Probabilistic evaluation of three-dimensional seismic active earth pressures using sparse polynomial chaos expansions

Computers and Geotechnics 129 (2021) 103869

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Research Paper

Probabilistic evaluation of three-dimensional seismic active earth pressures using sparse polynomial chaos expansions Zheng-Wei Li , Qiu-Jing Pan *, Rui-Zheng Fei School of Civil Engineering, Central South University, Changsha, Hunan 410075, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Probabilistic analysis 3D seismic active earth pressures Sparse polynomial chaos expansion (SPCE) Kinematical approach of limit analysis

This work presents a probabilistic analysis of three-dimensional (3D) seismic active earth pressures acting on rigid retaining walls that are designed to support cohesive-frictional soil masses. The analysis is implemented using sparse polynomial chaos expansions (SPCE) to represent the physical model responses. The deterministic evaluation of 3D seismic active earth pressures is conducted with the application of the kinematic approach of limit analysis combined with a pseudo-static method. All of the input parameters are deemed as random vari­ ables. After the SPCE metamodels are determined, the method of Monte Carle simulations is applied to perform the probabilistic analysis of 3D seismic active earth pressures. The effects of uncertainty levels and correlation relationships of input parameters are investigated. Several illustrative examples of the reliability-based design of retaining walls are shown. Results indicate that uncertainties in input parameters possess a notable effect on 3D seismic active earth pressures, suggesting the need to investigate the problem from a stochastic perspective. In most cases, the soil cohesion, internal friction angle, and seismic coefficient are found to be three of the pa­ rameters that affect the final response the most significantly. It is also indicated that considering 3D effects and soil cohesion has a favorable effect on increasing the stability of retained slopes.

1. Introduction Retaining walls are generally constructed to resist active forces. As such, the determination of active earth pressures is a core research issue in the design of retaining walls. The knowledge body regarding this topic has grown enormously and continued to do so (Chen and Liu, 1990; Soubra and Macuh, 2002; Vahedifard et al., 2015; Li and Yang, 2019; Qian et al., 2020). The majority of previous studies focused on devel­ oping deterministic computational models, whereas little attention was given to probabilistic analyses. Although these deterministic models can provide reasonable predictions in most cases, their accuracy is highly dependent upon whether input parameters can be determined in a suf­ ficiently accurate way. In practice, uncertainties in input parameters unavoidably exist (Zhang and Goh, 2013; Zhang and Goh, 2016; Wang et al., 2020). These uncertainties may result from inherent soil vari­ ability, measurement error, model uncertainty, and so forth. Deter­ ministic analyses fail to account for these uncertainties. In order to better understand the effects of these uncertainties, it is necessary to analyze problems from a stochastic viewpoint. It is well recognized that probabilistic analysis is a rational method

because it can provide a mean of quantifying the impact of input parameter uncertainties on the final response. Many different ap­ proaches have been developed for probabilistic analysis of geotechnical problems. The crudest and direct method is the Monte Carlo simulation (MCS), which accounts for the influence of input parameter un­ certainties on the final response by (i) generating a sufficiently large number of samples of random variables that follow a specified distri­ bution, (ii) then using the generated samples as input parameters to obtain the corresponding physical responses, and (iii) finally counting the number of samples falling in the failure zone to estimate failure probability (Malkawi et al., 2001; Wang et al., 2010). The main advantage of the MCS lies in its versatility and robustness, making it a useful tool to validate other probabilistic approaches. However, the MCS method is often criticized due to its low computational efficiency. The first-order and second-order reliability methods (FORM/SORM), which, respectively, adopt the first order and second order of a Taylor series expansion of the performance function, have been widely used to esti­ mate the required information regarding the physical response, such as the mean value and variance (Mollon et al., 2009; Low, 2014). Compared with the SORM, the FORM has received much more attention

* Corresponding author. E-mail address: [email protected] (Q.-J. Pan). https://doi.org/10.1016/j.compgeo.2020.103869 Received 15 January 2020; Received in revised form 10 September 2020; Accepted 10 October 2020 Available online 4 November 2020 0266-352X/© 2020 Elsevier Ltd. All rights reserved.

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because of its simplicity. The use of the SORM and FORM requires an explicit performance function, which, in most cases, seems impossible to achieve. Under this circumstance, the response surface method (RSM) is commonly adopted, which frequently makes use of a polynomial around the design point to represent the performance function (Mollon et al., 2009; Li et al., 2015). Among them, quadratic forms without cross-terms are the most widely used. After an explicit function (also referred to as the metamodel) that surrogates the actual model response is obtained, the FROM or SROM can be used to obtain the required information. Recently, a more powerful RSM termed the collocation-based stochastic surface response method (CSSRM), which uses polynomial chaos ex­ pansions (PCE) or sparse polynomial chaos expansions (SPCE) to represent a complex physical response, has gained much attention in the literature (Mollon et al., 2010; Pan and Dias, 2017a; Pan and Dias, 2018; Xu and Wang, 2019; Pan et al., 2020). The main advantage of this method is that it enables to simulate complicated physical models in a sufficiently accurate way. To date, only a limited number of probabilistic analyses on retaining walls have been performed. For example, Fenton et al. (2005) presented a probabilistic approach for reliability analysis of retaining walls in spatially variable soils based on the MCS method; a random finite element model was developed to conduct the deterministic evaluation in that paper. Low (2005) proposed a convenient procedure that can be operated in the widely used spreadsheet platform; for the reliability design of retaining walls by means of the Hasofer-Lind index and FORM. Zevgolis and Bourdeau (2010) applied the MCS method to conduct a probabilistic analysis of the external stability of cantilever retaining walls. Gao et al. (2019) made an improvement of the MCS for the full probabilistic design of retaining walls by using a generalized subset simulation, which requires significantly less computational efforts than the traditional MCS. The deterministic models adopted in the aforementioned studies have been historically performed under 2D conditions. However, the 2D analyses neglect the actual boundary conditions of geotechnical struc­ tures, resulting in an inaccurate response, especially for a retaining wall with a small width-to-height ratio. Therefore, in several circumstances, it is advisable to conduct a 3D analysis to obtain a more accurate and realistic prediction. Note that the deterministic model plays a very important and indispensable role in the probabilistic analysis. The progress of deterministic approaches can also promote the development of probabilistic analyses. Currently, several attempts have been made to ¨rden and investigate 3D active earth pressures (Huder, 1972; tom Wo Achmus, 2013; Ant˜ ao et al., 2016), of which all were conducted assuming backfills to be purely frictional. More recently, Yang and Li (2018) developed a kinematic limit analysis-based approach for computing 3D seismic active earth pressures based on a 3D rotational failure mechanism of Michalowski and Drescher (2009). In that work, the contribution of soil cohesion was considered. The method of Yang and Li (2018) is adopted in this work to conduct the deterministic analysis. The focus of this paper is on developing a probabilistic analysis of 3D seismic active earth pressures and an approach for the reliability design of retaining walls. For guidance, this paper is organized as follows. In the next section, a brief introduction about the PCE and SPCE methods, and the SPCE-based probabilistic analysis is given, followed by building a deterministic model for calculating 3D seismic active earth pressures. On this basis, an SPCE-based probabilistic analysis on 3D seismic active earth pressures is conducted, and parametric effects are discussed. This paper ends with a reliability-based seismic design of a retaining wall and remarking conclusions.

