response surface methods for slope reliability...

Engineering Geology 203 (2016) 3–14

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Engineering Geology

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Response surface methods for slope reliability analysis: Reviewand comparison

Dian-Qing Li a, Dong Zheng a, Zi-Jun Cao a,⁎, Xiao-Song Tang a, Kok-Kwang Phoon b

a State Key Laboratory ofWater Resources andHydropower Engineering Science, Key Laboratory of RockMechanics inHydraulic Structural Engineering (Ministry of Education),WuhanUniversity,8 Donghu South Road, Wuhan 430072, PR Chinab Department of Civil and Environmental Engineering, National University of Singapore, Blk E1A, #07-03, 1 Engineering Drive 2, Singapore 117576, Singapore

⁎ Corresponding author at: State Key Laboratory of WaEngineering Science, Wuhan University, 8 Donghu South R

E-mail address: [email protected] (Z.-J. Cao).

http://dx.doi.org/10.1016/j.enggeo.2015.09.0030013-7952/© 2015 Elsevier B.V. All rights reserved.

a b s t r a c t
a r t i c l e i n f o
Article history:Received 1 June 2015Received in revised form 9 August 2015Accepted 15 September 2015Available online 24 September 2015

Keywords:Slope stabilityUncertaintyReliability analysisResponse surface methodComputational efficiencyAccuracy

This paper reviews previous studies on developments and applications of response surface methods (RSMs) indifferent slope reliability problems. Based on the review, four types of soil slope reliability analysis problemsare identified from the literature, including single-layered soil slope reliability problem ignoring spatial variabil-ity, single-layered soil slope reliability problem considering spatial variability, multiple-layered soil slope reliabil-ity problem ignoring spatial variability, and multiple-layered soil slope reliability problem considering spatialvariability, which are referred to as “Type I–IV problems” in this study. Then, the computational efficiency andaccuracy of four commonly-used RSMs (namely single quadratic polynomial-based response surface method(SQRSM), single stochastic response surface method (SSRSM), multiple quadratic polynomial-based responsesurface method (MQRSM), and multiple stochastic response surface method (MSRSM)) are systematicallycompared for cohesive and c–ϕ slopes, and their feasibility and validity in the four types of slope reliabilityproblems are discussed. Based on the comparison, some suggestions for selecting relatively appropriate RSMsin slope reliability analysis are provided: (1) SQRSM is suggested as a suitable method for the single-layeredsoil slope reliability problem ignoring spatial variability (i.e., Type I problem); (2) MQRSM is applicable tothe multiple-layered soil slope reliability problem ignoring spatial variability (i.e., Type III problem); and(3) MSRSM is suggested to solve slope reliability problems (including single-layered and multiple-layeredslopes) considering spatial variability (i.e., Type II and IV problems).

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

Reliability analysis of soil slopes has gained considerable attention inthe geotechnical reliability community over the past few decades(e.g., Baecher and Christian, 2003; Low and Tang, 2004; Cho, 2007,2009, 2010, 2013; Fenton and Griffiths, 2008; Ching et al., 2009; Wanget al., 2010, 2011; Ji and Low, 2012; Ji, 2014; Zhang et al., 2013a,b;Jiang et al., 2014, 2015; Li et al., 2011, 2014, 2015a). Many reliabilitymethods have been proposed for slope reliability analysis in literature,such as the first-order second moment method (FOSM) (e.g., Christianet al., 1994; Hassan and Wolff, 1999; Duncan, 2000; Xue and Gavin,2007; Suchomel and Mašin, 2010), first-order reliability method(FORM) (e.g., Low and Tang, 1997, 2004; Low, 2007; Cho, 2007; Hongand Roh, 2008; Ji, 2014; Zeng and Jimenez, 2014), second-order reliabil-ity method (SORM) (e.g., Cho, 2009; Low, 2014), and Monte Carlo Sim-ulation (MCS) (e.g., El-Ramly et al., 2002, 2005; Griffiths and Fenton,2004; Hsu and Nelson, 2006; Cho, 2007, 2010; Huang et al., 2010,2013; Tang et al., 2015; Li et al., 2015c) and its advanced variants(e.g., Ching et al., 2009; Wang et al., 2010, 2011; Li et al., 2015d).

ter Resources and Hydropoweroad, Wuhan 430072, PR China.

In addition to the aforementioned reliability methods, response sur-facemethods (RSMs) have been used for slope reliability problemswithimplicit performance functions (e.g. Wong, 1985; Xu and Low, 2006; Jiand Low, 2012; Zhang et al., 2011b, 2013b; Jiang et al., 2014, 2015; Liet al., 2015a,b; Li and Chu, 2015). RSMs have been proved to be an effi-cient method for slope reliability analysis. For instance, Wong (1985)applied RSM to evaluate the reliability of a homogeneous slope. Xuand Low (2006) used RSM to approximate the performance functionof slope stability in slope reliability analysis, in which the response sur-face is taken as a bridge between stand-alone numerical packages andspreadsheet-based reliability analysis. Recently, several researchers(e.g., Zhao, 2008; Li et al., 2013; Samui et al., 2013) proposed a supportvectormachine (SVM)-based RSM to approximate implicit performancefunction using a small set of actual values of the performance function.Taking the radial basis function neural network (RBFN) as an approxi-mate response surface function for the actual performance function,Tan et al. (2011) discussed similarities and differences between RBFN-based RSMs and SVM-based RSMs, which indicated that there is no sig-nificant difference between them. To reduce the number of evaluationsof the actual performance function, Tan et al. (2013) proposed two newsampling methods and a hybrid RSM. Similar to SVM-based RSM, rele-vance vector machine (RVM)-based FOSM is adopted to build a RVMmodel to predict the implicit performance function and evaluate the

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partial derivatives with sufficient accuracy (Samui et al., 2011). In addi-tion, the artificial neural network (ANN) technique (e.g., Cho, 2009;Chen et al., 2011), the Gaussian process regression (Kang et al., 2015),Artificial bee colony (ABC) algorithm optimized support vector regres-sion (SVR) (Kang and Li, 2015), the high dimensionalmodel representa-tion (HDMR) (Chowdhury and Rao, 2010) and neural networks (NN)-based RSM (Piliounis and Lagaros, 2014) can be also used to establisha relationship between the factor of safety and soil parameters. Luoet al. (2012a,b) and Zhang et al. (2011a) adopted Kriging-based re-sponse surface to simulate performance functions and demonstratedits applications in solving several geotechnical reliability problems(Zhang et al., 2013a). Yi et al. (2015) indicated that the particle swarmoptimization–Kriging model has a good curve fitting performance.

Recently, an important advance in slope reliability analysis usingRSM is that multiple response surface methods were proposed to eval-uate system reliability of slope stability. For example, Zhang et al.(2011b) first proposed the multiple response surfaces method forslope reliability analysis. Ji and Low (2012) constructed a group of strat-ified response surfaces corresponding to the most probable failuremodes, and slope system reliability is evaluated based on these strati-fied response surfaces. Using the quadratic polynomial-based responsesurface method, Zhang et al. (2013b) extended the Hassan and Wolffmethod into a practical tool for system reliability analysis.

The aforementioned studies have not considered the spatial variabil-ity when the RSMs are used to evaluate slope reliability problems.

