Yagi slope

SLOPE STABILITY ENGINEERING VOLUME 1

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PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON SLOPE STABILITY ENGImERING - IS-SHIKOKU99/MATSUYAMA/SHIKOKU/ JAPAN/8- 11 NOVEMBER 1999

Edited by

Norio YagiEhime Universio,Japan

Takuo Yamagami & Jing-Cai JiangUniversity of Tokushima,Japan

VOLUME 1

U

A. A. BALKEMA/ R OTTERDAM BROOKFIELD/ 1999

The texts of the various papers in this volume were set individually by typists under the supervision of each of the authors concerned.

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Published by A.A. Balkema, PO. Box 1675,3000 BR Rotterdam, Netherlands Fax: +3 1.10.413.5947; E-mail: balkema@ balkema.nl; Internet site: www.balkema.nl A.A. Balkema Publishers, Old Post Road, Brookfield, VT 05036-9704, USA Fax: 802.276.3837; E-mail: [email protected] For the complete set of two volumes, ISBN 90 5809 079 5 For Volume 1 , ISBN 90 5809 080 9 For Volume 2, ISBN 90 5809 08 1 7

0 1999 A.A. Balkema, Rotterdam Printed in the Netherlands

Slope Stability Engineering, Yagi, Yamagami & Jiang 0 1999 Balkema, Rotterdam, ISBN 90 5809 079 5

Table of contents

Preface Organization

XIII

xv

Special lectureFlow-type failure of slopes based on behavior of anisotropically consolidated sand K. Ishihara, YTsukamoto & S Nakayama3

Keynote lecturesThe limit analysis for slopes: Theory, methods and applications Z Chen Using limit equilibrium concepts in finite element slope stability analysis D. G. Fredlund & R. E. G.Scoular Stability of geosynthetic reinforced steep slopes D. Leshchinsky The mechanisms, causes and remediation of cliff instability on the western coast of the Black Sea M. Popescu Design of slope stabilizing piles H. G.Poulos15 31 49 67

83

1 Geological and geotechnical site investigationsGeoenvironmental factors influencing the deterioration of shale in a rockslope A. M. Elleboudy Weathering mechanism and slope failures of granitic rocks in Southwest Japan Effect of hydrothermal activities R. Kitagawa-

103

109

Site investigation of weathered expansive mudrock slopes: Implications for slope instability and slope stabilization R.J. MahurujV

1 15

Investigation of cut slope consisting of serpentinite and schist H. Kitarnura, M.Aoki, TNishikawa, TYarnamoto, M. Suzuki & TUmezaki Using multibeam sonar surveys for submarine landslide investigations J. Locat, J.KGardner, H. Lee, L. Mayer, J. E. Hughes Clarke & E. Karnrnerer Automatic measurement of pore water pressure in the hard-rock slope and the sliding weathered-rock slope - Field survey in mountainous region in Shikoku Island, Japan E.Tamura & S. Matsuka Field measurement of suction in soil and rainfall in Kagoshima Prefecture R. Kitarnuru, K.Jomoto, K. Yamamoto, TTerachi, H.Abe & T Iryo Application of acoustic emission method to Shirasu slope monitoring T.Fujiwara, K. Monrna & A. Ishibashi Acoustic emission technique for monitoring soil and rock slope instability A. Kousteni, R. Hill, N Dixon & J. Kavanagh Hydraulic fracturing as a mechanism of rapid rock mass slides S. Hasegawa & T Sawadu Evolution of ridge-top linear depressions and a disintegration process of mountains K. Mokudai & M. Chigira Geological characteristics of landslides of the soft rock type, Central Japan 7:Fujita Study of configuration, scale and distribution of landslides S. Ueno Geodynamics and spatial distribution of properties of sea cliff colluvium E. Dembicki & WSubotowiczA mineralogical study of the mechanism of landslide in the serpentinite belt K.Yokota, R. Yatabe & N. Yagi

121 127 135

141 147

151157 163 169 175 181 187 193 199

Detailed geotechnical study in Modi Khola Hydroelectric Project, Western Nepal VDangol & 7:R. Puudel Local instability in saturated colluvial slopes in southern Brazil WA. Lacerdu

2 Soil slope stability analysesA new theory on instability of planar-sliding slope - Stiffness effect instability theory Qin Siqing Ultimate state of a slope at non-linear unsteady creep and damage SA ElsouJiev Application of FEM on the basis of elasto-viscoplastic model to landslide problems H. Fujii, S. Nishirnura, T.Hori & K. Shimuda Coupled excavation analyses of vertical cut and slopes in clay T.Hoshikawa, 2: Nakai & Y NishiVI

207 213 219225

Effects of a deep excavation on a potentially unstable urban hllside in San Marino G.Gottardi, G.Marchi, L.Tonni & F: Bianchi Displacements of a slope in the Euganean Hills induced by quarrying S.Cola & RSirnonini Stability evaluation of sliding failure along thin mudstone deposit due to excavation Y Nakarnura, J. Kojirna, S. Hanagata, K. Narita & YOhne Appraisal of Bishops method of slope stability analysis G.L. Sivakurnar Babu & A. C Buoy A convenient alternative representation of Taylors stability chart R. Baker & YTanaka Influence of stress-strain curves on safety factors and inter-slice forces in FEM A. Mochizuki, J. Xiong & M. Mikasa Slope stability analysis considering the deformation of slices YTerado,H. Hazarika, TYarnazaki & H. Hayarnizu Slope stability analysis using a spring attached to inter-slice planes K. Kondo & S. Hayashi Three-dimensional stability analysis of locally loaded slopes X.Q.Yang, S.X. He & 2 D. Liu . A lower-bound solution of earth pressure of cohesive backfill with inclined slope surface M. Luan, 7:Nian, C.E Lee, K.T. Law, K. Ugai & Q.Yang Shear band formation and propagation in clay slopes L. E.Vallejo Progressive failure analysis of slopes based on a LEM TYarnagarni,M.Taki, J.-CJiang & S.Yarnabe Progressive failure analysis based on a method of non-vertical slices TYarnagarni,YA.Khan & J. -C.Jiang Back analysis of unsaturated shear strength from a circular slope failure J. -CJiang, TYarnagarni & Y Ueta A back analysis of MC-DP model parameters based on FEM and NLSSQP method T.Q.Feng, TYarnagarni & J.-C.Jiang An FE analysis of anisotropic soil slopes and back analysis for its parameters T.0, Feng TYarnagarni & J.-CJiang

233 239 245 249 253 259 265 27 1 277 28 1 287 293 299 305 31 1 3 17

3 Rock slope stability analysesAn upper bound wedge failure analysis method ZYChen, YJ.Wang,X.G.Wang & J.Wang Stability analysis of rockfill dam and retaining wall constructed on dip bedrock S. S Chen & X.S. Fang325 329

VII

Soil-water coupling analysis of progressive failure of cut slope using a strain softening model 333 TAdachi, E Oka, H. Osaki, H. Fukui & E Zhung A back analysis in assessing the stability of slopes by means of surface measurements S. Sakurai & 7:Nakayama Numerical simulation of excavation of the permanent ship lock in the Three Gorges Prqject Y Zhang & K. Yin Numerical simulation of the buckling failure in rock slopes I!Hu & H. -G.Kempfert Fuzzy-based stability investigation of sliding rock masses NO.Nawari & R. Liang Stability evaluation of discontinuous rock slope K. Kawarnura & M. Nishioka Earthquake and seepage effects on the mobilised shear strength of closely jointed rock M.J. Pender339 345 349 355 36 1 367

4 Effects of rainfall and groundwaterDesign chart for cut slope in unsaturated residual soils R. Subrarnaniam & E H.Ali Factors affecting on water retention characteristic of soils K. Kawai, D. Karube & H. Seguchi Suction profiles and stability of residual soil slopes E. C.Leong, B. K. Low & H. Rahardjo Effects of perched water table on slope stability in unsaturated soils L. 7:Huat, E H.Ali, S. Mariappan & l? K. Soon Field suction variation with rainfall on cut slope in weathered sedimentary residual soil L. 7:Huat, E H.Ali & S. Mariappan Study of slope stability for Pleistocene cemented sandy sediments in Singapore (Old Alluvium) K. K. Poh, l? B. Ng & K. Orihara Influence of pore water pressures in partly submerged slopes on the critical pool level E.N. Brornhead, A.J. Harris & l D.J. Watson ? Role of pore water and ar pressures on slope stability in reservoir for pumped storage i power plant TSato, N.Nishizawa, M. Wakarnatsu,I Hiraiwa & I. Kurnazaki ! Seepage characteristics of decomposed granite soil slope during rainfall S. Sasaki, S.Araki & K. Nishida Relation between slope stability and groundwater flow caused by rainfalls M. Enoki & A.A. Kokubu375 38 1 387 393 399 405

41 1 417

423 429

Vlll

Salient aspects of numerical analyses of rainfall induced slope instability C.-H.Wang Centrifuge model tests and stability analysis on mobilizing process of shear strength of decomposed granite soil slope S.Yushituk & KOnitsuka Centrifuge tests on slope failure during water infiltration H. G. B.Allersrna Reinforcements effects in the tank-model prediction of slope failures due to rainfalls M. Shirnizu Investigation of danger rainfall prediction system for natural and cut slopes H. Miki, A. Fujii & M. Furuta Predicting ramfall-induced slope failures from moisture content measurement M. Nishigaki, A. Tohari & M. Kornatsu Analytical study on the slope stability during ramfall and the rainfall indexes A. Togari-Ohta, TSugiyama, T Nara & S. Yarnazaki Evaluation of critical rainfall with logit model I:Sugii, K.Yarnada & T Uno Strategy for prevention of natural disaster due to slope failure R. Kitarnura, T Iryo, H.Abe, H. Yakabe & K. Yarnarnoto Relationships between rainfalls and landslides after forest damages by typhoons S. Murata, H. Shibuya & K. Hayashi Threshold rainfall for Beragala landslide in Sri LankaA K. Dissanayake, Y Sasaki & N H. Seneviratne

