Full probabilistic design of slopes in spatially variable soils using simplifiedreliability analysis methodTe Xiao, Dian-Qing Li, Zi-Jun Cao and Xiao-Song Tang

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, People’s Republic of China

ABSTRACTA simplified reliability analysis method is proposed for efficient full probabilistic design of soil slopesin spatially variable soils. The soil slope is viewed as a series system comprised of numerouspotential slip surfaces and the spatial variability of soil properties is modelled by the spatialaveraging technique along potential slip surfaces. The proposed approach not only providessufficiently accurate reliability estimates of slope stability, but also significantly improves thecomputational efficiency of soil slope design in comparison with simulation-based fullprobabilistic design. It is found that the spatial variability has considerable effects on the optimalslope design.

ARTICLE HISTORYReceived 29 February 2016Accepted 16 October 2016

KEYWORDSSlope; reliability-baseddesign; system reliabilityanalysis; spatial variability;spatial averaging

Notation

FORM first-order reliability methodMCS Monte Carlo SimulationPNET probabilistic network evaluation techniqueQExp squared exponential (auto-correlation

function)RBD reliability-based designRSS representative slip surfaceSExp single exponential (auto-correlation

function)Pf,T target failure probabilityPf,sys system failure probabilitynSS number of potential slip surfacesnRSS number of RSSsRnSS × nSS correlation matrix corresponding to all

potential slip surfacesRnRSS × nRSS correlation matrix corresponding to RSSsSp p-th slip surfaceFSp factor of safety for SpGp performance function for Spβp reliability index for GpVpk segment of Sp in the k-th soil layerLpk length of Vpkc cohesion at the point levelf friction coefficient at the point levelCpk spatially averaged cohesion along VpkFpk spatially averaged friction coefficient

along Vpkγpk variance reduction factor of Cpk (or Fpk)

ρpq correlation coefficient between Gp and Gqρcf,k cross-correlation coefficient between c and

f at the same location of the k-th soil layerρpq,k auto-correlation coefficient between Cpk

and Cqk (or Fpk and Fqk)ρ0 correlation coefficient threshold in PNETX point uncertain soil property (e.g. c and f )XA spatially averaged uncertain soil property

(e.g. C and F)

1. Introduction

In the past few decades, reliability-based design (RBD)has been gaining increasing attentions in geotechnicalengineering. For example, the international standardISO2394:2015 (general principles on reliability of struc-tures) contains an Annex D (reliability of geotechnicalstructures) that highly advocates the need to developgeotechnical RBD methods (Phoon and Retief 2015).For practical purpose, simplified RBD methods (orsemi-probabilistic design methods) are usually adoptedin current geotechnical designs, such as load and resist-ance factor design (e.g. Paikowsky 2004) and multipleresistance factor design (e.g. Phoon, Kulhawy, and Gri-goriu 2003) in North America, and Eurocode 7 (e.g.Bond and Harris 2008; Orr 2012) in Europe. These sim-plified RBD methods have been successfully applied topiles and foundations, but their applications to soilslope designs are rather limited (e.g. Christian 2013; Pan-telidis and Griffiths 2014; Salgado and Kim 2014). This

© 2016 Informa UK Limited, trading as Taylor & Francis Group

CONTACT Dian-Qing Li dianqing@whu.edu.cn State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8Donghu South Road, Wuhan 430072, People’s Republic of China

GEORISK, 2017VOL. 11, NO. 1, 146–159http://dx.doi.org/10.1080/17499518.2016.1250279

mailto:dianqing@whu.edu.cnhttp://www.tandfonline.com

can be attributed to, at least partially, two reasons: (1)correlation between load and resistance in slope stabilityanalysis, and (2) existence of numerous correlated failuremodes due to the inherent spatial variability of soil prop-erties, which constitute two distinctive elements in RBDof soil slopes.

These two challenges can be easily addressed by fullprobabilistic design, in which the reliabilities of possibledesigns are evaluated using reliability analysis methods.Theoretically, all existing reliability analysis methodscan be directly used in full probabilistic design withtrial-and-error procedure (e.g. Tang, Yucemen, andAng 1976; Javankhoshdel and Bathurst 2014; Gonget al. 2015; Low and Phoon 2015). In addition to thedirect manner, simulation-based reliability analysismethods (e.g. Monte Carlo Simulation (MCS) and SubsetSimulation) have been applied to developing morespecialised full probabilistic design approaches, such asequivalent design approach (e.g. Ching 2009; Wu et al.2011) and expanded RBD (e.g. Wang, Au, and Kulhawy2011; Wang 2013; Wang and Cao 2013; Li et al. 2016a).

Although simulation-based reliability analysismethods are robust and feasible to slope reliability analy-sis, particularly when spatial variability of soil propertiesis considered (e.g. Griffiths and Fenton 2004; Wang, Cao,and Au 2011; Kasama and Whittle 2016; Li et al. 2016b,2016c; Xiao et al. 2016), their computational efforts aregenerally expensive when they are applied to RBD ofsoil slopes. Consider a series of possible slope designs.Among these designs, the feasible ones usually have rela-tively small failure probabilities (i.e. less than target fail-ure probability Pf,T). To ensure a desired accuracy ofreliability estimates, a considerable number of samplesare usually needed in simulation-based methods. More-over, the required number of samples increases withthe number of possible designs, leading to a significantincrease in computational efforts.