Table 1Summary of applications of RSMs in soil slope reliability analyses.

PaperID

Authors Year Types of response surfaces Singleresponsesurface

Mresu

1 Wong 1985 Quadratic polynomial √2 Xu and Low 2006 Quadratic polynomial without cross terms √3 Zhao 2008 SVM-based response surface √

4 Cho 2009 ANN-based response surface √5 Chowdhury

and Rao2010 High dimensional model representation √

6 Chen et al. 2011 SVM-based response surface √7 Tan et al. 2011 RBFN and SVM-based response surface √8 Samui et al. 2011 RVM-based response surface √9 Samui et al. 2013 LSSVM-based response surface √10 Luo et al. 2012a Kriging-based response surface √11 Luo et al. 2012b Kriging-based response surface √12 Ji et al. 2012 Quadratic polynomial without cross terms √13 Ji and Low 2012 Quadratic polynomial without cross terms √14 Zhang et al. 2011a Kriging-based response surface √15 Zhang et al. 2011b Quadratic polynomial without cross terms √16 Zhang et al. 2013a Classical RSM without cross terms, quadratic

polynomial without cross terms,Kriging-based response surface

√

17 Zhang et al. 2013b Quadratic polynomial without cross terms √18 Li et al. 2013 Updated SVM-based response surface √

19 Tan et al. 2013 Quadratic polynomial √20 Piliounis

and Lagaros2014 NN-based response surface √

21 Jiang et al. 2014 Hermite polynomial chaos expansion √22 Jiang et al. 2015 Hermite polynomial chaos expansion √23 Li et al. 2015a Quadratic polynomial without cross terms √24 Li and Chu 2015 Quadratic polynomial without cross terms √25 Yi et al. 2015 PSO Kriging based response surface, classical

RSM without cross terms√

26 Kang et al. 2015 GPR-based response surface √27 Kang and Li 2015 ABC-SVR response surface √

Note: SVM= support vector machine; ANN= artificial neural network; RBFN= radial basis fuvectormachine; NN=neural networks; PSO=particle swarm optimization; GPR=Gaussian= finite difference method. ABC-SVR = artificial bee colony algorithm optimized support vect

Ji et al. (2012) made a first attempt to solve slope reliability with spatialvariability by the RSM with second-order polynomial approximatefunction without cross terms. Based on the stochastic response surfacemethod (SRSM), Jiang et al. (2014) proposed a non-intrusive stochasticfinite element method for slope reliability analysis considering spatialvariability in shear strength parameters, by which the system reliabilityof soil slopes considering spatial variability is evaluated by the multiplestochastic response surface method (e.g., Jiang et al., 2015; Li et al.,2015a; Li and Chu, 2015).

Based on the above studies, it can be seen that significant advanceshave been made in applications of RSMs in soil slope reliability analysis.Essentially, the RSM uses a computationally efficient model to approxi-mate the original analysis model (e.g., limit equilibrium analysis orfinite element analysis). Then, slope reliability analysis is carried outbased on the explicit performance function represented by the RSM.Table 1 summarizes the applications of RSM-based reliability methodsin soil slope reliability analyses. These references are listed in achronological order. The soil slope reliability problems concerned in theprevious studies on RSMs can be divided into four categories accordingto the probabilistic model of soil properties (e.g., random variable or ran-dom field models) and slope types (e.g., single-layered or multiple-layered): (1) single-layered soil slope reliability problem ignoringspatial variability (i.e., Type I problem); (2) single-layered soil slope reli-ability problem considering spatial variability (i.e., Type II problem);(3) multiple-layered soil slope reliability problem ignoring spatial

ultiplesponserfaces

Spatialvariability

Slope type Deterministic slopestability analysis

No Yes Single-layered Multiple-layered

√ √ FEM√ √ LEM (Spencer), FEM√ √ √ LEM (Simplified Bishop,

Spencer)√ √ FDM√ √ LEM (Simplified Bishop,

Janbu,Morgenstern–Price,Spencer, GLE)

√ √ LEM (Morgenstern–Price)√ √ FEM√ √ LEM (Simplified Bishop)√ √ LEM (Simplified Bishop)√ √ √ FDM√ √ √ FEM

√ √ LEM (Spencer)√ √ LEM (Ordinary, Spencer)√ √ FDM√ √ √ LEM (Morgenstern–Price)√ √ LEM (Simplified Bishop)

√ √ LEM (Simplified Bishop)√ √ LEM (Simplified Bishop,

Spencer)√ √ LEM (Morgenstern–Price)√ √ LEM (Simplified Bishop)

√ √ LEM (Morgenstern–Price)√ √ LEM (Simplified Bishop)√ √ √ LEM (Simplified Bishop)√ √ LEM (Ordinary)

√ √ FDM

√ √ LEM (Simplified Bishop)√ √ LEM (Simplified Bishop)

nction neural network; RVM= relevance vector machine; LSSVM= least square supportprocess regression; LEM= limit equilibriummethod; FEM= finite element method; FDMor regression.

Table 2Four types of soil slope reliability problems and the corresponding adopted RSMs.

Problemtype

Response surfaces Uncertaintymodels

Types of response surfacemethods

Uncertaintypropagationmethods

I Quadratic polynomial (Wong, 1985) Randomvariable

Single response surface, multipleresponse surface

MCS, FOSM,FORMSVM-based response surface (Zhao, 2008)

RVM-based response surface (Samui et al., 2011)Quadratic polynomial without cross terms (Zhang et al., 2011a)Kriging-based response surface (Luo et al., 2012a, b)LSSVM-based response surface (Samui et al., 2013)NN-based response surface (Piliounis and Lagaros, 2014)

II Hermite polynomial chaos expansion (Jiang et al., 2014, 2015) Randomfield


MCS, FORMQuadratic polynomial without cross terms (Ji et al., 2012; Li et al., 2015a)

III Quadratic polynomial without cross terms (Xu and Low, 2006; Zhang et al., 2011b, 2013a,b; Jiand Low, 2012; Tan et al., 2013)

Randomvariable


MCS, FOSM,FORM

SVM-based response surface (Zhao, 2008; Chen et al., 2011; Tan et al., 2011)ANN-based response surface (Cho, 2009)High dimensional model representation (Chowdhury and Rao, 2010)RBFN-based response surface (Tan et al., 2011)Kriging-based response surface (Zhang et al., 2011a, 2013a; Luo et al., 2012a,b)Quadratic polynomial with cross terms (Tan et al., 2013)Classical RSM without cross terms (Zhang et al., 2013a)Updated SVM-based response surface (Li et al., 2013)PSO Kriging based response surface, Classical RSM without cross terms (Yi et al., 2015)GPR based response surface (Kang et al., 2015)ABC-SVR response surface (Kang and Li, 2015)

IV Quadratic polynomial without cross terms (Li et al., 2015a; Li and Chu, 2015) Randomfield

Multiple response surface MCS

Note: MCS = Monte Carlo Simulation; FORM = first-order reliability method; FOSM = first-order second moment.

5D.-Q. Li et al. / Engineering Geology 203 (2016) 3–14

variability (i.e., Type III problem); and (4)multiple-layered soil slope reli-ability problem considering spatial variability (i.e., Type IV problem), asshown in Table 2.

