Engineering Geology 187 (2015) 60–72

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Engineering Geology

j ourna l homepage: www.e lsev ie r .com/ locate /enggeo

A multiple response-surface method for slope reliability analysisconsidering spatial variability of soil properties

Dian-Qing Li a,⁎, Shui-Hua Jiang a,b, Zi-Jun Cao a, Wei Zhou a, Chuang-Bing Zhou b, Li-Min Zhang c

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 School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, PR Chinac Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

⁎ Corresponding author. Tel.: +86 27 6877 2496; fax: +E-mail address: dianqing@whu.edu.cn (D.-Q. Li).

http://dx.doi.org/10.1016/j.enggeo.2014.12.0030013-7952/© 2014 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 28 May 2014Received in revised form 7 December 2014Accepted 8 December 2014Available online 27 December 2014

Keywords:SlopesReliability analysisSpatial variabilityRandom fieldAutocorrelation functionResponse surface

This paper proposes a multiple response-surface method for slope reliability analysis considering spatially vari-able soil properties. The scales of fluctuation of soil shear strength parameters are summarized. The effect of the-oretical autocorrelation functions (ACFs) on slope reliability is highlighted since the theoretical ACFs are oftenused to characterize the spatial variability of soil properties due to a limited number of site observation data avail-able. The differences infive theoretical ACFs, namely single exponential, squared exponential, second-orderMar-kov, cosine exponential and binary noise ACFs, are examined. A homogeneous c–ϕ slope and a heterogeneousslope consisting of three soil layers (including a weak layer) are studied to demonstrate the validity of the pro-posedmethod and explore the effect of ACFs on the slope reliability. The results indicate that the proposedmeth-od provides a practical tool for evaluating the reliability of slopes in spatially variable soils. It can greatly improvethe computational efficiency in relatively low-probability analysis and parametric sensitivity analysis. The ex-tended Cholesky decomposition technique can effectively discretize the cross-correlated non-Gaussian randomfields of spatially variable soil properties. Among the five selected ACFs, the squared exponential and second-order Markov ACFs might characterize the spatial correlation of soil propertiesmore realistically. The probabilityof failure associatedwith the commonly-used single exponential ACFmay be underestimated. In general, the dif-ference in the probabilities of failure associated with the five ACFs is minimal.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The inherent spatial variability of soil properties has been consid-ered as one of the major sources of uncertainty in geotechnical proper-ties (i.e., Phoon and Kulhawy, 1999; Cho, 2007, 2014; Ahmed andSoubra, 2012; Ching and Phoon, 2013; Zhao et al., 2013; Cao andWang, 2014; Low, 2014), which is often characterized by an autocorre-lationmodel in geotechnical practice. In the literature (e.g., Li and Lumb,1987; Jaksa, 1995; Phoon et al., 2003; Uzielli et al., 2005; Kim and Sitar,2013; Lloret-Cabot et al., 2014), theoretical ACFs such as single expo-nential and squared exponential ACFs are often employed to capturethe correlation structure of soil properties. As pointed out by Li andLumb (1987), Tanahashi (1998) and Kasama et al. (2012), however,the rationale behind the commonly-used single exponential ACF shouldbe explained to clarify the assumption made in geotechnical practice.

Direct Monte Carlo Simulation (MCS) is a conceptually simple androbust tool for analyzing the reliability of slopes in spatially variablesoils (Cho, 2010; Huang et al., 2010; Wang et al., 2011; Li et al., 2013a,2014). However, the MCS suffers from a lack of efficiency at small

86 27 6877 4295.

probability levels. As an alternative, the response surface method(RSM) has been widely applied to slope reliability problems (e.g., Liet al., 2011, 2013b; Jiang et al., 2014a). However, this traditional RSMmay not efficiently solve slope reliability problems considering spatialvariability of soil properties (Huang et al., 2010). A more efficient reli-ability method for slope reliability analysis considering spatially vari-able soil properties should be developed. Additionally, to explore theeffect of various ACFs on slope reliability when the spatial variabilityand cross-correlation of soil shear strength parameters are incorporat-ed, an effective and general approach is to performa series of parametricsensitivity studies. Theoretically, direct MCS can be used for such pur-pose. However, it could be time-consuming for parametric sensitivityanalyses because numerous similar slope stability analyses are carriedout repeatedly.

The objective of this study is to propose a multiple response-surfacemethod for slope reliability analysis considering spatially variable soilproperties. First, the typical ranges of scales of fluctuation underlyingsoil shear strength parameters are extensively summarized. Thereafter,five 2-D ACFs are compared systematically. Then, a multiple response-surface method is presented, and an implementation procedure is de-veloped step by step. Finally, a homogeneous c–ϕ slope and a heteroge-neous slope consisting of three soil layers (including a weak layer) are

61D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

investigated to illustrate the proposed method. A parametric sensitivitystudy is also carried out to explore the effect of ACFs on the slope reli-ability. The slope reliability associated with the commonly-used singleexponential ACF is highlighted.

