Reliability analysis of slopes considering spatial variability of soil properties based on efficiently identified representative slip surfaces

Bin Wang, Leilei Liu, Yuehua Li, Quan Jiang

a State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences, Wuhan,430071,China

b University of Chinese Academy of Sciences, Beijing,100049, China

c Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring,Ministry of Education, School of Geosciences and Info-Physics, Central South University, Changsha, 410083, China

d Hunan Key Laboratory of Nonferrous Resources and Geological Hazards Exploration, Changsha, 410083, China

e School of Geosciences and Info-Physics, Central South University, Changsha, 410083, China

Abstract Slope reliability analysis considering inherent spatial variability (ISV) of soil properties is timeconsuming when response surface method (RSM) is used, because of the “curse of dimensionality”.This paper proposes an effective method for identification of representative slip surfaces(RSSs)of slopes with spatially varied soils within the framework of limit equilibrium method (LEM), which utilizes an adaptive K-means clustering approach. Then, an improved slope reliability analysis based on the RSSs and RSM considering soil spatial variability, in perspective of computation efficiency, is established. The detailed implementation procedure of the proposed method is well documented, and the ability of the method in identifying RSSs and estimating reliability is investigated via three slope examples. Results show that the proposed method can automatically identify the RSSs of slope with only one evaluation of the conventional deterministic slope stability model. The RSSs are invariant with the statistics of soil properties, which allows parametric studies that are often required in slope reliability analysis to be efficiently achieved with ease. It is also found that the proposed method provides comparable values of factor of safety (FS) and probability of failure (Pf) of slopes with those obtained from direct analysis and literature.

2020 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Slope reliability analysis Spatial variability Representative slip surfaces (RSSs)Response surface method (RSM)Random field simulation

1. Introduction

Soil properties often exhibit uncertainties of various kinds,such as inherent spatial variability (ISV), because of complex depositional and post-depositional processes of soil deposit (e.g.Li et al., 2016a; Zhu et al., 2019). These uncertainties can be transferred to the subsequent slope stability analysis, and thus affect the assessment of slope stability. As such, it is essential to quantify the effects of these uncertainties on the slope stability analysis. Reliability approaches based on statistics and probability theory provide a rational and effective tool for such purpose, and have gained an increasing popularity in geotechnical engineering field over the past few decades (e.g.Vanmarcke, 1977; Chowdhury and Xu, 1995; Sivakumar Bubu and Mukesh, 2004; Cho, 2007; Griffiths et al., 2009; Hicks and Spencer, 2010; Wang et al., 2011; Ji, 2013; Kim and Sitar, 2013;Li et al., 2015; Pantelidis et al., 2015; Liu et al., 2018, 2019).However, it is not a trivial task to advance the application of slope reliability analysis in geotechnical engineering practice,because a commonly used yet the most robust slope reliability analysis approach, Monte Carlo simulation (MCS), generally requires an expensive cost of computation resources, especially when the ISV is considered (e.g. Griffiths and Fenton, 2004;Zhang et al., 2011). Therefore, it is of practical significance to enhance the efficiency of slope reliability analysis considering ISV of soil properties.

In general, within the framework of MCS, the probability of failure (Pf) of a slope is calculated as

whereNmcsdenotes the number of MCS samples,Xidenotes theith vector of randomly generated variables,FSidenotes theith factor of safety (FS) of the slope given Xi, andI½$denotes an indicator function that is equal to unity whenFSiXi<1 and zero otherwise(e.g. Ang and Tang, 1984). Thus, there are two main components that dominate the computation efficiency of slope reliability analysis within the framework of MCS:the evaluation of FS and the MCS sample size. In other words, the efficiency of slope reliability analysis can be well improved by enhancing the efficiency for each component. For example, many efforts have been made to reduce the MCS sample size in slope reliability analysis in order to improve the computation efficiency (e.g. Wang et al., 2010, 2011; Jiang and Huang, 2016; Li et al., 2016b; Huang et al., 2017; Liu et al., 2017,2020). In addition, several tens of surrogate models or response surface methods (RSMs) have been widely proposed to approximate and replace the time-consuming slope stability analysis model to improve the computation efficiency of FS(e.g.Kang et al.,2016; Li et al., 2016a; Liu et al., 2019). Furthermore, dimension reduction methods are also employed to overcome the “curse of dimensionality”, thus improving the computation efficiency when ISV is considered (Pan and Dias,2017; Li et al., 2019).

The above methods, however, can be further improved because many computation resources are spent in calculating FSs for numerous trivial failure modes, which is a kind of waste. This inefficiency problem can be solved by representative slip surfaces(RSSs)-based approaches, which have been widely used for system reliability analysis of slopes(Zhang et al.,2011;Li et al.,2013;Jiang et al., 2015, 2017; Zeng et al., 2015; Ma et al., 2017), risk assessment of slope failure (Li and Chu, 2016; Zhang and Huang,2016) and probabilistic analysis of pile-stabilized slopes (Zhang et al., 2017). In RSSs-based approaches, the RSSs are firstly identified by some unique techniques, and then the afore-mentioned RSMs and direct MCS are performed directly on the RSSs. Since FSs are evaluated only for the RSSs that are generally in a small number, the computation time can be substantially reduced. This situation is especially prominent when limit equilibrium method(LEM) is used for slope stability analysis. Nevertheless, the RSSsbased approaches still suffer from several disadvantages that are to be resolved. Firstly, the identification of RSSs in the available methods is relatively complex, which requires not only the calculation of reliability index along each failure surface but also the correlation coefficients between the FSs of different slip surfaces. Meanwhile, determination of RSSs in previous studies requires performing a series of parametric studies on either a threshold correlation coefficient(Zhang et al.,2011;Li et al.,2014)or the number of subdivision groups of potential slip surfaces(PSSs) (Li et al., 2013; Jiang et al., 2017), which could be a challenging task for inexperienced engineers. Secondly, repetitive identifications of RSSs are required for different cases of statistics of soil properties, which might be inefficient for slope reliability analysis that requires a series of parametric studies (Jiang et al.,2015).