This paper aims to review previous studies on applications of RSMs inthe four types of slope reliability problems and to investigate the capacityand validity of four commonly-used RSMs in solving different slope reli-ability problems. The computational accuracy and efficiency of the fourRSMs are systemically compared using cohesive and c–ϕ slope examples.To achieve such a goal, the paper is organized as follows. In Section 2, fourtypes of soil slope reliability problems are briefly introduced. Then, singleresponse-surface methods and multiple response-surface methods arereviewed in Section 3. In Section 4, four benchmark reliability problems(including two cohesive slopes and two c–ϕ slopes) are investigated toexamine the capacity and validity of the selected RSMs. Finally, severalrecommendations for selecting relatively appropriate RSMs for differentslope reliability problems are provided.

2. Major soil slope reliability analysis problems

Soils are natural materials and their properties are affected by severalfactors during formation process, such as parent materials, weatheringand erosion processes, transportation agents, and sedimentation condi-tions. Thus, soils exhibit a large degree of uncertainty (e.g. Cao andWang, 2013, 2014a; Ching and Phoon, 2014; Le, 2014; Li et al., 2014;Lloret-Cabot et al., 2014; Jamshidi Chenari and Alaie, 2015). To effectivelycharacterize the uncertainty in soil properties, probabilistic models areoften applied to represent soil properties in slope reliability analyses(e.g., Christian et al., 1994; Griffiths and Fenton, 2004; Srivastava andBabu, 2009; Srivastava et al., 2010; Suchomel and Mašin, 2010; Cao andWang, 2014b). Probabilistic models mainly include random variablesmodel and random fields (e.g., Chowdhury and Xu, 1995; Phoon andKulhawy, 1999; El-Ramly et al., 2002; Zhang et al., 2011a,b, 2013a,b;Jiang et al., 2014, 2015; Li et al., 2011, 2014, 2015a; Li and Chu, 2015).As shown in Table 2, the problems involved in soil slope reliability analy-ses using RSMs are divided into four categories, which are briefly de-scribed in the following four subsections, respectively.

2.1. Type I problem: single-layered soil slope ignoring spatial variability

Type I problem is a relatively simple case where the reliability anal-ysis of a homogeneous soil slope is concerned. For such a reliabilityproblem, Wong (1985) adopted the quadratic polynomial-based RSMfor evaluating the reliability of the single-layered soil slopewithout con-sideration of spatial variability. Subsequently, the Type I problem hasbeen extensively investigated (e.g., Zhao, 2008; Samui et al., 2011,2013; Zhang et al., 2011a; Luo et al., 2012a,b; Piliounis and Lagaros,2014) using different RSMs, such as SVM-based RSM, RVM-basedRSM, quadratic polynomial without cross terms, Kriging-based RSM,and NN-based RSM. For a homogenous soil slope, the slope failureis dominated by the deterministic critical slip surface because thefactors of safety (FSs) of other slip surfaces are highly correlatedwith the FS of the deterministic critical slip surface. Hence, the sys-tem failure probability of slope stability can be well approximatedby the failure probability of the critical slip surface (Zhang et al.2011b).

2.2. Type II problem: single-layered soil slope considering spatialvariability

Inherent spatial variability (ISV) is one of the primary sources ofgeotechnical uncertainties, which significantly affects the slope sta-bility (e.g., Phoon and Kulhawy, 1999; Griffiths and Fenton, 2004;Cho, 2007, 2010; Srivastava and Babu, 2009; Srivastava et al., 2010;Huang et al., 2010; Hicks and Spencer, 2010; Wang et al., 2011; Jiet al., 2012, Ji and Low, 2012; Jiang et al., 2014, 2015; Li et al.,2015a; Li and Chu, 2015). When the spatial variability of soil proper-ties is considered, there exist multiple failure modes, which will sig-nificantly influence the failure probability of slope stability (Wanget al., 2011). Estimating the reliability of slope stability consideringspatial variability is, hence, more difficult than that for a single-layered soil slope ignoring spatial variability. Jiang et al. (2014,2015) made use of the Hermite polynomial chaos expansion-basedRSM to solve the reliability of a single-layered soil slope considering


spatial variability (i.e., Type II problem). Similarly, Li et al. (2015a)used the quadratic polynomial without cross terms-based RSM toanalyze the Type II problem.

2.3. Type III problem: multiple-layered soil slope ignoring spatialvariability

Unlike the aforementioned two types of problems, the Type III prob-lem focuses on the reliability of a multiple-layered soil slope ignoringthe spatial variability in each soil layer. Such a reliability problem hasbeen extensively investigated in literature, as shown in Table 2. Unlikethe single-layered soil slope, multiple failure modes often exist in themultiple-layered soil slope (e.g., Zhang et al., 2011b, 2013b; Low et al.,2011; Ji and Low, 2012; Cho, 2013; Zeng and Jimenez, 2014; Kanget al., 2015; Kang and Li, 2015). Only using the deterministic criticalslip surface to estimate slope failure probability leads to underestima-tion of the slope failure rate because the systemeffects of all the possiblefailure modes are not taken into account (e.g., Oka and Wu, 1990;Chowdhury and Xu, 1994, 1995; Zhang et al., 2011b). Therefore, allthe representative failure modes of a multi-layered soil slope shall betaken into account to achieve a more realistic evaluation of system reli-ability of a multi-layered soil slope.

2.4. Type IV problem: multiple-layered soil slope considering spatialvariability

Type IV problem is the most complicated case among the fourmajor problems of soil slope reliability analyses. Such a problem con-cerns with the reliability of a multiple-layered soil slope consideringspatial variability. Li et al. (2015a) and Li and Chu (2015) appliedquadratic polynomial without cross terms to deal with this problem.Li and Chu (2015) found that when the spatial variability is ignored,the number of representative slip surfaces is equal to the number ofsoil layers. The number of multiple response surfaces also highly de-pends on the spatial variability of soil properties. Li et al. (2015a)proposed a multiple response surface method in which the extendedCholesky-decomposition technique is used for discretizing the cross-correlated non-Gaussian random fields. For a multiple-layered soilslope considering spatial variability, the number of potential failuremodes and the locations of representative slip surfaces depend onthe spatial variability and the stratification of the slope. Multiple po-tential failure modes resulted from spatial variability and stratifica-tion in the Type IV problem render the slope reliability analysis amore challenging task.

3. Response surface methods (RSMs)

The aforementioned four types of slope reliability problems and thecorresponding RSMs used in literature are summarized in Table 2. It isshown that only the quadratic polynomial without cross terms-basedRSM (e.g., Xu and Low, 2006; Zhang et al., 2011a,b, 2013a,b; Ji andLow, 2012; Tan et al., 2013; Li et al., 2015a; Li and Chu, 2015) is usedto solve all the four types of slope reliability analysis problems in litera-ture. In addition, theHermite polynomial chaos expansion-based RSM isused to analyze the Type II problemby Jiang et al. (2014, 2015) though itcan also be applied to the other three types of problems in theory. Theremaining part of this paper hence focuses on comparing the feasibilityand validity of the two polynomial-based response surfaces (i.e., thequadratic polynomial without cross terms and the Hermite polyno-mial chaos expansion, as shown by bold font in Table 2) in solvingthe Type I–IV problems.

