Probabilistic stability study of optimally designed retaining structures against nonlinear soil backfills

Abstract

This study adopts a reliability-based optimization approach for the failure mechanism analysis and design of the retaining wall considering nonlinear soil backfills. The assumed failure mechanism is represented by rigid blocks within a kinematically admissible framework in a rotational coordinate system. Then the active and passive earth pressures are derived from the optimization procedure. A convenient way for incorporating seepage effects is proposed and implemented in the nonlinear upper bound analysis. Finally, a novel response surface method is employed to calculate the failure probability considering different probabilistic scenarios and distribution types with high calculation efficacy. The accuracy of the proposed method is evaluated using the Monte Carlo simulations with 1 million trials. Sensitivity analysis indicated that soil unit weight and initial cohesion are the critical factors dominating the failure probability of passive and active mechanism, respectively. The reliability-based design can be performed to obtain the safe range of the lateral force against nonlinear soil backfills with a target failure probability.

Graphical abstract

Keywords

reliability analysis / retaining structure / improved response surface method / nonlinear failure / optimal earth pressure

1 Introduction

Predicting the lateral force exerted by a soil mass on a retaining structure is one of the most fundamental and classical problems in geotechnical engineering. Both Coulomb’s [1] and Rankine’s [2] theories are commonly used in the design of simpler retaining structures, with the assumption that backfills obey a linear and pressure-dependent Mohr–Coulomb failure criterion. However, the presence of water in soil significantly influences the lateral earth pressure on soil retaining structures. This situation has been treated in Terzaghi theory [3], along with proposals for appropriate drainage systems that help reduce those effects. Various numerical approaches have been employed to investigate the failure behaviors of the retaining structures [4–7]. During the construction of retaining walls, the influence of vertical seepage water on earth pressure was explored without considering the drainage system [8]. Engineers place significant emphasis on identifying the failure mechanism when evaluating limit earth pressure for retaining structures, through the application of the upper bound theorem for this purpose. Different from the previous work, this study proposes a novel procedure to calculate the lateral thrust based on upper bound limit analysis, which is highly efficient in estimating the most likely failure mechanism. The upper bound limit analysis is a good method for identifying the failure mechanism of the geotechnical engineering problem [9,10]. In theoretical analyses of the retaining structure, it is commonly assumed that the strength of the backfill soil follows the linear yield condition (such as Mohr–Coulomb yield rule). However, soil exhibits inherently nonlinear behavior [11–13], posing a significant impact on the failure mechanism of geotechnical structures. Therefore, this paper aims to conduct the general failure mechanism analysis of retaining walls in nonlinear soil, which lays the foundation for probabilistic stability study of retaining structures. The approach follows that of Zhang and Smith [14], who used upper bound theory to analyze geotechnical stability of soil with nonlinear failure criterion.

Probabilistic stability analysis enables the prediction of a safe range for applied earth pressures when a specified acceptable failure probability is given. Reliability-based design (RBD) has gained increasing attention in geotechnical engineering [15], and the theory of RBD optimization is widely recognized as an advanced and effective approach in geotechnical applications [16]. Growing emphasis has been placed on conducting reliability analysis during the design of geotechnical engineering projects. Reliability analysis seeks to evaluate the probabilistic behavior of a system with uncertain design parameters, such as loadings and material properties. Hasofer and Lind [17] proposed to assess the failure state of system performance using reliability index, which represents the measured distance between the mean value of the random variables and the limit state surface. By varying random variables in high dimensional space, the failure probability can be estimated. In this study, a RBD for safe earth pressures on retaining structures is carried out to provide guidance for practical engineering in complex construction scenarios. A range of approaches has been developed to implement probabilistic analysis for the retaining structure stability problems. For instance, Basha and Babu [18] used a first-order reliability method (FORM) for load and resistance factor design of retaining walls. Mandali et al. [19] utilized the second-order reliability method to assess the failure probability for various geotechnical and structural failure modes in a typical counterfort retaining wall. Yu and Bathurst [20] illustrated the application of the response surface method (RSM) for conducting a probabilistic assessment of specific performance characteristics of geosynthetic-reinforced segmental retaining walls during operational process. Nonetheless, these techniques face significant challenges in managing complex and implicit performance functions, sampling key points, and tackling sophisticated numerical models. Therefore, a key innovation of this work is the development of an effective approach that integrates optimization-based limit analysis method with improved RSM reliability analysis. The limit analysis method is employed for characterizing the performance function in complex systems, i.e., the retaining structures against nonlinear soil backfills, while the improved RSM is conducted to approximate the implicit performance function, thereby enabling more accurate reliability assessments.

