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AN IMPROVISED THREE-DIMENSIONAL SLOPE STABILITY ANALYSIS
BASED ON LIMIT EQUILIBRIUM METHOD BY USING PARTICLE SWARM
OPTIMIZATION
ROOHOLLAH KALATEHJARI
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Doctor of Philosophy (Civil Engineering)
Faculty of Civil Engineering
Universiti Teknologi Malaysia
FEBRUARY 2013
iii
DEDICATION
I dedicated this thesis to my beloved father and mother for their support and encouragement.
iv
ACKNOWLEDGMENTS
I would like to express my gratitude to my supervisor, Dr. Nazri Bin Ali for
his guidance, advice, constructive feedback, and critical review of this thesis. For his
support, time and help, they are the motivations for me to strive harder in my
academic journey. I also would like to thank Professor Dr. Khairul Anuar Kasim,
Assoc. Prof. Ir. Dr. Hj. Ramli Nazir, and Assoc. Prof. Dr. Nurly Binti Gofar for their
critical feedbacks on my research. I wish to thank University Teknologi Malaysia
for giving me the apportunity to do my research in a supportive academic
environment. I also appreciate Professor D. Leshchinsky, Professor C.C. Huang, and
Assoc. Prof. Dr. Edy Tonnizam bin Mohamad for their valuable help and guidance.
I would like to thank my wife for her understanding and love during the past
few years. Her support and encouragement was in the end what made this
dissertation possible. My parents, receive my deepest gratitude and love for their
patience, prayers, and the many years of support during my studies that provided the
foundation for this work.
Special thanks go to my dear friends Mr. Mohsen Hajihassani, Mr. Ehsan
Gatavi, Mr. Mehrdad Kholghi Fard, Mr. Houman Sohaei, and Mr. Amin Kheirandish
for their valuable helps.
I always thank the almighty God for removing my weaknesses and answering
to my prayers.
v
ABSTRACT
The stability of slopes is a major concern in the field of geotechnical
engineering. Two-dimensional (2D) limit equilibrium methods are usually
implemented in this field due to their simplicity. However, these methods ignore the
features of the third dimension of slopes. Although three-dimensional (3D) methods
tried to remove the previous limitation, most of them assumed the direction of
sliding, simplified or ignored the intercolumn forces, and avoided to search for
location and shape of three-dimensional critical slip surface. This study was
performed to overcome the mentioned limitations. In the present study, a new slope
stability method was established based on the force and moment equilibrium in two
vertical directions that was able to find the unique direction of sliding. Moreover, a
modified Particle Swarm Optimization was developed by replacing the worst particle
of each swarm with the previous global best particle and using a dynamic inertia
weight to determine the 3D critical slip surface. Then, a computer program was
established to model 3D slopes and perform the required calculations. Several
benchmark problems were re-analyzed to verify the results of the study and good
agreements were achieved with the results of previous studies when different failure
mechanisms as ellipsoid, cylindrical, and composite slip surfaces were successfully
applied in the analysis. The results indicated that the 3D factor of safety of a slope is
always greater than its corresponding 2D factor. Moreover, the end effect in 3D
analysis was found to be more significance in the problems with lower ratio of length
to the width of the sliding mass. It was also found that the presence of water and
weak layer enlarged this effect. Through the verification study, it was observed that
different sliding directions produce different factors of safety, while the lowest value
of factor of safety and 3D critical slip surface is only reachable through the real
direction of sliding. Finally, case studies of actual stability problems were analyzed
to find their critical slip surfaces. Achieving the minimum factor of safety of 0.977
for the critical slip surface of a failed slope demonstrated the validity of performance
of presented computer code. Based on the obtained results, this study successfully
overcame the mentioned limitations of the previous methods. The results of this
study provided a better understanding of the actual failure mechanism and helped to
enhance the safety and reduced the economic and health costs due to slope failure by
a more detailed analysis than before.
vi
ABSTRAK
Kestabilan cerun kini menjadi kebimbangan utama dalam bidang
kejuruteraan geoteknik. Penggunaan had dua dimensi (2D) yang merupakan kaedah
keseimbangan biasanya dijalankan kerana penyerderhanaan pengaplikasian. Walau
bagaimanapun, kaedah ini mengabaikan dimensi ciri ketiga cerun. Kaedah tiga
dimensi (3D) dihasilkan bagi menghapuskan keterbatasan sistem aplikasi
sebelumnya. Kebanyakan pengkaji menganggap arah gelongsor, dipermudah atau
diabaikan daya intercolumn, menghalang untuk mencari lokasi dan bentuk tiga
dimensi permukaan slip critical. Kajian ini telah dijalankan untuk mengatasi
keterbatasan tersebut. Dalam kajian ini, satu kaedah baru telah diperolehi
berdasarkan kestabilan cerun dalam mengimbangi tenaga dua arah menegak dapat
mencari hala tuju unik gelongsor. Pengubahan zarah sekumpulan pengoptimuman
dibangunkan bagi menggantikan kekurangan kumpulan zarah terdahulu, dengan
penggunaan berat inersia dinamik untuk menentukan permukaan slip critical 3D.
Program komputer telah diperolehi untuk model cerun 3D dalam melaksanakan
pengiraan yang tepat. Beberapa masalah penanda aras dianalisis untuk mengesahkan
keputusan kajian dan satu kesepakatan telah dipersetujui berkaitan dengan penemuan
kajian terdahulu, akibat terdapat kegagalan mekanisma yang berbeza di permukaan
slip ellips, silinder, dan kegagalan permukaan komposit telah berjaya digunakan
dalam analisis kajian. Keputusan menunjukkan bahawa faktor keselamatan cerun 3D
adalah sentiasa lebih besar berbanding faktor 2D. Hasil akhir dalam analisis kajian
3D didapati lebih penting dalam penyataan masalah dengan nisbah yang lebih rendah
jisim panjang gelongsor. Kajian juga mendapati bahawa kehadiran air yang
diperbesarkan akibat dari kesan lapisan yang lemah. Melalui kajian pengesahan,
Dapat diperhatikan bahawa gelongsor arahan berbeza menghasilkan keselamatan
faktor yang berbeza, manakala faktor keselamatan yang lebih rendah dan 3D
kegagalan permukaan kritikal hanya dapat dicapai melalui gelongsor arah sebenar.
