Efficient system reliability analysis of soil slopes using multivariate adaptive regression splines-based Monte Carlo simulation

Computers and Geotechnics 79 (2016) 41–54

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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Efficient system reliability analysis of soil slopes using multivariate adaptive regression splines-based Monte Carlo simulation Lei-Lei Liu, Yung-Ming Cheng ⇑ Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong

a r t i c l e

i n f o

Article history: Received 21 January 2016 Received in revised form 23 March 2016 Accepted 1 May 2016

Keywords: Slope stability System reliability analysis Monte Carlo simulation Multivariate adaptive regression splines Response surface method

a b s t r a c t System effects should be considered in the probabilistic analysis of a layered soil slope due to the potential existence of multiple failure modes. This paper presents a system reliability analysis approach for layered soil slopes based on multivariate adaptive regression splines (MARS) and Monte Carlo simulation (MCS). The proposed approach is achieved in a two-phase process. First, MARS is constructed based on a group of training samples that are generated by Latin hypercube sampling (LHS). MARS is validated by a specific number of testing samples which are randomly generated per the underlying distributions. Second, the established MARS is integrated with MCS to estimate the system failure probability of slopes. Two types of multi-layered soil slopes (cohesive slope and c–u slope) are examined to assess the capability and validity of the proposed approach. Each type of slope includes two examples with different statistics and system failure probability levels. The proposed approach can provide an accurate estimation of the system failure probability of a soil slope. In addition, the proposed approach is more accurate than the quadratic response surface method (QRSM) and the second-order stochastic response surface method (SRSM) for slopes with highly nonlinear limit state functions (LSFs). The results show that the proposed MARS-based MCS is a favorable and useful tool for the system reliability analysis of soil slopes. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Soil properties are uncertain due to the complex geologic origins of soil deposits and the various methods of exploration and testing. Thus, a conventional factor of safety (FS), which is usually calculated based on a set of deterministic soil strength parameters, does not accurately reflect the state of slope stability. For example, approximately 5% of the stabilized slopes in Hong Kong that are designed based on classical deterministic slope stability analyses, without consideration of the uncertainty in soil parameters, eventually fail [15]. For this problem, probabilistic methods provide a systematic and quantitative tool to address the effect of this uncertainty [31]. Fruitful contributions to slope reliability analyses have been achieved in recent decades, including studies by Li and Lumb [35], Christian et al. [10], Hassan and Wolff [19], Griffiths and Fenton [18], Low [38], Jimenez-Rodriguez et al. [27], Cho [7], Low et al. [39], Zhang et al. [49], Li et al. [32], Wang [44], Li et al. [36], Xu et al. [47], Gong et al. [17], Li et al. [33] and Jiang et al. [25]. These studies indicate that Monte Carlo simulation (MCS) is the most popular approach as it is conceptually simple and can provide ⇑ Corresponding author. E-mail address: [email protected] (Y.-M. Cheng). http://dx.doi.org/10.1016/j.compgeo.2016.05.001 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved.

unbiased reliability results. However, MCS is inefficient when directly evaluated based on the original deterministic stability model. To achieve an acceptable accuracy in MCS, a minimum of 104 direct deterministic stability analyses are required [42]. Hours and days are indispensable to performing a MCS, even for modern computer architecture, which is tremendously inconvenient for routine engineering design. This finding explains the low acceptance of slope reliability analysis by engineers. The response surface method (RSM) integrated with MCS is considered to be a suitable alternative for slope reliability analysis. The merit of RSM is the construction of an explicit closed-form expression between the input variables and the corresponding output responses to accurately approximate the implicit limit state functions (LSF) of a slope. Thus, time-consuming deterministic stability models are not required. Instead, MCS is performed based on the constructed RSM, which is expected to be significantly more efficient without sacrificing the evaluation accuracy. Regarding the application of RSM in slope reliability, previous studies have addressed applications such as the polynomial-based RSM [37,46], Kriging methodology [40,48,51], the artificial neural network (ANN) [6] and the support vector machine (SVM) [30]. The polynomial-based RSM has been the most prevalent application, and Wong [45] was the first researcher to establish a linear RSM

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to approximate the finite element model of a homogenous slope. However, nonlinear stability problems are prevalent in slope engineering. Thus, the quadratic response surface method (QRSM) is becoming popular for more complicated system reliability analyses, of which the validity and capacity of the method have been extensively demonstrated by published studies [24,33,34,37,46,49]. In addition, an extension of the QRSM based on the Hermite polynomial chaos expansion, which is referred to as the stochastic response surface method (SRSM), has been proven to be a powerful approximation to the LSF of a slope [23,25,31]. A recent and more detailed review of the RSM is given by Li et al. [34]. A common feature of the QRSM and the SRSM is that they pertain to the category of parametric regression methods, which must be employed with a prior assumption on the order and type of polynomials. If a response surface (e.g., a QRSM) is erroneously assumed to approximate the true LSF, which may be a multimodal function with several peaks and troughs, and the corresponding results would not be credible as expected. For a specific slope system, the degree of nonlinear property is usually an unknown for the engineers or researchers, and the QRSM is often adopted based on the rule of thumb or even intuition, which may cause unsatisfactory estimation of the probability of failure (Pf). As reported by Zhang et al. [51], the traditional QRSM may not accurately approximate the LSFs for the system reliability analysis of soil slopes. High-order SRSMs seem to perform better than their loworder counterparts when small Pf (lower than or on the order of 10
3) values are involved, as demonstrated by Li et al. [31]. Therefore, an acceptable RSM should be automatically determined by lending itself to available data samples without potentially wrong assumptions; this model is referred to as the data-driven RSM. The objective of this paper is to suggest a data-driven RSM for the system reliability analysis of soil slopes, and MARS is selected for this purpose. The application of MARS to geotechnical engineering is a relatively recent development that primarily involves soil liquefaction assessment [28,53], tunnel convergence prediction [1], and other geotechnical problems [52]. To the best of our knowledge, MARS as a tool for slope reliability analysis has not been well appreciated. It is adopted to automatically establish the response surface to approximate the implicit LSF of a slope based on a limited number of training samples generated by Latin hypercube sampling (LHS). A hundred randomly generated testing samples are obtained to validate the MARS model. Then, MCS is performed based on the constructed MARS for the system reliability analysis of soil slopes. Four examples are examined to demonstrate the capacity and validity of the proposed approach. For convenient comparison purposes, the results obtained by the proposed approach are also compared with those evaluated by QRSM, SRSM and several other available reliability approaches. The remainder of this paper is organized as follows: In Section 2, we will provide a brief review of the most popular RSMs: QRSM and SRSM. In Section 3, a detailed theory of MARS is presented. MARS-based MCS for system reliability analysis of soil slopes is introduced in Section 4. In Section 5, the applicability and validity of the proposed MARS-based MCS are illustrated using four practical examples from literatures. The conclusions are presented in Section 6.

