Miguel Angel Vivas Mefle

8/10/2019 Miguel Angel Vivas Mefle

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MSc Geotechnical Engineering

STUDY OF THE ACCURACY OF LIMITSTATE: GEO

AND COMPLIANCE WITH EUROCODE 7

IN SLOPE STABILITY ANALYSIS

Miguel ngel Vivas MefleID: 1378375

Thesis submitted is in partial fulfilment of the degree of Master of Science

School of Civil Engineering

UNIVERSITY OF BIRMINGHAMAugust 2014


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Study of the accuracy of LimitState: GEO and compliance with Eurocode 7 in slope

stability analysis

Vivas Mefle, Miguel ngel

ABSTRACT

LimitState: GEO has supposed the introduction of the first software for slope stability analysis

based on discontinuity layout optimization. However,due to its novelty, there are still many

doubts about its accuracy and how wellEurocode 7has been implemented.

In this study, throughout more than 5,000 calculations, a total of 872 cases were analyzed by

using the power of thestability numbers, obtained by iterative processes using the cohesionas

the key parameter, and involving a broad range of cases of rotational and translational slides

under drained and undrained conditions.

Bishops Simplified Method implemented in Oasys Slope software, the Classic Formula of

Limit Equilibrium for translational slides, Taylors Chart and Analytical Methods of Limit

Analysis were used to calculate the stability numbers for each case and then by introducing

these values in the models created in LimitState:GEO, the accuracy of the program was

estimated.

The implementation of EC7 in LimitState: GEO was also studied, firstly by calculating the

stability numbers for EC7-DA1 (Eurocode 7- Design Approach 1), combinations 1 and 2

using LimitState: GEO and Bishops Simplified Method adapted to EC7, and then, by

comparing the most restrictive EC7-DA1combination obtained in each case from both

methods. This study was also complemented with the calculation of the Equivalent Global

Factor of Safety, when designing in EC7 with LimitState:GEO, by bounding the stability

numbers previously calculated from EC7-DA1 with other stability numbers calculated to be

equivalent to a definite range of global FoS.

The results showed a good agreement between LimitState: GEO and the other methods of

analysis for rotational and drained slides, with an accuracy ranging from exact to 5%, whereas

for rotational and undrained slides, although the agreement was also good, the accuracy was

slightly worse varying from exact to 10%.


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As far as translational slides is concerned, the accuracy was considerably poor in cases with

steep slopes with inclinations of 1/1.5 or higher, with some cases showing differences in the

FoS between 30% and 50% for drained conditions and between 15% and 30% for undrained

analysis respectively.

The agreement for the most restrictive combination obtained for each case, between

LimitState: GEO and the other methods, was good in general, however, LimitState: GEO

considered, in some cases with low angles of friction, C2 as more restrictive than C1 in

discrepancy with other limit equilibrium methods.

Finally, the Equivalent Global FoS, when calculating in EC7 with LimitState: GEO for

drained rotational slides was in all the cases the same as the obtained by the other methodsand similar, but with slight variations, for undrained rotational slides studies. However, for

translational slides cases, the values of the global FoS varied from equal to limit equilibrium

methods, for cases with no steep slopes, to FoS in the order of 40% to 60% more conservative

for steep slopes.

Key words: LimitState: GEO, Eurocode 7, Stability Number, Equivalent Global FoS and

Accuracy


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LIST OF CONTENTS

ABSTRACTi

LISTOF CONTENTS......iii

LIST OF FIGURES.....ix

LIST OF TABLESx

1. INTRODUCTION 1

1.1.Background and Context..1

1.2.

Aims and objectives...2

1.3.Outl ine Of Dissertation.....3

2. LITERATURE REVIEW 5

2.1.Introduction...5

2.2.Background...5

2.3.

Slope Stabil ity Analysis: Fundamentals and Principles..8

2.3.1. Limit Analysis....................................................8

2.3.2. Limit Equilibrium Methods.....12

2.3.3. Numerical Methods..16

2.3.3.1. Discontinuity Layout Optimization...17

2.4.The Concept of the Stabil i ty Number.19

2.5.Taylor s Chart.20

2.6.

Eurocode 7: A new approach.222.7.Geotechnical Software: L imi tState: GEO and Oasys Slope..29

3. METHODOLOGY 32

3.1.Rotational Slides under Drained Conditi ons:....32

3.1.1. Study of the accuracy of LimitState: GEO in rotational slides for drained

conditions...32


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3.1.1.1. LimitState: GEO vs Analytical methods of Limit Analysis..32

3.1.1.2. LimitState: GEO vs Oasys Slope...37

3.1.2. Study of the most restrictive combination in EC7-DA1 for drained conditions

and rotational slides when using LimitState: GEO and Oasys Slope....423.1.2.1. LimitState: GEO study..42

3.1.2.2. Oasys Slope study..45

3.1.3. Equivalent Global FoS for rotational slides when calculating in EC7-DA1.47

3.2.Rotational Slides under Undrained Conditi ons....49

3.2.1. Study of the accuracy of LimitState: GEO software in rotational slides for

undrained conditions ..49

3.2.1.1.

Accuracy of LimitState: GEO vs Taylors Charts493.2.1.2. Accuracy of LimitState: GEO vs Bishops Simplified Method...51

3.2.1.3. Accuracy of LimitState: GEO vs Analytical methods of Limit

Analysis.....53

3.2.2. Study of the most restrictive combination in EC7-DA1 for rotational slides

under undrained conditions when using LimitState: GEO and Oasys Slope...54

3.2.3. Equivalent Global l FoS for rotational slides under undrained conditions when

calculating in EC7-DA1..56

3.3.Translati onal Slides under Dr ained Conditions57

3.3.1. Study of the accuracy of LimitState: GEO software in translational slides for

drained conditions57

3.3.2. Study of the most restrictive combination in EC7-DA1 for translational slides

under drained conditions when using LimitState: GEO and classic methods..62

3.3.3. Equivalent Global FoS for translational slides when calculating

in EC7-DA166

3.4.Translati onal Sli des under Undrained Conditions....67

3.4.1. Study of the accuracy of LimitState: GEO software in translational slides for

undrained conditions...67

3.4.2. Study of the most restrictive combination in EC7-DA1 for translational slides

under undrained conditions when using LimitState: GEO and Classic

Methods..69


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3.4.2.1. LimitState: GEO study..70

3.4.2.2. Classic methods study....71

3.4.3. Equivalent Global FoS for translational slides when calculating in EC7-

DA1..71

4. DISCUSSION 73

4.1.Rotational Slides under Dr ained Conditions.73

4.1.1. Study of the accuracy of LimitState: GEO and Oasys Slope software in

rotational slides for drained conditions...73

4.1.1.1.

Limit Stat: Geo vs Analytical Limit Analysis ..........734.1.1.2. LimitState: GEO vs Oasys Slope...74

4.1.2.

Study of the most restrictive combination in EC7-DA1 when using

LimitState: GEO and Oasys Slope in rotational slides under drained

conditions...74

4.1.2.1. LimitState: GEO study..75

4.1.2.2. Oasys Slope study......75

4.1.3. Equivalent Global FoS for rotational slides and drained conditions when

calculating in EC7-DA1..76

4.1.3.1. LimitState: GEO study for depth factor 1 study....76

4.1.3.2. LimitState: GEO study for depth factor 2 study

4.2.Rotational Slides under Undrained Conditions.77

4.2.1. Study of the accuracy of LimitState: GEO software in rotational slides for

undrained conditions...77

4.2.1.1. Accuracy of LimitState: GEO vs Taylors chart ..77

4.2.1.2. Accuracy of LimitState: GEO vs Bishops simplified method.....77

4.2.1.3. LimitState: GEO Analytical method of Limit Analysis78

4.2.2.

Study of the most critical combination in EC7-DA1 for rotational slides

under undrained conditions when using LimitState: GEO and Oasys Slope78

4.2.2.1. LimitState:Geo.78

4.2.2.2. Oasys Slope......79


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4.2.3. Equivalent Global Fos In Limitstate:GEO For Rotational Slides Under

Undrained Conditions When Calculating In EC7 DA1.79

4.3.Translational Sli des Under Drained Conditions..79

4.3.1.

Study Of The Accuracy Of Limitstate:GEO Software In Translational SlidesFor Drained Conditions: When Compared With The Limit Equilibrium Classic

Formula..79

4.3.2. Study Of The Most Restrictive Combination In EC7 DA1 When Using

Limitstate:GEO And The Classic Method In Translational Slides Under Drained

Conditions..80

4.3.2.1. LimitState:GEO.80

4.3.2.2.