3.1. Single response surface method (SQRSM & SSRSM)

For a potential slip surface of a soil slope, the relationship betweenthe FS and the input parameters can be approximated by a quadratic

polynomial function (e.g., Bucher and Bourgund, 1990; Zhang et al.,2011b, 2013b):

FSQ Xð Þ ¼ a0 þXn

i¼1

bixi þXn

i¼1

cix2i ð1Þ

where FSQ(X) is the FS for the given slip surface estimated from thequadratic polynomial response surface; X = (x1,⋯, xi,⋯, xn) is thevector of input random variables, in which n is the number ofinput random variables or the number of random field elements;a = (a0, b1,⋯, bn, c1,⋯, cn)T is the vector of unknown coefficients.To determine the unknown coefficients, the FS of a slip surface isevaluated at (2n + 1) points: {μx1, μx2,⋯, μxn}, {μx1 ± kσx1, μx2,⋯, μxn},{μx1, μx2 ± kσx2,⋯, μxn},⋯, and {μx1, μx2,⋯, μxn ± kσxn}, where k is setas 2 in this study. A regression-based approach is used to compute theunknown coefficients a (e.g., Li et al., 2015a).

Under the Hermite polynomial chaos expansion, the FS for a givenslip surface is calculated by (e.g. Huang et al., 2009; Li et al., 2011;Al-Bittar and Soubra, 2013; Jiang et al., 2014, 2015):

FSH ξð Þ ¼ a0Γ0 þXn

i1¼1

ai1Γ1 ξi1� �þ

Xn

i1¼1

Xi1

i2¼1

ai1 i2Γ2 ξi1 ; ξi2� �

þXn

i1¼1

Xi1

i2¼1

Xi2

i3¼1

ai1 i2 i3Γ3 ξi1 ; ξi2 ; ξi3� �þ⋯

þXn

i1¼1

Xi1

i2¼1

Xi2

i3¼1

⋯Xin−1

in¼1

ai1 i2 ;⋯; inΓn ξi1 ; ξi2 ;⋯; ξin

� � ð2Þ

where n is the total number of random variables, a=(a0,ai1,⋯ ,ai1i2, ⋯ ,in)is the unknown coefficients to be evaluated; ξ=(ξi1,ξi2,⋯ ,ξin) is thevector of independent standard normal variables representing the un-certainties in the input parameters; and Γn(⋅) is the multi-dimensionalHermite polynomials of order p, in which p is taken as 2 in this study.Eq. (2) is often referred to as the stochastic response surface thatapproximates the FS.

For a single quadratic response surface method (SQRSM), the surro-gate performance function given by Eq. (1) is constructed between theminimum factor of safety (FSmin) among all potential slip surfaces andthe original random variables. The single stochastic response surfacemethod (SSRSM) shown in Eq. (2) is also a surrogate model for slopestability analysis that provides an explicit and approximate perfor-mance function to predict the FS of slope stability (e.g., Huang et al.,2009; Li et al., 2011; Jiang et al., 2014, 2015). In the SSRSM, the surrogateperformance function is constructed between the FSmin among allpotential slip surfaces and the independent standard normal randomvariables ξ.3.2. Multiple response surface methods (MQRSM & MSRSM)

Unlike the single response surface method, the multiple responsesurface methods are often used to solve the slope reliability problemwith multiple failure modes. The multiple response surface methodsconsist of the multiple quadratic response surface method (MQRSM)and multiple stochastic response surface method (MSRSM), which arebriefly introduced as below.

MQRSM makes use of multiple quadratic response surfaces to ap-proximate the relationship between FSs of all the potential slip surfacesand random variables in their original space, each of which correspondsto the original performance function of one potential slip surface. Afterobtaining multiple quadratic response surfaces, NMC realizations ofrandom fields XNG,F are generated using the extended-Choleskydecomposition technique (e.g., Li et al. 2015a). If spatial variability ofsoil properties is not considered, NMC random variable samples XNG,V

are obtained using the Nataf transformation (e.g., Lebrun and Dutfoy,2009; Li et al., 2011; Tang et al., 2013). Substituting NMC realizations of


random fields or random variables into multiple quadratic response

surfaces yields NMC estimates (i.e., minj¼1;2;⋯;Ns

FSQj ðXNGÞ) of FSmin, where

Ns is the number of potential slip surfaces. Then, the number of FSmin

values less than 1.0 is determined and is denoted as NF. The systemfailure probability of slope stability is calculated by Pf = NF ∕ NMC.

Unlike the MQRSM, the MSRSM is constructed between FS values ofall potential slip surfaces and the input parameters ξ in independentnormal space. To constructNs surrogate performance functions betweenFS values of all the potential slip surfaces and the input parameters ξwhen spatial variability of soil properties is considered, Np realizationsof independent or cross-correlated random fields are generated usingthe Karhunen–Loève expansion technique (e.g., Ghiocel and Ghanem,2002; Phoon et al., 2002; Ghanem and Spanos, 2003; Vořechovský,2008; Cho, 2010). Note that the correlation function is needed in gener-ating random field. For example, the squared exponential (SQX) auto-correlation function (ACF) (e.g., Cao and Wang, 2014a; Li et al., 2015a)is adopted to model spatial variability in this study. Then the limit equi-libriummethods (e.g., simplified Bishopmethod) of slope stability withtheNp realizations of random fields are applied to calculate the FS valuesof each potential slip surface. The unknown coefficients a in Eq. (2) aredetermined based on the Np random samples ξ and the correspondingFS values. The above procedure is repeated for Ns potential slip surfaces.Then, the surrogate performance function is constructed between the FSvalues and the input parameters ξ for each potential slip surface. If thespatial variability of soil properties is not considered, the input soil pa-rameters used for limit equilibrium analysis are obtained via the Nataftransformation. After obtaining MSRSM, MCS is carried out to generateNMC realizations of independent standard normal samples ξ. Substitut-ing each realization of input samples ξk, k = 1, 2,⋯, NMC, into multiple

stochastic response surfaces leads toNMC values of FSmin ¼ minj¼1;2;⋯;Ns

FSHj

ðξkÞ. Similar toMQRSM, the number (i.e.,NF) of FSmin values less than 1.0is determined, and Pf is then calculated as NF ∕ NMC.

4. Soil slope reliability analysis using RSMs

This section will examine the capacity of the aforementioned fourRSMs (i.e., SQRSM, SSRSM, MQRSM, and MSRSM) in solving differentslope reliability problems (i.e., Type I–IV problems). The computationalaccuracy and efficiency of the RSMs are compared and discussed usingfour soil slope examples, namely Examples #1, #2, #3 and #4. Foreach soil slope example, the mean value P ̅

f of slope failure probabilityfor each RSM is taken as the average value of Pf obtained from 20independent runs of the method. The mean value of P f is an importantindex in evaluating the accuracy of a given RSM and indicates whetherthe method is biased or not. The coefficient of variation of Pf, COV[Pf],is also obtained using the results from 20 independent runs. The UnitCOV (i.e., Δ) is used to quantify the efficiency of RSMs in this study,

Fig. 1. The geometry of the

and it is calculated as (e.g., Schuëller and Pradlwarter, 2007; Au et al.,2007):

Δ ¼ COV P f� ��

ffiffiffiffiffiffiNe

pð3Þ

where Ne is the equivalent number of evaluations of the original perfor-mance function (e.g., limit equilibrium analysis) in each run, which iscalculated by.

