robust design of rock slopes with multiple failure modes...
TRANSCRIPT
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Robust Design of Rock Slopes with Multiple Failure Modes 1
– Modeling Uncertainty of Estimated Parameter Statistics with Fuzzy Number 2
Changjie Xua, Lei Wang
b*, Yong Ming Tien
c, Jian-Min Chen
c, C. Hsein Juang
b 3
aCenter of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China 4 bGlenn Department of Civil Engineering, Clemson University, Clemson, SC 29634, USA 5
cDepartment of Civil Engineering, National Central University, Jhongli City, Taoyuan County 32001, Taiwan 6 7
*Corresponding author (email: [email protected]) 8
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Abstract: The variability of shear characteristics of rock discontinuities is often difficult to 10
ascertain. Thus, even with the reliability-based design (RBD) approach, which allows for 11
consideration of the uncertainty of input parameters, the design of a rock slope system may be 12
either cost-inefficient (overdesign) or unsafe (under-design), depending on whether the 13
variation of input parameters is overestimated or underestimated. The uncertainty about the 14
variation of input parameters is a critical issue in a RBD. This paper presents a feasible 15
approach to addressing this problem using robust design concept. First, the uncertainty of the 16
estimated statistics of input parameters (such as rock properties) is represented by fuzzy sets, 17
which requires only the knowledge of lower and upper bounds of the estimated statistics. 18
Then, the robust design concept is implemented to ensure that the final design is insensitive to, 19
or robust against, the uncertainty of the estimated statistics of input parameters. The design 20
methodology is demonstrated with an application to the design of a rock slope system with 21
multiple failure modes. This design methodology, termed Robust Geotechnical Design (RGD), 22
aims to achieve a certain level of design robustness, in addition to meeting safety and cost 23
requirements. In this paper, the RGD framework is realized through a multi-objective 24
optimization, as it involves three requirements, safety, cost, and robustness. The significance 25
of the design methodology is demonstrated with an example of rock slope design. 26
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Key words: Uncertainty; Reliability; Rock slope; Failure probability; Robust design; Fuzzy 28
sets; Optimization. 29
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1. Introduction 31
The design of rock slopes is often carried out in the face of uncertainty especially 32
regarding rock properties. Earlier design approaches using the deterministic methods rely on 33
factor of safety (FS) that is calibrated to engineering experience to compensate for the 34
uncertainty. More recently, various probabilistic approaches, including reliability-based 35
design methods, have been adopted to aid in the design of rock slopes (Hoek 2006; 36
Jimenez-Rodriguez et al. 2006; Jimenez-Rodriguez and Sitar 2007; Low 2007; Low 2008; 37
Penalba et al. 2009; Li et al. 2011; Lee et al. 2012; Park et al. 2012a&b; Tang et al. 2012). 38
However, the results of the reliability-based design of rock slopes are often affected by the 39
accuracy of the statistical characterization of the rock properties. However, the fully accurate 40
statistical characterization of rock properties along the rock discontinuities often requires a 41
high cost in site investigation and laboratory testing, which may not be feasible for each 42
design project. In this paper, we mainly focus on how to make rational decision with only 43
very limited data, which is a common scenario due to budget constraint in practice. 44
The variation of shear resistance properties of rock discontinuities may be affected by 45
the inherent variability of its strength and roughness, limited availability of quality samples 46
and testing, uncertainty in the adopted transformation models, and so on. With only very 47
limited data, the variation of rock properties, in terms of the coefficient of variation (COV), 48
often can only be estimated and expressed in a range based on published literature and 49
engineering judgment (Duncan 2000; Feng and Hudson 2004; Hoek 2006). 50
In a typical reliability analysis of the stability of rock slopes, a fixed value is often 51
arbitrarily assumed for the COV of each strength parameter within the perceived range (Lee et 52
al. 2012). However, the outcome of the reliability-based design is often very sensitive to the 53
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assumed COVs, and as a result, the design can be either cost-inefficient or unsafe depending 54
on whether these COVs are under- or over-estimated. A possible solution to this dilemma is to 55
adopt the robust design concept with which the design robustness against the variation of the 56
estimated COVs is achieved. In other words, the effect of the variation of the estimated COVs 57
is minimized through robust design (Juang and Wang 2013; Wang et al. 2013). It should be 58
noted that the design robustness may be measured in various ways (Taguchi 1986; Chen et al. 59
1996; Lagaros et al. 2010), and in this paper, it is measured in terms of the variation of the 60
system response (in terms of failure probability of the system). Thus, the design is considered 61
more robust if the variation of the system response is getting smaller. 62
Historically, the robust design concept was originated from Industry Engineering for 63
quality control (Taguchi 1986; Chen et al. 1996). This concept has since been applied to many 64
design fields, including structural design and mechanical design (Marano and Quaranta 2008; 65
Lagaros et al. 2010; Lee et al. 2010). The writers have also adapted this concept to formulate a 66
robust geotechnical design (RGD) methodology for geotechnical engineering problems (Juang 67
and Wang 2013; Wang et al. 2013). The RGD methodology seeks to achieve a certain level of 68
design robustness, in addition to meeting safety and cost requirements. In this paper, we 69
adapted this methodology to design rock slope systems with multiple failure modes. Herein, a 70
rock slope design is considered “robust” if the variation in the failure probability is insensitive 71
to the variation of noise factors (mainly uncertain COVs of rock properties). To evaluate the 72
variation in the failure probability of the designed rock slope, which is used as a means of 73
measuring the design robustness, the uncertain COVs of rock properties are modeled using 74
fuzzy numbers (see Section 2.1). These uncertainties in the estimated COVs, modeled as 75
fuzzy numbers, are propagated through the entire analysis process, rendering a measure of the 76
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variation of the response of the rock slope system. 77
In this paper, the robust design of a rock slope is to find an optimal design that is 78
insensitive to the variation of noise factors by carefully adjusting design parameters. The 79
noise factors refer to the parameters that cannot be controlled by the designer. For the rock 80
slope problem, the noise factors mainly refer to the uncertain rock properties. The design 81