Classic method...804.3.3. Estimation Of The Equivalent Global Fos For Translational Slides And

Drained Conditions When Calculating In EC7 DA1.....80

4.4.Translational Sli des Under Undrained Conditions..81

4.4.1. Study Of The Accuracy Of Limitstate:GEO Software In Translational Slides

For Undrained Conditions..81

4.4.2. Study Of The Most Restrictive Combination In EC7 DA1 For Translational

Slides Under Undrained Conditions When Using Limitstate:GEO And Classic

Method Of Analysis......81

4.4.3. Equivalent Global Fos In Limitstate:GEO For Translational Slides Under

Undrained Conditions When Calculating In EC7 DA1.81

5. CONCLUSION 82

6. REFERENCES 84


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APPENDICES

A. Calculation of the Stability Number using Upper Bound limit analysis theory for

failure planes passing through the toe and failure planes passing below the toe.

B. Finite Element Limit Analysis (FELA) Applied To Slope Stability Analysis

C. Mathematical approach of the Stability Number.

D. Stability Numbers for EC7-DA1, Combinations 1 and 2 when designing in

LimitState: GEO and Oasys Slope for rotational slides under drained conditions.

E. Stability Numbers for the calculation of the Equivalent Global FoS for rotational

slides under drained conditions.

F. Charts for the Equivalent Global FoS in rotational slides under drained conditions,

when designing in EC7-DA1 with LimitState: GEO and Oasys Slope.

G. Stability Numbers for EC7-DA1, Combinations 1 and 2, for rotational slides under

undrained conditions when designing in LimitState: GEO and Oasys Slope.

H. Stability Numbers for the calculation of the Equivalent Global FoS for rotational

slides under undrained conditions.

I. Stability Numbers for EC7-DA1, Combinations 1 and 2, when designing in the Classic

Method and LimitState: GEO for translational slides under drained conditions.

J. Mathematical demonstration of Classics Formulas for the Stability Numbers when

designing in EC7-DA1, Combinations 1 and 2, for translational slides under drained

conditions.

K. Stability Numbers for the calculation of the Equivalent Global FoS for translational

slides under drained conditions.

L. Charts for the Equivalent Global FoS in translational slides under drained

conditions.

M. Verification of the Accuracy of LimitState: GEO through 22 Benchmark Problems.


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LIST OF FIGURES

2.1. Stress strain relationship for ideal and real soils. (Chen 1975)....9

2.2. Coulomb yield criterion represented by two straight lines. Limit analysis and soilplasticity. (Chen 1975).....9

2.3. Typical slice and forces for the methods of slices. (U.S. Army corps of engineers

2003)..13

2.4. Method of Slices....14

2.5. Stages in DLO procedure...18

2.6. Compatibility at a node Gilbert (2007)..18

2.7. Elements of the Friction Circle method.21

2.8. Taylors Charts for undrained conditions. (Barnes 1995)..21

2.9. Statistical approach for the verification of the Limit States...22

2.10. Factors on actions EC7. (Bond and Harris 2008)23

2.11. Factors on material properties. EC7. (Bond and Harris 2008).....24

2.12. (Bond and Harris 2008)....26

2.13. Working description of DLO...31

3.1. General case for Analytical methods of Limit Analysis calculations....33

3.2. Example of models created for LimitState:GEO...33

3.3. Example of the model created for LimitState:GEO, depth factor 1...42

3.4. Example of the model created for LimitState:GEO, depth factor 2...45

3.5. Example of the model created for translational slides in LimitState:GEO,...63

3.6. Chart representing the Equivalent Global FoS when calculating in EC7 with LimitState:

GEO for translational slides under undrained conditions..73


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LIST OF TABLES

2.1. Stability numbers for homogeneous simple slopes by several methods from Taylor

(1948)...7

2.2. Stability numbers for homogeneous simple slopes by limit equilibrium and upper bound

limit analysis. (Chen 1975)....10

2.3. Number of unknowns and equations for limit equilibrium methods from the U.S. Army

Corps of Engineers (2003).13

2.4. Limitations of limit equilibrium methods from the U.S. Army Corps of Engineers

(2003).14

2.5. Unknowns and equations of the Bishops simplified method from the U.S. Army Corps of

Engineers (2003)15

2.6. Set of partial factors for EC7 design approach 1 from Bond and Harris (2008)...25

3.1. Stability numbers for homogeneous simple slopes by analytical method of Limit Analysis

for rotational slides with =0 and drained conditions....35


for rotational slides with =20 and drained conditions..35


for rotational slides with =30 and drained conditions..35

3.4. Accuracy of LimitState: GEO in relation to analytical method of limit analysis for

homogeneous simple slopes with rotational slides, =0 and drained conditions.36

3.5.Accuracy of LimitState: GEO in relation to analytical method of limit analysis for

homogeneous simple slopes with rotational slides, =20 and drained conditions...37


homogeneous simple slopes with rotational slides, =30 and drained conditions...37

3.7. Stability numbers obtained from Oasys Slope for homogeneous simple slopes with

rotational slides, ru=0 and drained conditions...39


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rotational slides, ru=0.3 and drained conditions39


rotational slides, ru=0.5 and drained conditions40

3.10.Acuracy of LimitState: GEO in relation to Oasys Slope for homogeneous simple slopes

with rotational slides, ru=0 and drained conditions...40

3.11. Accuracy of LimitState: GEO in relation to Oasys Slope for homogeneous simple slopes

with rotational slides, ru=0.3 and drained conditions....41


with rotational slides, ru=0.5 and drained conditions41

3.13. Stability numbers and critical DA1 Combination in Oasys Slope and LimitState: GEO

for homogeneous simple slopes with rotational slides, ru=0, depth factor 1 and drained

conditions...44


for homogeneous simple slopes with rotational slides, ru=0.3, depth factor 1 and drained

conditions...44



conditions...45


for homogeneous simple slopes with rotational slides, ru=0, depth factor 2 and drained

conditions...46



conditions...46


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conditions...47

3.19. Stability numbers from Taylors chart for homogeneous simple slopes with rotational

slides under undrained conditions..50

3.20. Accuracy of LimitState: GEO in relation to Taylors chart for homogeneous simple

slopes with rotational slides under undrained conditions..51


rotational slides under undrained conditions.52


with rotational slides under undrained conditions53

3.23. Stability numbers obtained from analytical methods of Limit Analysis for homogeneous

simple slopes with rotational slides under undrained conditions...53


homogeneous simple slopes with rotational slides and undrained conditions...54


for homogeneous simple slopes with rotational slides under undrained conditions..56

3.26. Stability numbers obtained from the classic formula for translational slides, ru=0 and

drained conditions..59

3.27. Stability numbers obtained from the classic formula for translational slides, ru=0.3 and


3.28. Stability numbers obtained from the classic formula for translational slides, ru=0.5 and


3.29. Accuracy of LimitState: GEO in relation to the classic formula for translational slides,

ru=0 and drained conditions...61


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ru=0 under drained conditions.61


ru=0.5 and drained conditions ..62

3.32. Stability numbers and critical DA1 Combination in the classic method (CM) and

LimitState: GEO (LS) for translational slides, ru=0 and drained conditions...64


LimitState: GEO (LS) for translational slides, ru=0.3 and drained conditions....65


LimitState: GEO (LS) for translational slides, ru=0.5 and drained conditions....65

3.35. Stability numbers obtained from the classic formula for translational slides under

undrained conditions..69

3.36. Accuracy of LimitState: GEO in relation to the classic formula for translational slides

under undrained conditions....69

3.37. Stability numbers and critical DA1 Combination in LimitState: GEO for translationalslides under undrained conditions..70

3.38. Stability numbers and critical DA1 Combination in the classic formula for translational

slides under undrained conditions..71

3.39. Stability numbers for the Equivalent Global FoS for translational slides under undrained

conditions...72


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1. INTRODUCTION

1.1.Background and Context

First attempts to develop a solution for slope stability analysis were realized in 1773 by

Coulomb and since then, a lot of research has been done based principally in two different

approaches: Limit Equilibrium and Limit Analysis . Limit equilibrium technique relies

exclusively on static equilibrium equations where it is only considered the equilibrium of the

soil as a rigid body, whereas limit analysis considers the soil as a plastic material and the

failure of the soil is based on yielding criteria.

Limit Equilibrium (LE) methods of analysis have been used for many years in the industrydue to its relatively simplicity in comparison with limit analysis based methods, in addition,

they have proved to be accurate enough for solving the majority of slope stability problems.

On the contrary, limit analysis was not normally used for the calculation of real cases on a

daily basis but it was mostly considered for research because of their complexity and the fact

that they were only able to solve simple problems.

The introduction of computers supposed a progress for LE methods with the application of

modern theories for searching the critical slip surfaces, but mostly, it boosted the developmentof limit analysis methods since more complex mathematical calculations could be quickly

done, in particular finite element limit analysis.

