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Simplified Robust Geotechnical Design 1
of Soldier Pile-Anchor Tieback Shoring System for Deep Excavation 2
3
Wenping Gong a,b*
, Hongwei Huang a, C. Hsein Juang
b,c, and Lei Wang
d 4
5
Abstract 6
The shoring system that consists of soldier piles and anchor tiebacks is often used in 7
deep excavations in sandy deposits. However, uncertainties often exist in the design of such 8
shoring systems. Here, a simplified robust geotechnical design (RGD) method is proposed to 9
account for these uncertainties in the shoring system design. Specifically, for a given deep 10
excavation, uncertain soil parameters and surcharges are treated as noise factors, and the 11
parameters of soldier piles and tieback anchors are treated as design parameters. Robust 12
design is then implemented as a multi-objective optimization problem, in which the design 13
robustness is sought along with cost efficiency and safety requirements. A tradeoff between 14
design robustness and cost efficiency exists and the optimization usually leads to a Pareto 15
front. By applying the knee point concept, the most preferred design that meets the safety 16
requirements and yields the best compromise between design robustness and cost efficiency 17
can be identified on the Pareto front. Improvements made to the existing RGD method include 18
an efficient formulation of the design robustness and a new procedure for finding the most 19
preferred design in the design pool. The new simplified RGD method is illustrated with a 20
real-world excavation case study. 21
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Keywords: Deep Excavation; Design Robustness; Knee Point; Optimization; Pareto Front; 23
Shoring System; Uncertainty. 24
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_______________________ 26 27 a Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China. 28
29 b Glenn Department of Civil Engineering, Clemson University, Clemson, SC 29634, USA. 30
31 c Department of Civil Engineering, National Central University, Taoyuan 320, Taiwan. 32
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d Department of General Engineering, Montana Tech of the University of Montana, Butte, MT 59701, 34
USA. 35
36 * Corresponding author (Email: [email protected]) 37
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1. Introduction 39
In the traditional design of shoring systems using the deterministic methods, the 40
design must meet the safety or stability requirement, in terms of limiting factors of safety 41
against failures of the ground and shoring system, and the serviceability requirement, in terms 42
of limiting maximum wall and/or ground deformation for preventing damages to adjacent 43
structures (JSA 1988; PSCG 2000; TGS 2001; Kim and Lee 2005; Ou 2006; Kim et al. 2012; 44
Goh and Mair 2014). However, the deterministic methods do not allow for an explicit 45
consideration of uncertainties in the input parameters. Because of these uncertainties, however, 46
the computed system response (i.e., the response of a designed system) is subject to variation. 47
Thus, the design decision has to compensate for variation or uncertainty in the computed 48
system response, which often leads to a cost-inefficient design. 49
To explicitly account for the uncertainties in the input parameters, the probabilistic 50
design approach may be adopted (Harr 1987; Baecher and Christian 2003; Zhang et al. 2005; 51
Ang and Tang 2007; Fenton and Griffiths 2008; Juang et al. 2013b; Gong et al. 2014c). 52
However, a meaningful probabilistic design requires an accurate statistical characterization of 53
input parameters such as soil parameters, which is often difficult to attain in a typical 54
geotechnical project due to budget constraint. Furthermore, use of the probabilistic approach 55
to deal with uncertain soil parameters in the design of a complex shoring system based on 56
finite element method (FEM) is often computationally prohibitive. 57
In this paper, the robust geotechnical design (RGD) method is adopted to deal with the 58
uncertainties in the design of shoring systems for deep excavations. The robust design concept 59
is originated in the field of industry engineering for product optimization (Taguchi 1986), and 60
has attracted increasing interests from many other engineering disciplines (Phadke 1989; 61
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Doltsinis et al. 2005; Beyer and Sendhoff 2007). The robust design concept has also been 62
applied in various geotechnical problems (Juang et al. 2012a; Juang et al. 2013a; Wang et al. 63
2013; Juang et al. 2014; Gong et al. 2014a). In the context of robust design, a design is 64
deemed robust if the system response of concern, such as the maximum wall deflection in the 65
case of deep excavations, is insensitive to the variation of uncertain input parameters (termed 66
noise factors herein). Thus, the goal of robust design is to seek a safe and cost-efficient 67
optimal design, the system response of which is robust against, or insensitive to, the 68
unforeseen variation of input parameters; such an optimal design may be achieved through a 69
proper selection of design parameters (i.e., those that can be specified by the designer). 70
This paper presents an application of RGD method to the design of a shoring system 71
for an excavation. Note that while the RGD method has been reported in various geotechnical 72
applications (Juang et al. 2012a; Juang et al. 2013a; Wang et al. 2013; Khoshnevisan et al. 73
2014; Juang et al. 2014; Gong et al. 2014a; Gong et al. 2014b), this paper has three new 74
features. First, a new gradient-based robustness measure is adopted, which, as demonstrated 75
later, eliminates the need of computing design robustness based on probabilistic analyses of 76
the system response, a step that becomes computationally prohibitive when the system 77
response can only be analyzed using numerical methods such as FEM. Second, the RGD 78
method is usually implemented as a multi-objective optimization problem (Juang et al. 2013a; 79
Juang et al. 2014), and the results are usually presented as a Pareto front (Deb 2001). When 80
the results are presented as a Pareto front, the knee point can usually be identified on this front, 81
which offers the best compromise solution or the most preferred design in the design pool. A 82
new procedure is developed in this study to find the most preferred design in the design pool; 83
this procedure involves a series of single-objective optimizations, rather than a 84
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multi-objective optimization, and thus greatly reduces the computation efforts. In this paper, 85
the RGD implemented with these new procedures is termed the simplified RGD method. 86
Third, the simplified RGD method is applied to the design of a relatively complex shoring 87
system for a deep excavation project, in which the system response of concern is analyzed 88
with a commercially available computer program TORSA (Sino-Geotechnics 2010), and a 89
comparison with the original design that was selected by an engineering firm is made. 90
Through this real-world application, the advantages of the new simplified RGD method are 91
demonstrated. 92
This paper is organized as follows. First, the formulation of a new robustness measure 93
based on the gradient of the system response is presented. Second, the framework of the 94
simplified RGD method is presented, in which a new procedure to find the most preferred 95
design in the design pool is developed. Third, the results of a case study of a real-world deep 96
excavation project using the simplified RGD method are presented. Finally, results of a series 97
of parametric analyses to demonstrate the advantages of the simplified RGD method are 98
presented, which are followed by the concluding remarks. 99
100
2. Gradient-Based Design Robustness Measure: Sensitivity Index (SI) 101
In the construction of a deep excavation project in an urban area, the maximum wall 102
deflection is often monitored and used to assess the performance of the ground-wall-shoring 103
system. In general, an excessive wall deflection would signal poor stability and serviceability 104
performance of the shoring system, which poses a risk to the excavation project and adjacent 105
utilities (Juang et al. 2012b). Although the wall deflection is correlated well with the ground 106
settlement (Clough O’Rourke 1990; Kung et al. 2007), the former is much easier to monitor in 107
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the field, and is easier to analyze with better accuracy. Thus, the performance of the shoring 108
system can be monitored, and remedial actions can be taken as needed, through the wall 109
deflection monitoring during the construction phase. In this paper, the maximum wall 110
deflection is adopted as the system response of concern in the context of robust design; in fact, 111
use of the maximum wall deflection as the system response in the robust design has been 112
reported (Juang et al. 2014; Wang et al. 2014). 113
2.1. Gradient as a robustness measure 114
For a design of a shoring system for a deep excavation project with design parameters 115
of d and noise factors of as inputs (note: both are vectors), the system response of concern, 116
in terms of the maximum wall deflection (y), can be represented as a function of d and , 117