435441

447 453 459 465 47 1 477 483 489 49550 1

The importance of the groundwater regime studies of unstable slopes - An example of investigations on the landslide Plavinac, Yugoslavia G. Rasula & M. Rasula Landslides induced by rainstorm in the Poun area of Chungchongbukdo Province D. Hun & K. Kim Characteristics of Cretaceous granite slopes that failed during heavy rainfall TYarnarnoto, M. Suzuki, N. Matsurnoto & X Sehara Seepage analyses of embankments on Tokaido-Shinkansen in long term rainfalls K. Kato & S. Sakajo Instability analyses of embankments on Tokaido-Shinkansen in heavy rainfalls S. Sakajo & K. Kato Chemical effect of groundwater from acid rain on slope evolution Z X u & R. Huang Slope failures triggered by an earthquake and a heavy rain in Chiba S.Yasuda, XYoshida, I:Kobayashi & TMizunaga

509 515 521 527 533 539

IX

Numerical evaluation of the effects of drainage pipes TYamagami, K. Nishida & J.-CJiang Effects of horizontal drains on ground water level and slope stability RCai & K. Ugai

545

55 1

5 Effects of seisrnicityCollapse of high embankment in the 1994 far-off Sanriku Earthquake KShioi & S. Sutoh Slope instability of large embankments in residential areas caused by the Hyogoken-Nanbu Earthquake, 1995 T.Kamai, I:Kobayashi & H. Shuzui Analysis of toppling failure of mountain slope caused by the Hyogoken-NanbuEarthquake TOkimura,NYoshida & NTorii Stress condition and consequence of liquefaction on weathered granitic sands ZOkada, K.Sassa & H. Fukuoka Effects of density, stress state and shear history on sliding-surfaceliquefaction behavior of sands in ring-shear apparatus G.Wang & K. Sassa Real seismic-waveloading ring-shear test on the Nikawa landslide EWWang, K. Sassa & H. Fukuoka Dynamic properties of fine-grained soils in pre-sheared sliding surfaces M.Yoshimine, R. Kuwano, J. Kuwano & K. Ishihara Dependence of pore pressure generation on frequency of loading at sliding surface D.A. Vankov & K. Sassa On-line earthquake response tests on embankments founded on saturated sandy deposits T.Fujii, M. Hyodo, I Nakata, KYahuki & S. Kusakabe : Dynamic centrifuge tests of embankments on sloped ground and their stability analyses J. Koseki, 0.Mutsuo, K. Kondo & S. Nishihara Evaluation of liquefaction potential for loose minefill slopes ? Kudella Runout distances of earthquake-inducedlandslides I:Kobayashi Evaluation of measured vertical and horizontal residual deformation at crest of rockfill dam under earthquake T. Okamoto Displacements of slopes subjected to seismic loads R. L. Michalowski & L.You Permanent displacement analysis of circular sliding block during shaking H. R. Razaghi, E.Yanagisawa & M. KazarnaX

559 565

57 1 577 583

589 595 60 1 607 613 619 625 63 1

637 641

Dynamic analyses of slopes based on a simple strain-softening model of soil A.Wakui & K. Ugai Slope instability due to rainfall and earthquake K. Shirnada, I Fujii, S. Nishirnura, ?:Nishiyarna & ir: Morii 3 Shaking table tests of concrete block retaining walls S. Mori, ir:Matsuyarna & ?:Ushiro Shakedown analysis of soil foundations under varied loads M. Luan, Z: Cao & K. Ugai Author index

647 653 657 663

669

XI


Slope Stability Engineering, Yagi, Yamagami & Jiang (c) 1999 Balkema, Rotterdam, ISBN 90 5809 0795

Preface

It is of a great concern to civil, geotechnical, and environmental engineers to overcome different problems caused by natural disasters, human errors and geo-environmental problems, whch are related directly or indirectly to the soil and rock properties. Although significant progress in the field of geotechmcal engineering has been made in past few decades, there are still a number of problems that arise in geotechnical analyses, designs, and specifications to prevent the possible damages due to unexpected disasters like landslides, debris flows, earthquakes, etc. So, figuring out these problems and tackling them very professionally are the main challenges at present-day world of geotechnical engineering. With this objective, the International Symposium on Slope Stability Engineering: Geotechnical and Geo-environmental Aspects - IS-Shikoku99 was held at Matsuyama, Ehime from November 8 to 1I , 1999. The symposium was sponsored by the Japanese Geotechnical Society on its 50th anniversary under the auspices of the technical committee on landslides (TC-11) of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE) and the Japan Landslide Society. The aim of the symposium was to bring different professionals from different disciplines and backgrounds together into a place to broaden the knowledge and understand the problems all over the world from various perspectives. This symposium covers a broad range of topics such as site investigation, seismic effect, soil strength parameters, damage assessment, remediation techniques, land development, waste disposal, landslide hazard, simulation, analysis, etc on slope stability engineering. The main themes of the symposium are as follows: 1. Site investigation; 2. Stability analysis of soil and rock slopes; 3. Effects of seismicity and rainfall; 4. Design strength parameters of natural slopes; 5. Effect of land development; 6. Slope stability of waste materials; 7. Stability of landfills; 8. Stabilization and remedial works; 9. Reinforced steep slopes; 10. Probabilistic slope stability; 1 1. Landslide inventory and landslide hazard zonation; 12. Simulation and analysis of debris flow.

After reviewing the abstracts and manuscripts of 246 full papers from over 4 0 countries by the organizingcommittee, a total of 221 papers has been accepted for the presentation in the symposium and publication in the proceedings volumes. The chairman, on behalf of the organizing committee, would like to extend his deep gratitude to the special speaker, Prof. Kenji Ishihara, President of ISSMGE and the keynote speakers, Dr. Zuyu Chen, Prof. Delwyn G.Fredlund, Prof. Dov Leshchinsky, Prof. Mihail Popescu, and Prof. Harry G.Poulos. Thanks are also due to the professionals who made this symposium a grand success by submitting and presenting the papers in different topics in the field of slope stability engineering. All participants without whom the symposium would not have been a lively discussion forum are greatly acknowledged for their active participation. Special thanks from the chairman go to all the session chairpersons and to Prof. Yamagami, Prof. Mochizuki, Prof. Yatabe, Dr Jiang and the members of local and international advisory committee for their active involvement in accomplishing the symposium. Finally, the Ministry of Education, Science, Sports, and Culture that financially supported the symposium under the Grant-in-Aid for publication of Scientific Research Results is highly appreciated. Norio Yagi Chairman of the International Symposium on Slope Stability Engineering - IS-Shlkoku99 Professor of Ehime University, Japan November 1999

XIV

Slope Stability Engineering, Yagi, Yamagami & Jiang 0 1999Balkema, Rotterdam, ISBN 905809 079 5

Organization

INTERNATIONAL ADVISTORY COMMITTEE Prof. T.Adachi, Japan Prof. K.Arai, Japan Prof. A.Asaoka, Japan Prof. R. Baker, Israel Dr R. K. Bhandari, India Prof. C. Bonnard, Switzerland Prof. E. N. Bromhead, UK Dr Zuyu Chen, China Prof. M.Chigira, Japan Prof. R.Chowdhury, Australia Prof. D. M.Cruden, Canada Prof. J. M. Duncan, USA Prof. M.Enoki, Japan Prof. R. M. Faure, France Prof. D.G. Fredlund, Canada Dr H.FuJita, Japan Prof. T.Furuya, Japan Prof. J. N. Hutchinson, UK Prof. Y. Ichlkawa, Japan Prof. K. Ishihara, Japan Prof. H. Kawakami, Japan Prof. Sang-Kyu Kim, Korea Prof. T. Kimura, Japan Prof. R. Kitamura, Japan Prof. Y. Kobayashi, Japan Prof. 0.Kusakabe, Japan Prof. W.A. Lacerda, Brazil Prof. K.T. Law, Canada Prof. C. E Lee, Hong Kong Prof. D. Leshchinsky, USA Prof. J. Locat, Canada Prof. M. Maksimovic, Yugoslavia Prof. T. Matsui, Japan Prof. R. L. Michalowski, USA Dr H.Miki, Japan Prof. T. Mitachi, Japan Prof. S. Miyauchi, Japan Prof. H. Nakamura, Japan Prof. K.Narita, Japan Prof. M. Nishigaki, Japan Prof. H.Ochiai, Japan Prof. Y.Ohrushi, Japan Prof. H.Ohta, Japan Prof. K.Okada, Japan Prof. TOhmura, Japan Prof. S.Okuzono, Japan Prof. M. J. Pender, New Zealand Dr D. J. Petley, UK Prof. L. Picarelli, Italy Prof. M. Popescu, Romania Prof. H.G. Poulos, Australia Prof. S.Sakurai, Japan Prof. Y.Sasaki, Japan Prof. D.Schreiner, South Africa Prof. R. L. Schuster, USA Prof. H.Sekiguchi, Japan Prof. K. Senneset, Norway Prof. ETatsuoka, Japan Dr Gongxian Wang, China Prof. S.G.Wright, USA Prof. E.Yanagisawa, Japan Prof. S.Yasuda, Japan Dr H.Yoshimatsu, Japan

xv

ORGANIZING COMMITTEE

Chairman Prof. N.Yagi General Secretary Prof. T.Yamagami Secretaries Dr J.-C. Jiang Prof. A. Mochizuki Prof. R.Yatabe Members Dr S.Akutagawa Dr S. Hasegawa K. lshikawa E Kamada K. Koumura Prof. T. Muro H. Nishda Assoc. Prof. M.Ogura Dr H.OhtsuProf. K. Sassa Dr N. Shimizu Y. Shono Dr A. Suemine M.Takeyama Prof. 1.Towhata Prof. K.Ugai M.Yamamoto A.YZiISIanaka

