock open pit case study 372,375; porphyry copper open pitcase study 360

bentonite 104, 122, 123Berkeley Pit, Montana 322beta distribution 17Bishop circular failure analysis

188; Hoek–Brown strength100

blasting: bench height 249; blocksize 391; buffer holes 257;burden, mechanism of rockfracture 250; burden,pre-shear 258; burden, wallcontrol 261; choke blasting257; circular failure case study354; costs 247, 258; cushionblasting 354; damagethresholds 265; damage torock faces 285; decouplingratio 260; delay interval 255;detonation sequence(production blast) 253;detonation sequence (wall

control) 261; disturbancefactor, D 95; ditch excavation352; drill off-sets 259; effecton stability 11, 84; exampleproblem 273; explosive load,wall control 260; explosiveproperties 248; flyrock 270;fragmentation 255; geologyeffects on vibrations 267;ground vibrations 264; guideholes 259; hole depth,pre-shear 259; hole diameter251; hole spacing, productionblasting 253; hole spacing,wall control 261; humanresponse 269; joint aperture388; noise, air blast 270;objectives 245; peak particlevelocity 264; powder factor255; pre-blast survey 269;production 248; results 256;rock damage 119, 345; rockfracture mechanism 246; rockproperties 251; scaled distance264; shot-in-place buttress307; slope movement 323;slope stabilization 307, 345;stemming 252, 253; stemming,wall control 260; stiffness ratio253; structural damage control262; sub-drill 252; topplingcase study 355; trim blast308; vibration 258; vibrationfrequency 267; vibrations inconcrete 267; wall control257; wet blast holes 109

block shape 59; rock fall hazardrating 282

block size 59block size/shape: ISRM procedure

391

426 Index

bond length (rock anchor) 296Brazil: iron ore deposits 361breakback angle (bench face) 360bridge abutment 348buckling failure 237; coal mine

case study 364bulk modulus 52burden: pre-shear blasting 258;

production blast 250; wallcontrol blasting 261

buttress 304; reinforced concrete140; waste rock 141

California 110, 141, 309;Northridge earthquake 147;rock falls 277; seismic factor144

Canada 111, 424; coal mine casestudy 361; Hope Slide 49;rock falls 278

case study: circular failure 352;folded coal deposits 361;Hong Kong plane failure 334;pit in altered rock 368; pit incompetent rock 372; planefailure, granite 342; porphyrycopper case study 357;toppling failure 354; wedgefailure 348

catenary 350cement: piezometer 122, 123;

rock anchorage 296centrifuge: toppling failure 207,

214Chile 306China: inventory of rock slopes

4; Three Gorges Project 116China clay pit slope 185Chuquicamata mine 239, 306circular failure 5; altered rock

open pit case study 371;Bishop method 190; case study353; computer programs 195;conditions for failure 177;critical surface 178, 184;example problem 197; failuresurface shape 177; highway186; Hoek–Brown strengthcriterion 193; Janbu method190; limit equilibrium analysis178; movement monitoring322; movement vectors 332;numerical analysis 196; openpit coal mine 203; photograph177; porphyry copper open pit

case study 361; stabilityanalysis charts 180;three-dimensional analysis196; weathered granite 185

cleavage: definition 54coefficient of thermal expansion

25cohesion: back analysis 91;

circular failure 191; circularfailure charts 180; direct sheartest 90; Hoek–Brown strength100; infilling 79, 85; limitequilibrium analysis 11; limitstates design 20;Mohr–Coulomb 79;probabilistic analysis 148;residual strength 86; rock mass79, 353; weak, massive rock79; wedge analysis 159, 172

Colorado 315compression wave 51, 263compressive strength: point load

test 104; rock mass 95, 100concrete: ground vibrations 267;

rock fall barriers 313; shearkey 287

conjugate joint set 5, 154construction access 285conversion factors 408core recovery 71corrosion: rate calculation 293;

rock anchors 292Cosserat theory 222cost: slides 3, 276; slope

stabilization 285creep 242; landslides 324;

numerical analysis 216, 222;open pit slope movement 320

creep test: rock anchors 298

damage (blasting) 262Darcy’s Law 111, 127database: rock slopes 279data logger 326daylight joints 5; bench stability