an explicit expression between the FS and soil shear strength parameters to approximate the implicit LSF of a slope using quadratic polynomials. The quadratic polynomials without cross terms, which are commonly utilized by researchers [3,33,37,49], are defined as

FSðXÞ ¼ a0 þ

p X

ai xi þ

i¼1

p X aiþp x2i

ð1Þ

i¼1

where FS(X) is the FS for a given vector of input variables X = (x1, x2, . . . , xp); p is the number of variables; and a = (a0, a1, . . . , a2p)T is the vector of unknown coefficients. To calibrate Eq. (1), discrete training data sets must be obtained from direct evaluation of the true LSF, and the number of training samples should not be less than 2p + 1. Suitable training samples should be as small as possible and contain effective information (e.g., inflexion) about the true LSF. Design of experiments (DOE), such as central composite design and LHS can be employed as references. The calibrated QRSM can serve as a substitution of the true LSF, with probabilistic methods such as MCS, to perform the probabilistic analysis of slope stability. 2.2. SRSM Unlike the QRSM, the premise of the SRSM is to approximate the potential relationship between the FS and the soil strength parameters in terms of random variables by a polynomial chaos expansion [21,31]. The expansion usually proceeds in the following steps: First, input variables are represented by selected random variables, such as standard normal variables. Second, the FS is expressed in the form of polynomial chaos expansion with the foregoing random variables. Third, unknown coefficients in the polynomial chaos expansion are determined as described in the QRSM. Last, a reliability analysis of slope stability is performed based on the SRSM using a probabilistic approach, such as MCS. A Hermite polynomial chaos expansion, which is the most common expansion in terms of independent standard normal variables, is usually adopted. The SRSM is expressed as

FSðnÞ ¼ a0 C0 þ

i1 p p X X X ai1 C1 ðni1 Þ þ ai1 i2 C2 ðni1 ; ni2 Þ i1 ¼1

i1 ¼1 i2 ¼1

i1 X i2 p X X þ ai1 i2 i3 C3 ðni1 ; ni2 ; ni3 Þ i1 ¼1 i2 ¼1 i3 ¼1

þ  þ

i1 X i2 p X X



ip
1 X ai1 i2 ;...;ip Cp ðni1 ; ni2 ; . . . ; nip Þ

i1 ¼1 i2 ¼1 i3 ¼1

ð2Þ

ip

where FS(n) is the FS for a given vector of independent standard normal variables n ¼ ðni1 ; ni2 ; . . . ; nip Þ that represents the uncertainty in the input variables; p is the total number of random variables; a ¼ ða0 ; ai1 ; . . . ; ai1 i2 ;...;ip ÞT is the vector of unknown coefficients to be determined; and Cp() is the multi-dimensional Hermite polynomial of order p (assumed to be two by default in this study), which is expressed as 1 Tn

Cp ðni1 ; ni2 ; . . . ; nip Þ ¼ ð
1Þp e2n

@p 1 T e
2n n @ni1 @ni2 . . . @nip

ð3Þ

2. Review of QRSM and SRSM

3. MARS-based RSM

2.1. QRSM

3.1. Basic theory of MARS

For conventional engineering design, the FS of a slope is usually calculated using a limit equilibrium method (LEM) or a finite element method (FEM), which only requires the definitions of the strength parameters. The basic idea of the QRSM is to establish

MARS was introduced by Friedman [14] as a flexible statistical strategy to represent the relationship between a group of input variables and their dependent outputs. Generally, this underlying functional relationship is identified by lending itself to and

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completely depending on the observed training data sets instead of relying on specific prior presuppositions about its degree of nonlinearity, which provides greater flexibility with this method. Theoretically, MARS is established in the form of an expansion in product splines (basis functions (BFs) based on piecewise polynomials), in which the number of BFs and the associated parameters with respect to each of them is automatically identified by the training data sets as presented in this paper. The approximation to a true function f(X) using MARS is expressed as

^f ðXÞ ¼ a0 þ

M X

1

am Bm ðXÞ

ð4Þ

m¼1

where ^f ðXÞ is the MARS predictor; X = (x1, x2, . . . , xp) is the vector of input variables; am is the coefficient of the mth term in Eq. (4) obtained by the least squares method; and Bm(X) is the mth BF or spline consisting of a product of several bkm(), as shown in Eq. (5).

Bm ðXÞ ¼

which a group of submodels are generated. Then, a less computationally expensive method of generalized cross-validation (GCV) is adopted as an evaluation criterion to assess the goodness of fit of each submodel. The best MARS model is identified as the submodel that has the lowest value of GCV in the previously mentioned model subsets. GCV is defined as the mean-square residual error divided by a penalty, which is dependent on the model complexity; for a training data set with N points, it is calculated as [20]

Km Y bk;m ðxv ðk;mÞ jpk;m Þ

ð5Þ

GCV ¼ Nh

PN

i¼1 ½yi

2


^f ðX i Þ i2

ð8Þ

1
MþdðM
1Þ=2 N

where M is the number of BFs; yi is the true value at Xi; ^f ðX i Þ is the predictive value at Xi; and d is a penalizing factor with a default value of three adopted in this study according to Friedman [14]. Note that the denominator is the foregoing penalty that indicates the change of the model variance with respect to the model complexity.

k¼1

where Km is the number of bk,m(), which is a two-sided truncated power function that is referred to as the spline basis function (SBF) in the form of

bk;m ðxv ðk;mÞ jpk;m Þ ¼ ½sk;m  ðxv ðk;mÞ
t k;m Þqþ ¼ maxð0; sk;m  ðxv ðk;mÞ
t k;m Þq Þ

ð6Þ

where sk,m is the truncation direction with the value +1 or
1, xv(k,m) is the input variable that corresponds to the kth truncated SBF in the mth term of Eq. (4); tk,m is the knot that marks the end of an interval and the beginning of another interval with respect to the input variable xv(k,m); q is a non-negative parameter that is the power of SBF reflecting a different degree of smoothness of the resulting MARS estimation. To simplify the process, only piecewise linear (q = 1) and piecewise cubic (q = 3) functions are examined in this study. Note that Bm(X) can be a single SBF and the product of two or more spline basis functions; thus, MARS can approximate highly nonlinear problems, as will be subsequently illustrated. Regarding the implementation of MARS, it is achieved by a twophase process: forward selection and backward pruning. The forward phase starts with only the basis function B0(X) = 1 in the MARS model. Then, the model automatically singles out knots for each variable from the training data sets and a pair of BFs can be defined based on these knots. The candidate knots are determined from the paired BFs that yield the largest decrease in training error when they are added to the current model. Therefore, a pair of BFs is added at each step. Assume a current model with M BFs; after two BFs are added, the next model would be updated as