XVI

Special lecture


Slope Stability Engineering, Yagi, Yamagami 8 Jiang 0 1999 Balkema, Rotterdam, ISBN 90 5809 079 5

Flow-type failure of slopes based on behavior of anisotropically consolidated sandK. Ishihara, YTsukamoto & S. NakayamaDepartment of C v l Engineering, Science University oj' Tokyo,Japan ii

ABSTRACT: Soil deposits in natural slopes are subjected to an initial shear stress as well as confining stress which are induced by the gravity. To evaluate effects of the initial shear stress on the behaviour of sand undergoing large deformation, a series of laboratory tests were performed, using the biaxial test apparatus, on saturated samples of Toyoura sand consolidated anisotropically under various Kc-conditions. The results of the tests were examined to determine the initial stress conditions distinguishing contractive and dilative behaviour in undrained application of shear stress. It was found that the major effective principal stress at the time of anisotropic consolidation is a parameter controlling dilative or contractive behaviour of the sand under otherwise identical conditions. Based on this conclusion, it was pointed out that the most appropriate way to normalize the residual strength of anisotropically consolidated sand is by the use of the major principal stress at consolidation. The outcome of the test results as above was used to address a method or criterion by which to identify whether or not a given sandy soil deposit under a slope will have a potential to develop the flow type failure with large deformation. INTRODUCTION In the conventional analysis of slope stability, a potential sliding plane is assumed and the shear stress expected to occur is compared against the shear strength that can be mobilized along the sliding plane. It has been customary to take up the magnitude of peak shear stress to define the shear strength. In the case of saturated loose sandy soils, the peak stress is mobilized at a relatively small shear strain of the order of 2 - 5%. Thus, even when the peak shear stress is passed over by some external forces, the resulting deformation may not be large enough, if there is no strain-softening taking place in the soils. In this case, cracking or small amount of deformation may be manifested on the surface of soil deposits and damage would be minor. However, if the soils are loose enough to induce strain-softening due to contractive nature of deformation, the shear strain of the order of 10 - 20% can easily be generated leading to flow type deformation. In terms of field behaviour, the soil in the slope is envisaged to move largely downstream giving rise to destructive damage there. Thus, the factor of safety against sliding of slopes can be defined in two ways, namely, (1) the factor of safety for triggering the slide against the peak strength, and (2) the factor of safety for the flow failure against the residual strength. Generally, it is a difficult task to determine the factor of safety for the slide triggering, because of uncertainty in quantitatively identifying the slideinducing external force to be applied to the soil element in addition to the gravity-induced shear stress. This external force could be seismic shaking or additional weight by rainfall. In contrast, the factor of safety for the flow slide can be determined rather easily primarily because the gravity-induced shear stress is the major driving force to be compared against the residual strength of the soils, and there is no need to identify other external forces. i The am of the present study is to indicate a basic concept for determining the residual strength for sandy soils that can be used to determine the factor of safety for flow type failure of slopes. In this type of analysis, no matter what is the slide-triggering driving force, the consequence is recognized as more important and there is no need to seek for the cause of the slide. The only force to be considered is the force induced by the gravity and this makes the analysis simple and straightforward. BASIC CONCEPT For the sake of simplicity, let a potential sliding plane be located in parallel to the surface of the slope as illustrated in Figure 1. Then, from the equilibrium of

3

forces amongst the weight of a soil mass W, and normal and tangential forces N and S acting on the potential sliding plane, the stresses o, and T are , obtained asCT

, e

N =-=y HCOS~CX... (1)

S z = - = y Hsina. coscx ,

e

Then, given the values of stress components, o, and z, as above it is possible to locate a point B in the , diagram of o, and 7, as illustrated in Figure 2. The direction of the line OB indicates the angle of obliquity of stress application, a , or the angle of stress mobilization. By drawing a half circle through the point B so that it is tangential to the line OB, it becomes possible to identify the points of the minor and major principal stresses o, and o3on the Mohr diagram. Then, from geometrical consideration, the following relations are obtained.

where y is the unit weight of the soil, a is the angle of the sliding plane, and H is the height of the soil mass being considered.

o1= 0 , + (tan

+

-lz, a cos

1

( ~ = C, 3T

1 + (tana - -)za a cos

I1

... (2)

Introducing Eq. (1) into Eq. (2), one obtains

o, = y H( l+sina)o3= y H( 1-sina)

... (3)

Thus, the ratio between the minor and major principal stresses is obtained as 1-sina KC= CT~/CT, = l+sina

... (4)

Figure 1. Forces acting on the soil element above a sliding plane in a slope.

The relation of Eq. (4) is displayed in Figure 3. It is known that the majority of natural slopes consisting of relatively soft soils have an angle ranging = approximately between a O and a=45". Thus, the ratio, Kc, between the two principal stresses has a value between 0.2 and 1.0.

Figure 3. Relation between Kc-value and angle of slope. Figure 2. Mohr circle to determine 0, and and 7,.

o3from o,

4

BACKGROUND OF LABORATORY TESTS When attempting to identify mechanism of failure of soils underneath sloping surface by virtue of laboratory tests, it has been a usual practice to subject a soil specimen to the stress changes which are similar to those expected to take place in the field. The principle of duplication of in-situ conditions as above would be executed in the laboratory tests by applying under drained conditions the principal stress CT, and C T ~ and then by shearing the soil specimen under undrained conditions. It would be argued that the undrained conditions may not prevail in shallowly seated partially saturated soil deposits where sliding could frequently take place. However, the change in void ratio of the soil during large deformation leading to sliding may be deemed not so much appreciable that the constant volume condition may be maintained approximately to a tolerable level of accuracy. In addition, it may as well be assumed that, even though the soil is partially saturated, the deformation behaviour is considered to be represented approximately by that of a fully saturated sample, if its volume stays little changed. With the assumptions as above multiple series of triaxial tests were conducted by subjecting sand specimens to a stress system with varying Kc-values defined as Kc = C T /~c T , ~ ~ where o , ~ C T ~stands for, respectively, the and ~ effective major and minor principal stresses at the time of consolidation. After the specimens were consolidated anisotropically, they were subjected to shear stress under undrained conditions by increasing the major principal stress CT].

TYPICAL PATTERN OF DEFORMATION The typical pattern of undrained deformation of anisotropically consolidated specimens is schematically illustrated in Figure 4 in terms of stress path and stress-strain curve. In Figure 4 (a), the abscissa indicates the mean principal effective stress defined by p=(0,+20,)/3 and the ordinate represents the shear stress defined by q In Figure 4, point A indicates an initial state of Kcconsolidation whereupon undrained shear stress application starts. When the specimen is loose, it shows an increase in shear stress, q, to a point B at peak strength and then a decrease down to a point C corresponding to the phase transformation. The bentover in the stress path takes place at point C and the shear stress increases to a point D where large deformation starts to occur without any change in the effective mean stress p and shear stress q. This state is called the steady-state. When the specimen is loose, the minimum shear stress is encountered, concomitant with fairly large deformation, at point C where the phase transformation take place from contractive to dilative behaviour. Thus, the residual strength should be defined by the shear stress qas which is mobilized at point C. The residual strength thus defined is called the strength at quasi-steady state. When the specimen is medium dense to dense, the stress drop does not appear and the shear stress at the phase transformation does not produce large deformation. In such a case, the residual strength should be defined as the shear stress mobilized at the steady-state, namely the point D. In the present study, attention will be drawn to the state of stress at the quasi-steady state, that is, the point C in Figure 4. No matter what is the strength at the steady state at point D, of practical importance in

Figure 4. Typical stress-path and stress-strain relation for loose sand.

5

Figure 5. Stress path and stress-strain relation of anisotropically consolidated sand with Kc=0.5.

Figure 7. Stress path and stress-strain relation of anisotropically consolidated sand with Kc=0.7.

Figure 6. Stress path and stress-strain relation of anisotropically consolidated sand with Kc=0.6.6

Figure 8. Stress path and stress-strain relation of isotropically consolidated sand with Kc=l .O.

loose sands would be the shear stress that can be mobilized at the point C in the state of phase transformation. In t h s context, the strength at the ultimate steady state is beyond the scope of the present study . OUTCOME OF TESTS The results of undrained compression tests on samples with void ratios ranging between 0.882 and 0.993 are displayed in Figure 5 where the shear stress q=(o,-o,)/2 is plotted versus the effective confining stress defined as p=(o,+o,)/2. The saturated samples were consolidated with a vertical stress of oI,=196kPa and a lateral stress of o,,=98kPa producing an initial state of Kc=0.5. It may be seen in Figures 5(a) and 5(b) that the dilatant behaviour is exhibited when the sample is prepared with a void ratio less than about 0.90, but otherwise the sample is contractive. It is to be noticed that the sample with e=0.912 has reached a steady-state with a shear stress of q=30kPa which is smaller than the initially applied shear stress of q=SOkPa. It is seen in Figure 5(b) that large deformation began to occur at an early stage of load application and continues further until an axial strain of 20% developed. The smallness of the shear stress at the quasi-steady state as compared to the shear stress at the outset would be regarded as a criterion for an unstable condition where flow-type deformation could be triggered if the peak shear stress is passed over by application of a slight agitation at the beginning. Another series of tests with the same initial lateral stress of 03,=98kPa but with an increased Kc-value of 0.6 is demonstrated in Figure 6 for samples with various void ratios where the general tendency is seen to be the same as the results of the tests shown in Figure 5. Still other series of the tests with a further increased value of Kc are displayed in Figure 7 where it may be noted that the sample with a void ratio of 0.900 has reached a steady-state where the shear stress is about q=SOkPa which is much larger than the initial shear stress of q=20kPa. In such a condition, the flow type deformation would not be induced because of the gain in shear strength as compared to the initially applied shear stress. The last series of the tests with Kc=l.O are demonstrated in Figure 8 where it is apparently noted that the specimen with e=0.884 exhibits delative behaviour. In comparison amongst the cases of Kc=0.5 through 1.0, it is noted that the sample changes its behaviour from contractive to dilative with increasing Kc-values even if the void ratio is kept at a constant value of e=0.900. This means that, with an increasing degree of anisotropy at the time of consolidation, the sample becomes more contractive and susceptible to triggering of the flow failure.