denoted as y = f(d, ). Here, the noise factors are referred to uncertain soil parameters and 118
surcharge on the ground surface behind the wall, and the design parameters are referred to the 119
dimensions, sizes, and layouts of the soldier piles and tieback anchors. Here, the system 120
response function may be constructed using an analytical method, empirical model or finite 121
element method (FEM). In this paper, a special-purpose FEM code TORSA 122
(Sino-Geotechnics 2010) is adopted. TORSA is an FEM code based on the 123
beam-on-elastic-foundation theory and has been used in hundreds of real-world excavation 124
projects. 125
Mathematically and intuitively, the sensitivity of the system response, f(d, θ), to the 126
noise factors can be captured by its gradient f, which is defined as the variation of the 127
system response caused by one unit change in the noise factors. In reference to Figure 1, two 128
different designs (referred to herein as d1 and d2) are examined to illustrate the gradient-based 129
robustness concept. Here, d2 is seen more robust than d1 against the variation of noise factors, 130
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as the variation of the system response of d2 is much smaller than that of d1. Furthermore, the 131
gradient of the system response to the noise factors is lower in d2 than d1. That is to say, the 132
variation of the system response (due to the perceived but uncharacterized uncertainties of 133
noise factors) is proportional to the gradient of the system response to the noise factors. Thus, 134
the design robustness can be effectively represented with the gradient of the system response 135
to the noise factors. Symbolically, the gradient of the system response to the noise factors, f, 136
at a checkpoint of noise factors, denoted as ' , can be computed as follows (Gong et al. 137
2014b): 138
1 2
( , ) ( , ) ( , ), , ,
n
f f ff
'
' ' '
d d d (1) 139
where n is the number of noise factors. In a deterministic design of the shoring system for 140
deep excavation, only the nominal values of noise factors, denoted as θn, are characterized 141
and could be available to the engineer. Thus, the characterized nominal values of noise factors 142
can reasonably be assigned as the checkpoint in Eq. (1): ' = θn. 143
Since no explicit form of the system response function could be obtained in FEM 144
solutions, the closed-form solution of the partial derivatives in Eq. (1) cannot be obtained. In 145
this study, the numerical approach is adopted and implemented, within which, the partial 146
derivative of the system response to a noise factor (θi) can be approximated as: 147
( , ) ( , ) ( , )
2
i i i i
i i
f f ' d f ' d
d
' '
d d d (2) 148
where id represents a step increment of the ith noise factor, which is taken as 0.1 i' in this 149
study. It is noted that in the computation of ( , )i if ' d d and ( , )i if ' d d , the values of 150
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all other noise factors are kept at their checkpoints ( ' ). The step increment of 0.1 i' is 151
obtained through a trial and error analysis, which yields converged results. This parameter 152
might be problem-specific and should be verified for a future problem. 153
Though formulated in a deterministic manner aiming at improving the computation 154
efficiency, a design (or system) with a lower gradient generally exhibits a lower variation in 155
the computed system response (regardless of the levels of relative variation among the noise 156
factors) and thus it yields a more robust design, and vice versa, as illustrated in Figure 1. 157
Therefore, the gradient of the system response to the noise factors is an effective measure of 158
the design robustness, and is suitable for use in a robust design optimization. 159
2.2. Formulation of sensitivity index (SI) 160
While the gradient f, defined in Eq. (1), is shown as an effective indicator of the 161
design robustness, two problems need to be resolved before the robust design optimization 162
can be implemented. First, the gradient is an n-dimensional vector; as the units of noise 163
factors are different, the mathematical operation of this vector could be a problem. Second, 164
the gradient is a vector rather than a scalar; it is not as convenient and effective as a scalar to 165
use for screening candidate designs in the design pool. 166
To solve the first problem, each partial derivative in the gradient vector,( , )
i
f
'
d, 167
is multiplied by a scaling factor of i' so that the effect of the units of noise factors on the 168
formulation of design robustness can be eliminated. Indeed, scaling the gradient term to avoid 169
the problem caused by inconsistent units is not uncommon (Gonsamo 2011). As such, the 170
gradient vector in Eq. (1) is re-written as follows, which is defined herein as the normalized 171
gradient vector (J): 172
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1 2
1 2
( , ) ( , ) ( , ), , ,
n
n
' f ' f ' f
' ' '
d d dJ
(3) 173
To solve the second problem, the Euclidean norm of the normalized gradient vector, 174
which indicates the length of the normalized gradient vector, is adopted and defined herein as 175
the sensitivity index (SI): 176
TSI J J (4) 177
Eq. (4) yields a single value representation of the normalized gradient vector, in terms of the 178
sensitivity index (SI). As can be seen, a higher SI value signals lower design robustness, as it 179
indicates a greater variation of the system response in the face of the uncertainty in noise 180
factors. It is found that although gradient concept might be viewed as “local”, the sensitivity 181
index defined in this paper is sufficient for the proposed robust design optimization. For more 182
complex geotechnical systems that exhibit significant nonlinear behaviors, use of advanced 183
sensitivity index such as Sobol index (Marrel et al. 2009) may be worth pursuing. 184
It is noted that while the gradient-based robustness measure was introduced in a 185
previous paper (Gong et al. 2014b), three new features are introduced in this study. First, the 186
evaluation of design robustness is coupled with the deterministic analysis of the system 187
response. Second, the nominal values of noise factors are assigned as the checkpoint for 188
computing the gradient of the system response to the noise factors. Third, a full statistical 189
characterization of noise factors is not required. Whereas, in the previous study (Gong et al. 190
2014b), the system response was analyzed using probabilistic methods; the most probable 191
point (MPP) of noise factors that yields the failure of the system was taken as the checkpoint 192
for computing the gradient of the system response to the noise factors; and, the full statistical 193
information of noise factors is a prerequisite for conducting the robust design. 194
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In general, different noise factors may have different levels of variation; and a robust 195
design should be less sensitive (or more robust) to the noise factor that has higher variability. 196
However, in the deterministic analysis and design, the uncertainty or variation of noise factors 197
is recognized but uncharacterized. Indeed, if the statistics of noise factors are available, the 198
reliability-based robust geotechnical design method (Juang et al. 2013a; Wang et al. 2013) can 199
readily be used. While the reliability-based robust geotechnical design method is theoretical 200
sounder, it requires the knowledge of the statistics of noise factors and probabilistic analyses 201
of the system response. In this paper, although the variation levels of noise factors are not 202
included in the formulation of the normalized gradient vector and sensitivity index, the new 203
robustness measure presented herein offers a practical alternative to the reliability-based 204
robustness measures, as it does not require the knowledge of the variation levels of noise 205
factors and probabilistic analyses of the system response can be avoided. The results of the 206
case study and the sensitivity analysis presented later confirm the effectiveness of this new 207
gradient-based robustness measure. 208
209
3. Simplified Robust Geotechnical Design (RGD) 210
3.1. Multi-objective optimization setting for robust design optimization 211
One unique feature of the RGD method is that it treats the geotechnical design as a 212
multi-objective optimization problem that considers explicitly and simultaneously the safety 213
requirements, cost efficiency, and design robustness (Juang et al. 2013a). Following the RGD 214
framework developed by Juang et al. (Juang et al. 2013a), the safety requirements are 215
considered as compulsory design constraints that must be satisfied, while the design 216
robustness and cost efficiency are treated as design objectives to be optimized simultaneously. 217
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The robust design optimization algorithm for the design of a shoring system for a given deep 218
excavation can be set up as follows (modified after Juang et al. 2014): 219
1
2
factor
Find:
of safety
(design p
against push-in
arameters)
Subject to: (des
failur
ign pool)
Fs 1. e
factor of safety against basal
5 ( )
Fs 1.5 (
>
>
d
d S
)
( , ) 0.7% ( )
Objectives: min (sensitivity of
heave failure