370; ground water pressure134; numerical analysis 233;plane failure 75, 129; planefailure case study 345; slopemovement 324; stereonet 37;wedge failure 39, 154

decibel 271decision analysis 284decision sight distance 281decomposition weathering 58

delay constant: blasting 255Delphi panel 16detonation sequence (blasting)

254detonation sequence (wall

control) 261diamond drill: dimensions (rods,

casing) 67; operations 68dip: definition 26dip direction: definition 26direct shear test 88, 349, 358;

altered rock open pit case study370; coal mine case study 362;example problem 106; graniteslope case study 342; infilling85; normal, shear stiffness 90;wedge case study 349

discharge area 114discontinuity: daylight 38, 64;

definition 22; dispersion 64;effect on slope stability 25;length (probabilitydistribution) 66; orientation26; orthogonal 31, 54; sets31; spacing (probabilitydistribution) 61, 66

discontinuity orientation: ISRMprocedure 386

discontinuity type 53disintegration weathering 58displacing force 18disturbance factor, D 95, 101ditch: circular failure case study

353; effectiveness 281; rockfall catchment 312, 354

dowel (fully grouted,untensioned) 139

Downie Slide (Canada): drainage306; movement monitoring324

drainage 1; altered rock open pitcase study 370; competentrock open pit case study 375;Hong Kong case study 340;methods 304; open pit 7;tunnel 306

drilling: blast hole 247, 250,251; core orientation 71;diamond drill 68;down-the-hole hammer 347;equipment access 285; groundwater studies 124; hole sealing297; horizontal drains 305;investigation plan 46; ISRMlogging 381; logs 71;movement monitoring 328;

Index 427

mud 69; percussion 289;photograph of drill 46;reconnaissance 48; rotary289; secondary blast holes 256

driving force: limit equilibriumanalysis 12; plane failure 132;plane failure case example338; rock anchor design 138;wedge failure 156

earthquake: Arias intensity 142;attenuation 144; circularfailure case study 353;displacement analysis(Newmark) 145; displacementthresholds 148; effect on slopestability 1, 4, 141; groundmotion intensity 143; groundmotion time history 144;numerical analysis 244; planefailure case study 337, 344;rock falls 278; seismic source143; slope movement 320;topographic amplification 142;toppling failure 206; topplingfailure case study 354

electronic distance measurement(EDM) 326, 342

equipotential line 114evaporation 109example problem: blast damage

control 275; blast design 273;circular failure 197; controlled(final wall) blasting 275; directshear test 106; ground water127; plane failure 150; pointload test 107; stereo net 43;toppling failure 216

explosive properties 248

factored load/resistance 20factor of safety: acceptable values

10; effect of site conditions 11;limit equilibrium 12;numerical analysis 219; openpit 1; probabilistic design 150

fault: altered rock open pit casestudy 369; definition 53;earthquake source 143;geophysics 51; gouge 59;ground motion 144; hydraulicconductivity 111, 120; infillingshear strength 79, 85; mapping383; piezometer installation120; shear strength 85; slope

movement 332; stereo net 31;wedge failure case study 349

finite element analysis: grouteddowel 139

FISH functions 223FLAC 221, 322; circular failure

analysis 233; ground wateranalysis 229

FLAC3D 221flow line 144; refraction 117flow net 113; heterogeneous rock

117flyrock 270folding: coal mine case study 361foliation: definition 53fracture index 70fragmentation: blasting 255freeze–thaw cycles 4, 277freezing (ice in cracks) 109friction angle 77; back analysis