Kajian kes masalah kestabilan sebenar telah dianalisis bagi mencari kegagalan
permukaan kritikal. Faktor minimum keselamatan 0.977 di kegagalan permukaan
kritikal kegagalan cerun menunjukkan kesahihan prestasi kod komputer dipaparkan.
Berdasarkan keputusan yang diperolehi, kajian ini telah berjaya mengatasi
keterbatasan aplikasi yang terdahulu. Hasil keputusan kajian memberikan
pemahaman yang lebih baik terhadap mekanisme kegagalan sebenar dan dapat
membantu untuk meningkatkan keselamatan bagi mengurangkan kos ekonomi dan
kesihatan akibat kegagalan analisis cerun secara lebih terperinci berbanding
sebelumnya.
vii
TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGMENTS iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES xiiii
LIST OF FIGURES xv
LIST OF ABBREVIATIONS xx
LIST OF SYMBOLS xxi
LIST OF APPENDICES xxvii
1 INTRODUCTION 1
1.1 Background of the Study 1
1.2 Statement of the Problems 3
1.3 Objectives of the Study 5
1.4 Research Questions 5
1.5 Significance of the Study 6
1.6 Scope of the Study 7
1.7 Expected Outcomes 8
1.8 Limitations of the Study 9
1.9 Definition of Terms 10
2 LITERATURE REVEIW 11
2.1 Introduction 11
viii
2.2 Slope Stability Analysis based on Limit Equilibrium
Method 12
2.2.1 Two-Dimensional Limit Equilibrium Methods 14
2.2.1.1 Two-Dimensional Method of Slices 15
2.2.1.2 Common Methods of Slices 18
2.2.1.3 Two-Dimensional Methods of Slices 21
2.2.2 Three-Dimensional Limit Equilibrium Methods 24
2.2.2.1 Anagnosti’s Method 26
2.2.2.2 Baligh and Azzouz’s Method 27
2.2.2.3 Hovland’s Method 32
2.2.2.4 Chen and Chameau’s Method 33
2.2.2.5 Dennhardt and Forster’s Method 36
2.2.2.6 Leshchinsky, Baker and Silver’s
Method 38
2.2.2.7 Ugai’s Method 39
2.2.2.8 Leshchinsky and Baker’s Method 42
2.2.2.9 Hungr’s Method 44
2.2.2.10 Gens, Hutchinson, and Cavounidis’
Method 46
2.2.2.11 Leshchinsky and Mullet’s Method 47
2.2.2.12 Xing’s Method 48
2.2.2.13 Hungr, Salgado, and Byrne’s
Method 50
2.2.2.14 Leshchinsky and Huang’s Method 51
2.2.2.15 Cavounidis and Kalogeropoulos’
Method 53
2.2.2.16 Lam and Fredlund’s Method 54
2.2.2.17 Yamagami and Jiang’s Method 56
2.2.2.18 Huang and Tsai’s Method 58
2.2.2.19 Huang, Tsai, and Chen’s Method 60
2.2.2.20 Chen, Mi, Zhang, and Wang’s
Method 62
2.2.2.21 Cheng and Yip’s Method 64
2.2.2.22 Zheng’s Method 67
ix
2.2.2.23 Sun, Zheng, and Jiang’s Method 67
2.3 Critical Slip Surface 69
2.3.1 Finding the Critical Slip Surface 70
2.4 Optimization Techniques 72
2.4.1 Global Optimization Techniques 72
2.5 Summary and Discussion 76
3 RESEARCH METHODOLOGY 81
3.1 Introduction 81
3.2 Procedure of the Research 82
3.3 Data Sources 84
3.4 Instrumentation 84
4 DEVELOPMENT OF INNOVATED THREE-
DIMENSIONAL LIMIT EQUILIBRIUM ANALYSIS 86
4.1 Introduction 86
4.2 Initial Considerations on Slope Stability Analysis 89
4.2.1 Total and Effective Stresses 89
4.2.2 Tension Crack 91
4.2.3 Earthquake 92
4.2.4 Local Loads 93
4.2.5 Three-Dimensional Failure Mode 93
4.3 Three-Dimensional Equation of Factor of Safety 93
4.3.1 Directional Factor of Safety in X-Direction 100
4.3.2 Directional Factor of Safety in Y-Direction 103
4.3.3 Intercolumn Forces 106
4.3.4 Overall Factor of Safety 110
4.3.5 Determinacy of the Problem 111
4.3.6 Solving Process 112
4.3.7 Special Considerations 114
4.4 Summary and Discussion 114
5 APPLICATION OF PARTICLE SWARM
OPTIMIZATION 117
x
5.1 Introduction 117
5.2 Parameters of Particle Swarm Optimization 118
5.3 Swarm of the Particle Swarm Optimization 121
5.3.1 Topology of the Swarm 122
5.3.2 Size of the Swarm 125
5.4 Termination Criteria 125
5.5 Application of Particle Swarm Optimization in Slope
Stability Analysis 126
5.5.1 Coding of the Particles 126
5.5.2 Fitness Function 128
5.5.3 Modification of Particle Swarm Optimization 129
5.6 Sensitivity Tests of Particle Swarm Optimization 130
5.6.1 Optimum Swarm Size 132
5.6.2 Optimum Coefficients of Velocity Equation 135
5.6.3 Optimum Inertia Weight of Velocity Equation 137
5.7 Solving Process 139
5.8 Summary 141
6 THREE-DIMENSIONAL SLOPE MODELLING 143
6.1 Introduction 143
6.2 Framework of the Code 144
6.2.1 Modelling Process 144
6.2.2 Optimization Process 148
6.2.3 Calculation of Three-Dimensional Factor of
Safety 152
6.3 Testing of the Code 153
6.3.1 Calculation of Total Area 156
6.3.2 Calculation of Total Volume 157
6.3.3 Calculation of Total Weight 158
6.3.4 Discussion on the Value of Grid Width 159
6.4 Summary 161
7 VERIFICATION OF THE RESULTS AND VALIDATION
OF THE COMPUTER PROGRAM 162
xi
7.1 Introduction 162
7.2 Verification of the Results 163
7.2.1 Problem 1 163
7.2.1.1 Geometry and Material Properties of
Problem 1 164
7.2.1.2 Results and Discussion of Problem 1 165
7.2.2 Problem 2 167
7.2.2.1 Geometry and Material Properties of
Problem 2 168
7.2.2.2 Results and Discussion of Problem 2 169
7.2.3 Problem 3 175
7.2.3.1 Geometry and Material Properties of
Problem 3 176
7.2.3.2 Results and Discussion of Problem 3 176
7.2.4 Problem 4 182
7.2.4.1 Geometry and Material Properties of
Problem 3 182
7.2.4.2 Results and Discussion of Problem 4 183
7.3 Validation of the Computer Program 188
7.3.1 Case Study 1 188
7.3.1.1 Geometry and Material Properties of
the Case Study 1 190
7.3.1.2 Results and Discussion of the Case
Study 1 192
7.3.2 Case Study 2 199
7.3.2.1 Geometry and Material Properties of
the Case Study 2 200
7.3.2.2 Results and Discussion of the Case
Study 2 202
8 CONCLUSIONS AND RECOMMENDATIONS 206
8.1 Conclusions 206
8.2 Recommendations 210
xii
REFERENCES 212
Appendices A-H 227-315
