Efficient System Reliability Analysis of Slope Stability inSpatially Variable Soils Using Monte Carlo Simulation

Shui-Hua Jiang1; Dian-Qing Li, M.ASCE2; Zi-Jun Cao3; Chuang-Bing Zhou4; andKok-Kwang Phoon, F.ASCE5

Abstract: Monte Carlo simulation (MCS) provides a conceptually simple and robust method to evaluate the system reliability of slope sta-bility, particularly in spatially variable soils. However, it suffers from a lack of efficiency at small probability levels, which are of great interest ingeotechnical design practice. To address this problem, this paper develops a MCS-based approach for efficient evaluation of the system failureprobabilityPf ,s of slope stability in spatially variable soils. The proposed approach allows explicit modeling of the inherent spatial variability ofsoil properties in a system reliability analysis of slope stability. It facilitates the slope system reliability analysis using representative slip surfaces(i.e., dominating slope failure modes) andmultiple stochastic response surfaces. Based on the stochastic response surfaces, the values ofPf ,s areefficiently calculated using MCS with negligible computational effort. For illustration, the proposed MCS-based system reliability analysis isapplied to two slope examples. Results show that the proposed approach estimates Pf ,s properly considering the spatial variability of soils andimproves the computational efficiency significantly at small probability levels.With the aid of the improved computational efficiency offered bythe approach, a series of sensitivity studies are carried out to explore the effects of spatial variability in both the horizontal and vertical directionsand the cross-correlation between uncertain soil parameters. It is found that both the spatial variability and cross-correlation affectPf ,s significantly.The proposed approach allows more insights into such effects from a system analysis point of view. DOI: 10.1061/(ASCE)GT.1943-5606.0001227. © 2014 American Society of Civil Engineers.

Author keywords: Slope stability; System reliability; Monte Carlo simulation (MCS); Representative slip surface; Stochastic responsesurface; Spatial variability.

Introduction

Uncertainties are unavoidable in slope engineering, such as the in-herent spatial variability of geotechnical properties (Vanmarcke1977a, b) and uncertainties associated with deterministic calculationmodels for estimating the safety margin of slope stability. Amongthese uncertainties, the inherent spatial variability of geotechnicalproperties has been considered to be one of the major sources of

uncertainties in geotechnical properties (Christian et al. 1994; Phoonand Kulhawy 1999; Ching and Phoon 2013; Lloret-Cabot et al.2014). It significantly affects the slope stability (Griffiths and Fenton2004; Cho 2007, 2010; Huang et al. 2010; Wang et al. 2011; Ji et al.2012; Jiang et al. 2014a; Tabarroki et al. 2013; Li et al. 2014). Oneprimary effect of the inherent spatial variability on slope stability isthat the critical slip surface with minimum factor of safety (FSmin)varies spatiallywhen the spatial variability is considered (Wang et al.2011). Identification of the critical slip surface among a large numberof potential slip surfaces is an elementary step in slope stabilityanalysis. From a system analysis point of view, the slope can beviewed as a series system (Chowdhury and Xu 1995; Li et al. 2009,2011b), in which each potential slip surface is a component and thecritical slip surface is the weakest one. System failure occurs as theslope slides along the critical slip surface. Under a probabilisticframework, slope reliability analysis is then defined as a systemreliability analysis problem, in which the overall failure probability(or system failure probability, Pf ,s) of a slope considering numerouspotential slip surfaces is of interest and it is greater than the failureprobability of any individual potential slip surface because of systemeffects (Cornell 1967;Ditlevsen 1979). System reliability analysis ofslope stability has been gaining increasing interest during the pastdecade (Oka and Wu 1990; Chowdhury and Xu 1995; Ching et al.2009; Low et al. 2011; Zhang et al. 2011, 2013; Ji and Low 2012).These previous studies have demonstrated that there might existmultiple dominating failuremodes (or slip surfaces) in slope stabilityanalysis problems, and these failure modes are considered rationallyduring the system reliability analysis of slope stability. However,most of the previous studies focused on various slope failure modes(i.e., slip surfaces) caused by stratification (i.e., layered soils), andthe inherent spatial variability of soil properties in each soil layer israrely considered.

1Graduate Student, State Key Laboratory of Water Resources andHydropower Engineering Science, Key Laboratory of Rock Mechanics inHydraulic Structural Engineering (Ministry of Education), Wuhan Univ.,Wuhan 430072, P.R. China.

2Professor, State Key Laboratory of Water Resources and HydropowerEngineering Science, Key Laboratory of Rock Mechanics in HydraulicStructural Engineering (Ministry of Education), Wuhan Univ., Wuhan430072, P.R. China (corresponding author). E-mail: dianqing@whu.edu.cn

3Associate Professor, State Key Laboratory of Water Resources andHydropower Engineering Science, Key Laboratory of Rock Mechanics inHydraulic Structural Engineering (Ministry of Education), Wuhan Univ.,Wuhan 430072, P.R. China.

4Professor, State Key Laboratory of Water Resources and HydropowerEngineering Science, Key Laboratory of Rock Mechanics in HydraulicStructural Engineering (Ministry of Education), Wuhan Univ., Wuhan430072, P.R. China.

5Professor, Dept. of Civil and Environmental Engineering, NationalUniv. of Singapore, Singapore 117576.

Note. This manuscript was submitted on January 14, 2014; approved onSeptember 24, 2014; published online on October 14, 2014. Discussionperiod open until March 14, 2015; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Geotechnical andGeoenvironmental Engineering, © ASCE, ISSN 1090-0241/04014096(13)/$25.00.

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To explicitly incorporate the inherent spatial variability into asystem reliability analysis of slope stability, Griffiths and Fenton(2004) and Huang et al. (2010) applied a random FEM and MonteCarlo simulation (MCS) to evaluate the system reliability of slopestability in spatially variable soils, and Cho (2010) combined ran-dom field theory and MCS to assess Pf ,s based on limit-equilibriummethods [e.g., simplified Bishop method (Duncan and Wright2005)]. Although MCS is a conceptually simple and robust methodfor probabilistic analysis (Wang et al. 2011; Wang 2013), it suffersfrom a lack of efficiency at small probability levels, which are ofparticular interest in slope engineering practice. This drawback ofMCS becomes even more profound for slope reliability analysisin spatially variable soils, because the critical slip surface variesspatially and needs to be located for each random sample generatedduring the MCS. To improve the efficiency of MCS-based slope re-liability analysis, Wang et al. (2011) used an advanced MCS methodcalled subset simulation to calculatePf ,s, andLi et al. (2013) developeda MCS-based risk deaggregation approach for the system reliabilityanalysis of slope stability. In their studies, the inherent spatial vari-ability along the depth is modeled explicitly using a one-dimensional(1D) randomfield. Despite these efforts, how to efficiently evaluate thesystem reliability of slope stability remains a difficult problem, par-ticularly when multidimensional [e.g., two-dimensional (2D) or three-dimensional] spatial variability is considered.

This paper develops an efficient system reliability analysis ap-proach based onMCS and limit-equilibriummethods to evaluate thePf ,s of slope stability in spatially variable soils. The proposed ap-proach allows explicit modeling of the inherent spatial variability inthe system reliability analysis of slope stability and improves thecomputational efficiency of slope system reliability analysis usingrepresentative slip surfaces (RSSs) and multiple stochastic responsesurfaces. The paper starts with probabilistic modeling of the inherentspatial variability of soil properties in both the horizontal and verticaldirections and cross-correlation between soil properties, followed bythe system reliability analysis of slope stability and construction ofstochastic response surfaces. Then, implementation procedures forthe proposed approach are described. Finally, the proposed approachis illustrated through two slope examples with one single soil layer.A series of sensitivity studies are also performed using the twoexamples to explore the effects of the inherent spatial variability andcross-correlation on the system reliability of slope stability.