In recent years, the difficulty associated to the design of the slopes was incremented by the

introduction of EC7 in 2010 which supposed a new approach, as the global factor of safety

gave way to the use of partial factors. As a consequence, the existing software had to be

adapted to EC7 and it caused that new concerns about geotechnical software are now focused

on how well implemented is EC7 on them.

A new theory was recently developed in limit analysis termed Discontinuity Layout

Optimization (DLO). This theory was recently developed by Smith and Gilbert in 2006 and

shortly after in 2007 the first commercial software was introduced in the market:

LimitState:GEO.


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Before LimitState: GEO was released, the program was subjected by the developers

(LimitState Ltd)to a variety of tests that compared the performance of the program in an array

of cases, to the results obtained by the analytical method of limit analysis proposed by Chen,

(1975), and Bishops Simplified method. (www.limitstate.com/geo/validation). The test wasconducted using the default coarse nodal density and the results were presented in terms of

change in input parameters required for the LimitState: GEO solution to match the benchmark

solution. In general, even when using the coarse nodal density, accuracy in most of the tests

varied from exact to ~5%. The tests were carried out for 15 different cases of rotational slides.

A recent comparative study between the accuracy of LimitState: GEO in relation to LE

techniques was carried out by Leshchinsky (2013) for rotational slides, comparing for 5 cases

of complex slopes the results of the FoS obtained and also the critical failure surfaces

determined by the two methods. From the study, Leshchinsky deduced a good agreement in

the results obtained between LE and DLO which were mostly equal but more critical for DLO

in some cases.

Both studies were conducted for a small number of cases, so they were appropriate in order to

derive the general behavior of the program but translational slides were not included on them

and neither was addressed the implementation of EC7. As there is no previous experience in

other DLO based software, there is still little knowledge on the behavior of LimitState: GEO,

in addition to the previous studies where only a limited number of cases were analyzed,

further studies including more cases by varying the slope angles, water level and depth

factors, and also incorporating translational slides, would contribute to enhance the mentioned

knowledge. Moreover, as designs must be based on EC7, since it became compulsory in 2010,

and due to the fact that no published research was previously done for LimitState: GEO on

this topic, it would be also equally important to check how well EC7 has been implemented in

the software in comparison with others well EC7 implemented techniques.

1.2.Aims and Objectives

The principal aims of the research described in this dissertation are:

(i) Checking the accuracy of LimitState: GEO

(ii) Checking how well implemented is EC7 in LimitState: GEO


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Objectives:

(i) Comparative study of the accuracy of LimitState: GEO in relation to other

accurate techniques such as Limit Equilibrium, Analytical Methods of LimitAnalysis, Taylors Chart and Limit Equilibrium Classic Formulas.

(ii) Comparative study between the most critical EC7-DA1 combinations deduced for

each case by LimitState: GEO and the results obtained from other limit

equilibrium techniques adapted to EC7 approach.

(iii) Comparative study between the Equivalent Global FoS obtained from LimitState:

GEO and Limit Equilibrium techniques when calculating on EC7.

1.3.Outl ine Of Di ssertation

This dissertation contains seven core chapters:

Chapter 1provides an introduction as well as a brief outline of subsequent chapters.

Chapter 2gives an overview of the development over time of the different approaches in

slope stability analysis including limit equilibrium methods, analytical limit analysis

methods and numerical methods covering Finite Element Limit Analysis and Discontinuity

Layout Optimization. Then, the mathematical and physical concepts of these theories are

described giving a more detailed explanation to Bishops simplified method and

Discontinuity Layout Optimization. The concept of the stability number is also addressed

and its importance in the development of charts, Taylors chart was also introduced

together with the friction circle method in which it is based, then, an explanation is made

about the new EC7 approach and how it affects to slope stability calculations including

how classic methods are EC7 adapted. Finally LimitState: GEO and Oasys Slope are

introduced with an explanation of their main features.

Chapter 3 offers an explanation of the methodology followed, in order to achieve the

objectives of this dissertation.

Chapter 3.1. addresses the study of LimitState: GEO when calculating in rotational

slides under drained conditions. First, the accuracy of LimitState: GEO in relation to

the analytical method of limit analysis and Bishops simplified method is investigated

by making use of the stability numbers. Then, a study was done on the EC7


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implementation by comparing the most restrictive EC7-DA1 combination for each

case in LimitState: GEO with the results obtained in Oasys Slope using Bishops

simplified method. Finally, the equivalent global FoS for each case when calculating

in EC7 is obtained in LimitState: GEO and also compared with the results obtained inOasys Slope.

Chapter 3.2 focuses on the study of LimitState: GEO when calculating in rotational

slides under undrained conditions. First, the accuracy of LimitState: GEO in relation

to Taylors chart, Bishops simplified method and Analytical methods of Limit

Analysis is obtained by making use of the stability numbers, then, the same study as

before for drained conditions, was done on the EC7 implementation in LimitState:

GEO for undrained conditions. Chapter 3.3 is centered on the study of LimitState: GEO when calculating in

translational slides under drained conditions. First, the accuracy of LimitState: GEO in

relation to the classic drained limit equilibrium formula for translational slides GEO is

analyzed by making use of the stability numbers, then, the same study as for rotational

slides but using the limit equilibrium classic formula instead of Bishops simplified

method, was done on the EC7 implementation in LimitState: GEO for drained

translational slide.

Chapter 3.4 comprises the study of LimitState: GEO when calculating in translational

slides under undrained conditions. First, the accuracy of LimitState: GEO in relation

to the classic undrained limit equilibrium formula for translational slides is estimated

by making use of the stability numbers and then the same study as for the previous

case was done on the EC7 implementation in LimitState: GEO for undrained

translational slide

Chapter 4provides a discussion of the results obtained from the calculations carried out for

the cases described in chapters 3.1, 3.2, 3.3 and 3.4.

Chapter 5summarizes the key conclusion of this dissertation.


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2. LITERATURE REVIEW

2.1.Introduction

Slope Stability could be described as one of the main concerns in Geotechnical engineering

due to the risk it supposes not only for the stability of structures and buildings but to the

safety of the population. As a consequence, a lot of effort has been dedicated along time in

researching of more accurate methods of analysis.

Nowadays, a large number of methods for calculations are offered to the Geotechnical

engineers with also an extensive array of different software, based on distinct physical and

mathematical approaches. In addition, the introduction of EC7 approach in 2010, bringing inthe use of partial factors for more detailed designs, has supposed a new challenge for the

designers.

Considering that the physical and mathematical principles that govern slope stability analysis

are paramount, in order to understand how the slope stability programs work, an overview of

them is presented, including: Limit Equilibrium, Limit Analysis and Numerical Methods,

covering for this last one, continuum, discontinuum and hybrid methods. A mathematical

explanation of the slope stability number is also given together with an introduction of itsrelevance in the slope stability field, since it introduces the use of charts which improve the

ease of the calculations.

Finally, EC7 design methodology is explained with a description of its new approach and the

implementation of the conventional methods to EC7, along with an overview of the two

programs that will be used in the development of this dissertation: LimitState: GEO and

Oasys Slope.

2.2.Background

Limit equilibrium methods were appointed as the first techniques used to analyze slope

stability, with the assumption of the soil considered as a rigid material and an approach based

on static equilibrium equations.


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Coulomb (1773) was the first to develop a limit equilibrium method to calculate the

maximum height for excavations based on statics equilibrium equations.

Francais (1820) and later on, Culmann (1866) were the first to develop a limit equilibrium

method of analysis based on static equilibrium equations in order to calculate the maximum

excavation depth in steep cut slopes, under the assumption of a plane surface of rupture.

However, yearslater Taylor (1948) considered that they were only approximately correct for

steep slopes.

In 1922, the Swedish National Commission established that in most of the slides, the lines of

failure were similar to the circumference of a circle; this study was based on a substantial

amount of failure cases and just a few years later, this study gave birth to Fellenius Method in1927, consisting of splitting the soil into vertical sections with interslice forces parallel to the

base, then, failure is assumed to occur by rotation of a block of soil on a cylindrical slip

surface and by examining global moment equilibrium, an expression for the FoS is obtained.

A different approach was introduced by Rendulic (1935), appointing logarithmic spiral as the

rupture surface instead of circles which nowadays constitutes the base of modern methods of

calculus. The main advantage of this method was that all the intergranular forces are directed

towards the centre of the spiral and therefore the analysis is statically determinate without anassumption relative to the pressure distribution. (Taylor 1948).

Taylor (1937) came up with the friction circle method, assuming that the kinematical function

represents an arc, but it was not until Taylor (1948) when it had major application with the

development of slope stability charts which speeded the hand calculations and still are in use

on early stages of the design process. Over the years, Bishop and Morgenstern (1960),

Terzaghi and Peck (1967), Spencer (1967), Hunter and Schuster (1968), Janbu (1968), Bray

(1977) and Duncan (1987) among others, introduced new charts associated to the new

upcoming methods.

