NON-CIRCULAR SLOPE STABILITY ANALYSIS USING THE GENERALIZED WEDGE METHOD WITH MODIFICATIONS AND

EXTENSIONS FOR APPLICATION IN ROCK ENGINEERING

Atiia Zaki

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Civil Engineering

University of Toronto

O Copyright by Atila Zaki 1999

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ABSTRACT

NON-CIRCULAR SLOPE STABILITY ANALYSIS USING THE GENERALIZED

WEDGE METHOD WITH MODIFICATIONS AND EXTENSIONS FOR APPLICATION

IN ROCK ENGNEERiNG

Master of Applied Science

Atila Zaki 1999

Graduate Department of Civil Engineering

University of Toronto

The stability of a slope, either natural or artificial, is of considenble importance. The purpose

of this thesis is to find a method that is suitable to analyze slopes using a non-circula failure

surface, where the failure mechanism is govemed by structural weaknesses of the dope

material. The Generalized Wedge Method was modified and extended to handle the

requirements posed by rock slopes. The method was updated and several features were added

to improve the application to stability analysis problems. Once implemented, the method was

tested on a large number of test problems to validate its accuracy with success. The modified

method was found to be accurate in modeling and analyzing problems commonly

encountered in rock dope engineering.

ACKNOWLEDGMENTS

The author would like to take the oppomuiity to thank for the support and assistance

provided by his supervisor, Dr. J.H. Curran, who made possible the development of this

thesis with his knowledge and guidance. A special thanks goes to al1 the members of the

Rock Engineering Group at the University of Toronto and to Rocscience Inc. for providing

the opportunity to learn more about the issues involved in rock slope stability analysis.

The author gratefùlly acknowledges the financial support provided by Rocscience

Inc. and NSERC through the IPS Scholarship.

The author wishes to recognize the help from Dr. I.B. Donald at Monash University

for the numerous materials provided from the original research work and his time spend on

persona1 communication with the author.

And finally, the author wishes to express his gratitude to his parents and family and to

a special someone, without them the whole project would have been impossible to

accomplish.

iii

TABLE OF CONTENTS

*.. List of Tables ........................................................................................ viit List of Figures ............................................................................................ x

CHAPTER 1 - INTRODUCTION

1.1 General .................. .... .................................................................. 1 3 1.2 Scope of the Thesis ............................................................................. - 3 .................................................. 1.3 The Modifications to the Original Method -

CHAPTER 2 - WVIEW OF THE CURRECT STATE OF SLOPE STABILITY ANALYSIS

2.1 General ............................................................................................ 4 2.2 Limit Equilibriurn Methods ............................................................... 5

................... 2.2.1 Circular Failure Swface Methods .............................. ... 5 ............................................ 2.2.2 Non-Circular Failure Surface Methods 7

2.3 Energy Methods ............................................................................. 8 ...*...*..*.......... ......***.........* 2.4 Finite Element and Finite Difference Methods .. 9

..................... 2.5 Optimization Techniques for Search of the Critical Failure Surface 9

CHAPTER 3 - THE GENERALIZED WEDGE METHOD

........................... 3.1 Generd ................ ,.....*............. .............. I I

3.2 Theoretical Foundation of the Generalized Wedge Method ............................. 12 Basis of the Method .................................................................. 12 Force Equilibrium .................................................................... 15 Moment Equilibrium ................................................................. 18 Multi-Layered Medium .............................................................. 21

................................................ Pore and Surface Water Definition 24 Extemal Surface Loads ............................................................. 2 6 Tension Cracks ........................................................................ 30

3.3 Equivalence of the Generalized Wedge Method with the Energy Method Upper-

Bound ..................................................................................... 30 3.4 Deficiencies of the Original Generalized Wedge Method ............................... 31

CHAPTER 4 . MODIFICATIONS AND EXTENSIONS TO THE GENERALIZED W D G E METHOD

General ...................... .. ............................................................... 33 Improved Numerical Method for the Evaluation of the Factor of Safety .............. 34 Automatic Generation of the Wedges from the Slope Profile Line and the

Defined Failure Surface ...................................................................... 39 Automatic Genention of a Wedge System to Find the Optimal Solution ............. 42 Modification of the Governing Force Equilibrium Equations to be in Accordance

with the Donald-Chen Kinematic Admissibiiity Criterion ............................... 43 The Effect of a Weak Seam on the Global Factor of Safety ............................. 53

................................................................ Constrained Node Movement 55 ................................. 4.7.1 Horizontal and Vertical Constmint ... .......... 5 6

................................................... 4.7.2 Constraint Along a Defined Line 57 Global Unconstrained Methods of Optimization .......................................... 58

.......................................................... 4.8.1 Hooke and Jeeves' Method 60 .............................................................. 4.8.2 Roseabrock's Method 63

4.8.3 PoweN'sMethod ...................................................................... 65 ........................ Chen's Random Trial S d a c e Method ... ....................... 67

........................................................... 4.10 Modeling of Reinforced Material 70 ....................................... 4.1 1 Tension Cracks - übiquitous and Non-Ubiquitous 71

................... 4.12 Discrete Water Pressure Distribution Definition on Failwe Surface 72 .......................................... 4.13 Discussion of the Modifications and Extensions 73

C W T E R 5 . WNFICATION OF T m MODIFIED GENERALIZED WEDGE METHOD

5.1 General .......................................................................................... 74 5.2 Introduction to ACADS Exarnple Problems ............................................... 75

......................................................... . Acads 1 Example Problem 75 5.2.1.1 Discussion of the Results of Andysis for Acads 1 .a ................... 75 5.2.1.2 Discussion of the Results of Analysis for Acads 1 . b ................... 80 5.2.1.3 Discussion of the Results of Analysis for Acads 1.c ................... 83 5.2.1.4 Discussion of the Results of Analysis for Acads 1 . d .................... 87 Acads 2 . Example Problem .......................................................... 90 5.2.2.1 Discussion of the Results of Analysis for Acads 2.a ................... 92 5.2.2.2 Discussion of the Results of Analysis for Acads 2.b ................... 95

......................................................... . Acads 3 Example Problem 95 5.2.3.1 Discussion of the Results of Analysis for Acads 3.a .................... 96 5.2.3.2 Discussion of the Results of Analysis for Acads 3.b ................... 98 Acads 4 . Example Problem and Discussion of the Results of Analysis ...... 99

..... Acads 5 . Example Problem and Discussion of the Results of Analysis 102 Discussion of the Overall Eficiency and Validity of the GWM for the

........................................................................... ACADS Exarnpies 1 06

Discussion of the Overall Ef'ficiency and Validity of the Modified GWM for the ACADS

Eumples

CHAPTER 6 . APPLICATIONS OF THE METHOD

6.1 General .................. .... .................................................................. 107

...................................... ..................... 6.2 Chen and Shao's Example ... 108 .............................. ......................... 6.3 Yarnagami and Utea's Example .. 109

............................................................................. 6.4 Dartmouth Dam 110 ................................................. 6.5 Constrained Node Movement Examples 114 ............................................... 6.6 Discussion of the Validity of the Analysis 115

CHAPTER 7 . CONCLUSIONS

7.1 General ......................................................................................... 116 ................................................................................... 7.2 Conclusions 117

.............................................................. 7.3 Remarks on Further Research 117

..................................................................... APPENDlX A O Acads Results 123

vii

LIST OF TABLES

.......................................................... . Table 5.1 Acads l a Results user defined 76 ........................................................... Table 5.2 Acadsla Results - subdivision 78

.............................................. Table 5.3 Acads 1 a Results - progressive refinement 79 ................................................. Table 5.4 Acads 1 a Results - random trial surface 79

........................................................... Table 5.5 Acads I b Results - user defmed 81 ........................................................... Table 5.6 Acadsl b Results - subdivision 82

............................................. Table 5.7 Acads 1 b Results - progressive refinement 83 ................................................. Table 5.8 Acadslb Results - random trial surface 83

.......................................................... Table 5.9 Acads 1 c Results - user defined 84 .......................................................... Table 5.10 Acads l c Results - subdivision 86

............................................ Table 5.1 1 Acads 1 c Results - progressive refinement 86 ................................................ Table 5.12 Acadsl c Results . random trial surface 87