In contrast, simplified reliability analysis methods(e.g. first-order second-moment method and first-orderreliability method (FORM)) can evaluate slope reliabilityefficiently. It is, however, not a trivial task to incorporatespatial variability into slope reliability analysis using sim-plified reliability analysis methods (e.g. Vanmarcke 1977;Li and Lumb 1987; Christian, Ladd, and Baecher 1994;El-Ramly, Morgenstern, and Cruden 2002; Cho 2007;Ji, Liao, and Low 2012). The complexity in direct spatialvariability modelling, such as high-dimensional problem,defeats many existing simplified reliability analysismethods (Schuëller, Pradlwarter, and Koutsourelakis2004). This issue may be addressed by the spatial aver-aging technique (e.g. Vanmarcke 2010). Since slope fail-ure is more likely to occur when the average shear

strength along a slip surface is insufficient to resist thedriving force along the slip surface (El-Ramly, Morgen-stern, and Cruden 2002), it is more reasonable to per-form spatial averaging along slip surfaces rather thanover the whole slope domain (e.g. Suchomel and Mašín2010). Nevertheless, spatial variability can only becharacterised in vertical and horizontal directions dueto the testing procedures and limited data in practice(e.g. Fenton 1999; Uzielli, Vannucchi, and Phoon 2005;Cao and Wang 2013, 2014; Lloret-Cabot, Fenton, andHicks 2014; Wang, Cao, and Li 2016). It is impossibleto directly characterise the spatial variability along slipsurfaces through site investigation because the actuallocation of failure slip surface is unknown before slopefails. Furthermore, soil slope may have many potentialslip surfaces and shall be viewed as a series system com-prised of a number of failure modes (e.g. Oka and Wu1990; Low, Zhang, and Tang 2011; Zhang, Zhang, andTang 2011; Cho 2013; Li, Wang, and Cao 2014; Jianget al. 2015). The spatial variability of soil propertieswould lessen the correlation among failure modes andstrengthen the system effect of slope. How to properlyevaluate the correlation among different failure modeswith consideration of spatial variability and to incorpor-ate it into slope system reliability analysis using simpli-fied reliability analysis methods remain open questions.

This paper proposes a simplified reliability analysismethod for full probabilistic design of slopes in spatiallyvariable soils. The spatial variability of soil properties isexplicitly modelled using random field theory, in whichspatial averaging is performed along slip surfaces. Corre-lation among different failure modes is analyticallyderived accounting for spatial variability and is properlyincorporated into slope system reliability analysis. Thepaper starts with the full probabilistic slope design frame-work, followed by the development of the simplifiedreliability analysis method, including system reliabilityanalysis for a given slope design and spatial averaging ofsoil properties along slip surfaces. Finally, the proposedmethod is illustrated using two soil slope design examples.

2. Full probabilistic design framework of soilslopes using simplified reliability analysismethod

Figure 1 schematically illustrates the full probabilistic soilslope design framework using simplified reliability analy-sis method. The space of slope design parameters (e.g.slope height and angle) is deliberately discretized into aseries of possible designs. For a given design, the simpli-fied reliability analysis method is performed to evaluatethe system failure probability Pf,sys of slope stability,

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which is founded onmultivariate Gaussian approximation(Hohenbichler and Rackwitz 1982) and the probabilisticnetwork evaluation technique (PNET) (Ang and Tang1984). There are two major elements in the systemreliability analysis, namely, evaluating reliability for eachslip surface and estimating correlation among differentslip surfaces. The former can be computed using theFORM (Low, Zhang, and Tang 2011; Breitung 2015),and the latter is calculated according to correlations ofsoil properties. Note that both of them are related to thespatial variability of soil properties. To address the high-dimensional problem in spatial variability modelling, thespatial averaging technique along slip surfaces is adoptedbased on random field theory (Vanmarcke 2010). After allpossible slope designs are assessed, the optimal design isselected from the feasible designs satisfying the reliabilityrequirement (i.e. Pf,sys≤ Pf,T) and with a minimum con-struction cost (e.g. excavated volume).

Before moving to the development of the proposedapproach, two major assumptions on soil propertiesare given below:

(1) Only spatial variability of shear strength (i.e. cohe-sion (c) and friction coefficient ( f )) and variabilityof external load are considered. The variability ofunit weight is ignored due to its naturally smallvariability and relatively large spatial averaging area.

(2) Soil properties in different soil layers are assumedindependent, while c and f within the same soillayer are assumed to possess the same auto-corre-lation structure and scale of fluctuation (i.e. consist-ent spatial variability).

All the discussions in the remaining part of this paperare based on the above two assumptions.

3. System reliability analysis for a given slopedesign

3.1. System reliability approximation based onrepresentative slip surfaces (RSSs)

Consider, for example, a soil slope with nSS potential slipsurfaces, that is, S1, S2,… , SnSS, as shown in Figure 2. Fora series system, the system failure occurs if any com-ponent fails. Hence the system failure probability Pf,sysof slope stability can be written as

Pf ,sys = P[(G1 , 0)< (G2 , 0)< . . .< (GnSS , 0)]= 1− P[(G1 ≥ 0)> (G2 ≥ 0)> . . .> (GnSS ≥ 0)],

(1)

whereGp, p = 1, 2,… , nSS, is the performance function ofSp. The joint probability in second equality can be eval-uated using multivariate Gaussian approximation (orGaussian Copula) (Hohenbichler and Rackwitz 1982;Tang et al. 2013; Zeng and Jimenez 2014):

Pf ,sys = 1−F(b1, b2, . . . , bnSS ; RnSS×nSS ), (2)where βp = ϕ

−1[P(Gp≥ 0)], p = 1, 2,… , nSS, is thereliability index of Gp; RnSS × nSS = [ρpq] is the correlationmatrix corresponding to all potential slip surfaces; ρpq= correlation coefficient between Gp and Gq; ϕ(·) andΦ(·) are univariate and multivariate standard Gaussiancumulative distribution functions, respectively. How-ever, it is not easy to evaluate high-dimensional multi-variate cumulative distribution function, particularlyfor numerous potential slip surfaces (e.g. nSS > 5000).Moreover, performance functions of some slip surfacesmay be highly correlated (i.e. ρpq is close to unity), result-ing in the singularity of RnSS × nSS. An example can befound in Low, Zhang, and Tang (2011), as shown inTable 1. As a result, it is impractical to apply Equation(2) to slope system reliability analysis directly.

To address this issue, PNET (Ang and Tang 1984) isadopted to identify a few (e.g. nRSS) relatively indepen-dent RSSs of the slope system. In the context of PNET,the slip surface with minimum reliability index isselected as a RSS, and those slip surfaces highly corre-lated with the RSS (i.e. ρpq≥ prescribed correlation coef-ficient threshold ρ0) are represented by the RSS and

Figure 1. Schematic slope design framework using simplifiedreliability analysis method.