In view of the two issues, this paper aims to propose a simple but effective method for identification of RSSs within the framework of LEM for slope stability analysis by using an adaptiveK-means clustering approach, and thus for an efficient reliability analysis of slopes in spatially varied soils. The proposed method shares the advantage that the RSSs identified for a slope by the proposed method are invariant for different statistics of soil properties so that parametric studies that are often required in slope reliability analysis can be efficiently achieved. As such, the mentioned limitations of the available RSSs-based methods can be released with ease.

2. Proposed method for slope reliability analysis considering inherent spatial variability

A key prerequisite of LEM for slope stability analysis is that a great number of PSSs must be predefined such that the FS of the slope can be localized.However,it is recognized that the FS values of most PSSs are correlated with each other,and it is not necessary to search for the FS of the slope among all PSSs(Zhang et al.,2011).Thus, the main idea of the proposed method is to adaptively identify the RSSs of a slope within the LEM framework using a classic clustering method,i.e.K-means clustering, so that the FS of the slope is searched for among only the RSSs instead of all PSSs.Then, RSMs are established for the RSSs to further improve the computation efficiency of slope reliability analysis considering ISV.Therefore,the proposed method consists of four parts:(1)random field simulation of ISV;(2)identification of RSSs;(3)construction of RSMs based on RSSs;and(4)MCS for reliability analysis.Each of the four parts is introduced as follows.

2.1. Simulation of inherent spatial variability by random field theory

Mathematically,the ISV of soil property is often characterized by an autocorrelation function (ACF) (Li et al., 2015). For example, a single exponential ACF used in this study is considered as

where x is a randomly generated standard normal matrix with a dimension ofne2; L1and L2are the lower triangular matrices decomposed respectively from matrices C and R by using the Cholesky decomposition method (CDM) as

Additionally, it is worthwhile to point out that the single exponential ACF in Eq.(2)is just an assumption that has not been validated systematically, and it is a weakness of most reliability analyses in the literature, and a barrier for implementing reliability analysis in engineering practice. To release such a limitation in the literature, reliability analysis should be conducted with ISV being reasonably quantified from practical in situ and/or laboratory testing data, although the data quantity is often not enough. For this, a recently developed new random field generator(Wang et al.,2018;Hu et al.,2019), which is able to simulate random field samples directly from sparse measurements without any assumption of the function form of ACF,can be used.The new random field generator is also able to generate non-Gaussian and non-stationary random fields (e.g. Montoya-Noguera et al., 2019; Wang et al., 2019), which are often the case for soil properties.

2.2. Identification of representative slip surfaces using K-means clustering method

Generally, it can be known from natural landslides that slopes often fail along a relatively thick slip band (or failure zone) rather than a perfect slip surface without any thickness.Slip surfaces that are predefined in the slip band, therefore, probably have very similar FSs because these slip surfaces share almost the same soil properties and loads. In addition, a common feature of these slip surfaces is that they share almost the same sliding volume.Therefore,it would be a waste of time to search for the minimum FS of the slope among all slip surfaces in a similar potential failure zone.Hence,inspired by these typical observations on slope failure,an empirical sliding volume-basedK-means clustering method is proposed to identify the RSSs of slope to avoid redundant search of the FSs on similar PSSs, with the RSSs being identified from different clusters of slip surfaces. In the following, the fundamentals of the proposed sliding volume-basedK-means clustering method and how the RSSs are identified from the corresponding clustering results are described in detail (Bishop, 2006; Huang et al., 2013).

The rationale of theK-means method is to clusterNobservations intoKclusters (KN) in which each observation belongs to the cluster with the nearest mean, by minimizing a squared error function as

where ckk1; 2; /;Kdenotes the center of thekth cluster;dii1; 2; /;Ndenotes theith observation which is anmdimensional real vector;anddenotes the observation that belongs to thekth cluster,withLkbeing the size of thekth cluster.The whole clustering process consists of the following steps:

(1) Randomly selectKobservations from all theNobservations as the initial cluster centers ofKclusters,which is denoted as

(2) Calculate the distance (e.g. the Euclidean distance) between each observation diand each cluster center at current step,and assign dito the cluster whose cluster center is the nearest to it.

(3) Evaluate the value of the squared error function (i.e.Jin Eq.(8)) for the clustering results at the current step, and then update each cluster center by means ofas

(4) Stop the process ifJconverges orotherwise,makejjþ 1 and repeat Steps (2) and (3).

In this study, as mentioned above, the sliding volume (or sliding mass in two-dimensional slope stability analysis) of a slip surface serves as the observation dii1; 2; /;Nwithm1, andNis the number of PSSs predefined in an LEM slope stability model. In addition, as a prerequisite for theK-means method, theKvalue should be predefined, which, however, is not a trivial task, because different clustering results would be obtained for differentKvalues. To this end, this study introduces a classic index,DUNN(Dunn,1974), to assess the clustering results,with a largerDUNNvalue indicating a better clustering result.TheDUNNindex has been widely and successfully used in evaluating the clustering results of geographic data in the literature and is described as

whereDUNNKdenotes theDUNNindex for the situation where the observations are classified intoKclusters; Cidenotes theith cluster;denotes the minimum distance between two observations respectively in Ciand Cj; anddenotes the diameter of thepth cluster.