The purpose of this study is to develop a reliability-based optimization approach for the failure mechanism analysis and design of the retaining wall considering nonlinear soil backfills. The RBD can be performed to obtain the safe range of the lateral force against nonlinear soil backfills with a target failure probability. The organization of this paper is outlined as follows. In Section 2, a review of the proposed reliability approach for assessing the failure probability of retaining structures is provided, for effective reliability design implementation. Section 3 introduces a novel optimization-based limit state function designed to conduct stability analysis of retaining structures, emphasizing the optimization of both earth pressures and failure mechanisms with consideration of nonlinear yield conditions. Section 4 offers a detailed examination of the proposed probabilistic approach for failure analysis, emphasizing how the failure probability varies with different distribution types. Additionally, it also addresses both active and passive failure mechanisms to validate the effectiveness of the proposed method in accurately determining failure probabilities. Section 5 is devoted to conclusions.

2 Overview of improved response surface method using vector projection technique

The determination of the performance function formulation is crucial for RBD. However, the performance functions are frequently not clearly articulated in the context of complex engineering challenges. RSM is an efficient technique for approximating the actual performance function by utilizing a simplified explicit function based on various sampling techniques. The reliability of a structure is typically characterized by two distinct indicators: the reliability index β and the failure probability P_f, as defined in Eqs. (1) and (2), where x is the vector of random variables, μ is the mean vector of the random variables x, and Φ is the cumulative distribution function of the normal distribution evaluated at the reliability index β

(1)\beta =\underset{x\in F}{\mathrm{m}\mathrm{i}\mathrm{n}}\sqrt{{(x-\mu )}^{\mathrm{T}}{C}^{-1}(x-\mu )},

(2)P_{\mathrm{f}}\cong 1-\mathrm{\Phi } (\beta ).

Improved RSM is suggested [21,22] to identify the suitable positions of sampling points by virtue of the gradient projection technique. In contrast to traditional sampling point methods, the gradient projection technique allows for fitting points to be positioned more accurately in proximity to the original limit state. The comparison of conventional sampling points and vector projected sampling points is illustrated in Fig.1. And the algorithm for approximating the performance function in this study is presented as

(3)\overline{g}(X)= a_0+\sum _{i=1}^n{a}_i{x}_i,

where x_i(i= 1,2,. ..,n) is the basic random variable, n is the number of the random variable, a_i(i= 0,1,. ..,n) is the coefficient for describing the performance function. In this algorithm, squared terms were excluded to mitigate potential estimation errors that could arise from an inaccurate nonlinear shape derived from the limited information about the original limit state. Through the vector projection technique, an initial response surface is established by choosing a new set of sampling points. The FORM is then applied to estimate the reliability index \beta using the explicit limit state function obtained by RSM. The initial values of the coefficients are established by selecting sampling points centered at the initial central point x_i^0 and positioned in each direction at

(4)x_i^0\pm h\sigma x_i,i= 1,2,. ..,n.

The value of h is typically chosen between 1 and 2. The initial central points are often set at the mean values {\sigma }_{x_i} but can be shifted toward the failure domain to speed up convergence.

In this study, the improved RSM is adopted for the implementation of the reliability analysis for the retaining structures against backfills with a nonlinear failure criterion. The procedure for the reliability analysis can be outlined as follows: The process begins by selecting sampling points using the central composite design, which includes both high and low-level points. In the second step, the performance function remains undefined until the sampling point is evaluated within the failure mode, yielding multiple sets of parameters. In the third step, tests are performed to gather system responses based on the sampling points and the corresponding performance functions derived in Step 2. Then, the initial approximate performance function is refined through iterative solving in the fourth step. This iterative procedure continues to cycle through Step 2 to Step 5 until the termination criterion is met. In this study, the termination criterion is defined by checking if the relative error is smaller than 10⁻⁶.