xiii
LIST OF TABLES
TABLE NO. TITLE PAGE
2.1 Available equations of typical 2D slope
( Fredlund and Krahn, 1977) 17
2.2 Available unknowns of a typical slope
( Fredlund and Krahn, 1977) 17
2.3 Comparison of the most commonly used methods of slices 19
2.4 Components of moment and force equations in the methods of
slices (Fredlund and Krahn, 1977) 22
2.5 Comparison of metaheuristic methods (Kitagawa et al., 2004) 75
2.6 Summary and characteristics of 3D limit equilibrium methods 77
4.1 Unknowns, equations and assumptions of the present method 111
5.1 Sample coded swarm of the current study 128
5.2 Properties of soil layers in sensitivity analysis 131
5.3 Results of the sensitivity analysis on swarm size 133
5.4 Relationship between coefficients of different tests 135
5.5 Inertia values of the sensitivity tests 137
6.1 Position of particles in a sample swarm 149
6.2 Results of the calculation analysis 154
7.1 Properties of soil of problem 1 (Huang et al., 2002) 164
7.2 Results of different studies for problem 1 165
7.3 Properties of layers in problem 2 (Xing, 1988a) 169
7.4 Properties of produced models in different tests 169
7.5 Overall results of analysis of problem 2 171
7.6 Comparison between 2D and 3D results of problem 2 173
7.7 Comparison between the results of 3D simplified methods for
problem 2 174
xiv
7.8 Comparison between the results of 3D rigorous methods for
problem 2 174
7.9 Properties of slope in problem 3 (Alkasawneh et al., 2008) 176
7.10 Minimum 3D FOS of problem 3 180
7.11 Properties of slope in problem 4 (Yamagami and Jiang, 1997) 183
7.12 Minimum 3D FOS of problem 4 186
7.13 Classification of soil layers in considered area of KUR
(HCE, 2005) 191
7.14 Properties of access trenches of line 2 of KUR
(DKPCE, 2005) 192
7.15 Properties of soil layers of case study 2 (Liew et al., 2003) 201
7.16 Groundwater level of case study 2 (Liew et al., 2003) 202
xv
LIST OF FIGURES
FIGURE NO. TITLE PAGE
2.1 Forces in the method of slices ( Fredlund and Krahn, 1977) 16
2.2 Forces acting on a thin vertical slice in Anagnosti’s method
(Anagnosti, 1969) 26
2.3 The combined slip surfaces of Baligh and Azzouz’s method
(Baligh and Azzouz, 1975) 28
2.4 Slip surfaces of Azzouz and Baligh’s method including a) 2D
and b) 3D problems (Azzouz and Baligh, 1983) 31
2.5 A typical column of Hovland’s method (Hovland, 1977) 32
2.6 The slip surface of Chen and Chameau’s method
(Chen and Chameau, 1983) 34
2.7 Chen and Chameau’s method after applied assumptions
(Chen and Chameau, 1983) 35
2.8 Three-dimensional problem of Dennhardt and Forster’s
method (Dennhardt and Forster, 1985) 37
2.9 Three-dimensional model of Leshchinsky et al.’s method
(Leshchinsky et al., 1985) 39
2.10 Three-dimensional slip surface of Ugai’s method (Ugai, 1985) 40
2.11 Developed model of method of slices by Ugai (Ugai, 1988) 41
2.12 Three-dimensional model of Leshchinsky and Baker’s method
(Leshchinsky and Baker, 1986) 42
2.13 Three-dimensional model of Baker and Leshchinsky’s method
(Baker and Leshchinsky, 1987) 43
2.14 Columns system of Hungr’s method (Hungr, 1987) 45
2.15 Three-dimensional slip surface of Gens et al.’s method
(Gens et al., 1988) 46
xvi
2.16 Log-spiral failure mechanisms of Leshchinsky and Mullet
(Leshchinsky and Mullet, 1988b) 48
2.17 Three-dimensional model of Xing’s method (Xing, 1988) 49
2.18 Assumed rotation axes in Hungr et al.’s method
(Hungr et al., 1989) 51
2.19 Discretized slip surface of Leshchinsky and Huang
(Leshchinsky and Huang, 1992b) 52
2.20 Slip surface of Cavounidis and Kalogeropoulos’ method
(Cavounidis and Kalogeropoulos, 1992) 53
2.21 Discretization of slip surface in Lam and Fredlund’s method
(Lam and Fredlund, 1993) 55
2.22 Typical slip surface and a column in Jiang and Yamagami’s
method (Jiang and Yamagami, 2004) 57
2.23 Three-dimensional model of Huang and Tsai’s method
(Huang and Tsai, 2000) 59
2.24 Component of base shear strength at the base of column and
their projection on x-y plane (Huang et al., 2002) 61
2.25 Typical column of Chen et al.’s method a) before, and b) after
applying the assumptions (Chen et al., 2003) 63
2.26 Forces acting on a typical column of Cheng and Yip’s method
(Cheng and Yip, 2007) 65
2.27 Triangular mesh and projection of slip surface in Sun et al.’s
method (Sun et al., 2009) 68
2.28 Classification of optimization techniques 73
3.1 Flowchart of the research procedure 82
4.1 Plan view of discretized sliding mass (Huang et al., 2002) 94
4.2 Internal and external forces on a typical soil column, modified
after Huang et al. (2002) 94
4.3 Base angles and sliding angle of a typical soil column
(Huang and Tsai, 2000) 95
4.4 Boundaries normal forces in x- and y-directions
(Huang et al., 2002) 102
4.5 Forces on central section of column (i,j) in x-direction,
modified after Huang et al. (2002) 107
xvii
4.6 Forces on central section of column (i,j) in y-direction,
modified after Huang et al. (2002) 109
4.7 Process to find the overall FOS 113
5.1 Standard flow chart of PSO (Kennedy and Eberhart, 1995) 119
5.2 Schematic structure of a particle in PSO 122
5.3 Sample of (a) local and (b) global topologies in 2D space 124
5.4 Projection of rotated ellipsoids on x-y plane 127
5.5 Schematic structures of a particle in the present study 127
5.6 Projection of disqualified slip surfaces on y-z plane 130
5.7 Three-dimensional slope model of sensitivity analysis 131
5.8 Convergence processes of swarms with different sizes 133
