An analytical method for quantifying the correlation among slope failure modes in spatially variable soils

Transcription

1 Bull Eng Geol Environ (2017) 76: DOI /s ORIGINAL PAPER An analytical method for quantifying the correlation among slope failure modes in spatially variable soils Dong Zheng 1 Dian-Qing Li 1 Zi-Jun Cao 1 Xiao-Song Tang 1 Kok-Kwang Phoon 2 Received: 13 April 2016 / Accepted: 16 July 2016 / Published online: 28 July 2016 Ó Springer-Verlag Berlin Heidelberg 2016 Abstract An efficient analytical method for quantifying the correlation between performance functions of different slope failure modes in spatially variable soils is proposed, and its performance in slope system reliability analysis is investigated. First, a new correlation coefficient is proposed to evaluate the correlation among slope failure modes considering spatial variability. For comparison and verification, the simulation-based correlation coefficient is also presented. Second, appying these two types of correlation coefficients, the effects of soil spatial variability on the representative slip surfaces (RSSs) and the system probability of slope failure are investigated using different system reliability methods, including a probabilistic network evaluation technique, a risk aggregation approach, and a bimodal bounds method. A single-layered cohesive slope is investigated to illustrate the validity of the proposed NCC. The results indicate that the proposed NCC can efficiently and accurately quantify the correlation among slope failure modes considering soil spatial variability. The number of RSSs indicated by the NCC is in good agreement with the number obtained using the SCC. The system failure probabilities of slope stability obtained with the SCC and the NCC using a risk aggregation approach are generally comparable. Also, the system reliability bounds of slope stability obtained using the NCC & Dian-Qing Li dianqing@whu.edu.cn 1 2 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan , China Department of Civil and Environmental Engineering, National University of Singapore, Blk E1A, #07-03, 1 Engineering Drive 2, Singapore , Singapore are relatively close together and comparable to those obtained using the SCC. Thus, the NCC shows good performance when evaluating the correlation among slope failure modes, and was effectively applied to analyze a single-layered cohesive slope considering soil spatial variability. Keywords Slope Spatial variability Correlation coefficient Failure mode System reliability Introduction Geotechnical problems often involve multiple failure modes; that is, there may be several potential modes of failure, in which the occurrence of any one of the potential failure modes will cause nonperformance of the system (Ang and Tang 1984; Tangetal.2015). Multiple failure modes can be found in several research fields, i.e., the stability of slopes (Fenton and Griffiths 2008; Huang et al. 2010, 2013), retaining walls (Li et al. 2015a), piles (Fan et al. 2014), and foundations (Li et al. 2015b). More importantly, different failure modes are often correlated (Chowdhury and Xu 1994, 1995; Low et al. 2011; JiandLow2012; Zhangetal.2011, 2013; Cho 2013; Lietal.2014; Zeng and Jimenez 2014; Zeng et al. 2015). For instance, if we consider two different failure modes of a soil slope, their factors of safety (FSs) are mainly influenced by the strength parameters (e.g., undrained shear strength S u, cohesion c, or friction angle /) at the bottom of each slip surface. Note that soil properties are spatially correlated and vary spatially (e.g., c and /), i.e., they differ in value from one point in space to another due to differences in geological deposition history and human activities (Elkateb et al.