2. Autocorrelation function and scale of fluctuation

The soil parameters at a particular location are random variables dueto the spatial variation, but are more similar to those at adjacent loca-tions than those far away (Vanmarcke, 2010). As reported in the litera-ture (Jaksa, 1995; Fenton and Griffiths, 2003; Kasama et al., 2012; Wuet al., 2012; Ji, 2014; Jiang et al., 2014b,c; Salgado and Kim, 2014), thesoil properties in a statistically homogeneous soil layer are consideredto be weakly stationary random fields. In this case, the means and vari-ances of soil properties remain the same at every point within the re-gion of the random field, and an ACF governs the degree of correlationbetween the residuals of any two points regardless of their absolute co-ordinateswithin the randomfield. However, the randomfields are oftennonstationary for multi-layered soil slopes. The method reported in Luand Zhang (2007) and Cho (2012) can be adopted to characterize andmodel the spatial variability of multi-layered soil properties. The simu-lation domain is partitioned intomultiple non-overlapping subdomains.The soil behavior is assumed to be statistically stationary within eachsubdomain, which implies that the covariance between any two pointsdepends only on their separate distance rather than their locations. Thecovariance between any two points in different regions is assumed to bezero. Therefore the soil medium is globally nonstationary. The ACF of astationary random field within each subdomain, H(x, y), is defined asthe correlation between the spatial quantities at two locations(Vanmarcke, 2010; Wu et al., 2012; Salgado and Kim, 2014) as follows:

ρ H xi; yið Þ;H xj; yj

� �h i¼

COV H�xi; yi

�;H xj; yj

� �� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar H xi; yið Þ½ �p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var H xj; yj

� �h ir ð1Þ

where (xi, yi) and (xj, yj) are the coordinates of two locations in a 2-Dspace; Var(.) denotes the variance; COV(.,.) denotes the covariance.

According to Vanmarcke (2010), the scale of fluctuation (SOF) is aconvenient measure for describing the spatial variability of a soil prop-erty in a random field. It is a measure of the distance within whichpoints are significantly correlated. A small SOF implies that the soilproperty fluctuates rapidly about the mean value. On the contrary, alarge SOF indicates that the soil property is significantly correlatedover a large spatial extent. It is important to estimate the SOF accuratelybecause the SOF plays a key role in characterizing spatial variability ofsoil properties at a site. Although several methods, such as the methodbased on goodness-of-fitting of different theoretical ACFs to the sampleACF estimated from available site observation data (Uzielli et al., 2005),and the method based on information from conditional random fields(Lloret-Cabot et al., 2014), can be used to estimate the SOF, thesemethods may not be easily implemented because a large amount ofdata is required. To determine appropriate values of SOFs for a specificsoil, an extensive literature review on the values of SOFs underlyingsoil shear strength parameters is performed in this study. The valuesof vertical and horizontal SOFs reported in the literature are summa-rized in Table 1. The horizontal and vertical SOFs for clays typically fallwithin the ranges of 10–92.4 m and 0.1–8.0 m, respectively. For sandsand silts, the horizontal and vertical SOFs are about 12.7–75 m and0.14–3.0 m, respectively.

As mentioned earlier, the autocorrelation between spatial quantitiesis usually described by an ACF which falls off rapidly as the separationdistance between them increases. The ACF can be estimated using themethod of moments (e.g., Jaksa, 1995; Phoon et al., 2003) or the maxi-mum likelihoodmethod (DeGroot and Baecher, 1993). However, deter-mining a sample ACF in geostatistics is also not easy because acquisition

of a large quantity of statistical data needs much effort especially whenthe available sample size is small (Phoon et al., 2003). Thus, some theo-retical ACFs are usually used to characterize the spatial correlation ofsoil properties. For instance, the single exponential (SNX) ACF hasbeen widely used to model the inherent spatial variability of soil prop-erties in probabilistic analysis of slope stability (Griffiths and Fenton,2004; Cho, 2010; Huang et al., 2010; Wang et al., 2011; Ahmed andSoubra, 2012; Ji et al., 2012; Li et al., 2013c, 2014; Salgado and Kim,2014). The squared exponential (SQX) ACF is usually selected tomodel spatially varying soil properties in the series expansion methodsince the least expansion terms are required to achieve a desired accu-racy (e.g., Jiang et al., 2014b,c). The cosine exponential (CSX) ACF hasbeen used by Cafaro and Cherubini (2002) to fit the experimental datafor estimating the vertical SOF of the shear strength of Taranto clay.The binary noise (BIN) ACF has been employed by Cheng et al. (2000)to estimate the SOFs ofmore than 300 different soils of over 10 differentcities in China.

Table 2 summarizes five 2-D theoretical ACFs for the characteriza-tion of spatial correlation of soil properties. These five ACFs are investi-gated to explore the effect of various ACFs on slope reliability. In Table 2,lags τx and τy are the distances between two locations in the horizontaland vertical directions, respectively; δh and δv are the horizontal andvertical SOFs, respectively; a, b, c, d and e are model parameters relatedto the scale of fluctuation. When the model parameters are adjusted toproduce unit horizontal and vertical SOFs, the resulting surfaces of five2-D ACFs are plotted in Fig. 1. Generally, the differences among thesesurfaces are significant. Since the SNX, CSX and BIN ACFs are non-differentiable functions, the corresponding surfaces exhibit four edgeangles and a sharp corner near the origin. On the contrary, the SQXand SMK ACFs are differentiable near the origin, and the correspondingsurfaces are isotropic and very smooth. Additionally, the surface of theSNX ACF decays more quickly with the increase of absolute values ofτx and τy in comparison with the other four ACFs.

3. Multiple response-surface method

3.1. General concept of multiple response-surface method

A quadratic polynomial chaos expansion (PCE) without cross termsis adopted to establish a response surface function (RSF) between thefactor of safety for each potential slip surface and input original randomvariables (such as shear strength parameters) involved in slope stabilityanalysis. To correctly identify the critical slip surface underlying a slope,Ns different potential slip surfaces are randomly generated to cover theentire failure domain of the slope, in which the value of Ns is frequentlyin the order of magnitude of 103–104 (e.g., Zhang et al., 2011; Li et al.,2013c). In this way, the jth quadratic RSF is expressed as follows(Jiang et al., 2014a):

FSj Xð Þ ¼XNc

i¼1

ai; jΨi; j Xð Þ ¼ a1; j þXni¼1

bi; jxi þXni¼1

ci; jx2i ð2Þ

where FSj(X), j=1, 2,⋯,Ns, is the factor of safety for the jth potential slipsurface; X = (x1, ⋯, xi, ⋯, xn)T is the vector of input random variables inthe physical space, in which n is the number of input random variables;aj = (a1,j, b1,j, ⋯, bn,j, c1,j, ⋯, cn,j)T is the vector of unknown coefficientswith a size of Nc = 2n + 1; Ψi,j(⋅) is a general PCE.