Ne ¼ Npe þ t0∕t ð4Þ

in which Npe is the number of evaluations of the original perfor-mance functions needed to construct single or multiple response sur-faces; t is the computational time needed to perform one evaluation ofthe original performance function; and t′ is the computational time ofsubsequent MCS based on response surfaces that have been construct-ed. A relatively small value of Unit COV indicates a relatively highcomputational efficiency.

For the soil slope reliability analysis, the limit state function (LSF) isusually defined as g(X)= FS(X)− 1=0. The FS(X) is the factor of safetycalculated by a deterministic slope stability analysis method, e.g., limitequilibriummethod (LEM), which is used to calculate FS values for con-struction of response surfaces in the four examples. In other words, theresponse surfaces in the four examples are calibrated using the limitequilibrium analysis of slope stability.

4.1. Applications of RSMs to reliability analysis of two cohesive slopes

4.1.1. Type I problem: a single-layered cohesive slope (Example #1, no ISV)The profile of the undrained slope (Example #1) in a uniform layer is

shown in Fig. 1, which has been studied by Cho (2010) and Jiang et al.(2015). The cohesive slope has a height of 5 m, a slope angle of 26.6°and a total unit weight of 20 kN/m3. The undrained shear strength cuis assumed to be lognormally distributed with a mean cu of 23 kPa anda COV of 0.3. Based on the mean value of the undrained shear strengthcu, the FSmin is obtained as 1.356 using simplified Bishop method,which is consistent with the value reported by Cho (2010) and Jianget al. (2015). In the calculation, a total number of 4851 potential slipsurfaces at different locations are generated using the “Entry and Exit”method in SLOPE/W (GEO-SLOPE International Ltd., 2012), as shownin Fig. 1.

Table 3 summarizes the results of Example #1 for the Type Iproblem. It can be seen that the average values P ̅

f of the slope failureprobability are 0.173, 0.173, 0.172 and 0.172 for SQRSM, MQRSM,SSRSM and MSRSM, respectively. These values agree well with 0.173obtained from direct MCS in this study and are slightly lower than thevalue 0.186 reported by Cho using FORM (2010). Since the values ofP ̅

f obtained from the four RSMs are in good agreementwith that obtain-ed using direct MCS and simplified Bishop method, all of the four RSMs

slope (Example #1).

Fig. 3. The geometry of three-layered slope and 14,896 potential slip surfaces (Example #2).

Fig. 2. A typical realization of random field (Example #1).


provide reasonably accurate estimates ofP f of the Type I problem in thisexample. The SQRSMhas the smallest value of Unit COV=0.003 amongthe four RSMs. Comparedwith the SQRSM, the computational efficiencyof theMCS is low, which produces a relatively high value of Unit COV=0.26. Though the four RSMs have comparable accuracy, the SQRSM hasthe highest calculation efficiency among the four RSMs. Therefore, it isrecommended to be used for the Type I problem with a single-layeredcohesive slope ignoring spatial variability.

4.1.2. Type II problem: a single-layered cohesive slope (Example #1,consider ISV)

In this section, the slope profile of the undrained slope shown inFig. 1 is used again. Unlike the Type I problem in Example #1, the spatialvariability of undrained shear strength cu is considered in the Type IIproblem. In other words, the Type II problem in Example #1 focuseson evaluating the reliability of the undrained slope considering the spa-tial variability of undrained shear strength. The statistical properties ofthe other soil parameters remain the same as those in Type I problem.The spatial variability of cu is modeled using a 2D lognormal stationaryrandom field with a horizontal autocorrelation distance θln,h of 20 m,and a vertical autocorrelation distance θln,v of 2.0 m. The random fieldcan be simulated using the mid-point method (e.g., Srivastava et al.,2010; Suchomel and Mašin, 2010; Wang et al., 2011), in which the

Fig. 4. A typical realization of ra

cohesive soil layer is discretized into 910 elements with a side lengthof 0.5 m. Fig. 2 shows a typical realization of the random field of cu.

Table 4 presents the results of the Type II problem in Example #1.When the four types of RSMs are used as surrogate models for theimplicit LSF for slope reliability analysis, the mean values of slope failureprobability P f are 0.276, 0.082, 0.081 and 0.079 for the SQRSM, MQRSM,SSRSM and MSRSM, respectively. The values of P ̅

f obtained fromMQRSM, SSRSM and MSRSM generally agree well with 0.076 reportedby Cho (2010) (see Table 5), 0.083 from reported by Jiang et al. (2015)(see Table 5), and 0.078 obtained from direct MCS and simplified Bishopmethod. On the other hand, using SQRSM leads to significant overestima-tion of slope failure probability in the Type II problem of this example.This indicates that the SQRSM cannot capture the feature of LSF for asoil slope when the spatial variability of undrained shear strengthcu is considered, which subsequently results in a misleadingestimate of slope failure probability. Note that both the MQRSMand MSRSM produce relatively small values of Unit COV of 0.446and 0.327, which are significantly smaller than 2.860 by directMCS. Compared with the MQRSM and MSRSM, the SSRSM leads toa slightly higher Unit COV of 0.694. Based on these results, theMQRSM and MSRSM are more efficient than SSRSM and direct MCS.Thus, both the MQRSM and MSRSM are recommended to be usedfor dealing with the Type II problem.

ndom field (Example #2).

Fig. 5. The geometry of the slope and 5491 potential slip surfaces (Example #3).



4.1.3. Type III problem: a three-layered cohesive slope (Example #2, no ISV)The first example, namely Example #1, focuses on the single-layered

cohesive slope. It is well-known that multiple failure modes may be in-volved in amultiple-layered slope. For this reason, a three-layered cohe-sive slope (Example #2) is used as the second example to investigatethe capacity of the four RSMs in solving the Type III problem. Example#2 is adopted from Feng and Fredlund (2011), Zhang et al. (2013b), Liand Chu (2015) and Kang et al. (2015). The geometry of the slope isshown in Fig. 3 and the soil parameters are summarized in Table 6.The undrained shear strength Su1, Su2 and Su3 of three clayed layersare modeled as independent random variables. Based on mean valuesof the undrained shear strength, the FSmin is obtained as 1.282 usingsimplified Bishop method, which is the same as that reported by Kanget al. (2015). As shown in Fig. 3, a total number of 14,896 potentialslip surfaces are randomly generated to cover the whole potentialfailure area using the “Entry and Exit” method in SLOPE/W.

Fig. 7. The geometry of the slope and 25,94

Table 7 presents the results of the Type III problem in Example #2.For comparison, Table 8 also summarizes the reliability results reportedin literature. The mean estimates of slope failure probability P f are0.216, 0.184, 0.219 and 0.240 for the SQRSM, MQRSM, SSRSM, andMSRSM, respectively. In general, they compare favorably with results(around 0.19) obtained by Zhang et al. (2013b) and Kang et al.(2015), and 0.197 using direct MCS in this study though the estimatesof P f from SQRSM, SSRSM, and MSRSM are slightly higher. It seemsthat all the four RSMs can provide reasonably accurate estimates ofslope failure probability for the Type III problem. Further examinationon the Unit COV in Table 7 for the four RSMs shows that the SQRSMand MQRSM have higher computational efficiency than SSRSM andMSRSM. Therefore, the SQRSM and MQRSM are suggested to be usedto solve the Type III problem in cohesive soils.