2 1344 D. Zheng et al. 2003; Lietal.2016d). The correlations among these soil parameters at corresponding points at the bottoms of the slip surfaces could also partly explain the correlation between the FSs corresponding to the two failure modes. In other words, the inherent spatial variability of soil properties is due to correlations among basic random variables associated with soil properties in space, which also influences the correlation among different failure modes (Li et al. 2011, 2016a, b, c). Considering the complexity of the correlation among failure modes and the wide applications of this correlation in geotechnical engineering, it is necessary to accurately and efficiently quantify the correlation among failure modes. The correlation among slope failure modes can be approximately evaluated using the method proposed by Chowdhury and Xu (1994, 1995). The approximate correlation coefficient (ACC) between two performance functions G k and G l corresponding to failure modes k and l, respectively, is given by q A ¼qðG k ; G l Þ¼ r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; 2 P m j¼1 P m j¼1 og k r 2 X j og k P r m Xj j¼1 r Xj ð1þ in which m is the number of input random variables; ogðþ=s the partial derivative of G() with respect to the basic random variable X, and r X is the standard deviation of X. It is noteworthy that the ACC is derived based on the assumption that the basic random variables X = (X 1, X 2,, X m ) are uncorrelated (Chowdhury and Xu 1994). If the ACC is used, the correlation among slope failure modes may be biased when the basic random variables (i.e., X 1, X 2,, X m ) associated with soil properties are correlated for spatially variable soils. Hence, in theory, Eq. 1 cannot be used to evaluate the correlation among failure modes in spatially variable soils. Alternatively, the correlation coefficient can be obtained by a simulation-based method (Chu et al. 2015). Based on the factors of safety (FSs) of two slip surfaces (i.e., FS k and FS l ) selected from a large number of samples, the simulation-based correlation coefficient between two failure modes is calculated by (Chu et al. 2015) P NMC q S ¼ ðg ki G k ÞðG li G l Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NMC ðg ki G k Þ 2 P ; ð2þ NMC ðg li G l Þ 2 where G ki and G li are the performance function values and G ki = FS ki - 1, G li = FS li - 1 for the ith slope stability analysis; G k and G l are the mean values of G obtained from N MC slope stability analyses. Theoretically, if the number of simulation samples N MC is large enough, the SCC can accurately characterize the correlation among slope failure modes for spatially variable soils. However, a large number of samples are usually needed to approximate the true correlation among failure modes. Unlike the above two methods, the FORM-based correlation coefficient between two performance functions (i.e., G k and G l ) can be computed based on the tangent planes that pass through the design points u* and n* (Ang and Tang 1984; Low et al. 2011; Zhang et al. 2011) as follows: q F ¼ ut k u l ¼ nt k R 1 n l ; ð3þ b k b l b k b l where b k and b l are the reliability indices of failure modes k and l, respectively, and R is the correlation matrix of the basic random variables. The use of this method to evaluate the correlation among failure modes in spatially variable soils is hindered for three reasons. (1) The high-dimensional problem caused by soil spatial variability. A large number of random variables are involved when considering soil spatial variability. (2) Difficulty in transforming the random variables from the original space (i.e., X-space) to u-space. This step involves complicated mathematical manipulation that most users are not familiar with. (3) The design points u* orn* obtained from FORM could be unstable, since they can be affected by the starting point and the degree of robustness of the algorithms used to find the design points. In this study, a new method of analytically quantifying the correlation among slope failure modes considering soil spatial variability is proposed. This method is deemed efficient and practical for four reasons. (1) No transformation from the original correlated basic random variables X to uncorrelated standardized normal variables u is required, which is practically convenient. (2) No optimization tools are involved, so the results (i.e., correlation coefficients among failure modes) are stable. (3) The need for a large number of samples is avoided, greatly reducing the computational burden. (4) Taking soil spatial variability into account enhances the applicability of the proposed method for evaluating correlation among failure modes. To explore the performance of the proposed method and investigate its applicability to slope system reliability analysis, this paper is organized as follows. In Correlation between slope failure modes in spatially variable soils, the proposed method for evaluating correlation coefficients among slope failure modes is presented. Two major concerns in the system reliability analysis of slope failure using NCC are then discussed in The two main concerns when using the correlation among different failure modes in slope system reliability analysis. Investigative procedures are summarized in Applications to the identification of RSSs and the