As such, given the sliding volume values associated with all PSSs, the bestKvalue and the associated clustering results on the slip surfaces here can be adaptively identified from a variety ofKvalues, i.e. [Kmin;Kmax], by comparing theDUNNvalues corresponding to all possibleKvalues based on the exhaustive method.Then, RSSs are identified from each cluster of the slip surfaces.However, it is worthwhile to point out that it is not the cluster centers that are selected as the RSSs herein, which is often the case inK-means clustering method. Instead, it is the slip surface that has the minimum FS among all slip surfaces in each cluster being the RSS of that cluster, where the FSs associated with all PSSs are evaluated at the mean values of soil properties.It should be noted that, since the clustering process is performed based on the sliding volume or mass values that are predefined before slope stability analysis, and the identification of RSSs is based on the FS results of all PSSs obtained using mean values of soil properties and the above clustering results, the proposed method is independent of the soil ISV. As such, the proposed method is efficient and particularly suitable for slope reliability analysis involving many parametric studies. This observation is, however,different from those in the literature that the RSSs are correlated with the ISV (Li et al., 2014; Jiang et al., 2015). The two contradictory conclusions lie in the fact that the rationales for the proposed method and the methods in the literature are different.In the current study, the correlations among different slip surfaces are quantified based on the sliding volume/mass and are implicitly considered in the clustering results which are independent of the soil ISV. By contrast, previous studies as shown above considered the correlations among different slip surfaces by FS, which might vary with the soil ISV. Obviously, the current and previous studies identify the RSSs from two different perspectives. Generally, the statistical and mechanical fundamentals for the previous studies are more rational than those for the current study in terms of identification of RSSs. The proposed method, however, provides a different insight into the identification of RSSs, which evaluates the slope stability from a small number of PSSs selected from different clusters at some sacrifice of accuracy and can be considered as an empirical approach. The effectiveness of the proposed method for RSSs identification will be illustrated and validated by three slope examples in Sections 4e6, respectively.

2.3. Response surface method based on representative slip surfaces

As previously mentioned, within the framework of LEM, many computation resources are spent in searching for the minimum FS of slope among all PSSs. This situation becomes more profound when reliability analysis is performed, because thousands of executions of an LEM program are often required.Although it has been highlighted that it might not be necessary to search for the FS among all PSSs,but among only the RSSs,direct evaluation of the FS for each RSS still requires many iterations, which can be further improved.To this end,a multiple RSM,which has been successfully applied to numerous slope reliability analysis problems (Li et al.,2015, 2016b; Li and Chu, 2015; Liu et al., 2018), is utilized to explicitly approximate the limit state function of LEM for slope stability analysis in order to improve the computation efficiency in FS evaluation. A commonly used quadratic polynomial without cross terms is used to construct the RSM in this study and it is given as

whereFSiXis the FS of theith RSS;Nris the number of RSSs,which is equal to the number of clustersK;Xfx1;x2;/;xngTis a random variable vector, e.g. a set of random field realization ofcand;a1i;bijandcijare the 2nþ1 unknown coefficients, which can be easily calibrated by central composite design method (Bucher and Bourgund, 1990) by solving a system of 2nþ1 linear algebraic equations with the following 2nþ1 samples:andmx1;mx2;/;mxnmxn,where mxiandxiare the mean and standard deviation of theith random variable, respectively; andmis a coefficient for generating sample points,which is generally taken as 2(Zhang et al.,2011).

2.4. Slope reliability analysis based on representative slip surfacesbased response surface method

This section introduces the MCS for slope reliability analysis based on the above established RSSs-based RSM(RSSs-RSM).Since the evaluations of FSs in the MCS are conducted on the RSM rather than on the original LEM model,slope reliability analysis herein can be efficiently achieved. Consider, for example,Nsimrandom field samples generated from Eq. (4). The probability of failurePfof a slope is generally calculated as

whereIf$g is an indicator function which is equal to unity ifand zero otherwise;anddenotes the FS calculated based on thejth response surface given theith random field sample.The accuracy ofPfis assessed by its coefficient of variation (COV) as

Therefore, a large value ofNsimis often desired in order to ensure a high resolution ofPf,which,however,in turn determines the efficiency of the reliability analysis.

3. Implementation procedure

Fig.1 plots the flowchart of the proposed method to facilitate its understanding and application. Generally, the flowchart consists mainly of the following seven steps:

(1) Define necessary parameters for slope reliability analysis,such as shear strength and geometrical parameters. Meanwhile,identify the probabilistic parameters and estimate the associated statistics,for example,the means,COVs,ACFs,and ACDs.

(2) Discretize the slope domain into finite random field elements. Then, export the coordinates of the centroid of each element, and use the CDM described in Section 2.1 to generateNsimsets of random field samples of the underlying spatially varied soil properties for subsequent MCS use.

(3) Use an in-house code or commercial software to build a deterministic slope stability model by LEM (e.g. Bishop’s simplified method (BSM)), including but not limited to predefining the number and locations of the PSSs, and run the LEM model based on the mean values of soil properties.Then,save the information such as FS and sliding volume or mass associated with each PSS for furtherK-means based RSSs identification purpose.

(4) Define a potential range (i.e. [Kmin,Kmax]) for the cluster numberKof the PSSs and performK-means clustering on differentKvalues to obtain the best cluster numberKoptbased on theDUNNindex,as shown in Eq.(10).Then,identify RSSs from the optimal clusters of PSSs using the suggested method in Section 2.2.

(5) Run the deterministic slope stability model for the central composite design samples based on the RSSs to evaluate the corresponding FSs, and use quadratic polynomials without cross terms to establish a relationship between the FS and random field samples for each RSS.

(6) Substitute theNsimsets of random field samples generated in Step 2 into the response surface models established in Step (5) to evaluate FS values corresponding toNsimsamples,which are then used to calculatePfof the slope and the associatedCOVPfaccording to Eqs. (12) and (13),respectively.

(7) Check if theCOVPfreaches the required accuracy of MCS.If it is less than a target resolution (e.g. less than 0.1),then stop the program; otherwise, enrich the MCS samples and go back to Step (6) until it satisfies the accuracy demand.