CONSIDERATION FOR TEST RESULTS It has been shown by Chern (1985), and, Vaid Chern (1985) that the relation between the void ratio and the minor effective stress at phase transformation G,, determined almost uniquely irrespective of the is Kc-condition at the time of anisotropic consolidation. This conclusion has been proved to be valid as well for Toyoura sand as indicated by the data shown in Figure 9 where four test data are plotted for the cases of K,=0.5, 0.6 and 0.7. The specimens with an initial void ratio of ei=0.892 were consolidated to vertical stresses of o,, =60, 70 and 12OkPa and sheared undrained in the triaxial compression mode.

Figure 9. Relation between void ratio and major principal stress G,, at the state of phase transformation. The minor effective stress G ~ , at phase transformation obtained in the tests was multiplied by a factor, ( 1+sin$,)/( 1-sin$,), to obtain the corresponding major principal stress, o,,, and this value of is plotted versus the void ratio in Figure 9, together with the consolidation curve for the initial void ratio of 0.892. It was then possible to draw a curve amongst the data points to establish a correlation between the void ratio and as indicated in Figure 9. Note that there are some scatters in the data, but the scatters become less and less as the becomes large. It may be consolidation pressure o,, seen in Figure 9 that for the two specimens with o ,=60 and 70kPa, dilative responses were observed throughout shear stress application, but for other two tests with o 120kPa, specimens exhibited contractive behaviour with limited deformation.7

Figure 10. Plots of initial states of specimens in terms of void ratio and ollC determine the Initial Dividng to Line for anisotropically consolidated sand.

Thus, the threshold condition differentiating between contractive and dilative behaviour would be obtained as marked in the diagram of Figure 9. In looking at the diagram in Figure 9, it is to be noticed that a unique set of curves are obtained for the consolidation and phase transformation, if the effective major principal stress, GI c and G are used to plot the test data of Kc - consolidated samples. Thus, it may be mentioned that, the deformation behaviour of Kc consolidated sand is dominated by the effective major principal stress ol. In order to examine the characteristic features of undrained deformation as above, the major principal stress c f I C at consolidation is plotted in Figure 10 for each of the test results with varying Kc-values. Note that each point in the figure indicates the void ratio and olc at initial stages before application of undrained shearing. It may be seen in Figure 10 that the Initial Dividing Line (ID-line) defined as a threshold curve differentiating between conditions of flow and non-flow can be established uniquely for anisotropically consolidated sample, if 0 ,1c is chosen as a parameter to indicate confinement of the sample at the initial state. Superimposed in Figure 10 is the quasi-steady state line established previously in Figure 9. According to the study by Kato et al. (1999), the QSS-line was shown to be determined uniquely also for anisotropically consolidated sand, if d I C chosen as a parameter to indicate initial is confinement.

RESIDUAL STRENGTH OF ANISOTROPICALLY CONSOLIDATED SAND It has been customary to define the residual strength, Sus, by referring to the minimum shear stress at the QSS which is mobilized at the state of phase transformation for sands exhibiting contractive behaviour. By denoting the deviator stress at this state by qs=o1s-o3s, the residual strength is expressed as (Ishihara, 1996, p. 268) sus = 4 M cos@ = -cos@ .p 2 2 s s

... (6)M=- 6sin@,3 - sin@,

where ps is the confining stress at the quasi-steady state as defined by ps=(oIs+20,,)/3 and M is a parameter related with the angle of phase transformation in the p-q plot. When normalizing the residual strength, Sus, there are three methods that are conceived to properly represent the strength. In the previous study (Ishihara, 1993) dealing with isotropically consolidated samples of sands, the mean effective stress at the time of consolidation, pc=(~lc+203c)/3, been used as a variable to has represent the degree of confinement at the state of

8

consolidation. However, when dealing with the anisotropically consolidated samples of sand, it may not be convenient to utilize the mean effective stress p,'. The other options would be to adopt the confining stress fjC=(o',,+ci',,)/2 or to use the major effective confining stress o',,.The three options are summarized as follows. P'C'0' 1c + 2 d 3c

36 , +d3c 1

'

-

Pc =

... (7)~

0' IC = d 1 c

Using the three confining stresses, the normalized residual strength is obtained variously as follows.sus - M -- -cos$, P'c 2-

1

r,

1i

I-

... (9)

... (10)

11 (a). Those data from denser samples exhibiting dilative behaviour are displayed with open circles and those shown by solid circles indicate that samples exhibited contractive behaviour. The boundary separating conditions of contractive and dilative behaviour is indicated by a vertical straight line in Figure 11 (a). It can be seen that the threshold initial state ratio differentiating between contractive and dilative behaviour remains almost unchanged with variation of Kc-values. Thus, it is considered appropriate to assume that the threshold initial state ratio, r,', takes a constant value which is equal to rc'= 1.2 for Toyoura sand. The same data set is expressed alternatively in Figure ll(b) now in terms of the Kc-value plotted versus the initial state ratio, ic defined by Eq. (9). It , may be seen that the threshold value of fctends to increase with an increasing value of K ~ = o ' ~ c / o ' , ~ . The other approach was adopted to arrange the data set in terms of the initial state ratio, rc=pc'/ps', defined by Eq. (8). The data plotted in Figure 1l(c) versus the Kc-value indicate as well that the threshold r,-value differentiating conditions between contractive and dilative behaviour tends to increase with increasing Kc-values. It is to be noticed in Figure 1O(c) that the value of r,=2.1 corresponding to Kc=l.O condition is approximately equal to the value of r,=2.0 determined in the previous study (Ishihara, 1993). Based on the observation as above, it may be assumed that the initial state ratio, r,', defined by Eq. (10) is to be taken as a fundamental parameter to indicate the threshold condition between the contractiveness and dilativeness of sand no matter whatever the anisotropic condition would be at the initial state. It may also be concluded that for Toyoura sand the threshold initial state ratio takes a value of r,'=1.2 for all the Kc-conditions employed in the tests. The relationship between rc', ?,and r, can be derived from their definitions as follows,C = - (1 2Kc)(2M + 3)

The ratio of the confining stress at the initial state to that at the quasi-steady state, rc, was introduced in the previous study (Ishihara, 1999, p269) as an important parameter to represent the degree of contractiveness in undrained loading on isotropically consolidated sand. It was referred to as the initial state ratio. The initial state ratio ?,and r,' are newly introduced in the present study as defined by Eqs. (9) and (10). In order to examine effects of Kc-consolidation on the value of the initial state ratio, the effective confining stress at the state of phase transformation was read off from all the test data such as those shown in Figures 5 through 8. The value of rc'=o,,'/ols' as defined by Eq. (10) was calculated first for all the test data on Toyoura sand and plotted versus the value of Kc =o,c'/~,c' shown in Figure as

r rc'-

1 9

+

... (1 1)

2M+3 = (1 + Kc)M+6 rc where M=(0'

3(0' 1s- 3s 1 - 6sin Qs 6 +2d3, ) 3 -sin$,

The results of extensive tests in the previous studies (Ishihara, 1993) have shown that for Toyoura sand the value of M takes a value of 1.24 and Qs=31" . In the subsequent study, this value proved to be valid as

9

well for anisotropically consolidated samples of Toyoura sand with various Kc-values. Introducing this value into Eq. (1 l), one obtains 2=0.61(1+2Kc)rc

It has been known in the above that the threshold initial state ratio rc=o,c/oIstakes a value of rc7=l.2, as demonstrated in Figure 11(a). Introducing this value, Eq. (12) can be rewritten as, rc = 0.73( 1+ 2Kc)

% = 0.76( I + Kc)r C

-

... (12)TC =0.91(1+Kc)~

J

1

. . . ( 13)

Figure 1 1. Relation between Kc-value and variously defined initial state ratios.10

Figure 12. Relation between Kc-value and variously defined normalized residual strength.

These relations are displayed in Figures 1 l(b) and

1 I(c). It may be seen that the relations of Eq. (13) areconsidered to hold true with a reasonable level of coincidence to mark the boundary lines differentiating between conditions of contractive and dilative behaviour of Toyoura sand, if the initial state ratio, rc and ?,are to be used to obtain the normalized residual strength through the use of Eqs. (8), (9) and (10). The values of the normalized residual strength can be determined for all the test data obtained in the present study based on the three expressions indicated by Eqs. (8) (9) and (10). The normalized residual strength obtained using Eq. (10) is displayed in Figure 12(a). Since the threshold value of rc is known to take a constant value of 1.2, the normalized residual strength is determined uniquely independent of the Kc-value. As indicated in Figure 12 (a), the normalized residual strength takes a threshold value of Sus/0,,=0.24 which is the upper limit amongst a number of data corresponding to the condition f C 1.2. It is to be noticed that the test data indicated 2 by open circles all belong to the state of phase transformation in dilative samples and the normalized residual strength in this region is not the minimum value of the strength. The ultimate strength in the region of fc51.2 needs to be determined by considering the ultimate state (steady state in dilating samples). The ultimate strength at the steady state in the dilative sand is generally higher and beyond the scope of the present study . The normalized residual strength Sus/p, and Sus/pc determined by Eqs. (8) and (9), respectively, is also demonstrated in Figure 12. The threshold value of the strength bounding the upper limit of any of the strength values in contractive sand is obtained by simply introducing Eq. (13) into Eqs. (8) and (9), as follows. S u s / d I c 0.24 =

magnitude of the residual strength is equal to or smaller than that of the shear stress induced by the gravity force. It is to be mentioned here that, no matter whatever may be the genetic cause of the slide, the gravity-induced shear stress would be the main force driving the soils mass moving downhills. If the soil deposit is in a loose state exhibiting the contractive behaviour with a residual strength which is smaller than the gravity-induced shear stress, then the soil mass would continue to move downwards leading to the flow-type of slide. As mentioned above, the degree of susceptibility to the flow slide depends also on the initial state of shear stress as expressed in terms of the Kc-values. Thus, it would be of interest to examine how the initial state will affect the potential for the flow slide if the soil is in the initial state under the slope as illustrated in Figure 1.