Probabilistic Modeling of Spatially Variable Soils

Random field theory (Vanmarcke 1977a, b) is applied in this studyto explicitly model the inherent spatial variability of relevant soilproperties, such as the undrained shear strength cu for cohesive soilsand the cohesion c and friction anglef for c-f soils. LetH denote theuncertain soil property of interest, and its inherent spatial variabilityin the horizontal and vertical directions can be modeled by a 2Dlognormal stationary random field Hðx, yÞ with a mean mH and SDsH , where x is the horizontal coordinate on a bounded domain V,and it varies from 0 to Lx, i.e., 0# x# Lx; y is the vertical coordinateon a bounded domainV, and it ranges from 0 to Ly, i.e., 0# y#Ly;and Lx and Ly are the lengths of V in the horizontal and verticaldirections, respectively, and they are greater than zero by definition.Because the statistics of statistically homogeneous random fieldsonly depend on relative distance, rather than absolute locations, Vcan be defined in several alternative manners [e.g., V5 fðx, yÞ:2Lx=2# x# Lx=2;2Ly=2# y# Ly=2g] provided that the dimen-sions (i.e., Lx and Ly) in the horizontal and vertical directions remainthe same. In this study, V is defined as fðx, yÞ: 0# x# Lx; 0# y# Lyg only for mathematical convenience. In addition, it shall be

noted that a lognormal random field is applied in this study, becausethe lognormal random variable is a continuous variable and strictlynonnegative, which complies with the physical meaning of mostgeotechnical parameters (e.g., cu, c, and f). The lognormal randomfield is frequently used to model the inherent spatial variability ofgeotechnical parameters and has been shown to perform well in thegeotechnical literature (Griffiths et al. 2002; Griffiths and Fenton2004; Cho 2010; Tabarroki et al. 2013).

In the context of random field theory, the spatial correlation be-tween the variations of the considered soil property at differentlocations is described by a spatial autocorrelation structure. In thispaper, the autocorrelation structure is taken as a squared exponentialautocorrelation function, and the autocorrelation coefficient be-tween the logarithm [i.e., lnðHÞ] of H at different locations [e.g.,(x1, y1) and (x2, y2)] is calculated as (Fenton and Griffiths 2008)

rln½ðx1, y1Þ, ðx2, y2Þ� ¼ exp

(2

"�jx12 x2juln,h

�2

þ�jy12 y2j

uln,v

�2#)

(1)

where uln,h and uln,v 5 horizontal and vertical autocorrelation dis-tances, respectively. With Eq. (1), the random field Hðx, yÞ can beefficiently simulated using series expansion methods, such asKarhunen-Loève (K-L) expansion (Ghanem and Spanos 2003;Phoon et al. 2002; Cho 2010; Jiang et al. 2014a). For practicalimplementation and concise representation,Hðx, yÞ is truncated upto the order of M. Then, under K-L expansion, the lognormalrandom field Hðx, yÞ is expressed as

Hðx, yÞ ¼ Hðx, y; uÞ ¼ exp

�mN,H þ PM

j51sN,HxjðuÞ

�, ðx, yÞ 2V

(2)

where mN,H 5 lnðmHÞ2s2N,H=2 and sN,H 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnð11 s2

H=m2HÞ

p5 mean and SD of lnðHÞ, respectively; xjðuÞ 5

ffiffiffiffiffilj

pfjðx, yÞjjðuÞ,

j5 1, 2, . . . ,M 5 set of spatially correlated standard normal ran-domvariables (i.e., a standard normal randomfield);u5 coordinate inthe decomposed outcome space; jjðuÞ, j5 1, 2, . . . ,M 5 set of in-dependent standard normal random variables; and lj and fjðx, yÞ,j5 1, 2, . . . ,M 5 eigenvalues and eigenfunctions of the autocorre-lation function [Eq. (1)], respectively. Specifically, lj and fjðx, yÞ canbe obtained by solving the homogeneous Fredholm integral equationof the second kind (Phoon et al. 2002; Cho 2010). The determinationof M in Eq. (2) relies on the desired accuracy and autocorrelationfunction [Eq. (1)] adopted in random fields. In this study, M is de-termined to ensure that ɛ5

PMi51li=

P‘i51li is greater than 95%, in

which theli are sorted in a descending order, so that the total varianceof Hðx, yÞ within the considered domain V is basically preserved(Marzouk and Najm 2009; Laloy et al. 2013; Jiang et al. 2014a).Interested readers are referred to Marzouk and Najm (2009), Laloyet al. (2013), and Jiang et al. (2014a) formore details and explanationson choosing the value of M.

Note that only the autocorrelation of a single soil propertyH (e.g.,cu, c, or f) is explicitly modeled in Eq. (2). In practice, two or evenmore soil properties might be considered as uncertain parameters ineach soil layer. Consider, for example, c and f in a c-f soil layer.Their standard normal random fields xcðuÞ5fxc, jðuÞ, j51,2, . .. ,Mgand xfðuÞ5fxf, jðuÞ, j51,2, . . . ,Mg that account for the autocor-relation of each parameter can be generated first. Then, the correlationbetween c and f, namely, cross-correlation, is taken into account by(Fenton and Griffiths 2003; Jiang et al. 2014a)

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c ðx, yÞ ¼ c ðx, y; uÞ� exp

"mN,c þ

PMj51

sN,cxc, jðuÞ#, ðx, yÞ 2V

(3)

fðx, yÞ ¼ fðx, y; uÞ� exp

(mN,f þ PM

j51sN,f

hxc, jðuÞrc-f,N

þ xf, jðuÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 r2c-f,N

q i), ðx, yÞ 2V

(4)

wheremN,c 5 lnðmcÞ2s2N,c=2 andsN,c 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnð11s2

c=m2cÞ

p5mean

and SD of lnðcÞ, respectively; mN,f 5 lnðmfÞ2s2N,f=2 and sN,f

5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnð11s2

f=m2fÞ

q5mean and SD of lnðfÞ, respectively; mc and

sc 5mean and SD of c, respectively; mf and sf 5mean and SD off, respectively; and rc-f,N 5 cross-correlation coefficient betweenlnðcÞ and lnðfÞ, which can be calculated from the cross-correlationcoefficient (i.e., rc-f) between c and f (Fenton and Griffiths 2008).Note that Eqs. (3) and (4) account for both the autocorrelation of cand f and the cross-correlation between them. They allow for theefficient simulation of the cross-correlated lognormal random fieldsof c and f in the MCS-based system reliability analysis of slopestability, which is discussed in the “MCS-Based System ReliabilityAnalysis of Slope Stability” section.