^f ðXÞ ¼ a0 þ

M X

To validate the proposed MARS, a quadratic function and another complicated function (from [16]) with one variable are approximated by MARS using the same training data sets. The two illustrative functions are expressed as follows:

y ¼
x2 þ x þ 0:5 ð0 < x < 1Þ y ¼ e10xðx
1Þ 2 x cosð12pxÞ ð0 < x < 1Þ

ð7Þ

^Mþ1 and a ^Mþ2 are estimated by the least squares method and where a Bl(X) is the formerly determined BF with 0 6 l 6 M. This forward process of adding BFs continues until the predefined maximum number of terms or the threshold of the training error is reached. Generally, this process will produce a very complex and overfitted model which may poorly predict other new points; however it fits the training data sets [4]. To enhance the predictive ability of the MARS model, the backward pruning phase can be employed to delete the redundant BFs that have the smallest contribution to the model. At each step, the least effective BF in the current model will be deleted, which produces a submodel with one BF less than the current model. This process is repeated until no BF is available to be deleted, with

ð9Þ ð10Þ

Fig. 1(a) and (b) presents the fitting results obtained by MARS, the results from the true analytical expressions and the results obtained by QRSM and SRSM for comparison. As show in both sub-figures, very high coefficients of determination of MARS are produced, which indicates that MARS is capable of fitting functions with very high accuracy. The QRSM performs very well when approximating Eq. (9) but performs poorly for Eq. (10), which is expected due to the underlying principle of the QRSM. The SRSM is unable to obtain satisfactory results using the second-order SRSM. Although a high-order SRSM (i.e., the sixth-order) can improve the results for the relatively simple function in Eq. (9), the results of the more complex function in Eq. (10) using the sixth-order SRSM remain unsatisfactory. Theoretically, a highorder SRSM can improve the fitting results; however, the closedform expression for very a high-order SRSM is difficult to obtain, as reported by Li et al. [31]. In this respect, MARS appears to perform well and with a relatively easy implementation. The fitting capacity of the MARS is illustrated by a more complicated case with two variables as

f ðx; yÞ ¼ sinð0:8pxÞ cosðpyÞ ðx; y 2 ½
1; 1Þ

^Mþ1 Bl ðXÞ maxð0; xj
tÞ am Bm ðXÞ þ a

m¼1

^Mþ2 Bl ðXÞ maxð0; t
xj Þ þa

3.2. Validation of MARS

ð11Þ

Fig. 2(a) shows the real surface plot based on Eq. (11), whereas Fig. 2(b)–(e) presents the surface plots fitted by the MARS, QRSM and SRSM. Each RSM is calibrated by 500 samples generated from LHS, and tested by 1000 randomly generated samples. As shown in these subplots, the coefficients of determination for the MARS with both linear and cubic BFs are very large (0.9991 and 0.9993); however, they are relatively low for the QRSM and the SRSM (0.0005 and 0.0015, respectively), even for the sixth-order SRSM (0.2760). Thus, this finding demonstrates the advantage of MARS for fitting high dimensional nonlinear problems. 4. MARS-based MCS for system reliability analysis of slopes With the uncertainty in the soil properties that are incorporated in a slope stability analysis, numerous potential slip surfaces may

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meaningful and quantitative index Pf can be adopted to assess the stability state of a slope, which is usually defined as

0.75

Z

0.70

Pf ¼ PðgðXÞ 6 0Þ ¼

Z Z

 gðXÞ60

0.65

y

2

y=-x +x+0.5 2 MARS with 10 BFs (linear splines), R =0.9998 2 MARS with 10 BFs (cubic splines), R =1.0000 2 QRSM, R =1.0000 2 Second-order SRSM, R =0.8066 2 Sixth-order SRSM, R =0.9976

0.50 0.45 0.40 0.0

0.2

0.4

x

0.6

0.8

gðXÞ ¼ min FSi ðXÞ
1 i¼1;2;...;n

1.0

(a) Curve fitting for Eq. (9)

Pf 

10x(x-1) x

y=e 2 cos(12πx) 2 MARS with 22 BFs (linear splines), R =0.9969 2 MARS with 22 BFs (cubic splines), R = 0.9988 2 QRSM, R =0.0093 2 Second-order SRSM, R =0.0960 2 Sixth-order SRSM, R =0.4186

1.0



 min FSi ðXÞ 6 1

i¼1;2;...;n

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
Pf COVPf ¼ NP f

0.0 -0.5 -1.0 0.0

N 1X I N i¼1

0.2

0.4

x

0.6

0.8

ð14Þ

where I[] is an indicator function that is equal to a unit when mini¼1;2;...;n FSi ðXÞ 6 1 and zero otherwise and N is the number of MCS samples. The estimation accuracy of Pf is highly dependent on the number of samples and it is assessed by the coefficient of variation of Pf as

y

0.5

ð13Þ

where FSi(X) is the FS of the ith potential slip surface that is obtained by a deterministic slope stability analysis method. According to Eq. (12), direct integration of the p-fold integral is not a trivial matter. As MCS can provide an unbiased estimation of Pf in a systematic manner [2], it is adopted in this study and described as

2.0 1.5

ð12Þ

where fX(X) is the joint probability density function (PDF) of the uncertain input variables X and g(X) is the LSF of the considered slope, which is formulated such that the slope is unstable if gðXÞ 6 0 and stable otherwise, as defined bellow

0.60 0.55

f X ðXÞdX

1.0

(b) Curve fitting for Eq. (10) Fig. 1. Curve fitting for Eqs. (9) and (10) using MARS, QRSM and SRSM.

exist in a slope reliability analysis. An event of slope failure occurs when a slope slides along any individual slip surface, which can be considered as one failure mode of the slope. Thus, multiple failure modes may exist for a slope due to the potential existence of numerous possible slip surfaces. The total probability of the failure of a slope should be calculated by considering all the possible failure modes in a systematic manner. In this study, this probability is referred to as the system failure probability [9,12,41]. As reported by Cornell [11], the system failure probability of a slope can exceed the probability of failure of the slope that slides along any single slip surface. Therefore, the reliability of a slope should be evaluated in a systematic manner, especially for layered soil slopes [49]. In this section, the proposed MARS will be integrated with MCS to address the system reliability problem. The fundamentals of MCS and the procedure of the proposed MARS-based MCS are detailed in the next sections. 4.1. Probabilistic slope stability analysis based on MCS In contrast to the traditional deterministic slope stability analysis, which considers the soil shear strength parameters as constants, probabilistic slope stability analysis considers the random properties of these parameters. Therefore, the shear strength parameters comprise a vector of random variables X = (x1, x2, . . . , xp) that is subjected to specific distributions (e.g., normal and lognormal distributions), and p is the number of random variables that are considered. Based on the effect of uncertainty, the