Figure 2. Illustration of soil slope system.

148 T. XIAO ET AL.

excluded from further consideration. This procedure isrepeatedly performed for the remaining slip surfacesuntil all slip surfaces are represented. By this means,the original slope system with nSS potential slip surfacescan be represented by the simplified slope system withnRSS RSSs, and Equation (2) can be approximately esti-mated as

Pf ,sys ≈ 1−F(b1, b2, . . . , bnRSS ; RnRSS×nRSS ), (3)where RnRSS × nRSS = [ρpq] is the correlation matrix corre-sponding to RSSs. Note that all ρpq values inRnRSS × nRSS for p ≠ q are smaller than ρ0, which makesRnRSS × nRSS more likely nonsingular. Applying PNETwith ρ0 = 0.8, Table 2 provides a simplified slope systemexample for that shown in Table 1. Using Equation (3) toevaluate slope system failure probability needs values ofβp and ρpq (p = 1, 2,… , nSS; q = 1, 2,… , nRSS), which arecalculated in Sections 3.3 and 3.4, respectively. To facili-tate understanding, the performance function for a singleslip surface is provided in the next subsection prior tocalculating βp and ρpq.

3.2. Performance function for a single slip surface

In this study, the performance function of slope stabilityis determined using Ordinary Method of Slices (Duncanand Wright 2014). Based on the total stress analysismethod, the factor of safety FSp of a circular slip surfaceSp is calculated as

FSp =∑nsp

i=1 [c pil pi + f pi(Wpi + Qpi) cosa pi]∑nspi=1 (Wpi + Qpi) sina pi

, (4)

where nsp = number of slices shown in Figure 3(a); cpiand fpi = tanϕpi are cohesion and friction coefficientalong the base of the i-th slice, respectively;Wpi = weight

of the i-th slice;Qpi = qbpi is external load acting on top ofthe i-th slice; q = unit load; bpi = width of the i-th slice; lpi= length of the i-th slice along slip surface; αpi = anglebetween the base of the i-th slice and the horizontalplane. By setting FSp = 1 as the limit state, the perform-ance function Gp of Sp can be written as

Gp =∑nspi=1

[c pil pi + f piWpi cosa pi + f piQpi cosa pi

−Wpi sina pi − Qpi sina pi].(5)

Note that cpi and fpi vary along the slip surface whenspatial variability is considered in slope stability analysis(Cho 2007; Ji, Liao, and Low 2012). Hence direct spatial

Table 1. Illustrative example for singular slope system (after Low, Zhang, and Tang (2011)).Slip surfaces S1 S2 S3 S4 S5 S6 S7 S8

RnSS × nSS S1 1 0.454 1 0.456 0.454 1 1 0.531S2 1 0.454 1 1 0.454 0.454 0.996S3 1 0.456 0.454 1 1 0.531S4 1 1 0.456 0.456 0.996S5 1 0.454 0.454 0.996S6 1 1 0.531S7 sys 1 0.531S8 1

β 2.795 2.893 2.837 2.943 2.902 3.047 3.112 3.539

Note: Pf,sys cannot be obtained using Equation (2).

Table 2. Simplified slope system for illustrative example.Slip surfaces S1 S2

RnRSS × nRSS S1 1 0.454S2 0.454 1

β 2.795 2.893

Note: Pf,sys = 0.44% using Equation (3). Figure 3. Slope stability analysis using Ordinary Method of Slices.

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variability modelling using random field theory willresult in high-dimensional difficulty in reliability analy-sis. To avoid this problem, the spatial averaging tech-nique is adopted in this study. Herein, it is worthwhileto note that the spatial variability of soil propertiesmay lead to non-circular slip surfaces, which can beincorporated in slope reliability analysis using more rig-orous probabilistic analysis with direct modelling ofspatial variability (Li and Lumb 1987; Griffiths and Fen-ton 2004; Cho 2007; Ji, Liao, and Low 2012; Tabarroki etal. 2013; Li et al. 2016c). Spatial averaging along non-circular slip surface is more complicated than thatalong circular slip surface. For simplicity, only circularslip surface is considered in this study. Such a simplifica-tion may lead to some errors in estimated slope failureprobability due to ignoring non-circular slip surfaces,but this is a good starting point to develop the efficientslope RBD method based on simplified reliability analy-sis with consideration of spatial variability. Effects ofnon-circular slip surface on slope failure probabilityhave been explored in literature. Interested readers arereferred to Ching, Phoon, and Hu (2010), Tabarrokiet al. (2013), and Bahsan et al. (2014).

As shown in Figure 4, each slip surface is partitionedinto several segments by soil stratification, for example,segment Vpk is a part of Sp located in the k-th soil layer.Likewise, the performance function for a slip surfacecan be treated as the summation of the performance func-tion for each segment, for example, Gpk for Vpk. Considerthat, for example, resistance from all cpi values along Vpkcan be written as

∑(c pil pi) = Cpk

∑l pi = CpkLpk, in

which Cpk = spatially averaged cohesion along Vpk andLpk =

∑l pi is length of Vpk. Analogous spatial averaging

is also performed over all fpi values, which leads to Fpk, thespatially averaged friction coefficient along Vpk. By thismeans, the originalnsp slices aremerged intonlp slices cor-responding to the soil layer as shown in Figure 3(b), andEquation (5) can be rewritten as

Gp =∑nlpk=1

Gpk =∑nlpk=1

[CpkLpkRp + FpkWpkyWpk

+ FpkQpkyQpk −WpkxWpk − QpkxQpk],(6)

where nlp = number of soil layers; Wpk =∑Wpi and Qpk=∑Qpi = qBpk are summations of soil weight and externalload acting on Vpk, respectively; Rp = radius of Sp; Bpk=∑bpi is width of Vpk; xWpk =

∑WpiRp sina pi/Wpk,

xQpk =∑

QpiRp sinapi/Qpk, xQpk =∑

QpiRp sinapi/Qpkand yQpk =

∑QpiRp cosapi/Qpk are horizontal and verti-

cal forcemoment arms ofWpk andQpk to the centre of slipsurface, as shown in Figure 3(b). Since all Qpk valuesdepend on a uniform q, random variables in Equation(6) only contain Cpk, Fpk (k = 1, 2,… , nlp) and q. Thespatial averaging technique significantly reduces thenumber of random variables for Gp from 2nsp + 1 to2nlp + 1. This lessens the computational difficulty in eval-uating βp and ρpq.