It is pointed out that the above procedures can be used as a“black box” so that geotechnical engineers who are not familiar with the proposed method and probability theory can use the procedure with ease, because they only need to focus on the deterministic slope stability analysis that they are most familiar with. In addition, it shall be noted that, since FS of a slope is evaluated based on a small number of RSSs with the aid of RSM in the proposed method,the method is expected to be very efficient,and slope reliability analysis based on a robust MCS can be efficiently performed while maintaining an acceptable accuracy.Another key advantage of the proposed method is that the identification of RSSs depends only on the predefined PSSs and the corresponding FS values under the mean values of soil properties,so it is straightforward to use the method to conduct various parametric studies without extra cost of computation resources,which is generally required by some available approaches (Jiang et al., 2015). In the following three sections, the effectiveness and efficiency of the proposed approach are illustrated and validated.

Fig.1. Flowchart for the proposed method.

4. Illustrative example #1: Application to an undrained cohesive slope

This section uses the proposed method to evaluate the reliability of an undrained cohesive slope.The slope has been widely studied in the literature (e.g. Cho, 2010; Jiang et al., 2015; Liu et al., 2018),from which the results can be conveniently used as references for validation of the proposed method.The cross-section of the slope is plotted in Fig. 2, which has a height of 5 m and an angle of 26.6.The slope is comprised of a clay layer with a thickness of 10 m and a unit weight of 20 kN/m3. The undrained shear strengthsuis assumed to be a lognormal random field with the meanCOVandlvvalues of 23 kPa, 0.3, 20 m and 2 m,respectively,which are the same as those used by Cho(2010),Jiang et al. (2015) and Liu et al. (2018), so that the results in these references can be compared.

4.1. Deterministic slope stability analysis

Following the proposed procedure,the slope geometry is firstly divided into 910 random field elements. Then, random field simulation is conducted to model the ISV ofsu. As a reference, Fig. 3 shows the discretization of the random field elements for this slope and a typical realization of the random field ofsu. It is observed from the figure thatsusignificantly varies with locations,which is signified by different colors e darker color indicating largersuvalues and lighter color representing smallersuvalues.suchanges more rapidly in vertical direction than it does in horizontal direction,becauselhis larger thanlv(i.e.20 m vs.2 m),suggesting a reasonable simulation for the ISV ofsu.

A deterministic slope stability analysis model is subsequently established for the slope based on BSM with an assumption of 4851 predefined PSSs,as shown in Fig.2.To validate the effectiveness of the established stability model,suis firstly taken as a deterministic value of msu.Then,the FS is calculated as 1.355,which is very close to the results reported in the literature(Cho,2010;Jiang et al.,2015;Liu et al., 2018). Furthermore, both the FS and associated critical deterministic slip surface, which are localized among 4851 predefined PSSs,are also shown in Fig.2.It is found that the results are almost consistent with those given by Jiang et al. (2015) and Liu et al. (2018). Therefore, the similar results verify the accuracy and feasibility of the deterministic slope stability model.Obviously,this deterministic slope stability model searches for the FS among a large number of PSSs, which, to some extent, wastes some computation resources and lacks of efficiency. This inefficiency would enlarge significantly when MCS is involved. Therefore, to enhance the efficiency for slope reliability analysis, this model is mainly used here as a baseline model to identify the RSSs and provide reference results for validation of the proposed method.For convenience purpose and to avoid confusion in the following analysis, the original model that uses PSSs to search for FS is simplified as PSSs model, whereas the modified model that uses RSSs to search for FS is referred to as the RSSs model.

4.2. Identification of representative slip surfaces

This subsection identifies the RSSs from the PSSs model established above using the proposed adaptiveK-means clustering method, which is realized using MATLAB R2014a software. As mentioned before, to adaptively identify the bestKvalue for clustering the PSSs, a range forKvalue should be determined in advance,i.e.[Kmin;Kmax].The value ofKmincan be easily identified as a small value (larger than 3 here) because there are often multiple failure modes in a slope.By contrast,determination ofKmaxis more complex.In theory,Kmaxcan be as large as the number of the PSSs,i.e.4851 here,which,however,is computationally expensive.In view of this dilemma,Kmaxis determined from experiences that 200 RSSs can be enough for characterizing the failure modes of a slope with spatially varied soils (Jiang et al., 2015), and it is finally determined as 200 to maintain a balance between the accuracy and efficiency.

Fig.4 plots theDUNNvalues for variousKvalues.It is seen from the figure that theDUNNvalue varies withK, which shows thatKhas a significant influence on the clustering results. It can also be observed from Fig. 4 that theDUNNindex reaches the maximum value(i.e.5.21103)whenKis selected as 113,signifying the best cluster number for PSSs.Then,within each cluster,the slip surface with the minimum FS evaluated at the mean value ofsuis identified as the RSS in that cluster,and finally there are a total of 113 RSSs for the undrained cohesive slope, as shown in Fig. 5. Note that the critical deterministic slip surface is also included in the RSSs.Furthermore, it is also worth noting that previous studies (Zhang et al., 2011; Li et al., 2013, 2014; Jiang et al., 2017) might identify only one RSS for the studied slope system,because the RSSs in these studies are identified by a different statistical and mechanical theory, i.e. the correlation between FS values. Therefore, from the perspective of the number of RSSs herein, the proposed method might be less efficient than the previous ones,but the identification process of the RSSs in the current study is more efficient.Moreover,as mentioned before, the proposed method is independent of the soil ISV, which facilitates greatly its application to parametric studies where ISV of soil properties is difficult to be accurately quantified.

To gain more insights into the proposed method,typical clusters of RSSs forK113 are plotted in Fig.6,which shows that different clusters are comprised of different numbers of slip surfaces and the failure modes in a same cluster are almost the same. For example,the slip surfaces in the 2nd cluster are nearly all in shallow failure modes, while those in the 50th cluster belong to deep failure modes.This indicates the reasonability of the proposed method for the identification of RSSs.

Fig.2. Cross-section of the undrained cohesive slope with 4851 potential slip surfaces(modified after Liu et al., 2018).

Fig.3. Discretization of random field elements for the undrained cohesive slope and a typical realization of the random field of su.

Fig. 4. Variation of DUNN with K for the undrained cohesive slope.