maximum wall deflection
ma wall deflection to noise factors )
min (cost)
ximum
f y = f < H
SI
C
d
(5) 220
where Hf is the final excavation depth. The design constraint of ( , ) 0.7% fy = f < Hd is a 221
limiting condition that prohibits the maximum wall deflection (y) from exceeding 0.7% fH , as 222
per PSCG (2000). Here, the robust geotechnical design is to seek an optimal design 223
(represented by design parameters d) in the design pool (S) such that the cost (C) is 224
minimized and the design robustness is maximized (which is achieved by minimizing the 225
sensitivity index SI), while the safety requirements with respect to both stability (in terms of 226
Fs1 and Fs2) and serviceability (in terms of maximum wall deflection y) are satisfied. It is 227
noted that while there are many other potential failure models in deep excavations, such as the 228
failure of the anchor systems and the retaining structure, they are not studied in this paper for 229
simplicity. 230
In reference to the optimization setting denoted as Eq. (5), the design robustness is 231
formulated in a deterministic manner and the safety requirements are evaluated with the 232
deterministic methods. In this formulation, there is no need to perform a detailed statistical 233
characterization of noise factors and a full probabilistic analysis of the system response. Note 234
that while the uncertainties of noise factors are not explicitly considered in this optimization 235
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setting, the design robustness of the system response against the variation of noise factors is 236
optimized. Indeed, this optimization setting can raise the design robustness by minimizing the 237
sensitivity index and thus reduces the variation of the system response. This is the essence of 238
the proposed new robustness measure, referred to herein as the simplified RGD method. 239
3.2. Pareto front and knee point 240
In general, the desire to maximize the design robustness (or to minimize the sensitivity 241
index) and the desire to minimize the cost are two conflicting design objectives in the robust 242
design optimization. Thus, a utopia design that is optimal with respect to both design 243
objectives simultaneously is not attainable; rather, a set of non-dominated designs can be 244
identified in the design pool that are superior to all others in the design pool, but within which, 245
none of them is superior or inferior to others. These non-dominated solutions collectively 246
form a Pareto front (Deb 2001), which shows a tradeoff between design robustness and cost 247
efficiency. This Pareto front may be obtained using genetic algorithms such as 248
Non-dominated Sorting Genetic Algorithm version II, NSGA-II (Deb et al. 2002). 249
As a tradeoff between design robustness and cost efficiency, the Pareto front is an 250
effective design aid. For example, either the least cost design that is above a target robustness 251
level or the most robust design that falls within a target cost level can be selected as the most 252
preferred design in a given design pool. The determination of an appropriate target level of 253
design robustness or cost, however, is problem-specific. When no such a design preference is 254
specified by the owner or client, the knee point on the Pareto front, which yields the best 255
compromise between design robustness and cost efficiency, may be taken as the most 256
preferred design in the given design pool (Branke et al. 2004; Deb and Gupta 2011; Juang et 257
al. 2014; Gong et al. 2014b). Figure 2 shows a conceptual illustration of the utopia design, 258
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Pareto front and knee point in a bi-objective optimization problem. Note that the feasible 259
domain is defined as a domain that consists of feasible designs that meet the safety 260
requirements. 261
Oftentimes, only a single knee point can be obtained from a multi-objective 262
optimization, as shown in Figure 2(a). However, multi-knee points may be possible for some 263
complex problems, as illustrated in Figure 2(b) (Branke et al. 2004; Deb et al. 2006; Bechikh 264
et al. 2011). In such a circumstance, the global knee point, rather than the local knee point, is 265
taken herein as the most preferred design in the design pool. In this paper, the term “knee 266
point” is referred to the global knee point if no specific interpretation is given. 267
3.3. Existing approaches to identify the knee point 268
Three existing approaches to identify the knee point on the Pareto front, the reflex 269
angle approach, normal boundary intersection approach, and marginal utility function 270
approach, are briefly summarized herein to set the stage for the proposed new procedure. In 271
these methods, the objective functions are often normalized into a value ranging from 0.0 to 272
1.0, denoted as n ( )f d , through the following transformation: 273
n min
max min
( ) ( )( )
( ) ( )
i i
i
i i
f ff
f f
d dd
d d (6) 274
where max
( )if d and min
( )if d are the maximum and minimum values, respectively, of the 275
ith objective function, ( )if d . 276
Reflex angle approach 277
The reflex angle at a non-dominated design on the Pareto front indicates the bend of 278
the Pareto front from its left to right and hence can be used as a measure of the gain-sacrifice 279
in the tradeoff relationship. The knee point is identified as the non-dominated design on the 280
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Pareto front that yields the maximum reflex angle, as shown in Figure 3(a). It is noted that the 281
reflex angle is computed using only two neighboring points, thus it can correspond to a local 282
property and may not yield a global knee point in some cases (Branke et al. 2004; Deb and 283
Gupta 2011). 284
Normal boundary intersection approach 285
To alleviate the possible local knee point using the reflex angle approach, the normal 286
boundary intersection approach can be used. As shown in Figure 3(b), the two extreme 287
designs of a Pareto front, denoted as A and B, are used to construct a straight boundary line; 288
and the non-dominated design on the Pareto front that has the maximum distance from the 289
constructed boundary line is identified the knee point (Deb and Gupta 2011; Juang et al. 290
2014). 291
Marginal utility function approach 292
As can be seen, the reflex angle approach and the normal boundary intersection 293
approach are valid and efficient only for the bi-objective optimization problem. To deal with 294
multi-objective optimization that involves more than two design objectives, the marginal 295
utility function approach (Branke et al. 2004; Gong et al. 2014b) may be used. The marginal 296
utility function, denoted as ,U' d , is formulated as: 297
, min , , , ( )i j iU' U U i j d d d (7) 298
where ,U d is a linear utility function of the design objectives, defined as: 299
n, ( )i iU f d d (8) 300
where i is a weighting parameter with a value ranging from 0.0 to 1.0, and i = 1.0; and 301
n ( )if d is the ith normalized objective function to be minimized. By means of Monte Carlo 302
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simulations (MCS), different random values of i can be generated with an assumption of 303
uniform distribution, and then the expected marginal utility function can be computed for each 304
non-dominated design on the Pareto front. The non-dominated design with the maximum 305
expected marginal utility is taken as the knee point on the Pareto front. 306
3.4. New procedure to identify the most preferred design 307
Oftentimes, the owner or client may be only interested in the most preferred design in 308
the design pool, and not the Pareto front per se. In fact, if the most preferred design in the 309
design pool can be identified directly, use of multi-objective optimization algorithms such as 310
NSGA-II would not be needed; and thus, the RGD can be more readily adaptable for general 311
applications. In this paper, a new procedure to identify the most preferred design in the design 312
pool is developed. The new procedure is summarized in the following steps: 313
Step 1: Conduct a single-objective optimization with respect to the ith design objective 314
of concern, ( )if d , which might be set up as follows: 315
Find: (design parameters)
Subject to: (design pool)
(design pool)
Safety requirements
Objective: ( ) min ( )
i
i
i i i
f f
d *
d * S
d S
d * d
(9) 316
where id * represent the optimal design based on the ith
design objective, which meets the 317
safety requirements and yields the minimum of the ith objective function, denoted as 318
min
( )if d = ( )i if d * = min ( )if d . 319
By repeating the single-objective optimization in Eq. (9) for each and every design 320
objective, a utopia point of { 1 min( )f d , 2 min
( )f d , …, min
( )mf d } can be constructed in the 321
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design pool, where m represents the number of design objectives to be optimized. As implied 322
by its name, the utopia design is in reality not attainable if the design objectives are 323
conflicting with each other. Note that the utopia design defined this way may also violate the 324
safety requirements or even fall outside the design pool. 325
Step 2: Determine the maximum value of each objective function among the designs 326
of {1d * , 2d * , …,