3 An analytical method for quantifying the correlation among slope failure modes in spatially 1345 evaluation of the system reliability of a slope. In A single-layered cohesive slope, a single-layered cohesive slope is used as an example to explore the performance of the proposed correlation coefficient (i.e., NCC) and the effect of soil spatial variability on slope system reliability. Finally, in the Summary and conclusions section, our work is summarized and conclusions are drawn based on our results. Correlation between slope failure modes in spatially variable soils In this section, a new method of estimating the correlation coefficient between two slope failure modes considering soil spatial variability is analytically derived. According to the definition of a linear correlation coefficient, the correlation coefficient between two slope failure modes, q N,is given by (Chowdhury and Xu 1994; Ang and Tang 1984) q N ¼ CovðG k; G l Þ ; ð4þ r Gk r Gl where Cov(G k,g l ) is the covariance between G k and G l, the performance functions corresponding to the kth and lth slip surfaces, respectively (k, l = 1, 2,, N); N is the number of potential slip surfaces of the soil slope; and r G is the standard deviation (SD) of the corresponding performance function G. To obtain the correlation coefficient q N in Eq. 4, the covariance Cov(G k,g l ) between two performance functions needs to be determined first, which can be expressed as CovðG k ; G l Þ¼EðG k G l Þ EðG k ÞEðG l Þ; ð5þ where E(G k ), E(G l ), and E(G k G l ) are the expected values of G k, G l, and G k G l, respectively. Thus, the problem of evaluating the correlation coefficient is converted into the calculation of the expected values (i.e., E(G k ), E(G l ), and E(G k G l )) and SDs (i.e., r Gk and r Gl ) of the corresponding performance functions. Note that the performance function of a potential slip surface, G s (s = k, l), can be expanded in a Taylor series at the design point x* (G(x*) = 0), which is given by G s ¼ G s ðx 1 ; X 2 ;...; X m Þ G s ðx 1 ; x 2 ;...; x m ÞþXm ðx i x i Þ og s ox i ; ð6þ in which X = (X 1 ; X 2 ;...; X m ) is the vector of input basic random variables of soil properties; x* = (x 1 ; x 2 ;...; x m )is the vector representing the design point on the failure surface (G(x*) = 0); ogðþ=s the partial derivative of the performance function with respect to each random variable X i (i = 1, 2,, m) at the design point x i on the failure surface (e.g., Ang and Tang 1984). The expected value and standard deviation of G s are expressed as EG ð s Þ ¼ Xm ðeðx i Þ x i Þ og s ð7þ ox i vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X m X m u r Gs ¼ t og s og s q Xi X j r Xi r Xj ; ð8þ j¼1 in which r Xi and r Xj are the standard deviations of random variables X i and X j (i, j = 1, 2,, m), respectively; q Xi X j is the correlation coefficient between X i and X j. Based on Eqs. 6 and 7, the expected values can be derived as follows: EðG k G l Þ¼ Xm X m j¼1 EðX i X j Þ og k x i EðX jþ og k þx i x j EðX i Þx j og k og k ð9þ EðG k ÞEðG l Þ¼ Xm X m EðX i ÞEðX j Þ og k ox j¼1 i EðX i Þx og k j x i ox EðX jþ og k j þ x og k i x j : ð10þ Substituting Eqs. 9 and 10 into Eq. 5 results in the following equation: CovðG k ; G l Þ¼ Xm X m EðX i X j Þ og k ox j¼1 j EðX i ÞEðX j Þ og k : ð11þ The covariance between X i and X j, Cov(X i, X j ), can be expressed as Cov(X i ; X j ) ¼ q Xi Xjr Xi r Xj ¼ EðX i X j Þ EðX i ÞEðX j Þ: ð12þ Replacing the right-most bracket in Eq. 11 with Eq. 12 gives CovðG k ; G l Þ¼ Xm X m og k q Xi Xjr Xi r Xj : ox j¼1 j ð13þ Finally, substituting Eqs. 8 and 13 into Eq. 4, the correlation coefficient between failure modes k and l corresponding to the kth and lth slip surfaces, respectively, is derived as