5.9 Total consumed time by CPU for different swarm sizes 134
5.10 Results of sensitivity tests of (a) unequal coefficients, and
(b) equal coefficients 136
5.11 Results of sensitivity analysis on inertia weight 138
5.12 Results of the tests for dynamic inertia weight 138
5.13 Process to find the critical slip surface by particle swarm
optimization 140
6.1 Graphical user interface of basic inputs 145
6.2 Graphical user interface of section plotter 146
6.3 Three-dimensional surface of a sample slope 146
6.4 Graphical user interface of layers properties 147
6.5 Three-dimensional model of a sample slope with layers and
piezometric surface 148
6.6 Slip surfaces of a sample swarm 149
6.7 Intersection area of a sample problem 150
6.8 Discretized sliding body of a sample problem 150
6.9 Slip surfaces of qualified particles 151
6.10 Matrix calculation compare with conventional loops 152
6.11 Cropping method to reduce the time of calculations 153
6.12 Assumed model for calculation test 154
6.13 Projection of columns on x-y plane 155
6.14 Number of columns versus different grid widths 156
6.15 Percentage of area error in different tests 157
xviii
6.16 Percentage of volume error in different tests 158
6.17 Percentage of weight error in different tests 159
6.18 Total weight and total effective weight in different tests 159
6.19 Calculation time of different tests 160
6.20 Errors of total area and volume versus grid width ratio 160
7.1 Vertical cut of problem 1 (Huang and Tsai, 2000) 164
7.2 Generated model of problem 1 166
7.3 Central section of problem 2 (Xing, 1988a) 167
7.4 Generated model for case 6 of problem 2 170
7.5 Three-dimensioal FOS versus volume of the sliding mass for
problem 2 172
7.6 geometry of 2D slope in problem 3 (Alkasawneh et al., 2008) 176
7.7 Two-dimensional and 3D views of model of problem 3 by (a)
and (c) Plaxis 3D and (b) and (d) current study, respectively 177
7.8 Randomly generated cylindrical slip surfaces in first swarm
of PSO for problem 3 in (a) cross sectional, and (b) 3D view 178
7.9 Average and maximum fitness versus iterations for problem 3 179
7.10 Minimum FOS versus iterations for problem 3 180
7.11 Exaggerated displacement vectors of Plaxis 3D result for
problem 3 180
7.12 (a) cylindrical shape, (b) sliding mass, and (c) sliding mode of
the 3D critical slip surface for problem 3 181
7.13 (a) half-plan view and (b) central cross section of geometry of
problem 4 (Yamagami and Jiang, 1997) 182
7.14 Generated model of problem 3 by current study in
(a) plan view, (b) central cross section, and (c) 3D view 184
7.15 (a) cross sectional and (b) 3D view of randomly generated
ellipsoid slip surfaces in the first swarm of PSO for problem 4 185
7.16 Average and maximum fitness versus iterations for problem 3 185
7.17 Minimum FOS versus iterations for problem 3 186
7.18 (a) ellipsoid shape, (b) sliding mass, and (c) sliding mode of
the 3D critical slip surface for problem 4 187
7.19 Location of Karaj city in Iran (sited as point A) 188
7.20 Karaj urban & suburban railway map (HCE-HPCE, 2010) 189
xix
7.21 Top views of (a) T1, and (b) T2 access trenches to the main
tunnel in part 2, phase I of line 2 of KUR 190
7.22 Geological map of part 2, phase I of line 2 of KUR
(DKPCE, 2005) 191
7.23 Geometry of trenches in (a) end section, (b) top view, and
(c) central cross section (HCE, 2005) 193
7.24 A sample ignored slip surface in conventional 3D methods 194
7.25 Cross sectional and 3D view of generated models of T1
and T2 195
7.26 Randomly generated slip surfaces for the first swarm 196
7.27 Process of PSO to achieve the maximum fitness for T1
and T2 197
7.28 Trend of minimizing FOS by PSO for T1 and T2 197
7.29 Critical slip surfaces of (a) T1 and (b) T2 obtained by the
present code 198
7.30 Location of the site of case study 2 (sited by red rectangular) 199
7.31 Front view of failed slope (Liew et al., 2003) 200
7.32 Boreholes and Instrumentation for case study 2
(Liew et al., 2003) 201
7.33 Cross section plan of case study 2 (Liew et al., 2003) 202
7.34 Cross section and 3D view of generated model of case
study 2 203
7.35 Process of PSO toward maximum fitness for case study 2 204
7.36 Trend of minimizing FOS by PSO for case study 2 204
7.37 Critical slip surfaces of case study 2 (a) cross section and
(b) 3D 205
xx
LIST OF ABBREVIATIONS
2D - Two-Dimensional
3D - Three-Dimensional
DP - Dynamic Programming
FOS - Factor of Safety
FEM - Finite Element Method
GA - Genetic Algorithm
KUR - Karaj Underground Railway
LEM - Limit Equilibrium Method
NATM New Austrian Tunneling method
PSO - Particle Swarm Optimization
RNG - Random Number Generation
SRM - Strength Reduction Method
xxi
LIST OF SYMBOLS
a - direction of sliding on base of column
a’ - direction of sliding on x-y plane
ax - inclination of column base along x-directions
ay - inclination of column base along y-directions
ac - total number of active columns
aL - distance of left side water force from center of rotation
aR - distance of right side water force from center of rotation
Ae - percentage of error of area calculation
Ahalf - area of half-sphere
Ai,j - base area of column
At - calculated area of half-sphere by the code
AL - resultant water force at the left side
AR - resultant water force at the right side
b - horizontal width of slice
bc - width of tension crack
bgn(i) - best position in the swarm so far
bi,j - width of the column on x-y plane
bnm(i) - best position of mth
neighbour of nth particle so far
bpn(i) - best position of nth