......................................................... Table 5.13 Acadsld Results - user defined 88 .................................. ...........*..*. Table 5.14 Acadsld Resuits - subdivision .. 89

............................................ Table 5.1 5 Acads l d Results . progressive refinement 90 ............................................. Table 5.16 Acads l d Results - random trial surface 9 0

...................................................... Table 5.17 Acads2a Results - user defined 92 .......................................................... Table 5.18 Acads2a Results - subdivision 94

................... ............... Table 5.19 AcadsZa Results - progressive refmement ... 9 4 .............................................. Table 5.20 Acads2a Results - random trial surface 94

......................................................... Table 5.2 1 Acads3a Results - user defined 96 ........................ .........*... Table 5.22 Acads3a Results - progressive refinement .. 9 7

viii

Table 5.23 Acads3a Results . random trial surface ............................................... 98 . ......................................................... Table 5.24 Acads4 Results user defmed 100 . ............................................ Table 5.25 Acads4 Results progressive refmement 101

Table 5.26 Acads4 Results . random trial surface ..........................,................. 102 ......................................................... Table 5.27 Acads5 Results . user defined 103

. Table 5.28 Acads5 Results subdivision ................... ,., .. ,., .............................. 105

. ....................... Table 5.29 Acads5 Results progressive refinement ... .................. 105

. ............................................... Table 5.30 AcadsS Results random trial surface 106 .................................................. Table 6.1 Dartmouth dam results - user defined 112

LIST OF FIGURES

Figure 3.1 Mohr-Coulomb criterion with the concept of the factor of safety ................... 13 Figure 3.2 A homogeneous dope with a phreatic surface subdivided in to 3 wedges ........ 13 Figure 3.3 Free body diagrams for the wedge system shown in Figure 2.2 .................... 14 Figure 3.4 Force equilibrium for a wedge in a heterogeneous media ........................... 16 Figure 3.5 Moment equilibrium on a typical wedge ............................................... 20 Figure 3.6 Calculation of average cohesion for a wedge face crossing three materiais ...... 21 Figure 3.7 Calculation of the average fiction angle for a face crossing three materials ..... 22 Figure 3.8 Water modeled by phreatic surface and ponded water ............................... 24 Figure 3.9 Pore water pressure defined by a piezometric line .................................... 25 Figure 3.10 Pore water pressure defined by pressure grid and a ponded water ce11 ........... 26 Figure 3.1 1 Integrated Boussinesq problem ........................................................ 27

............................................................. Figure 3.12 Integrated Cerruti problem 28 .................................................................... Figure 3.1 3 Stress transformation 29

Figure 3.14 Region of stress influence of a line load .............................................. 29 Figure 4.1 Problem geometry for root-finding analysis ........................................... 35 Figure 4.2 Factor of safety function for the probiem shown on Figure 4.1 ..................... 35 Figure 4.3 The fitted line w d in the new approach ............................................... 36 Figure 4.4 intercept of the fitted line 6 t h the horizontal axis .................................... 36 Figure 4.5 The method of fmding the new estimate for the root ................................. 37 Figure 4.6 Automatic generation of wedge system by bisecting the angles .................... 40 Figure 4.7 Crossing lines in the wedge generation process ....................................... 40 Figure 4.8 Degenetate polygon fonned due to interface passing through air ................... 41

Figure 4.9 Rules for generation of the three 3-wedge systems ................................... 42 Figure 4.10 Case 1 . Lefi wedge moves upward relative to the right one ....................... 44 Figure 4.1 1 Case 2 . Left wedge moves downward relative to the right one .................... 45 Figure 4.12 Al1 the possible relative movements between wedges .............................. 46 Figure 4.13 Case 1 for the relative movement ...................................................... 47 Figure 4.14 Case 2 for the relative movement ...................................................... 47 Figure 4.1 5 Case 3 for the relative movement ...................................................... 48 Figure 4.16 Case 4 for the relative movement ...................................................... 48 Figure 4.1 7 Test case for the investigation of the effect of the weak searn on the factor

...................................................................................... of safety 5 3 Figure 4.18 Variation of the factor of safety for the test case shown in Figure 4.17 .......... 54 Figure 4.19 Constrained node movement in the horizontal direction ............................ 56 Figure 4.20 Constrained node movement in the vertical direction .............................. 56 Figure 4.2 1 Constrained node movement along a defmed direction ............................ 57 Figure 4.22 Horizontal degree of fieedom of a node ............................................... 62

................................................. Figure 4.23 Vertical degree of freedom of a node 62 ........................................... Figure 4.24 Angular degree of freedom of an interface 62

................................. Figure 4.25 Degrees of freedom for the optimization process ... 63 Figure 4.26 Chen's random failuse surface generation scheme .................................. 68

............................................. Figure 4.27 Application of a reinforced material ce11 70 Figure 4.28 Concept of' the ubiquitous tension crack zone ........................................ 71 Figure 4.29 Concept of the non-ubiquitous tension crack zone .................................. 72

..................................................... Figure 4.30 Discrete water pressure definition 73 Figure 5.1 Acads l a problem geometry and material properties ............................... 76 Figure 5.2 Acads 1 a problem - plot of optimized failure surfaces ............................... 77 Figure 5.3 Acads la problem - automatic selection of the best 3-wedge system .............. 78 Figure 5.4 Acadslb problem geometry and material properties .................................. 80 Figure 5.5 Acads 1 b problem - plot of optimized failure surfaces ................................ 81 Figure 5.6 Acadsl b problem - automatic selection of the best 3-wedge system .............. 82 Figure 5.7 Acads 1 c problem geometry and matetial properties ....................... ... ..... 84 Figure 5.8 Acadslc problem - plot of optimized failure surfaces ............................... 85

Figure 5.9 Acads lc problem . automatic selection of the best 3-wedge system .............. 86 Figure 5.10 Acads ld problem . plot of optimized failure surfaces .............................. 88 Figure 5.1 1 Acads ld problem . automatic selection of the best 3-wedge system ............. 89 Figure 5.12 Acads2 problem geometry and material properties ................................. 91 Figure 5.13 Acads2a problem . plot of optimized failure surfaces .............................. 93 Figure 5.14 Acads2a problem . automatic selection of the best 3-wedge system ............. 93 Figure 5.15 Acads3 problem geometry and material properties ................................. 95 Figure 5.16 Acads3a problern . plot of optimized failure surfaces .............................. 97 Figure 5.1 7 Acads3a problem . automatic selection of the best 3-wedge system ............. 98 Figure 5.18 Acads4 problem geometry and material properties ................................. 99

.............................. Figure 5.19 Acads4 problem - plot of optimized failure surfaces 100 Figure 5.20 Acads4 problem - automatic selection of the best 3-wedge system ............. 101 Figure 5.2 1 AcadsS problem geometry and material properties ................................ 103 Figure 5.22 Acads5 problem - plot of optimized failure surfaces .............................. 104 Figure 5.23 AcadsS problem - automatic selection of the best 3-wedge system ............. 105

......................... ................*..*.*.............. Figure 6.1 Chen & Shao's problem .. 108 Figure 6.2 Solution to Yamagarni and Utea's example . subdivision .......................... 109

........... Figure 6.3 Solution to Yamagami and Utea's exarnple . progressive refinement 110 ......................... Figure 6.4 Geometry and material properties for the Dartmouth dam 111

............................ Figure 6.5 Dartmouth dam . user defined 6-wedge failure system 112

............................ Figure 6.6 Dartmouth dam . user defined 7-wcdge failure system 113 Figure 6.7 Dartmouth dam - user defmed 8-wedge failure system ............................ 113

................................ Figure 6.8 Acads3a example with constrained node movement 114 ............................... Figure 6.9 Acads4 example with constrained node movement 115

Figure A . 1 Acads la user defied 3-wedge system with pattern search optimization ....... 124 ....... Figure A.2 Acads 1 a user defined 4-wedge system with pattern search optirnization 124

Figure A.3 Acads 1 a user defined 5-wedge system with pattern search optimization ....... 125 Figure A.4 Acads 1 a user defined 6-wedge system with pattern search optimization ....... 125