4 1346 D. Zheng et al. P m P h i m q N kl ¼ j¼1 q og X i ;X j r Xi r k Xj rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m j¼1 q P : ð14þ X i ;X j r Xi r m Xj j¼1 q X i ;X j r Xi r Xj P m og k og k P m To distinguish this correlation coefficient from the correlation coefficients calculated by Eqs. 1 and 2, the correlation coefficient obtained using Eq. 14 is denoted the new correlation coefficient here. The NCC can fully account for the correlation among basic random variables X = (X 1 ; X 2 ;...; X m ), effectively and reasonably characterizing the correlation among slope failure modes in spatially variable soils. Note that all of the manipulations (e.g., the expansion of the performance function as a Taylor series) and calculations (e.g., the calculation of the partial derivatives og s = ) in Eqs are performed in X-space, rather than in the reduced space (i.e., n-space). Hence, the accuracy of the correlation coefficient given by Eq. 14 relies only on the linearity of the performance function in the original space, regardless of the relationship between the original variables X and the reduced variables n. If the performance function in the original space is linear, the partial derivatives evaluated at the design point and the mean values of X are identical. Hence, the partial derivative obtained at the mean values of X can be used in Eq. 14 to calculate the NCC, which avoids the need to determine the design point. The two main concerns when using the correlation among different failure modes in slope system reliability analysis In this section, we discuss the two main concerns the use of correlation among failure modes in slope system reliability analysis that accounts for soil spatial variability. Typically, in system reliability analysis of soil slopes, representative failure modes and slope system reliability are of interest to geotechnical engineers when carrying out probabilistic analysis of slope stability. To obtain representative failure modes and the system reliability of the slope, the correlation among potential failure modes needstobeincorporatedintotheidentificationandevaluation in a rational manner (Chowdhury and Xu 1994, 1995; Lowetal.2011; JiandLow2012; Zhang et al. 2011; Cho2013; Lietal.2013, 2014; LiandChu 2015a; ZengandJimenez2014; Zengetal.2015). Along with the correlation among failure modes, the two main concerns (i.e., representative failure modes and the slope system reliability) in slope stability analysis are explained step by step in the following two subsections. Representative failure modes As noted by Zhang et al. (2011), we can use the failure probability of the most critical slip surface to denote the failure probability of the group of slip surfaces. The most critical slip surface (i.e., that with the highest probability of failure) among a group of highly correlated slip surfaces is called the representative slip surface (RSS). Identifying representative failure modes associated with the RSSs in spatially variable soils is important for two main reasons. (1) It can help to reduce the computational cost of evaluating the system reliability of slope failure. A soil slope is usually formulated as a series system (Zhang et al. 2011; Cho 2013; Li and Chu 2015a), i.e., the whole slope will fail if sliding occurs along any individual slip surface. Theoretically, the series system consists of infinite potential slip surfaces (Hicks and Samy 2002; Hong and Roh 2008; Cho 2013; Zhang et al. 2011, 2013), which means that considerable computational effort is needed to evaluate system reliability based on these infinite slip surfaces. However, different failure modes associated with these slip surfaces are often correlated. Such correlations are mainly attributable to correlations among input soil parameters in space. This correlation among failure modes can help us to identify representative failure modes of a soil slope, thus greatly reducing the computational cost of evaluating slope system reliability, as the evaluation can be based on a finite number of RSSs. (2) The risk of slope failure can be quantitatively evaluated based on this finite number of representative slip surfaces (RSSs), which is also beneficial for risk mitigation (Li and Chu 2016). Different failure modes are associated with different risks, since the sliding volume varies with each. The consequence of each slope failure along this RSS can be quantified based on the volume of sliding mass associated with each RSS. The risk of slope failure is de-aggregated into each RSS by calculating the contribution of each RSS to the risk for the whole slope. Overall, given the resulting improvement in the