91; base of toppling block 206;circular failure 191; directshear test 88; exampleproblem 107; fault 349;granite 342, 355; granite sheetjoint 338; Hoek–Brownstrength 99; infilling 79, 85;limit equilibrium analysis 11,39; limit states design 20;Mohr–Coulomb 79;peak/residual values 83;probabilistic analysis 148;residual strength 86; rock mass79, 353; rock type 81; roughsurface 56, 62, 79, 82; seismicmotion 145; shale 5; toppling14, 39, 204, 208; toppling FS211; wall strength 57;weathered granite 185; wedgeanalysis 174; wedge kinematictest 155

friction cone 39

gabion 313geofabric rock fall barriers 313Geological Strength Index 95geophone 265geophysics 49global positioning system (GPS)

327Goodman 64, 71, 147, 153; key

block theory 29; topplingfailure 202; tunnel inflow 306

grain size: ISRM procedure 381granite: competent rock open pit

case study 373; Hong Kong

case study 334; plane failurecase study 342; toppling casestudy 354

gravel (energy absorption) 319great circle 27, 33; hand plot

377Griffith crack theory 92ground vibration: attenuation

264; concrete 269; control269; effect of geology 267;electronic equipment 269;frequency 266; humanresponse 269; monitoring264; structural damagethresholds 265

ground water: altered rock openpit case study 370; anisotropicrock 118; case study 337;circular failure case study 353;circular failure charts 180;coal mine case study 363;competent rock open pit casestudy 374; drainage 304;effect of geology 118; effect onstability 109; example problem127; flow net 113; fracturedrock 114; Hong Kong casestudy 335; horizontal drains304; hydraulic conductivity111; hydraulic conductivitymeasurement 123; hydrologiccycle 110; ice 110;investigation plan 48; jointaperture 59; numericalanalysis 229; piezometer 120;plane failure 133; plane failurecase study 344; porosity 113;porphyry copper open pit casestudy 359; pressuredistributions 13; rock fallhazard rating 282; seismicstability 142; toppling casestudy 355; toppling failure213; wedge analysis 157;wedge failure case study 350

Hawaii 76hazard rating/required action

282highway: aesthetics 285; case

study 341; cut in shale 5;geologic mapping 49;kinematic analysis 40; rock fall9; rock fall ditch 312; rockslope stability 1; wedge failureanalysis 170

428 Index

Hoek–Brown strength criterion92; altered rock open pit casestudy 370; circular failure193; circular failure example193; competent open pit rockcase study 373; generalized95; numerical analysis 222

hole spacing: wall controlblasting 261

Hong Kong: drainage 310; planefailure case study 334; slopedrainage 304

Hope Slide, Canada 49horizontal drain: altered rock

open pit case study 370;competent rock open pit casestudy 374; porphyry copperopen pit case study 359

human response (blasting) 269hydraulic conductivity 111;

anisotropic rock 118;anisotropic test 124;conversion factors 112;discontinuities 115; pump test126; rock types 113; variablehead test 124

hydrodynamic time lag 120hydrologic cycle 110

ice 128inclined plane (sliding) 81inclinometer 328; altered rock

open pit case study 369;competent rock open pit casestudy 375

infilling: definition 59; hydraulicconductivity 116; ISRMprocedure 389; shear strength85

inflow rate (tunnel) 306initial response (slope movement)

322in situ stress: competent rock

open pit case study 373;numerical analysis 226

International Society of RockMechanics: mapping 54, 381

inter-ramp slope angle 5; alteredrock open pit case study 372;competent rock open pit casestudy 372

inventory of rock slopes 279, 282investigation program 46Italy 4, 85

Janbu: Hoek–Brown strengthexample 193; lambdacoefficient 224

Janbu circular failure analysis188

Japan 418; cost of slides 4; rockfall barriers 315; rock shed318

joint: definition 53joint compressive strength (JCS)

83joint formation 24joint roughness coefficient (JRC)

83; plane failure 137

Kalsbeek net 31, 377key block theory 29kinematic analysis 22, 37; coal

mine case study 363

landslide: aerial photograph 49;definition of features 8;seismically induced 141

laser imaging 327limestone 75, 81, 91; friction

angle 80; geological map 31;plane failure 5; porosity 113;weathering 58

limit equilibrium 10; circularfailure 178; comparison withnumerical analysis 218, 220;design principle 12; planefailure 132; porphyry copperopen pit case study 361;toppling failure 206; wedgefailure 156