MCS-Based System Reliability Analysis ofSlope Stability

Slope failure occurs when the slope slides along any slip surface. Intheory, there may exist an infinite number of potential slip surfaces.Evaluating the overall failure probability (i.e., system failure prob-ability, Pf ,s) along all the potential slip surfaces is a mathematicallyformidable task (El-Ramly et al. 2002). Therefore, Pf ,s is frequentlycalculated using a large, but finite, number of potential slip surfaces.Let S1, S2, . . . , SNs denote Ns potential slip surfaces that are con-sidered in the limit-equilibrium analysis of slope stability. Then, theslope can be considered as a series system that consists of Ns

components (i.e., S1, S2, . . . , SNs ). The system failure occurs whenany component (i.e., S1, S2, . . . , SNs ) fails. The system failure prob-ability Pf ,s can be calculated using MCS (Li et al. 2013)

Pf ,s ¼ 1Nt

PNt

k51IðFSmin , 1Þ (5)

where Nt 5 total number of samples generated during MCS; FSmin

5minimumFS value amongFS values of theNs potential slip surfacesfor a given set of random samples of uncertain parameters X (i.e., cu, c,and f) involved in slope stability analysis; and If × g 5 indicatorfunction. For a given random sample, IðFSmin , 1Þ is taken as thevalue of 1 when FSmin , 1 occurs. Otherwise, it is equal to zero.

Note that the slip surface with FSmin (i.e., critical slip surface)needs to be located amongNs potential slip surfaces for each randomsample generated during MCS, and its corresponding FSmin is cal-culated using a deterministic slope stability analysis method, such aslimit-equilibrium methods [e.g., simplified Bishop method (DuncanandWright 2005)]. To ensure that the critical slip surface is properlylocated, a large number of potential slip surfaces are usually con-sidered in deterministic slope stability analysis, i.e., Ns is large. Thevalue of Ns is frequently on the order of magnitude of 103e104

(Zhang et al. 2011; Li et al. 2013). Performing MCS for slope

reliability analysis based on such a large number of potential slipsurfaces requires extensive computational efforts, particularly inspatially variable soils, because the critical slip surface varies spatiallywhen spatial variability is considered (Wang et al. 2011).

Several previous studies (Zhang et al. 2011, 2013; Ji and Low2012; Li et al. 2013; Cho 2013) have suggested using some RSSs,which dominate the slope failure, as a surrogate for the large numberof potential slip surfaces to evaluate FSmin for each random sample.The number Nr of RSSs is generally much less than that (i.e., Ns) ofpotential slip surfaces (Zhang et al. 2011; Li et al. 2013). Therefore,the critical slip surface can be identified among RSSs with relativeease, and FSmin can be calculated more efficiently duringMCS. Thissubsequently leads to efficient evaluation of the system reliability ofslope stability. In this study, the RSSs are used in the MCS-basedsystem reliability analysis of slope stability, and a stochastic re-sponse surface is constructed for each RSS to calculate the FS of theRSS during MCS, by which the computational efficiency of theMCS-based system reliability analysis of slope stability is furtherimproved, as discussed in the “Multiple Stochastic Response Sur-faces for RSSs” section.

Multiple Stochastic Response Surfaces for RSSs

To facilitate the calculation of FSmin in the MCS-based system re-liability analysis of the slope, this study applies Hermite polynomialchaos expansion to construct a stochastic response surface for eachRSS.Under theHermite polynomial chaos expansion,FS for a givenRSS is calculated as (Huang et al. 2009; Li et al. 2011a; Al-Bittar andSoubra 2013; Jiang et al. 2014b)

FSjr�j�¼ a0G0 þ PN

i151ai1G1

�ji1

�þ PNi151

Pi1i251

ai1,i2G2�ji1 , ji2

�

þ PNi151

Pi1i251

Pi2i351

ai1,i2,i3G3�ji1 , ji2 , ji3

�þ � � � (6)

where jr 5 1, 2, . . . ,Nr; N5M3NF 5 number of random varia-bles in standard normal space; NF 5 number of random fields in-volved in slope reliability analysis; M 5 order of the series used inthe K-L expansion of random fields [Eqs. (2)–(4)]; a0, ai1 , ai1,i2 ,ai1,i2,i3 , . . .5 unknown coefficients;Gjpð × Þ, jp 5 1, 2, 3, . . .5Hermitepolynomials with jp degrees of freedom (Ghanem and Spanos 2003;Huang et al. 2009; Li et al. 2011a); and j 5 ðj1, j2, . . . , jNÞ 5 setof independent standard normal random variables correspondingto those used to discretize the random fields by K-L expansion[Eq. (2)]. From a physical point of view, the stochastic responsesurface given by Eq. (6) is a surrogate for the slope stabilityanalysis model that involves uncertain geotechnical parameters,and provides an explicit and approximate functional relationship topredict theFS of slope stability considering the uncertainties (Huanget al. 2009; Li et al. 2011a).

For the nHPCEth-order Hermite polynomial chaos expansion,there are a total of ðN1 nHPCEÞ!=ðN!3 nHPCE!Þ unknown coef-ficients (i.e., a0, ai1 , ai1,i2 , ai1,i2,i3 , . . .) in Eq. (6), which need to bedetermined for the construction of the stochastic response surfaces.Determination of these unknown coefficients requires Np realiza-tions of the random variables (or random fields) involved in slopereliability analysis and their corresponding Np FS values for eachRSS estimated from the conventional deterministic slope stabilityanalysis. The Np realizations of random fields can be generated byLatin hypercube sampling (LHS) using Eq. (2) or Eqs. (3) and (4)(Choi et al. 2006). Based on the Np random samples and theircorresponding FS values for a given RSS, Np linear equations are

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obtained using Eq. (6). Then, the unknown coefficients are determinedby solving theseNp linear equations. After that, the stochastic responsesurface for the RSS concerned is obtained. The accuracy of the sto-chastic response surface relies on the order (i.e., nHPCE) of the Hermitepolynomial chaos expansion, the number (i.e., N) of random variablesinvolved in the reliability analysis, and the accuracy of the estimatedcoefficients (i.e., a0, ai1 , ai1,i2 , ai1,i2,i3 , . . .). The accuracy of the esti-mated coefficients in this study also depends on the number (i.e.,Np) ofrealizations of random fields used to determine the unknown coef-ficients, random field element mesh, and deterministic slope stabilityanalysis method used to estimate FS of slope stability. The effects ofthese three factors on the results obtained from this study are furtherdiscussed in the “Discussion” section.

The procedure described previously is repeated for the Nr RSSsto obtain their respective stochastic responses surfaces. DuringMCS,the Nr stochastic responses surfaces are, collectively, used as a sur-rogate for the conventional deterministic slope analysis to explicitlyand efficiently evaluate FSmin for each random sample. In this way,the computational cost used for each random sample is substantiallyreduced and so are the total computational costs of MCS.

Note that, for the construction of the Nr stochastic responsesurfaces, there needs to be a selection of Nr RSSs among all thepotential slip surfaces and a determination of the Np FS values foreach RSS. These are achieved simultaneously in this study througha simple and practical procedure. The procedure starts by generatingNp realizations of the random fields involved in the slope reliabilityanalysis. For each of the Np realizations of random fields, a con-ventional deterministic analysis (e.g., simplified Bishop method) isperformed to calculate theFS values for all the potential slip surfacesand to locate the critical slip surface with FSmin. This is repeated Np

times, leading to Nr critical slip surfaces. These Nr critical slipsurfaces are used as RSSs in this study, whereNr is less than or equalto Np, because different realizations of random fields might result inthe same critical slip surface. In addition, the repeated calculationsfor theNp realizations of random fields simultaneously lead toNp FSvalues for each RSS. These Np FS values for the RSS are sub-sequently used to construct its corresponding stochastic responsesurface using Eq. (6). By these means, the selection of RSSs and thedetermination of the stochastic response surface for each RSS areachieved simultaneously, and no additional computational effortsare required to identify the RSSs for the system reliability analysis ofslope stability.