md * }. Symbolically, the maximum value of the ith objective function is 327
obtained as follows: 328
max
( ) max ( ) =1, 2, , i i jf f j m d d * (10) 329
Step 3: Normalize all the objective functions into a value ranging from 0.0 to 1.0 330
using the transformation described in Eq. (6). As a result, the coordinates of the normalized 331
utopia point are all equal to 0.0. 332
Step 4: Compute the distance from the normalized utopia point to the normalized 333
objective functions for each candidate design in the design pool. The design that meets the 334
safety requirements and yields the minimum distance is regarded as the most preferred design 335
in the design pool, as shown in Figure 3(c). 336
Note that the computed distance from the utopia point to a candidate design can be 337
interpreted as an additional “price”, in terms of a combination of the design robustness and 338
cost efficiency, one has to pay to select this candidate design, in lieu of the utopia point 339
(design), as the final design. This idea is inspired by the marginal utility concept (Branke et al. 340
2004) and the compromise programming approach (Chen et al. 1999). In reality, the utopia 341
design is not attainable. Thus, the design that meets the safety requirements and yields the 342
minimum distance (thus incurring the least additional price) can be treated as the most 343
preferred design in the design pool. As will be shown later, the most preferred design 344
17
identified with this new procedure is virtually the same as the knee point on the Pareto front 345
obtained through a multi-objective optimization. 346
Note that with this new procedure, the multi-objective optimization is solved with a 347
series of single-objective optimizations. Here, the single-objective optimization can be easily 348
performed using Matlab function fmincon (MATLAB 2010) or Microsoft Excel Solver that 349
returns the minimum of a constrained nonlinear multivariable function. Thus, use of 350
multi-objective optimization algorithms such as NSGA-II is no longer required in the RGD 351
implemented with this new procedure. It should be noted that many outstanding studies on 352
knee points have been published (Deb et al. 2006; Bechikh et al. 2011). However, the 353
development of this new procedure described herein places a high premium on simplicity, 354
aiming to enable the simplified RGD method as a practical geotechnical design tool. Further, 355
it is noted that this new procedure is applicable to the multi-objective optimization with more 356
than two design objectives, although only a bi-objective optimization problem is illustrated in 357
Figure 3(c). For convenience of description, the RGD implemented with the new robustness 358
measure and the new knee-point seeking procedure is referred to hereinafter as the simplified 359
RGD method. 360
361
4. Robust Geotechnical Design of Shoring System for Deep Excavation Case Study 362
4.1. Brief summary of the deep excavation case studied 363
To illustrate the simplified RGD method, a real-world deep excavation project in 364
Taiwan is studied herein as a demonstrative example. The selected deep excavation case is 365
located in Taipei, Taiwan, which is designed for the student dormitory and underground 366
parking of Wesley Girls High School. The plan layout of the excavation site is approximately 367
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a rectangular shape with a length of 140 m and a width of 45 m. The excavation depths range 368
from 6.9 m to 11.9 m, as shown in Figure 4. For illustration purposes, the robust design 369
optimization is only focused on section D, although other sections can be designed using the 370
same simplified RGD method. With respect to section D, the excavation is carried out in 371
layered soils with sands and rock. The soil profiles and soil properties for each soil layer are 372
shown in Figure 5 and listed in Table 1, respectively. The effect of groundwater is negligible 373
as the groundwater table is well below the excavation depth. Based on the local experience of 374
similar projects, a soldier pile wall that consists of reinforced concrete piles and timber 375
laggings is used as the retaining structure, while the 4-strand anchor (each with diameter of 376
12.7 mm) is used as the tieback of the retaining structure. 377
4.2. Deterministic model for assessing the performance of the shoring system 378
In this study, TORSA, a commercially available computer program developed by 379
Trinity Foundation Engineering Consultants (TFEC), is adopted as the deterministic model for 380
evaluating the performance or response of the shoring system for the deep excavation. Of 381
course, other special-purpose computer programs such as DeepXcav (Deep Excavation 2014) 382
or general FEM computer programs such as PLAXIS (Likitlersuang et al. 2013; Plaxis 2014) 383
can be used. TORSA has been validated as an effective design tool in analyzing the stability 384
(i.e., push-in and basal heave) and serviceability (i.e., wall deflection) of the shoring system 385
for deep excavation through numerous projects in Taiwan (Sino-Geotechnics 2010). 386
As noted previously, TORSA is an FEM code based on the beam-on-elastic-foundation 387
theory. The retaining structure, in terms of the soldier pile wall studied in this paper, is 388
modeled as an elastic beam. The pressure acting on the back of the soldier pile wall is 389
assumed to be the active earth pressure and the resistance of soil inside the excavation is 390
19
modeled as soil springs (Ou 2006; Sino-Geotechnics 2010). The water pressure, surcharge, 391
tieback anchor (or internal strut) stiffness and preload, and the passive earth pressure caused 392
by the plastic deformation of soil inside the excavation can be considered. A limit equilibrium 393
model that considers all the forces and soil springs applied on the soldier pile wall is 394
formulated. This equilibrium model is then solved via the finite element method (FEM) 395
implemented in TORSA. With TORSA, the factor of safety against push-in failure (Fs1), the 396
factor of safety against basal heave failure (Fs2), and the maximum wall deflection (y) can 397
readily be obtained. 398
4.3. Noise factors characterization for the robust design optimization 399
For the deep excavation project illustrated in Figure 5, the main soil parameters 400
affecting the stability and serviceability behavior of the shoring system are the cohesion (c), 401
friction angle (φ) and modulus of horizontal subgrade reaction (kh). These soil parameters, 402
along with the surcharge behind the wall (q0), are the controlling input parameters. Possible 403
high variation in these parameters necessitates their treatment as noise factors in the robust 404
design. The variation of soil parameters might be attributed to the inherent variability, 405
measuring error, and transformation uncertainty (Phoon and Kulhawy 1999), while the 406
variation of the surcharge behind the wall may come from the nearby transportation and 407
heaping. 408
With limited data availability at a given site in a typical geotechnical practice, an 409
accurate statistical characterization of the aforementioned noise factors may not be possible; 410
rather, these noise factors are usually characterized with their nominal values based on the 411
available data and local experience (in a deterministic manner). Listed in Table 1 are the 412
characterized nominal values of these soil parameters in the case studied (Figures 4 and 5). 413
20
Similarly, the surcharge behind the wall (q0) is characterized with a nominal value of 1.0 414
ton/m2. Based on the characterized nominal values of noise factors, the deterministic analysis 415
and design of the shoring system for the deep excavation can readily be conducted. Note that 416
the nominal values, listed in Table 1, are the only information available to the engineer in the 417
context of deterministic design. 418
4.4. Design pool selection for the robust design optimization 419
Comparing to the hard-to-characterize (or hard-to-control) noise factors, the diameter 420
of the concrete soldier pile (D), length of the concrete soldier pile (L), interval of concrete 421
soldier piles or center-to-center spacing of soldier piles (I), vertical spacing of tieback anchors 422
(V), horizontal spacing of tieback anchors (H), and installed angle of tieback anchors with 423
respect to the horizontal direction () are the easy-to-control input parameters. These input 424
parameters can be selected by the designer; they are classified as design parameters in the 425
robust design of the shoring system for the deep excavation. 426
Apart from the aforementioned noise factors and design parameters, the stability and 427
serviceability behaviors of the shoring system can also be affected by other input parameters, 428
such as the preload of tieback anchors and the length of tieback anchors. For the limited scope 429
of this paper, the preload of tieback anchors are all set up as 20 ton/tieback, and the free 430
length and fixed length of tieback anchors are both designed as 8.0 m based on the local 431
design experience. 432
For ease of construction, a discrete design pool is often considered based on the local 433
experience. For example, the discrete design pool (S), tabulated in Table 2, may be adopted 434
for the robust design optimization of this demonstrative excavation case study. In the selected 435