4.1.4. Type IV problem: a three-layered cohesive slope (Example #2,consider ISV)

In the Type IV problem, the undrained shear strength Su1, Su2 and Su3of three clay layers of Example #2 ismodeled by three two-dimensionallognormal stationary random fields, respectively. The horizontal auto-correlation distance θln,h and vertical autocorrelation distance θln,v aretaken as 20 m and 2.0 m, respectively. The three random fields of Su1,Su2 and Su3 are assumed to bemutually independent, and are discretizedinto 495, 939 and 1080 elements, respectively, with a side length of0.5 m as shown in Fig. 4.

Table 9 summarizes the reliability results of the Type IV problem inExample #2. The mean estimates of slope failure probability P f are0.027, 0.057, 0.088 and 0.031 for the SQRSM, MQRSM, SSRSM, andMSRSM, respectively. Taking the slope failure probability 0.057 obtain-ed fromdirectMCS and simplified Bishopmethod as an “exact” solution,the four RSMs provide estimates of slope failure probability that aregenerally comparable with that obtained from direct MCS. The SQRSMand MSRSM may slightly underestimate the slope failure probability,while the SSRSM may slightly overestimate the probability of failure.The slight differences in estimates of slope failure probability betweenthe four RSMs and direct MCS may be attributed to two aspects. First,the potential failure modes for the multiple-layered slopes consider-ing spatial variability becomemore complicated. Second, the SQRSM,SSRSM and MSRSM may not completely capture the feature of LSFsfor the potential slip surfaces. It is noted that the MSRSM has thelowest value of Unit COV (i.e., 0.69) among the four RSMs, whichindicates that the MSRSM is more efficient than the other threeRSMs. It is, therefore, recommended that the MSRSM is used for theType IV problem.

4.1.5. Suggested RSMs for reliability analysis of cohesive slopeTaking two cohesive soil slopes as examples (i.e., Examples #1 and

#2), four types of slope reliability problems (i.e., Type I–IV problems)

7 potential slip surfaces (Example #4).


Table 5Results for Type II problem obtained from the references (Example #1, ISV).


are investigated using the SQRSM, MQRSM, SSRSM and MSRSM. Therecommendations for selecting the relatively appropriate RSMs foreach type of problem are summarized in Table 10. Note that the RSMsmarked with “√” indicate that they can provide reasonably accurate es-timates of slope failure probability in the corresponding slope reliabilityproblems. As mentioned previously, the Unit COV is an important indi-cator for evaluating computational efficiency. The computational effi-ciency associated with the four RSMs is ranked from 1 to 4. As therank number increases from 1 to 4, the corresponding RSM becomesless efficient. Based on the trade-off between computational accuracyand efficiency, the appropriate RSMs are recommended for differentslope reliability problems, as shown in Table 10. For the single-layeredcohesive soil slope reliability problem ignoring spatial variability,the SQRSM is recommended. For the single-layered cohesive soil slopereliability problem considering spatial variability, both the MQRSMand MSRSM can be applied. In addition, both the SQRSM and MQRSMare applicable to the Type III problem in cohesive soils, i.e., multiple-layered cohesive soil slope reliability problem ignoring spatial variabil-ity. For multiple-layered cohesive slope reliability analysis consideringspatial variability, the MSRSM can be adopted.

Table 3Results for Type I problem: a single-layered cohesive slope (Example #1, No ISV).

Methods Equivalent number ofperformance functionevaluations of each run

Meanvalue of P f

COV[Pf] UnitCOV

Source

SQRSM 3.02 0.173 0.002 0.003 This studyMQRSM 4.47 0.173 0.003 0.006SSRSM 3.01 0.172 0.022 0.038MSRSM 3.57 0.172 0.022 0.042MCS 100.00 0.173 0.03 0.26

Note: The methodmarked by bold font is the recommended RSM. This usage is also validfor the results shown in Tables 3, 4, 7, 9, 12, 13, 16 and 17.

Table 4Results for Type II problem: a single-layered cohesive slope (Example #1, ISV).


Meanvalue of P f

COV[Pf] UnitCOV

Source


4.2. Applications of RSMs to reliability analysis of two c–ϕ slopes

4.2.1. Type I problem: a single-layered c–ϕ slope (Example #3, no ISV)The above two examples (e.g., Examples #1, #2) are under

undrained condition, where ϕ is taken as 0. Hence, simplified Bishopmethod is identical to the ordinary slice method for Examples #1 and#2. In such a case, the performance functionG for a potential slip surfaceis linear (Chowdhury and Xu, 1994). For further illustration, a single-layered c–ϕ slope (Example #3) shown in Fig. 5 with a height of 10 mand a slope angle of 45° is explored, which has been analyzed by Cho(2010) and Jiang et al. (2015). Statistics of soil parameters are shownin Table 11. Based on the mean values of soil parameters, the FSmin iscalculated using simplified Bishop method, and it is equal to 1.206,which agrees well with those (i.e., 1.206 and 1.204, respectively)reported by Jiang et al. (2015) and Cho (2010).

Table 12 summarizes the results of the Type I problem in Example#3 illustrated by a single-layered c–ϕ slope ignoring spatial variability

Method Pf Source

MCS (100,000 samples) 0.076 Cho (2010)MSRSM 0.079 Jiang et al. (2015)MCS (1000 samples) 0.083 Jiang et al. (2015)

Table 6Statistical properties of soil parameters in Example #2.

Slope layers Variable Unit weight, γ (kN/m3) Undrained strength, Su (kPa)

Mean COV Distribution

Clay 1 Su1 18 18 0.3 LognormalClay 2 Su2 18 20 0.2 LognormalClay 3 Su3 18 25 0.3 Lognormal

Table 7Results for Type III problem: a three-layered cohesive slope (Example #2, No ISV).


Meanvalue of P f

COV[Pf] UnitCOV

Source


Table 8Results for Type III problem obtained from the references (Example #2, No ISV).

Methods Pf Source

MCS with 10,000 samplings 0.187 Zhang et al. (2013b)Extended Hassan and Wolff method with 10,000samples

0.184 Zhang et al. (2013b)

MCS using GPR-based RSM with 10,000 samples 0.186 Kang et al. (2015)MCS using GPR-based RSM with 20,000 samples 0.185 Kang et al. (2015)MCS using GPR-based RSM with 100,000 samples 0.186 Kang et al. (2015)

Table 9Results for Type IV problem: a three-layered cohesive slope (Example #2, ISV).


Meanvalue of P f

COV[Pf] UnitCOV

Source



Parameter Mean COV Distribution Correlation coefficient

Cohesion, c (kPa) 10 0.3 Lognormalr = −0.7

Friction angle, ϕ (°) 30 0.2 LognormalUnit weight, γ (kN/m3) 20 ― ― ―

Note: The symbol “―” denotes the result is not applicable.