5 An analytical method for quantifying the correlation among slope failure modes in spatially 1347 efficiency of system reliability evaluation and benefit to risk mitigation in slope stability analysis, the utilization of failure mode correlation is highly rational and convenient when attempting to identify representative failure modes associated with RSSs, as the correlation acts as a useful bridge between different failure modes. In terms of methods that can be used to identify the underlying RSSs for a soil slope, the probabilistic network evaluation technique (PNET) and its variants have been widely applied in the literature (Ang and Tang 1984; Zhang et al. 2011; Li et al. 2014). In the present study, the PNET method was applied to identify the representative failure modes underlying a slope. The RSSs were identified based on the reliability index of each potential slip surface and the correlation coefficients among different potential slip surfaces, as well as a prescribed threshold value for the correlation coefficient q 0 between two slip surfaces. The correlation coefficients among different slope failure modes were calculated using the method proposed in Correlation between slope failure modes in spatially variable soils (i.e., NCC). The process of using PNET to determine RSSs can be briefly summarized as follows: 1. Calculate the failure probabilities of all potential failure modes (i.e., potential slip surfaces). 2. Determine the RSS with the largest failure probability. 3. Evaluate the correlation coefficients between the RSS identified in step (2) and other remaining potential slip surfaces. 4. Exclude the slip surfaces with correlation coefficients that are larger than q 0, and these slip surfaces are not considered anymore. 5. Perform steps 2 4 repeatedly until no potential slip surface is left. System reliability of slopes For a slope with several representative failure modes considering soil spatial variability, the overall probability of failure will depend on the reliability of individual representative modes of failure, as well as the correlations among different failure modes. The above subsection ( Representative failure modes ) discussed the methods for obtaining RSSs, then combing with the correlation coefficients among failure modes, the system reliability of slope failure can be obtained. In this study, after the RSSs were determined, the system probability of slope failure was estimated using a risk aggregation approach (Li et al. 2013, 2014). In addition, the bimodal bounds of the system failure probability of slope stability can be calculated using the correlation coefficient between slope failure modes (Ditlevsen 1979; Chowdhury and Xu 1994, 1995; Low et al. 2011; Cho 2013). For a slope involving p representative failure modes, the bimodal bounds of the slope system failure probability are given by (Ditlevsen 1979; Low et al. 2011; Cho 2013) " ( ) # P F1 þ max Xp P Fi Xi 1 PðF i F j Þ ; 0 P F Xp Xp i¼2 i¼2 max PðF i F j Þ; j\i j¼1 P Fi ð15þ where P Fi is the probability of slope failure corresponding to failure mode i and P(F i F j ) is the joint probability of failure modes i and j. A procedure for evaluating P(F i F j ) based on Gaussian probability distributions of the basic random variables was developed by Ditlevsen (1979). Applying this procedure, the lower and upper bounds of P(F i F j ) are expressed as max½a; bšpðf i F j Þa þ b ðq ij 0Þ ð16aþ 0 PðF i F j Þmin½a; bš ðq ij 0Þ; h. qffiffiffiffiffiffiffiffiffiffiffiffiffii ð16bþ in which a ¼ Uð b i ÞU ðb j q ij b i Þ h. qffiffiffiffiffiffiffiffiffiffiffiffiffii 1 q 2 ij and b ¼ Uð b j ÞU ðb i q ij b j Þ 1 q 2 ij :Here, U() is the cumulative function of the standard normal distribution; b i and b j are the reliability indices of failure modes i and j, respectively; and q ij is the correlation coefficient between the ith and jth failure modes, which can be calculated by Eqs. 1, 2, and 14. Applications to the identification of RSSs and the evaluation of the system reliability of a slope In the work reported in this paper, we performed a series of studies to investigate the performance of the proposed correlation coefficient, and the effects of soil spatial variability on slope system reliability analysis (including the identification of RSSs and the evaluation of the slope system failure probability and its bimodal bounds) based on the NCC. Figure 1 shows a schematic of the investigation procedures employed. The investigation procedure consisted of five steps, which can be summarized as follows: Step 1: Construct a deterministic slope stability model with the mean values of the random variables. Generate

6 1348 D. Zheng et al. Construct deterministic slope stability model with N potential slip surfaces Generate N MC realizations of random fields using mid-point method for each specific λ k, k = 1, 2,, 5 Select 5 illustrative potential slip surfaces (S 1~S 5) Evaluate SCCs and NCCs between the deterministic slip surface and the 5 illustrative slip surfaces Identity RSSs based on correlation coefficients SCC and NCC using PNET method Calculate the system probability, P F,SYS, within risk aggregation approach Obtain single probability of slope failure, P Fi, of each RSS by direct MCS Calculate the bi-modal bounds of system probability of failure (P FU and P FL) with correlation coefficients (i.e., SCCs and NCCs) and P Fi of each