design pool, a total number of 38,500 discrete candidate designs is possible (see Table 2) and 436
21
they will be analyzed and the optimal design will be derived accordingly. Given a specified 437
vertical spacing of anchor tiebacks, the excavation steps can be determined accordingly, as 438
shown in Figure 5. In Figure 5, the location of the tieback anchor is set at approximately 0.8 439
m above the excavation depth at each excavation stage, except for the last excavation stage. 440
4.5. Cost estimate of the shoring system for the deep excavation 441
As formulated in Eq. (5), cost efficiency and design robustness are the two design 442
objectives in the robust design optimization of the shoring system for the deep excavation 443
project. For a given excavation project, the site dimensions and excavation depth are fixed 444
according to either the structural or architectural requirements. The major cost of concern in 445
the robust design optimization is the cost of the shoring system (C), which is the sum of the 446
cost for the soldier pile wall (Cw) and the cost for the tieback anchor systems (Ct), expressed 447
as: 448
w tC C C (11) 449
Using the geotechnical practice in Taiwan as an example, the cost for the soldier pile 450
wall (Cw) may be estimated as (Hsii-Sheng Hsieh, personal communication 2013): 451
2
10.6
w
SL DC L t
I
(12) 452
where SL is the perimeter of the excavation site (m). In this study, SL is taken as 54.08 m as 453
only section D in Figure 4 is considered. The parameter t1 is the unit price (US dollar) for a 454
0.6 m diameter pile, and in this study, t1= 66 US dollars. 455
Similarly, the cost for the tieback systems (Ct) may be estimated as (Hsii-Sheng Hsieh, 456
personal communication 2013): 457
2 2 3 t
SLC VL t l t
H
(13) 458
22
where VL is the number of vertical levels of the tieback anchor systems, which are determined 459
from Figure 5. The parameter t2 is the unit price (US dollar) for the unit length of the anchor, 460
l2 is the length of the anchor, and t3 is the unit price (US dollar) for the anchor head. In this 461
study, these unit prices are: t2 = t3 = 66 US dollars (note: it is a coincidence that t1 = t2 = t3 in 462
this design example). Note that the cost formulated above (Eqs. 11-13) is the construction cost 463
and the material cost is already included in which. 464
It should be noted that Eqs. (12) and (13) represent a rule of thumb used by Trinity 465
Foundation Engineering Consultants in its cost estimation for the design of the shoring system 466
for the excavation project that is being re-analyzed in this paper. It is adopted herein as an 467
example for the cost estimate of the robust design optimization. Any reasonable cost estimate 468
model that is applicable to the local excavation project can be used. 469
4.6. Robust design optimization results: Pareto front and knee point 470
Based on the robust optimization setting presented previously (see Eq. 5), the robust 471
design of the shoring system for the deep excavation can be conducted. For each of the 472
candidate designs in the specified design pool (see Table 2), the safety requirements (i.e., 473
factor of safety against push-in failure Fs1, factor of safety against basal heave failure Fs2, and 474
maximum wall deflection y), cost (Eq. 11), and design robustness (in terms of sensitivity 475
index SI) can be readily evaluated. Then, a Pareto front that consists of 61 non-dominated 476
designs is established using the sorting algorithm of NSGA-II (Deb et al. 2002), as shown in 477
Figure 6(a). It is noted that all the non-dominated designs on the Pareto front are superior to 478
all others in the feasible domain. Among the non-dominated designs on the Pareto front, 479
however, none of them is superior or inferior to others when all objectives are considered. 480
Thus, a tradeoff exists between design robustness and cost efficiency, as depicted in Figure 481
23
6(b). By applying the normal boundary intersection approach (Deb and Gupta 2011; Juang et 482
al. 2014), a knee point is identified as shown in Figure 6(b). Although not shown herein, the 483
same knee point is obtained using the marginal utility function approach (Branke et al. 2004; 484
Gong et al. 2014b). 485
On the left side of the knee point, a slight reduction in the cost leads to a drastic 486
increase in the sensitivity index SI, indicating a drastic reduction in design robustness. On the 487
other side of the knee point, a slight reduction in SI requires a large increase in the cost. Thus, 488
the knee point can be treated as the most preferred design in the design pool if no design 489
preference is specified by the owner or client. 490
For comparison, two other designs on the Pareto front, one representing the least-cost 491
design and the other the most robust design (which is also the most costly), are also shown in 492
Figure 6(c) along with the knee point and the original design. The original design is the one 493
designed by an experienced engineering firm (Hsii-Sheng Hsieh, personal communication 494
2013) without the knowledge of robust design. While the original design appears to be a 495
sound engineering practice, offering a compromise between the least cost design and the most 496
robust design in this case, it is inferior to the knee point on the Pareto front, as the latter is 497
more robust and cost less. Of course, the knee point is the most preferred design obtained with 498
a Pareto front that was obtained through robust design optimization. The design parameters, 499
safety performance (i.e., factor of safety against push-in failure Fs1, factor of safety against 500
basal heave failure Fs2, and maximum wall deflection y), sensitivity index (SI), and cost (C) 501
of these four designs are compared in Table 3. 502
While all four designs listed in Table 3 meet the safety requirements, such as Fs1 >1.5, 503
Fs2 > 1.5, and y < 0.7%Hf = 8.33 cm, different designs have different levels of design 504
24
robustness and cost efficiency. In this excavation problem, the least-cost design among the 505
pool of feasible designs is also the least robust design, in which a slight variation in noise 506
factors can result in a highest variation in the system response, which is not desirable. On the 507
other hand, the most robust design corresponds to the least cost efficiency, which is not 508
desirable, either. The knee point on the Pareto front, or the most preferred design in the design 509
pool, is shown as the best compromise between these two extremes on the Pareto front. 510
Compared with the original design that was selected by the local engineering firm without 511
using robust design, the knee point is cheaper and much more robust. 512
4.7. Simplified RGD with new procedure to identify the most preferred design 513
Although the RGD method shows a clear advantage of being able to find the best 514
compromise solution, or most preferred design, in the design pool, it requires use of genetic 515
algorithms such as NSGA-II (Deb et al. 2002) for the multi-objective optimization. For a 516
robust geotechnical design that involves FEM modeling in a multi-objective optimization 517
framework, the required computation effort can be overwhelming for a routine project. The 518
simplified RGD method with the proposed new procedure can find the most preferred design 519
in the design pool through a series of single-objective optimizations, without the need to use 520
genetic algorithms such as NSGA-II. 521
Using the simplified RGD method, the most preferred design in the design pool is 522
obtained without developing and sorting the Pareto front, as depicted in Figure 7. The design 523
parameters of this most preferred design, denoted as d2-2, are listed in Table 4. As a 524
comparison, the knee point obtained by the original RGD method implemented with the 525
normal boundary intersection approach (or the marginal utility function approach) is also 526
listed in Table 4. The latter two approaches within the original RGD actually yield the same 527
25
design, which is denoted as d2-1. The difference between the design parameters of d2-2 and that 528
of d2-1 is quite negligible. The results show that the most preferred design obtained with the 529
simplified RGD method is practically the same as the knee point obtained with the existing 530
RGD method that requires a two-step solution (developing a Pareto front by multi-objective 531
optimization using algorithms such as NSGA-II, and then searching for knee point using, for 532
example, the marginal utility function). This confirms the validity of the simplified RGD 533
method that is based on the new procedure for finding the most preferred design. 534
Table 5 compares the computation efforts of these three approaches. The computation 535
time is recorded using the same Windows 7® PC equipped with a 8.0 GB RAM and an Intel® 536
Core™ i7-3630QM CPU running at 2.40GHz. The simplified RGD method with the proposed 537
new procedure for knee point is seen to have drastically reduced the computation time. The 538
savings in the computation time here is mainly attributed to the use of the proposed procedure 539
to find the most preferred design in the design pool. 540
541
5. Further Discussions on the Simplified RGD Method 542
5.1. Sensitivity of the system response with respect to noise factors 543