Monte Carlo simulations (MCS) method is also used to evaluate the accuracy of the failure probability and reliability index obtained by improved RSM. The failure probability of structure is estimated as

(5)P_{\mathrm{f}}=\frac{1}{N}\sum _{i=1}^{n}I(x_i),

where N is the number of samples.

3 Nonlinear upper bound analysis for optimization of earth pressure

The nonlinear Power-type yield rule is widely used nowadays due to its simple mathematical form in calculus. It was developed to provide a replacement for the linear (Mohr–Coulomb) failure criterion based on the observation that the yield envelopes of many soils are not linear [11,13,14], particularly in the range of small normal stresses. In general, a nonlinear Power-Law yield rule is written as follows

(6)\tau =c_0(a+{\sigma }_{\mathrm{n}}/{\sigma }_{\mathrm{t}}{)}^{1/m},

where {\sigma }_{\mathrm{n}} and \tau are normal and shear stresses on the failure surface, respectively, m is nonlinear parameter, c_0 is initial cohesion, {\sigma }_{\mathrm{t}} is axial tensile stress, and a is a shear strength coefficient.

Fig.2 and Fig.3 depict an elemental failure block represented by two coordinate systems: a global system (x, y) and a local system (ξ, η). The block is characterized by dimensions dξ and dη. The failure block at a point (x, y) in the global coordinate system, due to a rotational conversion can be expressed as

(7)\left[\begin{array}{l}\xi \\ \eta \end{array}\right]= \left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right] \left[\begin{array}{l}x\\ y\end{array}\right],

According to Chen [23], it is assumed that the soil is perfectly plastic and follows the associative flow rule

(8){\dot{\epsilon }}_{ij}=\dot{\lambda }\mathrm{\partial }\mathrm{\Psi }\left({\sigma }_{ij}^{\prime }\right)/\mathrm{\partial }{\sigma }_{ij}^{\prime },\dot{\lambda }⩾0,

where {\dot{\epsilon }}_{ij} is the strain rate tensor of the soil, {\sigma }_{ij}^{\prime } is the tensor of effective stress, \mathrm{\Psi } \left({\sigma }_{ij}^{\prime }\right)is the plastic potential and \dot{\lambda } is a nonnegative multiplier. And {\dot{\epsilon }}_{ij} is the product of the differential of the plastic potential with respect to effective stress tensor {\sigma }_{ij}^{\prime }, and multiplier \dot{\lambda }. To account for the seepage effect, the work rate generated by seepage forces is included in the total external work rate in this study. Consequently, the complete form of the upper bound formulation is given by

(9)\begin{array}{rl}D=& {{\displaystyle \int }}_V{\sigma }_{ij}\cdot {\dot{\epsilon }}_{ij}\mathrm{d}V⩾{{\displaystyle \int }}_S{T}_i\cdot v_i\mathrm{d}S+{{\displaystyle \int }}_V{X}_i\cdot v_i\mathrm{d}V\\ & +{{\displaystyle \int }}_V-\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}u\cdot v_i\mathrm{d}V,\end{array}

where D is the internal energy dissipation, {\sigma }_{ij} is the stress tensor, T_i represents the limit load applied to the failure block, S denotes the length of velocity discontinuity, X_i refers to the weight of the failure block, V is the volume of the failure soil blocks, v_i is the velocity of the block, -\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}u is the excess pore-water pressure. This study proposes a novel way to calculate the lateral thrust in nonlinear upper bound limit analysis, considering seepage effects. From the perspective of energy, the solutions to limit lateral thrust could be obtained by incorporating the corresponding summation of work rates of pore water pressure in the upper bound theorem, which is very different from the conventional kinematic approach for linear Mohr–Coulomb material.