Limit States Design 20line drilling 258line mapping 53line of intersection 34; hand plot

378; wedge failure 156load and resistance factor design

10, 20lognormal distribution 67long-term slope creep 324

magnetic declination: correction54

mapping: bias 65; data sheets396; fault identification 143;hazard 276; investigation plan47; ISRM procedure 381;line/window mapping 53;number of readings 23;

objectives 22; outcrops 52;reconnaissance 48; stereophotographs 53

margin of safety 18material constant mi 95mechanical anchor 295mesh (draped) 317method of slices (circular failure)

190mode of slope instability 35modulus of deformation 25;

Hoek–Brown strength 99; rockmass 99

Mohr–Coulomb 11; circularfailure 191; curved strengthenvelope 100; curvilinearstrength envelope 95; factoredstrength 20; Hoek–Brownstrength 99; numerical analysis221

Mohr diagram 25Monte Carlo analysis 18, 20, 150montmorillonite 85, 104movement monitoring: altered

rock open pit case study 369,372; borehole probe 328;competent rock open pit casestudy 374; crack width 325;displacement plot 332; failuremechanism 332; failureprediction 331; GPS 327;inclinometer 328; laserimaging 327; movement/timeplots 331; numerical analysis240; regressive/progressivemovement 323; surveying326; synthetic aperture radar327; tiltmeter 327;time-domain reflectometry328; velocity plot 332; wireextensometer 325

mud (drilling) 69

negative exponential distribution66

net (wire rope, rock fall) 316Newmark analysis 145nitrogen 126normal distribution 17, 66, 148;

margin of safety 18number of discontinuity sets 59;

ISRM procedure 390numerical analysis 11; 3DEC

220; advantages 218;boundary conditions 228;

Index 429

buckling failure 237; circularfailure 232; continuum/discontinuum models 220,225; creep/strain 222, 242;earthquakes 244; excavationsequence 230; factor of safety219; FLAC 221; ground water229; Hoek–Brown failurecriterion 222; initial conditions226; in situ stress 226, 241;joint material models 221;movement monitoring 240;plane failure 233; rock anchor237; rock mass failure 231;rock mass material model 221;slope curvature 224;three-dimensional analysis224; toppling failure 234;UDEC 220; wedge failure234; zone size 226

observation well 121open pit: altered rock case study

368; competent rock open pitcase study 372; kinematicanalysis 40; probabilisticdesign 15; slope components 5

orientation of discontinuity 54orthogonal joints 22, 31, 53;

coal mine case study 361;granite plane failure case study342; toppling case study 354

packer test 125Palabora Mine 1, 322peak particle velocity (PPV) 264;

case study 354peak shear strength 80performance test (rock anchor)

298persistence 56; effect on stability

25; hydraulic conductivity119; ISRM procedure 390

pH 292piezometer 120, 306; competent

rock open pit case study 375;electrical transducers 122;multiple standpipe 122;Multi-port (MP) 122;pneumatic 122; porphyrycopper open pit case study358; standpipe 121

plane failure 5; altered rock openpit case study 372; anchorreinforcement 297; block

geometry 14; case study,granite slope 342; case study,Hong Kong 335; coal minecase study 365; conditions forfailure 129; critical slide plane136; example problem 44,150; joint persistence 25;kinematic test 37; movementmonitoring 322; numericalanalysis 233; open pit 42;photograph 22, 129; porphyrycopper open pit case study360; probability of failure 20;rough plane 137; stabilityanalysis 130; tension crack134; tensioned anchors 138

plunge: definition 26point load test 57, 105; example

problem 107Poisson’s distribution 32Poisson’s ratio 25, 52pole contours 377pole density 31pole projection 28polymers (drilling mud) 68porosity 113, 119porphyry copper: open pit case

study 357powder factor 255pre-blast survey 269primary/secondary hydraulic

conductivity 112probabilistic design 10;

acceptable probability offailure 17; circular failure196; design principles 15;distribution functions 17;geological distributions 65;ground water modelling 117;margin of safety 18; MonteCarlo analysis 19; plane failureexample 148; seismic groundmotions 143; slopestabilization 284; structuralgeology 64

probability of failure 20, 150proof test (rock anchor) 298pseudo-static seismic analysis