gathered in an input vector ξ = [ξ1 , ξ2 , ..., ξM ] and a scalar response Y. The model response Y can be represented by the PCE as follows (Sudret, 2008): ∑ Y = T(ξ) ≅ kα ψ α (ξ) (1) α∈NM

where ψ α (ξ) represents a set of multivariate polynomials, ψ α (ξ) = ∏M i=1 Hαi (ξi ) with Hαi (ξi ) as a univariate polynomial; α is M− tuple that contains a set of non-negative integers, α = (α1 , ..., αi , ..., αM ); αi ∈ N denotes the degree of the univariate polynomial; and kα denotes a set of unknown coefficients of the PCE needed to be determined. If the input random variables are normally distributed, the family of multivariate Hermite polynomials can be adopted as the basis of the expansion. A multivariate Hermite polynomial can be determined as the product of several univariate Hermite polynomials. Because any distribution of input random parameters can be recast as an independent standard normal distribution, the family of multivariate Hermite polynomials is applied in the present paper. For practical application, Eq. (1) should be truncated so that only a finite number of PCE terms are retained. The common truncation scheme is frequently used for a small number of random variables. In this truncation scheme, only those polynomials whose total degree does not exceed a specified order, p, are retained, namely |α| = ‖α‖1 =

M ∑

αi ⩽p

(2)

i=1

Then, the number of terms in the PCE retained, P, holds: P=

(M + p)! M!p!

(3)

The coefficients of the PCE, kα , are determined using regression methods (Sudret, 2008; Pan and Dias, 2018). For this purpose, let us choose a set of N realizations of design of experimental (DoE) denoted by { }T ζ = ξ(1) , ... , ξ(N) and compute their corresponding physical re­ { }T sponses represented by Y = Y (1) , ... , Y (N) . According to the DoE ζ and the model responses Y, a linear system is built. Then, the PCE co­ efficients can be determined using the least-square minimization method, which are: ( )− 1 ̂ k = ΨT Ψ ΨY (4) where Ψ is a N × P data matrix, calculated by: ( ) Ψij = ψ αj ξ(i) , i = 1, ... , N, j = 0, ... , P − 1

(5)

The size of the DoE, N, should guarantee the accuracy of the leastsquare minimization method, and the Latin hypercube sampling is used here to select the DoE. For computing the model response, each DoE set in the standard space should correspond to a point in the physical space. To this end, the independent standard normal variables in standard space are trans­ formed into non-normal variables in physical space using the concept of isoprobability transformation. Moreover, the correlated variables can be uncoupled by Nataf transform or Cholesky transform. 2.2. SPCE It should be emphasized that not all terms in a PCE are necessary for accuracy. Instead of the common truncation scheme, a hyperbolic truncation scheme was developed by Blatman and Sudret (2011) to reduce the PCE terms. In this scheme, a so-called q-quasi-norm is defined, and the q-quasi-norm is not bigger than the PCE order p, namely:

2. SPCE metamodeling 2.1. PCE Let us begin with a mathematical model T with M input variables 2

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( ‖α‖q =

M ∑

Computers and Geotechnics 129 (2021) 103869

)1/q

αqi

⩽p (0 < q < 1)

(6)

i=1

Such a norm penalizes those terms with high-order interactions. Note that the case of q = 1 corresponds to the common truncation scheme. A lower value of q leads to fewer terms retained in the PCE. Several studies have shown that a value of q in the range of 0.7–0.9 can provide a good balance between accuracy and sparsity (Blatman and Sudret, 2010; AlBittar and Soubra, 2013; Pan et al., 2017). Even though the use of the hyperbolic truncation scheme can partly reduce the terms in the PCE, some of the remaining terms still play a negligible role on the accuracy of the model, which should be discarded. Considering this fact, a hybrid procedure is proposed here to construct an SPCE. The PCE in Eq. (1) is first truncated by the hyperbolic trun­ cation scheme as the candidate SPCE basis, followed by the stepwise regression technique adopted to further select those remarkable terms from the candidate basis. In this procedure, four indices, including target accuracy Q2tgt , cut-off value εcut , degree pmax , and norm q, are involved.