A sample design method using (2n + 1) combinations proposed byBucher and Bourgund (1990) is adopted to determine the unknown co-efficients in Eq. (2). The factor of safety for the jth potential slip surface,j = 1, 2,⋯, Ns, is first evaluated at Nc = 2n + 1 samples as below:

μx1 ; μx2 ; ⋯; μxn

n o, μx1 � kσ x1 ; μx2 ; ⋯; μxn

n o, μx1 ; μx2 � kσ x2 ; ⋯; μxn

n o,…,

and μx1 ; μx2 ; ⋯; μxn � kσ xn

n o, where k is a coefficient for generating

the sampling points, and k = 2 is used in this study (e.g., Zhang

Table 1Summary of scales of fluctuation for soil shear strength parameters.

Soil property Soil type ACF Scale of fluctuation (m) References

Horizontal(δh)

Vertical(δv)

cu Marine clay, Japan SNX – 1.3–2.7 Matsuo (1976)cu New Liskeard varved clay – 46 5.0 Vanmarcke (1977)cu (DST) Clay SQX 92.4 1.19–1.23 Ronold (1990)c Shanghai silty clay SNX, CSX – 0.31–0.42 Gao (1996)ϕ – 0.32–0.47cu (VST) Clay – 46–60 2.0–6.2 Phoon and Kulhawy (1999)cu (qc) Sandy soil SQX, CSX, BIN – 0.1–1.0 Cheng et al. (2000)cu (qc) Clay – 0.1–1.8cu (qc) Soft clay – 0.2–2.0cu (qc) Taranto clay CSX – 0.287–0.401 Cafaro and Cherubini (2002)cu Clay SNX – 0.25–2.5 Hicks and Samy (2002)c Yan'an silty clay – – 1.47 Ni et al. (2002)ϕ 1.44c Jiangzhang silty clay – – 6.47ϕ 2.96c Tongguan silt – – 7.19ϕ 1.2cu Sensitive clay, soft clay SNX 20–80 2.0–6.0 El-Ramly et al. (2003)c (DST) Taiyuan silty clay BIN 36.2–41.7 0.37–0.58 Li et al. (2003)ϕ (DST) 36–41.4 0.35–0.49c (DST) Taiyuan silt 41.5–45.1 0.6–0.84ϕ (DST) 41.8–45.5 0.54–0.92c (DST) Hangzhou silty clay 40.5–45.4 0.52–0.75ϕ (DST) 40.4–45.2 0.49–0.71c (DST) Hangzhou clay – 0.5–0.77ϕ (DST) – 0.59–0.73cu (qc) Sand, clay SMK, CSX – 0.13–1.11 Uzielli et al. (2005)c, ϕ Clay SNX 2–30 0.1–5.0 Hsu and Nelson (2006)cu Chicago clay – – 0.79–1.25 Xie (2009)cu Saturated clay, Japan – – 1.25–2.86cu (qc) Tianjin port clay – 8.37 0.132–0.322 Yan et al. (2009)cu (qc) Tianjin port silty clay – 9.65 0.095–0.426cu (qc) Tianjin port silt – 12.7 0.140–1.0cu (qc) Silty clay SNX – 0.8–6.1 Haldar and Sivakumar Babu (2009)c, ϕ Clay SNX 10–40 0.5–3.0 Suchomel and Mašin (2010)c, ϕ Alluvial soil SNX 30–49 0.2–0.9 Wu et al. (2011)c, ϕ Ocean and lake sedimentary soils 40–80 1.3–8.0c, ϕ Moraine soil – 2.0c, ϕ Aeolian soil – 1.2–7.2cu (VST) Clay SNX 46–60 2.0–6.2 Ching et al. (2011)cu, c, ϕ In situ soils SNX 30–60 1.0–6.0 Ji et al. (2012)c(qc), ϕ (qc) Clay SNX 10–62 1.3–4.0 Salgado and Kim (2014)c(qc), ϕ (qc) Sand SNX 35–75 2.2–3.0 Salgado and Kim (2014)

Note: cu, c,ϕ are undrained shear strength, cohesion and friction angle, respectively; cu (VST) is undrained shear strength from vane shear test; cu (DST), c(DST),ϕ (DST) are shear strengthparameters from direct shear test; cu (qc), c(qc), ϕ (qc) denote the SOFs of shear strength parameters and are referred to those of cone tip resistance qc from cone penetration test. Thesymbol “–” denotes the results are not available.

62 D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

et al., 2011, 2013); μxi andσxi are the mean and standard deviation ofthe ith variable, respectively. In this way, a system of Nc linearalgebraic equations can be established for the selected samples interms of the unknown coefficients aj. Then, a regression-basedapproach is used to compute the unknown coefficients aj. Afterthat, a quadratic RSF for the jth slip surface is obtained. Applying

Table 2Common autocorrelation functions for geostatistical analysis.