parameters are the parameters that can be determined and controlled by the designer, such as 82
the geometry of the design for the rock slope problem. Here, the RGD framework is realized 83
through multi-objective optimization, in which the safety requirement is satisfied by the 84
reliability constraint and the design is optimized with respect to cost and robustness. For the 85
rock slope design, the objectives of cost and robustness are usually conflicting with each other, 86
and the multi-objective optimization generally yields a Pareto Front (Cheng and Li 1997), 87
which is a collection of multiple non-dominated optimal designs, as opposed to a single best 88
design. The Pareto Front essentially describes a trade-off relationship between robustness and 89
cost for all non-dominated designs that satisfy the safety requirement, thus it enables selection 90
of the best design according to a desired cost range or a target robustness level. A design 91
example of a rock slope system is used to demonstrate the significance of the proposed RGD 92
approach. The example is a hypothetical rock slope with two potential unstable blocks, which 93
involves multiple failure modes. 94
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2. Framework for Reliability-Based RGD Using Fuzzy Sets 96
2.1 Modeling parameter statistics using fuzzy numbers 97
The variation of the failure probability of the rock slope system is naturally affected by 98
the variation of the unknown COVs of input parameters (such as rock strength parameters). 99
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However, it generally difficult to characterize accurately the variation of these COVs given 100
limited quality data and spatial variability. Nevertheless, the engineer can usually characterize 101
these COVs as a range, as often reported in the literature. Thus, a reasonable compromise is to 102
construct a fuzzy number (Zadeh 1965; Zadeh 1978; Juang et al. 1992; Juang et al. 1998; Luo 103
et al. 2011) using the estimated range. For example, if the COV is estimated with a range of [a, 104
b], where a is the lowest conceivable value (or low bound) and b is the highest conceivable 105
value (or upper bound), then this COV (denoted as x in Figure 1) may be represented by a 106
fuzzy set (or fuzzy number) as shown in Figure 1. 107
A fuzzy set (Zadeh 1965) is mathematically defined as a set of ordered pairs, 108
[ , ( )]x x . The membership grade ( )x , ranging from 0 to 1, is used to characterize the degree 109
of belief that a member x belongs to this set. A fuzzy number is a fuzzy set that is normal and 110
convex, in which the shape of the membership function is single-humped and has at least one 111
value with a membership grade (or degree of belief) of 1. A triangular fuzzy number is often 112
used in the geotechnical applications, which is characterized with three values: a low bound, 113
an upper bound, and a mode. The mode has a membership grade of 1, the highest possibility, 114
to represent COV of rock properties (parameter x in Figure 1). As the value of the parameter 115
departs from the mode, the degree of belief for this value to represent the parameter x 116
decreases, and when the value reaches the lower bound (or the upper bound), the degree of 117
belief is reduced to zero (Juang et al. 1992; Juang et al. 1998; Luo et al. 2011). 118
Figure 1(a) shows a fuzzy number, in which the highest membership grade (degree of 119
belief) occurs at x equal to the mode m. In this fuzzy number, the lowest membership grade 120
(degree of belief) occurs when x is equal to either lower bound a or upper bound b. Note that 121
if m = (a+b)/2, the fuzzy number will be symmetric. On the other hand, if the mode (m) is not 122
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equal to (a+b)/2, the fuzzy number will be non-symmetric. A symmetric fuzzy number is the 123
most preferred choice to represent an uncertain variable when only knowledge of lower and 124
upper bounds is known (Juang et al. 1992). In this paper, the uncertainty in the estimated 125
COV of an input parameter is represented by a symmetric fuzzy number. 126
It should be noted that the above approach of modeling the uncertainty of the 127
estimated COV is not a limitation of the proposed robust design methodology but rather a 128
choice that works well with the stability problem of rock slopes with discontinuities. 129
2.2 Reliability-Based RGD Approach Using Fuzzy Sets 130
The reliability-based robust geotechnical design (RGD) methodology developed by 131
the authors (Juang and Wang 2013; Wang et al. 2013) is adapted for rock slope design 132
considering multiple failure modes in this paper. In particular, the uncertainty of the estimated 133
COVs of rock properties is modeled with fuzzy numbers as available rock properties data are 134
generally very limited, which makes it difficult to characterize the uncertainty with a 135
probability distribution function, and the formulation for the entire analysis process is 136
modified accordingly. The modified reliability-based RGD framework is summarized in six 137
steps as follows (in reference to Figure 2): 138
(1) Establish the deterministic model for stability analysis of rock slope system. 139
For a given slope, a proper deterministic model for rock slope analysis can be selected 140
based on the sliding mechanism and the number of removable (unstable) blocks. The 141
removable blocks are referred to as the rock masses geometrically isolated by discontinuity 142
planes (Giani 1992). In this paper, for the design case of a rock slope composed of two 143
blocks separated by a vertical tension crack as shown in Figure 3, the deterministic model 144
proposed by Jimenez-Rodriguez et al. (2006) is employed, which is summarized in 145
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Appendix (Appendix A, ESM only). 146
(2) Classify the input parameters and specify the design domain for the rock slope. 147
In the context of Robust Geotechnical Design (RGD), the input parameters are 148
classified into two categories: design parameters and noise factors. The design parameters 149
refer to the parameters that can be easily and accurately controlled by the designer and can 150
be treated as fixed values, while the noise factors mainly refer to inherent properties of 151
geo-materials that are difficult to control by the designer. In this paper, noise factors mainly 152
refer to uncertain rock properties (e.g., shear properties of discontinuity). 153
For the design of rock slope, there are generally two categories for design and 154
remedial measures. One is to reduce the slope height and slope face angle; the other is to 155
reinforce the slope by rock bolts and anchors. As noted by Hoek (2006), the design of rock 156
slope by reducing the height and face angle of slope is generally more cost-efficient than that 157
by reinforcing by rock bolts and anchors. Furthermore, the rock bolts may endure significant 158
strength reduction due to long-term corrosion, creep and deterioration effects, which are quite 159
difficult for long-term maintenance (Wang et al. 2013). In this study, for illustration purpose, 160
the design of rock slope is realized by manipulating the two design parameters, slope height H 161