Table 12Results for Type I problem: a single-layered c–ϕ slope (Example #3, No ISV).

MethodsEquivalent number ofperformance functionevaluations of each run

Meanvalue of P f

COV[Pf] Unit COV Source

SQRSM 5.04 0.043 0.005 0.011

This studyMQRSM 8.25 0.042 0.005 0.014SSRSM 6.01 0.040 0.752 1.843MSRSM 7.91 0.079 1.824 5.129MCS 250.00 0.039 0.131 2.067


of c and ϕ. The average values of failure probability P f obtained fromSQRSM, MQRSM and SSRSM are 0.043, 0.042 and 0.040, respectively,which agree well with 0.039 obtained using direct MCS and simplifiedBishopmethod. However, the probability of failure (i.e., 0.079) obtainedfromMSRSM is obviously higher than that of directMCS (e.g., 0.039). Inaddition, it is shown that SQRSM has the smallest Unit COV among thefour RSMs (see column 5 of Table 12). It is recommended to be usedfor the Type I problem in c–ϕ soil (i.e., single-layered c–ϕ slope ignoringsoil variability) though MQRSM and SSRSM also provide reasonablyaccurate slope failure probabilities for this type of problem.

4.2.2. Type II problem: a single-layered c–ϕ slope (Example #3, considerISV)

In the Type I problem of Example #3, the ISV of c and ϕ is not takeninto account. In the Type II problem, their ISV is characterized by twocross-correlated 2D lognormal stationary random fields c and ϕ. θln,hand θln,v of both c and ϕ are taken as 20 m and 2.0 m, respectively. Asshown in Fig. 6, the random fields are discretized into 1210 elementswith a side length of 0.5 m.

Table 13 summarizes the results of the Type II problem in Example#3. Table 14 also summarizes the reliability results obtained from theother references for comparison. Compared with results (i.e., 0.0041)obtained fromdirectMCS in this study and those (around0.004) report-ed by Cho (2010) and Jiang et al. (2015), SSRSM and MSRSM providereasonably accurate estimates (i.e., 0.0052 and 0.0051, respectively) ofslope failure probability. By contrast, both SQRSM and MQRSM overes-timate the failure probability for the Type II problem in Example #3.This ismainly attributed to the fact that SQRSMandMQRSMcannot rea-sonably approximate the LSF of slope stability for a soil slope systemwhen the spatial variability of c and ϕ is considered. In the view of com-putational accuracy, SSRSM and MSRSM are appropriate methods forthe Type II problem in c–ϕ soils. Note that computational accuracy

Table 10Recommendations of selecting appropriate RSMs for four types of reliability problems of a coh

Problem type Indicator SQRS

Single-layered slope ignoring spatial variability (I)Accuracy √Efficiency 1

Single-layered slope considering spatial variability (II)AccuracyEfficiency 2

Multiple-layered slope ignoring spatial variability (III)Accuracy √Efficiency 1

Multiple-layered slope considering spatial variability (IV)Accuracy √Efficiency 4

takes precedence over computational efficiency in choosing suitableRSMs among the four ones. Although the values of Unit COV of SSRSMand MSRSM are 2.56 and 3.57, respectively, which are higher thanthose of SQRSM and MSRSM, they still have distinct superiority overdirect MCS in the view of computational efficiency. Based on thecomputational accuracy and efficiency, both of SSRSM and MSRSM aresuggested methods for dealing with the Type II problem in c–ϕ soils.

4.2.3. Type III problem: a multiple-layered c–ϕ slope (Example #4, no ISV)Fig. 7 shows the geometry of the last slope example (Example #4),

i.e., the Congress Street Cut, which has been analyzed by Oka andWu (1990), Chowdhury and Xu (1994, 1995), Ching et al. (2009),Chowdhury and Rao (2010), and Ji and Low (2012). The cross sectionof the slope consists of an upper layer of sand at the top and three claylayers, each of which has its corresponding c and ϕ. Table 15 presentsthe statistics of shear strength parameters (i.e., c andϕ) and unit weight(γ), which are adopted from Oka and Wu (1990) and Chowdhury andXu (1994). Based on the mean values of soil properties, the FSmin iscalculated using simplified Bishop method, and it is 1.392.

Table 16 shows the results of the Type III problem in Example #4.Compared with the result (i.e., 0.010) obtained from direct MCS andsimplified Bishop method, MQRSM provides an un-biased estimate ofPf in the sense that the P ̅

f from MQRSM is equal to 0.010. As shown inTable 16, MQRSM provides the most accurate estimate of slope failureprobability among the four RSMs. From the perspective of computation-al accuracy, the estimates of slope failure probability from SQRSM,SSRSM and MSRSM are somewhat biased. Note that the value of UnitCOV of MQRSM (i.e., 0.04) is much smaller than those of SRSM,MSRSM, and direct MCS (i.e., 4.20, 5.57, 9.98, respectively). This indi-cates that MQRSM can efficiently evaluate the slope failure probabilityfor the Type III problem in c–ϕ soils compared with SRSM, MSRSM,and direct MCS. Based on both computational accuracy and efficiency,MQRSM is recommended to be used to solve the Type III problem inc–ϕ soils.

esive slope.

M MQRSM SSRSM MSRSM Recommended method

√ √ √SQRSM

2 3 4√ √ √

MQRSM, MSRSM3 4 1√ √ √

SQRSM, MQRSM2 3 4√ √ √

MSRSM2 3 1

Table 13Results for Type II problem: a single-layered c–ϕ slope (Example #3, ISV).


Meanvalue of P f

COV[Pf]UnitCOV

Source

SQRSM 4541.61 0.0828 0.01 0.96

This studyMQRSM 4547.58 0.0112 0.06 3.71SSRSM 1000.23 0.0052 0.08 2.56MSRSM 1020.06 0.0051 0.11 3.57MCS 2500.00 0.0041 0.22 10.99

Table 14Results for Type II problem obtained from the references (Example #3, ISV).

Methods Pf Source

MCS (50,000 samples) 0.0039 Cho (2010)MSRSM 0.0049 Jiang et al. (2015)MCS (10,000 samples) 0.0044 Jiang et al. (2015)MCS (50,000 samples) 0.0039 Jiang et al. (2015)

Table 17Results for Type IV problem: a three-layered c–ϕ slope (Example #4, ISV).


Meanvalue of P f

COV[Pf]UnitCOV

Source

SQRSM 6918.84 0.0543 0.02 1.93

This studyMQRSM 7064.21 0.0021 0.11 9.31SSRSM 1000.07 0.0072 2.16 68.42MSRSM 1013.77 0.0015 0.07 2.15MCSa 70,000 0.0027 0.07 19.40

Note:a A single MCS with 70,000 samples.


4.2.4. Type IV problem: amultiple-layered c–ϕ slope (Example #4, considerISV)

This subsection focuses on the multiple-layered c–ϕ soil slope con-sidering spatial variability (Type IV problem in Example #4). Theexample of Congress Street Cut is, again, used while the spatial variabil-ity of c and ϕ in each soil layer is explicitly modeled using a 2D lognor-mal stationary random field with a horizontal autocorrelation distanceθln,h of 20 m and a vertical autocorrelation distance θln,v of 6.0 m. Asshown in Fig. 8, the random fields in three soil layers are discretizedinto 434, 773 and 522 elementswith a side length of 0.5m, respectively.