....... Figure A S Acads 1 a user defmed Fwedge system with pattern search optimization 126

....... Figure A.6 Acads 1 a user defmed 8-wedge system with pattern search optimization 126

....... Figure A.7 Acads 1 a user defmed 9-wedge system with pattern search optirnization 127

xii

Figure A.8 Acadsla Cwedge system with pattem search optimization,

subdivision scheme .......................................................................... 127 Figure A.9 Acads l a 5-wedge system with pattem search optimization,

subdivision scheme ......................................................................... 128 Figure A. 10 Acads la 6-wedge system with pattern search optimization,

.......................................................................... subdivision scheme 128 Figure A. 1 1 Acadsla 7-wedge system with pattern search optimization,

subdivision scheme .......................................................................... 129 Figure A. 12 Acads 1 a 8-wedge system with pattem search optimization,

subdivision scheme .......................................................................... 129 Figure A. 1 3 Acads 1 a 9-wedge system with pattem search optimization.

subdivision scheme .......................................................................... 1 30 Figure AS4 Acadsl a 4-wedge system with pattern search optirnization, progressive

refinernent scheme ......................................................................... 130 Figure A. 15 Acadsl a 5-wedge system with pattern search optimization, progressive

refinement scheme ........................................................................... 13 1 Figure A. 16 Acads 1 a 6-wedge system with pattern search optimization, progressive

........................................................................... refinement scheme 1 3 I Figure A. 17 Acadsl a 7-wedge system with pattem search optimization, progressive

refinement scheme ........................................................................... 1 33 Figure A. 1 8 Acads 1 a 8-wedge system with pattern search optimization, progressive

re finement scheme ........................................................................... 1 32 Figure A. 19 Acads 1 a 9-wedge system with pattem search optimization, progressive

........................................................................... refinement scherne 1 3 3 Figure A.20 Acadsla 3-wedge system with pattern search optimization, Chen's random

.......................................................................... surface generation 133 Figure A.2 1 Acads l a 4-wedge system with pattern search optimization, Chen's random

......................................................................... surface generation 134 Figure A.22 Acadsl a 5-wedge system with pattern search optimization, Chen's random

............................... ..................*.........*........... surface generation ... 134 Figure A.23 Acadsl b user defmed 3-wedge system with pattern search optimization ...... 135

Figure A.24 Acads 1 b user defined 4-wedge system with pattern search optimization ...... 1 35 Figure A.25 Acadsl b user defmed 5-wedge system with pattern search optimization ...... 136 Figure A.26 Acads 1 b user defined 6-wedge system with pattern search optimization ...... 136 Figure A.27 Acadsl b user defined 7-wedge system with pattern search optimization ...... 137 Figure A.28 Acadsl b user defined 8-wedge system with pattern search optimization ...... 137 Figure A.29 Acads l b user defined 9-wedge system with pattern search optimization ...... 138 Figure A.30 Acads 1 b 4-wedge system with pattern search optimization.

subdivision scheme .......................................................................... 138 Figure A.3 1 Acads 1 b 5-wedge system with pattem search optimization.

subdivision scheme .......................................................................... 139 Figure A.32 Acads 1 b 6-wedge system with pattem search optimization.

subdivision scheme .................... ,, ............................................... 139 Figure A.33 Acadsl b 4-wedge system with pattern search optimization. progressive

refinement scheme ......................................................................... 140 Figure A.34 Acads 1 b 3-wedge system with pattem search optimization. Chen's random

surface generation ........................................................................ 140 Figure A.35 Acads 1 b 4-wedge system with pattern search optimization. Chen's random

surface generation ........................................................................ 141 Figure A.36 Acads 1 b 5-wedge system with pattem search optimization. Chen's random

surface generation .......................................................................... 141 ...... Figure A.37 Acadslc user defmed 3-wedge system with pattern search optimization 142 ...... Figure A.38 Acads lc user defined 4-wedge system with pattern search optimization 142 ...... Figure A.39 Acads 1 c user defmed 5-wedge system with pattern search optimization 143

Figure A.40 Acads lc user defined 6-wedge system with pattern search optimization ...... 143 ...... Figure A.41 Acads lc user defmed 7-wedge system with pattern search optirnization 144 ...... Figure A.42 Acads 1 c user defmed 8-wedge system with pattern search optimization 144

Figure A.43 Acads 1 c Cwedge system with pattern search optimimtion.

.................................................. subdivision scheme ................... .. 145 Figure A.44 Acadslc 5-wedge system with pattern search optimization.

............................................ .................... subdivision scheme .... 145

Figure A.45 Acads lc 6-wedge system with pattern search optimization.

.......................................................................... subdivision scheme 146 Figure A.46 Acads lc 7-wedge system with pattem search optimization.

............................................................... subdivision scheme 146 Figure A.47 Acads lc 8-wedge system with pattern search optimization,

subdivision scheme ......................................................................... 147 Figure A.48 Acads 1 c 9-wedge system with pattern search optimization.

.......................................................................... subdivision scheme 147 Figure A.49 Acads 1 c 4-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 148 Figure AS0 Acadslc 5-wedge system with pattem search optimization. progressive

........................................................................... refinement scheme 148 Figure A S 1 Acads lc 6-wedge system with pattem search optimization. progressive

........................................................................... refinement SC heme 149 Figure AS2 Acads lc 7-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 149 Figure AS3 Acads lc 8-wedge system with pattern search optimization. progressive

........................................................................... refinernent scheme 150 Figure AS4 Acadsl c 9-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 150 Figure A.55 Acadslc 3-wedge system with pattem search optimization. Chen's random

..............................*.. ...................*.......*....... surface genention .. 151 Figure AS6 Acads lc Cwedge systern with pattem search optirnization. Chen's random

........................................................................... surface generation 151 Figure AS7 Acads 1 c 5-wedge system with pattem search optirnization. Chen's random

........................................................................... surface generation 152 Figure A.58 Acads ld user defmed 3-wedge system with pattern search optimization ...... 152 Figure A.59 Acadsld user defined 4-wedge system with pattern search optimization ...... 153 Figure A.60 Acadsld user defmed 5-wedge system with pattern search optimization ...... 153 Figure A.61 Acads Id user defined 6-wedge system with pattern search optimization ...... 154 Figure A.62 Acads 1 d user defined 7-wedge system with pattern search optimization ...... 154

Figure A.63 Acadsld user defined 8-wedge system with pattern search optirnization ...... 155 Figure A.64 Acads 1d 4-wedge system with pattem search optimization.

.......................................................................... subdivision scheme 155 Figure A.65 Acadsld 5-wedge system with pattem search optimization.

.......................................................................... subdivision scheme 156 Figure A.66 Acads ld 6-wedge system with pattern search optimization.

.......................................................................... subdivision scheme 156 Figure A.67 Acads 1 d Fwedge system with pattern search optimization.

.......................................................................... subdivision scherne 157 Figure A.68 Acads 1 d 4-wedge system with pattern search optimization. progressive

........................................................................... refinement SC heme 157 Figure A.69 Acads 1 d 5-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 158 Figure A.70 Acadsld 6-wedge system with pattem search optimization. progressive

........................................................................... refinement scheme 158 Figure A.7 1 Acads Id 7-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 159 Figure A.72 Acads 1 d 8-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 159 Figure A.73 Acads l d Qwedge system with pattern search optimization. progressive

........................................................................... refinement scheme 160 Figure A.74 Acadsld 3-wedge system with pattem search optimization. Chen's random

........................................................................... surface generation 160 Figure A.75 Acads 1 d 4-wedge system with pattem search optimization. Chen's random

............................................. ....................... surface generation ... 161 Figure A.76 Acadsld 5-wedge system with pattern search optimization. Chen's random

......................................................................... surface generation 161 ......................................................... Figure A.77 AcadDa material properties 162

...... Figure A.78 Acads2a user defined 3-wedge system with pattern search optimization 162

...... Figure A.79 Acads2a user defmed 4-wedge system with pattern search optimization 163

Figure A.80 Acads2a user defined 5-wedge system with pattern search optimization ...... 163 Figure A.81 Acads2a user defmed 6-wedge system with pattern search optimization ...... 164 Figure A.82 AcadsZa user defmed 7-wedge system with pattern search optimization ...... 164 Figure A.83 Acads2a user defined 8-wedge system with pattern search optimization ...... 165 Figure A.84 AcadsZa user defined 9-wedge system with pattern search optimization ....... 165 Figure A.85 Acads2a Cwedge system with pattern search optirnization.