ð15Þ

For example, if the system reliability of a slope is 0.01, N should be at least on the order of 104 to obtain an ideal estimation with a coefficient of variation of Pf (COVPf ) less than 0.1 [42], which would require a minimum of 107 runs of the deterministic stability analysis if 1000 potential slip surfaces are involved. However, directly running the deterministic stability model (particularly for a FEM model) for such a large number would be prohibitively timeconsuming and expensive. Thus, MARS-based MCS is suggested to improve the efficiency in this study. 4.2. MARS-based MCS for system reliability analysis of slopes The underlying principle of MARS-based MCS is easily understood and is implemented in a two-phase process. First, MARS is adopted to construct a RSM to approximate the implicit relationship between the FS and the shear strength parameters of a slope. An explicit LSF of a slope can be formed and employed as a surrogate of the original deterministic stability model. Second, MCS is performed by evaluating the deterministic slope stability analysis based on the MARS-based RSM instead of the original deterministic stability model. Compared with the direct MCS, the evaluation time that is required by MARS-based MCS is significantly reduced as only a small number of runs of the original deterministic stability analysis (e.g., FEM) are required for establishing the MARSbased RSM. Thus, MARS-based MCS can be considered as an improvement to the direct MCS, which can be a useful tool for many engineers. To implement MARS-based MCS for the system reliability analysis of a soil slope, the procedure is summarized in the flowchart in Fig. 3 and detailed procedures are as follows: Step 1: Set parameters. Identify the random variables and determine their stochastics, such as distribution types, means and coefficients of variation (COVs). Define the deterministic parameters, including the slope geometry and the unit weight of soil.

L.-L. Liu, Y.-M. Cheng / Computers and Geotechnics 79 (2016) 41–54

(a) Surface plot based on Eq. (11)

(c) Surface plot based on the MARS with 53 BFs (cubic splines)

(e) Surface plot based on the second-order SRSM

45

(b) Surface plot based on the MARS with 53 BFs (linear splines)

(d) Surface plot based on QRSM

(f) Surface plot based on the sixth-order SRSM

Fig. 2. Curve fitting for Eq. (11) using MARS, QRSM and SRSM.

Step 2: Establish a deterministic stability analysis model. An initial deterministic slope stability analysis model is constructed using the mean values of the input parameters based on Bishop’s simplified method (BSM) with an in-house software or a commer-

cial program. Then, this model is saved as an input file named ‘‘BSM-FS.xml” for subsequent use in step 4. This file contains all the information required by the deterministic slope stability model and can be read and written by a text editor or note pad.

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Fig. 3. Flowchart of the MARS-based MCS for system reliability analysis of slopes.

Note that the other LEMs can also be employed depending on different situations. An extensive review of LEMs that are suitable for a slope stability analysis is provided elsewhere [13]. Step 3: Generate training samples. This step is aimed at producing training samples from the p-dimensional variable space (assume that the number of random variables is p) using LHS in combination with their stochastics. For example, if only one random variable existed, the purpose of this step is to sample several x-coordinates that would be employed to determine their corresponding y values to constitute the training data sets (x, y) in step 4 for calibrating the MARS-based RSM in step 5. As the number of training samples is critical to the accuracy and computational time of the RSM, the ideal situation is to maintain an acceptable balance between them. Specific guidelines for determining the sample size have not been established. As a rule of thumb, the sample size Nt = 10p or 15p suggested by Silvestrini et al. [43] can be referenced. To ascertain a

predefined accuracy, however, the sample size Nt is determined in an iterative manner in this study and will be described in step 6, which is considered to be an improvement to the suggestion by Silvestrini et al. [43]. Step 4: Prepare the training data sets for MARS-based RSM. Substitute the Nt generated training samples to the slope stability program to obtain the corresponding FSs and form Nt training data sets. This process is achieved based on the foregoing file ‘‘BSM-FS.xml” and with the help of the program WinBatch for batch processing, which is briefly described as follows: First, replace the mean values of the p random variables in ‘‘BSMFS.xml” with new values in the Nt generated training samples for Nt times, which produces Nt sequential files, that is, ‘‘BSMFS-i.xml” (i = 1, 2, . . . , Nt). Second, a WinBatch program is coded to control the slope stability program to ensure that it can automatically run these Nt files in a sequential manner, which will yield Nt FSs saved in the result files ‘‘BSM-FS-i.fac” (i = 1, 2, . . . , Nt). Last, extract Nt FSs from the result files ‘‘BSMFS-i.fac” (i = 1, 2, . . . , Nt). Specific information about the use of WinBatch to control the master program is provided by Jiang et al. [23]. Step 5: Calibrate the MARS-based RSM. As previously mentioned, MARS is a data-driven process that can identify the optimal knots and piecewise spline functions and is completely dependent on the training data sets. The least squares method is adopted to calculate the unknown parameters. MARS can be considered as an RSM to approximate the LSF of a slope. Step 6: Validate the MARS-based RSM. To verify the predictive ability of the MARS-based RSM, Nv randomly generated testing samples are substituted in the explicit MARS-based RSM and the original deterministic model to obtain their corresponding predictions and true FSs, respectively. Comparisons are made, and the value of the coefficient of determination (R2) is selected as an assessment index for the accuracy of the MARS-based RSM. As mentioned in step 3, a predefined accuracy of 0.95 is assumed in this study. If R2 is greater than or equal to 0.95, the established MARS-based RSM is considered to be suitable. Otherwise, it is inaccurate, and we should return to step 3 to increase the training sample size to obtain a more accurate RSM. Step 7: MCS for computing Pf. Generate N samples according to the stochastics of the random variables and substitute them to the well-established MARS-based RSM instead of the original deterministic slope stability model to obtain the predictive FSs. Count the number of failure events in which FS is less than or equal to the unit, which is denoted as Nf, and divide by N to obtain Pf according to Eq. (14). Step 8: Check the accuracy of the MSC estimation. COVPf is selected as an index to measure the confidence of the estimation by MCS; a COVPf of 0.15 is assumed in this study. If COVPf is smaller than or equal to 0.15, the estimation of MCS is acceptable. Otherwise, it is unacceptable and we should return to step 7 to increase the value of N. 5. Illustrative examples To verify the capacity and validity of the proposed approach, four multi-layered soil slopes that are characterized by system effects are considered in this section, which are numbered as Examples #1, #2, #3 and #4. Each of the four slope examples indicates a different type of research problem. Examples #1 and #2 are multi-layered cohesive slopes; the former has a general failure probability (GFP) and the latter has a relative small failure probability (SMP) on the order of magnitude of or less than 10
3. Although Example #3 and Example #4 represent the GFP problem

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and SMP problem, respectively, they also represent multi-layered c–u slope problems.

Table 1 Statistics of soil parameters (Example #1).