0.48s u s / Pc =

1... (14)

Figure 13. Residual strength versus the gravityinduced initial stress. For each of the results of the tests on loose samples with void ratios ranging between e=0.880 and 0.92 1, the value of shear stress qQsat the state of phase transformation was read off and its ratio to the initially applied shear stress qo was obtained as plotted in the ordinate of the diagram in Figure 13. The definition of qQsand qo is illustrated in the inset of Figure 13. Plotted in the abscissa of Figure 13 is the Kc-value in each of the anisotropically consolidated sample. Also plotted in the figure in the value of the slope angle, a, as obtained from the chart in Figure 3. It may be mentioned that if the ratio, qQs/qO less than unity, there would be a potential for is the flow-type slide being induced in the soil and otherwise the soil will be safe and free from being

0.72 S,,/p = ____ 1 + 2Kc

1

The relations of Eq. (14) are also displayed in Figures 12(b) and 12(c) where it may be seen that the normalized residual strength as determined by E.( 14) could represent the upper limit of the strengths if the residual strength is to be normalized by p, and P,. POTENTIAL FOR FLOW SLIDE As mentioned in the foregoing, the flow-type failure will be induced in loose sandy deposits, if the11

involved in the catastrophc slide due to flow-type deformation. Interpreted in this context, it may be inferred from the data in Figure 13 that , if the Toyoura sand exists in a slope with a void ratio of e=0.880 and 0.921, the slope with an angle of inclination greater than about 12.5 (Kc 50.65) would be considered to have a danger of being involved in the flow slide. It is to be noticed that the relation as shown in Figure 13 depends upon the density and material properties of sandy soils and more test data will need to be accumulated before any conclusion is drawn. CONCLUSIONS

Okuhara, students of the Civil Engineering Department, Science University of Tokyo. The authors wish to express their gratitude to these persons. REFERENCES Chern, J. C. 1985. Undrained Response of Saturated Sands with Emphasis on Liquefaction and Cyclic Mobility. Ph. D. Thesis, University of British Columbia, Vancouver. Ishihara, K. 1993. Liquefaction and Flow F d u r e during Earthquakes. Geotechnique, Vol. 32, NO. 3 : 351 -415. Ishihara, K. 1996. Soil Behaviour in Earthquake Geotechnics. Oxford University Press. Kato, S., K. Ishihara & I. Towhata 1999. Undrained Shear Characteristics of Saturated Sand under Anisotropic Consolidation. submitted to Soils and Foundations. Vaid. Y. P. & J. C. Chern 1985. Cyclic and Monotonic Undrained Response of Saturated Sands. Advances in the Art of Testing Soils under Cyclic Conditions, Proc. ASCE Convention in Detroit, Michigan: 120 - 147.

A series of undrained triaxial compression tests were conducted on saturated specimens of Toyoura sand with various densities to investigate effects of anisotropic consolidation on undrained behaviour distinguishing between contractive and dilative characteristics. The outcome of the tests indicated that the major at the time of anisotropic principal stress consolidation is a governing factor to uniquely determine the initial dividing line and quasi-steady state line in the plot of void ratio and confining stresses. This means that neither the mean principal stress defined by p=(o,,+2o,,)/3 nor p=(o1c+o3c)/2is an appropriate parameter to specify the initial state of confinement in the consolidated sand. Based on the above conclusion, the residual strength of the sand normalized each to different initial p stresses, i.e., o],, and Tj, was examined, with the result that the normalization by oYlcis most appropriate to define the normalized residual strength. It was also shown that the residual strength ~ normalized by o ,takes a value of 0.24 as an upper lirmt beyond which the residual strength can not be defined because of the sand becoming dilative with increasing density. To evaluate whether the Kc-consolidated sand is susceptible to flow-type failure, the value of residual strength was compared with the shear stress applied at the time of the anisotropic consolidation for loose samples with a void ratio between e=0.880 and 0.921. The outcome of such assessment indicated that for a loose deposit of Toyoura sand, there would be a potential for the flow failure to be triggered, if the angle of slope becomes greater than 12.5 and otherwise there would be no danger for such catastrophic failure.ACKNOWLEDGMENTS The laboratory tests described herein were performed by the help of Mr. T. Yoshimura and Mr. M.12

Keynote lectures


Slope Stability Engineering, Yagi, Yamagami & Jiang ( 1999 Balkema, Rotterdam, ISBN 90 5809 079 5 0

The limit analysis for slopes: Theory, methods and applicationsZuyu ChenChina Institute of WaterResources und Hydropower Research, Beijing, Peoples Republic of Chinu

ABSTRACT: The solution of a slope stability problem can be approached by its least upper bound and maximum lower bound. The limit equilibrium methods that employ vertical slices, such as those proposed by Bishop (1955), Morgenstern and Price (1965), imply a lower bound of the factor of safety. Those that employ slices with inclined interfaces, such the methods proposed by Sarma (1979), Donald and Chen (1997), give an upper bound approach to the stability analysis. In most cases the gap between the two bounds is very small and the rigorous solutions are indeed obtainable. However, care must be taken of the possible two directions of shear between the adjacent slices when the upper bound approach is used. The concept of upper bound and lower bound principles has been extended to wedge slide analysis. A number of case histories regarding the slope engineering of Chinas hydropower construction, including those of the Three Gorges and Xiaolangdi projects, have been reviewed which indicated that an understanding of the Bound Theorems will help to obtain reliable and economical solutions to slope stability problems. 1INTRODUCTION The limit equilibrium method, or in a broader sense, the limit analysis method (Chen, 1975), is an approach that has been extensively used in solving various practical problems concerned with slope stability analysis. In spite of its successfbl applications in geotechnical engineering for both soil and rock slopes there have been some critical issues needed to be discussed. The limit equilibrium method has been regarded sometimes as an empirical approach since some assumptions were introduced when establishing the governing equations and since the displacement of the soil or rock mass is not properly considered in the method. Another issue related to this method is that the method is well developed and understood. More work in updating the method seems not to be highly demanded. As a branch of applied science, Soil Mechanics and Rock Mechanics benefit from the recent developments in the Classic Mechanics and Computer Science. The former offers a theoretical background, such as the upper bound and lower bound theorems of Plasticity, which enables us to establish a modern system of limit analysis based on the traditional method of slices. The latter makes the application of the theory to practical geotechnical problems possible. In this paper, the author wishes to give a general review of the theoretical background of the limit analysis method, demonstrate its accuracy, bring some critical issues that have not yet been discussed in literature and report its successhl applications in some important projects in China. 2THEORETICAL BACKGROUND2.1 Fundamentals

The procedures of solving slope stability problems is similar to that for solid mechanics. For a specified load system, it is required to find a stress fielder,,, and its associated displacement field U,, which satisfy the following conditions (expressed in tensors).(1) Force equilibriumnq.,

=

*,

with the boundary conditions:

15

in which W, is the body force, T, the tractions in the boundary S and nl is the directional derivatives of the surface S. The force equilibrium conditions can be expressed in a formulation employing the virtual work principle.

I, crJ dv = .i.,

W , . ri, dv

+IF,

U,

LJS

(2.3)

stability problems. However, rock mass is highly discontinuous, non-homogeneous, anisotropic and nonlinear, which exhibits complicated deformation behavior at failure, such as dilatancy, strain softening and large displacements, Finding the solution by some simplified methods is an approach actually employed by many practitioners in their consulting work.

where li is a compatible displacement increment field assigned on each force. The left side of (2.3) is sometimes called energy dissipation. (2) Compatible displacement filed A compatible displacement filed requires that the strain at any point follows the definition:

(3) Constitutive law The constitutive law relates the force equilibrium and deformation compatibility requirements and represents the material behavior. It includes both deformation and strength requirements.

where Cllk, a matrix representing elastic or elastois plastic relationships expressed in tensors. For Eq. (2.6), Mohr-Coulumn' s failure criterion is generally employed, which states as, - o,,ig$ - c 4 0 r

(2.7)

or

where on and 'tr are normal and shear strength on the failure surface, while c and 4, shear strength parameters respectively. For rock and soil material, we also restrict the presence of tensile stress, i. e.,20

(2.9)FIG. 2.1 Slope stability analysis by an upper bound approach. (a) a general case; (b) the multi-slice failure mode: (3) the multi-block failure mode.

where 0 3 is the minor Principle stress at any Point of the media.

2.2 The upper bound and lower. borrnd theorems of PlasiiciiySatisfying all the conditions stated in Section 2.1 will lead to a real or rigorous solution to slope

(1) The lower bound theorem The lower bound approach starts from the force equilibrium condition and states that any stress field that satisfies Eq. (2.1). (2.2) and (2.7) or (2.8) will

16

be associated with an external load that is lower than or equal to the real load that brings the failure. (2) The lower bound theorem The upper bound approach starts from an increment of displacement, generally referred to as velocity U,, , in the plastic zone Q* and the slip surface r*.It states that the load calculated by (2.3) and (2.8) will be either greater than or equal to the real load associated with a real failure mechanism Q and r (Refer to Fig. 2.1 (a)). The left part of Eq. (2.3) consists of two parts, becoming

r

2.3 Definition ofthe factor of safety Traditionally, the theorems of Plasticity employ a loading factor q that brings a structure to failure. Donald and Chen (1997) discussed the unique and monotonic relationship between the loading factor 7 and factor of safety F which, in order to bring the structure to failure, reduces the available shear strength parameters to new values asC,

=CIF

(2.11) (2.12)

tanp, = tanp, I F

L o ~.zi,,JdQ+IdD=dW;.Uldv+IT,zi,ds ,~

(2.10)

where D is the energy dissipation developed on the slip surface r. The limit analysis renders the solution by approaching the real ultimate load from lower bound and the upper bound, trying to find the least upper bound and the maximum lower bound. If the difference between the two bounds is small, we may conclude that the rigorous solution is actually obtained. The advent and rapid development of computers and the associated various numerical algorithms have enabled a practicable procedure to find the extreme for geotechnical problems and confirm that the two bounds are indeed very close. In explaining this concept, Pan Jiazheng (1980) summarized the following principles: (1) Among many possible slip surfaces, the real one offers the minimum resistance against failure ( Principle of minimum); (2) For a specified slip surface, the stress in the failure mass as well on the slip surface will be reorganized to develop the maximum resistance against failure ( Principle of maximum). The author has given a formal demonstration to Pans principle based on the Bound Theorems of Plasticity and Druckers postulates (Chen, 1998). In fact, Pans Principle is identical to the Bound Theorems but expressed in a more understandable way. Following the Bound Theorems or Pans Principles, performing slope stability analysis generally includes the following two steps: (1) For a specified failure mechanism, find a stress distribution that satisfies Eq. (2.1) with the constraints of (2.7) or (2.8), and search for a distribution that offers the maximum value of factor of safety. (2) Among all possible failure mechanism, find the one that has the minimum factor of safety.