........................................................................... subdivision scheme 166 Figure A.86 AcadsZa 5-wedge system with pattem search optimization.

.......................................................................... subdivision scheme 166 Figure A.87 Acads2a 6-wedge system with pattern search optimization.

.......................................................................... subdivision scheme 167 Figure A.88 Acads2a Fwedge system with pattern search optimization.

.......................................................................... subdivision scheme 167 Figure A.89 Acads2a 8-wedge system with pattern search optimization.

.......................................................................... subdivision scheme 168 Figure A.90 Acads2a 9-wedge system with pattern search optimization.

.......................................................................... subdivision scheme 168 Figure A.9 1 AcadsZa Cwedge system with pattem search optimization. progressive

........................................................................... re finement scheme 169 Figure A.92 Acads2a 5-wedge system with pattern search optimization. progressive

................................. ........*..*.....*....*.....*.*......*. refinement scheme .. 169 Figure A.93 Acads2a 6-wedge system with pattem search optimization. progressive

........................................................................... refinement scheme 170 Figure A.94 Acads2a 7-wedge system with pattem search optimization. progressive

................................. ....*.**......*..*............*...... refinement scheme ... 170 Figure A.95 Acads2a 8-wedge system with pattern search optimization. progressive

......................................................................... refinement scheme 171 Figure A.96 Acads2a 9-wedge system with pattem search optimization. progressive

.......................................................................... refinement scheme 171 Figure A.97 Acads2a 3-wedge system with pattern search optimization. Chen's random

.................................................... ................... surface generation .. 172

Figure A.98 Acads2a 4-wedge system with pattem search optimization. Chen's random

............................................................................ surface generation 172 Figwe A.99 Acads2a 5-wedge system with pattem search optimization. Chen's random

........................................................................... surface generation 173 Figure A . 100 AcadsZb 6-wedge system with pattern search optimization .................... 173 Figure A . 10 1 Acads3a user defined 3-wedge system with pattern search optimization .... 174 Figure A . 102 Acads3a user defined 4-wedge system with pattern search optimization .... 174 Figure A . 103 Acads3a user defined 5-wedge system with pattern search optimization .... 175 Figure A . 104 Acads3a user defined 6-wedge system with pattern search optimization .... 175 Figure A . 1 OS Acads3a user defined 7-wedge system with pattern search optimiziition .... 176 Figure A . 106 Acads3a user defined 8-wedge system with pattern search optimizatiotion .... 176 Figure A . 107 Acads3a user defined 9-wedge system with pattern search optimization .... 177 Figure A . 108 Acads3a 4-wedge system with pattern search optirnization.

............................................... ...................... subdivision scheme .. 177 Figure A . 109 Acads3a 4-wedge system with pattern search optirnization. progressive

........................................................................... refinement scheme 178 Figure A . 1 10 Acads3a 5-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 178 Figure A.111 Acads3a 6-wedge system with pattern search optimization. progressive

........................................................................... refinement SC heme 179 Figure A . 1 12 Acads3a 7-wedge system with pattern search optimization. progressive

........................................................................... refinement scheme 179 Figure A . 1 13 Acads3a 8-wedge system with pattem search optimization. progressive

........................................ .......................... refinement scheme .... 180 Figure A . 1 14 Acads3a 9-wedge system with pattem search optimization. progressive

........................................................................... refmement scheme 180 F i g w A . 1 15 Acads3a 3-wedge system with pattern search optimization. Chen's random

...................................................... ................... surtace generation ... 181 Figure A.116 Acads3a 4-wedge system with pattem search optimization. Chen's random

........................................................................... surface generation 181

Figure A . 1 17 Acads3a 5-wedge system with pattem search optirnization. Chen's random surface generation ............................................................................ 182

Figure A . 1 18 Acads3 b user defmed 3-wedge system with pattern search optimization .... 182 Figure A . 1 19 Acads4 user defmed 3-wedge system with pattern search optimization ...... 183 Figure A . 120 Acads4 user defined Cwedge system with pattern search optimization ...... 183 Figure A . 12 1 Acads4 user defined 5-wedge system with pattern search optimization ...... 184 Figure A . 122 Acads4 user defined 6-wedge system with pattern search optimization ...... 184 Figure A . 123 Acads4 user defined 7-wedge system with pattern search optimization ...... 185 Figure A . 124 Acads4 user defined 8-wedge system with pattern search optimization ...... 185 Figure A . 125 Acads4 user defined 9-wedge system with pattern search optimization ...... 186 Figure A . 126 Acads4 4-wedge system with pattern search optimization.

subdivision scheme ...................................................................... 186 Figure A . 127 Acads4 4-wedge system with pattem search optimization. progressive

refinement scheme ........................................................................ 187 Figure A . 128 Acads4 5-wedge system with pattern search optimization. progressive

refinement scheme .......................................................................... 187 Figure A . 129 Acads4 6-wedge system with pattern search optimization. progressive

re finement scheme ........................................................................... 188 Figure A . 130 Acads4 3-wedge system with pattern search optimization. Chen's random

............................................................................ surface generation 188 Figure A . 13 1 Acads4 4-wedge system with pattern search optimization. Chen's random

surface generation ............................................................................ 189 Figure A.132 Acads4 5-wedge system with pattem search optimization. Chen's random

surface generation ........................................................................... 189 ............................. Figure A.133 AcadsS problem geometry with water pressure grid 190

...... Figure A . 134 AcadsS user defmed 3-wedge system with pattern search optimization 190

...... Figure A . 135 AcadsS user defmed 4-wedge system with pattern search optimization 191

...... Figure A . 136 AcadsS user defmed bwedge system with pattern search optimization 191

...... Figure A . 137 AcadsS user defmed dwedge system with pattern search optirnization 192

...... . Figure A 138 AcadsS user defmed 7-wedge system with pattern search optimizittion 192

...... Figure A . 139 AcadsS user dehed 8-wedge system with pattern search optimization 193

Figure A. 140 AcadsS user defined 9-wedge system with pattem search optimization .... . . 193 Figure A. 141 AcadsS Cwedge system with pattem search optimization,

subdivision scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Figure A. 142 Acads5 5-wedge system with pattern search optimization,

subdivision scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . 194 Figure A. 143 AcadsS 6-wedge system with pattern search optimization,

subdivision scheme . . . . . . . . . . , . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . , . . . . . . . 195 Figure A.144 Acads5 7-wedge system with pattem search optimization,

subdivision scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Figure A. 145 Acads5 8-wedge system with pattem search optimization,

subdivision scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *. . . . . . . . . . . .. . . . . . . . . . . . . 196 Figure A.146 Acads5 9-wedge system with pattem search optimization,

subdivision scheme . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Figure A. 147 AcadsS Cwedge system with pattem search optimization, progressive

refinement scheme . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Figure A.148 AcadsS 5-wedge system with pattern search optirnization, progressive

refinement scheme . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . 197 Figure A. 149 Acads5 6-wedge system with pattern search optimization, progressive

refinement scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , . . . . . . . . . . . . . . . . . . . . . 198 Figure A. 150 AcadsS 7-wedge system with pattem search optimization, progressive

re finement scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Figure A. 15 1 AcadsS 8-wedge system with pattern search optimization, progressive

refinement scheme .... .. ..... ....... ... .. . ...... ...... . .. . . . . . . ........ . . 199 Figure A. 152 AcadsS Fwedge system with pattern search optimization, progressive

ce finement scheme . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199 Figure A.153 AcadsS 3-wedge system with pattem search optirnization, Chen's random

surface generation . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Figure A. 154 Acads5 Cwedge system with pattern search optimization, Chen's random

surface generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Figure A. 155 AcadsS 5-wedge system with pattem search optimization, Chen's random

surface generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .20 1

CHAPTER 1 - INTRODUCTION

The failure of a slope, either natural or artificial was documented since the ancient times of

the human history. However, due to the lack of scientific knowledge of the people of those

times, the basic pnnciples of a slope failure were not understood. Only the past two centuries

brought understanding of the principles goveming the failure of a slope. Even though the

advances made on several scieniific fronts, the area of dope stability analysis is an ongoing

research topic.