Taylor (1948) carried out a comparison of the results obtained by using plane, circular and log

spiral failure surfaces, study that concluded stating that the logspiral slip surface is the most

critical for homogeneous slopes. Table 2.1. illustrates the results obtained by Taylor (1948)


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Table 2.1. Stability numbers for

homogeneous simple slopes without

seepage-by several methods. Taylor

(1948).

Bishop (1954) presented an improved version of Fellenius method, where the vertical forces

equilibrium conditions yielded to more accurate results. Depending on the assumptions madeon the equilibrium equations and interslice shear forces, a lot of different variants to this

method appeared during the following years, the difference among them being based on the

validity of the characteristics applied to each case.

Drucker and Praguer, (1952) appeared with a totally different approach for slope stability

analysis, termed Limit Analysis. In this method the soil is assumed to have an ideally plastic

behavior with an associated flow rule. (Chen 1970).

Limit analysis has been proved to be an accurate and very useful alternative for assessing the

stability of slopes. By utilizing the lower and upper bound theorems of plasticity, rigorous

lower and upper bound solutions are analytically obtained. However, this analytical way of

obtaining solutions is only possible for simple problems, as for more complicated cases it is

necessary a previous identification of the geometrical form of the solution, resulting on an

unachievable analytical approach. To overcome this issue, matrix approach and series

representations can be utilized (Dewhurst & Collins 1973), together with the method of


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characteristics (Sokolovski 1965, Martin 2003, Smith 2005). Though, these methods are still

not suitable for regular engineering praxis. (Gilbert and C. Smith 2010).

A great effort has been carried out to obtain numerical limit analysis procedures capable to

run more complex scenarios and easy to use. Amongst different numerical procedures, Finite

Element Limit Analysis (FELA) has been the most investigated since it was introduced by

Lysmer (1970) and further improved by Sloan (1988) and Makrodimopoulos & Martin

(2006). This method is highly dependent on the layout of the mesh especially in areas with

singularities.

However, it is possible to formulate and solve a simpler problem based on a discontinuum

approach instead of continuum mechanic problem of FELA. Alwis, W.A.M. (2000),introduced a model based on triangular elements with restricted deformations and separated

by potential discontinuities; nevertheless, it led to a very restricted search space and poor

solutions due to the fact that the discontinuities can only be positioned at the bounds of the

elements located in a fixed mesh. In order to overcome this, Smith& Gilbert (2007), proposed

a new computational limit analysis procedure denominated Discontinuity Layout

Optimization (DLO).

2.3.Slope Stabil ity Analysis: Fundamentals and Pr inciples

2.3.1. Limit Analysis

Hill (1950) and Drucker&Prager (1952) are known as the creators of the theoretical

framework for limit analysis.

One of the main requirements for limit analysis is considering the soil as an ideally plastic

material where the strain softening in stress-strain relationship is ignored (Drucker et al 1952,

Chen 1975), as shown on Figure 2.1.


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Figure2.1. Stress strain relationship for

ideal and real soils. (Chen 1975).

A second condition applies, being this the assumption that the soil satisfies the Coulomb yield

criterion where plastic flow occurs and when the tensional state of the soil touches the Mohr

Coulomb envelope. Within the framework of these assumptions, the limit analysis approach

for slope stability analysis is rigorous and as a result, the technique is competitive with those

of limit equilibrium. (Chen 1975).

Figure2.2. Coulomb yield criterion,

represented by two straight lines, limit

analysis and soil plasticity. (Chen 1975).

Limit Analysis theory is built upon two limit theorems applied to soils with elastic-perfectly

plastic behavior:

1. Lower bound theorem, which states that if a statically admissible stress distribution

can be found, uncontained plastic flow will not occur at a lower load. This technique

considers only equilibrium and yield, giving no consideration to soil kinematics.

2. Upper bound theorem on the contrary, states that the loads determined by equating the

external rate of work to the internal rate of dissipation satisfying velocity boundary


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conditions, and strain and velocity compatibility conditions, are not less than the

actual collapse load.

With the proper choice of stress and velocity fields, the above two theorems thus enable the

collapse parameters of the slope to be bracketed as closely as it seems necessary for the

problem under consideration.

In relation to the two bound theorems, and in order to bind the true solution, it is necessary

a mechanism of failure in the form of a velocity field or flow pattern so that an upper bound

solution can be obtained, and a stress field meeting the conditions imposed by the lower

bound theorem to find a lower bound solution. (Chen 1968)

Finding a statically admissible stress field was proved to be very difficult and that is the

reason why limit analysis was mostly developed on the upper bound method (Chen 1975) and

subsequently, the majority of the research in slope stability analysis was carried out in this

field: (Chen and Giger 1971).

Based on the upper bound theorem and considering logspiral surfaces as failure mechanisms,

the critical height of the slopes for different values of , and were obtained and thesolutions were given as stability factors (1/stability number), published in tables. (Chen1975).

By comparing these results against limit equilibrium methods using circular and logspiral slip

surfaces, it was demonstrated a very good agreement in the results, as it is shown on Table2.2.

Table 2.2. Comparison of stability

factor , by methods ofLimit Equilibrium and Upper

Bound Limit Analysis. (Chen

1975).


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From the comparison shown above, it could be concluded that the solutions obtained by using

Limit Equilibrium methods were very similar to those obtained by using Upper Bound limit

analysis. As a consequence, Lower Bound limit analysis solutions were considered too

conservatives and very few studies in this field were developed for slopes with the exceptionof some authors as Lysmer (1970), Chen (1975), Basudhar (1976) and Singh (1993).

Refer to Appendix A for an insight into the calculation of the stability number using upper

bound limit analysis theory for failure planes passing through the toe and failure planes

passing below the toe.

Based on these results, it can be concluded that upper bound solutions may be used as a good

reference for other methods of slope stability analysis and they are currently used as aframework for benchmark problems when checking the accuracy of new slope stability

software as it is the case of LimitState: GEO. ( www.limitstate.com/geo/validation). However,

lower bound solutions can also be useful as far as a finding a conservative solution is

concerned.

In case that both, upper and lower bound solutions could be obtained; the true collapse

mechanism can be bracketed. However, it is important to point out that limit analysis is based

in a perfect plastic behavior of the soil and therefore, although it is an important tool inobtaining an estimation of the true mechanism of collapse, it cannot be considered as

providing exact solutions. (Yu.H.S. et al. 1998).

The solutions obtained from upper bound limit analysis are still considered accurate and used

to check the validation of new geotechnical software, as it was done for the release of

LimitState: GEO. (www.limitstate.com/geo/validation).

Analytical limit analysis approach was a valuable tool for obtaining more accurate slopestability analysis calculations based on a strong physical and mathematical theory. However,

due to the complexity of its calculations, it was only suitable for simple slopes. Afterwards,

the appearance of computers permitted its use for more complex cases.


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2.3.2.Limit Equilibrium Methods.

Limit equilibrium analysis represent a totally different approach to limit analysis. It is atechnique based on calculation of forces (stresses or moments) and a comparison of those

causing stability (resisting forces) with those causing instability (disturbing forces). By

assuming a relationship between the shear strength and normal stress on the slip surface, that

can be linear (Mohr-Coulomb) or nonlinear, a FoS can be provided as a ratio of the available

shear resistance with the needed for equilibrium. (Chen 1975).

The main characteristic of limit equilibrium approach is that it relies exclusively on static

equilibrium equations and it does not take into consideration the plastic flow rule of the soil in

opposition to limit analysis. Compatibility, between limit analysis and limit equilibrium

approaches, is impossible since the collapse mechanism resulting by a limit equilibrium

approach would be kinematically inadmissible in case that the soil was considered to meet the

assumptions of limit analysis approach (perfectly plastic behavior of the material and plastic

flow rule associated).

H.S.yu (1998). Back in 1974, Collins (1974) and Chen (1975) , got to the conclusion that by

no means these two methods could be regarded as equivalent. This conclusion was reinforced

by Michalowski (1994) who demonstrated that upper bound solutions obtained by limit

analaysis approach, also comply with limit equilibrium equations but not all the limit

equilibrium solutions can be considered as upper bound limit analysis solutions. And again,

Yu. H.S. et al (1998) refered that in limit equilibrium the stress field does not satisfy the static

admissibility due to the assumptions made to solve the static indeterminacy, and as a result,

only global equilibrium conditions are accomplished instead of equilibrium conditions at

every point in the soil, in opposition to limit analysis.

Among all the limit equilibrium methods available nowadays, the Method of Slices is the

most popular, where discretization of the soil into slices is performed previously to applying

the equilibrium conditions. The factor of safety obtained varies depending on the method

chosen as these are based on different assumptions and equilibrium requirements.