particle so far
B - height of the side of column in y-direction
c' - average effective cohesion of the soil
- cohesion of the soil in terms of effective stress
d - distance of the external lineear load from the center of rotation
D - height of the side of column in x-direction
Ds - dimension of the particles of swarm
e - distance of the center of slice from the center of rotation
ei,j - effective weight of column
E - deformation module
xxii
Es - normal interslice force
Ex - normal intercolumn force in x-direction
Ey - normal intercolumn force in y-direction
EL - normal interslice forces at left side of slice
ER - normal interslice forces at right side of slice
Fw - horizontal force induced by water in tension crack
f - perpendicular distance of normal force and center of rotation
f(x) - interslice force function of Morgenstern and Price’s method
f1, f2, f3 - unit vectors for base shear force
f(X) - objective function of optimization techniques
Feh - horizontal force induced by earthquake
Fehx - horizontal force induced by earthquake in x-direction
Fehy - horizontal force induced by earthquake in y-direction
Fev - vertical force induced by earthquake
Fm - overall moment equation
Fs - factor of safety of column
Fsx - directional factors of safety in x-directions
Fsy - directional factors of safety in y-directions
Ft - overall force equation
Fit - fitness value of particles
FOS - factor of safety
g1, g2, g3 - unit vectors for base normal force
Gx - normal intercolumn forces along x-directions
Gy - vertical shear intercolumn forces along x-directions
G - resultant of intercolumn forces along x-directions
h1 - height of right side of the column
h2 - height of left side of the column
hc - depth of tension crack
hi,j - central height of column
h - thickness soil layers
hp - depth of point from water level
hx - moment arm of normal intercolumn force in x-direction
hy - moment arm of normal intercolumn force in y-direction
hE - height of acting point of normal interslice force
Hu - coefficient of inclination of piezometric line
xxiii
Hx - horizontal intercolumn shear force in x-direction
Hy - horizontal intercolumn shear force in y-direction
k - coefficient of seismic dynamic force
K - number of the neighbours in swarm
Kh - pseudo-static horizontal seismic coefficient
Kv - pseudo-static vertical seismic coefficient
l - external linear load on slope
L - horizontal length of sliding surface
Lz - external vertical load
Lx - external horizontal load in x-direction
Ly - external horizontal load in y-direction
N - total normal force at base of column
N` - effective normal force base at base of column
Ns - swarm size of particle swarm optimization
Nx,Ny,Nz - components of base total normal force in x- y- and z-directions
p - horizontal shear intercolumn forces
P - total normal force at base of slice
Pv - vertical external force
Px - moment of external forces in x-direction
Py - moment of external forces in y-direction
q - normal intercolumn forces along z-directions
Q - resultant of intercolumn forces
r - radius of sphere
ru - pore water ratio
R - radius of moment arm of shear force
Rc - cohesion part of intercolumn shear force
Rx,Ry,Rz - semi-radiuses of ellipsoid in x- y- and z-directions
R - cohesionless parts of intercolumn shear force
S - mobilized shear strength at base of column
S’ - projection of mobilized shear strength on x-y plane
Sf - available shear strenght of soil
Sm - mobilized shear strenght at the base of slice
Sx,Sy,Sz - components of base shear strength along x- y- and z- directions
Sxz, Syzi - components of base shear strength in x-z and y-z planes
T - base shear strength of column
xxiv
Ta/F - resultant of resisting forces
Tx,y - resultant of base shear strength in x-y plane
Tz - shear strength along z-direction
- pore air pressure
- pore water pressure
u(0, ) - vectors of random numbers for velocity equation of swarm
U - pore water force at the base of column
v - velocity of particle
Ve - percentage of error of volume calculation
Vhalf - volume of half-sphere
V - vertical intercolumn shear force
Vt - calculated volume of half-sphere by the code
w - inclination of the external linear load
W - weight of column
xh - horizontal distance of center of slice and center of rotation
xv - design variables in optimization techniques
xvmin - lower bound of design variables in optimization techniques
xvmax - upper bound of design variables in optimization techniques
x - position of particle in article swarm optimization
Xs - vertical interslice shear force
Xc,Yc, Zc - coordinates of center of ellipsoid in x- y- and z-directions
Xv - vector of design variables in optimization techniques
Xx - vertical intercolumn shear force in x-direction
Xy - vertical intercolumn shear forces in y-direction
X - vertical intercolumn shear forces
XR - interslice shear forces at right side of slice
XL - interslice shear forces at left side of the slice
s - inclination of base of slice
1 - inclinations of the left side of column base
2 - inclinations of the right side of column base
α - projection of direction of sliding at base of column
α’i - direction of sliding on x-y plane
αxz - inclination of column base along x-direction
αyz - inclination of column base along y-direction
β - angle of resultant intercolumn force along x-direction
xxv
- poisson’s ratio
w - unit weight of water
t - total unit weight of overburden
- unit weight of soil layer
s - saturated unit weight of soil layer