The stability of a slope has a considerable effect on the swounding area of the slope,

because very ofien, human lives are in danger or significant material darnage results if a

siope is unstable or fails. Thus the slope stability analysis is one of the most important areas

of practical geotechnical and rock dope engineering.

This thesis reviews the current state of slope stability methods and concentrating on a

particular method, examines its strengths and weaknesses and develops modifications and

extensions to the method, while focusing on the method's implementation and its application

to rock slope stability analysis through practical examples drawn fiom various sources.

1.2 Scope of the Thesis

The prirnary focus of the thesis is to investigate the suitability of the Genenlized Wedge

Method (Giam, 1989) to rock slope stability analysis applications, while examining the

theoretical background of the Generalized Wedge Method and modifiing the method in order

to suite the issues commonly encountered in rock dopes. There are a few modifications made

to the method itself, such as improving the calculation of the factor of safety for a single

failure surface, generation of the wedges from a defined failure surface and a soi1 profile line,

automatic generation of a set of failure surfaces to capture the global minimum factor of

safety with the corresponding critical failure surface and modifications to the governing

equations to address the correct implernentation of the kinematic admissibility criterion.

The modified method is implemented in a form of a cornputer prograrn and ngorously

tested for accuracy and validity on examples drawn from various literature reported sources

to the cornplete ACADS dope stability program survey examples totaling over 150 examples

discussed in the thesis and show in the Appendix.

1.3 The Modifications to the Original Method

The Generalized Wedge Method developed by P. S. K. Giam (Giam, 1989) is a non-circular

limit equilibriurn method of siope stability analysis that uses the hlly mobilized strength on

al1 of the faces including the slip surface and the inter-wedge faces.

The solution of the factor of safety of a single failure surface mechanism without

optirnization is of iterative nature. Starting with an assumed factor of safety, the method

calculates a new, improved factor of safety at the end of the calculation cycle. Howeuer, as

discussed later in the thesis, this process is complicated due to the highly non-linear and

discontinuous nature of the factor of safety function, thus a new method is presented to

decrease the computational tirne and steps required for finding the solution.

The definition of a wedge mechanism, particularly if the number of wedges is large,

can be a tedious process. Thus a method was developed to address this issue by simplimng

the creation method ushg only the slope profile and the failure surface.

The Generalized Wedge Method uses force equilibrium to calculate the factor of

safety. In the initial work by Giam (Giam, 1989) only one relative movement was considered

between two adjacent wedges. Subsequently (Zhao, 1995) it was considered that two possible

relative movements between the two adjacent wedges are possible. This thesis M e r

examines the mechanism of the movernents between two adjacent wedges and defines a

movement-history-based approach to identify, track and apply the correct forces on each

wedge.

Other minor modifications such as ubiquitous and non-ubiquitous tension cracks,

reinforced material and constrained node movement are added to the mode1 to make it more

versatile and applicable to rock dope stability anaiysis.

CHAPTER 2

REVIEW OF THE CURRECT STATE OF SLOPE STABILITY

ANALYSIS

2.1 General

The methods available to solve for the factor of safety of a given slope are classified into

categories based upon the underlying theory. This thesis highlights the most commonly

encountered approaches and describes the merits and shortcomings of each method.

The most commonly used methods are grouped under lirnit equilibrium methods, in

which the failure surface is treated that it is at its limit of equilibrium, thus the failure is

imminent. If the stress-strain history of the slope is of importance to the analyst, fmite

element or fuite difference methods should be employed in the anafysis. These methods are

significant if progressive failure is anticipated.

Using the principle of virtuai work and the conservation of energy, methods can be

formuiated to solve for the stability of the dope.

Al1 of the above methods deal with a single failure surface that is fixed in the spatial

domain. However, a number of these failure surfaces must be investigated in order to fmd the

one, which yields the minimum factor of safety. Various mathematical procedures are

applied to aid the search, ranging from simple empincal rules to complex mathematical

formulations.

2.2 Limit Equilibrium Methods

The earliest investigatoa and researchers used the concept of equilibrium of a body to solve

for the measure of stability of a slope or structure.

It is said that the condition of limit equilibrium is reached when the resisting and

driving forces or moments are equd, resulting in a state when the failure is imminent.

The limit equilibrium methods can be divided into a number of categories, starting

fiom the fûnction of the factor of safety being linear or non-linear, to the shape of the fidure

surface being circulat or non-circular and to the position of the inter-slice faces being vertical

or inclined and ending with whether the method satisfies force equilibrium or moment

equilibrium or both. This thesis categorizes the methods based on the shape of the failure

surface, thus the division is made to classify methods as circular or non-circular, putting

more emphasis on the latter, since the non-circular methods are more suitable to problems

encountered in rock engineering. It must be noted that some methods are general enough to

be capable of solving problems involving both circular and non-circular failure surfaces.

2.2.1 Circular Failure Surface Methods

The assurnption of a circular failure surface at first sight seerns rather simplistic. However,

many of the documented slope failws in soils, especially in homogeneous materials, are

circular in nature.

Arnong the earliest techniques developed are the method used by Collin in 1846

where the failure surfaces are based on actual observations, but it was not until 1916, when

Petterson formulated the earliest forrn of the Swedish Circular Method of Slices, that the

actual concept of failure was understood. The W e r developrnent of the Swedish method by

Fellenius in 1928 prompted the first circular method that used moment equilibriurn of the

whole failure surface to express the factor of safety in the terms of resisting moments over

driving moments. Fellenius' factor of safety function was linear, thus relatively easy to solve

for. The resulting factor of safety was conservative, underestimating the tme factor of safety;

thus the error was on the safe side.

The first circular method of analysis, that has a well-defïned foundation, is attributed

to Bishop (Bishop, 1955). He proposed a rigorous method that satisfies both force and

moment equilibriurn by t a h g moments at the center of the circle. The solution for the factor

of safety was obtained by force equilibrium on each vertical slice and by the overall moment

equilibrium. In Bishop's Simplified Method an assumption was made that the difference in

the vertical inter-slice forces is zero. With this assumption, the factor of safety is eiisier to

determine. Bishop was the first to introduce the factor of safety concept based on the ratio of

available shear strength to that required to reach the limit equilibrium. The analysis satisfies

overall moment equilibrium but not for each slice. Bishop's Simplified Method of Slices is

the most widely used in the geotechnical field due to its simplicity while maintaining

sufficient accuracy for practical purposes, where the failure surface is circular.

The assumption of zero inter-slice forces is too restrictive for some purposes. Spencer

in 1967 and consequently in 1973 developed a rnethod in which the resultants of the inter-

slice forces are inclined at an angle 8 that is constant throughout the slope. For a circular slip

surface taking moments about the center of rotation, the goveming equations cm be defined

for overall moment and force equilibrium. Using a set of values for the inclination angle 8,

the factor of safety is calculated both €rom the force and moment equilibriurn equations

resulting in two curves. The intersection of the two denotes the factor of safety where both

force and moment equilibriurn is satisfied.

The following methods are applicable to both circular and non-circular failure

surfaces; thus here on1 y the issues applicable to circular failure surfaces are hi ghlighted.

In the rigorous method by Janbu, the factor of safety caiculations are developed fkom

force equilibrium for each slice with an assumption on the location of the horizontal force on

the side of each slice. The solution is iterative and complicated for practical use. It is Janbu's

Simplified Method that is widely used by the profession. It must be noted that Janbu's

Simplified Method satisfies force equilibrium on each slice and moment equilibrium on the

whole failure surface.

Other methods that have applications to circular failure surfaces are Morgenstem &

Pnce, Sarma, Fredlund & Krahn. However, these methods achieve the circular shape by

definition of the failure surface to be circular and are only a simplified subset of the methods'

for non-circular failure surfaces discussed in the following section.