Implementation Procedure

Fig. 1 shows a flowchart for the implementation of the proposedMCS-based system reliability analysis approach schematically.In general, the implementation procedure involves five steps. Thedetails of each step and its associated equations are summarizedas follows.1. Determine input information for the system reliability anal-

ysis of slope stability, including but not limited to the slopegeometry and statistics (e.g., mean, SD, marginal distribution,cross-correlation coefficient, autocorrelation function, and au-tocorrelation distance) of the soil properties.

2. Generate Np realizations of random fields by LHS according toprescribed statistical information using Eq. (2) for independentrandom fields or Eqs. (3) and (4) for cross-correlated randomfields.

3. Perform a limit-equilibrium analysis (e.g., simplified Bishopmethod) of slope stability with the Np realizations of randomfields generated in Step 2 to determine Nr RSSs and calculateNp FS values for each RSS.

4. Construct Nr stochastic response surfaces [Eq. (6)] for the Nr

RSSs based on theNp random samples and their correspondingNp FS values.

5. Perform the MCS to generate Nt random samples in theindependent standard normal space to estimate the systemfailure probability Pf ,s using Eq. (5), where FSmin for eachrandom sample is determined using theNr stochastic responsesurfaces obtained in Step 4.

The proposed MCS-based system reliability analysis approachfor slope stability and its previously described implementationprocedures can be programmed as a software package with the aidof commercial software packages for deterministic slope stabilityanalysis, such as SLOPE/W 2007. The software package can bedivided into three modules: the deterministic slope stability analysismodule (e.g., SLOPE/W), the stochastic response surface module,and the MCS module. By these means, the deterministic slopestability analysis is deliberately decoupled from the reliabilityanalysis (including the construction of stochastic response surfacesandMCS for uncertainty propagation), so that the reliability analysisof slope stability can proceed as an extension of the deterministicslope stability analysis in a nonintrusive manner. This effectivelyremoves the hurdle of a reliability computational algorithm andallows geotechnical practitioners to focus on the deterministic slopestability analysis that they are familiar with, which will facilitatewider implementation of the proposed approach.

Fig. 1. Flowchart for implementation

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In addition, it is worthwhile to emphasize that the Nr stochasticresponse surfaces used in the MCS are explicit functions betweenuncertain input parameters and FS, and hence, the computationaleffort required for calculating FSmin during MCS is minimal. Afterthe Nr stochastic response surfaces are obtained, Pf ,s is calculatedefficiently using MCS. The main computational effort used for theproposed approach is that spent on the construction of Nr stochasticresponse surfaces, which is generally much less than the compu-tational effort needed for direct implementation of a MCS withconventional deterministic slope stability analysis in spatially var-iable soils, particularly at small probability levels. This is furtherillustrated through a cohesive slope example and a c-f slope ex-ample in the “Illustrative Example I: Cohesive Slope” and “Illus-trative Example II: c-f Slope” sections, respectively.

Illustrative Example I: Cohesive Slope

This section applies the proposed MCS-based system reliabilityanalysis approach to analyze a saturated cohesive slope exampleunder an undrained condition, which has been analyzed by Cho(2010) to demonstrate the probabilistic analysis of slope stability inspatially variable soils. As shown in Fig. 2, the cohesive slope hasa height of 5 m and a slope angle of 26.6�. The cohesive soil layerextends to 10m below the top of the slope and has a total unit weightof 20 kN=m3. In the cohesive soil layer, the short-term strength ischaracterized using the undrained shear strength cu, and the inherentspatial variability of cu is explicitly modeled using a 2D lognormalstationary random field cu with ameanmcu of 23 kPa, a coefficient ofvariation (COV) COVcu of 0.3, a horizontal autocorrelation distanceuln,h of 20 m, and a vertical autocorrelation distance uln,v of 2.0 m.These statistics are the same as those adopted by Cho (2010). Asshown in Fig. 2, the random field cu is discretized into 910 elements

with a side length of 0.5 m for its realization, and its random samplesare then generated using Eq. (2) at the centroid of each element, inwhichM is taken as 15 to ensure that the ɛ-value is larger than 95%in this example. As a reference, the nominal value of FSmin thatcorresponds to the case of all the cu being equal to its mean value(i.e., 23 kPa) is calculated using the simplified Bishop method(Duncan and Wright 2005), and it is equal to 1.356, which is identicalto the value reported by Cho (2010). In addition, the critical slip surfacein the nominal case is also located and is shown in Fig. 2. It is referredto as the critical deterministic slip surface (CDSS) in this example.

For the identification of RSSs and the construction of theirrespective stochastic response surfaces, 1,000 (i.e., Np 5 1,000)realizations of cu are simulated by LHS using Eq. (2), and theircorresponding critical slip surfaces are determined using the sim-plified Bishop method and are taken as RSSs in this example,resulting in a total of 71RSSs, as shown in Fig. 3.Note that theCDSSidentified in the nominal case (see the dashed line in Fig. 3) is alsoincluded in the 71 RSSs. After the 71 RSSs are obtained, onestochastic response surface is constructed for each RSS using thesecond-order Hermite polynomial chaos expansion according to theprocedure described in the “Multiple Stochastic Response Surfacesfor RSSs” section. This results in 71 stochastic response surfaces.For example, the stochastic response surface for the CDSS in thisexample is given by

FSCDSS�j�¼ a0G0 þ P15

i151ai1G1

�ji1

�þ P15i151

Pi1i251

ai1,i2G2�ji1 , ji2

�(7)

The values of coefficients a5 fa0, ai1 , ai1,i2g in Eq. (7) are given inTable 1. The 71 stochastic response surfaces can be verified bycomparing the FSmin of slope stability obtained from the stochastic

Fig. 2. Example of a cohesive slope

Fig. 3. The cohesive slope example’s 71 RSSs (COVcu 5 0:3)

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Table 1. Coefficients of the Stochastic Response Surface for the CriticalDeterministic Slip Surface in Example I

Number a Value

0 a0 1.37621 a1 20.18402 a2 20.12543 a3 0.08124 a4 20.02675 a5 0.02256 a6 20.03007 a7 0.01178 a8 20.00149 a9 0.000410 a10 0.001011 a11 20.000812 a12 20.001013 a13 0.006314 a14 0.022715 a15 20.002216 a1,1 0.013517 a2,1 0.017318 a2,2 0.013719 a3,1 20.006820 a3,2 20.013821 a3,3 0.010022 a4,1 0.001223 a4,2 0.004924 a4,3 20.008725 a4,4 0.005526 a5,1 20.003427 a5,2 0.002928 a5,3 20.000529 a5,4 20.000930 a5,5 0.004231 a6,1 0.003832 a6,2 20.002033 a6,3 20.001134 a6,4 20.000235 a6,5 0.001936 a6,6 0.002837 a7,1 038 a7,2 20.000339 a7,3 0.001540 a7,4 20.003541 a7,5 0.000142 a7,6 0.000243 a7,7 0.002744 a8,1 20.000545 a8,2 20.001746 a8,3 20.001147 a8,4 20.001948 a8,5 0.000749 a8,6 0.000350 a8,7 0.000351 a8,8 0.001952 a9,1 0.000153 a9,2 0.001054 a9,3 20.001855 a9,4 20.000956 a9,5 20.001157 a9,6 20.000658 a9,7 20.0004

Table 1. (Continued.)