The minimum and maximum loading factors are directly related to the minimum and maximum factor of safety respectively. Therefore, all the statements related to the bound Theorems can be expressed in terms of factor of safety. In the following presentations, the subscription e appeared for all variables would invariably mean that the related c and 4 values are reduced by (2.1 l), (2.12). 2.4 Significance ofthe Bound Theorems Before proceeding with the details, we present the following three examples indicating that a proper implementation of the bound theory will help us find the solution in a very simple way with high accuracy. Further more, it will offer better understanding to some basic rock mechanics concepts which otherwise could hardly be well interpreted. Example I The upper bound approach used for solving structural problems. Fig. 2.2 shows an example taken from the textbook (Wang, et. al, 1992). The frame is subjected to a set of external load. Although modern Mechanics of Structure has provided well defined methods to obtain the ultimate external load that brings the structure to failure, use of the Bound Theorems could lead to the following very simple and direct solution. We know that the structure collapses in a failure mode that involves 4 hinges. Fig. 2.2 shows 4 possible such modes. For each of the failure modes, we assign a virtual rotation 8 and establish the equation for energy and work balance. For example, in mode (a), a virtual rotation 8 will cause the external vertical load 2P to do work with a magnitude of 10, and develop an internal energy dissipation 011 hinges 2,3,4. Equating the work and energy dissipation gives

which leads to17

Similarly, the ultimate loads for mode (b), (c), (d) are P = M/21 ,P = 5 M/81, P = 5M/41 respectively. According to the upper bound theorem, the real ultimate load is the one that gives the lowest P, which is mode (b) with P = M/21. Performing the rigorous procedures of Structural Mechanics will give the same solution but in a much complicated way. This example indicates that if we are only interested in the ultimate loads and do not care about the failure process and the information about the stress and deformation during loading, there exists a straight forward and easy way to obtained the solution. This concept has been adopted to solve slope stability analysis problems as shown in the next example.

Examzple 2 A classical problem with the closed form solution. Fig. 2.3 shows uniform slope subjected to a vertical surface load. Sokolovski (1954) gave a closed-form solution with the assumption that the weight of the soil is neglected.-For this particular example in which c=98 kPa, $=30", the closed-form solution for the ultimate load T is 111.44 kPa. Associated with this load, we started with a four slice mechanism as shown in Fig. 2.3(a). Using Sarma's method, it is easy to find that the value of factor of safety is F=l.047. Sarma's method assumes that failure develops on both the slip surface and the inclined inter-slice faces. Therefore this solution can be regarded as the one that realizes Pan's principle of maximum. Following Pan's principle of minimum, we tried to find a failure mode that gives the minimum value of F as shown in Fig. 2.3(b) with a solution F,, =1.013. If the failure mass is divided into 16 slices, we obtained a failure mode almost identical to the one suggested by the closed-form solution as shown in Fig. 2.3(c), associated with F,], = 1.006. It is clear that with the theoretical support of the Bound Theorems, we are able to offer this example a solution for the ultimate load as accurate as the close-form solution.

FIG. 2.2 An example explaining a simple way to solve the ultimate loads using the upper bound theorem

FIG. 2.3 Example 2, an example describing the upper bound approach. (a) A four slice failure mode, initial estimate, F,=1.047; (b) Results of the optimization search, F,"=l.013; (c) Result of the optimization search using 16 slices, F = 1.006. ,

18

Examule 3 An issue regarding the wedge failure analysis Fig, 2.4 shows the forces applied on the two failure surfaces of a typical wedge. When establishing the force equilibrium equations, we noticed that the resultant forces PI and P,applied on the two failure surfaces involve six unknowns, i.e., their components in XJ,Z .directions. The factor of safety adds one more. The number of available force equilibrium equations for the wedge block is three. Mohr-Coulumn failure criterion on the failure surfaces added another two equations. Therefore, two assumptions must be made to render the problem statically determinate. The traditional method presented in Textbook implies an assumption that the shear forces on the failure surfaces are parallel to the line of intersection of the two failure surfaces. Pan (1980) argued on the theoretical background of making such assumptions. H e believed that among all the solutions satisfiing force equilibrium equations. the real solution should be related to the one that gives the maximum factor of safety. It is after the observation of this critical issue Pan put forward his Principles of Maximum and Minimum. On a separate paper published in this Symposium Proceedings (Chen et. al. 1999), the author and his associates presented an example which showed that the factors of safety obtained by the conventional and the upper bound approaches were 0.870 and 1.136 respectively. This indicates that even in a very simple area of rock mechanics, there are still some fundamental concepts for which a critical study is needed.2.5 Numerical supports - the method o optimization f

Fig. 2.4 The wedge failure analysis, (a) Sketch; (2) Forces applied on the two failure surfaces: (3) Co-ordinate system.

Fig. 2.5 Search for the critical failure mode by the method of optimization, 1: the original estimated ; 2. the critical

Use of the Bound Theorems or Pan's Principles essentially leads to a mathematical problem of finding the minimum of the factor of safety, which is associated with the input geometry of the failure mode, given the strength parameters for the material The method of optimization renders a powerfbl tool to find the minimum for geotechnical problems that involve complicated slope profiles and material properties. The task of an optimization operation is to find F,,, the minimum of the objective function F associated with the variable ZT=( z2,...,z,J z,, which represents the failure mode. In slope stability problems, the slip surfacey(x) is discretized by ni number of points A,, A2,,..., A,, (Fig. 2.5), whose coordinate values are ZI (i=l,2, ... m): (2.13)

To simulate this curve, we connect these points by either straight lines or smooth curves. Once this discretization mode is specified, factor of safety can be expressed as a function of x,,y,, x, y,, ... .xi,,y,, In , the upper bound method, the inclination of an interface 6, should also be included in the variable. We have

We start with an initial estimated failure mode, ., represented by A,,A2,....,A,, and 6,, 62,.. 6v,,which is associated with an initial value of F.Implementing the optimization routine, we eventually obtained a new mode represented by B',,B,,....,Bi,( refer to Fig. 2.5, n1=6 here), and a new set of 6,. 62 ,..., 6,, associated with the minimum value of F. A variety of optimization methods are available (Celestino and Duncan, 1981; Chen and Shao, 1988). Chen and Shao (1988) discussed the applications of the Simplex method, Negative gradient method and DFP method. While these methods on many occasions functioned well in finding the minimum factors of safety, they19

sometimes suffered from not being able to find the global minimum. A random search technique was consequently developed (Chen, 1992; Greco, 1996) which greatly enhances the efficiency of the search. 3SIMPLIFIED LOWER BOUND APPROACHTHE METHOD OF VERTICAL SLICES 3.1 Theoretical back ground As a simplified approach, our profession has a long history of employing the method of slices to solve various practical problems of geotechnical engineering. Early approach divides the failure mass into a series of slices with vertical interfaces. The method proposed by Morgenstern and Price (1965), as well as by others (Bishop, 1955; Janbu, 1973), imply a lower bound approach since the solutions are associated with a force distribution satisfying Eq. (3.1) on the slip surface, and (2.7) or (2.8) on the interfaces. (1) To allow the satisfaction of Eq. (2.1) for each slice, the force and moment equilibrium equations are formulated as (Chen and Morgenstern, 1983)-dG _

The factor of safety will be obtained by solving the relevant boundary conditions based on the assumptions made for the distribution of p(x). (2) To satisfy (2.7), or (2.8), it is required that on the interfaces shear and tensile failure not occur, i. e.

dxand

dP tan w -G = - p ( x ) sec t y

dx

d Gsinp=-y-(Gcosp)

dx

+-(y G a p ) + 7-h d x t d x t

d

d w

(3 4

in whichdW . p(x) = -sin@:dx

-a) +qsin@i -a) -7, --.seuy.singljdx

dW

(3.3)

G = the total interslice force; y, = y value of the point of application of the interslice force; a= inclination of the slice base; p=inclination of the interslice force; dW/& = weight of the slice per unit width; q= vertical surface load; q= the coefficient of horizontal seismic force, h, = distance between base and the horizontal seismic force, rt,= pore pressure coefficient (refer to Fig. 3.1(a)). Eq. (2.1) is obtained by projecting all the forces applied on a slice onto the line A-A' (Fig. 3.l(a)) which inclined at an angle of $e to the base of the slice. In that case the resultant of the normal force N and its contribution of the shear force on the base of the slice N tan$,, denoted as P , would be perpendicular to A-A' and not appear in Eq. (3.1).