and slope angle . The feasible ranges for these design parameters must be chosen and these 162
parameters are usually modeled as discrete values in the design space. 163
(3) Model the uncertainty in the estimated statistics of noise factors using fuzzy numbers. 164
In routine geotechnical exploration, the mean values of geotechnical parameters can 165
usually be adequately estimated even with a small sample of data (Wu et al. 1989). However, 166
the parameter statistics such as the coefficient of variation (COV) and correlation coefficient 167
(ρ) of rock properties are quite difficult to ascertain with a small sample of data; the COV 168
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and correlation coefficient of rock properties are usually estimated based on the typical 169
ranges reported in the literature and with engineering judgment (Hoek 2006). Under these 170
circumstances, the statistics (e.g., COV and ρ) of rock properties (these are noise factors in 171
the context of robust design) should be characterized logically with a range. For an uncertain 172
parameter that is characterized with only a lower bound and an upper bound, a fuzzy set 173
approach can be employed (Juang et al. 1992; Juang et al. 1998; Sonmez et al. 2004). In the 174
proposed RGD framework, the uncertainty in the estimated COVs and correlation 175
coefficients of rock parameters is modeled as a symmetric fuzzy number, which requires 176
only the knowledge of the estimated range [a,b], as depicted in Figure 1. 177
(4) Process the fuzzy data to obtain the variation of failure probability using fuzzy point 178
estimate method for robustness evaluation. 179
As noted previously, the robustness of a rock slope design is measured in this paper by 180
the variation (in terms of standard deviation) of the failure probability caused by variations of 181
the statistics of noise factors, and the smaller variation in the failure probability indicates the 182
greater robustness. If the statistics of rock strength properties (noise factors) can be 183
ascertained, the computed failure probability will be a fixed value. However, when the 184
statistics of noise factors cannot be ascertained (for example, can only be described with a 185
range and modeled with fuzzy number), the computed failure probability will be uncertain. 186
In this step, the variation of the failure probability for each of the totally Y possible 187
designs in the design space will be computed, as depicted with the 3rd
loop in Figure 2. In this 188
paper, the system reliability algorithm (Jimenez-Rodriguez et al. 2006; Jimenez-Rodriguez 189
and Sitar 2007) is used to compute the failure probability of the rock slope given a fixed set of 190
statistics of noise factors. The uncertainty in the computed probability of failure caused by the 191
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uncertainty of input statistics of noise factors (expressed as fuzzy numbers) is then evaluated 192
using the fuzzy set method. 193
Considering a performance function y = g (x1, x2, ..., xM), in which M is the number of 194
input uncertain statistics of noise factors (M equals seven in the case study presented later) 195
and y is the outcome of an algorithm for evaluation of system failure probability. For each 196
uncertain statistical parameter with only the knowledge of a range [a,b], a fuzzy number is 197
assigned as shown in Figure 1 (a). After the fuzzy numbers are defined, the vertex method 198
(Dong and Wong 1987; Juang et al. 1998) is used for propagation of the set of fuzzy numbers 199
through the adopted system reliability algorithm. 200
The vertex method is based on the α-cut concept, in which a fuzzy number is 201
discretized into a group of α-cut intervals. By drawing a horizontal line at a selected 202
membership value ( )x = k , an interval of two points can be obtained as shown in Figure 1 203
(b). For example, when is set as 0.2 for α ranged from 0 to 1, six different α-cut levels 204
can be obtained (α = 0, 0.2, 0.4, 0.6, 0.8 and 1.0). The step size of = 0.2 is generally 205
adequate for the fuzzy set-based reliability analysis (Luo et al. 2011). 206
At each α level, the intervals for each input variable are obtained, and the 207
combinations of vertexes (i.e., the lower bound and the upper bound for each input variable at 208
a given α level) are determined. For an analysis with M input variables, the number of 209
combination for vertexes (denoted as Z in Figure 2) is 2M
. Each combination of input 210
parameters is entered into the adopted system reliability algorithm (also referred to herein as 211
the performance function), which yields a solution (in this case, a failure probability). 212
Repeating this process for all combinations, a set of 2M
solutions is obtained. According to the 213
vertex method (Dong and Wong 1987), the maximum value and the minimum value of 214
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obtained 2M
solutions are taken as the resulting interval of the performance function y at the 215
specific α level (see the 1st loop of Figure 2). Repeating the above analysis for each of the six 216
α levels (N = 6), six corresponding intervals for y are obtained. These six α levels and the 217
corresponding intervals define the final fuzzy number that represents the output or the system 218
response (see the 2nd
loop of Figure 2). 219
With the resulting fuzzy number that represents the system response, the fuzzy-based 220
point estimate method (Fuzzy-based PEM) proposed by Dodagoudar and Venkatachalam 221
(2000) can be used to evaluate the mean and standard deviation of this system response. First, 222
denote the lower bound and the upper bound of the resulting fuzzy number as k
g and
k
g , 223
k = 1, 2,…, N, where N is the number of α cut levels (N = 6 in this case). The sum of the 224
function values at each α level considering the correlation effect between variables is 225
calculated as (Dodagoudar and Venkatachalam 2000): 226
k k k
r r rw p g p g (1) 227
where p and p are weighting factors, given by the following expression (Dodagoudar 228
and Venkatachalam 2000): 229
1
1 1
1
2
2
)1(1
1M
i
M
ij M
i
i
ijp
(2) 230
where M is the number of fuzzy input variables; ij is the correlation coefficient between 231
fuzzy variables ix and jx ; and i)1( is the skewness coefficient of the fuzzy variable ix . 232
The rth
moment of the function is calculated as: 233
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1[ ]k
Nr
kr k
w
E WN
(3) 234
With the first two moments of the output fuzzy number (the system response, or in this 235
case, the failure probability) being obtained by Eq. (3), the mean and standard deviation of the 236
failure probability can be determined. Specifically, the mean and standard deviation of the 237
failure probability (denoted as p and p ) are calculated as follows: 238
239
[ ]p E W (4) 240
2 2[ ] { [ ]}p E W E W (5) 241
The standard deviation of the failure probability computed by Eq. (5) is used herein as 242
a measure of design robustness. Greater design robustness is achieved with a smaller standard 243
deviation of the evaluated failure probability. It should be noted that, although the traditional 244
sensitivity analysis using upper and lower bounds of the input parameters can give us some 245