RSS Yes, then k = k + 1 k < 5 End Fig. 1 Flowchart of the investigation procedure Investigation of the performance of NCC Identification of RSSs Evaluation of system reliability of the slope N potential slip surfaces to cover the entire potential failure domain. Step 2: Obtain N MC sets of samples through the discretization of the random fields X (e.g., S u, c, or/) using the mid-point method (Wang et al. 2011; Haldar and Babu 2008; Srivastava et al. 2010) according to prescribed statistical information. The single exponential autocorrelation function (Wang et al. 2011) was adopted to describe the spatial correlation of soil probability X. The correlation coefficient q Xi X j between the values of X at the depths of z i and z j (i.e., X i and X j ) is then given by q Xi X j ¼ exp 2z i z j =k ; in which k is the scale of fluctuation. Step 3: Investigate the performance of the NCC. 1. Select five illustrative slip surfaces (denoted S 1 S 5 ) based on the SCC. Evaluate the SCCs between the critical deterministic slip surface and other potential slip surfaces at the minimum scale of fluctuation k min. Divide these potential slip surfaces into five groups according to correlation coefficient ranges (0, 0.2], (0.2, 0.4],, (0.8, 1.0]. If the correlation coefficient associated with a slip surface falls within a specific range, the slip surface is assigned to the corresponding group. Select the slip surface corresponding to the minimum reliability index, b min, from each group and denote them S 1,S 2,, S 5. Note that five slip surfaces are enough to illustrate the performance of the NCC, so only five illustrative slip surfaces were selected in this study, although it is also acceptable to use more than five illustrative slip surfaces. Note that the accuracy and trend in the correlation (which vary with soil spatial variability) among the different failure modes are verified by increasing the scale of fluctuation k from the minimum (i.e., k min ) to the maximum (i.e., k max ) under the five fixed illustrative slip surfaces. 2. Calculate the SCC and NCC between the critical deterministic slip surface and the five illustrative slip surfaces for different scales of fluctuation k (ranging from k min to k max ). Step 4: Identify the RSSs. Identify the RSSs using the PNET method based on the two different correlation coefficients (SCC and NCC) for five different values of k, and record the number of failure samples n i (i = 1, 2,, p) for each RSS at different values of k. Step 5: Evaluate the system reliability of the slope. 1. Calculate the system probability of slope failure, P F,SYS, within the framework of the risk aggregation approach, P F,SYS = (n 1? n 2?? n p )/N MC. To achieve computational efficiency without losing accuracy, the threshold value q 0 is taken as 0.9 according to Li et al. (2014); this threshold value is used to determine whether a slip surface from a potential slip surface library (PSSL) is excluded, based on the correlation coefficients between the new representative slip surface and the remaining slip surfaces in the PSSL during each iteration. 2. Calculate the system probability of slope failure, P F,MC, by direct Monte Carlo simulation (MCS). Record the number of failure samples n i,mc (i = 1, 2,, p) for N MC slope stability analyses of each RSS. Then, the probability of slope failure, P Fi, of each RSS is calculated by P Fi = n i,mc /N MC. 3. Use Eq. 15 to calculate the slope system failure probability bounds [P FL,P FU ] based on the P Fi values and different correlation coefficients (including SCC and NCC). A single-layered cohesive slope We then considered a single-layered cohesive soil slope that has been studied by Wang et al. (2011), Li et al. (2014), and Li and Chu (2015a). The slope geometry is shown in Fig. 2. Following Li et al. (2014), the cohesive

7 An analytical method for quantifying the correlation among slope failure modes in spatially 1349 Elevation (m) 20 Critical deterministic slip surface S 1 10 soil slope had a height of 10 m and a slope angle of 26.6, corresponding to an inclination ratio of 1:2. The unit weight of soil was 20 kn/m 3. The cohesive soil was underlain by a firm stratum at 20 m below the top of the slope. The spatial variability of the undrained shear strength S u was modeled by a lognormal stationary random field with a mean of 40 kpa and a coefficient of variation (COV) of To investigate the effect of the scale of fluctuation on the slope failure modes and system reliability, the scale of fluctuation k was varied from 1 to 1000 m. A total of 15,170 potential slip surfaces were generated to cover the entire failure domain using the Entry and Exit method in SLOPE/W (GEO-SLOPE International Ltd. 2008). The five selected illustrative slip surfaces (i.e., S 1,S 