Table 17 shows results of the Type IV problem in Example #4. Asshown in Table 17, the values (i.e., 0.0543 and 0.0072, respectively) ofslope failure probability obtained using SQRSM and SSRSM are higherthan that (i.e., 0.0027) from direct MCS and simplified Bishop methodin this study. On the other hand,MQRSM andMSRSMprovide estimates(i.e., 0.0021 and 0.0015, respectively) of slope failure probability thatare generally comparable with that from direct MCS. In addition, theUnit COV (i.e., 2.15) of MSRSM is much smaller than that (i.e., 9.31) ofMQRSM. Hence, it has a higher computational efficiency. Based on thetrade-off between the computational accuracy and efficiency, MSRSMis suggested to solve the reliability of slope stability for the Type IVprob-lem in c–ϕ soils.


Slopelayers

Unit weight, γ(kN/m3)

Cohesion, c (kPa)Angle of internal friction,ϕ (°)

Mean COV Distribution Mean COV Distribution

Sand 21 0 ― ― 30 ― ―Clay 1 19.5 55 0.37 Lognormal 5 0.2 LognormalClay 2 19.5 43 0.19 Lognormal 7 0.21 LognormalClay 3 20 56 0.2 Lognormal 15 0.24 Lognormal

Table 16Results for Type III problem: a three-layered c–ϕ slope (Example #4, No ISV).

MethodsEquivalent no. ofperformance functionevaluations of each run

Meanvalue of P f

COV[Pf]UnitCOV

Source

SQRSM 13.04 0.057 0.005 0.02

This studyMQRSM 23.19 0.010 0.009 0.04SSRSM 28.03 0.086 0.793 4.20MSRSM 37.82 0.106 0.743 4.57MCSa 20,000 0.010 0.071 9.98

Note:a A single MCS with 20, 000 samples.

4.2.5. Suggested RSMs for reliability analysis of c–ϕ slopeSimilar to the recommendations shown in Section 4.1.5, the recom-

mendations for selecting the relatively appropriate RSMs for slope reli-ability problems in c–ϕ soils are summarized in Table 18. The SQRSM isrecommended for the single-layered c–ϕ soil slope reliability problemignoring spatial variability. For the single-layered c–ϕ soil slope reliabil-ity problem considering spatial variability, both SSRSM andMSRSM canbe applied. The MQRSM is applicable to the multiple-layered c–ϕ soilslope reliability problem ignoring spatial variability. For the multiple-layered c–ϕ soil slope reliability problem considering spatial variability,the MSRSM is recommended.

5. Summary and conclusions

This paper reviewed previous studies on applications of RSMs in thedifferent slope reliability problems and systematically investigated thecapacity and validity of four response surface methods (i.e., SQRSM,SSRSM, MQRSM, and MSRSM) to solve the reliability of slope stability.Based on the review, four types of soil slope reliability analysis problemswere identified from the literature, including single-layered soil slopereliability problem ignoring spatial variability, single-layered soil slopereliability problem considering spatial variability, multiple-layered soilslope reliability problem ignoring spatial variability, and multiple-layered soil slope reliability problem considering spatial variability.These problems are referred to as “Type I–IV problems” in this study, re-spectively. For each type of the problem, the computational accuracyand efficiency of SQRSM, SSRSM,MQRSM, andMSRSMare systematical-ly explored and compared for cohesive and c–ϕ soil slopes, respectively.Based on the comparison, some recommendations for selecting relative-ly appropriate RSMs for different slope reliability analysis problems inslope engineering practice are provided, as shown in Table 19.

In summary, for a single-layered homogenous soil slope ignoringspatial variability (i.e., Type I problem), the critical deterministic slipsurface is the dominating slope failure mode. Thus, the SQRSM providesreasonably accurate estimate of slope failure probability and has a highcomputational efficiency. When the spatial variability of soil propertiesis taken into account in reliability analysis of a single-layered soil slope(i.e., Type II problem), the number of potential failure modes increasescompared with those of the homogenous soil slopes. In such a case,theMSRSM can be applied to solve the slope failure probability. In addi-tion, stratification of soils also affects the potential failuremodes of a soilslope and usually leads to multiple failure modes in the Type III prob-lem, for which the MQRSM is recommended. When both stratificationand spatial variability are involved in soil slope reliability analysis(i.e., Type IV problem), the MSRSM is the suggested method that cancapture the performance functions well and has a good computationalefficiency.

Acknowledgments

This work was supported by the National Science Fund for Distin-guished Young Scholars (Project No. 51225903), the National Basic

Table 18Recommendations of selecting appropriate RSMs for four types of reliability problems of a c–ϕ slope.

Problem type Indicator SQRSM MQRSM SSRSM MSRSM Recommended method

Single-layered slope ignoring spatial variability (I)Accuracy √ √ √

SQRSMEfficiency 1 2 3 4

Single-layered slope considering spatial variability (II)Accuracy √ √

SSRSM, MSRSMEfficiency 1 4 2 3

Multiple-layered slope ignoring spatial variability (III)Accuracy √

MQRSMEfficiency 1 2 3 4

Multiple-layered slope considering spatial variability (IV)Accuracy √ √

MSRSMEfficiency 1 3 4 2

Table 19Selected RSMs for soil slope reliability problems.

Problem typeRecommendedmethod

Single-layered slope ignoring spatial variability (I) SQRSMSingle-layered slope considering spatial variability (II) MSRSMMultiple-layered slope ignoring spatial variability (III) MQRSMMultiple-layered slope considering spatial variability (IV) MSRSM


Research Programof China (973 Program) (ProjectNo. 2011CB013506),the National Natural Science Foundation of China (Project No.51329901, 51409196) and the Natural Science Foundation of HubeiProvince of China (Project No. 2014CFA001).

Appendix A. List of symbols

Symbol

RFOFOSOMSQMSSMSVRRAHNFSFSISXXNNNNNNPfPI{ξCΔt′tLESRLSθl

Description

SM
response surface method SM first-order second moment method RM first-order reliability method RM second-order reliability method CS Monte Carlo Simulation RSM single quadratic response surface method QRSM multiple quadratic response surface method RSM single stochastic response surface method SRSM multiple stochastic response surface method M support vector machine
BFN
radial basis function neural network VM relevance vector machine NN artificial neural network DMR high dimensional model representation N neural networks
factor of safety
min minimum FS among Ns potential slip surfaces V inherent spatial variability NG,F non-Gaussian random field in original space NG,V non-Gaussian random variable in original space MC number of Monte Carlo Simulation s number of potential slip surfaces p number of realizations of random fields e equivalent number of evaluations of the original performance function pe number of original performance function evaluations in a run F number of FSmin values less than 1.0
probability of failure

f
average value of failure probability obtained from 20 independent runs ·} indicator function
vector of independent standard normal variables
OV coefficient of variation
Unit COV
computational time of MCS procedure computational time of one original performance function evaluation
M
limit equilibrium method M strength reduction method F limit state function n,h horizontal autocorrelation distance n,v vertical autocorrelation distance θl
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