144; Hong Kong case study337; wedge case study 351

pump test 126

quarry 91

railway: circular failure case study352; toppling case study 354

Raleigh wave 51, 263ramp: open pit 5random number 19recharge area 114reconnaissance mapping 46, 48reference sphere 27regressive/progressive slope

movement 323residual shear strength 80residual soil 58resin anchor 295resisting force 12, 19resloping/unloading 308; Hong

Kong case study 339rippability 52, 105riprap 247rock anchor: acceptance criteria

300; allowable bond stress296; anchored slope(photograph) 1; bond length296, 347; bond type 295;cement anchor 296;comprehensive wedge analysis399; corrosion 292; corrosionprotection 347; drilling 289;grouting drill hole 347; holelayout 297; limit equilibriumanalysis 13; materials 290;mechanical anchor 295;numerical analysis 237;optimum dip angle 14;optimum orientation (wedge)174, 407; optimum plunge(plane) 138, 152; plane failureanalysis 138; plane failure casestudy 338, 345; reaction wall301; resin anchor 295; tensiontesting 297; toppling 206,210, 213; toppling case study355; total anchor length 296;waterproofing drill hole 297;wedge analysis 174; wedgecase study 351; workedexample 150

rock fall attentuator (mesh) 317rock fall hazard rating system

(RHRS) 280rock fall history 282rock fall modelling 310rock falls: causes 277; hazard

276; volume 278rock mass characterization: ISRM

procedure 381rock mass rating 11, 93

430 Index

rock mass strength: back analysis91; circular failure case study353; Hoek–Brown criterion 95

rock properties: blasting 251rock shed 318rock strength: investigation plan

48rock strength classification 57rock type 54; ISRM procedure

381roughness 56; ISRM procedure

386RQD 69; porphyry copper open

pit case study 359rubber tires (energy absorption)

319

sandstone: benched slope 310;circular failure analysis 193;coal mine case study 361; flownet 117, 118; hydraulicconductivity 59; jointmeasurements 66; plane failure5; shear strength 87; wedgecase study 348

satellite imaging 327scaled distance (blasting) 264scale effect: JRC 84scaling (stabilization) 308schistosity: definition 54Schmidt hammer test 57, 83;

bedding 24Seed 144, 147seepage 59; ISRM procedure

392; mapping 32seismic coefficient 144; Hong

Kong case study 334; planefailure case study 344; wedgecase study 350

seismic hazard analysis 143seismic investigation (geophysics)

49selection of stabilization measures

283sensitivity analysis 14, 337serviceability limit state 20service life 20shale: benched slope 310; coal

mine case study 361;degradation 102; flow net117; friction angle 80;highway cut 5; horizontallybedded 5; hydraulicconductivity 59; topplingfailure 201; Vaiont Slide 85;wedge case study 349

shape factor: conductivity test124

shear key 287shear modulus 52shear strength: back analysis

187; displaced/undisplaceddiscontinuities 87; effect ofwater 88; granite joints 335;Hoek–Brown criterion 99

shear strength reductiontechnique 219

shear wave 51, 263sheet jointing 335shotcrete: reinforcement 301;

strength tests 302; wet/dry mix302

shot-in-place buttress 306silica fume (shotcrete) 302sinusoidal wave 264slake durability 102slickenside 388slope curvature (numerical

analysis) 224slope design: coal mine case study

365; design factors 2; designmethods 8; factor of safety 10;probability of failure 15

slope displacement: earthquake146; regressive/progressive 323

slope height/angle relationship 4slug test 124socioeconomics (consequences of

slides) 3South Africa 224, 322spacing of blast holes 253spacing of discontinuity 54;