insights of the rock slope problem, however, such sensitivity analysis using upper and lower 246
bounds of the input parameters cannot tell us whether a design is robust, which requires the 247
knowledge of the variation of the system response. Processing fuzzy number data through the 248
analytical model allows us to evaluate the variation of the system response. The process 249
involves rigorous mathematics but in a straightforward and systematic way. 250
(5) Establish the Pareto Front optimized with respect to both robustness and cost. 251
Geotechnical designs often involve multiple criteria, including the requirements for 252
safety, cost and robustness. In this paper, a multi-objective optimization is performed to obtain 253
the optimal designs. The mean failure probability obtained by Eq. (4) is first compared with a 254
target (acceptable) failure probability to screen out the unsatisfactory designs. Once the safety 255
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constraint is satisfied, all acceptable designs are optimized with the other two objectives on 256
cost and robustness. 257
In this paper, the cost for a given design is simplified as the volume of rock mass to be 258
excavated, and the robustness is measured with the standard deviation of the failure 259
probability. The volume of rock mass to be excavated is determined by the difference between 260
initial volume of rock slope and volume of rock slope after remediation (Wang et al. 2013). 261
The multi-objective optimization is carried out using Non-dominant sorting genetic algorithm 262
(NSGA-II), developed by Deb et al. (2002). No single best design can be obtained if the 263
objectives are conflicting with each other. Thus, the multi-objective optimization often leads 264
to a group of non-dominated designs that are optimal to all objectives, which collectively 265
defines a Pareto Front (Cheng and Li 1997; Deb et al. 2002). As will be shown later, the 266
obtained Pareto Front offers a trade-off relationship between cost and robustness. This 267
trade-off relationship enables to reach a more informed decision especially when a desired 268
cost or robustness level is selected. 269
(6) Determine feasibility robustness to aid in the decision making. 270
In addition to Pareto Front, an index called “feasibility robustness” (Parkinson et al. 271
1993) may be used as a decision-making aid. For the rock slope design, the feasibility 272
robustness may be defined as the confidence probability that the failure probability of slope, 273
as designed, meets the constraint of a target failure probability. Symbolically, the feasibility 274
robustness is expressed as follows (Wang et al. 2013): 275
0[( ) 0]f TP p p P (6) 276
where fp is the computed failure probability of the slope system, which is not a fixed value 277
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because of the uncertainty in the statistics of noise factors, as reflected in step 4; Tp is the 278
target failure probability (in this paper, Tp 0.0062, which corresponds to a target reliability 279
index T 2.5, is adopted as per Low 2008); and 0P is an acceptable confidence 280
probability (say, 85%) specified by the designer. 281
To compute the feasibility robustness, [( ) 0]f TP p p , an equivalent counterpart in 282
form of [( ) 0]TP , where is the reliability index that corresponds to the failure 283
probability fp and T is the target reliability index ( T = 2.5), may be evaluated (Wang et 284
al. 2013). In fact, Eq. (6) can be re-written as: 285
0[( ) 0] ( )TP P (7) 286
where Φ is the cumulative standard normal distribution function, and is defined as: 287
2.5T
(8) 288
where and are the mean and the standard deviation of computed by the 289
Fuzzy-based PEM described and formulated previously. 290
Thus, instead of using the confidence probability, the index , referred to herein as 291
the feasibility robustness index, is used to measure the design robustness. A feasibility 292
robustness index of 0 corresponds to a confidence probability level of 50% that the reliability 293
(or failure probability) target will be satisfied. A feasibility robustness index of = 1 294
corresponds to a confidence probability level of 84.13% that the reliability (or failure 295
probability) target will be satisfied, and = 2 will raise the confidence probability to 296
97.72%. As will be shown in the illustrative example later, the feasibility robustness index 297
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( ) provides an easy-to-use measure of robustness to aid in the decision-making. 298
299
3. Example: Robust Design of Rock Slope with Multiple Failure Modes 300
3.1 Rock slope with multiple failure modes 301
Rock slopes are often composed of several potentially unstable blocks. In such cases, 302
the rock slope should be modeled as a rock slope system, as it may involve multiple failure 303
modes. For illustration purposes, a simple rock slope composed of two blocks separated by a 304
vertical tension crack (Jimenez-Rodriguez et al. 2006) is used as an example to illustrate the 305
rock slope with multiple failure modes. The position of tension crack separating two blocks is 306
considered random, which may be located at the slope top or slope face (see Figure 3). The 307
rock slope is considered safe if the factor of safety of block A is grater than unity using the 308
stability model by Hoek and Bray (1981) with a slight modification to account for the 309
interaction between blocks. Based on the condition of interaction between the two blocks, two 310
distinct scenarios are possible. In scenario 1, block B is stable by itself and there is no 311
interaction between the two blocks. In scenario 2, block B is unstable and tends to slide such 312
that an interaction force will be imposed on block A (Jimenez-Rodriguez et al. 2006). Detailed 313
formulation for factor of safety for both blocks is summarized in Appendix (Appendix A, 314
ESM only). 315
The rock slope composed of two removable blocks depicted in Figure 3 is used to 316
demonstrate the reliability-based robust geotechnical design. The initial geometry of the slope 317
is defined by a slope height of H = 25 m and a slope face angle of = 50°. The location of the 318
slip surface is assumed to be certain with a dip angle = 32°, and the unit weight of rock 319
(325 kN/m ) is considered a fixed value. 320
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As noted by Jimenez-Rodriguez et al. (2006), the parameters describing rock 321
properties along the slip surface, as well as the position of tension crack and water depth 322
should be considered as random variables in the reliability analysis of rock slope with two 323
removable blocks. Thus, totally seven random parameters, listed in Table 1, are considered in 324
the design analysis. Specifically, cohesion along the slip surface of block A and block B (cA 325
and cB), friction angle along the slip surface of block A and block B ( A and B ), as well as 326
friction angle along the contact surface between two blocks ( AB ) are assumed as truncated 327
normal random variables (Hoek 2006). The ratio of the tension crack depth filled with water 328
wz is assumed to follow the exponential distribution with a mean of 0.25 and truncated to the 329
range [0, 0.5] (Low 2007). The location of tension crack BX (see Figure 3) is modeled as a 330