2.2.2 Non-Circular Failure Surface Methods

The non-circular methods have an inherent property that they cm assume a roughly circular

approximation if the optimized critical failure surface is close to a circle. The non-circular

methods are very usehl in situations where the rock mass contains weak searns that have

significantly lower strength parameters than the surrounding materials or an interface

between a hard stratum such as bedrock and the deposited soils on the top.

Spencer's method is capable of handling non-circular slip surfaces. It requires that the

derivation of the goveming equations be based on the moment taken around the midpoint of

each slice. As previously described, a set of inter-slice inclination angles is evaluated and the

factor of safety is calculated based on force and moment equilibrium. Again the intersection

of the two curves obtained is the m e factor of safety.

Morgenstem and Pnce (Morgenstem & Price, 1965) developed a method based on

force and moment equilibrium for each slice with an assumption that the inter-slice forces'

inclination takes a fùnctional value. From constant value to skusoidal to parabolic and

trapezoidal functional relationships exist for this rnethod.

The method developed by Sama uses a very different approach to solve for the factor

of safety (Sarma, 1979). It is the first method noted in this review that cm handle non-

vertical slices and the strength of the material on the inter-slice faces is included in the

analysis. The basis of the method is the application of a uniforrn horizontal acceleration

factor to the slope. This factor is a fraction of the weight of each slice that produces a body

force in each wedge. The critical acceleration factor that causes the slope to be unstable c m

be calculated teadily. Once, if this acceleration factor is reduced to zero, the static factor of

safety can be easily calculated.

The Generalized Wedge Method (Donald & Giarn, 1989) that is quite similar to

Sarma's method, but it does not use the horizontal acceleration factor to obtain the factor of

safety. Rather the factor of safety is obtained by force equilibrium on each wedge through an

iterative approach, the moment equilibrium is also checked.

2.3 Energy Methods

Analyses using the principle of virtual work and energy considerations where static

equilibrium of the material enclosed by the failure surface were developed. These methods

use the conservation of energy taking into consideration the kinematics of the failing mass.

There is a well-established theoretical background for these methods and their proper usage

results in a ûue upper bound solution. With the inclusion of kinematic considerations a non-

circular failure surface cannot form a tme failure mechanism without some form of interna1

mobilized failure surface.

The Energy Method Upper (EMU) (Giam & Donald, 1989) is one such method based

on the upper bound theorem of classical plasticity. It is a multi-wedge failure mechanism

where the energy dissipation dong the slip surface and the interfaces is included in the

analysis. Consequently (Donald & Chen, 1995) the method was modified to accommodate

curved slip surfaces and the relative movement between two adjacent wedges.

Kara1 developed other energy methods, in 1977, where the stability analysis is based

on energy balance via the theorems of plasticity. The factor of safety is being defined as the

ratio of total energy dissipation to the total work done by extemal forces.

2 a 4 Finite EIement and Finite Difference Metbods

The application of the limit equilibrium methods gives an insight of the stability of the dope

at the state of failure and gives no information about the stress-strain history of the slope

prior and after failure has occurred. The limit equilibrium methods generally do not satisfy

the stress equilibrium at any given point in the slope at any given time, thus the methods are

inappropriate to mode1 progressive failure mechanisms. Finite element and finite difference

methods cm mode1 the deformation of the dope and the stress caused by the deformations

throughout the faiiure. There are some computer programs based on these methods that cm

solve such problems, however these methods still require an interpretation of the results of

analysis, and it have not been widely used for general dope stability analysis. However, with

the advance of computer technology and interactive visualization of the results of such

analyses, the methods have a place among the general methods used in dope stability

analysis.

2 5 Optimization Techniques for Search of tbe Critical Failure Surface

Every method discussed in the previous sections in this review solves for a factor of safety

for a given failure surface. The general aim of the slope stability analysis is to find minimum

factor of sakty and the conesponding critical failure surface. Therefore, a number of failure

surfaces must be analyzed in order to find the minimum. This process c m be a considerable

task for some rnethods.

We start with the simplest case for both circular and non-circular failure sutfaces,

where the exact location of the failure surface is known; thus no optimization is required.

Increasing in dificulty are the circular methods, where the unknowns in the

optimization are the center and the radius of the critical circle. A nurnber of mathematical

and quasi-mathematical methods exist, such as the Grid Search, Tangent Search, Line Search

and Star Search to more complex mathematical, such as the Simplex Method (Nguyen,

1985), Pattern Search (Swann, 1972), and Modined Alternathg Variable (Li & White, 1987).

For more complex failure mechanisms encountered in non-circular methods, where the

number of unknowns is more than three, the following mathematical methods gained

acceptance in the geotechnical profession: Pattern Search (Hooke & Jeeves, 1961), Powell's

Method (Powell, 1964), Simplex (Nedler & Mead, 1964)' Rosenbrock's Method

(Rosenbrock, 1960) and the Random Trial Method (Chen, 1992).

The aim of al1 these methods is to minimize the factor of safety function that has n

variables in a n + l dimensional space. The goal of al1 of these is to fmd the minimum of the

function and thus the variables representing the critical failure surface. Some of the above

methods are suitable for a certain type of problem or work well in a range of variables while

others can successfully solve for al1 nurnber of variables.

This thesis considers only a few of these methods that are proven to be successfÙ1 to

address the solution of a system with a large nurnber of unknowns as discussed in C hapter 4.

CHAPTER 3 - THE GENERALIZED WEDGE METHOD

3.1 General

The purpose of this chapter is to introduce the theoretical foundation of the Generdized

Wedge Method, the basis of the method and the solution procedure dong with the inherent

capabilities of the method as presented by the method's original author, Dr. P. Giam (Giarn

1989), in his doctoral thesis. This chapter slightly touches on the equivalence of the

Generalized Wedge Method with the Energy Method Upper Bound to prove the upper-bound

nature of both methods' results. Finally, a section at the end of this chapter summarizes the

deficiencies of the original Generalized Wedge Method by Giam and lays the foundation for

the modifications and extensions presented by in Chapter 4.

3.2 Theoretical Foundation of the Generalized Wedge Method

The Generalized Wedge Method is a simple and complete limit equilibrium method that uses

multiple wedges, which c m have non-vertical interfaces like Sarnia's method. In the

Generalized Wedge Method both force and moment equilibrium are satisfied dong with

kinematic conditions, because fully mobilized strengths are used both on the failure surface

and the inter-wedge faces. Moreover, the kinematic admissibility is achieved since neither

separation nor overlap of the wedges is allowed. Though the initial wedge-based methods

were intended to address the solution of a problem with long linear segments in the failure

surface, the Generalized Wedge Method cm solve for any type of failure surface. The

constitutive mode1 on which the Generalized Wedge Method is based is Mohr-Coulomb,

which is adequate for most soi1 dope stability analyses and various types of rock slope

stability analyses.

3.2.1 Basis of the Method

Since the original definition by Bishop, the well-accepted factor of safety is defined as the

amount by which the available shear strength on the failure surface must be reduced to bring

the dope into a state of limiting equilibrium.

Using this definition, the shear stress on a failure surface for a Mohr-Coulomb

material can be expressed as shown on Figure 3.1 or in ternis of equations (Das, 1990) using

the effective stress concept:

1 t = -(c'+a' tan)')

F

s = cllv +o'- tan l',.

where

Figure 3.1 Mohr-Coulomb criterion with the concept of the factor of safety

Consider a siope made up of a homogeneous material with a phreatic surface, discretized into

3 wedges, as shown on Figure 3.2. The free body diagrams for individual wedges and the

forces acting on them are show in Figure 3.3. It is evident from Figure 3.3 that the

mobilized shear and fnctional strength is used in the definitions of the forces acting on each

wedge.

Wedge #3

Intemal Face

Figure 3.2 A hornogeneous slope with a phreatic surface subdivided hto 3 wedges

Sign Convention for Angles

Figure 3.3 Free body diagrarns for the wedge system shown in Figure 3.2

The individual symbols on the Figure 3.3 above are defned as:

where

i = side numbenng

1 = Iength of side

X i = mobilued cohesive component of shear strength at intemal interfaces

S i = mobilized cohesive component of shear strength at outer failure surface enclosing the

slipping mass

tan i dcv = mobilized coefficient of fiction

Qi and Ri = resultant of normal effective stress and mobilized fnctional component of shear

If the number of wedges is n, then the number of sides in an n wedge system is:

no. sides = 2.n-1

In the conventional wedge methods, the forces are plotted on a force polygon chart and if the

last point coincides with the first, force equilibriurn is achieved. The presented figures so f i

assumed a 3-wedge failure mechanism, however the next sections show the extension of the

method to an arbitrary number of wedges.