5.1. Applications of MARS-based MCS to system reliability analysis of multi-layered cohesive slopes

Clay 1 Clay 2 Clay 3

18.0 18.0 18.0

Undrained strength, cu (kPa) Mean

COV

Distribution

18 20 25

0.3 0.2 0.3

Normal Normal Normal

1.00 0.98 0.96 0.94 0.92 0.90

MARS with piecewise linear splines MARS with piecewise cubic splines QRSM SRSM

0.88 0.86 20

40

60

80

100

120

Clay 2 4.5

Clay 3 20

200

5.1.2. Example #2: a two-layered cohesive slope with SMP The second example is a two-layered cohesive slope, which was previously investigated by many researchers, such as Cho [8], Ching et al. [5], Low et al. [39] and Zhang et al. [51]. The crosssection of the slope is shown in Fig. 7, and the soil parameters are listed in Table 3. Based on the mean values of the shear strength parameters, the FS was calculated as 1.993 using the BSM; this value is smaller than the value of 1.997 obtained by Ji and Low [22] using the ordinary method of slice. The critical deter-

Critical deterministic slip surface (FS=1.282)

10

180

models are comparable to the results obtained by Kang et al. [29], Zhang et al. [50], and Li et al. [34] using different methods. Note that the results obtained by the QRSM and the SRSM in this study and the study of Li et al. [34] are similar as they are actually based on the same methods. This similarity exhibits the confidence in the correctness of the implementation during the calculations in this study. The MARS-based MCS can provide an acceptable estimation of the slope failure probability for this example.

9.0

0

160

Fig. 5. Variation of coefficient of determination with the training sample size (Example #1).

Clay 1

0.0

140

Number of training samples generated by LHS

13.5

Elevation (m)

Unit weight, c (kN/m3)

Coefficient of determination

5.1.1. Example #1: a three-layered cohesive slope with GFP The first example is a three-layered undrained cohesive slope that was investigated by Kang et al. [29] and Zhang et al. [50]. The cross-section of the slope is shown in Fig. 4. Table 1 lists the statistics of the shear strength parameters and the unit weight, that were employed by Zhang et al. [50] and Kang et al. [29]. Based on the mean values of the undrained strength parameters, the FS was calculated as 1.282 using the BSM, which is equivalent to the value obtained by Kang et al. [29]. The critical deterministic slip surface passes through the boundary between Clay 2 and Clay 3, which is shown in Fig. 4; 52,111 potential slip surfaces are also listed. In this example, the training samples were generated by LHS, and the initial training sample size was set to Nt = 10p = 30. MARS was initially constructed according to these training samples and subsequently validated by 100 testing samples generated by MCS. However, additional samples are necessary as the predefined accuracy (0.95) cannot be obtained. Fig. 5 shows the variation of the coefficient of determination (R2) with the training sample size based on the 100 testing samples. The accuracy of MARS significantly improves with an increase in the sample size and R2 can be asymptotically close to 1, whereas additional samples do not have a significant effect on the accuracy of the QRSM and the SRSM. To maintain a balance between the effectiveness of the MARS and the efficiency of the MARS, the optimal training sample size of 150 is selected for this example. Fig. 6 shows the fitting and predictive capability of the MARS based on 150 training samples and 100 testing samples. For comparison, the results for the QRSM and SRSM are plotted in Fig. 6. The coefficients of determination that stem from both the training samples and the testing samples are very high for the MARS and larger than the coefficients of determination obtained by the QRSM and SRSM, which demonstrates the capacity and validity of MARS. To perform MCS, 100,000 random samples were generated from a 99.73% probability interval [l
3r, l + 3r] according to Kang et al. [29] to avoid negative samples. The probabilistic analysis results from the proposed approach and the results from other references are listed in Table 2. The estimated Pf values are 0.2338, 0.2334, 0.1959 and 0.2175 for MARS with piecewise linear splines, MARS with piecewise cubic splines, the QRSM and the SRSM, respectively. For comparison, the probability of failure of this slope was also calculated by the QRSM with cross terms as 0.2183. The difference between these methods is not significant; from a conservative point of view, the results obtained by the proposed MARS

Soil layers

30

40

50

Horizontal distance (m) Fig. 4. The geometry of the slope and 52,111 potential slip surfaces (Example #1).

60

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1.8

2

MARS with linear splines, R =0.9991 2 MARS with linear splines, R =0.9987 2 QRSM, R =0.8770 2 SRSM, R =0.9349 1:1 line

Predictied factor of safety

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Calculated factor of safety

(a) 150 training samples 1.7

Predicted factor of safety

1.5

1.3

2

MARS with linear splines, R =0.9859 2 MARS with cubic splines, R =0.9867 2 QRSM, R =0.9163 2 SRSM, R =0.9543 1:1 line

1.1

0.9

0.7

0.5 0.5

0.7

0.9

1.1

1.3

1.5

1.7

Calculated factor of safety

(b) 100 testing samples Fig. 6. Comparison between FSs predicted by different RSMs and FSs calculated by the BSM in Example #1.

ministic slip surface was determined to pass through the bottom of Layer 2, and the location of the critical deterministic slip surface and 9261 potential slip surfaces are schematically depicted in Fig. 6. In this example, the initial training sample size was set to Nt = 10p = 20. MARS was constructed according to these training samples and subsequently validated by 100 randomly generated testing samples. However, the predictive ability of the MARS model is poor with a relatively low coefficient of determination (0.6816 and 0.6745 for linear splines and cubic splines, respectively). Thus, additional samples are needed to enrich the training samples. Fig. 8 shows the variation of coefficient of determination (R2) with the training sample size based on the 100 testing samples. The accuracy of MARS significantly improves with an increase in the sample size and R2 can be asymptotically close to 1, whereas additional samples do not have a significant effect on the accuracy of the QRSM and the SRSM. To maintain a balance between the effectiveness of MARS and the efficiency of MARS, the optimal training sample size of 150 is selected for this example. Fig. 9 shows the fitting and predictive capability of the MARS based on the 150 training samples and 100 testing samples. For comparison, the results by the QRSM and SRSM are plotted in Fig. 9. The coefficients of determination that stem from both the training samples and the testing samples are very high for the MARS and larger than the coefficients of determination obtained by the QRSM and SRSM, which demonstrates the capacity and validity of MARS. The probability analysis of this slope was subsequently performed based on the constructed MARS-based RSM; the results and the results from other references are listed in Table 4. According to Table 4, the estimation of the failure probability (4.60  10
3 for MARS with linear splines or 4.40  10
3 for MARS with cubic splines) using the MARS-based MCS in this study is consistent with the estimation (approximately 4.58  10
3 and 4.70  10
3) performed by Zhang et al. [50]. It also falls between the failure probability bounds obtained by Low et al. [39] and Ji and Low [22]. The system probability of failure of this slope was also evaluated using importance sampling by Ching et al. [5] and MCS by Cho [8]. The results (4.40  10
3 and 4.15  10
3 for IS and MCS, respectively), which can be considered as the exact solutions, indicate that the MARS-based MCS can accurately estimate the system failure probability of this slope. Conversely, the results obtained by the QRSM and the SRSM are not acceptable. The QRSM indicates that the failure probability of

Table 2 Probabilistic results obtained by different methods (Example #1). Methods