3.3. Reliability analysis for a single slip surface

Because the performance function defined in Equation(6) is relatively linear, the FORM can be used to evaluatethe component reliability index with high accuracyand efficiency, which is written as (Low, Zhang, andTang 2011; Cho 2013; Breitung 2015; Low and Phoon2015)

bp = min {‖U‖|Gp(U) = 0}, (7)where U = standard normal vector with independentcomponents and it is then converted to original randomvector XA (i.e. the group of Cpk, Fpk (k = 1, 2,… , nlp)and q) by inverse Nataf transformation during the cal-culation. In the case where point uncertain soil propertyX (i.e. c or f ) is non-normally distributed, the distri-bution of spatially averaged variable XA (i.e. Cpk orFpk) is technically unknown. Herein, it is assumedthat XA has the same distribution as X. Under thisassumption, the whole transformation procedure con-tains spatial averaging from X to XA, discussed in Sec-tion 4 later, and distribution transformation from XA toU.

To efficiently evaluate βp, gradient-based optimisationmethod (e.g. sequential quadratic programming method)is commonly used in the FORM. For performance func-tion defined in Equation (6), gradients in XA-space can

Figure 4. Spatial averaging along slip surfaces.

150 T. XIAO ET AL.

be analytically calculated as

∂Gp∂Cpk

= LpkRp,∂Gp∂Fpk

= WpkyWpk + qBpkyQpk,

∂Gp∂q

=∑nlpk=1

Bpk(FpkyQpk − xQpk).

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

They are converted to gradients in U-space according tochain rule of differentiation. Based on Equations (7) and(8), reliability index for each potential slip surface can beobtained efficiently.

3.4. Correlation estimation among different slipsurfaces

Consider, for example, performance functionsGp =

∑nlpk=1 Gpk and Gq =

∑nlqm=1 Gqm of slip surfaces

Sp and Sq, in which Gpk and Gqm are the performancefunction of segments Vpk and Vqm, respectively, andnlp and nlq are the respective number of soil layersthat Sp and Sq pass through. Their covariance Cov(Gp,Gq) can be expressed as

Cov(Gp, Gq) = Cov∑nlpk=1

Gpk,∑nlqm=1

Gqm

( )

=∑nlpk=1

∑nlqm=1

Cov(Gpk, Gqm). (9)

The value of Cov(Gpk, Gqm) is related to the statistics(e.g. standard deviation) of random variables (i.e. Cpkand Fpk along Vpk, Cqm and Fqm along Vqm, and q),auto-correlation coefficient ρpq,k for Cpk and Cqk (orFpk and Fqk), and cross-correlation coefficient ρcf,k forc and f in the k-th soil layer. Details of calculatingCov(Gpk, Gqm) can be referred to Appendix A. The var-iance of Gp can also be calculated through Equation (9)by setting Gq = Gp, that is, Var(Gp) = Cov(Gp, Gp).Then, the correlation coefficient ρpq between Gp andGq is given by definition,

r pq =Cov(Gp, Gq)

Var(Gp)Var(Gq)

√ . (10)Equation (10) gives the element of RnSS × nSS andRnRSS × nRSS. Based on Equations (9) and (10), the corre-lation coefficient between two performance functionsis analytically calculated from statistics (e.g. standarddeviation and correlation coefficient) of spatially aver-aged variables, which are estimated through the spatialaveraging technique in the next section.

4. Spatial averaging of soil properties alongslip surfaces

Recall that the slip surface is deliberately partitioned intoseveral segments in different soil layers as shown inFigure 4. Along one of these segments, Vpk of Sp forexample, the point uncertain soil property X (i.e. c andf ) is arithmetically averaged into the spatially averagedvariable XA (i.e. Cpk and Fpk). According to the randomfield theory (Vanmarcke 2010), XA owns a same meanand a reduced variance in comparison with X, that is,

mXA = mX , (11)

sXA = sX

gpk

√, (12)

where μX and σX are mean and standard deviation of X,respectively; μXA and σXA are mean and standard devi-ation of XA, respectively; γpk = variance reduction factorfor XA along Vpk and it is defined as (Vanmarcke 2010)

g pk =1L2pk

∫Lpk0

∫Lpk0

r1(Dl)dlidlj

= 1L2pk

∫Lpk0

∫Lpk0

r2(Dx, Dy)dlidlj,

(13)

where ρ1(·) = 1D auto-correlation function along the slipsurface with an unknown scale of fluctuation; ρ2(·) = 2Dauto-correlation function with measurable horizontal andvertical scales of fluctuation;Δl,Δx andΔy are, respectively,straight, horizontal and vertical distances between twolocations as shown in Figure 4. If a slip surface passesthrough a soil layer more than once, all parts in the layershall be integrated into one segment, for example, V (1)p,k−1and V (2)p,k−1 are assembled as Vp,k−1 in Figure 4. The corre-lation between XA values along V

(1)p,k−1 and V

(2)p,k−1 is also

incorporated into the calculation of variance reduction fac-tor. In this way,XA values of different segments on the sameslip surface, for example, Vp,k−1 and Vpk, are independent.