non-symmetric beta distribution, in order to represent the common observations that tension 331
cracks are more commonly presented at the top of the slope (Hoek and Bray 1981; 332
Jimenez-Rodriguez et al. 2006). The probability density function of this beta distribution 333
bounded by [a, b] with two shape parameters q and r is given by (Ang and Tang 2007): 334
1 1
11 1 1
0
( ) ( )( )
(1 ) ( )
q r
q r q r
x a b xf x a x b
x x dx b a
(9) 335
Furthermore, cohesion and friction angle are assumed to be negatively correlated with 336
correlation coefficients , ,A A B Bc c 0.35 (Lee et al. 2012). On the other hand, shear 337
strength parameters between the two blocks are assumed positively correlated with correlation 338
coefficients , , , ,A B A AB AB B A Bc c = 0.3 (Jimenez-Rodriguez et al. 2006). All other 339
random variables are assumed independent. The correlation structure of these variables is 340
shown in Table 2. 341
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3.2 System reliability approach for rock slope with multiple failure modes 342
For the reliability analysis of a rock slope system with multiple failure modes, a 343
disjoint cut-set formulation (Jimenez-Rodriguez et al. 2006; Jimenez-Rodriguez and Sitar 344
2007) may be employed. The slope system is modeled as a series of sub-systems each with 345
parallel components (see Figure 4). Each sub-system is a cut-set that addresses a particular 346
failure mode, and within a given cut-set, there are three parallel components each defined by a 347
limit state function ig (see Table 3). Note that cut-sets are disjointed (meaning that there is 348
no intersection in any two cut-sets). Symbolically, k lC C , for k l and 349
, 1,2, Ck l N , where kC and lC are two given cut-sets, and CN is the number of 350
cut-sets. 351
Based on the cut-set formulation, the failure probability of the complete rock slope 352
system can be obtained by taking the summation of individual failure probabilities of these 353
cut-set as follows (Jimenez-Rodriguez and Sitar 2007): 354
1
( )C
k
N
ii C
k
P E P E
(10) 355
where E represents the failure event of the entire rock slope system; P is the probability of a 356
given event (for example, ( )P E is the probability of event E); Ei represents the failure event 357
of ith
component of kth
cut-set Ck; the notation k
ii C
E
represents the intersection of the three 358
components of a cut-set Ck. 359
For a given cut-set (failure mode) Ck, the probability of failure can be calculated as 360
follows (Ditlevsen and Madsen 1996; Jimenez-Rodriguez 2004): 361
( , ) k
k
i Ci C
P E
β R (11) 362
17
where is the cumulative standard multi-normal distribution function; kCβ is the vector 363
composed of reliability index ( i ) of each component i in the cut-set Ck; R is the correlation 364
matrix between components within the cut-set Ck. 365
The reader is referred to Ditlevsen and Madsen (1996) for further details of this 366
formulation. The probability of failure for each of the four cut-sets (failure modes) can be 367
computed using Eq. (11), then the failure probability of the rock slope system can be obtained 368
using Eq. (10). As demonstrated by Jimenez-Rodriguez (2004), this approach is 369
computationally effective and provides a good approximation to the “exact” probability of 370
failure given by Monte Carlo simulation. 371
3.3 Traditional reliability-based design of rock slope system 372
With the defined statistics and correlation structure of input parameters listed in Table 373
1 and Table 2, the hypothetical rock slope system that is composed of two removable blocks 374
(Figure 3) can be analyzed with the traditional reliability-based design method. 375
The design space for design parameters should first be specified for the 376
reliability-based design. For the two design parameters of the hypothetical rock slope (see 377
Figure 3 and Tables 1 and 2), slope height H may be in the range of 20 m to 25 m, and slope 378
angle may be in the range of 40° to 50°. The upper bounds of design parameters are chosen 379
based on the initial slope condition and the lower bounds of design parameters are chosen 380
based on the geometry requirement of the deterministic model. For convenience in the rock 381
slope construction, H is rounded to nearest 0.2 m and is rounded to nearest 0.2°. Thus, H 382
can take from 26 discrete values and can take from 51 discrete values, with totally 1326 383
possible designs in the design space. 384
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If the statistics of the rock properties can be accurately characterized (so that they can 385
be described with fixed values as listed in Tables 1 and 2), the probability of failure of the 386
rock slope system for each possible design in the design space can be computed using the 387
system reliability approach (Jimenez-Rodriguez and Sitar 2007) discussed previously. The 388
results of these system reliability analyses are shown in Figure 5. As expected, the failure 389
probability increases with the increase of slope height and slope angle. The design with the 390
least cost while satisfying the reliability requirement (i.e., the computed failure probability is 391
less than a threshold value) is usually selected as the final design (Zhang et al. 2011). For 392
illustration purpose, the target failure probability Tp 0.0062 (as per Low 2008) is used as the 393
reliability constraint in this paper. The final design following the traditional reliability-based 394
design method yields a design with design parameters H = 25 m, = 45.6° and a design cost 395
of 43.8 units [note: 1 unit = unit cost ($) of excavated volume of rock mass in m3/m]. 396
However, the final design H = 25 m, = 45.6° is based on the assumption of fixed 397
parameter statistics (i.e., COVs and correlation coefficients) listed in Table 1. In practice, the 398
statistics of rock properties are usually difficult to ascertain. However, with the aid of 399
published COVs of rock properties and engineering judgment, these statistics may be 400
estimated as a range. For example, the coefficient of variation (COV) of cohesion c, denoted 401
as [ ]COV c , typically ranges from 10% to 30% (Lee et al. 2012); the COV of friction angle 402
, denoted as [ ]COV , typically ranges from 5% to 10% (Alejano et al. 2012). In addition, 403
the correlation coefficient between c and , denoted as ,c , typically ranges from -0.1 to 404
-0.6 (Lee et al. 2012). These published numbers can be used as a guide along with engineering 405
judgment to characterize the uncertainty of the statistics of these rock properties. 406
19
To examine the possible effect of the uncertainty in the estimated statistics of rock 407
properties on the final design of the hypothetical rock slope system, ,c is assumed as a 408
fixed value of -0.35, and [ ]COV c and [ ]COV are assumed to vary within their typical 409
ranges. Table 4 lists the final designs for various assumed COV levels obtained using the 410
traditional reliability-based design approach. It is found that the final design obtained with the 411
traditional reliability-based design approach is very sensitive to the levels of COVs of rock 412
properties. 413
Under the assumption of the lowest variability level ( [ ]COV c = 0.1 and [ ]COV = 414