The accuracy of the SPCE is evaluated using the leave-one-out crossvalidation, which can be found in detail in Blatman and Sudret (2010). The stepwise regression technique consists of two steps, the forward step and the backward step. The forward step is to select those significant terms resulting in a distinct increase in the coefficient of determination, R2 , i.e., greater than εcut , by introducing the candidate terms one by one; the backward step is to re-examine the significance level of all terms having been selected and discard those negligible terms if neglecting them results in an insignificant decrease in R2 , i.e., lower than εcut . This procedure is repeated until the target accuracy is met or the PCE degree reaches pmax . Herein, the target accuracy Q2tgt is set to 0.999, the cut-off

value εcut to 5 × 10− 5 , the maximum degree pmax to 7, and the norm q to 0.8. 2.3. Probabilistic analysis based on the SPCE 2.3.1. Probabilistic density function (PDF) and cumulative density function (CDF) After the coefficients of the SPCE are determined, an analytical SPCE metamodel is built to represent the actual physical response. On this basis, The MCS is used to obtain the PDFs and CDFs of the physical response. The Monte Carlo samples are obtained using the Latin hy­ percube sampling, and the size of the samples is set to 106 .

Fig. 1. 3D horn-like failure mechanism for the deterministic analysis: (a) a cross-section on the symmetry plane; (b) a 3D view (after Yang and Li, 2018 and Michalowski and Drescher, 2009).

2.3.2. Global sensitivity analysis The Sobol’ indices have been extensively adopted for global sensi­ tivity analysis, which can reflect the importance of different randomness of input parameters. The Sobol’ index of a variable indicates the contribution of the uncertainty in this variable to the total variance of the final response. A random variable with a greater Sobol’ index means a more significant effect on the final response. The Sobol’s indices are adopted in this work for global sensitivity analysis. The methodology for calculating the Sobol’ indices based on the SPCE method was reported in the work by Blatman and Sudret (2010). Note that this method is applicable only when the random variables are independent.

introduced, and seismic effects are then replaced by an inertia force whose magnitude is calculated as a fraction kh of the soil unit weight. The present paper adopts the kinematic limit-analysis method to conduct a deterministic evaluation of 3D active earth pressures. Due to its simplicity and ability to consider some complex factors in a simple manner, the approach has attracted much attention in stability analyses of geotechnical structures (Chen, 1975; Chen and Liu, 1990; Micha­ lowski and Drescher, 2009; Pan and Dias, 2017b; Yang and Li, 2018; Li and Yang, 2019; Qian et al., 2020). This approach yields the required solution from the equilibrium of the rates of external work and internal dissipation in an arbitrary valid failure mechanism. Therefore, the use of this approach concerns two main issues to be addressed: the construc­ tion of a kinematically admissible failure mechanism and the derivation of the work rate balance equation. These two issues are dealt with in the sequel.

3. 3D seismic active earth pressure determinations As shown in Fig. 1, a retaining wall with height H and width B is designed to support a slope with a sloping angle of α, and an inclination angle of β. The backfill has an internal friction angle of φ, a cohesion level of c, and a unit weight of γ. The soil-wall interface is considered to be rough with a friction angle of δ. For seismic design, seismic effects are included in the analysis by using a pseudo-static method. Because the vertical component of seismic effects has a significantly lower influence, it is neglected herein and only the horizontal component is considered. In the pseudo-static method, a horizontal seismic coefficient, kh , is

3.1. 3D horn-like failure mechanism This section is concerned with the construction of a valid failure mechanism used to capture the collapse of cohesive-frictional soil masses. In the framework of the kinematic approach, Michalowski and Drescher (2009) presented a horn-like failure mechanism for 3D 3

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stability assessment of slopes. This mechanism can capture slope failures more realistically and has inspired many subsequent investigations on 3D slope analyses (Nadukuru and Michalowski, 2013; Gao et al., 2014; Li et al., 2020). More recently, Yang and Li (2018) extended this mechanism to calculate 3D seismic active earth pressures in cohesive backfills. The mechanism is of rotational type about an axis passing through the rotation center, O, with an angular velocity of ω, as depicted in Fig. 1. In a rigid rotation, the kinematical admissibility requires that the ′ lower and upper contours of the mechanism, r and r , should be two logspiral curves. For the ease of description, a polar coordinate system (θ, ρ) is set up in the rotation center, O. Under this coordinate, the two logspiral curves take the following forms:

′

′

r = r0 e −

Be =

Ka =

where θ0 is the initial angle of the mechanism, and r0 and r0 are the ′ corresponding values of r and r at θ = θ0 . The mechanism is generated by revolving a circle defined by the two log-spiral curves. The center of the revolving circle lies in the center line of the mechanism, rm , which reads: ′

rm =

r+r 2

′

r− r 2

(9)

′

(10)

In cases where the width of a retained slope significantly exceeds its height, the 3D mechanism should be able to be simplified to a 2D one. To this end, the mechanism is enriched by incorporating a 2D plane-strain block into two halves of the 3D surface (Michalowski and Drescher, 2009; Nadukuru and Michalowski, 2013; Gao et al., 2014; Yang and Li, 2018; Li et al., 2020), as shown in Fig. 1(b). For a general 3D case, the width of the inserted block, B2D , is calculated by: ′

B2D = B − Bmax

A H/sinβ

(14)

2Pa γH 2 Be

(15)