Pf (%)

COV (%)

Source

MCS with 10,000 samples based on lognormal distributions Extended Hassan and Wolff method with 10,000 samples MCS using GPR-based RSM with 10,000 samples MCS using GPR-based RSM with 20,000 samples MCS using GPR-based RSM with 100,000 samples SQRSM MQRSM SSRSM MSRSM MCS MARS-based MCS with linear splines using 100,000 samples MARS-based MCS with cubic splines using 100,000 samples QRSM using 100,000 samples QRSM with cross terms using 100,000 samples SRSM using 100,000 samples

18.70 18.40 18.60 18.50 18.60 21.60 18.40 21.90 24.00 19.70 23.38 23.34 19.59 21.83 21.75

2.10 2.10 2.10 1.50 0.70 0.20 0.20 53.4 52.2 9.50 0.57 0.57 0.64 0.60 0.60

Zhang et al. [50] Zhang et al. [50] Kang et al. [29] Kang et al. [29] Kang et al. [29] Li et al. [34] Li et al. [34] Li et al. [34] Li et al. [34] Li et al. [34] This study This study This study This study This study

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28

Critical deterministic slip surface (FS=1.993)

Elevation (m)

24 20

Clay 1

16 12 8

Clay 2

4 0

0

4

8

12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92

Horizontal distance (m) Fig. 7. The geometry of the slope and 9261 potential slip surfaces (Example #2).

Table 3 Statistics of soil parameters (Example #2).

Clay 1 Clay 2

2

3.0

Unit weight, c (kN/m ) 3

Undrained strength, cu (kPa)

19.0 19.0

Mean

COV

Distribution

120 160

0.3 0.3

Lognormal Lognormal

1.00

2.8

Predicted factor of safety

Soil layers

3.2

2.4 2.2 2.0 1.8 1.6 1.4 1.2

0.95

1.0

0.90

0.8 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

Calculated factor of safety

0.85

(a) 150 training samples 0.80

0.70 0.65

3.2

MARS with piecewise linear splines MARS with piecewise cubic splines QRSM SRSM

0.75

0

50

100

150

200

250

300

350

400

450

500

Number of training samples generated by LHS Fig. 8. Variation of coefficient of determination with the training sample size (Example #2).


2

this slope is 1.56  10 , which significantly overestimated the system failure probability (4.40  10
3). To further evaluate the performance of the QRSM, a response surface based on the QRSM with cross terms and the same training data set has also been established to evaluate the probability of failure of this example. The corresponding estimation (2.70  10
3) of the system probability of slope failure seems to underestimate the system failure probability. However, the SRSM slightly overestimates the system failure probability by approximately 27.7% (5.40  10
3 vs. 4.15  10
3). The reason for the inaccurate estimation by these two methods may be that the complex exponential relationship between the FS and the strength variables (represented by standard normal variables) cannot be accurately reflected by the second-order RSM. Therefore, high-order SRSMs (such as the third- and the fourth-order) are employed to estimate the system failure probability of this slope, and the results (4.50  10
3 and

3.0 2.8

Predicted factor of safety

Coefficient of determination

2.6

MARS with linear splines, R =0.9986 2 MARS with cubic splines, R =0.9983 2 QRSM, R =0.8976 2 SRSM, R =0.9535 1:1 line

2.6 2.4

2

MARS with linear splines, R =0.9930 2 MARS with cubic splines, R =0.9949 2 QRSM, R =0.8550 2 SRSM, R =0.9558 1:1 line

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

Calculated factor of safety

(b) 100 testing samples Fig. 9. Comparison between FSs predicted by different RSMs and FSs calculated by the BSM in Example #2.

4.12  10
3) improved. Thus, the proposed MARS model can capture the potentially high nonlinearity between the FS and the strength variables as it is a data-driven process that can automatically identify the critical knots and provide a reasonable estimation of the system failure probability of a slope.

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Table 4 Probabilistic results obtained by different methods (Example #2). Methods

Pf
3

4.40  10 4.36  10
3 4.15  10
3 4.32–4.41  10
3 4.02–4.11  10
3 4.58  10
3 4.70  10
3 1.72  10
2 4.60  10
3 4.40  10
3 1.56  10
2 2.70  10
3 5.40  10
3 4.50  10
3 4.12  10
3

Importance sampling Multi-point FORM MCS Ditlevsen (1979) bounds Stratified RSM Kriging-based RSM Method suggested in Zhang et al. [49] QRSM with cross terms based-MCS with 100,000 samples MARS-based MCS with linear splines using 100,000 samples MARS-based MCS with cubic splines using 100,000 samples QRSM using 100,000 samples QRSM with cross terms using 100,000 samples SRSM using 100,000 samples The third-order SRSM using 100,000 samples The fourth-order SRSM using 100,000 samples

Elevation (m)

35

COV (%)

Source

15.0 4.70 4.60 4.30 4.64 4.76 2.51 6.11 4.28 4.72 4.92

Ching et al. [5] Cho [8] Cho [8] Low et al. [39] Ji and Low [22] Zhang et al. [50] Zhang et al. [50] Zhang et al. [50] This study This study This study This study This study This study This study

Critical deterministic slip surface (FS=1.405)

Layer 1

30

Layer 2 25

Layer 3 20 20

25

30

35

40

45

50

55

60

65

70

Horizontal distance (m) Fig. 10. The geometry of the slope and 9261 potential slip surfaces (Example #3).

Table 5 Statistics of soil parameters (Example #3). Soil layers

Layer 1 Layer 2 Layer 3

Unit weight, c (kN/m3)

19.5 19.5 19.5

Cohesion (kPa)

Friction angle (degree)

Mean

COV

Distribution

Mean

COV

Distribution

0 5.3 7.2

0.3 0.3

Normal Normal

38 23 20

0.2 0.2

Normal Normal

Note: Cross-correlation between cohesion and friction angle of each layer is neglected.