3.2.2 Force Equilibrium

The primary method of solution for the Generalized Wedge Method is force equilibriurn.

Considering a typical wedge and the forces acting on it as shown on Figure 3.4, the solution

proceeds fiom the first wedge, which has no side i to the last wedge, which has no side k. The

easiest way to present the equations for force equilibnum is to use matrix notation for the

solution of the unknowns.

By resolving the forces in the horizontal and vertical direction for each wedge:

cos A -sinB Q, [si. A - cor B ] { R k } = {zi}

Forces acting on a typical wedge Figure 3.4 Force equilibrium for a wedge in a heterogeneous media

The variables in the general matrix fot-m of force equilibrium in Equation 3.7 are defined as

follows:

Ci = W m + X k ~ ~ ~ 6 k - U k ~ i n 6 k - s j ~ i n a j - U j ~ ~ ~ a j

- X i ~ ~ ~ 6 i + U i ~ i n S i - R i ~ i n ( ~ i d e v - 6 i ) + ( other extemal vertical forces )

C 2 = X k ~ i n 6 k + U k ~ ~ ~ 6 k S j ~ ~ ~ a j + U j ~ i n a j

- X i ~ i ~ 6 i - U i ~ ~ ~ S i - R i ~ ~ ~ ( + i d e v - 6 i ) + ( other extemal horizontal forces )

Solving for Q j and R fiom the ma& equation we obtain:

The above equations h m 3.7 through 3.9 are valid for the lSt wedge to the (n-l)'h. However

wedge n has no side k, thus the factor of safety is obtained from the cohesional component S

as foltows:

cos A sinB Q, [sinA c o s B ] { s , } = { ~ : }

w here

C i = W m - U j ~ ~ ~ a j - X i ~ ~ ~ 6 i + U i ~ i n 6 i - R i ~ i n ( 4 i d e v - 6 i )

+ ( other extemal vertical forces )

Cz=+Uj~inaj-Xi~in6i-Ui~~~6i-Ri~~~(~idev-6i)

+ ( other extemal horizontal forces )

Solving for Q j and S j,

Once S j is found, the factor of safety can be calculated fiom the following equation,

However, if the last face of the failure surface entirely lies in a cohesionless material, a

substitution mut be made, since S , is zero, due to the cohesion appearing in Equation 3.1 3. Thus if S j = O is substituted into Equation 3.12,

but,

thus F can be obtained from Equations 3.14 and 3.15 with rearrangement.

3.23 Moment Equilibriun

The calculation of the moment equilibrium uses the same principal wedge, however

n-I assumptions must be made about the location of the inter-slice frictional forces Ri and Rk,

(the magnitudes of these forces were obtained from the force equilibrium calculations).

Assuming the positions of the forces can be determined fiom the vertical effective stress, the

moment equilibrium check proceeds as follows.

By taking moments about point 1 as shown on Figure 3.5, and taking

counterclockwise moments positive, the following equations can be established:

For Side i:

t in- t IR s ~ i = - u i [(xs-x I ) ~ + ( Y ~ I - Y i ) ] ~ i c o s + i ~ [ ( x 7 - x I ) ~ + ( Y , - Y 1) I (3.16)

For Side j :

For Side k:

where

The weight of the wedge

Surnrning the moments gives

Location of application of Q calculated Figure 3.5 Moment equilibrium on a typicai wedge

Solving Equation 3.21 for àx, the point of application of Q, is obtained from the following

relationship,

The above point must lie on the line between point I and point 2 as shown on Figure 3.5,

defined by coordinates (x ,,y ,) and (x z,y 2 ) in order to satisfy moment equilibrium. It is

advisable to calculate the point of application of Q, fiom numerical integration fiom the

vertical effective stress. The two points of application should be close. However, in cases of

complex pore water pressure distribution or large extemal loads, the moment equilibriurn is

not satisfied for certain wedges. In these cases, the point of application or either Ri or Rir or both should be adjusted to achieve moment equilibrium. Moment disequlibrium can also

occur if the initiai failure surface is poorly chosen or untealistic. Therefore, in the

optimization process for a critical failure surface, a moment equilibrium check is performed

for the tinal optimized failure surface only.

3.2.4 Multi-Layered Medium

Up to this point in the derivation of the method, materiai heterogeneity was introduced only

in the weight calculation of the wedges. Since the method uses both friction and cohesion

dong a part of the failure surface or the inter-wedge face, the case must be expanded if the

face crosses two or more materials. For these cases the average cohesion and fiction angle is

used in the force and moment equilibrium calculations.

The average cohesioa is obtained by weighting with respect to length in each stratum

as shown on Figure 3.6 and surnrnarized in Equations 3.22 through 3.24.

Slip Surface or Wedge Interface ,

Figure 3.6 Calculation of average cohesion for a wedge face crossing three matenals

cl -1, +c, 4, +cj e l , =C- L

there fore,

1 F = ( C , 01, +c, -1, +c, * I J -

F

where

The average coefficient of friction cannot be obtained by weighting the individual

material's angle of fnction over the length of the interface since the frictional strength is a

product of the tangent of the frictional angle and the normal stress in each layer, thus if the

normal stress is assumed to be proportionai to the vertical stress with a constant of

proportionality A, the average fiction angle cm be obtained using Equations 3.25 to 3.29

and referring to Figure 3.7.

Figure 3.7 Calculation of the average fnction angle for a face crossing three materials

Using the assumption that a,, = AG" the following can be defined fiom Figure 3.7:

where

ml = number of control points used along that portion of the interface which passes through

one soi1 layer

m2 = number of soil layers above a particular control point

The total f'ctional component T, dong the interface is given by

where m3 is the nurnber of soil layers through which the face is passing. Note that using the

component form along the interface,

and equating (3.27) and (3.28) we get

in which the constant of proportionality A cancels out.

In the derivation presented, the assumption of the proportionality of normal stress and

vertical stress is not unreasonable and it was shown by finite element calculations to be valid

(Giarn, 1989).

3.2.5 Pore and Surface Wa ter Defiaitioa

The presence of water in the slope is the largest destabilizhg factor in the slope stability

analysis. Therefore, the modeling and calculation of the various types of water pressure in or

on the slope is an important aspect of slope stability analysis. The pore water pressure

defulltion presented here are just some of the defuiitions fiom the original mode1 by Giarn.

However the ones listed here are the most commonly encountered in practical situations. The

special discrete pore water pressure definition is described in Chapter 4.

The simplest method is the use of ru values for individual materials. The pore water

pressure is defined as the weight of the material multiplied by the respective ru value for al1

the materials above the given interface or face of the failure surface to give the pore water

pressure as a force.

A more realistic approach is to define a phreatic surface line with or without ponded

water and the use of the Hu coefficient to simulate seepage For the case of heterogeneous

dopes, each material and its conesponding thickness is included in the surnmaiion. This

water definition is shown on Figure 3.8, where a phreatic surface and a ponded water ce11 are

defined.

Ponded Water

1 - =

Phreatic Surface

Figure 3.8 Water modeled by phreatic surface and ponded water

If the pore water pressure distribution was obtained using piezometers in the slope,

for one or multiple materials, the model is capable of handling the piezornetric lines for each

material separately in conjunction with ponded water. The pore pressure force on the faces is

calculated as the height above the control points to the piezometric line corresponding to the

material in which the control node is located. This method of pore water defuiition is shown

on Figure 3.9 for a slope made up of a homogeneous material without ponded water.