Type Autocorrelation function (1-D) Sc

Single exponential (SNX) ρ(τ) = exp(−aτ) δ

Squared exponential (SQX) ρ(τ) = exp[−(bτ)2] δ

Second-order Markov (SMK) ρ(τ) = exp(−cτ)(1 + cτ) δ

Cosine exponential (CSX) ρ(τ) = exp(−dτ)cos(dτ) δ

Binary noise (BIN)ρ τð Þ ¼ 1−eτ for τ≤1=e

0 otherwise

�δ

the similar method, Ns quadratic RSFs are constructed and taken assurrogate models of explicit functions between the factors of safetyand the original random variables. It can be observed that the con-struction of multiple RSFs using the quadratic PCEs without crossterms is relatively simple, which can be easily implemented bypractitioners.

ale of fluctuation Autocorrelation function (2-D)

= 2/a ρ τx; τy� � ¼ exp −2 τx

δhþ τy

δv

� �h i¼ ffiffiffi

πp

=b ρ τx; τy� � ¼ exp −π τ2x

δ2hþ τ2y

δ2v

� � = 4/c ρ τx; τy

� � ¼ exp −4 τxδhþ τy

δv

� �h i1þ 4τx

δh

� �1þ 4τy

δv

� �= 1/d ρ τx; τy

� � ¼ exp − τxδhþ τy

δv

� �h icos τx

δh

� �cos τy

δv

� �= 1/e

ρ τx; τy� � ¼ 1− τx

δh

� �1− τy

δv

� �for τx≤δh and τy≤δv

0 otherwise

(

Fig. 1. Common 2-D autocorrelation functions for geostatistical analysis (normalized to unit scales of fluctuation).

63D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

3.2. Discretization of cross-correlated non-Gaussian random fields using ex-tended Cholesky decomposition technique

For slope reliability problems considering spatial variability of soilproperties, cross-correlated non-Gaussian random fields are often in-volved, which should be discretized properly. The Karhunen–Loève(K–L) expansion (Cho, 2010; Jiang et al., 2014b,c) can be used todiscretize the cross-correlated non-Gaussian randomfields, but the pro-cess may be very complex especially when the eigenvalue problem ofthe Fredholm integral equation cannot be solved analytically. In the lit-erature (Haldar and Sivakumar Babu, 2009; Srivastava et al., 2010;

Suchomel and Mašin, 2010; Kasama et al., 2012; Wu et al., 2012), theCholesky decomposition technique has been widely used to discretizerandom fields of soil properties because it is conceptually simple andeasily implementable. However, the aforementioned Cholesky decom-position technique cannot deal with the cross-correlated non-Gaussian random fields. With this shortcoming in mind, this study ex-tends the Cholesky decomposition technique to discretize cross-correlated non-Gaussian random fields of spatially variable soilproperties.

Based on the centroid coordinates (xi, yi) of random field elements,in which i = 1, 2,⋯, ne, ne is the number of random field elements, and

Fig. 2. Flow chart of the multiple-response-surface method.

64 D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

the selected ACFs shown in Table 2, an autocorrelation matrix C isexpressed as

C ¼

1 ρ τx12 ; τy12� �

⋯ ρ τx1ne ; τy1ne

� �ρ τx12 ; τy12� �

1 ⋯ ρ τx2ne ; τy2ne

� �⋮ ⋮ ⋱ ⋮

ρ τx1ne ; τy1ne

� �ρ τx2ne ; τy2ne

� �⋯ 1

2666664

3777775 ð3Þ

Fig. 3. The stability model for

where ρ τxi j ; τyi j� �

is the autocorrelation coefficient between spatial

quantities at any two points, in which the lags τxi j ¼ xi−xj

��� ��� and τyi j ¼yi−yj

��� ��� are the absolute distances between the centroid coordinates of

the ith element and the jth element. Then, a cross-correlation matrixR is obtained as R= (ρk,l)m × m, in which ρk,l is the cross-correlation co-efficient between the kth random field and the lth random field under-lying the soil properties, and m is the number of the cross-correlatedrandom fields. The standard Cholesky decomposition algorithm isused to factor thematrices C and R in order to obtain the lower triangu-lar matrices L1 with a dimension of ne × ne and L2 with a dimension ofm × m, respectively:

L1L1T ¼ C and L2L2

T ¼ R: ð4Þ

The cross-correlated standard Gaussian random fields HG(x, y) isderived as

Hk;G x; yð Þ ¼ L1ξkL2T where k ¼ 1;2;…;Np ð5Þ

where ξk is a vector of independent standard normal random samples,which is partitioned intom vectors with a dimension of ne. The Eq. (5)is repeated forNp times. Thus, the components ofHG(x, y)with a dimen-sion of (ne×m) ×Np dependon the given randomfieldmesh, the select-ed ACF, the values of SOFs and cross-correlation coefficients amongrandom fields. Thereafter, the cross-correlated non-Gaussian randomfields HNG(x, y) in the physical space can be obtained via anisoprobabilistic transformation (e.g., Li et al., 2011) as follows:

Hk;NGXi

x; yð Þ ¼ F−1i Φ Hk;G

Xix; yð Þ

h in owhere k ¼ 1;2;…;Np; i ¼ 1;2;…;m

ð6Þ

where Fi−1(·) is the inverse function of marginal cumulative distribu-

tion of the non-Gaussian vector random field Xi; Φ(·) is the standardGaussian distribution function.

3.3. Probability of failure using multiple response-surface method

Having obtained the multiple quadratic RSFs for the factors of safetyand the cross-correlated non-Gaussian random fields in the physicalspace in Sections 3.1 and 3.2, respectively, the probability of slopefailure can be evaluated conveniently. A direct MCS with a total of Np

a c–ϕ slope (FS= 1.208).

Table 3Statistical properties of soil parameters for Example I.