- angle of resultant interslice forces in Spencer’s method
y - width of the column in x-direction
y - width of the column in x-direction
Ex - resultant normal intercolumn forces in x-direction
Ey - resultant normal intercolumn forces in y-direction
Hx - resultant horizontal intercolumn shear forces in x-direction
Hy - resultant horizontal intercolumn shear forces in y-direction
Xy - resultant vertical intercolumn shear forces in y-direction
Xx - resultant vertical intercolumn shear forces in x-direction
- required accuracy of result
- constant in velocity equation of particle swarm optimization
- angle of internal friction of soil in terms of effective stress
- angle between sides of column on base plane
xy - rotation angle of ellipsoid in x-y plane
x - total external forces in x-direction
y - total external forces in y-direction
- coefficient of interslice force function
- constant multiplier in velocity equation of swarm
ρ - direction of shear strength at base of column
- total normal stress
- effective normal stress
- coefficient related to degree of saturation
𝜓x - inclination between action points of normal intercolumn forces
𝜓y - inclination between action points of normal intercolumn forces
- inertia weight of particles in velocity equation
[ ] - calculator of integer part
xxvi
LIST OF APPENDICES
APPENDIX TITLE PAGE
A Equations of Three-Dimensional Methods of Slope Stability 227
B Derivation of Direction of Sliding and Force Components 231
C Derivation of Directional Factors of Safety 235
D Detail Results of Verification Problem 1 237
E Detail Results of Verification Problem 2 239
F Details of Validation Case 1 244
G Coding of the Study 251
H List of Publications 315
CHAPTER 1
INTRODUCTION
1.1 Background of the Study
The stability of slopes is always an important concern in geotechnical
engineering. Limit equilibrium method is commonly used in slope stability analysis
(Morgenstern and Price, 1965; Fredlund and Krahn, 1977; Baker, 1980; Chen and
Chameau, 1982; Fredlund, 1984; Ugai, 1995; Yu et al., 1998; Kim et al., 2002;
Krahn, 2003; Cheng et al., 2008; Askari and Farzaneh, 2008; Sun et al., 2012; and
Hongjun and Longtan, 2011). This method is established based on the principles of
kinematics that does not consider displacement within the soil mass. Consequently,
a kinematically admissible sliding surface is assumed for the failure. The soil mass
above the sliding surface is considered a free body. Then disturbing and resisting
forces along the sliding surface are estimated by using static equilibriums of force,
moment, or both within the failure mass. It is assumed that the free body of the soil
approaches the failure condition by reducing strength with a factor that is called the
factor of safety (FOS). Although this solution provides quantitative information
regarding the stability of slope as FOS, it is based on an assumed failure mass.
Therefore, it has to be repeated for a number of probable slip surfaces to find the
minimum FOS that is related to the critical slip surface. This surface is theoretically
the critical slip surface, but the real failure surface may differ if the sliding occurs.
This difference depends on how precisely the real slope idealized and what
assumptions were used in the analysis.
2
The major classifications of slope stability analysis are 2D and 3D methods.
In order to analyze the stability of a slope, 2D methods use its critical 2D section as a
plain-strain problem, while 3D methods utilize its 3D model by using an appropriate
mesh. These methods have similarities and differences in their basic theories and
assumptions regarding the shape of the slip surface and dealing with the internal
forces. The method of slices as a limit equilibrium method is commonly used to
analyze complicated condition of 2D slopes. The equivalent of these methods in 3D
analyses is called the method of columns. Since the internal forces of the failure
mass are mainly unknown in this method, usually the problem becomes statically
indeterminate. In this situation, the number of available equations to solve the
problem is less than the number of unknowns. In order to overcome the mentioned
indeterminacy, several assumptions and principles are applied in formulation of the
limit equilibrium analysis. These assumptions are mainly related to determination of
the shape and position of the sliding surface, and the magnitude, direction, and action
point of the forces inside the sliding mass. The difference in theory behind the slope
stability methods along with the applied assumption makes different processes to
analyze the stability of slopes.
A number of 3D methods have established to consider the third dimension of
slopes after the method of Anagnosti was proposed in 1969. However, the majority
of them are limited in practice because of assumptions and limitations. A major
assumption that is commonly made in 3D slope stability analyses is assuming a plane
of symmetry for the sliding mass (Hovland, 1977; Chen and Chameau, 1983;
Dennhardt and Forster, 1985; Leshchinsky et al., 1985; Ugai, 1985; Leshchinsky and
Baker, 1986; Baker and Leshchinsky, 1987; Hungr, 1987; Leshchinsky and Mullet,
1988; Ugai, 1988; Xing, 1988; Hungr et al., 1989; Leshchinsky and Huang, 1992;
Cavounidis and Kalogeropoulos, 1992; Lam and Fredlund, 1993; and Jiang and
Yamagami, 2004). Those methods that use the plane of symmetry assume the
direction of sliding and consider only half of the sliding mass.