0.05), all designs in the design space satisfy the reliability requirement. In this scenario, the 415
design that represents the initial slope system, which of course requires no excavation, is the 416
least cost design that satisfies the reliability requirement (i.e., the evaluated failure probability 417
is less than a threshold value of Tp 0.0062). However, under the assumption of the highest 418
variability level ( [ ]COV c = 0.3 and [ ]COV = 0.1), all designs in the design space, including 419
the one with the highest cost (H = 20 m, = 40° with design cost of 176.2 units), cannot 420
satisfy the reliability requirement. 421
The implication from the results shown in Table 4 is that the final design through 422
traditional reliability-based design approach is dependent on assumed variability levels of 423
rock properties. If the variation of rock properties is overestimated, the resulting final design 424
will not be cost-efficient. On the other hand, if the variation of rock properties is 425
underestimated (for example, when actual variation of rock properties is higher than the 426
assumed variation), the initially acceptable design (e.g., H = 25 m, = 45.6° based on the 427
assumed statistics listed in Tables 1 and 2) will no longer meet the reliability requirement. As 428
20
shown in Table 5, under the actual variation of rock properties ( [ ]COV c = 0.3 and [ ]COV = 429
0.1), the computed failure probability for the initially acceptable design (H = 25 m, = 45.6°) 430
is 0.0184, much higher than the target probability of failure ( Tp 0.0062). 431
3.4 Reliability-based robust design of rock slope with multiple failure modes 432
For the hypothetical rock slope composing of two removable blocks (Figure 3), the 433
design parameters are the slope height H and slope angle . As discussed previously, the 434
design space consists of 1326 possible designs with different combination of H and . For this 435
problem, the noise factors are mainly associated with uncertain rock properties, including cA, 436
cB, A, B, and AB. Here, seven statistical parameters, namely [ ]ACOV c , [ ]BCOV c , 437
[ ]ACOV , [ ]BCOV , [ ]ABCOV , ,A Ac , and ,B Bc , may be treated as fuzzy numbers. For 438
demonstration purpose, the typical ranges for [ ]COV c from 10% to 30%, [ ]COV from 439
5% to 10%, and ,c from -0.1 to -0.6, as reported in the literature, are used as the basis for 440
establishing fuzzy numbers that characterize these uncertain statistical parameters. Of course, 441
local experience and engineering judgment can play a significant role in selecting a proper 442
range to best characterize the uncertainty of these statistics. As an example, for the statistical 443
parameters [ ]ACOV c and [ ]BCOV c , the fuzzy number is constructed with only the 444
knowledge of lower bound (a = 0.1) and upper bound (b = 0.3), with the implied mode m = 445
0.2, as shown in Figure 1. 446
Following the RGD procedure outlined previously (in reference to Figure 2), the mean 447
and standard deviation of the failure probability for the hypothetical rock slope system, 448
denoted as p and p , can be obtained using Fuzzy-based PEM for each of the 1326 449
designs in the design space. For illustration purpose, the resulting mean and standard 450
21
deviation of selected designs with H = 20 m, 21 m, ... , 25 m are shown in Figure 6 and Figure 451
7, respectively (note: the initial geometry of the slope is H = 25 m and = 50°). Then, the 452
multi-objective optimization is set up as follows: 453
454
Find d = [H, ] 455
Subjected to: H {20m, 20.2m, 20.4m,…, 25m } and {40°, 40.2°, 40.6°,…, 50° } 456
0.0062p Tp 457
458
Objectives: Minimizing the standard deviation of failure probability ( p ) 459
Minimizing the cost for rock slope design. 460
461
Using the NSGA-II algorithm, 64 designs out of the 1326 designs are found 462
satisfactory with respect to all constraints and most optimal with respect to both objectives of 463
robustness (measured with p ) and cost. These 64 designs are more optimal than other 464
designs in the design space, but within the set of 64 designs, none of them is superior to any 465
others in all objectives. These 64 designs collectively form a Pareto Front, as shown in Figure 466
8. Recall that no design that belongs to the Pareto Front can be improved with respect to one 467
design objective without weakening the performance in the other objective (Deb et al. 2002). 468
Thus, the Pareto Front in this case offers a trade-off relationship between cost and robustness 469
(in terms of the standard deviation of the failure probability). 470
Figure 9 further depicts the relationship between cost and robustness for all designs on 471
the established Pareto Front (Figure 8), where the robustness is now measured by the 472
feasibility robustness index . Recall that every point on the Pareto Front is a non-dominated 473
design that satisfies the safety requirement (i.e., the mean failure probability must be smaller 474
than a target failure probability). If the least cost design is desired by the engineer, the final 475
22
design of H = 25 m and = 45.2° (the lowest point shown in Figure 9) will be selected, which 476
costs 48.1 units. This design has a feasibility robustness index of = 0.27, which is 477
equivalent to saying that there is a 60.57% chance (or confidence probability) that the design 478
will meet the safety requirement in face of the variation of the parameter statistics. 479
If the uncertainty of the estimated parameter statistics is unavoidable and an estimate 480
of this uncertainty (variation) represents the best knowledge we have, then it may be desirable 481
to select the final design based on a trade-off consideration using the established Pareto Front 482
or the cost- relationship derived from it. For example, an increase in the feasibility 483
robustness to = 1, which increases the chance of satisfying the safety requirement to 484
84.13%, will result in a final design of H = 25 m and = 42.6° (note: this data is not revealed 485
in Figure 9) that costs 77.6 units. 486
Thus, the final design may be selected based on a prescribed feasibility robustness 487
level (for example, = 1 or 2, and so on). In this case, we are confident that the final design 488
will have a certain percentage of satisfying the safety requirement (for example, 84.13% with 489
= 1; 97.72% with = 2, see Table 6) even with the existence of uncertainty in the 490
estimated parameter statistics. Alternatively, the designer can select the final design as the one 491
that gives the highest feasibility robustness under a certain cost. Either way, the Pareto Front 492
established using the reliability-based RGD methodology, or the cost- relationship derived 493
from it, provides a useful decision making tool that can aid in selecting the final design in a 494
more rational and easily-communicated way. 495
To help with selection of the final design using the developed Pareto Front (see Figure 496
8 or Figure 9), a further step can be undertaken to identify the knee point on the Pareto Front. 497
23
Based on the definition given by Deb and Gupta (2011), the knee point is the most preferred 498
design, since it requires a large sacrifice in one objective to gain a minor improvement in the 499