In the design of the shoring system for the excavation project analyzed previously, the 544
system response of concern is the maximum wall deflection. A good way to examine the 545
sensitivity of the system response to each of the noise factors is to compare the normalized 546
gradient vector (Eq. 3), which might be interpreted as the variation of the system response 547
caused by one percent change in each noise factor while maintaining constants for all other 548
noise factors. This gradient could also be viewed as a measure of the prediction error in the 549
system response caused by one percent variation in the estimated noise factor. 550
26
Figure 8 shows a comparison of the variation of the system response (i.e., maximum 551
wall deflection in this case) due to the relative variation in each noise factor in each of these 552
five designs listed in Tables 3 and 4 (designated as d1, d2-1, d2-2, d3, and d0 respectively). In 553
each of these five designs, the system response is most affected by the friction angles of the 554
top three soil layers and the surcharge behind the wall. This is quite reasonable since the 555
excavation work is mainly carried out in the top three soil layers. The implication is that the 556
prediction accuracy of the maximum wall deflection is greatly dependent upon the accuracy 557
of the characterization of the top soil layers in this excavation project. 558
Also revealed in Figure 8 is that through the use of RGD method, the derived final 559
design (i.e., either d2-1 from the existing RGD method or d2-2 from the simplified RGD 560
method) yields a significant reduction of the sensitivity of the system response to the noise 561
factors, comparing to the least-cost design (d1) and the original design (d0). Further reduction 562
in the sensitivity can be achieved, for example, by adopting the most robust design (d3) in the 563
design pool; however, the cost efficiency is a major concern for this most robust design. 564
5.2. Effect of the uncertainty in noise factors on the system response 565
It is often difficult to characterize noise factors with certainty in geotechnical practices 566
due to limited data availability at a given site. The uncertainty in the estimated noise factors 567
can lead to the variation in the predicted system response. To investigate the effect of the 568
robust design on the system response, all the non-dominated designs on the Pareto front 569
obtained with the existing RGD method (see Figure 6) are analyzed herein for their 570
feasibilities assuming that there is a 10% coefficient of variation (COV) in each and every 571
noise factor. For this series of analysis, the feasibility is defined as the probability that the 572
design remains meeting the serviceability requirement in the face of uncertainties, which is 573
27
denoted as Pr 0.7% fy H . For each non-dominated design on the Pareto front, 10,000 574
Monte Carlo simulations (MCS) are performed herein to sample the noise factors from the 575
joint distribution of noise factors; as such, the feasibility (or the probability of meeting the 576
design constraint of 0.7% fy H ) can be computed. 577
Figure 9 shows the feasibilities computed for all the non-dominated designs on the 578
Pareto front versus their corresponding sensitivity indexes. The feasibility of the design is 579
seen to decrease with the increase in the sensitivity index when there is a 10% variation in 580
noise factors. The implication is that a design with a lower sensitivity index (or higher design 581
robustness) has a higher probability to remain feasible in the face of the uncertainty in noise 582
factors. That is to say, a system might remain feasible even if the input parameters undergo 583
some variation, if the system is designed with high robustness against the unforeseen variation 584
of noise factors. Thus, the robustness measure in terms of the gradient-based sensitivity index, 585
proposed in this paper, is consistent with the well-accepted feasibility robustness measure 586
(Parkinson et al. 1993; Park et al. 2006; Juang et al. 2013a; Wang et al. 2013), which further 587
confirms the validity of Eqs. (1) through (4). 588
589
6. Concluding Remarks 590
In this paper, the authors present a simplified robust geotechnical design (RGD) 591
method. In the context of robust design, the safety requirements, cost efficiency, and design 592
robustness are simultaneously considered and implemented in a multi-objective optimization 593
framework. The results of such optimization are usually expressed as a Pareto font, as cost 594
efficiency and design robustness are conflicting design objectives. The knee point is then 595
identified that yields the best compromise design among all the non-dominated designs on the 596
28
Pareto front. This existing RGD method is improved in this paper in two aspects: 1) a more 597
efficient measure of design robustness based on gradient is developed, and 2) a new procedure 598
to identify the most preferred design in the design pool is introduced. The simplified RGD 599
method essentially turns the multi-objective optimization into a series of single-objective 600
optimizations. The following conclusions are reached based on the results presented: 601
1) The simplified RGD method is demonstrated as an effective and practical design tool 602
that considers the safety requirements, cost efficiency and design robustness 603
simultaneously. This simplified RGD method is formulated so that no reliability or 604
probabilistic analysis is required, and as such, is more readily adaptable for general 605
applications. The advantages of the simplified RGD method over the existing RGD 606
method are demonstrated in the example design of a shoring system for deep 607
excavation. 608
2) The new gradient-based robustness measure, in terms of sensitivity index, is shown to 609
be intuitive and effective, as it corresponds to the variation of the system response due 610
to a relative variation in noise factors. The design with a lower sensitivity index 611
always yields a lower variation of the system response regardless of the variation 612
levels of noise factors, and thus higher design robustness and a lower potential risk 613
(indicated by a higher feasibility) in the face of the uncertainty in noise factors. This 614
definition is easier to implement in a RGD and computationally more efficient than the 615
existing robustness measures. 616
3) With the proposed new procedure, the most preferred design in the design pool can be 617
identified directly through a series of single-objective optimizations, which does not 618
involve genetic algorithms such as NSGA-II, as such the simplified RGD method can 619
29
be easily implemented with Matlab function fmincon or Microsoft Excel Solver. The 620
proposed new procedure for finding the most preferred design in the design pool is 621
shown to be effective and computationally more efficient than the existing methods 622
that are based on the Pareto front obtained through a multi-objective optimization, 623
such as normal boundary intersection approach and marginal utility function approach. 624
4) Compared to the traditional deterministic design approach, the simplified RGD 625
method shows an advantage for being able to consider the effect of the uncertainty in 626
noise factors on the system response (through minimizing the sensitivity index of the 627
system response to the noise factors). This advantage allows for a more robust design 628
to be achieved with a balancing consideration of cost. Compared to the probabilistic 629
design approach, the simplified RGD method shows an advantage for being able to 630
achieve similar design results (i.e., to achieve a desired level of feasibility in the face 631
of uncertainties) without having to perform a probabilistic analysis of the system 632
response with a full statistical characterization of noise factors. The superiorities of the 633
simplified RGD method are demonstrated. 634
635
Acknowledgments 636
The study on which this paper is based was supported in part by the National Science 637
Foundation through Grant CMMI-1200117 (“Transforming Robust Design Concept into a 638
Novel Geotechnical Design Tool”) and the Glenn Department of Civil Engineering, Clemson 639
University. The results and opinions expressed in this paper do not necessarily reflect the 640
views and policies of the National Science Foundation. The TORSA code was provided by Dr. 641
Hsii-Sheng Hsieh of Trinity Foundation Engineering Consultants (TFEC), Taipei, Taiwan. Dr. 642
30
Hsieh is also thanked for his untiring effort in answering our countless questions regarding 643
TORSA and the engineering practice of braced excavations during the course of this study. 644
645
31
References 646
Ang, A.H.S., Tang, W.H., 2007. Probability Concepts in Engineering: Emphasis on 647
Applications to Civil and Environmental Engineering, 2nd edition, Wiley, New York, 648
USA. 649
Baecher, G.B., Christian, J.T., 2003. Reliability and Statistics in Geotechnical Engineering, 650
Wiley, New York, USA. 651
Bechikh, S., Said, L.B., Ghédira, K., 2011. Searching for knee regions of the Pareto front 652
using mobile reference points. Soft Computing 15(9), 1807-1823. 653
Beyer, H.G., Sendhoff, B., 2007. Robust optimization – A comprehensive survey. Computer 654
Methods in Applied Mechanics and Engineering 196(33), 3190-3218. 655