To further validate the effectiveness of the RSSs identified by the proposed method,Fig.7 compares the FSs obtained based on PSSs and RSSs for 10,000 random field samples ofsufor the statistics listed in Table 1. It is found from Fig. 7 that the FSs from the two models agree well with each other, suggesting that the proposed method for RSSs identification is reasonable and accurate in estimating the FS of the undrained cohesive slope. Therefore,compared with the PSSs model that consists of 4851 slip surfaces and is often adopted in the literature (Liu et al., 2018), the RSSs model searches for the FS of the slope among only 113 slip surfaces,indicating about an increase of 43 times in efficiency.Note that,in this part, FSs for both the PSSs and RSSs models are evaluated directly from the deterministic slope stability model,instead of the RSM,because the main purpose here is to validate the effectiveness of the RSSs identified by the proposed method. In the following,probabilistic slope stability analysis for this example is conducted to further validate the accuracy and efficiency of the proposed method.

4.3. Probabilistic slope stability analysis based on representative slip surfaces

Based on the RSSs identified above, RSM is then used to approximate the limit state function of this slope stability to further improve the computation efficiency in evaluating FS of the slope.Fig.8 shows the accuracy of the proposed RSSs-RSM for evaluating the FS of the slope by comparing the FSs evaluated based on RSSs model and those evaluated based on the RSSs-RSM for the abovementioned 10,000 random field samples ofsufor the statistics in Table 1.Similar to Fig.7,the FSs obtained from the two methods are nearly consistent with the 1:1 line, suggesting a good accuracy of the RSSs-RSM.Therefore,MCS can be performed directly based on the RSSs-RSM to evaluate the slope reliability.

Fig. 5. Schematic diagram of representative slip surfaces for the undrained cohesive slope.

Table 1 displays the reliability results of the slope obtained by different methods.The first four methods in the table are from this study, which include the PSSs-based MCS (PSSs-MCS), RSSs-based MCS (RSSs-MCS), PSSs-RSM-based MCS (PSSs-RSM-MCS), and RSSs-RSM-based MCS(RSSs-RSM-MCS).The other methods as well as the associated results are from literature(Cho,2010;Jiang et al.,2015;Liu et al.,2018).In general,the results from different methods are very consistent, which validates the accuracy of the proposed method. For example, thePfvalue estimated by the proposed method (i.e. RSSs-RSM-MCS) is about 7.21102, which matches well with that estimated by the direct MCS,for example,7.5102by the PSSs-MCS in this study, and 7.6102and 7.73102reported by Cho (2010) and Liu et al. (2018), respectively.

Fig.6. Typical clusters of potential slip surfaces for K113 for the undrained cohesive slope.

However, although the above methods have comparable accuracies,the proposed method might be superior to other methods in computation efficiency. To illustrate this, a dimensionless nominal indexNssusing the number of slip surfaces that are searched for evaluating the FS of the slope in the reliability analysis is proposed to measure the computation efficiencies of different methods,because the physical time for evaluatingPfby the methods from literature are difficult to be obtained and different computation tools might be used in different studies. Therefore,Nssis equal to 4851 for the PSSs model, and 113 for the RSSs model. In addition,since the efficiency of the proposed method relies not only on the time spent on searching for the FS, but also on the time for identifying the RSSs,and the time for the identification of the RSSs also needs to be characterized by an equivalentNss,which is obtained by dividing the actual physical time for the RSSs identification by that for evaluating the FS of a single slip surfaces. For example,it takes about 163 s to identify the RSSs of the slope by a desktop computer with 32 GB RAM and 12 processors of Intel(R) Core(TM) i7-8700 CPU clocked at 3.20 GHz in this study, while it requires about 0.001 s to evaluate the FS of a single slip surface with the same computer. This indicates that the equivalentNssfor the identification of RSSs is abut 163/0.001163,000. Then, the computation time for different methods can be effectively quantified by a same nominal index with the above method.Fig.9 plots the bar chart for the computation time of different methods.It is observed from the figure that the proposed RSSs-RSM-MCS method has the smallestNssvalue among all methods, suggesting a high efficiency of the proposed method in evaluating the reliability of the slope.Furthermore, compared with the PSSs-based methods, the RSSsbased methods are more efficient. Although this observation is expected,it indicates that many computation resources are wasted in the search for the FS among the non-trivial slip surfaces,and that it is necessary to identify the key failure modes of the slope within a reliability analysis framework.

Overall, since the RSM requires only 1821 evaluations of the RSSs model to calibrate the unknown coefficients and the evaluation of FS based on RSSs-RSM can be finished within few milliseconds,the proposed RSSs-RSM-MCS can be efficiently performed,as demonstrated above.In addition,there is no need to identify the RSSs and perform the RSM analysis again for different statistics of soil properties,and the proposed method can be conveniently used to conduct parametric studies, as will be shown in the next subsection.

Fig.7. Comparison of FSs obtained based on PSSs and RSSs for the undrained cohesive slope.

Table 1Reliability analysis results for the undrained cohesive slope.

4.4. Parametric studies

This subsection further investigates the influence of the ISV ofsuonPfusing the proposed method through a series of parametric studies with constantlhof 20 m but varyinglvbetween 0.5 m and 3 m. It should be noted that the main purpose here is to validate the effectiveness and robustness of the proposed method for estimatingPfconsidering the ISV, but not to investigate how the ISV affects the slope reliability results. Additionally, for simplicity, only the influence oflvis considered becauselhgenerally has little influence onPf.Fig.10 plots the variation ofPfwithlvfor different methods. From Fig.10,Pfobtained by the proposed method generally increases with increasinglv,which is consistent with those obtained by direct MCS(i.e.PSSs-MCS)and other methods in the literature(Jiang et al.,2015;Liu et al.,2018).This consistence among different methods thus suggests that the developed method is able to produce reasonable evaluations ofPffor different soil properties. Note that it is not needed to construct response surface for different ACDs, thus the proposed method is especially suitable for such parametric studies and is robust against the ISV ofsu.