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Limit equilibrium methods of slices, face the conditions for static equilibrium considering the

soil mass divided into an undetermined number of slices and imposing horizontal and vertical

equilibrium of forces in each of them and global equilibrium momentum. The forces acting in

each slice are represented in Figure 2.3.

Figure2.3. Typical slice with forces.

Method of Slices. (U.S. Army Corps of

Engineers 2003).

All the forces are unknown with the exception of the weight of each slice. The number of

unknowns and the number of equations for n slices are represented in the next table:

Table2.3. Unknowns and equations for limit equilibrium methods. (U.S. Army Corps of

Engineers 2003).

Since the number of unknowns (5n-2) outnumber the equilibrium equations (3n) it is

necessary to make some assumptions in order to achieve a statically determinate solution.


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Different assumptions were proposed by different authors to balance the number of unknowns

and the number of equations leading to the appearance of a large number of different

methods. Some of them do not satisfy all static equilibrium conditions as it occurs with

Fellenius, Bishops simplified method and the US Army Corps of Engineers, whereas otherssatisfy all the equilibrium conditions, (rigorous methods), as it is the case of Morgenstern and

Prices and Spencersmethods.

Several publications can be found with comparisons of the different limit equilibrium

methods, as for instance: Whitman and Bailey (1967), Duncan and Wright (1980) and

Fredlund and Krahn (1977).

The main limitations of the limit equilibrium methods were summarized in the next table

Table2.4. Limitations of limit equilibrium methods.(US Army corps of engineers 2003).

Among all the limit equilibrium methods, Bishops simplified method is considered one of the

most popular not only because its simplicity of use, but because it has been proven to be an

accurate method. It is still used on a daily basis for geotechnical engineers and also as a

benchmark for new stability methods of analysis.

Figure2.4. Method of Slices


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Bishops simplified method (1954), assumes that the tangential interslice forces are equal and

opposite, i.e. = , but the normal interslice forces are not equal (Figure2.4).These assumptions have been proved to yield little loss in accuracy. The FoS is assumed to be

the same for each slice and it is applied equally to the cohesion and the angle of internal

friction,and

, where F is the overall factor of safety applied to all of the slices. Mohr

Coulomb and equilibrium equation are combined together with the definition of the factor of

safety to obtain the forces on the base of the slice. Finally, moments are summed about the

center of the circular slip surface yielding the next expression for the FoS

[]

(2.1)

where W is the weight of each slice, c is the cohesion of the soil, u the pore water pressure at

the base of the slide and the rest of parameters are given in Figure2.4.

As F is located in both sides of the equation, this can only be solved by successive iterations

converging to the final result. (Barnes 1995)

Table2.5. Unknowns and equations of the Bishops simplified method (U.S. Army corps of

engineers)

It has been demonstrated by Whitman and Bailey (1967) and Fredlund and Krahn (1977) that

the FoS obtained by using Bishops simplified method is normally within 5% the FoS

calculated by other rigorous tecniques. Moreover, the procedure is simpler in comparison with

other rigorous solutions, due to the ease in computation, rapidness in execution and also

because of the facility on checking solutions by hand as they are not time consuming.

Limit equilibrium methods are currently the most used within the industry because of its

simplicity and its proved reliability along years


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2.3.3. Numerical Methods:

Analytical limit analysis methods for slope stability analysis were only applicable to simple

slope models until the introduction of computational limit analysis procedures that allowed

solving complex problems.

Numerical methods in limit analysis are based on discretization of the problem domain which

through a mathematical optimization technique allows the determination of the best

approximation for the discretization imposed and the limit strength of the soil for the slope

stability problem.

There are three different numerical techniques depending on the consideration of the domain:

Continuum, Discontinuum and Hybrid.

In relation to the continuum approach, special mention is given to the Finite Difference

Method (FDM), where nodes are distributed on a rectangular grid and where the quality of the

results relies on a well selection of the rectangular domains of the problem and the existence

of simple boundary conditions; the Linear Matching Method, (LMM), developed by Ponter

and Carter (1997), uses linear elastic methods to iteratively obtain upper bound solutions and

last, the Finite Element Limit Analysis (FELA) which utilizes constant strain elements and

has become an important tool in geotechnical engineering practice, being developed over time

by a large number of authors: Nagtegaal et al. (1974), Da Silva Vicente and Antao (2007), and

Makrodimopoulos and Martin (2007) among others.

As for discontinuum methods, there are two main types: The Rigid finite element method

(RFEM), that assumes deformations only along discontinuities at the boundaries of predefined

soil elements and where elements are not free to deform; and discontinuity layout

optimization (DLO) differing from (RFEM) in the fact that discontinuities are not restrictedonly to the boundaries of predefined solid elements but can connect any node to any other

node and optimization is used to determine the critical layout of the discontinuities. Therefore,

despite DLO makes use of the algorithms associated with RFEM, unlike RFEM, a large

number of potential mechanisms can be identified allowing the procedure to be largely mesh

independent. (Hawksbee.S.J. 2012) .


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Finally, Hybrid FELA first introduced by Lismer (1970) and later developed by Bottero et al

(1980), Sloan (1988), Martin (2006) and Muoz et al (2009), overcomes the limitations of

continuous methods when facing stress and strain singularities by incorporating

discontinuities between elements. However, this method has limitations since in order toachieve an effective mesh design; the form of the exact solution must to be known a priori.

Next, due to the relevance that DLO is increasingly having with the development of

geotechnical software LimitState: GEO, a more in detail explanation will be given to it in the

following section. Although software based on FELA is not considered in this dissertation

because of its importance a brief of the studies developed so far, is given inAppendix B.

2.3.3.1. Discontinuity Layout Optimization: DLO

DLO is a numerical technique applicable to discontinua where deformations are assumed to

occur only along discontinuities in the soil. This method was developed to tackle simpler

problems based on a discontinuum approach instead of continuum mechanic problem of

FELA.

DLO was introduced by Smith and Gilbert (2007) for plane strain. Unlike FEM, DLO is

entirely presented in terms of velocity discontinuities and involves the use of rigorous

mathematical optimization techniques. DLO supposes an advance over FEM as optimization

allows the determination of the critical layout of the discontinuities with the least upper bound

solution from among a large set of potential discontinuities. Although DLO uses the same

algorithms as hybrid FELA, a large number of potential mechanism can be identified and so

this procedure becomes mesh independent. Moreover, in DLO discontinuities are not

restricted to the boundaries of predefined solid elements but can connect any node to anyother node. (Gilbert &Smith 2010).

The DLO procedure for plane strain problems was presented by Gilbert et al (2010) in phases

as it is shown in the next figure:


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Figure2.5. Stages in DLO procedure

The first step of the procedure consists in the discretization of the problem approached by

distributing the nodes all over the body under consideration. Next, by establishing links from

each node towards every other node in the body, possible discontinuity lines are created.

Linear programming, with the aid of kinematic formulation, is then used to establish the

discontinuities that can occur in the critical failure mechanism which are constructed from

rigid blocks separated by discontinuities. Under the premise that a sufficient number of nodes

have been employed, a large number of potential mechanisms will be taken into

consideration. (Hawsbee 2012).

Compatibility of the displacements of the nodes necessary to obtain possible failuremechanisms is resolved by Gilbert (2007) as follows:

Figure2.6. Compatibility at a node

Gilbert (2007)


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In every discontinuity i, a shear jump and normal jump in displacement are permittedand compatibility of the mechanism is enforced by applying constraints to each node. Then,

the following summations must be accomplished at each node for compatibility

(2.2)

(2.3)In DLO, intersections or crossovers between potential discontinuities arise naturally at

locations other than the original nodes and millions of potential discontinuities are created

from which to obtain an accurate solution. Since in DLO, the total number of potential

discontinuities grows disproportionally with the number of nodes n, an adaptativeprocedure based on the philosophy of Gilbert and Tyas (2003), termed Adaptive Nodal

Connection, is necessary in order to obtain an efficient solution. With this procedure, the

number of potential discontinuities is reduced to those where the dual inequality constraint is

most violated.

2.4.The Concept Of The Stabil ity Number

The stability number is a specific feature for each slope stability problem which gives the only

possible relationship between cohesion, unit weight and height so that the slope is stable.

The stability number concept is key for the development of this dissertation as it comes across

as a powerful tool to check the performance of new geotechnical software, as if for a specific

case the stability number is calculated using a recognized accurate procedure, this result can

be used to check the precision of other methods.

In a slope, the component of the self-weightcauses instability and the cohesion contributesto stability. Therefore, the maximum height of a slope is directly proportional to the unitcohesion c and inversely proportional to the unit weight. In addition the maximum height is also related to the friction angle and the slope angle. This can be expressed as where the term is dimensionless and the equation dimensionally balanced.