Nevertheless, there exists a correlation among XAvalues along segments of different slip surfaces but inthe same layer, for example, Vpk and Vqk. Note thatcovariance Covpq,k between Cpk and Cqk (or Fpk andFqk) is defined as (Vanmarcke 2010)

Covpq,k = 1LpkLqk

∫Lpk0

∫Lqk0

Cov2(Dx, Dy)dlidlj, (14)

where Cov2(·) = 2D covariance function between twolocations. By the definition of covariance, the left-handand right-hand sides of Equation (14) can be, respect-ively, written as

Covpq,k = s pksqkr pq,k =

g pkgqk

√s2Xr pq,k, (15)

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Cov2(Dx, Dy) = s2Xr2(Dx, Dy), (16)where σpk and σqk are standard deviations of XA alongVpk and Vqk, respectively; ρpq,k = auto-correlation coeffi-cient between Cpk and Cqk (or Fpk and Fqk). SubstitutingEquations (15) and (16) into Equation (14) gives

r pq,k =1

LpkLqk

g pkgqk

√∫Lpk0

∫Lqk0

r2(Dx, Dy)dlidlj. (17)

Equation (17) describes the auto-correlation between thevalues of a soil property spatially averaged along differentsegments in the same layer. It is consistent with Equation(13) when Vpk =Vqk, that is, ρpq,k = 1. In this study,Equations (13) and (17) are integrated numericallyaccording to the slices partitioned in deterministicslope stability analysis, and are rewritten as

g pk =1Lpk

∑ns,pki=1

∑ns,pkj=1

r2(Dx, Dy)lilj, (18)

rpq,k =1

LpkLqk

gpkgqk

√∑ns,pki=1

∑ns,qkj=1

r2(Dx, Dy)lilj, (19)

where ns,pk and ns,qk are number of slices contained inVpk and Vqk, respectively. Using Equations (18) and(19), variance reduction factors for XA and auto-corre-lation coefficients between different XA values are evalu-ated, which are utilised to calculate the statistics of XA inEquations (11) and (12), reliability index of each slip sur-face in Equation (7), and correlation coefficients betweenperformance functions of different slip surfaces inEquations (9) and (10).

5. Illustrative example I: a two-layered soilslope

For illustration, this section applies the proposed methodto RBD of a two-layered soil slope. As shown in Figure 5,the slope has a height H of 5 m and an initial slope angleθ of 26.6° with external load acting on the top of groundsurface, which is taken as the nominal design scenario.The statistics of uncertain parameters are summarised

in Table 3, all of which are lognormally distributed.Among them, the cohesion and friction coefficient ofupper layer soil are considered to be negatively corre-lated with ρcf,1 =−0.5. Furthermore, the spatial variabil-ity of soil properties is modelled by single exponential(SExp) auto-correlation function, that is,r2(Dx, Dy) = exp (− 2|Dx|/dh − 2|Dy|/dv), with hori-zontal and vertical scales of fluctuation, δh and δv, of20 m and 2 m, respectively.

5.1. System reliability analysis for nominal designscenario

In this subsection, the proposed approach is applied toevaluating the system failure probability Pf,sys of thenominal design scenario (i.e. H = 5 m, θ = 26.6°). Thecorresponding results are validated against thoseobtained from MCS with 100,000 random samples.

For convenience, only three slip surfaces are con-sidered first, as shown in Figure 5. Their correspondingFS values are estimated as 1.484, 1.387 and 2.139,respectively, using Equation (4) based on the meanvalue of uncertain parameters. The spatial variability ofsoil properties is modelled through the spatial averagingtechnique in the simplified reliability analysis method,while it is modelled by direct spatial variability modellingalong those slip surfaces in MCS (Cho 2007; Ji, Liao, andLow 2012). Both of them are based on the same discre-tized locations with horizontal distance smaller than0.5 m.

In the context of the simplified reliability analysismethod, performance functions for all slip surfaces (i.e.Equation (6)) are first determined during the determinis-tic slope stability analysis. Spatial averaging is then per-formed, in which variance reduction factors for all thesegments (see Table 4 row 2) are calculated usingEquation (18) and substituted into Equations (11) and(12) to evaluate the statistics of spatially averaged soilproperties. Taking these statistics into the distributiontransformation, reliability indices of the three slip sur-faces are estimated using Equation (7), and they are3.394, 2.478 and 11.221, respectively. To enable a fair

Figure 5. Two-layered soil slope example.

152 T. XIAO ET AL.

comparison with MCS, they are converted to failureprobabilities by Pf = ϕ(−β) (see Table 4 row 5). Inaddition, correlation coefficients among spatially aver-aged soil properties along different segments (see Table4 row 3) are calculated using Equation (19). They aresubstituted into Equations (9) and (10) to evaluate corre-lation coefficients among different performance func-tions for slip surfaces (see Table 4 row 4). Note thatcorrelation coefficients among different performancefunctions can also be estimated through the FORM byreformulating the performance functions in an enlargedrandom space. For example, the correlation coefficientbetween G1 and G2 estimated by the FORM in theenlarged variable space is 0.84, which is consistent withthat obtained from Equation (10). This verifies theanalytical solution derived in this study. Finally, PNETis adopted with ρ0 = 0.8 to identify the RSSs, amongwhich S2 is the first RSS and S3 is the second one. ThePf,sys based on S2 and S3 is evaluated as 6.6 × 10

−3

using Equation (3). In contrast, variance reduction fac-tors and correlation coefficients in MCS are directly esti-mated through statistical analysis based on the randomsamples. As shown in Table 4, except the failure prob-ability of S3, which is too small to be captured by MCSwith 100,000 random samples, all the results obtainedfrom the simplified reliability analysis method agreewell with those from MCS. This validates the accuracyof the proposed approach.

In order to obtain more accurate Pf,sys of the nominaldesign scenario, a large amount of potential slip surfaces

(about 6000) are considered subsequently, as shown inFigure 6 with grey lines. The minimum FS obtainedfrom deterministic slope stability analysis is 1.371.Since the numbers of slip surfaces and the correspondingdiscretised locations are large in this case, direct spatialvariability modelling for every location along every slipsurface (see previous case) is technically formidable forMCS. Instead, the spatial variability is modelled using arandom field mesh with an element size of 0.5 m ×0.5 m using the mid-point method (Li et al. 2016b,2016c). The soil properties within a random fieldelement are assumed to be homogenous, no matterwhich slip surface passes through the element.