Soil parameters Mean COV Distribution Scale of fluctuation Correlation coefficient

c (kPa) 10 0.3 Lognormal δh = 40 m, δv = 4.0 m ρc,ϕ = −0.5ϕ (°) 30 0.2 Lognormal δh = 40 m, δv = 4.0 mγt (kN/m3) 20 – – – –

65D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

samples is performed to obtain the probability of slope failure asfollows:

p f ¼1Np

XNp

k¼1

I minj¼1;2;⋯;Ns

FS j Hk;NG x; yð Þh i

b 1:0�

ð7Þ

where I{.} is an indicator function. For a given random sample, I{.} istaken as the value of 1 for min

j¼1;2;⋯;Ns

FS j �ð Þb1:0. Otherwise, it is set to

zero. It should be pointed out that the evaluation of the factors of safetyherein does not require deterministic slope stability analyses again, butonly involves the evaluation of the algebraic expressions in Eq. (2).Hence, the computational cost of MCS for each random sample is re-duced substantially.

Note that the obtained RSFs between the factors of safety and theoriginal random variables do not rely on the correlation structures(i.e., ACFs) and the statistics (e.g., coefficient of variation, COV, andcross-correlation coefficient) of the soil properties. In other words, theRSFs remain unchanged for different ACFs and statistics, and there isno need to re-calibrate the RSFs during parametric sensitivity studies.Therefore, the proposed method provides an efficient way to performthe sensitivity analyses and to explore the influences of ACFs and statis-tics on slope reliability.

4. Implementation procedure of the multiple response-surfacemethod

To facilitate the understanding of the multiple response-surfacemethod, Fig. 2 shows a flow chart for the implementation procedure.This procedure consists of five steps. Details of each step are summa-rized as follows:

(1) Identify the uncertain parameters and determine their sta-tistics such as means, COVs, marginal distributions and cross-correlation coefficients among input random variables. Then, se-lect an appropriate ACF and the horizontal and vertical SOFs for a2-D random field model.

(2) Construct the slope stabilitymodelwith themean values of inputrandomvariables. Then,mesh random field elements and extractthe centroid coordinates (xi, yi) of each random field element, inwhich i=1, 2,⋯, ne. Thereafter, Ns potential slip surfaces are ran-domly generated to cover the entire failure domain of the slopewithin SLOPE/W (GEO-SLOPE International Ltd., 2010). The fac-tor of slope safety is calculated using Bishop's simplified methodin this study.

(3) Construct Ns quadratic RSFs between the factors of safety forNs potential slip surfaces and the original random variablesusing the sample design method in Section 3.1. A detailed

Table 4Comparison of the reliability results for Example I.

Methods δh (m) δv (m) μFS σFS COVFS pf Relativeerror of pf

This study 40 4.0 1.195 0.102 0.085 1.87 × 10−2 0.09Cho (2010) 1.199 0.106 0.088 1.71 × 10−2 –

This study 40 8.0 1.195 0.119 0.100 3.97 × 10−2 0.07Cho (2010) 1.202 0.126 0.105 3.70 × 10−2 –

This study 80 4.0 1.196 0.104 0.087 2.06 × 10−2 0.08Cho (2010) 1.200 0.109 0.091 1.91 × 10−2 –

implementation combined with SLOPE/W can be referred toJiang et al. (2014b).

(4) Perform MCS to generate the independent standard normal ran-dom sample vector ξ for Np times, and obtain Np realizations ofthe cross-correlated non-Gaussian random fields HNG(x, y) usingthe extended Cholesky decomposition technique in Section 3.2.

(5) Substitute each realization of HNG(x, y) into Eq. (7) to yield Ns fac-tors of safety. The minimum factor of safety among Ns factors ofsafety is identified as FSmin. In this way, Np FSmin values are obtain-ed. Taking the number of the FSmin value below 1.0 asNf, the prob-ability of slope failure is given by pf = Nf/Np.

5. Example I: Application to a c–ϕ slope

A hypothetic c–ϕ slope studied by Cho (2010) is suitable to illustratethe extended Cholesky decomposition technique for discretization ofcross-correlated non-Gaussian random fields. The results obtainedfrom Cho (2010) can be directly used to validate the proposed multipleresponse-surface method. Also, the effect of ACFs on slope reliabilityusing the proposed method is explored. A typical random field elementmodel for the considered slope is shown in Fig. 3. The slope has a heightof H = 10.0 m and an inclination of α = 45°. The random field meshconsists of 4-noded quadrilateral elements, which are degeneratedinto 3-noded triangular elements at the sloping mesh boundary. Therandom field mesh consists of 1210 elements (i.e., ne = 1210) and1281nodes. A total ofNs=2992 potential slip surfaces is randomly gen-erated to cover the entire failure domain of the slope.

Following Cho (2010), cohesion c and friction angleϕ are consideredas cross-correlated lognormal random fields to avoid negative values,and the single exponential ACF is used. The cross-correlation coefficientbetween c and ϕ is taken as ρc,ϕ = −0.5. The horizontal and verticalSOFs are δh = 40 m and δv = 4.0 m, respectively. These parametersare taken as the baseline case. Table 3 summarizes the statistical prop-erties of soil parameters for the considered slope. Based on the meanvalues of the shear strength parameters (c, ϕ), the minimum factor ofsafety is obtained as 1.208 using Bishop's simplified method, which is

Fig. 4. Comparison of the probabilities of failure obtained from two different methods.

Fig. 5. Typical realizations of random fields of the cohesion and friction angle and slope stability results (ρc,ф = −0.5, COVc = 0.3 and COVϕ = 0.2, δh = 40 m and δv = 4.0 m).

66 D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

Fig. 6.Variation of the probability of failure with cross-correlation coefficient between thecohesion and friction angle.

67D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

close to 1.204 using Bishop's simplified method as reported in Cho(2010). The corresponding critical failure surface is plotted in Fig. 3,which passes near the slope toe.