To simplify the problem, no shearing resistance will develop along the interface, only the supporting pressure F will be acting perpendicular to the wall, as shown in Fig.2 and Fig.3. Based on the analysis of characteristic lines describing passive and active failure mechanisms, the plane-strain failure wedge is defined as passive when the retaining wall moves toward backfill and active for the movement in the opposite direction. The geometry of the assumed sliding wedge is H in height, L in width, and the unit weight of the failure block \gamma. For the failure modes shown in Fig.2 and Fig.3, the failure surface secant is positioned at an angle θ relative to the positive x-axis and has a length of l.

Assuming the plastic potential, denoted as \mathrm{\Psi }, being coincident with the yield envelope [24]. Taking \tau as positive without loss of generality, the resulting expression is

(10)\mathrm{\Psi }=\tau -c_0{\left(a+{\sigma }_{\mathrm{n}}/{\sigma }_{\mathrm{t}}\right)}^{1/m}.

Based on the work of Ref. [14], the plastic strain rate is

{\dot{\epsilon }}_{\mathrm{n}}=\lambda \frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\sigma }_{\mathrm{n}}}= -\lambda \frac{c_0}{m{\sigma }_{\mathrm{t}}}{\left(a+{\sigma }_{\mathrm{n}}/{\sigma }_{\mathrm{t}}\right)}^{(1-m)/m},

(11){\dot{\gamma }}_{\mathrm{n}}=\lambda \frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }\tau } =\lambda ,

where \lambda is a scalar coefficient, {\dot{\epsilon }}_{\mathrm{n}} is the normal plastic strain rate and {\dot{\gamma }}_{\mathrm{n}} is shear plastic strain rate. By following a purely geometrical line of reasoning, the plastic strain rate components can be written in the form [14]

{\dot{\epsilon }}_{\mathrm{n}}=- \frac{v}{w}{\left[1+f^{\prime }{(\xi )}^2\right]}^{-\frac{1}{2}},

(12){\dot{\gamma }}_{\mathrm{n}}=\frac{v}{w}{f}^{\prime }(\xi ){\left[1+f^{\prime }{(\xi )}^2\right]}^{-\frac{1}{2}},

where f(\xi ) is used to describe failure block surface, v is the velocity of the failure soil body and w is the thickness of the plastic detaching zone.

The normal component of stress is

(13){\sigma }_{\mathrm{n}}=- a\cdot {\sigma }_{\mathrm{t}}+{\sigma }_{\mathrm{t}}{\left(\frac{c_0}{m{\sigma }_{\mathrm{t}}}\right)}^{\frac{m}{m-1}}{f}^{\prime }(\xi {)}^{\frac{m}{m-1}}.

and combined with Eq. (6),

(14)\tau =c_0{\left(\frac{c_0}{m{\sigma }_{\mathrm{t}}}\right)}^{\frac{1}{m-1}}{f}^{\prime }(\xi {)}^{\frac{1}{m-1}}.

In the context of impending failure, the internal dissipation energy at any point along the failure discontinuity, {\dot{D}}_i could be derived using the combinations of Eqs. (12)–(14)

(15)\begin{array}{rl}{\dot{D}}_i=& {\sigma }_{\mathrm{n}}{\dot{\epsilon }}_{\mathrm{n}}+\tau \cdot {\dot{\gamma }}_{\mathrm{n}}=\frac{v}{w}{\left[1+f^{\prime }{(\xi )}^2\right]}^{-\frac{1}{2}}\\ & \times \left[a\cdot {\sigma }_{\mathrm{t}}-{\sigma }_{\mathrm{t}}{\left(\frac{c_0}{m{\sigma }_{\mathrm{t}}}\right)}^{\frac{m}{m-1}}(1-m)f^{\prime }{(\xi )}^{\frac{m}{m-1}}\right].\end{array}

Therefore, the energy dissipation can be calculated by integrating {\dot{D}}_i over the interval [H\cdot \sin \alpha ,L\cdot \cos \alpha ],