Based on 100,000 random samples, Pf,sys consideringall potential slip surfaces is estimated as 3.1% usingMCS, which is taken as the “exact” solution as shownin Figure 7(b) with dashed line. In contrast, Pf,sysobtained from the simplified reliability analysis methodis evaluated based on RSSs identified by PNET, whichrelies on the selection of ρ0. Consider, for example, ρ0= 0.95. The failure probability of each identified RSSand system failure probability are demonstrated inFigure 7(a) and (b), respectively, by lines with circles.With the maximum RSS number nmax of 25 Pf,sys is esti-mated as 2.7% using the simplified reliability analysismethod, consistent with that (i.e. 3.1%) obtained fromMCS. The identified RSSs are plotted in Figure 6 withdark lines, among which the first RSS, also known asprobabilistic critical slip surface in literature, is colouredin red. Its failure probability is evaluated as 8.2 × 10−3,

Table 3. Statistics of uncertain parameters for two-layered soil slope example.Uncertain parameters Mean COV Distribution Spatial variability Cross-correlation

Soil layer 1 c1 (kPa) 5 0.3 Lognormal SExp [20, 2] −0.5f1 0.577 0.2 Lognormal SExp [20, 2]

Soil layer 2 c2 (kPa) 30 0.3 Lognormal SExp [20, 2] −q (kN/m) 20 0.1 Lognormal − −Note: SExp [20, 2] = SExp auto-correlation function with δh = 20 m and δv = 2 m.

Table 4. Results obtained from simplified reliability analysis method and MCS.Method Simplified reliability analysis method MCS

Variance reduction forsegments

Layer 1 Layer 2 Layer 1 Layer 2V1k 0.56 0.46 0.56 0.47V2k 0.59 0.50 0.59 0.51V3k 0.52 − 0.52 −

Correlation amongsegments

Layer 1 Layer 2 Layer 1 Layer 2V1k,V2k 0.71 0.84 0.74 0.84V1k,V3k 0.38 − 0.36 −V2k,V3k 0.50 − 0.48 −

Correlation among slipsurfaces

S1 S2 S3 S1 S2 S3S1 1 0.84 0.03 1 0.84 0.03S2 0.84 1 0.03 0.84 1 0.03S3 0.03 0.03 1 0.03 0.03 1

Component Pf S1 S2 S3 S1 S2 S33.4 × 10−4 6.6 × 10−3 1.6 × 10−29 3.2 × 10−4 6.2 × 10−3 0

System Pf,sys 6.6 × 10−3 6.2 × 10−3

GEORISK 153

much smaller than the estimated Pf,sys. Hence only con-sidering the probabilistic critical slip surface in RBDmaylead to unconservative slope design, especially for slopesin spatially variable soils. In addition to 0.95, other fourρ0 values, that is, 0.8, 0.85, 0.9 and 0.99, are also adoptedin PNET to determine a proper ρ0. As illustrated inFigure 7, both the number of RSS and system failureprobability increase with increasing ρ0, indicating thatboth the computational effort and accuracy for systemreliability analysis increase. For slope in spatially variablesoils, the spatial variability lessens the correlation amongdifferent slip surfaces and strengthens the system effect

of slope, thus requiring a relatively high ρ0 to producesatisfactory results. Trading off accuracy and efficiency,ρ0 = 0.95 with nmax = 25 appears a prude choice forsuch a case and it will be applied in the following studies.

With respect to the computational efficiency, the sim-plified reliability analysis method invokes performancefunction of each slip surface for nine times on averagein this example and takes 1.5 minutes to finish the sys-tem reliability analysis on a desktop computer with8GB RAM and one Intel Corei7 CPU clocked at3.4 GHz. The MCS, conversely, requires 100,000 calcu-lations for performance function of each slip surfaceand takes 1.5 hours on the same computer. The highcomputational efficiency of the proposed approach basi-cally satisfies engineering requirements in practice andcan significantly enhance the application of full probabil-istic design in soil slopes.

5.2. RBD for the slope

For RBD of the two-layered slope example, two target fail-ure probability levels, that is, Pf,T = 10

−3 and 10−4, are con-sidered in this study. For simplicity, only slope angle θ istaken as the design parameter and the excavated volumeof slope is taken as the quantity for measuring construc-tion cost. The design space of θ is discretised into 13 poss-ible designs ranging from 14° to 26° with an increment of1°. The number of potential slip surface varies fromaround 4000 to around 6000, due to the variation ofground length for different slope designs with differentslope angles. The uncertainties of soil properties are con-sistent with those in Table 3, and three different externalload cases, that is, μq = 10, 20 and 30 kN/m with anunchanged COVq = 0.1, are considered to explore theeffect of external load on RBD of the slope.

Figure 8 shows the slope failure probabilities and cor-responding construction costs for all possible designsusing the proposed approach. When Pf,T = 10

−3, the opti-mal designs for cases with μq = 10, 20 and 30 kN/m are22°, 18° and 15°, respectively. The optimal θ decreasesas external load increases. When Pf,T decreases to 10

−4,the optimal designs for cases with μq = 10 and 20 kN/m

Figure 6. Potential slip surfaces and RSSs identified by PNET (ρ0 = 0.95).

Figure 7. Effect of ρ0 in system reliability analysis using PNET.

154 T. XIAO ET AL.

reduce to 18° and 16°, respectively, and no feasible designamong the given range can be found for case with μq =30 kN/m. As the reliability requirement increases, thedesign alteration for the case with smaller external loadis more significant than that with larger external load.

5.3. Effect of spatial variability on optimal slopedesign

A sensitivity study is carried out to explore the effect ofspatial variability on RBD of the soil slope, where bothinfluences of auto-correlation function and scale of fluc-tuation are taken into consideration. Apart from theSExp auto-correlation function, the squared exponential(QExp) auto-correlation function, that is,r2(Dx, Dy) = exp [− p(Dx/dh)2 − p(Dy/dv)2], is alsoadopted in this subsection. Nine values of δh rangingfrom 10 to 50 m with an increment of 5 m and ninevalues of δv ranging from 2 to 10 m with an incrementof 1 m are considered, leading to a total of 81 cases forSExp and QExp auto-correlation functions, respectively.The possible θ values vary from 10° to 26° with an incre-ment of 1° for each case. In addition, the external load μq= 10 kN/m and the reliability requirement Pf,T = 10

−3.Figure 9(a) and (b) demonstrates the design charts for

SExp and QExp auto-correlation functions, respectively.Their contours are similar in shape, but quite differentin value. The optimal θ based on QExp auto-correlationfunction is 3°, on average, smaller than that based onSExp auto-correlation function. Using QExp auto-corre-lation function generates a more conservative design,which was also observed in previous studies (e.g. Li andLumb 1987; Li et al. 2015). In terms of scale of fluctuation,both horizontal and vertical scales of fluctuation affectoptimal design of slope, and the vertical one is more dom-inative. As scale of fluctuation increases (i.e. spatial corre-lation increases or spatial variability is less significant), theoptimal θ decreases. For the extreme case that soils are

perfectly correlated in space, the optimal θ drops to 10°.Therefore, the spatial variability, including both auto-cor-relation function and scale of fluctuation, has considerableeffects on RBD of soil slopes. Such effects can be properlyconsidered in RBD using the proposed approach.