Fig. 7. Variation of the probability of failure with coefficients of variation of shear strengthparameters (μc = 10 kPa and μϕ = 30°).

5.1. Reliability analysis results

Themultiple quadratic RSFs between the factors of safety associatedwith 2992 potential slip surfaces and 2420 original random variables(i.e., 1210 c and 1210 ϕ) are first constructed, which requires a total of4821 runs of deterministic slope stability analyses. The probability ofslope failure is then calculated using theMCSbased on themultiple qua-dratic RSFs shown in Eq. (2). The probabilities of failure, pf, and the sta-tistics of factor of safety (i.e., mean value μFS, standard deviation σFS,coefficient of variation COVFS) are listed in Table 4. For the consideredc–ϕ slope, Cho (2010) performed a probabilistic analysis using a MCSbased K–L expansion with 150 terms of eigenmode and 50,000 slopestability analyses. The results obtained from Cho (2010) are taken asthe “exact” solutions for this example. For three different sets of SOFsin Table 4, the probabilities of failure obtained from this study(e.g., 1.87 × 10−2, 3.97 × 10−2 and 2.06 × 10−2) are well consistentwith those (e.g., 1.71 × 10−2, 3.70 × 10−2 and 1.91 × 10−2) obtainedfrom Cho (2010), respectively. The resulting means and standard devi-ations of factor of safety appear to be similar. These results indicate thatthe proposed method can produce sufficiently accurate reliability re-sults with a consideration of spatially variable soil properties.

Fig. 8. Variation of the probability of failure with horizontal and vertical scales of fluctua-tion of shear strength parameters.

Fig. 9. Profile of a hypothetic slope with a weak layer and groundwater.

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Furthermore, Fig. 4 compares the probabilities of failure obtainedfrom this study and Cho (2010) for various cross-correlation coeffi-cients. As expected, the cross-correlation between c and ϕ has a signifi-cant influence on the probability of slope failure. The probability offailure obtained from this study increases from 4.94 × 10−3 to 0.12when ρc,ϕ varies from−0.7 to 0.5. It can be observed that the results ob-tained from this study also match well with those obtained from Cho(2010). These results indicate that the proposedmethod can accuratelyproduce the probability of slope failure at relatively low levels. Addi-tionally, it is emphasized that the required total number of deterministicslope stability analyses associated with the proposed method which is4841 keeps unchanged even though various cross-correlation coeffi-cients are used. The reason is that the obtained RSFs in the proposedmethod do not rely on the statistics (e.g., cross-correlation coefficientbetween c and ϕ) and do not need re-calibration when estimating theprobabilities of failure for different cross-correlation coefficients. Onthe contrary, themethod proposed by Cho (2010) needs to perform nu-merous similar slope stability analyses again for each different cross-correlation coefficient. Thus, the proposed method is more efficientthan that proposed by Cho (2010), and is adopted to explore the effectof ACFs on slope reliability in subsequent analyses.

5.2. Effect of ACFs on slope reliability

In this section, a parametric sensitivity study is carried out to explorethe influence of ACFs on slope reliability.With this purpose inmind, thecoefficients of variation, COVc and COVϕ, are set as [0.1, 0.7], and [0.05,0.20], respectively (Srivastava et al., 2010). Following Cho (2010) andTang et al. (2013), the range of ρc,ϕ is set as [−0.7, 0.5]. Based on the sta-tistical results of SOFs summarized in Section 2, the ranges of horizontaland vertical SOFs are selected as [10m, 60m] and [1.0m, 6.0m], respec-tively. These values are adopted to carry out the parametric sensitivitystudy. The case of ρc,ϕ = 0, COVc = 0.3, COVϕ = 0.2, δh = 40 m andδv = 4.0 m is taken as a nominal case in the parametric sensitivityanalysis.

Fig. 5 presents five typical realizations of random fields of cohesionand friction angle at ρc,ϕ = −0.5 corresponding to the SNX, SQX, SMK,

Table 5Statistical properties of soil parameters for Example II.

Soil type Parameters Mean COV

Sandy soil c1′ (kPa) 5 0.15ϕ1′ (°) 46 0.1γ1 (kN/m3) 20 –

Clay cu (kPa) 50 0.16γ2 (kN/m3) 20 –

Weak layer c2′ (kPa) 0 –

ϕ2′ (°) 10 0.15γ3 (kN/m3) 20 –

CSX and BIN ACFs. The corresponding slope stability results are alsoshown in Fig. 5. The darker and lighter shaded regions indicate areasof higher and smaller shear strength (c and ϕ), respectively. Note thatthe critical failure surfaces tend to pass near the slope toe. As expected,all simulated random fields of cohesion and friction angle exhibit a neg-ative correlation. The simulated random fields associated with the SQXand SMK ACFs show a smooth variation within the slope profile, whichmight characterize the spatially correlated soil properties more realisti-cally. In contrast, the simulated random fields associated with the SNX,CSX and BIN ACFs fluctuate roughly, especially in the vertical direction.

Fig. 6 shows the probabilities of slope failure associated with the fiveACFs for various cross-correlation coefficients. The probabilities offailure associated with the SNX ACF are smaller than those associatedwith the other four ACFs, which indicates that applying the commonly-used SNX ACF may produce unconservative probability of slope failure.The results underlying the SQX and CSX ACFs almost remain the same,which are slightly larger than those associated with the SMK and BINACFs. In addition, the difference in the probabilities of slope failure fordifferent ACFs increases as the negative cross-correlation between cand ϕ becomes stronger, but it still is less significant than one order ofmagnitude.