other objective. The normal boundary intersection method (Deb and Gupta 2011) may be used 500
to identify the knee point on the Pareto Front. A boundary line is first constructed by 501
connecting the two extreme points on the Pareto Front, and then the point on the Pareto front 502
that has the maximum distance to this boundary line is identified as the knee point (see 503
annotation in Figure 8 and Figure 9). Based on the normal boundary intersection method, the 504
same knee point is identified for Figure 8 and Figure 9, which is the design with H = 25 m and 505
θ = 40° that costs 110.2 units. Below this cost level, it requires a large sacrifice on robustness 506
to achieve a minor gain in cost-efficiency (reduction in cost). On the other hand, above this 507
cost level, it requires a large sacrifice in cost-efficiency to achieve a minor gain in robustness 508
improvement (i.e., increase of feasibility robustness or reduction of variation of failure 509
probability). This knee point concept provides an additional tool for selecting the most 510
preferred design on the Pareto Front. 511
512
4. Concluding Remarks 513
In this paper, a fuzzy set approach is incorporated into the reliability-based Robust 514
Geotechnical Design (RGD) framework to deal with the uncertainty in the estimated statistics 515
of rock properties for design of a rock slope system. Use of a fuzzy number to describe or 516
model the uncertainty in the estimated statistics (such as COV) is deemed appropriate, as the 517
amount of quality data is generally very limited. Construction of a fuzzy number for such 518
situation requires only the knowledge of a highest conceivable value and a lowest conceivable 519
value, which enables the engineer to quantify uncertainty based on the reported ranges from 520
24
literature and augmented with local experience and engineering judgment. It should be noted, 521
however, that use of the fuzzy set (or fuzzy number) approach within the RGD framework is 522
only for its practicality, and should not be viewed as a limitation of the reliability-based RGD 523
methodology. The proposed design approach has been demonstrated with an application to the 524
design of rock slope system considering multiple failure modes, and its effectiveness has been 525
demonstrated with the results presented in this paper. 526
In typical geotechnical practice, statistical parameters such as the coefficient of 527
variation and the correlation coefficient between noise factors (e.g., uncertain rock properties 528
in this paper), which are required in a reliability-based design, are difficult to ascertain. If 529
these statistics of noise factors are overestimated or underestimated, the final design obtained 530
from the traditional reliability-based design will be either cost-inefficient or unsafe. By 531
considering explicitly the robustness against the uncertainty in the estimated statistics of rock 532
properties, the RGD approach reduces the adverse effect of such uncertainty. In fact, by 533
considering three objectives, safety, robustness, and cost, in the design using a multi-objective 534
optimization within the RGD framework, a set of optimal, non-dominated designs, which 535
form a Pareto Front collectively, can be obtained. The established Pareto Front is shown to be 536
a useful decision-making tool, which has been demonstrated in the rock slope design example 537
presented in this paper. 538
It should be noted that the proposed RGD approach is mainly for improving decision 539
making under scarce and incomplete information. The RGD approach is not a methodology to 540
compete with reliability-based design, but is a complementary to reliability-based design 541
under scarce and incomplete information. If sufficient information is available for a 542
comprehensive assessment of all uncertainty involved in rock slope design (e.g., the 543
25
probability distributions of all input parameters can be fully and accurately characterized), the 544
traditional reliability-based design approach will suffice for rock slope design. Under the 545
scenario of scarce and incomplete information, the RGD approach often can reduce the 546
adverse effect of the uncertainty associated with the estimated statistical parameters. 547
548
ACKNOWLEDGMENTS 549
The study on which this paper is based was supported in part by National Science 550
Foundation through Grant CMMI-1200117. The results and opinions expressed in this paper 551
do not necessarily reflect the view and policies of the National Science Foundation. 552
26
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30
Lists of Tables
Table 1. Statistics of random variables for rock slope example with multiple failure modes
(modified after Jimenez-Rodriguez et al. 2006)
Table 2. Correlation structure of random variables for rock slope example with multiple
failure models (modified after Jimenez-Rodriguez et al. 2006)
Table 3. Physical interpretation of limit state functions that define the component performance
for analysis of rock slope system (modified after Jimenez-Rodriguez et al. 2006)
Table 4. Least-cost designs under various COV assumptions for rock properties
Table 5. Failure probability of a given design (H = 25.0 m, = 45.6°) under various COV
assumptions of rock properties
Table 6. Selected final designs at various feasibility robustness levels
31
Lists of Figures
Figure 1. An example of fuzzy number and α-cut interval (modified after Luo et al. 2011)
Figure 2. Flowchart for reliability-based robust geotechnical design of rock slope system
Figure 3. Geometrical definition of rock slope stability model with two removable blocks
(adapted from Jimenez-Rodriguez et al. 2006): (a) Tension crack at slope top; (b)
Tension crack at slope face
Figure 4. Disjoint cut-sets system formulation of rock slope system with two removable
blocks (adapted from Jimenez-Rodriguez et al. 2006)
Figure 5. Failure probabilities for selected designs obtained using system reliability approach
with fixed statistics for rock properties
Figure 6. Mean failure probabilities for selected designs
Figure 7. Standard deviation of failure probabilities for selected designs
Figure 8. Pareto Front based on objectives of cost and robustness
Figure 9. Cost versus feasibility robustness for all designs on the Pareto Front
32
Table 1. Statistics of random variables for rock slope example with multiple failure modes
(modified after Jimenez-Rodriguez et al. 2006)
Random variables Probability distribution Mean COV
Ac (kPa) Normal 20
0.2
Bc (kPa) Normal 18
0.2
A (°) Normal 36 0.075
B (°) Normal 32 0.075
AB (°) Normal 30 0.075
BX Beta distribution, q = 3, r = 4, a = 0, b = 1
wz Exponential with mean 0.25, truncated to the range [0, 0.5]
33
Table 2. Correlation structure of random variables for rock slope example with multiple
failure models (modified after Jimenez-Rodriguez et al. 2006)
Ac Bc A B AB BX
wz
Ac 1.0 0.3 -0.35 0.0 0.0 0.0 0.0
Bc 0.3 1.0 0.0 -0.35 0.0 0.0 0.0
A -0.35 0.0 1.0 0.3 0.3 0.0 0.0
B 0.0 -0.35 0.3 1.0 0.3 0.0 0.0
AB 0.0 0.0 0.3 0.3 1.0 0.0 0.0
BX 0.0 0.0 0.0 0.0 0.0 1.0 0.0
wz 0.0 0.0 0.0 0.0 0.0 0.0 1.0
34
Table 3. Physical interpretation of limit state functions that define the component performance
for analysis of rock slope system (modified after Jimenez-Rodriguez et al. 2006)
Limit state function Physical Interpretation Eqs.