Number a Value

59 a9,8 060 a9,9 0.001061 a10,1 20.000462 a10,2 20.001463 a10,3 0.001064 a10,4 20.000365 a10,5 0.000166 a10,6 20.000267 a10,7 20.001268 a10,8 0.000469 a10,9 070 a10,10 0.000971 a11,1 0.000372 a11,2 0.000173 a11,3 0.000374 a11,4 20.000675 a11,5 0.000276 a11,6 0.000577 a11,7 20.000178 a11,8 20.000579 a11,9 0.000580 a11,10 20.000381 a11,11 0.000482 a12,1 0.000983 a12,2 20.002284 a12,3 0.001885 a12,4 20.000886 a12,5 20.000487 a12,6 0.000988 a12,7 0.000289 a12,8 0.000590 a12,9 20.000191 a12,10 20.000192 a12,11 0.000193 a12,12 0.000294 a13,1 20.001095 a13,2 20.001496 a13,3 0.001597 a13,4 20.001198 a13,5 099 a13,6 20.0001100 a13,7 0.0002101 a13,8 20.0002102 a13,9 20.0004103 a13,10 0.0008104 a13,11 0105 a13,12 0.0002106 a13,13 0.0005107 a14,1 20.0023108 a14,2 20.0018109 a14,3 0.0033110 a14,4 20.0019111 a14,5 0.0008112 a14,6 20.0009113 a14,7 0.0006114 a14,8 0115 a14,9 20.0002116 a14,10 0.0004117 a14,11 0118 a14,12 0.0002

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response surfaces and the original deterministic analysis (e.g.,simplified Bishop method) of slope stability. Fig. 4 shows the FSmin

values obtained from the 71 stochastic response surfaces using 100sets of random samples versus those obtained from the simplifiedBishop method. The FSmin values obtained from the two approachesagreewellwith each other. This indicates that the stochastic responsesurfaces are good enough to obtain FSmin in this example.

For the 71 stochastic response surfaces, a MCS run withNt 5 500,000 random samples is performed to calculate Pf ,s, inwhich only the number (i.e., 15) of standard normal random vari-ables contained in each stochastic response surface and the coef-ficients of the 71 stochastic response surfaces are used as inputs.Although a relatively large number (i.e., 500,000) of randomsamples are generated during theMCS, the computational costs usedfor the MCS are minimal and negligible, because it is performedusing explicit functions (i.e., 71 stochastic response surfaces), bywhich FSmin can be solved instantaneously for each random sample.

System Reliability Analysis Results

Table 2 summarizes the reliability analysis results of this slopeexample obtained from this study and reported by Cho (2010). Thesystem failure probabilityPf ,s estimated from the proposed approachis 7:93 1022, which is in good agreement with that (i.e., 7:63 1022)obtained from the MCS using 100,000 random samples and thesimplified Bishop method by Cho (2010). Such good agreementvalidates the proposed approach and indicates that the reduced seriessystem comprised of 71 RSSs represents the slope reasonably well.Note that thePf ,s in this example is on the order ofmagnitude of 1022,for which the 1,000 LHS samples used to construct the stochasticresponse surfaces are sufficient to calculate Pf ,s. The value of Pf ,s

estimated from the 1,000 LHS samples is 8:33 1022 (Table 2) andagrees well with that obtained from the proposed approach. It does notseem necessary to construct the stochastic response surfaces andperform theMCSwith stochastic response surfaces to calculate Pf ,s inthis example. However, this is not true for the cases with relativelysmall values of Pf ,s, e.g., Pf ,s , 0:001. This is further discussed in the“Computational Efficiency at Small Probability Levels” section.

Computational Efficiency at Small Probability Levels

For example, as the COVcu decreases to 0.15 in this example, Pf ,s

decreases significantly and is calculated as 2:83 1024 using theproposed approach (Table 3). During the calculation, 1,000 randomsamples are again generated using LHS for the construction ofstochastic response surfaces, and a MCS run with 500,000 randomsamples is performed to obtain Pf ,s based on the stochastic responsesurfaces. Note that there is no failure sample among the 1,000random samples generated by LHS. In other words, 1,000 randomsamples are not sufficient to calculate Pf ,s when COVcu 5 0:15 inthis example, because Pf ,s (i.e., 2:83 1024) is relatively small. Tovalidate the value of Pf ,s obtained from the proposed approach, LHSis performed with 40,000 samples to recalculate Pf ,s, where thesimplified Bishopmethod is used to calculate FSmin for each randomsample. The resulting Pf ,s is 3:83 1024, as shown in Table 3. ThePf ,s value (3:83 1024) obtained from the LHS with 40,000 samplescompares favorably with that (e.g., 2:83 1024) estimated from theproposed approach.

Note that the computational efforts for the proposed approachcontain those used for both the construction of stochastic responsesurfaces and the subsequent MCS based on stochastic responsesurfaces. As mentioned previously, 1,000 realizations of randomfield are generated in this example. Subsequently, 1,000 de-terministic slope stability analyses are performed to estimate FSmin

using the simplified Bishop method and to determine the RSSs,which are used to construct stochastic response surfaces. In thisstudy, it takes approximately 38 s for each realization of the randomfield and the determination of its corresponding critical slip surfaceandFSmin on a desktop computer with 4-GB random-accessmemory(RAM) and one Intel Core i3 CPU clocked at 3.3 GHz. In addition,the total computational time for the MCS with 500,000 randomsamples using the stochastic response surfaces is approximately70 s, which is equivalent to that used for approximately 70=38� 2realizations of random field and the corresponding calculations ofFSmin using the simplified Bishop method. Therefore, the totalcomputational effort in the proposed approach is equivalent to thatused for 1,0001 25 1,002 realizations of random field and 1,002evaluations of FSmin using the simplified Bishopmethod. This effortismuch less than that used for the direct implementation of LHSwith40,000 realizations of cu and evaluations of FSmin using the sim-plifiedBishopmethod. The proposed approach calculates the systemfailure probability Pf ,s efficiently at small probability levels.

Table 1. (Continued.)

Number a Value

119 a14,13 0.0002120 a14,14 0.0004121 a15,1 20.0001122 a15,2 20.0003123 a15,3 0124 a15,4 0.0002125 a15,5 0126 a15,6 20.0004127 a15,7 20.0003128 a15,8 0.0009129 a15,9 20.0001130 a15,10 20.0001131 a15,11 20.0003132 a15,12 0.0001133 a15,13 0.0003134 a15,14 0.0001135 a15,15 0

Fig. 4.Validation of stochastic response surfaces in the cohesive slopeexample

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In addition, based on the stochastic response surface of the CDSSin the nominal case, the failure probability Pf ,CDSS of the CDSS isalso estimated using a MCS in this study, and it is equal to4:03 1022, which compares favorably with that (e.g., 3:163 1022)reported by Cho (2010). This further validates the proposed ap-proach. It is also worthwhile to note that Pf ,CDSS (i.e., 4:03 1022) isless than Pf ,s (i.e., 7:93 1022). The failure probability of slopestability is underestimated when only the CDSS is considered in theslope reliability analysis for the cohesive slope example.