Figs. 7(a) and (b) compare the probabilities of slope failure associat-ed with five ACFs for various values of COVc and COVϕ, respectively.Similar to Fig. 6, the SNX ACF leads to the smallest probability of failureamong the five ACFs. Different ACFs lead to different probabilities ofslope failure. In general, such difference is not very significant eventhough at the small coefficients of variation. Note that the probabilityof slope failure is more sensitive to the change in COVϕ. For the SNXACF, the probability of slope failure increases from 1.0 × 10−3 atCOVϕ = 0.05 to 7.0 × 10−2 at COVϕ = 0.2. The latter is 70 times theformer.

To investigate the effect of ACFs on slope reliability for differentSOFs, Figs. 8(a) and (b) show the probabilities of slope failure associatedwith five ACFs for various horizontal and vertical SOFs, respectively.Similar to Figs. 6 and 7, the probability of failure associated with thecommonly-used single exponential ACF may be underestimated, whilethe probability of failure associated with the CSX ACF may beoverestimated. The probabilities of failure associated with differentACFs differ slightly. As expected, the effect of the vertical SOF on theslope reliability is much more significant than that of the horizontalSOF, which is consistent with that observed in the literature (e.g., Cho,2010; Ji et al., 2012).

6. Example II: Application to a heterogeneous slope with a weaklayer

Real soils hardly exhibit stationary behavior. However, few attemptshave been made to study the reliability of a heterogeneous slope withmulti-layered soil properties modeled by random fields. Thus, a hetero-geneous slope example (see Fig. 9) that consists of three soil layers (in-cluding a weak layer with a thickness of 0.5 m and a centroid Y-coordinate of 3.75 m) is investigated to further demonstrate the validity

Distribution Scale of fluctuation Correlation coefficient

Lognormal δh = 40 m, δv = 4.0 m ρc1 0;ϕ1 0 = −0.5Lognormal δh = 40 m, δv = 4.0 m– – –

Lognormal δh = 40 m, δv = 4.0 m –

– – –

– – –

Lognormal δh = 40 m, δv = 4.0 m –

– – –

Fig. 10. Random field mesh and slope stability results (FS= 1.317).

69D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

of the proposedmethod. The groundwater with hydraulic heads of 11mand 5m on both sides is also considered. The statistical properties of thesoil parameters for the heterogeneous slope are summarized in Table 5.A typical random field element model of the heterogeneous slope withthree soil layers is shown in Fig. 10. Cohesion c1′ and friction angle ϕ1′

of the sandy soil in the embankment, undrained shear strength cu ofthe clay in the foundation and friction angle ϕ2′ of the weak layer arecharacterized by four different lognormal random fields. The cross-correlation coefficient between c1′ and ϕ1′ is taken as ρc1 0;ϕ1 0 = −0.5.The single exponential ACF with horizontal and vertical SOFs beingtaken as δh = 40 m and δv = 4.0 m, respectively, is used. The minimumfactor of safety based on the mean values of the shear strength parame-ters (c1′, ϕ1′, cu, ϕ2′) is obtained as 1.317 using Bishop's simplifiedmeth-od. The corresponding critical failure surface is plotted in Fig. 10, whichpasses through the bottom of the weak layer.

As mentioned in Section 2, the method reported in Lu and Zhang(2007) and Cho (2012) is adopted here to simulate globally nonstation-ary random fields of three-layered soil shear strength parameters. Basedon these, themultiple quadratic RSFs between the factors of safety asso-ciated with 2992 potential slip surfaces and 1820 original random vari-ables (i.e., 610 c1′, 610 ϕ1′, 540 cu and 60 ϕ2′) are first constructed,which requires a total of 3641 deterministic slope stability analyses.The probability of slope failure, pf, estimated from the proposedmethodis 7.1 × 10−4 (see Table 6). Such a low probability of failure is commonin slope engineering practice. For this problem, the MCS requires morethan fifteen thousand runs of slope stability analyses to produce accu-rate reliability results for a target coefficient of variation of pf, COVpf,below 30% since the least number of samples required for direct MCS,

Np, to evaluate pf accurately is calculated by Np≥ 1−pfpf COVpfð Þ2 for a target

COVpf (e.g., Ang and Tang, 2007). To further validate the proposedmethod, the Latin Hypercube Sampling (LHS) with 20,000 samples isperformed to re-calculate the pf, in which Bishop's simplified methodis used to calculate the minimum factor of safety for each random

Table 6Comparison of the reliability results for Example II.

Case Method Ncall μFS σF

1 This study 3641 1.270 0.0LHS 20,000 1.271 0.0

2 This study 18,205 1.268 0.0LHS 20,000 1.269 0.0

Note: Ncall denotes the number of deterministic slope stability analyses. Case 1: The location ofmodeled using a discrete random variable, Y.

sample. The resulting pf is 9.5 × 10−4, as shown in Table 6, which com-pares favorably with that (e.g., 7.1 × 10−4) from the proposed method.

It is challenging to identify detailed underground stratigraphy infor-mation with limited site data in geotechnical practice (Cao and Wang,2014; Tang et al., 2015). In other words, the location of the weak layerin the heterogeneous slope can be uncertain. Such an uncertaintyshould be taken into account in slope reliability analysis. As an illustra-tion, the location of theweak layer is characterized by a discrete randomvariable, Y. The probability mass function of Y is given by

p Yð Þ ¼

0:061; Y ¼ 2:75 mð Þ0:245; Y ¼ 3:25 mð Þ0:388; Y ¼ 3:75 mð Þ0:245; Y ¼ 4:25 mð Þ0:061; Y ¼ 4:75 mð Þ

8>>>><>>>>:

ð8Þ

where Y denotes the centroid Y-coordinate of the weak layer as shownin Fig. 9. The corresponding reliability results are listed in Table 6. Theresults obtained from the proposed method agree well with those ob-tained from the LHS method. The proposed method also accurately es-timates the probability of slope failure at small probability levelsincorporating the spatial variability of multi-layered soil properties. Itshould be pointed out that the computational time increases signifi-cantly when the uncertainty in the location of the weak layer is consid-ered because the RSFs need to be re-calibrated for each differentlocation of the weak layer.