1 (1 cot tan ) 0g z H Tension crack at top of slope (A-12)
2 1{ | 0} 1 0Bg FS g
Block B is unstable (without interaction
from A), given tension crack located at top
of slope
(A-7) &
(A-10)
3 1{ | 0} 1 0Bg FS g
Block B is unstable (without interaction
from A), given tension crack located at face
of slope
(A-7) &
(A-11)
4 1 2{ | ( 0, 0)} 1 0Ag FS g g
Block A is unstable given tension crack at
top of slope, block B is stable (no
interaction occurs)
(A-1) &
(A-5)
5 1 2{ | ( 0, 0)} 1 0Ag FS g g
Block A is unstable given tension crack at
top of slope, block B is not stable
(interaction occurs)
(A-13) &
(A-5)
6 1 3{ | ( 0, 0)} 1 0Ag FS g g
Block A is unstable given tension crack at
face of slope, block B is stable (no
interaction occurs)
(A-1) &
(A-6)
7 1 3{ | ( 0, 0)} 1 0Ag FS g g
Block A is unstable given tension crack at
face of slope, block B is not stable
(interaction occurs)
(A-13) &
(A-6)
35
Table 4. Least-cost designs under various COV assumptions of rock properties
[ ]COV c [ ]COV H (m) (°) Cost (units)
0.10 0.05 25.0* 50.0* 0.0
0.10 0.075 25.0 49.2 7.5
0.10 0.10 25.0 46.6 33.3
0.20 0.05 25.0 49.0 9.4
0.20 0.075 25.0 45.6 43.8
0.20 0.10 25.0 42.8 75.3
0.30 0.05 25.0 43.6 65.9
0.30 0.075 25.0 40.0 110.2
*Slope in its initial condition, which involves no excavation work.
36
Table 5. Failure probability of a given design (H = 25.0 m, = 45.6°)
under various COV assumptions of rock properties
[ ]COV c [ ]COV H (m) (°) fp
0.10 0.05 25.0 45.6 0.0008
0.10 0.075 25.0 45.6 0.0021
0.10 0.10 25.0 45.6 0.0046
0.20 0.05 25.0 45.6 0.0030
0.20 0.075 25.0 45.6 0.0060
0.20 0.10 25.0 45.6 0.0101
0.30 0.05 25.0 45.6 0.0079
0.30 0.075 25.0 45.6 0.0127
0.30 0.10 25.0 45.6 0.0184
37
Table 6. Selected final designs at various feasibility robustness levels
0P H (m) (°) Cost (units)
0.5 69.15% 25.0 44.4 56.9 1.0 84.13% 25.0 42.6 77.6 1.5 93.32% 25.0 40.6 102.4 2.0 97.72% 20.2 40.0 173.0
38
0.0
0.5
1.0
2000 2050 2100 2150 2200
Mem
ber
ship
- D
egre
e o
f su
pp
ort
a bm
Parameter x
(a) Triangular fuzzy number
0.0
1.0
2000 2050 2100 2150 2200
Mem
ber
ship
- D
egre
e o
f su
pp
ort
a bm
Parameter x
ix
ix
i
(b) -cut interval
0.0
1.0
2000 2050 2100 2150 2200
Mem
ber
ship
- D
egre
e o
f su
pp
ort
a bm
Parameter x
(b) -cut interval
k
kx
kx
(a)
(b)
Figure 1. An example of fuzzy number and α-cut interval (modified after Luo et al. 2011)
39
Figure 2. Flowchart for reliability-based robust geotechnical design of rock slope system
Yes
1st loop:
Yes
Yes
2nd loop: Repeat N times
Complete
repetitions for all
N α-levels?
Classify noise and design parameters and specify design domain
Discretize fuzzy numbers into N α-cut intervals
For each α-level, determine Z
vertex combinations
Determine the α-cut interval of
failure probability of rock slope system for each α-level
No Repeat Z times
START
Multi-objective optimization for Pareto Front considering safety, robustness, and cost
DESIGN DECISION
Establish deterministic model for
stability analysis of rock slope system
Complete Z repetitions for each
α-level?
Determine feasibility robustness for each design on Pareto Front
No
Represent uncertainty in the statistics of noise factors using fuzzy numbers
Compute failure probability for each of Z combinations using the
system reliability approach
Complete
repetitions for each
of Y designs?
3rd loop: Repeat Y times
Determine mean and std. dev. of failure probability using fuzzy-based PEM
No
40
H
θ ψ
zz H
hh H
FI AB
A
B
ww zz h
AA XX XBB XX X
cot pX H
(a)
H
θ ψ
zz H
hh HFI AB
A
B
AA XX XBB XX X
cot pX H
ww zz h
(b)
Figure 3. Geometrical definition of rock slope stability model with two removable blocks
(adapted from Jimenez-Rodriguez et al. 2006): (a) Tension crack at slope top; (b) Tension
crack at slope face
41
1 0g 1 0g 1 0g
1 0g
3 0g 3 0g 2 0g
2 0g
7 0g 6 0g 5 0g 4 0g
Failure Mode 1 Failure Mode 2 Failure Mode 3 Failure Mode 4
Figure 4. Disjoint cut-sets system formulation of rock slope system with two removable
blocks (adapted from Jimenez-Rodriguez et al. 2006)
42
0.000
0.005
0.010
0.015
40 42 44 46 48 50
Slope Angle, θ (°)
Pro
bab
ilit
y o
f F
ailu
re
H = 20 m H = 21 m
H = 22 m H = 23 m
H = 24 m H = 25 m
0.0062Tp
Figure 5. Failure probabilities for selected designs obtained using system reliability approach
with fixed statistics for rock properties
43
0.000
0.005
0.010
0.015
40 42 44 46 48 50
Slope Angle, θ (°)
Mea
n P
rob
abil
ity
of
Fai
lure
H = 20 m H = 21 m
H = 22 m H = 23 m
H = 24 m H = 25 m
0.0062Tp
Figure 6. Mean failure probabilities for selected designs
44
0.000
0.005
0.010
0.015
40 42 44 46 48 50
Slope Angle, θ (°)
Std
. D
ev.
of
Pro
bab
ilit
y o
f F
ailu
re 1
H = 20 m H = 21 m
H = 22 m H = 23 m
H = 24 m H = 25 m
Figure 7. Standard deviation of the failure probabilities for selected designs
45
0
50
100
150
200
0.000 0.001 0.002 0.003 0.004
Std. Dev. of Probability of Failure
Pareto Front
Knee Point
Boundary Line
Cost
(unit
s)
Figure 8. Pareto Front based on objectives of cost and robustness
46
0
50
100
150
200
0.0 0.5 1.0 1.5 2.0 2.5
Feasibility Robustness Level
Pareto Front
Knee Point
Boundary Line
Cost
(unit
s)
Figure 9. Cost versus feasibility robustness for all designs on the Pareto Front