Effects of Spatial Variability on System Reliabilityof Slope Stability

With the aid of the improved computational efficiency offered by theproposed approach, the effects of horizontal and vertical spatialvariability on the system reliability of slope stability are exploredthrough a series of sensitivity studies in this example. For the con-sideration of different spatial correlations, the sensitivity studies areperformed with uln,h varying from 10 to 40 m and uln,v varying from0.5 to 3.0 m. For each combination of uln,h and uln,v, the proposedapproach is applied to determine the RSSs and their correspondingstochastic response surfaces and then to estimate the system failureprobability Pf ,s and failure probability Pf ,CDSS of CDSS using theMCS and stochastic response surfaces.

Fig. 5(a) shows the variation of Pf ,s estimated from the proposedapproach as a function of uln,h by a line with squares. The results areobtained for uln,h varying from 10 to 40 m and uln,v 5 2:0 m. Fora given uln,v 5 2:0 m, Pf ,s increases from 5.9 to 8.9% as uln,hincreases from 10 to 40 m. Overestimation of the horizontal spatialcorrelation (i.e., underestimation of the horizontal spatial variability)leads to a slight overestimation of Pf ,s at small probability levels. Inaddition, Fig. 5(b) shows the variation of Pf ,s estimated from theproposed approach as a function of uln,v by a line with squares. Theresults are obtained for uln,h 5 20 m and uln,v varying from 0.5 to3.0 m. For a given uln,h 5 20 m, Pf ,s increases from 0.5 to 10.5% asuln,v increases from 0.5 to 3.0 m. Overestimation of the verticalspatial correlation (i.e., underestimation of the vertical spatial var-iability) leads to a significant overestimation of Pf ,s at smallprobability levels. From a physical point of view, overestimation ofthe vertical spatial correlation results in relatively slow variation inthe simulated values of soil strength parameters along the depth. Ifa weak point is generated during the random field simulation, thepoints above or below it have a greater chance to be weak. In otherwords, the probability of forming a weak zone during random fieldsimulation increases at small probability levels as the vertical spatial

correlation is overestimated. This subsequently leads to an over-estimation of Pf ,s. In addition, it is noted that the vertical spatialvariability has a much more significant effect on Pf ,s than thehorizontal spatial variability. Such observations on the effects ofspatial variability are consistent with those reported in the literature(Cho 2007; Ji et al. 2012).

From a system analysis point of view, the estimated Pf ,s relies ontwo factors: the numberNr of components (i.e., RSSs or dominatingfailure modes) in the reduced series system and the failure proba-bility of each component (i.e., each RSS). Generally speaking, thePf ,s increases as the number of RSSs and the failure probability of

Table 2. Reliability Analysis Results in Example I (COVcu 5 0:3)

Method Probability of failure PðFÞ Source

RSSs1 stochastic response surfaces1MCS 7:93 1022 This studyLimit-equilibrium method1LHS (1,000) 8:33 1022 This studyLimit-equilibrium method1MCS (100,000) 7:63 1022 Cho (2010)CDSS1 stochastic response surface1MCS 4:03 1022 This studyCDSS1 limit-equilibrium method1MCS (100,000) 3:163 1022 Cho (2010)

Table 3. Reliability Analysis Results in Example I (COVcu 5 0:15)

MethodProbability offailure PðFÞ Source

RSSs1 stochastic response surfaces1MCS 2:83 1024 This studyLimit-equilibrium method1LHS (1,000) 0 This studyLimit-equilibrium method1LHS (40,000) 3:83 1024 This study

Fig. 5. Effects of spatial variability on the system failure probability ofslope stability

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each RSS increase. The failure probability of a given slip surfaceusually increases with the increase of uln,v at small probability levels,because the variance of sliding resistance along the slip surfaceincreases as uln,v increases (Li and Lumb 1987; Cho 2007; Wanget al. 2011). For example, Figs. 5(a and b) show the respectivevariations of the failure probability Pf ,CDSS of CDSS in the normalcase as a function of horizontal and vertical autocorrelation dis-tances by lines with triangles. As both horizontal and verticalautocorrelation distances increase, Pf ,CDSS increases. For moredetails and theoretical explanations on the effects of vertical spatialvariability on failure probability for a given slip surface, the reader isreferred to Wang et al. (2011). On the other hand, Figs. 6(a and b)show the variations of Nr as a function of horizontal and verticalautocorrelation distances, respectively. As uln,h increases from 10 to40 m, Nr decreases from 115 to 40 [Fig. 6(a)], which tends to makePf ,s decrease. However, as shown in Fig. 5(a), Pf ,s increases as uln,hincreases. This indicates that the effects of the decrease inNr (i.e., thedecrease in the number of components of the series system) on Pf ,s

are dominated by those of the increase in the failure probability ofeach RSS on Pf ,s as uln,h increases. In addition, as uln,v increasesfrom 0.5 to 3.0 m, Nr is more or less approximately 65, as shown in

Fig. 6(b). Compared with the variation of Nr with the change inhorizontal spatial correlation [Fig. 6(a)], the variation of Nr with thechange in vertical spatial correlation is relatively minor [Fig. 6(b)].However, as shown in Fig. 5(b), Pf ,s increases significantly as uln,vincreases from 0.5 to 3.0 m. Such a significant increase in Pf ,s ismainly attributed to the increase of failure probability of each RSS atsmall probability levels as uln,v increases. Therefore, the spatialvariability in both horizontal and vertical directions affects thesystem failure probability and the failure probability of a given slipsurface in a consistent manner. The effects of spatial variability onthe failure probability of each individual slip surface dominate thoseon the number of RSSs and are subsequently reflected in the slopesystem failure probability.

Recommendations for Determining the Values ofAutocorrelation Distances in Slope Design

To incorporate the inherent spatial variability into slope design inpractice, there exists a need to determine uln,h and uln,v for a specificsite.When the values of uln,h and uln,v are available for a specific site,using their site-specific values in slope design is recommended.However, this is rarely the case, because site observation dataobtained from geotechnical site characterization are generally toosparse to generate statistically meaningful estimates of uln,h and uln,vfor a specific site. Although site-specific values of these parametersare rarely available, some possible values of them were reported inthe geotechnical literature according to global data. Based on a lit-erature review, the recommended ranges of uln,h and uln,v areobtained, and they are [10 m, 40 m] for uln,h and [0.5 m, 3.0 m] foruln,v (Phoon and Kulhawy 1999; El-Ramly et al. 2003; Ji et al. 2012;Salgado and Kim 2014). In case of no available information on uln,hand uln,v obtained from a specific site in practice, the recommen-dation is to use the values within their respective possible ranges(e.g., [10 m, 40m] and [0.5 m, 3.0 m] for uln,h and uln,v, respectively)reported in the literature to perform systematic sensitivity studies toexplore their effects on slope system reliability. For example, asensitivity study is performed in the “Effects of SpatialVariability onSystem Reliability of Slope Stability” section to explore the effectsof uln,h and uln,v on slope system reliability in this example. Theresults of the sensitivity study may further assist in choosing thevalues of uln,h and uln,v in slope design practice. As shown in Fig. 5,Pf ,s increases as uln,h and uln,v increase. In this case, the upper bounds(e.g., 40 and 3.0 m, respectively) of their possible ranges reported inthe literature are recommended to be used in slope design practice ifno site-specific information on uln,h and uln,v is available. This, ofcourse, subsequently leads to relatively conservative designs in thisexample.