It is expected that the sliding mass move in a cross-sectional direction in
symmetric slopes, but there is no guarantee to find the direction of sliding in
asymmetric slopes without precise calculation. Consequently, the application of a
3
large number of 3D methods is limited to symmetric problems. In order to
generalize the application of 3D methods some researchers have tried to find the real
sliding direction in asymmetric problems (Yamagami and Jiang, 1996 and 1997;
Huang and Tsai, 2000; Huang et al., 2002; and Cheng and Yip, 2007). However,
these methods still have limitations in basic theories and practice.
Accordingly, the slope stability analysis within the framework of limit
equilibrium needs to determine the critical slip surface. Although many 2D studies
were involved with the 2D critical slip surface, only a few well-known studies tried
to find the 3D critical slip surface (e.g. Yamagami and Jiang; 1997; Jiang,
Yamagami, and Baker, 2003; Mowen, 2004; and Mowen et al., 2011). All these
studies still have limitations and assumptions in their objective functions and/or in
the applied searching techniques that are discussed further. Consequently, an
effective method is required to be able to find the 3D critical slip surface.
1.2 Statement of the Problems
The slope stability is applied in 2D and 3D analyses; both of which have
shortcoming in modelling the real slope and sliding procedure. Two-dimensional
analyses mainly simplify the real condition of the slope. As a mutual assumption, all
2D methods consider infinite width for the slope. The significance of this
assumption is to reduce the unknowns related to the third dimension of the slope and
simplify the calculations.
In reality, natural slopes are generally limited in the third dimension and
performing a 3D analysis best presents the longitudinal changes of the slip surface,
especially when the geometry of the failure mass is complex. Consequently, the
results of 2D methods may largely differ from the real condition. Overestimated and
underestimated results as the probable outcomes of using inappropriate method may
cause economic and safety issues. These issues are more significant when
asymmetric slopes with complex geometries or failure mechanisms are considered by
2D methods. Moreover, the real direction and shape of sliding is not determinable in
4
2D analysis. Although, some quasi-3D approximately evaluates the 3D FOS based
on 2D factors, the accuracy of these methods is not guaranteed (Chen et al., 2006).
Therefore, performing 3D analysis provides closer results to the actual condition than
2D approaches.
The mentioned inadequacies of 2D analysis were partially solved by some
existing 3D slope stability analyses. However, many of the available methods cannot
provide an appropriate 3D model in asymmetric slopes, due to their assumptions and
simplifications. In addition, the existing 3D methods did not offer a well-defined
process to find the direction of sliding. Moreover, an effective search strategy is still
required to determine the 3D critical slip surface.
Researchers avoid using 3D analysis because it is difficult to consider
complex analytic equations, determine the static condition of the problem, and find
the 3D critical slip surface. In order to overcome the mentioned problems, the
limitations of the existing methods have to be removed to make 3D analysis feasible
for slopes stability analysis. The main constraints of existing limit equilibrium
methods in analyzing the 3D slopes are maintaining the required accuracy in
calculation, calculating the direction of sliding, making an appropriate model of the
real slope, and applying an effective search technique to find the 3D critical slip
surface. Limit equilibrium method contributes to the establishment of a stability
analysis technique. An optimization process can help to determine the position of
3D critical slip surface. This research attempted to eliminate the mentioned
limitations by developing a 3D limit equilibrium method, a PSO technique, and a
computer code. The proposed 3D method calculated the 3D FOS and direction of
sliding. The developed PSO technique determined the 3D critical slip surface.
Lastly, the developed computer code modeled the 3D slope and performed the
calculations.
5
1.3 Objectives of the Study
The present study was aimed at performing 3D slope stability analysis based
on limit equilibrium method by using PSO. In line with the main goals of the
research, the following are the objectives of the study:
i. To determine the limitations of the existing three-dimensional slope
stability analyses based on limit equilibrium methods
ii. To develop the equation of three-dimensional factor of safety based on
limit equilibrium method
iii. To develop a particle swarm optimization to determine the three-
dimensional critical slip surface
iv. To establish a computer code to model three-dimensional slopes and
perform the calculations of the slope stability analysis
v. To verify and validate the performance of the present study
1.4 Research Questions
In order to reach the objectives of the present research, the research questions
of the study were defined as following:
i. What are the limitations of existing three-dimensional slope stability
analyses based on limit equilibrium method?
ii. How to develop the equation of three-dimensional factor of safety based
on limit equilibrium method?
iii. How to develop a particle swarm optimization to find the three-
dimensional critical slip surface?
iv. How to establish a computer code to model three-dimensional slopes and
perform the calculations of the slope stability analysis?
v. How to verify and validate the performance of the present study?
6
1.5 Significance of the Study
In many cases of construction in urban areas or on industrial fields, the whole
project or a part of it has to be established on top of slope, on slope, or at the toe of
the slope. Consequently, the construction, design, remediation, and maintenance of
slopes have always been important to geotechnical engineers. On the other hand, the
importance of study on the stability of slopes is engaged with the safety and
economic aspects of human life. Over the past decades, the frequency and
consequences of landslides has increased significantly and this trend continues
(Petely et al., 2005; Petely, 2012). In addition to enormous economic losses
resulting from slope failures and landslides, considerable loss of human life and
injury also occur as a result of this events. Landslides were reported responsible for
more than 100,000 deaths for the period of 1980 to 2000 for the main continencial
area (Petely et al., 2005). Moreover, based on the recent published information more
than 32,000 people lost their lives around the world only in the period of 2004 to
2010 as the direct act of landslides (Petley, 2012). With increasing frequency and
adverse impact of slope failures, the increasing requirement of better understanding
of hilly urban areas and constructions related to slopes is revealed.