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Taylor (1937) expressed as a reciprocal of a dimensionless number called StabilityNumber (N)popularly called as Taylors stability Number. (S.V.Dinesh 2008).

(2.4)

(2.5)

Hence,

(2.6)

Taylor (1948) stated that the main value of this form of expression is that the stability numbermay be considered to be a composite variable which reduces the number of parameters in a

simple stability equation to three and thus allows the use of simple charts for representations

of stability relationships.

A mathematical demonstration of how to obtain the stability number is exposed in

Appendix C.

2.5.

Tayl or S Chart

Taylors stability chart is still in use for preliminary designs and is considered an accurate

method for total stress analysis (Duncan 1996). For this reason, it was considered in this

dissertation a suitable method in order to check the accuracy of LimitState: GEO and so some

basic principles on which this theory is based are exposed below.

Taylor in (1937) came up with the friction circle method, assuming that the kinematical

function represents an arc, but it was not until 1948 when it had major application with thedevelopment of slope stability charts which speeded the hand calculations. This solution was

strictly valid only for simple homogeneous finite slope with the types of cross sections shown

in Figure2.7.


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2.6.Ec7. A New Approach

Prior to the Eurocodes the concept of limit state design was common in disciplines like

Structural design but not in Geotechnical design. The traditional approach in Geotechnical

design was restricted to analytical methods incorporating estimated values of the load and

material parameters in order to obtain an ultimate value for the stabilizing and destabilizing

forces and moments. These values are then diminished by an overall factor of safety,

considered sufficient enough to mobilize safety strength values and cause acceptable

deformations. (Barnes 1995).

Partial factors were introduced into Danish geotechnical practice by Brinch Hansen (1953)

and now form the basis for limit state design in EC7. This draw attention to the separate

consideration of load conditions and material properties, providing a more robust approach

compared to the global factor of safety method. A statistical approach to their application is

shown on Figure2.9., illustrating the relationship between design loads and design resistances

for combinations 1 and 2 of design approach 1. (Barnes 1995)

Figure2.9. Statistical approach for the verification of the limit states.


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Partial factors are chosen to ensure that the risk of failure is minimal and they are applied

differently depending on the type of loads and material properties. When designing in EC7 the

application of partial factor must ensure the compliance of the following limit states involved:

Ultimate limit states(ULS) are defined as states associated with collapse

Serviceability limit state(SLS) are those that result in unacceptable levels of deformation

Before the appearance of EC7 the values for the global factor of safety only appeared as

recommendations in standards or were given to the election of the engineer. The Eurocode

brought about a new concept where the safety problem is analyzed based on the influence that

each parameter has in the calculus. These parameters are classified into three categories:

1. Actions: self weight and loads

2. Material properties: unit weight, angle of shearing resistance and cohesion

3. Resistance: overturning and resisting moments.

EC7 also discern between favorable or stabilizing and unfavorable or destabilizing actions, by

applying different values of the partial factors () depending on the type of action.

Unfavourable/destabilizing actions will be normally increased by the partial factor ( ) toobtain the design action whereas favorable/stabilizing actions will be decreased or leftunchanged ( ). (Bond and Harris 2008)

Figure2.10. Factors on

actions EC7. (Bond and

Harris 2008)


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Actions are also classified according to their variation in time as: permanent (G), variable (Q)

and accidental (Q), and partial factors will also depend on this. Representative actions are the

real expected actions and for the calculations are converted into design values by applyingdue partial factor () taking account of uncertainties in the magnitude of the action.

In the case of material properties, characteristic material properties () are divided by theircorresponding partial factors () to obtain the design values ()

Figure2.11. Factors on material properties.

EC7. (Bond and Harris 2008)

The approach in limit state design is to verify that the effects of the design actions (

), do

not exceed the design resistnce () so that the next inequality is accomplished

EC7 does not give any specific inequality to be satisfied for ULS neither a calculation model,

however, with regard to stability analyses of slopes, the UK National Annex,BS EN 1997-

1:2004, states that normally limit equilibrium methods will be applied, although alternatively,

it allows the use of Limit Analysis as the case of Finite Element Limit Analysis. It also

indicates that slope stability analysis should verify the overall moment and vertical

equilibrium of the sliding mass and in case that horizontal equilibrium is not checked, the

interslice forces, i.e. when using the method of slices, should be assumed to be horizontal.

This means that some slope stability analysis methods are not acceptable. (Bond.A.J et al.

2013).

Next is shown the EC7 compatibility of some of the most popular limit equilibrium method of

analysis. (Bond.A.J et al. 2013).


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1. Spencers method is acceptable because both moment and force equilibrium equations

are satisfied

2.

Bishops Simplified method is acceptable because moment equilibrium is satisfiedand, although force equilibrium is not satisfied, the interslice forces are horizontal

3. Janbus method is not acceptable as moment equilibrium is not satisfied

4. Fellenius method is not acceptable because, while moment equilibrium is satisfied,

forces equilibrium is not and the interslice forces are not horizontal

EC7 permits the adoption of three design approaches depending on different considerations of

the actions and resistances, being Design Approach 1 (DA1) the one adopted by the UK, as it

is stated in the UK national Annex. DA1 provides two combinations (1 and 2) of partial

factors on actions (A), material properties (M) and resistances (R). Design Approach 1 applies

partial factors to actions in Combination 1 (C1) and to unfavorable variable actions and

material properties in Combination 2 (C2). Therefore, for each problem two calculations will

have to be done by applying separately the set of partial factors corresponding to C1 and C2,

then, the final design will be given by the most critical solution of the two combinations.

The next table shows the set of partial factors to be applied to actions, material properties and

resistances for C1 and C2 in DA1, and also de design approach 2 and 3. (Barnes 1995)

Table2.6. Set of partial factors for

design approach 1. EC7. (Bond

and Harris 2008)


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Plane translational slides:

This dissertation checks the performance of LimitState: GEO for translational slides by

calculating the global factor of safety with no partial factors applied and with the

implementation of EC7. Next, a brief summary will be shown about the principles underlyingthe classic calculation of translational slides and the procedure followed for the

implementation of EC7.

For plane translational slides, considering a slope angle with the ground water table parallelto the ground surface and the same unit weight of the soil above and below the ground water

level, the global factor of safety is given by , Figure2.12, where is the shearing

resistance at the base of the planar slide given by , where c is theeffective cohesion, the effective normal stress and the angle of internal friction, and isthe tangential stress down the slope given by , where is the unitweight of the soil, H is the height of the slope and is the slope angle.

Figure2.12. (Bond and

Harris 2008)

For the general case, the global factor of safety is the ratio: shear strength of the soil/shear

strength mobilized as it is indicated in the Formula 2.7. corresponding to Figure 2.12.

with ru= (2.7)

This expression can be rearranged to give the stability number N necessary to provide an

Equivalent Global FoS


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Fsincos- (2.8)And in terms of partial factors:

)sincos- (2.9)

These expressions will be commonly used for the dissertation.

For design, EC7 requires that the inequality is satisfied. For the infinite slope case,the design effect of the actions is the tangential force down the slope, given by Formula 2.10:

=cos(2.10)

whereis the partial factor applied to the unit weight of the soil and the unit weight of thesoil.

On the other hand, the design resistance is given by:

(2.11)

where si the partial factor applied to cohesion, the partial factor applied to the angle ofinternal friction, the partial facto applied to the unit weight and the partial resistancefactor.

For slope stability problems, the limit state is dictated in most cases by the uncertainty in the

ground resistances rather than external forces, so for Design Approach 1, Combination 2 will

be the most representative in most cases.

If the slope behaves in an undrained manner, with , by substituting this on formula(2.7), the overall FoS is given by:

(2.12)where is the undrained cohesion. For limit state design, this would be expressed as


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(2.13)where is the partial factor applied to the undrained cohesion.

Finite rotational slides:

This dissertation tests the performance of LimitState: GEO for rotational slides by calculating

the global factor of safety with no partial factors applied and with the implementation of EC7.

Next, a brief summary will be shown about the principles underlying the classic calculation of

rotational slides and the procedure followed for the implementation of EC7.

Bishops simplified method has proved to yield accurate results, Krahn (1977), and is widely

applied for EC7 calculations. Next, the implementation of EC7 in the conventional method

will be shown.

Implementation of EC7 for drained analysis:

The global factor of safety F, for Bishops simplified method of slices, is equivalent to thepartial factor on the soil strength parameters with appropriate partial factors on the actions as

shown in the following equations, (Andrew J. Bond et al., 2013):

(7.6)

[]

(7.7)

In DA1-C1, is applied to permanent actions, including the soil weight force via thesoil weight density, and is applied to variable actions when analyzing the overallfactor of safety F, using the method of slices. Then it is checked that F, which is equal to

, is greater than or equal to 1.