Illustrative Example II: c-f Slope

For further illustration, this section applies the proposed approachto analyze a c-f slope example and to explore effects of cross-correlation between c and f on the system reliability of slope sta-bility. Fig. 7 shows a c-f slope with a height of 10 m and a slopeangle of 45�, which was analyzed by Cho (2010) usingMCS and thesimplified Bishop method. The c-f soil layer extends to 15 m belowthe top of the slope and has a total unit weight of 20 kN=m3. In thesoil layer, the inherent spatial variability of the cohesion and frictionangle is characterized by two cross-correlated 2D lognormal sta-tionary randomfields c andf. The randomfield c has amean value of10 kPa (i.e., mc 5 10 kPa) and COV of 0.3 (i.e., COVc 5 0:3), andthe random fieldf has amean value of 30� (i.e.,mf 5 30�) andCOVof 0.2 (i.e., COVf 5 0:2). Then, uln,h and uln,v in both c and f are

Fig. 6. Effects of spatial variability on the number of RSSs

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taken as 20 and 2 m, respectively. In addition, the cross-correlationcoefficient between c andf is taken as20:7. The statistics of c andfused in this study are consistent with those adopted by Cho (2010).For the realization of random fields, both c andf are discretized into1,210 elements with a side length of 0.5 m (Fig. 7), and their randomsamples are then generated using Eqs. (3) and (4) at the centroid ofeach element, whereM is taken to be 20 for each random field in thisexample. As a reference, the nominal value ofFSmin that correspondsto the case of all the c and f equal to their respective mean values(i.e., 10 kPa and 30�) is calculated using the simplified Bishopmethod, and it is equal to 1.206,which is almost identical to the value(i.e., 1.204) reported by Cho (2010). In addition, the critical slipsurface in the nominal case is also located and shown in Fig. 7. It isreferred to as the CDSS in this example.

For the identification of RSSs and the construction of their re-spective stochastic response surfaces, 1,000 realizations (i.e., Np

5 1,000) of c andf are simulated by LHS using Eqs. (3) and (4), andtheir corresponding critical slip surfaces are determined using thesimplified Bishop method and taken as RSSs in this example,resulting in a total of 79RSSs, as shown in Fig. 8. Note that theCDSSin the nominal case (see the dashed line in Fig. 8) is also included inthe 79 RSSs. After the 79 RSSs are obtained, one stochastic responsesurface is constructed accordingly for each RSS using the second-orderHermite polynomial chaos expansion, resulting in 79 stochasticresponse surfaces. Based on these stochastic response surfaces, aMCS run with 500,000 random samples is performed to calculatePf ,s. Similar to Example I, the computational costs used for theMCSareminimal and negligible, although a large number (i.e., 500,000) ofrandom samples are generated during the MCS.

System Reliability Analysis Results

Table 4 summarizes the reliability analysis results of this slopeexample obtained from this study and reported by Cho (2010). Thesystem failure probability Pf ,s estimated from the proposed ap-proach is 4:93 1023, which compares favorably with the result(i.e., 3:93 1023) obtained from the MCS using 50,000 randomsamples and the simplified Bishop method by Cho (2010). Theslight difference between the results might be attributed to the factthat different autocorrelation functions are used in this study andCho (2010). A squared exponential autocorrelation function [Eq. (1)]is used in this study. However, a single exponential autocorrelationfunction is used by Cho (2010). Although the single exponentialautocorrelation function is commonly used in the literature, it usuallynecessitatesmore terms (i.e., a larger value ofM) to ensure the desiredaccuracy of the truncated representation of the random field usinga K-L expansion compared with the squared exponential autocor-relation function. As M decreases, the computational efficiency of

random field generation using the K-L expansion improves. Hence,the squared exponential autocorrelation function is adopted in thisstudy, rather than the single exponential autocorrelation function. Inaddition, previous studies have demonstrated that the forms of au-tocorrelation functions have no significant effects on the reliability ofslope stability (Li and Lumb 1987), particularly when the scale offluctuation is relatively small comparedwith the dimension of the slipsurface. This is consistent with the observations obtained from thisstudy.

Note that Pf ,s in this example is on the order of magnitude of1023. It is also interesting to take a look at Pf ,s obtained from the1,000 LHS samples used to construct the stochastic response sur-faces at this order ofmagnitude. It is found thatPf ,s obtained from the1,000 LHS samples is 3:03 1023 (Table 4). This indicates that thereare only three failure samples among the 1,000 LHS samples, whichare too few to generate a meaningful estimate ofPf ,s. The 1,000 LHSsamples are not sufficient to calculatePf ,s at an order ofmagnitude of1023. To further validate the proposed approach, LHS with 10,000random samples is also performed to calculate the system failureprobability, where FSmin is directly evaluated using the simplifiedBishop method with a consideration of numerous potential slipsurfaces and a squared exponential autocorrelation function given byEq. (1) is used to simulate random fields. As shown in Table 4, thevalue of Pf ,s obtained from the LHS with 10,000 samples and thesimplified Bishop method is 4:43 1023, which agrees well withthe result (i.e., 4:93 1023) obtained from the proposed approach.The value of Pf ,s is efficiently calculated using the proposed ap-proach with reasonable accuracy.

Effects of Cross-Correlation on System Reliabilityof Slope Stability

To explore the effects of cross-correlation between c and f on thesystem reliability of slope stability, a sensitivity study is performedwith rc-f varying from 20:7 to 0. For each value of rc-f, theproposed approach is applied to determine the RSSs and theircorresponding stochastic response surfaces and to evaluate Pf ,s and

Fig. 7. Example of a c-f slopeFig. 8. The c-f slope example’s 79 RSSs

Table 4. Reliability Analysis Results in Example II

MethodProbability offailure PðFÞ Source

RSSs1 Stochastic response surfaces1MCS 4:93 1023 This studyLimit-equilibrium method1LHS (1,000) 3:03 1023 This studyLimit-equilibrium method1LHS (10,000) 4:43 1023 This studyLimit-equilibrium method1MCS (50,000) 3:93 1023 Cho (2010)

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Pf ,CDSS. Fig. 9(a) shows the variation of Pf ,s as a function rc-f bya line with squares. As rc-f varies from20:7 to 0, the estimated Pf ,s

increases from 0.49 to 8%. The cross-correlation between c and fsignificantly affects Pf ,s. Note that the cross-correlation between cand f is assumed to be negative (i.e., rc-f , 0) in this study. Then,the increase of c often corresponds to the decrease of f. As thenegative correlation between c and f becomes stronger (i.e., rc-fvaries from 0 to20:7), this trend becomes stronger, and the varianceof the soil total shear strength (including both c and f) decreases(Fenton and Griffiths 2003; Cho 2010). This may subsequently leadto a significant decrease in the failure probability of slope stability atsmall probability levels.Hence, ignoring thenegative cross-correlationbetween c and f (i.e., rc-f 5 0) leads to an overestimation of Pf ,s andconservative designs at small probability levels.

Similar to the effects of spatial correlation, the effects of cross-correlation onPf ,s can also be revisited from a system analysis pointof view and are divided into two aspects: the effects on the numberNr of RSSs and on the failure probability of each RSS. Based onprevious studies (Cho 2010;Wang et al. 2011; Li et al. 2011a, b), thefailure probability of a given slip surface generally increases at smallprobability levels as the negative cross-correlation becomes weaker(i.e., the negative value of rc-f increases). For example, Fig. 9also shows the variation of the failure probability Pf ,CDSS of theCDSS as a function of the cross-correlation coefficient by a line withtriangles. As the negative cross-correlation becomesweaker,Pf ,CDSS

increases. The effects of cross-correlation on the failure probabilityof a given slip surface are consistent with those on Pf ,s. In addition,Fig. 10 shows that, as rc-f varies from 20:7 to 0, Nr generallyincreases from approximately 80 to approximately 115, which alsotends to make Pf ,s increase. The effects of cross-correlation on thenumber Nr of RSSs and the failure probability of each RSS areconsistent for the c-f slope example, and they are combined togetherand reflected in the system reliability of slope stability as the cross-correlation varies.