Branke, J., Deb, K., Dierolf, H., Osswald, M., 2004. Finding knees in multi-objective 656
optimization. Parallel Problem Solving from Nature-PPSN VIII, 722-731. 657
Chen, W., Wiecek, M.M., and Zhang, J., 1999. Quality utility—A compromise programming 658
approach to robust design. Journal of Mechanical Design 121(2), 179-187. 659
Clough, G.W., O’Rourke, T.D., 1990. Construction induced movements of in situ walls. In: 660
Proceedings of Design and Performance of Earth Retaining Structure, GSP 25, ASCE, 661
New York, 439-470. 662
Deb, K., 2001. Multi-objective Optimization Using Evolutionary Algorithms. John Wiley and 663
Sons, New York, USA. 664
Deb, K., Gupta, S., 2011. Understanding knee points in bicriteria problems and their 665
implications as preferred solution principles. Engineering Optimization, 43(11), 666
1175-1204. 667
Deb, K., Sundar, J., Udaya Bhaskara Rao, N., Chaudhuri, S.. 2006. Reference point based 668
32
multi-objective optimization using evolutionary algorithms. International Journal of 669
Computational Intelligence Research 2(3), 273-286. 670
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., 2002. A fast and elitist multi-objective 671
genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 672
182-197. 673
Deep Excavation LLC, 2014. DeepXcav User’s Manual, 30-62 Steinway St. #188, Astoria, 674
New York. 675
Doltsinis, I., Kang, Z., Cheng, G., 2005. Robust design of non-linear structures using 676
optimization methods. Computer Methods in Applied Mechanics and Engineering 677
194(12), 1779-1795. 678
Fenton, G.A., Griffiths, D.V., 2008. Risk Assessment in Geotechnical Engineering. John 679
Wiley and Sons, New York, USA. 680
Goh, K.H., Mair, R.J., 2014. Response of framed buildings to excavation-induced movements. 681
Soils and Foundations 54(3), 250-268. 682
Gonsamo, A., 2011. Normalized sensitivity measures for leaf area index estimation using 683
three-band spectral vegetation indices. International Journal of Remote Sensing 32(7), 684
2069-2080. 685
Gong, W., Wang, L., Juang, C.H., Zhang, J., Huang, H., 2014a, Robust geotechnical design of 686
shield-driven tunnels. Computers and Geotechnics 56, 191-201. 687
Gong, W., Khoshnevisan, S., Juang, C.H., 2014b, Gradient-based design robustness measure 688
for robust geotechnical design. Canadian Geotechnical Journal 51(11), 1331-1342. 689
33
Gong, W., Luo, Z., Juang, C. H., Huang, H., Zhang, J., Wang, L. 2014c. Optimization of site 690
exploration program for improved prediction of tunneling-induced ground settlement in 691
clays. Computers and Geotechnics 56, 69-79. 692
Harr, M.E., 1987. Reliability-Based Design in Civil Engineering, McGraw-Hill Book, New 693
York, USA. 694
Japanese Society of Architecture (JSA), 1988. Guidelines of Design and Construction of Deep 695
Excavations. Japanese Society of Architecture, Tokyo, Japan. 696
Juang, C.H., Wang, L., Atamturktur, S., Luo, Z., 2012a. Reliability-based robust and optimal 697
design of shallow foundations in cohesionless soil in the face of uncertainty. Journal of 698
GeoEngineering 7(3), 75-87. 699
Juang, C.H., Luo, Z., Atamturktur, S., Huang, H., 2012b. Bayesian updating of soil 700
parameters for braced excavations using field observations. Journal of Geotechnical and 701
Geoenvironmental Engineering 139(3), 395-406. 702
Juang, C.H., Wang, L., Liu, Z., Ravichandran, N., Huang, H., Zhang, J., 2013a. Robust 703
Geotechnical Design of drilled shafts in sand – New design perspective. Journal of 704
Geotechnical and Geoenvironmental Engineering 139(12), 2007-2019. 705
Juang, C.H., Ching, J., Luo, Z., 2013b. Assessing SPT-based probabilistic models for 706
liquefaction potential evaluation: a 10-year update. Georisk: Assessment and management 707
of risk for engineered systems and geohazards 7(3), 137-150. 708
Juang, C.H., Wang, L., Hsieh, H.S., Atamturktur, S., 2014. Robust geotechnical design of 709
braced excavations in clays. Structural Safety 49, 37-44. 710
Kim, C., Kwon, J., Im, J.C., Hwang, S., 2012. A method for analyzing the self-supported 711
earth-retaining structure using stabilizing piles. Marine Georesources and Geotechnology 712
34
30(4), 313-332. 713
Kim, D.S., Lee, B.C., 2005. Instrumentation and numerical analysis of cylindrical diaphragm 714
wall movement during deep excavation at coastal area. Marine Georesources and 715
Geotechnology 23(1-2), 117-136. 716
Khoshnevisan, S., Gong, W., Wang, L., Juang, C.H., 2014. Robust design in geotechnical 717
engineering–an update. Georisk: Assessment and Management of Risk for Engineered 718
Systems and Geohazards 8(4), 217-234. 719
Kung, G.T.C., Juang, C.H., Hsiao, E.C.L., Hashash, Y.M.A., 2007. A simplified model for 720
wall deflection and ground surface settlement caused by braced excavation in clays. 721
Journal of Geotechnical and Geoenvironmental Engineering 133(6), 731-747. 722
Likitlersuang, S., Surarak, C., Wanatowski, D., Oh, E., Balasubramaniam, A., 2013. Finite 723
element analysis of a deep excavation: A case study from the Bangkok MRT. Soils and 724
Foundations 53(5), 756-773. 725
Marrel, A., Iooss, B., Laurent, B., Roustant, O., 2009. Calculations of Sobol indices for the 726
gaussian process metamodel. Reliability Engineering and System Safety 94(3), 742-751. 727
MATLAB version 7.10.0. Natick, Massachusetts: The MathWorks Inc; 2010. 728
Oka, Y., Wu, T.H., 1990. System reliability of slope stability. Journal of Geotechnical 729
Engineering 116(8), 1185-1189. 730
Ou, C.Y., 2006. Deep Excavation-Theory and Practice. Taylor and Francis, England. 731
Park, G.J., Lee, T.H., Lee, K.H., Hwang, K.H., 2006. Robust design: an overview. AIAA 732
journal 44(1), 181-191. 733
Parkinson, A., Sorensen, C., Pourhassan, N., 1993. A general approach for robust optimal 734
design. Journal of Mechanical Design 115(1), 74-80. 735
35
Phadke, M.S., 1989. Quality Engineering Using Robust Design. Prentice Hall, New Jersey, 736
USA. 737
Phoon, K.K., Kulhawy, F.H., 1999. Characterization of geotechnical variability. Canadian 738
Geotechnical Journal 36(4), 612-24. 739
Plaxis bv. PLAXIS-2D, 2014. User’s Manual, 2600 AN Delft, The Netherlands. 740
Professional Standards Compilation Group (PSCG), 2000. Specification for Excavation in 741
Shanghai Metro Construction. Professional Standards Compilation Group, Shanghai, 742
China. 743
Sino-Geotechnics, 2010. User Manual of Taiwan Originated Retaining Structure Analysis for 744
Deep Excavation. Sino-Geotechnics Research and Development Foundation, Taipei, 745
Taiwan. 746
Taguchi, G., 1986. Introduction to Quality Engineering: Designing Quality into Products and 747
Processes. Quality Resources, New York, USA. 748
Taiwan Geotechnical Society (TGS), 2001. Design Specifications for the Foundation of 749
Buildings. Taiwan Geotechnical Society, Taipei, Taiwan. 750
Wang, L., Hwang, J.H., Juang, C.H., Atamturktur, S., 2013. Reliability-based design of rock 751
slopes – A new perspective on design robustness. Engineering Geology 154, 56-63. 752
Wang, L., Juang, C.H., Atamturktur, S., Gong, W., Khoshnevisan, S., Hsieh, H.S., 2014. 753
Optimization of design of supported excavations in multi-layer strata. Journal of 754
GeoEngineering 9(1), 1-12. 755
Zhang, L.M., Li, D.Q., Tang, W.H., 2005. Reliability of bored pile foundations considering 756
bias in failure criteria. Canadian Geotechnical Journal 42(4),1086-1093. 757
758
36
Lists of Tables
Table 1. Basic soil properties adopted in the case study of deep excavation
Table 2. Design pool for the robust design optimization
Table 3. Analysis of the four designs of the shoring system for the deep excavation project
Table 4. Most preferred design (i.e., knee point) obtained with different approaches
Table 5. Comparison of the computational efficiency among the three approaches
37
Lists of Figures
Figure 1. Illustration of the robustness concept
Figure 2. Conceptual illustration of Pareto front and knee point in a bi-objective optimization
(a) Optimization problem with a single knee point
(b) Optimization problem with multi-knee points
Figure 3. Different approaches to identify the most preferred design (i.e., knee point)
(a) Reflex angle approach
(b) Normal boundary intersection approach
(c) New procedure proposed in this paper
Figure 4. Plan layout of the excavation site in the case study
Figure 5. Four possible tieback anchor layouts of the shoring system
(a) Vertical spacing, V = 2.0 m
(b) Vertical spacing, V = 2.5 m
(c) Vertical spacing, V = 3.0 m
(d) Vertical spacing, V = 3.5 m
Figure 6. Results of the robust design of the shoring system for deep excavation
(a) Pareto front obtained from robust design optimization
(b) Normal boundary intersection approach
(c) Preferred designs on the Pareto front and the original design
Figure 7. New simplified RGD procedure to find the most preferred design
Figure 8. Sensitivity of the system response of concern (i.e., maximum wall deflection) with
respect to noise factors (Note: 1 = φ of 1st soil layer; 2 = kh of 1
st soil layer; 3 = φ
of 2nd
soil layer; 4 = kh of 2nd
soil layer;5 = φ of 3rd
soil layer; 6 = kh of 3rd
soil
layer;7 = φ of 4th
soil layer; 8 = kh of 4th soil layer; 9 = su of 4
th soil layer; 10 =
q0 behind the wall; the designs d1, d2-1, d2-2, d3, and d0 are listed in Tables 3 and 4)
Figure 9. Design feasibility versus sensitivity index (Note: the noise factors are subject to a
10% variation)
38
Table 1. Basic soil properties adopted in the case study of deep excavation
Layer
No.