2,, S 5 ) are shown in Fig. 2. Applying the mean values of the undrained shear strength S u, the minimum FS of the cohesive soil slope was obtained as using the simplified Bishop method, which is consistent with the value of 1.18 reported by Wang et al. (2011). The corresponding critical slip surface is also plotted in Fig. 2. The 1D lognormal stationary random field of S u was discretized using the mid-point method. To do this, the 20-m cohesive soil layer was divided into 40 sublayers, each with a thickness of 0.5 m (Wang et al. 2011). The S u of each sublayer was randomized. In order to obtain sufficiently accurate system reliability results, 10,000 sets of random fields were generated. The number of MCSs, N MC, was set to 10,000. The threshold value q 0 was taken as 0.9. Performance during correlation evaluation S 2 S 4 S 5 S Distance (m) Fig. 2 Geometries of the cohesive slope and the five illustrative slip surfaces (S 1 S 5 ) Figure 3 compares the values of the two correlation coefficients (i.e., SCC and NCC) for various scales of fluctuation k. Note that both the SCC and the NCC increase with increasing k, which indicates that they both account reasonably well for the effect of spatial variability. Taking the SCC to be the true correlation coefficient among the failure modes, whenever the soil spatial variability is relatively weak (k = 10, 20, 1000 m) or strong (k = 1, 5 m), the difference between the ACC and the SCC is very small (i.e., negligible), and the NCC almost coincides with the SCC for various values of k. The NCC can reasonably Correlation coefficients S 1 S 1 S 2 S 2 S 3 S 3 S 4 S 4 S 5 S Scale of fluctuation, λ (m) Fig. 3 Correlation coefficients between the slip surfaces S 1 S 5 and the deterministic slip surface SCC The number of samples used in simulation Fig. 4 SCC values obtained using different numbers of samples at k = 1m account for the spatial variability, resulting in accurate and reliable correlation coefficients between different failure modes in comparison with the true results (i.e., the SCC results) obtained using a simulation-based method. Note that the SCC was calculated during the simulation, and its accuracy depended on the number of samples generated in the simulation, i.e., the number of evaluations of performance functions. For example, Fig. 4 shows the value of SCC versus the number of samples used in the simulation. It appears that at least 2000 samples (i.e., 2000 evaluations of the performance function for each slip surface) are needed to obtain reasonably accurate estimates of the SCC in this example. In contrast, when evaluating the NCC, the number of evaluations of the performance function relied only on the number (40 in this example) of random variables involved in the performance function. As indicated by Eq. 14, the partial derivatives of the S 1 S 2 S 3 S 4 S 5

8 1350 D. Zheng et al. Representative slip surface number, p Scale of fluctuation, λ (m) Based on SCC Based on NCC 1000 Fig. 5 Numbers of representative slip surfaces based on the correlation coefficients SCC and NCC a System probability of failure, P F,SYS b P F,MC P F,SYS P F,SYS Scale of fluctuation, λ (m) performance function over each random variable are needed to calculate the NCC. These can be calculated by the finite difference method, resulting in = 80 evaluations of the performance function for each slip surface. The number of evaluations of the performance function required to evaluate the NCC is much less than the number needed to calculate the SCC. Thus, using the NCC rather than the SCC saves considerable computational effort. Representative failure modes of a cohesive slope Soil spatial variability may affect the number of RSSs identified by the PNET approach. Figure 5 compares the number of RSSs for various values of k based on the two correlation coefficients (i.e., the NCC and SCC). When the soil spatial variability becomes weaker (i.e., k increases from 1 to 1000 m), the number of RSSs based on the SCC and the number of RSSs based on the NCC monotonously decrease. This means that when soil spatial variability increases (k decreases), the picture of the representative failure modes of the whole slope becomes more complex. Generally, the number of RSSs based on the NCC is almost the same as that based on the SCC. The trend seen in Fig. 5 that the number of RSSs based on either the SCC or NCC decreases with k is consistent with that observed by Jiang et al. (2015). Thus, the proposed NCC accounts reasonably well for the effect of spatial variability on the number of slope failure modes. System reliability of a cohesive slope As mentioned above, the number of RSSs can be affected by the soil spatial variability, which can also have an effect on the slope system reliability. Figure 6 shows