The application of the kinematic approach to active earth pressure calculations will yield lower bounds to the actual solution. Therefore, the maximum value of all possible outcomes, which is closest to the actual solution, is deemed as the optimal solution and searched using an optimization scheme. The optimization scheme involves three variables that are used to define the failure mechanism, i.e., angles θ0 and θh , and ′ ratio r0 /r0 . A mixed optimization algorithm is proposed to obtain the most critical solution in the Matlab platform. A genetic algorithm-based program is firstly coded to obtain the initial values of the optimization variables near the optimum point, followed by a more accurate NelderMead simplex algorithm used to locate the optimum point. After opti­ mization, the final solution is a function of parameters regarding soil properties (c, φ, δ, γ), seismic actions (kh ), and wall geometry (β, H, B/ H). Therefore, the objective function of the active earth pressure co­ efficients can be written in the following form: ) ( ′ Ka = f c, φ, δ, γ, kh , β, α, H, B/H | θ0 , θh , r0 /r0 (16)

and the circle radius, Rc , is calculated by: Rc =

(13)

where A is the area of the soil-wall interface covered by the failure mechanism. A detailed derivation for Be can be found in Yang and Li (2018). For the ease of practical use, the resultant of active earth pres­ sure is usually expressed using an unfactored coefficient, which is also referred to as the active earth pressure coefficient, Ka . In the current analysis, the coefficient is calculated by the following equation:

(8)

(θ− θ0 )tanφ

Wγ + Ws − D WPa /Pa

From Fig. 1(b), it is observed that the failure mechanism adopted in this work does not cover the whole soil-wall interface. For a more derivation, a new parameter termed as the equivalent width, Be , is introduced and defined as follows (Yang and Li, 2018):

(7)

θ0 )tanφ

r = r0 e(θ−

Pa =

4. Probabilistic analysis

(11)

where Bmax is the maximum width of the 3D portion.

4.1. Input random variables

3.2. Active earth pressure coefficient and optimization

In practice, the input parameters often possess various types of randomness and uncertainty. The model used to determine 3D seismic active earth pressure coefficients, as given in Eq. (16), involves nine input parameters. For more realistic consideration, all of these input parameters are deemed as random variables. For the ease of analysis, the most widely used distribution, normal distribution, is adopted in this work. This kind of distributions can be completely defined by the mean value (μ) and coefficient of variation (COV). Table 1 lists the mean values of the input parameters, which correspond to a cohesivefrictional backfill that requires support to maintain stability, where Ka = 0.3656. This work considers three kinds of scenarios, where different COVs of input parameters are adopted, as shown in Table 1. The representative ranges for the COVs of c, φ, γ and kh can be found in previous studies (Phoon and Kulhawy, 1999; Phoon and Kulhawy, 1999; Duncan, 2000; Basha and Babu, 2009), which are 0–40%, 5–15%, 3–7%, and 0–40%, respectively. The neutral uncertainty levels of c, φ, γ and kh correspond to the intermediate values of their representative ranges; the COVs of c, φ, γ and kh for the Optimistic Scenario (or Pessimistic Sce­ nario) are obtained by reducing (or increasing) by 10%, 5%, 2%, and 10%, respectively, from those values of the Neutral Scenario. The COV of δ is set to 10% for the Neutral Scenario, which is the same as that in Low (2005) and Gao et al. (2019); the COVs of δ for the Optimistic Scenario and Pessimistic Scenario are obtained by reducing and increasing by 5%, respectively, from this value. The remaining input parameters, i.e., β, α,

′

In the framework of the kinematic approach, the solution is obtained using the work-energy balance equation, which, in this work, takes the following form: Wγ + Ws − WPa = D

(12)

where Wγ , Ws , and WPa denote the rates of work created by the selfweight of the collapsing block, seismic forces, and wall reaction of the active earth pressure, and D is the rate of internal energy dissipation. The minus sign on the left-hand side of Eq. (12) indicates that the re­ action of active earth pressure resultant does negative work. It should be emphasized here that the resultant force of active earth pressure is assumed to act at a height of H/3 higher than the wall toe, which is a rather common assumption in active earth pressure calculations (Yang, 2007; Vahedifard et al., 2015; Yang and Li, 2018; Li and Yang, 2019; Qian et al., 2020). It was indicated that this assumption is reasonable for purely-frictional backfills and yields a more conservative outcome for cohesive backfills (Qian et al., 2020). The detailed expressions for these work rates can be found in several relevant publications (Michalowski and Drescher, 2009; Yang and Li, 2018). The resultant of active earth pressure can then be formulated as follows:

4

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Table 1 Mean values and COVs of input parameters used in the parametric study and Illustrative Example II. Input variable

μ

(a) 10 9

COV (%)

8 Neutral Scenario

Pessimistic Scenario

7

5 25

10 5

20 10

30 15

6

15

5

10

15

γ(kN/m )

18

3

5

7

kh

0.15

10

20

30

β(◦ )

90

1

3

5

α(◦ )

10

1

3

5

H (m) B/H γwall (kN/m3)

6 3 24

1 1 3

3 2.5 5

5 4 7

ca (kPa)

100

10

15

20

a (m) b (m)

0.4 –

1 1

3 3

5 5

c (kPa) φ(◦ ) δ(◦ )

3

PDF

Optimistic Scenario

SPCE MCS

Optimistic scenario Neutral scenario

Pessimistic scenario

5 4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

Ka

0.5

0.6

0.7

0.8

1

(b) 10

H, and B/H, are usually considered to have relatively low COVs, because they are easy to be measured. The COVs of these parameters for the three scenarios are taken from Li et al. (2012) and Pan et al. (2017).