5.2. Applications of MARS-based MCS to system reliability analysis of multi-layered c–u slopes 5.2.1. Example #3: a three-layered c–u slope with GFP The third example is a three-layered c–u slope, which was also examined by Ji and Low [22], Zhang et al. [50] and Kang et al. [29]. The cross-section of the slope is shown in Fig. 10, and the soil parameters are listed in Table 5. No water table is considered in this example. Based on the mean values of the shear strength parameters, the FS was calculated as 1.405 using the BSM. The same value was obtained by Kang et al. [29], and a value of 1.406 was reported in Ji and Low [22] using the Spencer method with a circular slip surface. The critical deterministic slip surface was determined to pass through all soil layers, whereas the surface located by Kang et al. [29] only passes through Layer 1 and Layer 2. The location of the critical deterministic slip surface and 9261 potential slip surfaces are shown in Fig. 10. In this example, the training samples were generated by LHS and the initial training sample size was set to Nt = 15p = 60. MARS

was initially constructed according to these training samples; the initial MARS model can attain the predefined accuracy (0.95). Therefore, no extra samples were required to enrich the training samples. Then, the MARS model was validated using 100 testing samples generated by MCS. Fig. 11 shows the fitting and predictive capability of the MARS based on 60 training samples and 100 testing samples. For comparative purposes, the results by the QRSM and SRSM are plotted in Fig. 11. The coefficients of determination that stem from both the training and testing samples are very high for the MARS and are slightly larger than the coefficients obtained by the QRSM and SRSM, which demonstrates the capacity and validity of MARS. The probabilistic analysis results of this slope based on the constructed MARS-based RSM and the results from other references are listed in Table 6. Note that the MCS samples were generated from a 99.73% probability interval [l
3r, l + 3r] according to Kang et al. [29] to avoid negative samples. The estimated Pf values are 0.0128, 0.0128, 0.0138 and 0.0155 for MARS with piecewise linear splines, MARS with piecewise cubic splines, the QRSM and

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1.7 2

MARS with linear splines, R =0.9946 2 MARS with linear splines, R =0.9947 2 QRSM, R =0.9725 2 SRSM, R =0.9884 1:1 line

Predicted factor of safety

1.5

1.3

1.1

0.9

0.7 0.7

0.9

1.1

1.3

1.5

1.7

the SRSM, respectively. In general, the estimation of failure probability using the four methods in this study are comparable with the estimation (approximately 0.0160) from Kang et al. [29], the estimation (0.0108–0.0153) from Ji and Low [22] and the estimation (0.0101, 0.0108 and 0.0133) from Zhang et al. [50]; however, the results obtained by the QRSM and the SRSM are slightly higher. Using the system failure probability value of 0.0133 obtained by Zhang et al. [50] as the exact solution, the relative errors of the proposed MARS-based MCS and the QRSM are almost equivalent (approximately 0.0376), which is smaller than the relative error of 0.1654 obtained by the SRSM. However, the QRSM and the SRSM may marginally overestimate the system failure probability. Using the same sample data sets, the probability of failure of this slope was also calculated by the QRSM with cross terms as 0.0148, which slightly overestimates the system failure probability (0.0133). The proposed MARS model, the QRSM and the SRSM can reasonably estimate the system failure probability of this slope.

Calculated factor of safety

(a) 60 training samples 2

MARS with linear splines, R =0.9661 2 MARS with cubic splines, R =0.9649 2 QRSM, R =0.9647 2 SRSM, R =0.9648 1:1 line

1.6

Predicted factor of safety

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Calculated factor of safety

(b) 100 testing samples Fig. 11. Comparison between FSs predicted by different RSMs and FSs calculated by the BSM in Example #3.

5.2.2. Example #4: a four-layered c–u slope with SMP The last example is the Congress Street Cut with four-layered soils, which has been analyzed by Chowdhury and Xu [9], Ching et al. [5], Zhang et al. [51] and recently by Jiang et al. [26] and Li et al. [34]. The cross-section of the slope is shown in Fig. 12. No water table is considered in this slope. The soil parameters that are listed in Table 7, are the same parameters utilized by Jiang et al. [26]. Note that only the uncertainty of the strength parameters of the bottom clay layers are considered in this study as the stability of this slope is not sensitive to the shear strength parameters of the first sand layer. Based on the mean values of the shear strength parameters, the FS was calculated as 1.485 using the BSM, which is similar to the value of 1.484 obtained by Jiang et al. [26] using the same method. The critical deterministic slip surface is tangential to the bottom of Clay 2; the location of the critical deterministic slip surface and 9261 potential slip surfaces are schematically depicted in Fig. 12. In this example, the initial training sample size was set to Nt = 10p = 60. MARS was constructed according to these training samples and subsequently validated by another randomly generated 100 testing samples. However, the predictive ability of the MARS model is poor with a relatively low coefficient of determination (0.9044 and 0.9034 for linear splines and cubic splines, respectively). Thus, additional samples were generated to enrich the training samples. Fig. 13 shows the variation in the coefficient of determination (R2) with the training sample size based on the

Table 6 Probabilistic results obtained by different methods (Example #3). Methods

Pf (%)

COV

Source

FORM system reliability bounds SORM system reliability bounds MCS based on stratified RSM with 50,000 samples Slide V6.0 with 20,000 samples MCS with 50,000 samples MCS on the most critical slip surface with 50,000 samples MCS on representative slip surfaces with 50,000 samples MCS using GPR-based RSM with 20,000 samples MCS using GPR-based RSM with 50,000 samples MCS using GPR-based RSM with 100,000 samples MCS using GPR-based RSM with 1,000,000 samples MARS-based MCS with linear splines using 100,000 samples MARS-based MCS with cubic splines using 100,000 samples QRSM using 100,000 samples QRSM with cross terms using 100,000 samples SRSM using 100,000 samples

1.08–1.30 1.35–1.53 1.34 1.40 1.33 1.01 1.08 1.61 1.59 1.59 1.59 1.28 1.28 1.38 1.48 1.55

3.80 4.40 4.30 5.50 3.50 2.50 0.80 2.78 2.78 2.67 2.58 2.52

Ji and Low [22] Ji and Low [22] Ji and Low [22] Ji and Low [22] Zhang et al. [50] Zhang et al. [50] Zhang et al. [50] Kang et al. [29] Kang et al. [29] Kang et al. [29] Kang et al. [29] This study This study This study This study This study

Note: Spencer method with circular slip surface is used in Ji and Low [22], while the BSM is adopted in the others.

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18 16

Critical deterministic slip surface (FS=1.485)

Sand

Elevation (m)

14 12

Clay 1

10 8

Clay 2

6 4 2 0

Clay 3 0

4

8

12

16

20

24

28

32

36

40

44

Horizontal distance (m) Fig. 12. The geometry of the slope and 9261 potential slip surfaces (Example #4).

Table 7 Statistics of soil parameters (Example #4). Soil layers

Unit weight, c (kN/m3)

Sand Clay 1 Clay 2 Clay 3

18.5 18.5 18.5 18.5

Cohesion (kPa) Mean

COV

Distribution

Mean

COV

Distribution

0 55 43 56

– 0.37 0.19 0.24

– Lognormal Lognormal Lognormal

30 5 7 8

– 0.20 0.21 0.21

– Lognormal Lognormal Lognormal

Coefficient of determination

0.98

0.96

0.94

0.92

MARS with piecewise linear splines MARS with piecewise cubic splines QRSM SRSM

0.90

0.88 60

110

160

Cross-correlation coefficient (q)

Friction angle (degree)

210

260

Number of training samples generated by LHS Fig. 13. Variation of coefficient of determination with the training sample size (Example #4).