6. Illustrative example II: a single-layered soilslope

The two-layered slope example shown in Figure 5 is tai-lored to illustrating the proposed approach in a relativelycomplicated design situation and has not been used byother investigators. For further validation, this sectionapplies the proposed approach to designing a relativelysimple slope shown in Figure 10, which was consideredby Wang, Cao, and Au (2011) to explore the effect ofspatial variability on system failure probability of slope.As shown in Figure 10, the simple slope example has aheight H of 10 m and a slope angle θ of 26.6°, and ithas only one single soil layer. The undrained shearstrength Su is modelled by 1D lognormal random fieldwith a mean of 40 kPa, COV of 0.25 and SExp auto-

Figure 8. RBD for two-layered soil slope example.

Figure 9. Effect of spatial variability on optimal slope design (Pf,T= 10−3).

GEORISK 155

correlation function with δv = 0.5 m (Wang, Cao, and Au2011).

Based on the mean value of Su, the deterministic criti-cal slip surface (see the solid line in Figure 10) with mini-mum FS is searched from about 12,000 potential slipsurfaces in this study, and the corresponding FS calcu-lated by Equation (4) is 1.180. Both the location andFS of the deterministic critical slip surface agree withthose (i.e. the dashed line in Figure 10 and 1.178)reported by Wang, Cao, and Au (2011). Using the pro-posed approach, the failure probability of deterministiccritical slip surface and system failure probability areestimated as 0.16% and 0.89%, respectively, also closeto those (i.e. 0.13% and 0.92%) reported by Wang,

Cao, and Au (2011) using Subset Simulation. Thisfurther validates the proposed approach.

For RBD of the simple slope example, both θ and Hare considered as the design parameters and the exca-vated volume of slope is used again to measure the con-struction cost. The possible θ values vary from 24.0° to26.5° with an increment of 0.5° and the possibleH valuesrange from 9 to 10 m with an increment of 0.2 m, leadingto a total of 36 designs. Figure 11(a) demonstrates thefailure probability of each design via a bubble chart, inwhich larger bubble size means higher failure prob-ability. As shown in Figure 11(a), the failure probabilityis more sensitive to H than θ in this example. Among the36 possible designs, 23 designs (i.e. 11 designs in greenand 12 designs in blue) are feasible for Pf,T = 10

−3, and12 designs (i.e. those in blue) are feasible for Pf,T =10−4. Further considering the construction cost demon-strated in Figure 11(b), the optimal design is determinedas H = 9.6 m and θ = 26° for Pf,T = 10

−3, and H = 9.2 mand θ = 26.5° for Pf,T = 10

−4. They can be obtainedusing the simplified reliability analysis method with ease.

7. Summary and conclusions

This paper proposed a simplified reliability analysismethod for full probabilistic design of slopes in spatiallyvariable soils. For a given design, the spatial variability ofsoil properties is incorporated into the proposed approachthrough the spatial averaging technique along potentialslip surfaces of the slope system. The statistics of spatiallyaveraged soil properties are then utilised in evaluating thereliability of each slip surface and the correlation amongdifferent slip surfaces, in which performance functionsfor slip surfaces are reformulated to facilitate the spatialaveraging. Subsequently, the RSSs are identified from allpotential slip surfaces using the probabilistic networkevaluation technique. The system failure probability ofslope design is finally estimated based on these RSSs bymultivariate Gaussian approximation.

Equations were derived for the simplified reliabilityanalysis method and were illustrated by two soil slope

Figure 10. Single-layered soil slope example.

Figure 11. RBD for single-layered soil slope example.

156 T. XIAO ET AL.

design examples in spatially variable soils. Results werevalidated against those obtained from MCS. The highcomputational efficiency provided by the simplifiedreliability analysis method satisfies engineering require-ments in practice and can significantly enhance theapplication of full probabilistic design in soil slopes.Finally, a sensitivity study was carried out to explorethe effects of spatial variability on RBD of soil slopes.It was found that the spatial variability, including bothauto-correlation function and scale of fluctuation, hasconsiderable effects on the optimal slope design. Sucheffects can be properly incorporated into full probabilis-tic design of soil slopes using the proposed approach.

In the end, it is worthwhile to point out that usingOrdinary Method of Slices generally results in conserva-tive designs for circular slip surfaces. The model uncer-tainty of Ordinary Method of Slices (e.g. Christian,Ladd, and Baecher 1994; Malkawi, Hassan, and Abdulla2000; Zhang, Zhang, and Tang 2009; Bahsan et al. 2014)may affect the optimal slope design more or less. How-ever, research is rare that calibrated the model uncertaintyof Ordinary Method of Slices in multi-layered slope stab-ility analysis considering spatial variability. More effortson calibrating the model uncertainty in such cases andincorporating it into RBD of soil slopes are still warranted.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by the National Science Fund for Dis-tinguished Young Scholars (Project No. 51225903), the NationalNatural Science Foundation of China (Project Nos. 51329901,51579190, 51679174), and the Natural Science Foundation ofHubei Province of China (Project No. 2014CFA001). The finan-cial support is gratefully acknowledged.