SPCE MCS

100

4.2. Parametric study

10-1

PDF

This section aims to investigate the effects of uncertainty levels of input parameters and correlation relationships on the 3D active earth pressure coefficient. For this purpose, a series of calculations are per­ formed using the proposed approach. Based on the obtained results, a parametric study is performed. Figs. 2 and 3 show the effects of uncertainty levels of random vari­ ables on the PDF and CDF curves of the 3D seismic active earth pressure coefficient. The curves are plotted using the kernel smoothing tech­ nique. Three different scenarios representing different uncertainty levels of input parameters, as given in Table 1, are all considered. To validate the proposed approach, the method of MCS is also used to obtain the PDF and CDF curves of the three scenarios, with the same number of samples. From Figs. 2 and 3, it is observed that the PDF and CDF curves calculated by the proposed approach are almost overlapped with those obtained using the MCS. This means that the proposed approach is feasible for probabilistic evaluation of 3D seismic active earth pressures. However, the proposed approach costs much less computational efforts than the direct MCS. As seen in Fig. 2, the uncertainty levels of input parameters significantly affect the shape of PDFs and CDFs. The PDF curves become wider and shorter as the uncertainty level increases. This suggests that a higher uncertainty level of input parameters leads to higher uncertainty in the final response, which corresponds to the PDF curves with lower peaks. With regard to CDFs, as shown in Fig. 3, a higher uncertainty level of input parameters results in a more increase in the CDF curves when Ka is lower than a certain value approximately equal to 0.37; this trend is opposite when Ka exceeds this value. Furthermore, it can also be found that different uncertainty levels of input parameters almost do not change the mean value of the final response. Different input parameters possess different effects on the final response. It can be expected that parameters c, φ, and kh are the three most important parameters that affect the active earth pressure co­ efficients. Fig. 4 shows the influences of the COVs of these three pa­ rameters on Sobol’ indices. In each graph, the uncertainty levels of other parameters remain the same as in the Neutral Scenario listed in Table 1, and only the Sobol’ indices of single variables are presented because the Sobol’ indices of groups of random variables are often negligible. It is observed that the change of the COV of one parameter affects Sobol’ indices of all random variables, that is, an increase in the COV of one parameter results in an increase in its own Sobol’ index, while a decrease

Pessimistic scenario

10-2 Neutral scenario

10

-3

Optimistic scenario

10-4 0.0

0.1

0.2

0.3

0.4

Ka

0.5

0.6

0.7

0.8

Fig. 2. PDF curves of 3D active earth pressure coefficients for the three sce­ narios in Table 1: (a) normal scale; and (b) log scale in Y-axis.

in Sobol’ indices of all remaining parameters. As elaborated previously, the Sobol’ indices can be used to identify the importance of input pa­ rameters, i.e., the effect order of input parameter on the final response. Taking a specific example, in Fig. 4(a) at COV (c) = 20%, the Sobol’ index of φ is greatest, suggesting that the uncertainty in φ has the highest influence on the active earth pressure coefficient, followed by c and kh . The Sobol’ indices of the remaining parameters are low, indicating that the effects of their uncertainties on the final response are small. They have the following order of the effect size: β > γ > H > α > B/H > δ. It should be noted that this ranking order is COV-dependent and may change with varying COVs. For example, parameter c appears to be the most important in Fig. 4(b) at COV (φ) = 5%. The above analysis is performed assuming that the input parameters are independent. However, several input parameters may exhibit somewhat correlation relationships. It is reported that there exists a negative correlation between c and φ (Cherubini, 2000; Li and Yang, 2018), and a positive correlation between φ and δ (Low, 2005; Gao et al., 2019). To investigate the influence of correlation relationships between random variables, Fig. 5 shows PDF curves for both correlated and in­ dependent cases. For the correlated case, we consider a negative cor­ relation coefficient of − 0.5 between c and φ and a positive coefficient of 0.8 between φ and δ. The COVs of input parameters are the same as those in the Neutral Scenario. From Fig. 5, it is observed that the correlation 5

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Computers and Geotechnics 129 (2021) 103869

(a) 0.8

0.3

0.4

Ka

0.5

0.6

0.7

0.8

0.36109 0.37509

0.0

(b) 100

10

φ

δ

γ

β

α

H

B/H

0.14384 0.09814

kh

0.67165

c

Fig. 3. CDF curves of 3D active earth pressure coefficients for the three sce­ narios in Table 1: (a) normal scale; and (b) log scale in Y-axis.

8E-5 0.00776

0.11207 0.07367 5.9E-4 0.00275 6.5E-4

0.12366

0.17904 0.12421

0.20169

5

9.1E-4 0.00448 0.00116

0.0

1.2E-4 0.01253

0.8

10

15

COV (φ) (%)

(c) 0.8 γ

H

B/H

5.1. Lumped safety factors in the design of retaining walls

0.2

Traditional methods for the design of retaining walls are usually based on a deterministic model. In this section, a reliability design method is developed for the design of a semi-gravity retaining wall based on the above analysis. As shown in Fig. 6, the wall has a top width of a and a bottom width of b. The wall is constructed using concrete with a unit weight of γwall . The base soil is stiff clay with a base adhesion of ca . In the design of a retaining wall, three types of possible failure are generally considered: overturning about the wall toe, sliding along the base of the wall, and bearing capacity failure of soils beneath the wall. Only the first two failures are generally considered in the analysis, as the same as in the previous studies (Low, 2005; Gao et al., 2019). The

0.1 0.0

10

20

COV (kh) (%)

0.32983 0.10028

0.4

0.1631

0.38413

0.5

0.3

kh

9E-5 0.01025

5. Reliability-based design of retaining walls

δ

α

0.17904 0.12421

Sobol' Indices

0.6

φ

β

9.1E-4 0.00448 0.00116

relationship has a significant effect on the distribution of active earth pressure coefficients. Considering the correlation relationships leads to a significantly more centralized distribution of the active earth pressure coefficient, whereas it has a negligible effect of the mean value of the active earth pressure coefficient. This observation indicates that the assumption of independent random variables provides a more conser­ vative design of retaining walls.