the system Probability of failure P F,MC P FU (Based on SCC) P FL (Based on SCC) P FU (Based on NCC) P FL (Based on NCC) Scale of fluctuation, λ (m) Fig. 6 System reliability analysis results based on the two correlation coefficients (i.e., SCC and NCC) in the cohesive slope example. a System probability of failure, P F,SYS. b Lower and upper bounds of the system probability of failure obtained using the SCC and NCC probability of failure and its bimodal bounds for various values of k. In Fig. 6a, the system probability of slope failure based on each correlation coefficient is evaluated using the risk aggregation approach (Li et al. 2013, 2014). For comparison, the results obtained using direct MCS are also shown in Fig. 6, and are taken as exact values. Note that the system probabilities of slope failure obtained using the NCC and SCC are slightly smaller than that obtained from direct MCS. This slight difference is attributed to the fact that only a limited number of RSSs (selected from a large number of potential slip surfaces using PNET) are used to estimate the slope system failure probability in the risk aggregation approach. Note that the slope failure probability calculated based on the selected RSSs using the NCC is almost identical to that obtained with the SCC, which further demonstrates that the NCC produces sufficiently accurate results.

9 An analytical method for quantifying the correlation among slope failure modes in spatially 1351 Figure 6b shows the bimodal bounds of the system probability of slope failure evaluated using the two correlation coefficients. For comparison, the system probability of slope failure evaluated using direct MCS is also shown in Fig. 6b. The system probability of slope failure obtained using direct MCS falls within the probability ranges obtained with the NCC and SCC, which indicates that the results afforded by the NCC and SCC are reasonable. Also, the bimodal bounds of the slope system failure probability based on the NCC and SCC are relatively close together, and appear to be similar for both the NCC and the SCC. This can be attributed to the similar numbers of RSSs obtained with the NCC and the SCC, as shown in Fig. 5. Thus, from the perspective of its applicability to soil slope reliability analysis accounting for spatial variability, the proposed NCC is clearly suitable for evaluating the system reliability bounds of slope stability. reasonably close together and accurate. These results demonstrate the accuracy and effectiveness of the NCC proposed in this paper for efficiently quantifying the correlation among slope potential failure modes. Also, when applied to a single-layered cohesive slope, the proposed NCC was found to yield representative slip surfaces and to allow the accurate evaluation of the probability of slope failure and its corresponding bimodal bounds. Indeed, this proposed NCC is generally applicable to other geotechnical structures (e.g., retaining walls, piles, foundations, etc.), although it was only applied to a slope in this study. Acknowledgments This work was supported by the National Science Fund for Distinguished Young Scholars (project no ), the National Natural Science Foundation of China (project nos , , ), and the Natural Science Foundation of Hubei Province of China (project no. 2014CFA001). Summary and conclusions In this paper, we have proposed an analytical method for quantifying the correlation among slope failure modes in spatially variable soils. The performance of the proposed correlation coefficient (i.e., NCC) was evaluated systematically. The effects of soil spatial variability on the RSSs and the system reliability of soil slopes were investigated for a single-layered cohesive slope. Several conclusions can be drawn from this study: 1. By directly comparing the values of the NCC and SCC under different scales of fluctuation, it was shown that the proposed NCC can accurately and reasonably quantify the correlation among slope failure modes considering soil spatial variability. With respect to the accuracy of the correlation among slope failure modes and the number of RSSs, using NCC in the system reliability analysis of cohesive slopes was found to yield reasonably accurate results. Using the NCC rather than the SCC in system reliability analysis considerably reduces the computational effort required to evaluate the correlation among different failure modes. 2. The number of RSSs obtained with the SCC and the NCC using the PNET method increases as the scale of fluctuation decreases (i.e., as the soil spatial variability becomes stronger). The whole picture for the representative slip surfaces becomes more complex as the degree of soil spatial variability is enhanced. 3. 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