FIG. 3.1 Slope stability analysis by the method of vertical slices, (a) the slope profile; (b) assumption for tan p ; (c) forces applied on a slice

[G' cosp' tanq:,, G'sinp'

+ cb,h] > F

(3.5)

20

G>O

(3.7)

Among a variety of assumptions for px, we () neglect those that produce results violating Eq. (3.5) or (3.7), and find one that gives the maximum factor of safety, according to the lower bound theorem. Solutions to the governing equations Chen and Morgenstern (1983) gave the solutions to the differential equations (3.1) and (3.2). They have been recently extended by Chen and Li (1997, 1998) to incorporate active earth pressure problems with the presence of a tension crack at the crown. The force and moment requirements take the form:

andf, (b) to be equal to the values of tanp at x=a and x=b respectively. J;,(x)is another function that has zero values at x=a and x=b. Fig. 3.l(b) shows an example that takesfix) as a sine function andf;,(x), a linear function that is zero at x=a and tan6 at x=a , where is the friction angle between the retaining wall and the soil. It is possible to find F (or P) and h from (3.8) and (3.9) by iterations. For details refer to Chen and Morgenstern (1983) or Chen and Li (I 998). 4SIMPLIFIED UPPER BOUND APPROACH THE METHOD OF INCLINED SLICES Theoretical background Sarma (1973) presented the method that employs slices of inclined interfaces. Therefore, the failure mode shown in Fig. 2.l(a) is simplified to a multiwedge system as shown in Fig. 2.1(b). We may understand the upper bound nature of Sarmas solution in the following two ways. (1) Since both the slip surface and the interfaces are assumed to be in a state of limit equilibrium condition, the solution means a mobilization of maximum resistance against failure. Estimation for the external load is thus either higher than or equal to the real load, according to Pans Principle of Maximum. (2) While Fig. 2.l(a) is simplified to Fig. 2.l(b), Eq.(2.3) in the upper bound approach is approximated as

(3.9) where s(x) = sec yE(x) (3.10) (3.11)

~ ( x=) [(sinp-cosptann)E-](nd{

(3.12) (3.13)

G,,, = P,,- PE(b)M,,, = P,,,h,,,- P[hCOS 6 + t(b)E(b)]i 7, -h,dx -

f

dW dx

(3.14)

in which P is the value of G(x) at x=b, or active earth pressure at the vertical wall. P, is the water pressure at x=a, i.e. P,, =G(a). h is the distance between the point of application of the active earth pressure and the bottom of the wall, i.e., the value of o/-y,) at x=b; h , the distance between the point of application of the water pressure and the bottom of the tension crack, i.e., the value of o/-y,) at x=a;6 is the value of p at x=b, i.e., the friction angle at the wall Eqs. (3.8) and (3.9) involve an unknown F (or P) and an unknown variable p () Chen and x, Morgenstern (1 983) suggested introducing an assumption defining /3 ( (Fig. 3.l(b)). x )

where the first and second terms of the left side of (4.1) refers to the energy dissipation developed on the interfaces and slip surface respectively. In the following discussion we will demonstrate that Eq. (4.1) is equivalent to the force equilibrium equations given by Sarma (1979). Therefore the factor of safety obtained by Sarmas method corresponds to an upper bound. It has been understood that for a material that obeys associated flow law and Mohr-Coulomb failure criterion, the plastic deformation produced by an increment in external load would incline at an angle $e to the shear band (Fig. 4.1 ), and the energy dissipation developed on the band is d D = (ccosp,, - usinpe)V

(4.2)

where U is the pore pressure applied on the shear surface (Donald and Chen, 1997). Let us examine a two block failure mode as,f ( ) is a linear function that allows the valueJ;,(a) x21

shown in Fig. 4.2. In Sarmas approach, MohrCoulomn criterion applies on both the left and right bases of the blocks as well as on the interface. The normal force P and its contribution of shear force Ptan@on each of the faces forms a resultant Pwhich inclines at an angle @e to the normal of the bases. Establishing force equilibrium equation, according to Sarmas concept, we have

formulated in a more efficient way by employing (4.5) the virtual work principle, with a set of virtual displacements, each inclined at an angle of &e to their respectively shear surfaces. (2) Since Eq. (4.5) is identical to (4.1) in this particular problem, the solution obtained by Sarmas method would be identical to that obtained by the upper bound method described in Section 4. I .

FIG. 4.1 The plastic deformation V and the energy dissipation developed on a shear band.

w, P,+Pi+c,,=0 +And

(4.3)

FIG. 4.2 A two block failure mode explaining the equivalence between Sarmas method and the energy approach.

Formulations o the upper bound solutions f (4.4) A brief introduction to this method is given as follows. For details, refer to Donald and Chen (1997). As explained in section 2.1, we start the upper bound solution by establishing a velocity field. For a pair of adjacent slices, the velocity of the left and right slices V, ,V, and the relative velocity form a closed triangle. Therefore we have (Refer to Fig. 4.3 and Fig. 4.4) sin(6,- 8,)

w, P,+Pi+c,,=0 +

for left and right slice respectively. In (4.3), W is the weight of the slice, C, is the shear force applied on the failure surface developed by cohesion. Now, we deliberately assign a set of virtual displacements V,, V,, y (Fig. 4.2) each inclined at an angle of @e to the shear surface. The work done P,, on V,, respectively is thus zero. P,, by P,, P,, Pi,as unknowns, disappear in the work and energy balance equation and Eq. (4.3) and (4.4) reduce to

< v,

v,. v,sin(@,.- 6, ) =sin(6, - 6,) v,= v,sin(6,. - 8,)

(4.6)

Alel COS@^,^^ +Arc,.cos@J,. + A,c,=

COS^,,^,

w,v, p, + wrvrcos p,. cos

(4.5)

(4.7)

where p is the angle between the weight vector and V The values V,., V, can be expressed as a linear function of V, , as will be given in the subsequent Section, and therefore are not unknowns. Eq. (4.5) remains only one unknown F which is implied in and is readily obtainable. We thus reach two conclusions: (1) Sarmas method, which typically involves a procedure of solving Eq. (4.3) and (4.4) can be

where 6 is the inclination of the interface with respect to the y axis. 8 is the angle of the velocity vector measured from the positive x axis. V, ,V, and V, of any slice can then be expressed as a linear function of the velocity of the left first slice V,. In general, the velocity of the wedge number k is determined byV=kV,22

(4.8)

where

Fig. 4.4 Velocity compatibility between adjacent slices. The left slice moves downward to the right one.

Fig. 4.3 Velocity compatibility between adjacent slices. The left slice moves upward to the right one. To enhance the numerical efficiency, we usually discretize a slip surface by several nodal points which are connected by smooth curves, as shown in Fig. 2.l(c). The velocity at any point of the slip surface can be integrated by the following equation.

V

= E(x)V,

(4.10)

where

E ( x ) = k exp[-

f cot(a-ro

-

p :

- Q ) -d 0 be satisfied with a definition of eJ by Eq. (4.14) while Case 2 requires the condition of @ - 8,< 0 or 8, - 8)> - E , with a definition of Eq. , (4.15) for 8/ , refer to Donald and Chen (1997). Example 5 A simple problems explaining the need for considering two directions of shear To support this statement, let us examine a simple case shown in Fig. 4.6. The two-block system is pushed by a horizontal force at the right side. It is not difficult to find the critical load of P that brings the system into a state of limit equilibrium by establishing the following equation:

z 8,=--6+p,,2

(4.14)

However, if V, lies lower than V,, as shown in Fig. 4.4, and consequently 0,.-i o,, the left slice would move downward with respect to the right one. This case, defined as case 2, occurs when the base of the left slice is a weak zone having lower friction angle compared to that of the right one, or when the base exhibits an abrupt decrease of a. i.e., 4,. -i or a,. -i a , . It can be easily found that if a downward 7 is assigned and consequently, Q, is defined as

The symbol '+' in ' 'is associated with case 1, k whereas '-' means case 2. For this example, the left block should move downward with respect to the right one since a,ta$+(ua

-U,

>ta@b > P

(2) The mobilized shear force, Sm, for each slice is calculated as the mobilized shear stress, zm, at the center of a slice multiplied by the base length, p.s r n

=

L P

(3)

The local factor of safety is defined as the ratio of the resisting shear force, S y , at a point along the slip surface divided by the mobilized shear force, Sm, at the same point,

(4)The resisting shear force, S y , and the mobilized shear force, Sm, are both calculated using the stresses computed in the finite element analysis. The normal stress, on, and shear stress, zm, can be "imported" as known values to the limit equilibrium analysis and the definition of both the overall and local factor of safety equations are linear.

A common set of coordinates is used to identify the center of a slice along a slip surface with respect to the surrounding finite element. The global coordinates for the center of the base are calculated in order to determine the location of the base center within the slope, and to determine which element is associated with the center of the base. The local coordinates of the center of the base are then calculated within the element that encompasses the center of the base (Fig. 3). The global coordinates for the center of the base of a slice are related to the global coordinates of the finite element nodal points through use of the shape functions.x=< N

>X {I

(5)

3.3 Element identification corresponding to the base of a sliceEach element must be checked to confirm that the center of the base of the slice is located within the element under consideration. Then the stresses calculated by the finite element analysis can be "imported" into the stability analysis. Once the element embracing the center of a portion along the slip surface is located, stress values from the Gauss points of the element can be transferred to the nodes of the element and consequently to the center of the base. The procedure is in accordance with the method described by Bathe (1982).35

y = {Y) (6) Where x = global x coordinates for the center of the base of a slice; y = global y coordinates for the center of the base of a slice; { ) = global x coordiX nates for the element nodal points; {Y) = global y coordinates for the element nodal points; and = matrix of shape functions. The shape functions are defined in terms of the local coordinates (r, s). Since the global coordinates for the center of the base of a slice and the nodes are known, the local coordinates can be obtained by solving Equations (5) and (6), simultaneously. The shape functions for a rectangular finite element with four nodes are as follows (Bathe 1982):1 NI = - ( l + r ) ( l + s ) 4( 7)

N,

=

1 -(1 -r)(l + s )

4

(8)

N, =

-

1 ( 1 - r)( 1 - s ) 4 1 4

(9)

N, = - ( 1 + r ) ( l - s )