c

7.5E-4 0.00364 9.8E-4

0.7

0.47077

0.7

0.20169

0.6

1.2E-4 0.01253

0.5

0.14408

Ka

0.54299

0.4

0.23529

0.3

1.3E-4 0.01468 0.05196

0.2

0.19607

2.25345E-4 0.01966

0.1

-4

0.1

0.2781

0.31323

0.3 0.2

SPCE MCS

Pessimistic scenario

0.4

0.00138 0.00688 0.00171

Neutral scenario

10-3

0.5

0.17985

Optimistic scenario

10-2

0.47077

0.6

Sobol' Indices

CDF

10-1

0.0

30

COV (c) (%)

(b) 0.8 0.7

10

20

7.4E-4 0.00357 9.3E-4

0.2

1.1E-4 0.01002

0.1

0.1

9.6E-4 0.0052 0.00123

0.0 0.0

0.3 0.2

SPCE MCS

0.1

0.4

0.17904 0.12421

Neutral scenario

0.2

0.5

0.05993

0.3

kh

9.1E-4 0.00448 0.00116

0.4

B/H

0.20169

0.5

γ

H

1.5E-4 0.01513

CDF

Sobol' Indices

Pessimistic scenario

0.6

δ

α

0.47077

0.6

Optimistic scenario

0.7

φ

β

0.21175 0.14701

0.8

c

0.55441

0.7

1.2E-4 0.01253

0.9

0.00111 0.00521 0.00137

(a) 1.0

30

Fig. 4. Effects of COVs of input parameters: (a) soil cohesion, c; (b) angle of shearing resistance, φ; and (c) horizontal seismic coefficient, kh , on Sobol’ indices for 3D seismic active earth pressure coefficients.

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(a) 7

Independent

6

Correlated

PDF

5 4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

Ka

0.5

1 (b) 10

0.7

0.8

Independent Correlated

100

PDF

0.6

10-1

10-2

10-3

10

Fig. 6. Schematic for the illustrative example: (a) a 3D view; (b) a cross-section.

-4

0.0

0.1

0.2

0.3

0.4

Ka

0.5

0.6

0.7

0.8

Pf1 =

Fig. 5. PDF curves of 3D active earth pressure coefficients with and without considering correlation between input parameters: (a) normal scale; and (b) log scale in Y-axis.

W1 Arm1 + W2 Arm2 + paV ArmaV paH ArmaH

+

1 3 Hcotβ,

1 3 H,

1 2 2 γH Ka .

1 3 Hcotβ,

pa cos(δ + β − 90 ), ArmaH = pa = The probability of overturning failure, Pf1 , is calculated by: ◦

(19)

where ca represents the adhesion between the wall and the base soil. The probability of sliding failure, Pf2 , is expressed as:

(17)

paV = pa sin(δ + β − 90◦ ), ArmaV = b +

bca paH

FS2 =

Pf2 =

where W1 and W2 are the two components of the retaining wall selfweight, with horizontal lever distances. Arm1 . and Arm2 , respectively, measured from the toe of the wall; paV and paH are the vertical and horizontal components of the active earth pressure resultant pa , with lever distances ArmaV and ArmaH , respectively. These parameters can be calculated as: W1 = 12γwall aH, Arm1 = 23 b + 23 Hcotβ − 13 a, W2 = 12γ wall bH, Arm2 = 2 3b

(18)

where N is the number of samples for the MCS, and I(x) is a function defined as: I(x) = 1 if × < 1.0, otherwise I(x) = 0. The lumped safety factor against the sliding failure takes the form as:

concept of lumped safety factors has been widely used in the geotech­ nical design. The lumped safety factors against overturning failure and sliding failure are defined below. Note that only per unit width of the retaining wall is considered in the sequel. The lumped safety factor against overturning can be written as fol­ lows: FS1 =

N 1 ∑ I(FS1 ) N i=1

N 1 ∑ I(FS2 ) N i=1

(20)

System probability of failure, Pf , is estimated as: Pf = Pf1 + Pf2 − Pf12

(21)

where Pf12 denotes that the probability that the overturning failure and sliding failure occur simultaneously, which can be calculated as:

paH =

Pf12 =

N 1 ∑ [I(FS1 )⋅I(FS2 ) ] N i=1

(22)

5.2. Illustrative Example I In this section, an example that has been considered in the prior 7

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Computers and Geotechnics 129 (2021) 103869

research is re-considered by the proposed SPCE method. A comparison analysis is then performed to show the validity of the proposed approach. Low (2005) adopted the FORM and direct MCS to perform a probabilistic analysis on active earth pressures in cohesionless soils under 2D static conditions. The Coulomb active coefficient was adopted as the deterministic model. The same example was performed in Gao et al. (2019) by using an efficient MCS method, called Generalized Subset Simulation (GSS). The situation considered in this work can be degenerated into the one in Low (2005) and Gao et al. (2019) by setting the following parameters: c = 0; kh = 0; and B/H→∞. The mean values of other parameters are kept the same as those in Low (2005). Two different cases are distinguished herein. In Case 1, as identified with Low (2005) and Gao et al. (2019), only uncertainties in φ, δ, and ca are considered,. In Case 2, uncertainties in all parameters are considered. The mean values and COVs of input parameters for the two cases are listed in Table 2. In the cases, a positive correlation coefficient of 0.8 is considered between φ and δ, and all of random variables are assumed to follow normal distributions. The number of samples for computing failure probabilities is set to 8 × 105 , which is the same as that in Low (2005). The failure probabilities for the two cases are calculated by the proposed SPCE approach and compared with those in Low (2005) and Gao et al. (2019), as given in Table 3. It is observed that the probabilities for Case 1 are almost the same as those in Low (2005) and Gao et al. (2019). This excellent agreement shows that the proposed SPCE method is feasible for the probabilistic analysis of active earth pressures and reliability design of retaining walls. The failure probabilities in Case 2 are found to be significantly greater than those in Case 1. The system failure probability in Case 2 is, for example, three times larger than that in Case 1. This implies that the results for Case 2 are more conservative. As stated earlier, the two cases differ in that the uncertainties in those parameters with small COVs, i.e., γ, β, α, H, a, b, and γ w , are considered in Case 2, whereas they are neglected in Case 1. The obtained results show that it is necessary to consider the uncertainties of the above seven parameters in reliability analyses of retaining walls for sake of more conservative results, although their COVs are significantly smaller than those of φ, δ, and ca . The results also show that considering the uncertainties in those parameters with small COVs leads to a more significant increase in the probability of overturning failure than that of sliding failure. To inves­ tigate the importance of input parameters on the two modes of failures, the Sobol’ indices of input parameters for FS1 and FS2 are calculated, as listed in Table 4. Because the formula for calculating the Sobol’ indices adopted by this work is only applicable when all random variables are independent, the correlation coefficient between φ and δ is set to zero herein. From Table 4, it is found that the sum of Sobol’ indices of those parameters with small COVs is more than 0.5 for FS1 , thereby implying that over 50% of the variance of FS1 is contributed by the uncertainties