ISRM procedure 389;true/apparent 60

stabilization: barriers 313;buttress 304; drainage 304;fences and nets 316; resloping308; rock anchor 287; rock fallattenuator 317; rock fall ditch312; rock shed 318; scaling308; shear key 287; shotcrete301; shot-in-place buttress 306

stabilization program (planning)279

standard deviation 17; margin ofsafety 18

stemming: production blast 252;wall control blasting 260

stereographic projection 27stereonet: angle between planes

35; contour plot 31; data

selection 32; discontinuity sets32; equal angle (Wulff net) 29;equal area (Lambert, Schmidt)net 29; example problem 43;friction cone 40; great circle27, 32; hand plotting 377;highway 40; identification ofmodes of stability 35; Kalsbeeknet 31, 377; kinematic analysis37; line of intersection 35;polar/equatorial projections28; pole contours 377; poledensity 31; pole plot 29; poleprojection 28; reference sphere27; wedge failure 34, 159, 171

stick-slip behavior 324stiffness: of joints, numerical

analysis 216; rock boltnumerical model 239

stiffness ratio (bench blasting)250

strain: onset of failure 10; rockanchor 295; slope movement322

strain gauge 122strain wave 246strand anchor 346strength (UCS): infilling 85;

ISRM procedure 382; jointwalls 57

strike: definition 26structural domain: coal mine case

study 363; porphyry copperopen pit case study 358

sub-drill (blasting) 252surface wave 263surveying (movement monitoring)

326synthetic aperture radar 327

tensile strength (rock mass) 95tensile stress 25tension crack 151, 399; circular

failure 184; plane failure 134;slope movement 322

tensioned cable bridge 350terrestrial photography 53Terzaghi correction 60texture: ISRM procedure 381Thistle Slide 3tiltmeter 327time-domain reflectometry 328topography: numerical analysis

228; seismic amplification 142toppling failure 5; altered rock

open pit case study 368; base

Index 431

plane angle 206; blockalignment test 205; blockgeometry 14, 204, 208; blockmodel 200; bolting 210;bridge abutment 216; casestudy (granite) 354; coal minecase study 365; creep 324;example analysis 211; exampleproblem 217; external forces213; factor of safety 211;flexural 201, 214; forcecalculations 209; geologicalconditions 25; highway cut40; inter-layer slip 204; jointspacing 55; kinematic test 39;limit equilibrium 206;movement monitoring 322,329; movement vectors 332;numerical analysis 236; openpit 42, 241; open pit coal mine203; secondary mechanism202; stereonet 36; tensioncrack 201; Vaiont Slide 200

trend: definition 26trim blast 308; plane failure case

study 347true/apparent spacing

(discontinuities) 60Tubex drill 289tunnel: drainage 305; mapping

65, 396; seepage (ISRMprocedure) 393; shear key287; slide by-pass 319

turbulent flow 113Turkey 77

UDEC 220, 221, 322;Chuquicamata Mine 239;competent rock open pit casestudy 375; plane failure 233;strain softening 223; toppling215, 236; typical analysis 231

ultimate limit state 20Utah 3

Vaiont Slide 4, 85, 200variable head test 124

warning fence 317waste rock disposal 285water sampling 122water table 393; circular failure

charts 181; Darcy’s Law 111;draw down 126; flow net 118;hydraulic conductivity test124; hydrologic cycle 110;mapping 59; plane failure133; seasonal 48; wedgeanalysis 399; worked example128

weathering 76; classification 58;effect on seismic stability 141

weathering classification: ISRMprocedure 382

wedge failure: altered rock openpit case study 372; bolting

174; case study 348;comprehensive analysis 171,350, 398; computer programs174; example analysis 175;example problem 44; externalforces 174; factor of safety156; geometry 154; highwaydesign example 170; kinematictest 37, 38, 154; line ofintersection 64; numericalanalysis 234; open pit 42;photograph 22, 153; porphyrycopper open pit case study360; removal 29; stabilityanalysis 157; stability charts160; stereonet 34

wedging action 157well sounder 122width of infilling 59, 389window mapping 53wire extensometer: altered rock

open pit case study 369; coalmine 331; movementmonitoring 325

X-ray diffraction 104

yield acceleration (seismic design)146

zero discharge drilling 46