(16)\begin{array}{rl}D=& {\displaystyle {\int }_{H\cdot \mathrm{s}\mathrm{i}\mathrm{n}\alpha }^{L\cdot \mathrm{c}\mathrm{o}\mathrm{s}\alpha }{\dot{D}}_{i}w\sqrt{1+f^{\mathrm{\prime }}{(\xi )}^2}\mathrm{d}\xi }\\ =& v{\displaystyle {\int }_{H\cdot \mathrm{s}\mathrm{i}\mathrm{n}\alpha }^{L\cdot \mathrm{c}\mathrm{o}\mathrm{s}\alpha }\left[a\cdot {\sigma }_{\mathrm{t}}-{\sigma }_{\mathrm{t}}{\left(\frac{c_0}{m{\sigma }_{\mathrm{t}}}\right)}^{\frac{m}{m-1}}(1-m)f^{\mathrm{\prime }}{(\xi )}^{\frac{m}{m-1}}\right]\mathrm{d}\xi }.\end{array}

Referred to Refs. [25,26], the gradient of pore pressure is

(17)-\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}u=- \frac{\mathrm{d}u}{\mathrm{d}y}={\gamma }_{\mathrm{w}}-r_{\mathrm{u}}\gamma .

By combining Eqs. (9) and (17), the explicit form of the collapse soil bodyweight work rate, considering seepage effects, is written as [26]

(18)\begin{array}{rl}{W}_{\mathrm{e}}=& v\gamma \left[\left(1-r_{\mathrm{u}}\right)\cos \alpha \right]\\ & \left[{\int }_{H\cdot \sin \alpha }^{L\cdot \cos \alpha }f(\xi )\mathrm{d}\xi +H\cos \alpha (L\cos \alpha -H\sin \alpha )\right]\\ & -v\gamma \left[\left(1-r_{\mathrm{u}}\right)\cos \alpha \right]\times \left[0.5l^2\sin (\theta +\alpha )\cos (\theta +\alpha )\right].\end{array}

It is widely known that failure occurs following the minimum energy principle. In essence, the problem is transformed to seek the expression of f(\xi ) which costs the minimum energy. To describe energy cost, an objective function \mathrm{\Lambda }, is established by balancing the internal and external work rate

(19)\begin{array}{rl}\mathrm{\Lambda }=& D-W_e=v{\int }_{H\cdot \sin \alpha }^{L\cdot \cos \alpha }{\psi }_1\left[f(\xi ),f^{\prime }(\xi ),\xi \right]\mathrm{d}\xi \\ & -v\gamma \left[\left(1-r_u\right)\cos \alpha \right]\times [H\cos \alpha (L\cos \alpha \\ & -H\sin \alpha )-0.5l^2\sin (\theta +\alpha )\cos (\theta +\alpha )],\end{array}

where

(20)\begin{array}{rl}\frac{{\psi }_1\left[f(\xi ),f^{\prime }(\xi ),\xi \right]}{\gamma H}=& \frac{a\cdot {\sigma }_{\mathrm{t}}}{\gamma H}-\frac{{\sigma }_{\mathrm{t}}}{\gamma H}{\left(\frac{c_0}{m{\sigma }_{\mathrm{t}}}\right)}^{\frac{m}{m-1}}(1-m)f^{\prime }(\xi {)}^{\frac{m}{m-1}}\\ & -\frac{\cos \alpha \cdot \left(1-r_{\mathrm{u}}\right)f(\xi )}{H}.\end{array}

By virtue of the Greenberg minimum principle, the optimal earth pressures can be obtained by minimizing the objective function \mathrm{\Lambda }, in the functional space of all the admissible velocity discontinuities f(\xi ). Consequently, the formulation of {\psi }_1 is transformed into Euler’s equation through the variational method, which is given by

(21)\frac{\mathrm{\partial }{\psi }_1}{\mathrm{\partial }f(\xi )}-\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\left[\frac{\mathrm{\partial }{\psi }_1}{\mathrm{\partial }{f}^{\prime }(\xi )}\right]=0.