Table 3 Comparison between the failure probabilities in the Illustrative Example I of this work and those in the previous studies. Failure mode

Overturning Sliding System

μ

Case 2

c (kPa) φ(◦ )

0 35

– 10

– 10

δ(◦ )

20

10

10

18

0

5

0

–

–

β(◦ )

90

0

3

α(◦ )

10

0

3

H (m) B/H

6 ∞

0 –

3 –

γ(kN/m3) kh

γwall (kN/m3)

24

0

5

100

15

15

a (m) b (m)

0.4 1.8

0 0

3 3

ca (kPa)

Gao et al. (2019)

Case 1

Case 2

FORM

Direct MCS

GSS

0.00608 0.00108 0.00694

0.02927 0.00183 0.03013

0.00637 0.000961 0.00637~0.00732

0.00633 0.00108 0.00715

0.00602 0.00113 0.00691

5.3. Illustrative Example II The previous example was considered under 2D static conditions. The aim of this example is to show the reliability design of retaining walls under 3D seismic conditions based on the proposed approach. All input parameters are deemed as random variables, and their mean values and COVs for the three scenarios are shown in Table 1. The base width of the wall, b, is chosen as the parameter to be designed. Note that all input parameters are deemed as independent random variables in the following analysis, and the number of samples used to compute failure probabilities is 107 . The failure probabilities are illustrated as a function of normalized wall base width b/H for the Neutral Scenario in Fig. 7. For comparison, the lumped safety factors for deterministic design calculated by Eqs. (17) and (19) are also included in the graph. Not surprisingly, increasing the base width of the wall leads to an increase in safety factors and a decrease in all failure probabilities. It can also be seen that, even when the lumped safety factors are greater than 1.0, the failure probabilities maybe still so high that cannot be accepted in engineering. This can be used to explain why the lumped safety factors are commonly required to exceed 1.0 to some extent in the design. The probability of system failure is not lower than those of overturning failure and sliding failure. At a small value of b/H (when b/H < 0.375), the probability of overturning failure is greater than that of sliding failure; this trend is opposite when b/H exceeds this value. The reliability index βHL required in Eurocode 7 (CEN (European Committee for Standardization), 2004) for the ultimate limit-state design is 3.8, which corresponds to a failure probability of 7.23 × 10− 5 . It is observed that the probability of overturning failure achieves this design level at a smaller value of b/H, suggesting that the design of retaining walls is controlled by the sliding failure. To investigate the uncertainty level of input parameters from a design respective, the system failure probability curves for the three scenarios listed in Table 1 are plotted as a function of normalized base width b/H, as shown in Fig. 8. It can be found that the system failure probability is greatly affected by the uncertainty level of input param­ eters. A higher uncertainty level of input parameters results in an in­ crease in the failure probability when b/H > 0.28. For example, the system failure probability increases from almost zero in the Optimistic Scenario to 2.43 × 10− 3 in the Neutral Scenario and 3.37 × 10− 2 in the Pessimistic Scenario at b/H = 0.4. However, when b/H < 0.28, this trend is opposite, that is, the failure probability decreases with increasing the uncertainty level. Such a phenomenon was also observed in probabilistic analyses of slope stability (Griffiths and Fenton, 2004; Pan et al., 2017); which was termed as a “bunching up” phenomenon. It can also be seen that in the case of a higher certainty level, reaching the target reliability level (i.e., βHL = 3.8) requires a greater value of b/H. For example, the value of b/H needed to reach the target reliability level is approximately 0.35 for the Optimistic Scenario; this value corresponds to 0.53 for the

COV (%) Case 1

Low (2005)

in those parameters. This value only corresponds to about 0.2 for FS2 . Therefore, considering the uncertainties in those parameters with small COVs is more necessary when computing the failure probability of overturning failure.

Table 2 Mean values and COVs of input parameters used in the Illustrative Example I. Input variable

This study

8

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Table 4 Sobol’ indices of input parameters for FS1 and FS2 in the Illustrative Example I. Variable

φ

δ

ca

γ

β

α

H

a

b

γwall

FS1

0.409

0.034

0.000

0.043

0.296

0.003

0.079

0.003

0.069

0.042

0.406

0.003

0.377

0.042

0.071

0.002

0.061

0.000

0.015

0.000

100 Pf

Failure probability

10

FS1

Pf1

-1

FS2

Pf2

4

10-2

3

10-3

2

10-4

lumped safety factors may need to be calibrated separately for different scenarios. However, it seems impossible to know a priori the suitable lumped safety factors. This is a main drawback of deterministic analyses, thus demonstrating the need to design retaining walls in a reliabilitybased way. In previous studies, the design of retaining walls has been historically conducted assuming a 2D failure condition and a cohesionless backfill (Fenton et al., 2005; Low, 2005; Zevgolis and Bourdeau, 2010; Gao et al., 2019). These assumptions are rather conservative. In practice, the collapse of retained soil masses always exhibits a somewhat 3D nature, and cohesive soil is also allowed as the fill material in some design codes (AASHTO, 2012; BSI, 2015). To clearly illustrate the 3D effect and the impact of the existence of soil cohesion, Figs. 9 and 10 present the curves of the failure probability as a function of normalized base width b/H with various levels of B/H and c, respectively. Note that the other pa­ rameters are set the same as those in the Neutral Scenario. From Figs. 9 and 10, it is observed that a smaller value of b/H is needed to reach the target reliability index at a smaller value of B/H and a higher level of soil cohesion. This suggests considering either the 3D effect or the existence of soil cohesion can result in significant savings in terms of retaining wall to be made, which should not be overlooked in the design for economic purpose.