Since the shape of slopes are naturally asymmetric, 2D analyses have to
simplify the real condition. This simplification is not satisfactory in many cases,
especially when a slope with structural asymmetry is considered. In order to perform
realistic analysis of these cases, three-dimensional slope stability studies are needed.
Furthermore, finding the critical slip surface is an important task in a 3D analysis,
because it defines the location and shape of the probable failure mass. Defining this
surface can lead to estimate of the most hazardous area of the slope. Another
important issue in stability analysis of the slope that is confined to 3D analyses is the
calculation of direction of sliding that determines the direction of probable failure.
The present research sets its goals to prepare a realistic model of 3D slopes, find the
direction of sliding together with the corresponding 3D FOS, and determine the 3D
critical slip surface of the slopes.
7
The outcomes of the present study help to better understanding the behavior
of soil slopes. This is also possible to find the probable instability of a 3D slope.
These significances can avoid the consequences of overestimated and underestimated
results in geotechnical designing or assessing of a slope that are the main causes of
economic lost and safety issues. Therefore, two direct benefits that can be achieved
from the results of this research are preventing the unnecessary stabilization costs
and enhancing the safety related to the slope environment.
1.6 Scope of the Study
In order to improve the 3D equation of FOS, this study choses limit
equilibrium method as the most common method among all soil slope stability
methods to focus on. Some of the reasons of this selection are as follows:
i. Limit equilibrium method is able to consider almost all of the engaged
conditions in a slope stability problem such as external and internal forces
of the soil mass, pore water pressure, and multi layered slopes
(Morgenstern and Price, 1965).
ii. Limit equilibrium method has a simple theoretical approach that considers
the major effective factors on the shearing resistance and is reliable in
modelling the practical cases (Fredlund and Krahn, 1977; Chen and
Chameau, 1982; and Askari and Farzaneh, 2008).
iii. The results of limit equilibrium method are similar to the results of more
rigorous methods, while the required input parameters are reduced and
achieved much easier than the mentioned methods (Spencer, 1967;
Wright, Kulhawy, and Duncan, 1973; Spencer, 1973; Yu et al., 1998;
Duncan, 1996; Hongjun and Longtan, 2011).
The scope of this research is limited to the soil slopes with both
homogeneous and layered material; however, the present method may be useful in
analyzing other soil-alike materials. The linear Mohr-Coulomb failure criteria was
adopted to model the behavior of material at the verge of failure. Moreover, the
8
linear relation of Terzaghi (1936) was adopted to determine effective stresses on the
sliding surface. Consequently, unsaturated soils and nonlinear relationships of
effective stresses was excluded from the scope of this research. The main failure
mechanism of the present study is rotational. A 3D ellipsoid slip surface was
adopted that includes the spherical shape as a specific condition. However, a
cylindrical slip surface also employed as a part of this study. The procedure of
finding the critical slip surface was established based on advance searching
optimization techniques. A particle swarm optimization was developed and
employed as a global search technique to find the critical slip surface. Finally,
Matlab coding language was used in this research due to its capabilities as:
i. Using matrix based calculation that saves the calculation time
ii. Providing great graphical tools to better understanding the results
iii. Providing powerful graphic user interface (GUI) to input data
iv. Providing advanced geometrical and mathematical functions
v. Co working with other databases such as Microsoft Excel to manage data
1.7 Expected Outcomes
The main outcomes of the present study are expected as following:
i. An equation of factor of safety based on limit equilibrium method that is
able to calculate the unique direction of sliding and factor of safety of 3D
slopes.
ii. A searching process based on particle swarm optimization that is able to
find the critical slip surface of 3D slopes.
iii. A computer program that is able to model 3D slopes and perform the
calculations of factor of safety and critical slip surface.
9
1.8 Limitations of the Study
There were some unavoidable limitations to carry out the present research.
Firstly, an elastic perfectly plastic behavior was adopted for the soil materials so the
linear Mohr-Coulomb failure criteria were used to determine the shear strength on
the slip surface. Secondly, the framework of limit equilibrium method was applied
in the analysis, thus the failure mass was considered as a rigid body and the static
equilibriums were used to establish the 3D equation of factor of safety. The third
limitation was related to the modelling of the slope. The method of columns with a
square grid was employed in the present study to discretize the sliding mass.
Therefore, the 3D failure mass was simulated by vertical columns.
As the next limitation, soil materials were assumed saturated by Terzaghi’s
(1936) linear equation of effective stress. Consequently, the effects of negative pore
pressure and nonlinear equations of effective stresses for unsaturated soils were not
included in the present study. Moreover, a hydrostatic water condition was assumed
to include the pore water pressure in the analysis, so the flow of moisture was not
considered. The fifth limitation was related to the materials of the slope. Although
the present study was able to handle different soil layers including weak layer and
bedrock, an isotropic and homogeneous condition was assumed for the behavior of
layers of the slope profile.
Finally, the mechanism of the failure was limited to 3D rotational or complex
rotational surfaces, so the translational sliding mode was not in the scope of the
analysis. Although a general ellipsoid shape was used in the analysis, other
rotational slip surfaces were also applicable within the framework of the present
study.
10
1.9 Definition of Terms
To provide a basis for discussion, following definitions are used in this study:
i. Method of slices: This method divides the sliding mass into a number of
vertical slices to establish the equation of factor of safety.
ii. Method of columns: This method can be considered as an extension of
method of slices into the third dimension that discretizes the failure mass
into a number of columns.
iii. Plane of symmetry: A hypothetical vertical plane that divides the sliding
mass into two symmetric parts.
iv. Graphical user interface: A graphic interface that is used to connect the
computer code and the user.
v. Optimization technique: A procees that uses different search techniques
among a series of probable answers to find the best suitable solution for a
problem.
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