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In the case of DA1-C2, is applied to permanent actions, including the soil weightforce via the soil weight density, and is applied to variable actions when analyzingthe overall factor of safety F, using the method of slices. Then it is checked that F, which is

equal to, is greater than or equal to 1.25. (Andrew J. Bond et al., 2013).

Implementation of EC7 for undrained analysis:

For undrained analysis the restoring moment is defined as:

() (7.8)where r is the radius of the slip circle, is the length of the slip surface beneath i and isthe undrained shear strength along the base of that slice.

The factor of safety F is defined as:

(7.9)

where is the angle between the base of the slice and the horizontal.

This equation can be rewritten in terms of EC7 design as follows:

{} (7.10)

where is the design self-weight of the slice i, any imposed surcharge acting on thatslice and the other terms are as defined above. (Bond and Harris, 2008).

2.7.Geotechnical Software: L imi tState: GEO And Oasys Slope

The first software for slope stability analysis was introduce in the 80s but it was not until the

mid 90s that they became popular, presenting an enhanced interface and easier usability.

Slope/W (1995) and Oasys Slope (1999) are two of the most popular limit equilibrium based

software and through updates are still successfully implemented in the market. As computers

became more and more powerful, the introduction of Plaxis (1998), supposed a new step in

slope stability software making use of finite element limit analysis. Recently, LimitState:

GEO (2008), was created by Smith and Gilbert (2007), based in a new limit analysis


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algorithm termed Discontinuity Layout Optimization, whereas three dimensional analysis

started to appear in the market such as SVSLOPE in 2010.

In the course of this dissertation, LimitState: GEO will be analyzed throughout a series ofexamples and in some cases a comparison against Oasys Slope will be carried out. Some of

the main features of both programs will be exposed next:

Oasys Slope

Oasys Slope represents one of the most popular conventional software based on two

dimensional limit equilibrium analyses (www.oasys-software.com).

Oasys Slope allows the use of three different limit equilibrium methods and its variants, for

drained and undrained conditions:

Swedish circle (Fellenius) method, Bishop's and Janbus method.

The three variants consist in considering horizontal interslice forces, parallel inclined

interslice forces and variably inclined interslice forces

The procedure followed to find the most critical slip surface consist in the creation of a mesh

with a density and location of points chosen by the designer and containing the centres of

circular slip surfaces. Then, for each centre the program calculate the FoS of the slope for

several values of the radio. The same procedure is followed for the remaining points of the

mesh and finally, the most critical case corresponding to the lowest FoS is the solution.

Several restrictions can be imposed to the program for searching the critical slip surface, as

points through which the critical surface must intercept or tangential surfaces. It also gives the

possibility to calculate a predefined slip surface. (Manual of Use, Oasys Software).

As far as the implementation of EC7 is concerned, Oasys Slope has been adapted to EC7

allowing the use of different set of partial factors.

LimitState: GEO
http://www.oasys-software.com/http://www.oasys-software.com/


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As per LimitState: GEO, this program also complies with EC7 and its different approaches,

aspect that makes it suitable to use under the new design codes in development. LimitState:

GEO is based on Discontinuity Layout Optimization (DLO) numerical procedure to tackle

geotechnical problems, entitling the designer to find the correct critical slope failuremechanism for every geotechnical situation presented. In the next illustration a quick

overview of how DLO works is presented. (www.limitstategeo.com)

Figure2.13 Working description of DLO.

LimitState:GEO, has been subjected to tests against Bishops simplified method of analysis

and other well known limit analysis solutions, in order to verify its precision,

(www.limitstate.com/geo/validation), resulting in an a accuracy from exact to 5%. Recently,

Leshchinsky (2013), compared the performance of LE, based on Spencers Method with

dynamic programming optimization, against LimitState: GEO for several examples of

complex slopes resulting in a good agreement between them, with LimitState: GEO providing

slightly lower FoS than Spencers method.

As far as the implementation of EC7 is concerned, LimitState: GEO has been built in EC7,

embracing the use of partial factors, compatible load descriptions (permanent, variable,

accidental), favourable and unfavourable load classifications, and also allows the user to

specify any partial factor set and multiple scenarios that can be solved together.

(www.limitstate.com).


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3. METHODOLOGY

In this section, a description of the procedures followed to check the accuracy of

LimitState:GEO, and how well implemented EC7 is, are presented next. In order to exposethe processes clearer, all the cases studied have been split up into four sections, each of them

containing cases with alike methodology: rotational and translational slides, under drained

and undrained conditions.

There is a common pattern in the methodology followed for all the sections, firstly a study of

the accuracy of LimitState: GEO was conducted comparing the FoS obtained for each case

against other accurate methods of analysis; secondly, the implementation of EC7 inLimitState: GEO was checked contrasting the most restrictive EC7-DA1 combination

resulting for each case with the results yielded by other reliable EC7 implemented methods of

analysis and finally a study of the Equivalent Global FoS deduced for the problems calculated

in EC7 with LimitState: GEO was carried out and also compared with other LE methods.

For each investigation carried out, the cases study, the assumptions made and the

methodology followed were described.

3.1.Rotational Slides Under Dr ained Conditions:

3.1.1. Study Of The Accuracy Of LimitState: GEO In Rotational Slides For Drained

Conditions

3.1.1.1. LimitState: GEO vs Analytical Methods of Limit Analysis:

In order to check the accuracy of LimitState:GEO, analytical methods of Limit Analysis were

used, as it is traditionally considered an accurate method of analysis. The models studied were

based on the general case for rotational slides with the slip surface passing through the toe and

the rigid layer at the bottom of the slope, Figure3.1, (Chen 1975), and in order to conduct the

analysis, all of them had to be modelled for LimitState: GEO as it is shown in Figure3.2.


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Figure3.1: General case

for Analytical methods of

Limit Analysis

calculations.

Figure3.2:.Example of models created for LimitState:GEO.

Cases study: the study was conducted for slopes created from all possible combinations

of the next parameters (Figure3.1):

Angles of the slopes : the angle of the slopes varied from to in steps of Angles of internal friction (degrees): 5,10,15,20,25,30,35 and 40.

Angle of the upper surface : , and

Assumptions:

o LimitState: GEO assumptions:

Depth factor of 1 with rigid layer at the bottom of the slope.


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The models created in LimitState: GEO consisted of slopes of homogeneous soil,

with unit weight of =1 KN/m3, and slope height of 1m, resulting incohesion=stability number.

All the models were calculated with a set of unity partial factors. The nodal density was medium in all cases.

No phreatic level.

o Analytical method of Limit Analysis assumptions:

Depth factor 1 with rigid layer at the bottom of the slope.

Homogeneous s.oil Critical failure surface passing through the toe.

Logarithmic spiral failure plane.

No phreatic level.

Methodology

1. Obtaining the stability numbers from the analytical method of Limit Analysis: due to

the complexity that calculating stability numbers implies, the solutions to the cases

study for analytical methods of Limit Analysis using algorithms for the upper bound

solution, were taken from Chen (1975) who calculated and compiled them in tables,

with the aid of a CDC 6400 digital computer and using the optimization technique

reported by Powell (1964), giving the results as stability numbers. Tables 3.1, 3.2 and

3.3 collect the mentioned results.


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Table3.1. Values of the stability numbers obtained from analytical methods of Limit Analysis

for rotational slides with =0and drained conditions.

Table3.2. Values of the stability numbers obtained from analytical methods of Limit Analysis

Table3.3. Values of the stability numbers obtained from Analytical methods of Limit Analysis

for rotational slides with =30and drained conditions.

DRAINED ANALYSISSTABILITY NUMBERS FROM NALYTICAL METHOD OF LIMIT ANALYSIS WHEN =0

SLOPE ANGLE 90 75 60 45 30 155 0.24 0.19 0.16 0.14 0.11 0.07

10 0.22 0.17 0.14 0.11 0.07 0.02

15 0.20 0.15 0.12 0.08 0.05 -

20 0.18 0.13 0.10 0.06 0.02 -

25 0.17 0.12 0.08 0.04 0.01 -

30 0.15 0.10 0.06 0.03 - -

35 0.13 0.09 0.05 0.02 - -

40 0.12 0.07 0.03 0.01 - -

DRAINED ANALYSIS

STABILITY NUMBERS FROM NALYTICAL METHOD OF LIMIT ANALYSIS WHEN =20SLOPE ANGLE

90 75 60 45 3020 0.19 0.14 0.10 0.07 0.0325 0.17 0.12 0.08 0.05 0.01

30 0.16 0.10 0.06 0.03 -

35 0.14 0.09 0.05 0.02 -

40 0.12 0.07 0.04 0.01 -

DRAINED ANALYSIS

STABILITY NUMBERS FROM NALYTICAL METHOD OF LIMIT ANALYSIS WHEN =30SLOPE ANGLE

90 75 60 4530 0.16 0.11 0.07 0.03

35 0.14 0.09 0.05 0.02

40 0.13 0.08 0.04 0.01


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2.Inputting of data in LimitState: GEO : creation of the same models in which Chens

tables are based and inputting the data with values of the cohesion equal to the stability

numbers taken from Chen.