(10)

be used to describe the change of a variable within an element in terms of nodal values. The finite element slope stability calculations require that stresses at the center of the base for each slice be within an element. This is achieved using the following procedure :

where r and s = local coordinates within the element. The local coordinates vary between -1 and +1 (Fig. 3). A knowledge of the local coordinates is crucial to identifying the element overlapping the center of the base of a slice. By definition, an element surrounds the center of the base of a slice if the following conditions are met: For a triangular element,

{oIn= < N > {F)

(15)

(0 5 r 21) and (0 5 s 21)For a rectangular element, (-1 5 r 21) and (-1 5 s 21)

(1 1)

(12) The center of the base is outside an element if the local coordinates are not within the above specified ranges. The search continues until an element is found that satisfies these conditions. 3.4 Transfer of element stresses to the center of the base of a slice Calculated stresses are stored within the computer software relative to the Gauss points of an element. Stresses must be transferred from the Gauss points of an element to the nodes of the element and then to the center of the base of a slice. The local coordinates of a point within a finite element are defined in relationship to the global coordinates at the nodes of the element by using the shape functions, as per Equations ( 5 ) and (6):

where oIn= stresses at the element node; = matrix of the shape functions; and {F) = stress values at the Gauss points. The local Gauss point integration coordinates are (0.577, 0.577), however, when the local Gauss point integration coordinates are projected outward to the element nodes, the local coordinates become (1.7320, 1.7320) (Fig. 5). This projection is carried out for each element and the values for the stresses from each contributing element are averaged at each node. Accordingly, the values of ox, q,, zxy can and be computed at each node of the finite element mesh. The nodal stresses, ox, q,, z, of an element are and , ) transferred to the center of the base of a slice along the slip surface.

(o.)=< >{oIn N

(16)

c $ where { = stresses at the center of the base of a slice. The stresses, ox, q,, and zxy, can now be computed at the center of the base for each slice.3.5 The normal and shear stresses at the center o a f slice Once the stresses, a,, q,, and zxy, are known at the center of the base for each slice, the normal stress, o n , the mobilized shear stress, z , can be caland , culated using Equations (17) and (1S), respectively (Higdon et al. 1976):+ox

on

+ oy2

0 ,

- oycos 2 e

= -

2ox- oysin 28

+ zxysin28(17)(18)

2,

=

~xycos2e -

2

Y =

N,N,N,N,

>

>

(14)

where x and y = global coordinate positions within the element that are known as the center of base of a slice (Fig. 4); X and Y = global coordinate at the element nodes; and N I , Nz, N3 and N4 = the shape functions defined in Equations 7 to 10. The stresses from a fmite element analysis are stored at the Gauss points. The shape functions can 36

where ox= total stress in the x-direction at the center of the base;oy = total stress in the y-direction at the center of the base; zxy= shear stress in the x- and ydirection at the center of the base; and e = angle measured from the positive x-axis to the line of application of the normal stress. The above steps provide the necessary information required to calculate the stability of a slope using the finite element stresses. The calculated values for the normal stress, on, and the mobilized shear stress,, ,z at the center of the base of a slice are entered into Equations (2) and ( 3 ) to give the resisting shear force

Figure 4. Location of the center of the base along the slip surface within a particular finite element.

Figure 5. Gauss point projections to the nodes of a finite element.

(strength) and the mobilized shear force (actuating shear), respectively. The local factor of safety is computed as the ratio of the resisting shear force to the mobilized shear force. The overall factor of safety is the sum of the shear force resistance values divided by the sum of the actuating shear forces along the slip surface.

4 PARAMETRIC STUDIES ON A SIMPLE 2:1 SLOPE A slope at 2 horizontal to 1 vertical is analyzed for 4 conditions (Scoular, 1997). The first case is a freestanding slope with zero pore-water pressures and the slope is referred to as a dry slope (Fig. 6). The second case is a free-standing slope with a piezometric line at three quarters of the slope height, and the slope is referred to as a wet slope (Fig. 6).37

Figure 6 . Selected 2: 1 free-standing slope with a piezometric line exiting at the toe of the slope.

The third case is a slope partially submerged in water with zero pore-water pressures in the slope (referred to as dry) (Fig. 7). The fourth case is a partially submerged slope with a piezometric line at one half of the slope height (referred to as wet) (Fig.7). The partially submerged slope is covered with water to one half of the slope height, providing support for the slope and increasing the factors of safety. The cohesion of the soil was varied from 10 to 40 kPa and the angle of internal friction was varied from 10 to 30 degrees for each slope type. 4.1 Limit equilibrium analysis The limit equilibrium analyses are performed using the General Limit Equilibrium method (GLE), (Fredlund & Krahn 1977) which provides a combined moment and force equilibrium solution. An empirical finite element interslice force function, based on an independent stress analysis (Fan et al. 1986) was used. The General Limit Equilibrium method along with a finite element interslice force function provides a method of comparison between the finite element based analysis and the limit equilibrium analysis. 4.2 Finite element stress analysis The finite element stress analysis was performed by switching-on gravity for the free-standing slope and for the partially submerged slope. The load of the water and the lateral support it provides to the

slope is simulated by point loads equal to the weight of water on the slope. The analyses are performed using Poissons ratios of 0.33 and 0.48, and a Youngs modulus equal to 20,000 and 200,000 kPa. The results showed that the stresses change with a changing poissons ratio, but are constant for changes in the Youngs modulus. This observation is consistent with the observations of Matos (1982). 5 RESULTS OF THE FINITE ELEMENT SLOPE STABILITY METHOD The local factors of safety differs along the overall slip surface (Fig. 8). Local factors of safety were computed for a 2: 1 (dry) slope with a cohesion equal to 40 kPa and an angle of internal friction equal to 30 degrees. While the local factors of safety differ along the slip surface, the overall finite element factors of safety fall within the range of the limit equilibrium factors of safety. The difference between the local factors of safety for Poissons ratios of 0.33 and 0.48, calculated using the finite element method, is reflected in Figure 8. The factor of safety computed by the limit equilibrium method and the finite element method appear to be very similar. The results appear to be within the limits of uncertainty associated with slope stability calculations. The finite element method incorporates the stress-strain characteristics of the soil when computing the shear strength and actuating shear force of the soil in the calculation of the factor of safety (Fig. 9).

38

Figure 7. Selected 2:1 partially submerged slope with a horizontal piezometric line at mid-slope.

The factor of safety results computed using the finite element method (i.e., F3 corresponding to a Poisson's ratio of 0.33, F4 corresponding to a Poisson's ratio of 0.48) are compared to the factors of safety computed using the limit equilibrium method (GLE) and are shown in Tables 1 and 2. To assess the variations in the factor of safety by each method of analysis, the results are grouped according to cohesion and angle of internal friction. The factors of safety grouped according to cohesion, c', are plotted versus the stability number, [(pVtan&)/cl, (Janbu, 1954). The factors of safety grouped according to the angle of internal friction, +', are plotted versus

the stability coefficient, (c Y y H ) (Taylor, 1937), where p is the unit weight of the soil, H is the height of the slope, $'is the angle of internal friction, and c' is the cohesion. The factors of safety are grouped according to the soil parameters and plotted versus the stability number and the stability coefficient. The greatest difference in factors of safety is noticed at high angles of internal friction, at low values of cohesion and at the maximum values of Poisson's ratio. The factors of safety for the (dry) free-standing slope, when grouped according to cohesion and plotted versus the stability number (Fig. 10) show a

Figure 8. Presentation of the local and global factors of safety for a 2:1 dry slope.

39

Table 1. 2: 1 &ee-standing slope Soil kPa Parameters degreeDry

10 20 10 40 20 10 40 20 40

10 10 20 10 20 30 20 30 30

GLE Finite element hnction 0.669 0.882 1.131 1.260 1.370 1.615 1.794 1.892 2.356

F3 p=O.330.662 0.867 1.125 1.230 1.352 1.639 1.765 1.884 2.324

F4p = 0.48

0.672 0.874 1.151 1.239 1.368 1.696 1.775 1.918 2.339

GLE Finite element function 0.488 0.677 0.782 0.995 1.021 1.102 1.335 1.374 1.741

Wet F3 p = 0.33

F4p = 0.48

0.456 0.634 0.745 0.930 0.969 1.077 1.260 1.287 1.627

0.467 0.647 0.755 0.953 0.988 1.101 1.293 1.312 1.661

Figure 9. Shear strength and shear force for a 2: 1 dry slope calculated using the finite element method.

Figure 10. Factors of safety versus stability number for a 2: 1 dry slope as a h c t i o n of cohesion.

40

Table 2. 2: 1 partially submerged slope SoilCI

kPa

Parameters Dry q Y GLE F3 degree Finite element ,U = 0.33 function 0.843 1.115 1.425 1.586 1.722 2.08 1 2.385 2.268 2.970

F4,U = 0.48

0.845 10 10 1.149 20 10 1.344 10 20 1.618 20 20 1.721 40 10 1.865 10 30 2.297 40 20 2.337 20 30 3.006 40 30 *n.s.a.: no solution achieved

0.827 1.OS5 1.422 1.575 1.691 n.s.a.* 2.368 2.204 2.899

Wet GLE Finite element function 0.649 0.886 1.050 1.318 1.322 1.482 13 0 0 1.783 2.303

F3,LI = 0.33

F4p = 0.48

0.635 0.874 1.046 1.314 1.296 1 SO5 1.774 1.763 2.260

0.641 0.880 1.068 1.343 1.316 1.530 1.795 1.786 2.274

Figure 11. Factor of safety versus stability coefficient for a 2: 1 dry slope as function of angle of internal friction.

Stability Number, [( yHtan4)lc'JFigure 12. Factor of safety versus stability number as a function of cohesion for a 2:l slope with the piezometric line at % of the slope height.

41

Figure 13. Factor of safety versus stability coefficient as a function of the angle of internal friction for a 2.1 slope with the piezometric line at % of the slope height.

Figure 14. Factor of safety versus stabili

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