5. Illustrative example #2: Application to a cohesivefrictional slope

Fig. 8. Comparison of FSs obtained based on RSSs and RSSs-RSM for the undrained cohesive slope.

This section checks the accuracy and efficiency of the proposed method for evaluating the reliability of a cohesive-frictional (c-)slope, which has also been widely studied in the literature (Cho,2010; Li et al., 2015; Liu et al., 2017, 2018). The cross-section of the slope is plotted in Fig.11, which has a height of 10 m and an angle of 45. The slope is comprised of a homogeneous cohesivefrictional soil layer with a thickness of 15 m and a unit weight of 20 kN/m3.The statistics of the shear strength parameters are given in Table 2,which are the same as those used by Cho(2010),Li et al.(2015) and Liu et al. (2017, 2018).

5.1. Deterministic slope stability analysis

Similar to the first example, the geometry of the slope is firstly divided into 1210 random field elements to reflect the ISV ofcand. The random field discretization of the slope is shown in Fig.12,which consists of 1190 quadrilateral elements and 20 triangular elements.Then,random field simulation is conducted based on the statistical parameters given in Table 2 and the centroids of the random field elements using the CDM described in Section 2.1 to model the ISV ofcand.Fig.12a and b shows a typical realization of the cross-correlated random fields ofcand, respectively. It is observed from Fig.12 that bothcandsignificantly vary with locations, which is signified by different colors e darker color indicating larger shear strength values and lighter color representing smaller shear strength values. The shear strengths change more rapidly in vertical direction than they do in horizontal direction,becauselhis larger thanlv(i.e. 20 m vs. 2 m), suggesting a reasonable simulation for the ISV ofcand.Comparing Fig.12a and b, it is found that when the value ofcis larger, the value ofbecomes smaller,and vice versa,becausecandhere are modeled as cross-correlated random fields with a negative correlation coefficient of0.5, as shown in Table 2.

Slope stability analysis is then performed for this slope example using the BSM with the mean values of shear strengths, and FS is calculated as 1.205. The result is very consistent with those reported in the literature (Cho, 2010; Li et al., 2015; Liu et al., 2018).The corresponding critical slip surface identified from 9261 PSSs is almost consistent with that given by Li et al. (2015) and Liu et al.(2018), as shown in Fig.11. The similar results thus verify the accuracy and feasibility of the deterministic slope stability model so that the model can be used for the following analysis.Note that this slope stability model searches for the FS among more PSSs than that in the previous example, which could be more timeconsuming, especially when MCS is involved. Again, to enhance the efficiency for slope reliability analysis,this model here is mainly used as a baseline model to identify the RSSs and provide reference results for validation of the proposed method. In addition, similar technique terms as those employed for the previous slope example are used again in the following analysis, i.e. PSSs model denoting the original model that uses PSSs to search for FS,and RSSs model signifying the modified model that uses RSSs to search for FS.

Fig. 9. Comparison of computation efficiency of different methods.

Fig.10. Influence of spatial variability of su on Pf.

5.2. Identification of representative slip surfaces

This subsection identifies the RSSs from the PSSs model established above using the proposed adaptiveK-means clustering method.Following the method described in Section 4.2,the values ofKminandKmaxare set as 3 and 200,respectively.Fig.13 plots theDUNNvalues for variousKvalues.It is seen from the figure that theDUNNvalue varies withK, which shows thatKhas a significant influence on the clustering results. It is also observed from Fig.13 that theDUNNindex reaches the maximum value (i.e.1.66104) whenKis selected as 14, signifying the best cluster number for PSSs.Then,within each cluster,the slip surface with the minimum FS evaluated at the mean shear strengths is identified as the RSS in that cluster,and finally there are a total of 14 RSSs for the slope, as shown in Fig.14. The critical deterministic slip surface is also included in the RSSs. To gain more insights into the proposed method,typical clusters of RSSs forK14 are also plotted in Fig.15,which shows that different clusters are comprised of different numbers of slip surfaces and the failure modes in a same cluster are almost the same. For example,the slip surfaces in the 12th cluster are nearly all in shallow failure modes,while the slip surfaces in the 13th cluster belong to deep failure modes. This indicates the reasonability of the proposed method for identification of RSSs.

Fig.11. Cross-section of the cohesive-frictional slope with 9261 potential slip surfaces(modified after Liu et al., 2018).

Table 2Statistics of shear strength parameters for example #2.

To validate the effectiveness of the RSSs identified by the proposed method,Fig.16 compares the FSs obtained based on PSSs and RSSs for 10,000 sets of cross-correlated random field samples ofcandfor the statistics in Table 2. It is found that the FSs from the two models match well with each other, suggesting that the proposed method for RSSs identification is reasonable and accurate in estimating the FS of the cohesive-frictional slope. Comparing the PSSs model that consists of 9261 slip surfaces and those often adopted in the literature(Liu et al.,2018),the RSSs model searches for the FS of the slope among only 14 slip surfaces,indicating about an increase of 661.5 times in efficiency. In the following, probabilistic analysis of the slope stability for this example is conducted to further validate the accuracy and efficiency of the proposed method.

5.3. Probabilistic slope stability analysis based on RSSs

Fig.17 shows the accuracy of the proposed RSSs-RSM for evaluating the FS of the slope by comparing the FSs evaluated based on RSSs and RSSs-RSM for the above-mentioned 10,000 sets of crosscorrelated random field samples ofcandaccording to the statistics of shear strengths in Table 2. It is seen from Fig.17 that the FSs from the two methods are nearly consistent with the 1:1 line,suggesting a good accuracy of the RSSs-RSM.Therefore,MCS can be performed directly based on the RSSs-RSM to evaluate the reliability of the slope.

Fig.12. Discretization of random field elements for the cohesive-frictional slope and a typical realization of the cross-correlated random fields of c and based on the statistics of shear parameters in Table 2.