3.Running the simulation and obtaining of results: obtaining a factor strength (FoS) of 1

means that the accuracy of the program in relation to analytical method of Limit

Analysis is 100%. Thus, the difference between factor strength 1 and the actual factor

strength obtained indicates the relative accuracy of the program. The results are

presented in Tables 3.4, 3.5 and 3.6. A positive value indicates that for the same case,

LimitState: GEO gives a lower FoS than the analytical method of limit analysis and

hence, a more conservative approach and vice versa.

Table3.4.Accuracy of LimitState: GEO in relation to Analytical methods of Limit Analysis forrotational slides with =0 and drained conditions.

DRAINED ANALYSIS

LIMIT STATE FoS IN COMPARISON WITH ANALYTICAL METHOD OF LIMIT ANALYSIS FOR =0SLOPE ANGLE

90 75 60 45 30 155 -0.2% -1.2% -1.9% -4.4% -9.8% -20.4%

10 -0.8% -1.7% -2.3% -2.9% -5% -3.4%

15 -0.5% -1.5% -2.5% -2.3% -2.3% -

20 -0.6% -2.2% -2.6% -2% -0.6% -

25 -0.8% -2% -2.1% -2% -0.4% -

30 -0.8% -1.4% -2.3% -1.7% - -

35 -1.3% -1.9% -2.3% -1.7% - -

40 -0.8% -2.4% -3.6% -1.2% - -


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Table3.5.Accuracyof LimitState: GEO in relation to Analytical methods of Limit Analysis for

rotational slides with =20 and drained conditions.

Table3.6.Accuracyof LimitState: GEO in relation to Analytical methods of Limit Analysis for

rotational slides with =20 and drained conditions.

3.1.1.2. LimitState: GEO vs Oasys Slope:

In order to back up the results obtained above, LimitState: GEO was also checked against

Bishops simplified method, implemented in Oasys Slope software.

Bishops simplified method was chosen for a second study as it is considered an accurate

method, giving normally factors of safety differing less than 5% from other rigorous limit

equilibrium methods.

Cases study: the study was conducted for slopes created from all possible combinations of

the next parameters:

Angles of the slopes : 1/1, 1/1.5, 1/2, 1/2.5, 1/3

DRAINED ANALYSIS

LIMIT STATE FoS IN COMPARISON WITH LIMIT ANALYSIS FOR=20SLOPE ANGLE

90 75 60 45 3020 -2.1% -2.3% -2.0% -1.5% -0.2%

25 -2.1% -3.0% -2.7% -1.7% -0.2%

30 -2.2% -3.1% -2.5% -1.7% -

35 -2.5% -2.6% -3.2% -0.9% -

40 -1.8% -2.2% -2.2% -0.9% -

DRAINED ANALYSIS

LIMIT STATE FoS IN COMPARISON WITH LIMIT ANALYSIS FOR=30SLOPE ANGLE

90 75 60 4530 -2.6% -2.7% -2.1% -1.0%

35 -2.2% -3.0% -2.5% -1.9%40 -2.3% -2.9% -3.0% -2.0%


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Angles of internal friction (degrees): 5,10,15,20,25,30,35 and 40.

Pore water pressure ratio: ru=0, ru=0.3 and ru=0.5.

Assumptions:

o LimitState: GEO assumptions:

The same as in 3.1.1.1.

o Oasys Slope:

Bishops simplified method with horizontal interslice forces option.

Depth factor 1 with rigid layer at the bottom of the slope

The models created in Oasys Slope were slopes of homogeneous soil, slope height

of 1m and unit weight of =20 KN/m3. (as the program did not allow=1 KN/m3and therefore N=cohesion/20)

All the models were calculated with the set of unity partial factors.

Methodology

1.

Obtaining the stability numbers from Oasys Slope: After creating the models describedin the case study, the stability numbers (N) were obtained by iterating the values of the

cohesion inputted in the program to reach a FoS equal to 1 and then applying the

formula , where =20KN/m3and H=1. The solutions are presented in tables x,yand z.


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Table3.7. Values of the stability numbers obtained from Oasys Slope for rotational slides,

ru=0 and drained conditions.


ru=0.3 and drained conditions.

DRAINED ANALYSIS. RU=0

STABILTY NUMBERS OASYS SLOPE

SLOPE 1/1 SLOPE 1/1.5 SLOPE 1/2 SLOPE1/2.5 SLOPE1/35 0.133 0.108 0.090 0.075 0.063

10 0.105 0.080 0.060 0.045 0.033

15 0.083 0.065 0.035 0.021 0.009

20 0.062 0.034 0.015 0.003

25 0.043 0.016 0.002

30 0.027 0.004

35 0.01540 0.005

DRAINED ANALYSIS. RU=0.3



10 0.125 0.100 0.077 0.064 0.054

15 0.113 0.083 0.058 0.044 0.034

20 0.098 0.063 0.043 0.027 0.014

25 0.083 0.050 0.028 0.011 0.001

30 0.065 0.037 0.014 0.001

35 0.055 0.024 0.003

40 0.045 0.014


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ru=0.5 and drained conditions.

2.Inputting of data in LimitState: GEO : creation of the same as models as in Oasys Slopeand inputting the data with values of the cohesion equal to the stability numbersobtained above.

3.Running the simulation in LimitState: GEO and obtaining of results: the same dynamic

as in section 3.1.1.1. is applied when interpreting the results obtained . The results are

presented in the tables 3.10, 3.11 and 3.12.

Table3.10.Acuracy of LimitState: GEO in relation to Oasys Slope for rotational slides with

ru=0and drained conditions.




10 0.138 0.110 0.090 0.078 0.065

15 0.130 0.098 0.077 0.063 0.050

20 0.123 0.088 0.065 0.050 0.037

25 0.115 0.075 0.053 0.035 0.023

30 0.108 0.065 0.040 0.024 0.010

35 0.100 0.058 0.030 0.013 0.001

40 0.093 0.048 0.020 0.003

DRAINED ANALYSIS. RU=0

LIMIT STATE FoS IN COMPARISON WITH OASYS SLOPE

SLOPE 1/1 SLOPE 1/1.5 SLOPE 1/2 SLOPE1/2.5 SLOPE1/35 -2.5% -2.9% -3.3% -2.0% -0.4%10 -1.8% -3.6% -2.7% -2.1% -1.4%

15 -3.0% -1.8% -2.4% -2.8% -0.7%

20 -3.1% -3.1% -2.7% -0.9% -

25 -2.2% -3.6% -0.6% - -

30 -2.2% -2.1% - - -

35 -2.3% - - - -

40 -1.9% - - - -


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Table3.11.Accuracy of LimitState: GEO in relation to Oasys Slope for rotational slides with

ru=0.3and drained conditions.

Table3.12.Accuracy of LimitState: GEO in relation to Oasys Slope for rotational slides withru=0.3and drained conditions.

(-) cases where the slope is stable



SLOPE 1/1 SLOPE 1/1.5 SLOPE 1/2 SLOPE1/2.5 SLOPE1/35 0.2% -2.2% 0.8% -1.3% -4.2%

10 -1.5% -3.1% 0.0% -1.7% -3.6%

15 -4.4% -4.9% 0.0% -1.4% -5.1%

20 -4.2% -1.8% -2.6% -2.4% -1.7%

25 -3.6% -3.4% -4.8% -0.9% -

30 0.8% -5.8% -2.8% 0.0% -

35 -0.8% -5.7% 0.0% - -

40 -1.1% -4.8% - - -

DRAINED ANALYSIS RU=0.5


SLOPE 1/1 SLOPE 1/1.5 SLOPE 1/2 SLOPE1/2.5 SLOPE1/35 0.0% -0.2% 1.3% -1.6% -1.4%

10 -0.5% 0.0% 0.0% -1.5% 0.0%

15 -2.0% -0.5% 0.0% -1.6% -3.1%

20 -3.8% -2.8% -1.0% -2.6% -4.5%25 -4.8% -1.6% -5.1% -0.8% -4.5%

30 -6.3% -2.0% -2.6% -2.9% -2.8%

35 -6.2% -6.4% -3.8% -2.2% -

40 -8.1% -7.8% -4.9% - -


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The nodal density was coarse in all cases, as LimitState Ltd carried out their

studies under these conditions which proved to

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