Recommendations for Determining the Value of theCross-Correlation Coefficient in Slope Design

It is also worthwhile to point out that the determination of rc-f fora specific site is not a trivial task in practice, because geotechnicalsite characterization usually provides a limited amount of c and f

data for the specific site. In case of no prevailing knowledge aboutrc-f for the specific site, the recommended values of rc-f can beobtained from the literature and the results of a sensitivity study.Similar to uln,h and uln,v, the recommended range of rc-f is obtainedfrom the literature, e.g., [20:7, 0] is used in this study (Lumb 1970;Yucemen et al. 1973; Tang et al. 2012). Then, using the values ofrc-f within the recommended range, a sensitivity study can beperformed to explore the effects of rc-f on the slope system re-liability. Based on the results of the sensitivity study (Fig. 9), somerecommended values of rc-f are obtained. For example, Fig. 9shows that Pf ,s increases as rc-f increases. When no prevailingknowledge on rc-f is available from the specific site, the recom-mendation is to take rc-f to be zero, i.e., ignore the negative cross-correlation. This subsequently leads to an overestimation of Pf ,s andrelatively conservative designs.

Discussion

The proposed approach facilitates the slope system reliabilityanalysis using RSSs andmultiple stochastic response surfaces. Theyare affected by the number (i.e.,Np) of realizations of random fieldsused to determine the unknown coefficients in the stochastic re-sponse surfaces, random field element mesh, and deterministic slopestability analysis method. The effects of these three factors on RSSsand the construction of stochastic response surfaces are discussed inthis section.

In the proposed approach, RSSs are simultaneously identifiedfrom potential slip surfaces during the construction of stochasticresponse surfaces. The number (i.e.,Np) of random field realizationsused to construct stochastic response surfaces affects the identifi-cation of RSSs. The determination of a reasonableNp is pivotal to theidentification of RSSs. In general,Np shall be at least greater than thenumber [i.e., ðN1 nHPCEÞ!=ðN!3 nHPCE!Þ in this study] of unknowncoefficients in the stochastic response surfaces. For example,N5 15and nHPCE 5 2 are adopted for the case with COVcu 5 0:3 in Ex-ample I, which leads to ð151 2Þ!=ð15!3 2!Þ5 136 unknowncoefficients. In the example,Np shall be greater than 136, and hence,Np 5 1,000 is adopted. As Np increases, the number of RSSs tendsto increase, because it is possible for slope failure to occur alongmore potential slip surfaces. Because each RSS corresponds to onestochastic response surface in the proposed approach, the number of

Fig. 9. Effects of cross-correlation on the system failure probability ofslope stability

Fig. 10. Effects of cross-correlation on the number of RSSs

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stochastic response surfaces used to estimate Pf ,s increases. Thissubsequently leads to a more accurate estimate of Pf ,s, whereas thecomputational efficiency deceases as Np increases. Then, the de-termination of Np becomes a trade-off between the computationalaccuracy and efficiency.

Note that random fields are simulated at an element level in thisstudy, by which the values of geotechnical parameters are simulatedat the centroid of random field elements and are then assigned tothese elements for slope stability analysis. As the random field el-ement mesh becomes finer (e.g., the element size becomes smaller),the spatial variability of soil parameters is simulated in a more ac-curate and realistic way in the sense that the soil parameters varyfrom one point to another in reality. This might subsequently lead tomore accurate estimates of the location of the critical slip surface anditsFS value for each realization of random fields. Because RSSs area collection of critical slip surfaces obtained during the constructionof stochastic response surfaces, the RSSs are also affected by therandom field element mesh. Using a finer random field elementmesh may give more accurate predictions of locations of RSSs andtheir respective FS values. This subsequently affects the accuracy ofstochastic response surfaces and Pf ,s as well.

In addition, it is also worthwhile to point out that the selectedRSSs and their FS values also rely on the deterministic slopestability analysis method. In this study, the limit-equilibriummethod with circular failure mechanism (i.e., simplified Bishopmethod) is applied to identify the RSSs and construct stochasticresponse surfaces. Therefore, all the RSSs are circular slip sur-faces in this study. However, it should be noted that, whena deterministic slope stability analysis method that allowscompound slip surfaces is applied, the compound failuremechanismmight be included in the selected RSSs.Moreover, fora given RSS, the stochastic response surfaces calibrated fromdifferent deterministic slope stability analysis methods are dif-ferent to some degree and may indicate different functionalrelationships between the system response concerned and inputuncertain parameters.

Summary and Conclusions

This paper developed an efficient system reliability analysis ap-proach based on MCS and limit-equilibrium analysis of slope sta-bility to efficiently evaluate the system failure probability Pf ,s ofslope stability in spatially variable soils. The proposed approachfacilitates the slope system reliability analysis using RSSs andmultiple stochastic response surfaces. Based on the RSSs and theircorresponding stochastic response surfaces, Pf ,s is evaluated usingMCS with negligible computation costs. The proposed MCS-basedsystem reliability analysis approach was illustrated through a co-hesive slope example and a c-f slope example. The results showedthat the approach estimates Pf ,s properly considering the spatialvariability of the soil properties and improves the computationalefficiency significantly at small probability levels. With the aid ofthe improved computational efficiency offered by the proposedapproach, a series of sensitivity studies were carried out to explorethe effects of spatial variability in both the horizontal and verticaldirections and cross-correlation between uncertain soil parameters.The results showed that both the spatial variability and cross-correlation significantly affect the system reliability of slope sta-bility in spatially variable soils. The proposed approach allowsinsights into the effects of spatial variability and cross-correlationonPf ,s from a system analysis point of view. Finally, the factors thataffect the accuracy of Pf ,s estimated from the proposed approachwere also discussed.

Acknowledgments

This work was supported by the National Science Fund for Dis-tinguished Young Scholars (Project No. 51225903), the NationalBasic Research Program of China (973 Program) (Project No.2011CB013506), and the National Natural Science Foundation ofChina (Project No. 51329901).

Notation

The following symbols are used in this paper:COVH 5sH=mH 5 coefficient of variation of H;

FSmin 5 minimum factor of safety;H 5 uncertain soil property of interest

(i.e., cu, c, or f);Hðx, yÞ 5 2D lognormal stationary randomfield;

Np 5 number of random samples used toconstruct stochastic response surfaces;

Nr 5 number of RSSs;Ns 5 number of potential slip surfaces;Nt 5 total number of random samples

generated during MCS;uln,h 5 horizontal autocorrelation distance;uln,v 5 vertical autocorrelation distance;mH 5 mean of H;

mN,H 5 mean of lnðHÞ;jjðuÞ, j5 1, 2, . . . ,M 5 set of independent standard normal

random variables;rc-f 5 cross-correlation coefficient between

c and f at the same location;rc-f,N 5 cross-correlation coefficient between

lnðcÞ and lnðfÞ at the same location;sH 5 SD of H;

sN,H 5 SD of lnðHÞ; andxjðuÞ, j5 1, 2, . . . ,M 5 set of spatially correlated standard

normal random variables.

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