Fig.13. Variation of DUNN with K for the cohesive-frictional slope.

Table 3 displays the reliability results of the cohesive-frictional slope evaluated from this study and reported by Cho (2010), Li et al. (2015) and Liu et al. (2018). According to the table, the result(i.e.1.95102)obtained by the proposed method matches well with the value of 1.6102estimated by MCS,and agrees well with the results(i.e.between 1.18102and 1.87102)reported by Cho (2010), Li et al. (2015) and Liu et al. (2018). These comparable results validate the accuracy of the proposed method.

To illustrate the efficiency of the proposed method, the same indexNssas that used in the previous slope example is calculated.The result is plotted in Fig.18,where the computation time of other methods is also quantified for comparison purpose. Again, it is observed from the figure that the proposed RSSs-RSM-MCS method has the lowestNssvalue among all methods, suggesting a high efficiency of the proposed method for evaluating the reliability of the slope.

5.4. Parametric studies

This subsection further investigates the influence of the ISV ofcandonPfusing the proposed method through a series of parametric studies withc,COVc,COVandlvvarying in[0.7,0.5],[0.1,0.7], [0.05, 0.2] and [0.5 m, 3 m], respectively. For simplicity and consistent comparison with the results in the literature (Li et al.,2015), in each parametric study, only one parameter is changed,whereas the others are kept the same as those in the nominal case,wherec0,COVc0:3,COV0:2,lh20 m andlv2 m.

Fig.14. Schematics of representative slip surfaces for the cohesive-frictional slope.

Fig.15. Typical clusters of potential slip surfaces for K 14 for the cohesive-frictional slope.

Fig.16. Comparison of factors of safety obtained based on potential and representative slip surfaces for the cohesive-frictional slope.

Fig.17. Comparison of factors of safety obtained based on representative slip surfaces(RSSs) and RSSs-based response surface method for the cohesive-frictional slope.

Table 3Reliability analysis results for the c- slope.

Table 3Reliability analysis results for the c- slope.

Note: RC is the reduced cohesion, and RFA is the reduced friction angle.

Method Pf COV (%) Source PSSs-MCS (10,000) 1.6 images/BZ_230_736_1371_757_1398.png 10images/BZ_230_797_1365_813_1384.png2 7.84 This study RSSs-MCS (10,000) 1.7 images/BZ_230_736_1406_757_1434.png 10images/BZ_230_797_1400_813_1420.png2 This study PSSs-RSM-MCS (10,000) 2.34 images/BZ_230_752_1442_773_1469.png 10images/BZ_230_813_1436_829_1456.png2 This study RSSs-RSM-MCS (10,000) 1.95 images/BZ_230_752_1478_773_1505.png 10images/BZ_230_813_1472_829_1491.png2 This study MCS (50,000) 1.71 images/BZ_230_752_1514_773_1541.png 10images/BZ_230_813_1507_829_1527.png2 3.39 Cho (2010)MRSM (50,000) 1.87 images/BZ_230_752_1549_773_1576.png 10images/BZ_230_813_1543_829_1563.png2 3.24 Li et al. (2015)MCS (10,000) 1.6 images/BZ_230_736_1585_757_1612.png 10images/BZ_230_797_1579_813_1598.png2 7.84 Liu et al. (2018)EQP þ RC þ MCS (10,000) 1.35 images/BZ_230_752_1621_773_1648.png 10images/BZ_230_813_1614_829_1634.png2 8.55 Liu et al. (2018)EQP þ RC þ MRSM þ MCS (10,000) 1.36 images/BZ_230_752_1656_773_1683.png 10images/BZ_230_813_1650_829_1670.png2 8.52 Liu et al. (2018)EQP þ RFA þ MCS (10,000) 1.18 images/BZ_230_752_1692_773_1719.png 10images/BZ_230_813_1686_829_1705.png2 9.15 Liu et al. (2018)EQP þ RFA þ MRSM þ MCS (10,000) 1.18 images/BZ_230_752_1728_773_1755.png 10images/BZ_230_813_1722_829_1741.png2 9.15 Liu et al. (2018)

Fig.18. Comparison of computation efficiency of different methods.

Fig. 19 plots the variation ofPfagainst the cross-correlation coefficient for different methods. Generally, the results obtained by the proposed method increase with the increase ofc,which is the same as those of other methods in the figure.Furthermore,the results obtained from the proposed method are in good agreement with those calculated by PSSs-MCS and reported in the literature,indicating the accuracy of the proposed method. Although the results at small values ofcare relatively large, they are still in the same magnitude of order. The results show that the proposed method is able to consider the effect ofconPfand is robust against the variation ofc.

Fig.19. Variation of Pf with c.

Fig. 20. Variation of Pf with (a) COVc and (b) COV.

Fig. 21. Influence of spatial variability on Pf.

Fig. 22. Geometry and random field mesh for the layered slope.

Fig. 20a and b shows the variations ofPfwithCOVcandCOV,respectively, for various methods. From the figure, the results obtained from the proposed method are generally consistent with those obtained from PSSs-MCS and reported in the literature.It is observed that the proposed method is still accurate enough at small probability levels (e.g. in the order of magnitude of 104). For example, whenCOV0:05,Pfis estimated as 9.8104using the proposed method,which is comparable with the “exact”value of 9.33104obtained by PSSs-MCS. Furthermore,Pfis much more sensitive toCOVthan toCOVc,which is the same as those in the literature.These observations thus validate the accuracy and robustness of the proposed method versus various values ofCOVandCOVc.

Fig. 21 shows the variations ofPfassociated with different methods with respect to various vertical ACDs. It is observed that the results evaluated by the proposed method match well with those obtained from PSSs-MCS and reported in the literature for the considered ranges oflv. This indicates that the proposed method is accurate in evaluatingPfof slopes characterized by different degrees of ISV. In addition, it is also captured by the proposed method that the vertical ACD is very sensitive toPf,which is in good accordance with those in the literature (Li et al.,2015). This thus validates the robustness of the proposed method against the ISV.