Soil
type
Depth
(m)
Weight, γt
(ton/m3)
cohesion, c (ton/m2)
Friction
angle, φ (°)
Modulus of horizontal
subgrade reaction, kh (ton/m3)
1 SM 3.35 1.80 0.0 28.0 500
2 GP 11.15 1.90 0.0 30.0 4000
3 GP 15.50 2.30 0.0 36.0 5000
4 Rock 20.0 2.40 2.0 34.0 5000
39
Table 2. Design pool for the robust design optimization
Design parameter Design pool (defined by ranges of design parameters)
Diameter of the solider pile, D
(m) {0.3 m, 0.4m, 0.5m, 0.6 m, 0.7 m}
Length of the solider pile, L
(m) {14 m, 15 m, 16 m, 17 m, 18 m, 19 m, 20 m}
Horizontal interval of the
solider pile, I (m) {D, D + 0.1 m, D + 0.2 m, …, D + 1.0 m}
Vertical spacing of tieback
anchors, V (m) {2.0 m, 2.5 m, 3.0 m, 3.5 m}
Horizontal spacing of tieback
anchors, H (m) {1.5 m, 2.0 m, 2.5 m, 3.0 m, 3.5 m}
Installed angle of the tieback
anchor, () {10, 15, 20, 25, 30}
40
Table 3. Analysis of the four designs of the shoring system for the deep excavation project
Resulting design Design parameters Safety performance Cost, C
(10,000 USD)
Sensitivity
index, SI D (m) L (m) I (m) V (m) H (m) () Fs1 Fs2 y (cm)
Least cost design, d1 0.5 17 1.3 3.0 3.5 10 4.46 2.75 8.20 8.66 19.06
Most preferred design, d2-1 0.6 18 1.6 3.0 2.0 10 5.67 2.96 3.48 13.12 4.28
Most robust design, d3 0.7 20 0.7 3.0 2.0 15 7.82 3.34 1.49 23.10 1.67
Original design, d0 0.5 17 0.6 3.0 2.5 20 4.46 2.75 4.13 14.42 11.22
Note: d1: least cost design on Pareto front; d2-1: most preferred design on Pareto front (identified by the marginal utility function
approach); d3: most robust design on Pareto front; and d0: original design in an actual project.
41
Table 4. Most preferred design (i.e., knee point) obtained with different approaches
Adopted approach Design parameters Safety performance Cost, C
(10,000 USD)
Sensitivity
index, SI D (m) L (m) I (m) V (m) H (m) () Fs1 Fs2 y (cm)
Normal boundary intersection
approach, d2-1 0.6 18 1.6 3.0 2.0 10 5.67 2.96 3.48 13.12 4.28
Marginal utility function
approach, d2-1 0.6 18 1.6 3.0 2.0 10 5.67 2.96 3.48 13.12 4.28
Proposed new procedure, d2-2 0.5 18 1.4 3.0 2.0 10 5.67 2.96 4.94 12.31 5.58
Note: d2-1: most preferred design on Pareto front (identified by the marginal utility function approach or the normal boundary
intersection approach); d2-2: most preferred design by the simplified RGD with the proposed new procedure.
42
Table 5. Comparison of the computational efficiency among the three approaches
Adopted approach
Computation time
Sorting Pareto front
using NSGA (sec)
Identifying knee
point (sec)
Total time
(sec)
Normal boundary intersection
approach, d2-1 110.6 0.5 111.1
Marginal utility function approach,
d2-1 110.6 0.8 111.4
Proposed new procedure,
d2-2 5.7 5.7
Note: d2-1: most preferred design on Pareto front (identified by the marginal utility function
approach or the normal boundary intersection approach); d2-2: most preferred design by the
simplified RGD with the proposed new procedure.
43
f (d1, )
Sensitive
Design response, f (d, )
Noise factors,
Robust
System with high gradient
f (d2, )
System with low gradient
Variation level 2
Variation level 1
Figure 1. Illustration of the robustness concept
44
Pareto front
Ob
ject
ive
2,
min
[f2(d
)]
Objective 1, min [f1(d)]
Feasible domain
Knee point
Utopia point
(a) Optimization problem with a single knee point
Local knee point
Pareto front
Ob
ject
ive
2,
min
[f2(d
)]
Objective 1, min [f1(d)]
Knee point
Utopia point
Global knee point
Local knee point
(b) Optimization problem with multi-knee points
Figure 2. Conceptual illustration of Pareto front and knee point in a bi-objective optimization
45
Pareto front
Ob
ject
ive
2,
min
[f2(d
)]
Objective 1, min [f1(d)]
B
AKnee point
Maximum reflex angle
(a) Reflex angle approach
Pareto front
Obje
ctiv
e 2,
min
[f2(d
)]
Objective 1, min [f1(d)]
Maximum distance
A
B
Knee point
Boundary line
(b) Normal boundary intersection approach
Feasible design
Ob
ject
ive
2,
min
[f2(d
)]
Objective 1, min [f1(d)]
Knee point
Minimum distance
Utopia point
(c) New procedure proposed in this paper
Figure 3. Different approaches to identify the most preferred design (i.e., knee point)
46
54.08 m
48. 90 m
28.74 m
AA
BB
DD
CC
41.79 m 27.68 m
Section E 8.5 m
Section D 11.9 m
Section C 11.4 m
Section B 9.15 m
Section A 6.9 m
Section Excavation depth
EE
Figure 4. Plan layout of the excavation site in the case study
47
Surcharge, q0
3.35 m
11.15 m
15.50 m
20.00 m
SM
GP
GP
Rock
10.30 m
8.30 m
6.30 m
4.30 m
2.30 m
11.90 m
4 . 8 0 m
S u r c h a r g e , q0
3 . 3 5 m
1 1 . 1 5 m
1 5 . 5 0 m
2 0 . 0 0 m
SM
GP
GP
Rock
2.30 m
11.90 m
7.30 m
9.80 m
(a) Vertical spacing, V = 2.0 m (b) Vertical spacing, V = 2.5 m
Surcharge, q0
3.35 m
11.15 m
15.50 m
20.00 m
SM
GP
GP
Rock
2.30 m
11.90 m
5.30 m
8.30 m
9 . 3 0 m
5 . 8 0 m
S u r c h a r g e , q0
3 . 3 5 m
1 1 . 1 5 m
1 5 . 5 0 m
2 0 . 0 0 m
SM
GP
GP
Rock
2.30 m
11.90 m
(c) Vertical spacing, V = 3.0 m (d) Vertical spacing, V = 3.5 m
Figure 5. Four possible tieback anchor layouts of the shoring system
48
5 10 15 20 250
4
8
12
16
20
24
Sen
siti
vit
y i
nd
ex,
SI
Cost, C (10,000 USD)
Pareto front
Feasible design
(a) Pareto front obtained from robust design optimization
5 10 15 20 250
4
8
12
16
20
24
Sen
siti
vit
y i
ndex
, SI
Cost, C (10,000 USD)
Pareto front
Maximum distance
Knee point
(b) Normal boundary intersection approach
5 10 15 20 250
4
8
12
16
20
24
Sen
siti
vit
y i
nd
ex,
SI
Cost, C (10,000 USD)
Pareto front
Preferred designs
Original design
Knee point on
the Pareto front Most robust design
Least cost design
(c) Preferred designs on the Pareto front and the original design
Figure 6. Results of the robust design of the shoring system for deep excavation
49
5 10 15 20 250
4
8
12
16
20
24
Sen
siti
vit
y i
nd
ex,
SI
Cost, C (10,000 USD)
Utopia point
Feasible design
Minimum distance
Most preferred design
Figure 7. New simplified RGD procedure to find the most preferred design
50
-16
-12
-8
-4
0
4
10
9876543
2
Sen
siti
vit
y v
ecto
r, J
i
d1
d2-1
d2-2
d3
d0
1
Figure 8. Sensitivity of the system response of concern (i.e., maximum wall deflection) with
respect to noise factors (Note: 1 = φ of 1st soil layer; 2 = kh of 1
st soil layer; 3 = φ of 2
nd soil
layer; 4 = kh of 2nd
soil layer;5 = φ of 3rd
soil layer; 6 = kh of 3rd
soil layer;7 = φ of 4th soil
layer; 8 = kh of 4th soil layer; 9 = su of 4
th soil layer; 10 = q0 behind the wall; the designs d1,
d2-1, d2-2, d3, and d0 are listed in Tables 3 and 4)
51
0 4 8 12 16 200.0
0.2
0.4
0.6
0.8
1.0
1.2
Non-dominated design
Fea
sib
ilit
y,
Pr[
y <
0.7
%H
f ]
Sensitivity index, SI
Most preferred design (d2-1
)
Most preferred design (d2-2
)
Figure 9. Design feasibility versus sensitivity index (Note: the noise factors
are subject to a 10% variation)