5

Safety factor

FS2

1

βHL = 3.8

0 10-5 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

b/H Fig. 7. Failure probabilities and safety factors as a function of normalized wall base width b/H.

6. Conclusions 10

0

5

This work developed a probabilistic analysis of 3D seismic active earth pressures acting on rigid retaining walls that are used to support cohesive-frictional backfills. The kinematic limit-analysis method in combination with a pseudo-static approach was adopted to perform the deterministic evaluation. An analytical metamodel used to represent the actual physical response was developed based on the SPCE method. All of the input parameters with respect to soil properties, seismic actions, and wall geometry were deemed as random variables. The direct MCS was used to validate the proposed approach. A comprehensive sensi­ tivity analysis was conducted to investigate the influence of uncertainty

Optimistic Scenario Neutral Scenario

Pessimistic Scenario

FS1

4

FS2 10-2

3

10-3

2

10-4

βHL = 3.8

Safety factor

System failure probability

10-1

100

1

B/H = 1

System failure probability

0 10-5 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

b/H Fig. 8. System failure probabilities for the three scenarios in Table 1.

Neutral Scenario, and as for the Pessimistic Scenario, it seems impossible to reach the target reliability level in the considered range of b/H up to 0.70. This suggests that the design of a retaining wall should carefully consider the COVs of input parameters. However, it should be noted that lumped safety factors are not affected by the uncertainty level of input parameters. Instead, the lumped safety factors are the same for the three scenarios, because the mean values of input parameters are kept the same in the three scenarios, thereby resulting in the same active earth pressure coefficient and lumped safety factors. The values of b/H required to reach the reliability level for different scenarios correspond to different lumped safety factors, i.e., a higher uncertainty level cor­ responding to higher lumped safety factors, which suggests that the

B/H = 2

10-1

B/H = 3 B/H = 5 B/H = ∞

10-2

10-3

10-4

βHL = 3.8

10-5 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

b/H Fig. 9. System failure probabilities for different values of wall width-toheight ratio. 9

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Computers and Geotechnics 129 (2021) 103869

existence of soil cohesion in the design.

100

System failure probability

c = 0.0

10

CRediT authorship contribution statement

c = 2.5 kPa

-1

c = 5.0 kPa

Zheng-Wei Li: Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Visualization. QiuJing Pan: Conceptualization, Resources, Supervision, Writing - review & editing. Rui-Zheng Fei: Writing - review & editing.

c = 7.5 kPa c = 10.0 kPa

10-2

Declaration of Competing Interest

10-3

10-4

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

βHL = 3.8

Acknowledgements

10-5 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

b/H

The funding support provided by the Fundamental Research Funds for the central universities of Central South University (2018zzts633) is greatly appreciated.

Fig. 10. System failure probabilities for different levels of soil cohesion.

level and correlation relationships of input parameters. On this basis, several illustrative examples regarding the reliability-based design of retaining walls were shown. The finding of this work might provide insight into the design of retaining walls in seismic zones. Based on the results of this work, the following conclusions are made: (i) The uncertainty level of input parameters significantly affects the distribution of the active earth pressure coefficient, which becomes more spread-out with increasing the uncertainty level. In most cases, Sobol’ indices of parameters c, φ, and kh are found to be among the top three, thus implying that uncertainties in these parameters affect the final response the most significantly. The Sobol’ index of one parameter also varies with the COVs of input parameters. An increase in the COV of one parameter will increase its own Sobol’ index, whereas it will reduce the Sobol’ indices of other parameters. (ii) The correlation relationships between c and φ, and φ and δ are observed to have a significant effect on the distribution of the final response. Considering the effects of the correlation relationships leads to a more centralized distribution, which suggests that the assumption of the independent variable is conservative for reliability analysis of retaining walls. (iii) An illustrative example about the reliability design of retaining walls considered in the prior research was re-considered in the paper. The failure probabilities calculated by the proposed SPCE method are almost the same as those in the previous studies, when the cases considered are the same. This means that the proposed approach is feasible. The results also show that considering uncertainties in several parameters with small COVs may greatly increase the failure probabil­ ities, thus implying that it is necessary to consider all input parameters as random variables to get more conservative results. (iv) The probability of system failure always exceeds those of over­ turning failure and sliding failure. To reach the target reliability level (a reliability index of 3.8 in Eurocode 7), the sliding failure needs a greater value of b/H than the overturning failure, indicating that the design of retaining walls is governed by the sliding failure. The uncertainty level of input parameters also greatly impacts the system failure probability of retaining walls. It is shown that a higher uncertainty level of input pa­ rameters requires a greater value of b/H to reach the target reliability level. The results indicate that, in the deterministic design, the lumped safety factors should be calibrated for different design circumstances. (v) The necessity of considering the 3D effect and the existence of soil cohesion is also shown. It is found that considering the 3D effect and the contribution of soil cohesion can significantly reduce the required b/H. This means that significant savings in terms of retaining walls to be constructed can be achieved with the inclusion of the 3D effect and the

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