Figure 3.9 Pore water pressure defmed by a piezometric line

The most accurate definition and calculation of the pore water pressure distribution is

achieved with the use of pressure gids or flow nets. A typical case of such a model is shown

on Figure 3.10, where a homogeneous slope is defined with a pore pressure grid and a

ponded water cell. The grids can be for total head, pressure head or pore-pressure. These

input gids are usually created and calculated using a seepage analysis program. In addition

to the pressure grid, ponded water and a phreatic surface can be added to the model. If a

phreatic surface is used, any pressure value calculated above the phreatic surface is

autornatically set to zero, thus the negative pore pressures can be filtered out. The solution of

the pore pressure value at a control node is obtained fiom the pressure grid using numerical

methods. Along any given face, the total pore pressure is calculated fiom the control nodes

using numencal Uitegration, such as Simpson's nile.

It is advisable to define the grid so that it extends over the anticipated area of search

and optimization in order to avoid the possibility of having parts of the failure surface well

outside the defined grid, where the pore pressures are not defined resulting in an erroneous

calculation of the factor of safety.

Figure 3.10 Pore water pressure defmed by pressure grid and a ponded water ce11

3.2.6 External Surface Loads

For complex slopes, in addition to the presence of water, the mode1 rnust be capable of

handling other surcharges. The most commody encountered type of loading considered in

this section is in the form of line loads and uniformly distributed loads. Line loads are used

when the loading is concentrated dong a line that is perpendicular to the plane of the two-

dimensional analysis, and distributed loads modeling a loading that is in the plane of the

analysis also extending normal to the plane. Both types of loading are incorporated into the

force and moment equilibrium equations to calculate the factor of safety. In addition their

inclusion into the general equations, the extemal loads have an effect on other wedges as

well. This effect can be incorporated using the loads and their effect as stresses in the

medium if the loads are resolved into components perpendicular and parallei to the slope

surface and the appropriate Boussinesq and Cermti equations are used to determine the

loads' contribution to the stress on a face of a wedge system.

Consider a load applied perpendicular to the slope suface as shown on Figure 3.1 1,

the appropriate Boussinesq equations for the stresses are:

where

R = JX2+Z? p per unit thinkness

Figure 3.1 1 integrated Boussinesq problem

Complementing the Boussinesq solution for a vertical load with a Cemti solution for a

horizontal load the following equations are used to obtain the stresses at a point in the

medium. The general problem is show on Figure 3.12:

S per unit thinkness

Figure 3.12 Integrated Cerruti problem

In the Equations 3.34-3.36, the definition of R is the same as for the Boussinesq solution. As

presented, the two solutions for stresses in the medium are for a set of axes that aligns with

the global horizontal and vertical axis system. Thus to find the stresses along an arbitrarily

inclined plane or line for the two-dimensional case, stress

used. The definition of variables is show on Figure 3.13

present the solution.

transformation equations must be

and the Equations 3.37 and 3.38

Plane of lnterest /

Figure 3.1 3 Stress transformation

The above equations for a Boussinesq and Cerruti problem represent a complete

solution for a line load, similarly, the distributed loads are converted into line loads and their

effect is added to the general equilibrium equations and to the stress calculations at the

control nodes.

The solution presented so far for the loads assumed an infinite half-space where the

load is applied on the boundary of the haif-space. However, the slope geometry is far from

being restricted to a plane, thus some restrictions must be made on the area of influence of

the load applied to the surface of a slope. This is achieved using a physical fact that load

cannot be transmitted through air. Therefore, the effect of a particular load is summed over

areas where the load directly exerts its effect as shown on Figure 3.14.

Due to Line-Lad P '-4'

Figure 3.14 Region of stress influence of a line load

3+2+7 Tension Cracks

Slopes with a factor of safety less than 1.20 usually exhibit tension cracks (Hoek, 1996).

Thus the incorporation of the tension crack into the mode1 can add a more detailed

description of the problem. Usually, tension cracks are modeled by providing an elevation of

a tension crack and whether the tension crack is dry or water filled.

Giam expanded on this by assuming a tension crack profile line across the slope and

using the whole material above the tension crack as having zero strength for both the failure

surface and the inter-wedge faces. The author of this thesis considers a tension crack zone

that is defined by an elevation, however adopts both approaches of the effect of the tension

crack; only on the failure surface and on al1 faces crossing the tension crack as discussed in

the next chapter.

3.3 Equivalence of the Generalized Wedge Method with the Energy Method Upper-

Bound

The limit equilibrium and upper bound procedures are based on entirely different derivations,

yet they yield identical results. This c o n f i s the validity of the methods of analysis.

Giam examined in detail the equivalence of the Generaiized Wedge Method and the

Energy Method Upper (Giam, 1989). We recdl that the Generalized Wedge Method is a true

limit equilibrium method and Energy Method Upper an energy method via the upper bound

theory of plasticity with M y associated flow d e .

It was shown by Giam on a two-wedge exarnple, that the factor of safety equations

take an identical fom for both the limit equilibrium and energy method. The equivalence of

the methods can also be shown for n number of wedges.

The limit equilibriurn method is more suitable for slope stability problems where

materials exhibit a piecewise non-linear behavior, due to the effect of the force and stress on

the strength parameters of the material in the calculations. in contrat the state of stress

cannot be obtained in the upper bound method of analysis.

However, in the derivation of the energy method if the materid is purely cohesional,

the solution for the factor of safety is obtaified without iteration.

3.4 Deficiencies of the Original Generalized Wedge Metbod

The previous sections in this chapter presented the theoretical basis and the implementation

of the Generalized Wedge Method dong with the capabilities of the rnodel. However, in the

author's view, a mode1 is only as good as its implementation. If the model's irnplementatiori

encounters dificulties in the practical use, the method might be rejected by end users.

In the original fom of the method, Giam presents one example how the factor of

safety is calculated for a three-wedge system documenting each step taken and the iteration

procedure. This example omits the detailed description of the factor of safety fwiction.

We exmined this fuaction in detail using a large number of examples and concluded

that the factor of safety funftien is highly discontinuous and the standard root-finding

procedures used in numerical analysis are prone to failure if used with a poor initial guess of

the factor of safety.

The Decision-Iteration scheme presented by Giam and subsequently highlighted and

modified by Donald (Donald, 1993) addresses the issue, but the Decision-Iteration scheme

starts at a predetermined value and if the factor of safety happens to be extremely high, it

might give erroneous results.

In order to use a particular method in a practical situation, the ease of the detinition of

problem geometry dong with loading and water level is crucial. Since the Generalized

Wedge Method uses a number of wedges resulting in a complex defuiition of geometry, the

method of generation of the wedges must be as simple as possible. Giam did not address this

issue in the original work.

Conside~g the kinematic admissibility cnterion and its implication that neither

separation nor ovedap of wedges can occur, the original definition of the method assumed

one type of relative movement between adjacent wedges. Subsequently it was modified to

accommodate two relative movements between two adjacent wedges.

However, we have discovered that the most recent definition does not apply the inter-

wedge forces correctly. This was confhned by Donald as well (Donald, 1998).

The following chapter addresses this issue and presents the full and detailed method

that is in accordance with the Donald-Chen kinematic admissibility critenon considering al1

possible relative movements between the wedges.

CHAPTER 4

MODIFICATIONS AND EXTENSIONS TO THE GENERALIZED

WEDGE METHOD

The previous chapter presented the general methodology and solution procedure for the

Generaiized Wedge Method and its extensions to handle a large variety of scenarios

commonly encountered is dope stability analysis. However, Section 3.4 summarized the

deficiencies of the method as seen fiom the point of view of the author. The purpose of this

chapter is to address these issues to present solution to them and to introduce new additions

to the method airned at making it a practicai tool in rock slope stability analysis problems.

Fust, a new solution method will be presented to find the factor of safety for a single

failure surface highlighting the solution dificuities. A general method will be presented to

adâress the need to generate the wedge system automatically fkom a given failure surface and

a siope profile iîne. Expanding on the automatic definition of wedges, a simple method will

be presented following Giam's approach, but simplified to generate a set of initial failure

surfaces that cover the analysis domain and capture the global minimum factor of safety.

The implementation of the Donald-Chen criterion for kinematic admissibility dong

with the expansion and redefinition of the goveming force equilibrium equations for the

factor of safety calculations will be presented.

The presence of a weak seam in a mostly homogeneous material to model joints or

faults will be examined and through an illustrative example, the factor of safety function will

be