vii
TABLE OF CONTENTS
CHAPTER
TITLE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
2
xiiii
LIST OF FIGURES
xv
LIST OF ABBREVIATIONS
xx
LIST OF SYMBOLS
xxi
LIST OF APPENDICES
1
PAGE
xxvii
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
LITERATURE REVEIW
11
2.1
11
Introduction
viii
2.2
Slope Stability Analysis based on Limit Equilibrium
Method
2.2.1
2.2.2
12
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
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
2.3
2.4
2.5
3
4
Sun, Zheng, and Jiang’s Method
Critical Slip Surface
69
2.3.1
70
Finding the Critical Slip Surface
Optimization Techniques
72
2.4.1
72
Global Optimization Techniques
Summary and Discussion
76
RESEARCH METHODOLOGY
81
3.1
Introduction
81
3.2
Procedure of the Research
82
3.3
Data Sources
84
3.4
Instrumentation
84
DEVELOPMENT OF INNOVATED THREEDIMENSIONAL 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
4.4
5
67
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
Summary and Discussion
114
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
5.5
Application of Particle Swarm Optimization in Slope
5.6
6
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
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
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
6.3
6.4
7
125
152
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
Summary
161
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
163
Problem 1
7.2.1.1
7.2.1.2
7.2.2
7.2.2.2
7.2.3.2
Results and Discussion of Problem 1
165
167
Geometry and Material Properties of
Problem 2
168
Results and Discussion of Problem 2
169
175
Geometry and Material Properties of
Problem 3
176
Results and Discussion of Problem 3
176
Problem 4
7.2.4.1
7.2.4.2
7.3
164
Problem 3
7.2.3.1
7.2.4
Problem 1
Problem 2
7.2.2.1
7.2.3
Geometry and Material Properties of
182
Geometry and Material Properties of
Problem 3
182
Results and Discussion of Problem 4
183
Validation of the Computer Program
188
7.3.1
188
Case Study 1
7.3.1.1
Geometry and Material Properties of
the Case Study 1
7.3.1.2
Results and Discussion of the Case
Study 1
7.3.2
Case Study 2
7.3.2.1
199
200
Results and Discussion of the Case
Study 2
8
192
Geometry and Material Properties of
the Case Study 2
7.3.2.2
190
202
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.
2.1
TITLE
Available equations of typical 2D slope
( Fredlund and Krahn, 1977)
2.2
PAGE
17
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)
7.14
191
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
2.1
Forces in the method of slices ( Fredlund and Krahn, 1977)
2.2
Forces acting on a thin vertical slice in Anagnosti’s method
(Anagnosti, 1969)
2.3
16
26
The combined slip surfaces of Baligh and Azzouz’s method
(Baligh and Azzouz, 1975)
2.4
PAGE
28
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)
2.7
Chen and Chameau’s method after applied assumptions
(Chen and Chameau, 1983)
2.8
35
Three-dimensional problem of Dennhardt and Forster’s
method (Dennhardt and Forster, 1985)
2.9
34
37
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)
2.13
42
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)
2.19
Discretized slip surface of Leshchinsky and Huang
(Leshchinsky and Huang, 1992b)
2.20
63
Forces acting on a typical column of Cheng and Yip’s method
(Cheng and Yip, 2007)
2.27
61
Typical column of Chen et al.’s method a) before, and b) after
applying the assumptions (Chen et al., 2003)
2.26
59
Component of base shear strength at the base of column and
their projection on x-y plane (Huang et al., 2002)
2.25
57
Three-dimensional model of Huang and Tsai’s method
(Huang and Tsai, 2000)
2.24
55
Typical slip surface and a column in Jiang and Yamagami’s
method (Jiang and Yamagami, 2004)
2.23
53
Discretization of slip surface in Lam and Fredlund’s method
(Lam and Fredlund, 1993)
2.22
52
Slip surface of Cavounidis and Kalogeropoulos’ method
(Cavounidis and Kalogeropoulos, 1992)
2.21
51
65
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)
4.3
Base angles and sliding angle of a typical soil column
(Huang and Tsai, 2000)
4.4
95
Boundaries normal forces in x- and y-directions
(Huang et al., 2002)
4.5
94
102
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
7.8
177
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
7.12
(a) cylindrical shape, (b) sliding mass, and (c) sliding mode of
the 3D critical slip surface for problem 3
7.13
182
Generated model of problem 3 by current study in
(a) plan view, (b) central cross section, and (c) 3D view
7.15
181
(a) half-plan view and (b) central cross section of geometry of
problem 4 (Yamagami and Jiang, 1997)
7.14
180
184
(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
7.22
Geological map of part 2, phase I of line 2 of KUR
(DKPCE, 2005)
7.23
190
191
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
ei,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