6. Illustrative example #3: Application to a layered slope

This section further applies the proposed method to the reliability analysis of a two-layered cohesive slope with a relatively low level of failure probability. The slope has been previously studied by Jiang and Huang (2016) and Jiang et al. (2017) considering different cases.The slope consists of two undrained cohesive soil layers, and the slope geometry is plotted in Fig. 22. As seen from the figure, the slope has a height of 24 m and an angle of 36.9. Following Jiang et al. (2017), both the undrained shear strengths (i.e.su1andsu2)of the two soil layers are assumed to be lognormally distributed random fields.The random field ofsu1has a mean value,msu1,of 120 kPa and a COV,COVsu1,of 0.3.The random field ofsu2has a mean value,msu2, of 160 kPa and a COV,COVsu2;of 0.3. Both the two soil layers have a unit weight of 19 kN/m3. As a reference, a deterministic slope stability analysis of the slope is firstly conducted using the BSM with the mean values of the undrained shear strengths.The minimum FS of the slope is calculated as 1.993, which is the same as that reported in Jiang et al. (2017).The critical slip surface,searched from 7436 PSSs,is also identified as the same as that located by Jiang et al. (2017), as plotted in Fig. 22.

The proposed method is then used to identify the RSSs of the slope. Results show that when the PSSs are classified into 197 clusters,theDUNNindex reaches a maximum,indicating 197 RSSs being identified by the proposed method. As a reference, the variation ofDUNNwithKand the 197 RSSs are plotted in Fig.23a and b,respectively. Following the same procedure implemented in the previous two examples, reliability analysis of this slope is subsequently performed based on the 197 RSSs. Note that, for a consistent comparison with the results obtained by Jiang et al.(2017),the random field mesh is the same as that used by Jiang et al.(2017),as shown in Fig.22;and the ACF is selected as the squared exponential ACF as

Fig. 23. Results of representative slip surfaces for the layered slope.

where dh40 m and dv4 m are the horizontal and vertical scale of fluctuations that have similar physical meanings aslhandlv, respectively. The failure probability estimated by the proposed method (i.e. RSSs-RSM-MCS) with the same MCS sample size (i.e.2107)used by Jiang et al.(2017)is 1107,which is consistent with the results of 1:38107, 1107, and 1:14107from subset simulation (SS), RSM-MCS and RSM-SS, respectively, which are provided by Jiang et al. (2017). This consistency thus verifies that the proposed method is also able to deal with slope reliability analysis problem with a relatively small level of failure probability.Regarding the computation efficiency, since the number of RSSs is much less than that of the PSSs, which reduces significantly the time for searching for the minimum FS for each time of running the deterministic slope stability model, the computation time for the training of RSM based on RSSs can be greatly reduced, compared with the training of RSM based on PSSs. In addition, the identification of the RSSs is very efficient.Therefore,it might be expected from the above quantitative analyses for the previous two examples that the proposed method would be more efficient than the PSSs-RSM-MCS and other direct simulation methods.

7. Summary and conclusions

This paper firstly proposed an efficient method to identify the RSSs of slopes with spatially variable soils using the sliding volumebasedK-means clustering approach.Then,RSM was combined with the RSSs to efficiently evaluate the reliability of slopes considering soil spatial variability. The fundamentals and implementation procedure of the proposed method were described. Three slope examples were used to illustrate and validate the proposed method for slope reliability analysis. The following conclusions can be drawn from this study:

(1) The proposed method can automatically identify the RSSs of slopes with only one evaluation of a deterministic slope stability model,and the RSSs are invariant with the statistics of different soil properties. Therefore, the proposed method is efficient and suitable for reliability analysis involving many parametric studies.

(2) The FS values evaluated based on the RSSs are generally consistent with those evaluated based on all PSSs,suggesting the effectiveness of the proposed method for identifying the RSSs.

(3) The FS values evaluated from the RSSs-RSM match well with those directly evaluated from the RSSs, and the RSM significantly increases the computation efficiency of slope reliability analysis. More importantly, the proposed method is more efficient than the commonly used RSM in the literature as many trivial PSSs are excluded when searching for the FS,which,however,would sacrifice some computation accuracy.

A series of parametric studies for slope reliability analysis considering soil spatial variability can be efficiently achieved by the proposed method, with the reliability results provided by the proposed method being comparable with those from literature and direct simulations.Therefore,the proposed method is efficient and accurate as well as robust. However, the proposed method is more complex than methods like direct MCS and RSM-MCS.Therefore, using which method for reliability analysis of a practical slope is case dependent.Regarding the proposed method,it is suggested to be used for reliability analysis of slopes with simple geometry and different levels of failure probability within the framework of LEM. Applications of the proposed method for complex slopes and combining it with other advanced numerical methods (e.g. finite element method) need further verifications,which shall be used with caution. Nevertheless, although the method is complex, it provides an alternate for slope reliability analysis and can be used with ease by engineers by considering it as a “black box”. In addition, the identified RSSs using the proposed method might be used as a reference to the location of key failure modes of a slope.

Overall, the proposed method provides a different insight into the identification of RSSs of slope, with evaluating the slope stability from a small number of slip surfaces (or RSSs) selected from different clusters of the PSSs at some sacrifice of accuracy, which belongs to an empirical approach. The RSSs identified by the proposed method, therefore, might not include several slope failure mechanisms that should be identified by fundamentally more rigorous methods (e.g. Li et al., 2017). Consequently, the RSSs identified by the proposed method shall be used with caution for further slope maintenance, which remains an open question for future study.

Declaration of Competing Interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

The work described in this paper was financially supported by the Natural Science Foundation of China (Grant Nos. 51709258,51979270 and 41902291), the CAS Pioneer Hundred Talents Program and the Research Foundation of Key Laboratory of Deep Geodrilling Technology, Ministry of Land and Resources, China(Grant No.F201801).