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Appendix A. Covariance between two performance functions

For simplicity, performance functions Gpk and Gqm (see Equation (6)) corresponding to segments Vpk and Vqm are rewritten as

Gpk = k0pk + k1pkX1pk + k2pkX2pk + k3pkX3pk + k4pkX2pkX3pkGqm = k0qm + k1qmX1qm + k2qmX2qm + k3qmX3qm + k4qmX2qmX3qm,

(A1)

where k0s = −WsxWs, k1s = LsRs, k2s =WsyWs, k3s = −xQs, k4s = yQs, X1s = Cs, X2s = Fs and X3s =Qs (s = pk, qm).The correlation between Gpk and Gqm comes from two parts, including auto-correlation for the same soil property and cross-

correlation for different soil properties. The auto-correlation (see Section 4) for X1pk and X1qm (C ), X2pk and X2qm (F) and X3pk andX3qm (Q) are ρpq,km, ρpq,km and 1, respectively, where ρpq,km = ρpq,k for k =m and ρpq,km = 0 for k ≠ m; ρpq,k = auto-correlation coef-ficient between Cpk and Cqk (or Fpk and Fqk) spatially averaged along Vpk and Vqk in the k-th soil layer, which can be calculated byEquation (19). The cross-correlation between c and f for different locations, denoted as a and b, within the k-th soil layer followsthe assumption that ρ(cak, fbk) = ρcf,k×ρab, where ρcf,k = cross-correlation coefficient between c and f at the same location of the k-thsoil layer; ρab = auto-correlation coefficient for same soil property between locations a and b. This assumption is consistent withprevious studies (Vořechovský 2008; Ji, Liao, and Low 2012) and can be extended to spatially averaged soil properties, that is, ρ(Cpk, Fqm) = ρcf,km×ρpq,km, where ρcf,km = ρcf,k for k =m and ρcf,km = 0 for k ≠ m.

Based on the abovementioned relationship, Cov(Gpk, Gqm) can be calculated as

Cov(Gpk, Gqm) = [Cov(Gpk, Gqm)]A + [Cov(Gpk, Gqm)]C

=

k1pkk1qmCov(X1pk, X1qm)+ k2pkk2qmCov(X2pk, X2qm)+k3pkk3qmCov(X3pk, X3qm)+ k4pkk4qmCov(X2pkX3pk, X2qmX3qm)+k2pkk4qmCov(X2pk, X2qmX3qm)+ k2qmk4pkCov(X2qm, X2pkX3pk)+k3pkk4qmCov(X3pk, X2qmX3qm)+ k3qmk4pkCov(X3qm, X2pkX3pk)

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

A

+ k1pkk2qmCov(X1pk, X2qm)+ k1qmk2pkCov(X1qm, X2pk)+k1pkk4qmCov(X1pk, X2qmX3qm)+ k1qmk4pkCov(X1qm, X2pkX3pk)

[ ]C

=

k1pkk1qmr pq,kmX̃1pkX̃1qm + k2pkk2qmr pq,kmX̃2pkX̃2qm + k3pkk3qmX̃3pkX̃3qm+k4pkk4qm(�X2pk�X2qmX̃3pkX̃3qm + r pq,kmX̃2pkX̃2qm �X3pk�X3qm + r pq,kmX̃2pkX̃2qmX̃3pkX̃3qm)+k2pkk4qmr pq,kmX̃2pkX̃2qm �X3qm + k2qmk4pkr pq,kmX̃2qmX̃2pk�X3pk+k3pkk4qmX̃3pkX̃3qm�X2qm + k3qmk4pkX̃3qmX̃3pk�X2pk

⎡⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

A

+ k1pkk2qmrcf ,kmr pq,kmX̃1pkX̃2qm + k1qmk2pkrcf ,kmr pq,kmX̃1qmX̃2pk+k1pkk4qmrcf ,kmr pq,kmX̃1pkX̃2qm�X3qm + k1qmk4pkrcf ,kmr pq,kmX̃1qmX̃2pk�X3pk

[ ]C

(A2)

where subscripts “A” and “C” denote auto-correlation and cross-correlation, respectively; hats “−” and “∼” denote mean and stan-dard deviation, respectively. In Equation (A2), the covariance items can be derived from the definition of covariance and the equal-ity that Cov(Y, Z ) = E(YZ)−E(Y )E(Z ), where E(·) = expectation operation. For instance,

Cov(X2pkX3pk, X2qmX3qm) = E(X2pkX3pkX2qmX3qm)− E(X2pkX3pk)E(X2qmX3qm)= E(X2pkX2qm)E(X3pkX3qm)− �X2pk�X3pk�X2qm�X3qm= [�X2pk�X2qm + Cov(X2pk, X2qm)][�X3pk�X3qm + Cov(X3pk, X3qm)]− �X2pk�X3pk�X2qm�X3qm= �X2pk�X2qmCov(X3pk, X3qm)+ �X3pk�X3qmCov(X2pk, X2qm)+ Cov(X2pk, X2qm)Cov(X3pk, X3qm)= �X2pk�X2qmX̃3pkX̃3qm + r pq,kmX̃2pkX̃2qm�X3pk�X3qm + r pq,kmX̃2pkX̃2qmX̃3pkX̃3qm

(A3)

Similarly,

Cov(X2pk, X2qmX3qm) = �X3qmCov(X2pk, X2qm) = r pq,kmX̃2pkX̃2qm�X3qm. (A4)

The other covariance items can be derived in a similar way. Although Equation (A2) appears complicated, the calculation canbe finished with relative ease since it is an explicit function of the statistics for uncertain parameters and those statistics have beendetermined before reliability analysis.

GEORISK 159

Abstract1. Introduction2. Full probabilistic design framework of soil slopes using simplified reliability analysis method3. System reliability analysis for a given slope design3.1. System reliability approximation based on representative slip surfaces (RSSs)3.2. Performance function for a single slip surface3.3. Reliability analysis for a single slip surface3.4. Correlation estimation among different slip surfaces

4. Spatial averaging of soil properties along slip surfaces5. Illustrative example I: a two-layered soil slope5.1. System reliability analysis for nominal design scenario5.2. RBD for the slope5.3. Effect of spatial variability on optimal slope design

6. Illustrative example II: a single-layered soil slope7. Summary and conclusionsDisclosure statementReferencesAppendix A. Covariance between two performance functions