Again, the effect of ACFs on slope reliability for different SOFs isexplored using the proposed method. For illustrative purpose, only thereliability results associated with the fixed weak layer (Y = 3.75 m,see Fig. 9) are presented. Figs. 11(a) and (b) show the probabilities ofslope failure associatedwithfiveACFs for various horizontal and verticalSOFs, respectively. Similar to Fig. 8, the probability of failure is under-estimated when using the commonly-used single exponential ACF;the probabilities of failure obtained using SQX, SMK, CSX and BIN ACFs

S COVFS pf Relative error of pf

91 0.071 7.1 × 10−4 0.2591 0.072 9.5 × 10−4 –

95 0.075 1.39 × 10−3 0.1696 0.075 1.65 × 10−3 –

weak layer is fixed (Y= 3.75 m) as shown in Fig. 9. Case 2: The location of weak layer is

Fig. 11. Variation of the probability of failure with horizontal and vertical scales of fluctu-ation of shear strength parameters.

70 D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

differ slightly. Note that only a total of 3641 runs of slope stability anal-yses are performed in these parametric sensitivity analyses. Similarly,the RSFs obtained in this case do not rely on the ACFs and SOFs.Hence, they remain unchanged as the ACFs and SOFs change.

Fig. 12. A typical realization of random fields (c1′, cu a

Unlike Fig. 8(a), the horizontal SOF has a more important effect onthe probability of failure for the heterogeneous slopewhen the horizon-tal SOF is less significant than 40 m, albeit not as much as the effect ofthe vertical SOF. Such finding is consistent with that reported in Jiet al. (2012). This is mainly attributed to the fact that a deep-seated fail-uremechanism occurs in the heterogeneous slope and themajor part ofthe failure surface horizontally passes through the foundation (seeFig. 12, FS = 0.976). Consequently, the horizontal spatial variability ofthe undrained shear strength of clay in the foundation affects theslope reliability. In contrast, for the homogeneous c–ϕ slope, only a shal-low “toe” failure mechanism is observed as reported in Cho (2010) andSection 5.

7. Conclusions

This paper has proposed a multiple response-surface method forslope reliability analysis considering spatially varying soil properties.The scales of fluctuation of soil shear strength parameters are summa-rized. The differences in five 2-D theoretical ACFs are compared system-atically. A homogeneous c–ϕ slope and a heterogeneous slope, whichconsists of three soil layers (including a weak layer) and is modeledby globally nonstationary random fields, are studied to demonstratethe validity of the proposed method and explore the effect of ACFs onthe slope reliability, respectively. Several conclusions can be drawnfrom this study:

(1) The proposed multiple response-surface method can efficientlyevaluate slope reliability considering spatially varying soil prop-erties, which provides a practical tool for solving slope reliabilityproblems with low failure risk which is of great interest in geo-technical practice. The extended Cholesky decomposition tech-nique can effectively discretize the cross-correlated non-Gaussian random fields of spatially variable soil properties,which is conceptually simple and easily implementable.

(2) The multiple quadratic RSFs between the factors of slope safetyand the original random variables do not rely on the correlationstructures (i.e., ACFs) and the statistics (e.g., coefficient of varia-tion and cross-correlation coefficient) of the soil properties. Theproposed method does not need additional deterministic slopestability analyses to estimate the probabilities of slope failurefor different ACFs and statistics. It results in very high computa-tional efficiency in the parametric sensitivity analysis and in ex-ploring the influences of ACFs and statistics on slope reliability.

(3) The random fields produced by the SNX, CSX and BIN ACFs fluc-tuate roughly, especially in the vertical direction.On the contrary,the random fields produced by the SQX and SMK ACFs are very

nd ϕ2′) and slope stability results (FS= 0.976).

71D.-Q. Li et al. / Engineering Geology 187 (2015) 60–72

smooth. In summary, the SQX and SMK ACFs might characterizethe spatial correlation of soil propertiesmore realistically. Amongthe selected five ACFs, the probability of failure associated withthe commonly-used single exponential ACF is underestimated.In addition, the difference in the probabilities of slope failure as-sociated with the five ACFs depends on the cross-correlation be-tween shear strength parameters, the COVs and SOFs of shearstrength parameters. Generally, such difference is minimal.

(4) The proposed method is developed based on the Choleskydecomposition, so it may be prone to numerical roundoff errorsbecause the construction of the autocorrelation matrix mayover-simplify the correlation structure between two finiteareas, especially when the random field elements are closelyspaced and irregular. Its computational cost will increase signifi-cantly if the uncertainties in the underground stratigraphy areconsidered and the number of random field elements increases.In addition, the slope deformation in spatially variable soils isnot taken into account in this study because only the ultimatelimit state (i.e., stability) of the slope is considered and the RSFsare constructed using the factors of safety calculated from thelimit equilibrium method, which provides no information onthe slope deformation. These limitations should be further stud-ied and removed in the future. How to determine the meanvalues, COVs, and SOFs of random fields and characterize the un-known locations and properties of weak layers for a specific sitewith limited site data also needs to be investigated in the future.

Acknowledgments

This work was supported by the National Science Fund for Distin-guished Young Scholars (Project No. 51225903), the National Basic Re-search Program of China (973 Program) (Project No. 2011CB013506),the National Natural Science Foundation of China (Project No.51329901) and the Natural Science Foundation of Hubei Province ofChina (Project No. 2014CFA001).

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