100 testing samples. The accuracy of the MARS significantly improves with an increase in the sample size, whereas additional samples do not have a significant effect on the accuracy of the QRSM and the SRSM. To maintain a balance between the effectiveness and the efficiency of the MARS, the optimal training sample size of 210 is selected for this example. Fig. 14 shows the fitting and predictive capability of the MARS based on the 210 training samples and 100 testing samples. For comparison, the results obtained by the QRSM and the SRSM are plotted in this figure. The coefficients of determination that stem from both the training samples and the testing samples are very high for the MARS and larger than the coefficients of determination from the QRSM and the SRSM, which demonstrates the capacity and validity of MARS.

–
0.7
0.7
0.7

Probability analysis of this slope was subsequently performed based on the constructed MARS-based RSM; the results are provided in Table 8. As the geometry of the slope and uncertainty of the shear strength parameters were considered differently in the previously mentioned references; however, the results obtained by Jiang et al. [26] are provided for reference and comparison. The system failure probability of the slope was calculated as 1.00  10
3 by direct LHS with 1000 samples, which can be considered as the exact solution of this example. According to Table 8, it can be observed that the estimation of failure probability (8.50  10
4 for MARS with linear splines or 8.20  10
4 for MARS with cubic splines) using the MARS-based MCS in this study is consistent with the estimation of failure probability (8.16  10
4 and 8.10  10
4) by Jiang et al. [26] and similar to the system failure probability (1.00  10
3) obtained by direct LHS in this study. This finding indicates that the MARS-based MCS can accurately estimate the system failure probability of this slope. Conversely, the results, i.e., 2.00  10
5 and 1.00  10
5, obtained by the QRSM and the SRSM, respectively, are unacceptable and underestimate the system failure probability by an order of magnitude. Additionally, the result of 1.31  10
3 from the QRSM with cross terms overestimates the failure probability by about 31%. This overestimation is probably attributed to the potentially highly nonlinear properties of this slope. These properties involve six stochastic variables that are subjected to correlated lognormal distributions, which cause a complex power relationship between the FS and the independent standard normal variables. Therefore, the second-order RSM (i.e., QRSM and SRSM) is not capable of approximating this type of high nonlinearity, which causes an inaccurate estimation of the system failure probability. To verify this deduction, higher order SRSMs (such as the third- and fourthorder) were employed to estimate the system failure probability; the results (8.150  10
3 and 8.10  10
4) were reasonable and consistent with the results obtained by Jiang et al. [26]. The proposed MARS model can capture the potentially high nonlinearity between the FS and the strength variables as it is a data-driven process that can automatically identify the critical knots and consider the interactions among the variables to provide a reasonable estimation of the system failure probability of a slope.

L.-L. Liu, Y.-M. Cheng / Computers and Geotechnics 79 (2016) 41–54

2.0

Predicted factor of safety

1.8

2

MARS with linear splines, R =0.9986 2 MARS with cubic splines, R =0.9988 2 QRSM, R =0.9358 2 SRSM, R =0.9892 1:1 line

1.6

1.4

1.2

1.0 1.0

1.2

1.4

1.6

1.8

2.0

Calculated factor of safety

(a) 210 training samples 2.0

Predicted factor of safety

1.8

2

MARS with piecewise linear splines, R =0.9648 2 MARS with piecewise cubic splines, R =0.9649 2 QRSM, R =0.9238 2 SRSM, R =0.9334 1:1 line

1.6

1.4

1.2

1.0 1.0

1.2

1.4

1.6

1.8

2.0

Calculated factor of safety (b) 100 testing samples Fig. 14. Comparison between FSs predicted by different RSMs and FSs calculated by the BSM in Example #4.

Table 8 Probabilistic results obtained by different methods (Example #4). Methods

Pf

COV (%)

Source

The third-order HPCE + MCS The fourth-order HPCE + MCS LHS with 1000 samples MARS-based MCS with linear splines using 100,000 samples MARS-based MCS with cubic splines using 100,000 samples QRSM using 100,000 samples QRSM with cross terms using 100,000 samples SRSM using 100,000 samples The third-order SRSM using 100,000 samples The fourth-order SRSM using 100,000 samples

8.16  10
4 8.10  10
4 1.00  10
3 8.50  10
4

10.30 10.50 3.160 10.80

Jiang et al. [26] Jiang et al. [26] This study This study

8.20  10
4

11.00

This study

2.00  10
5 1.31  10
3

70.7 8.73

This study This study

1.00  10
5 8.15  10
4

99.9 11.07

This study This study

8.10  10
4

11.11

This study

53

slopes. The novelty of the proposed approach is the application of the data-driven MARS with MCS for the system reliability analysis of layered soil slopes with different failure probability levels, which may be considered as an improvement to the direct MCS. The variation to the basic MCS using MARS appears to be novel with promising results. The proposed approach provides an alternate and approximate technique for efficient and accurate system reliability analysis. The proposed methodology can be easily implemented and coupled with any stand-alone software for deterministic slope stability analysis, which provides a practical tool for practitioners of reliability based design. In this study, the MARS-based MCS method has been applied to different cases (more cases in the internal studies), which include two types of multi-layered soil slopes (cohesive slope and c–u slope), each of which include two examples with different distributions and system failure probability levels. Several observations from this study are as follows: (1) The system failure probability of a soil slope—a cohesive soil slope or a c–u soil slope—can be addressed using the proposed MARS-based MCS method. The results obtained by the proposed MARS-based MCS are similar to the results obtained from direct MCS and are comparable to or better than the results of QRSM and SRSM, which indicates the validity of the proposed method. (2) For both cohesive and c–u soil slopes, the proposed method can provide reasonable and comparative estimations of the system failure probability for slopes with general failure probability levels. The method is proven to be more accurate than the QRSM and the SRSM when addressing slopes with highly nonlinear limit state functions. (3) The proposed method is suitable for slopes considering different distributions with or without cross-correlations. The method can be extended to other correlated non-normal variables using the Nataf transformation; however, only the normal and lognormal variables are considered in this paper. (4) The accuracy of the proposed approach is highly dependent on the calibrating samples. In this study, the sample size can be determined in an iterative manner to obtain a reasonable accuracy instead of the classical approach in which a prescribed number is employed. (5) The proposed method suffers from the limitation that a predefined degree of interactions between variables is required. However, the method can automatically identify the knots, whereas classical methods are unable to identify the knots in these conditions. Although the number of variables that are involved in a slope is not excessive, this limitation can be overcome by a simple trial and error approach as the approach applied to the four examples in this study. The results from the various cases have demonstrated the high accuracy of the proposed method. Compared with classic Monte Carlo simulation or other RSM methods, the proposed method has demonstrated the advantage of accuracy and the requirement of an acceptable amount of computations even for highly nonlinear problems.

6. Summary and conclusions

Acknowledgements

This paper has proposed a novel method, which is referred to as MARS-based MCS, for system reliability analysis of soil

This work was supported by The Hong Kong Polytechnic University through the account RU3Y and ZVCR.

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L.-L. Liu, Y.-M. Cheng / Computers and Geotechnics 79 (2016) 41–54

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