The detailed representation of the Euler’s equation corresponding to Eq. (20) is given by

(22)\left(1-r_{\mathrm{u}}\right)\cdot \cos \alpha +\frac{m{\sigma }_{\mathrm{t}}}{(m-1)\gamma }{\left(\frac{c_0}{m{\sigma }_{\mathrm{t}}}\right)}^{\frac{m}{m-1}}{f}^{\prime }(\xi {)}^{\frac{2-m}{m-1}}{f}^{″}(\xi )=0.

Equation (22) is a nonlinear second-order homogeneous differential equation. The expression of velocity discontinuity surface is accordingly expressed as

(23)\frac{f(\xi )}{H}=- \frac{k}{H}{\left(\frac{n_0}{\gamma \left(1-r_{\mathrm{u}}\right)\cdot \cos \alpha }-\xi \right)}^m+ \frac{n_1}{H},

where

(24)\frac{k}{H}=\frac{{\sigma }_t}{\gamma \left(1-r_{\mathrm{u}}\right)H\cdot \cos \alpha } {\left[\frac{\gamma \left(1-r_{\mathrm{u}}\right)\cdot \cos \alpha }{c_0}\right]}^m,

where n_0/r and n_1/H are the dimensionless parameters derived by geometric condition constrains including

(25)\begin{array}{rl}& f(\xi =H\cdot \sin \alpha )=-H\cdot \cos \alpha ,\\ & f(\xi =L\cdot \cos \alpha )=L\cdot \sin \alpha .\end{array}

The active failure mode can also be examined through variational calculus used in passive case. As shown in Fig.3 the expression of the failure surface should satisfy

(26)\begin{array}{rl}& f(\xi =H\cdot \sin \alpha )=H\cdot \cos \alpha \\ & f(\xi =L\cdot \cos \alpha )=-L\cdot \sin \alpha .\end{array}

Thus, the function of failure surface yields

(27)f(\xi )=-k {\left[\frac{n_2}{\gamma \left(1-r_{\mathrm{u}}\right)\cdot \cos \alpha }+\xi \right]}^m+n_3.

This study introduces a novel optimization approach aimed at maximizing F in active failure mode while minimizing it in passive situations. The optimization process is governed by two parameters {\psi }_{\mathrm{s}} and \theta, using Eq. (28). The full energy equation is

(28)\begin{array}{rl}F\kappa v\cos \left(\theta +\kappa {\psi }_{\mathrm{s}}\right)=& v\kappa \left[\frac{H^2\gamma }{2\tan \theta }-\kappa \hat{W}\left({\psi }_{\mathrm{s}},\theta ,l\right)\right]\\ & \times \left[\left(1-r_{\mathrm{u}}\right)\cos \alpha \right]+\hat{C}\left({\psi }_{\mathrm{s}},\theta ,l\right)v\cos {\psi }_{\mathrm{s}},\end{array}

where F can be determined by internal energy dissipation coefficient \hat{C}\left({\psi }_{\mathrm{s}},\theta ,l\right) based on Eq. (16) and external energy coefficient \hat{W}\left({\psi }_{\mathrm{s}},\theta ,l\right) based on Eq. (18). And \kappa =1 for the passive failure mechanism, \kappa =-1 for the active failure mechanism. The expressions for \hat{c} and \hat{W} can be written as follows

(29)\begin{array}{rl}\hat{C}\left({\psi }_{\mathrm{s}},\theta ,l\right)=& \frac{\kappa {\sigma }_t}{\cos {\psi }_{\mathrm{s}}}\left(\frac{m-1}{m+1}\right){\left[\frac{\gamma \left(1-r_{\mathrm{u}}\right)\cos \alpha }{c_0}\right]}^m\\ & [{\left(\frac{n_0}{\gamma \left(1-r_{\mathrm{u}}\right)\cos \alpha }-\kappa H\sin \alpha \right)}^{m+1}\\ & -{\left(\frac{n_0}{\gamma \left(1-r_{\mathrm{u}}\right)\cos \alpha }-\kappa L\cos \alpha \right)}^{m+1}]\\ & +a\cdot \frac{{\sigma }_t}{\cos {\psi }_{\mathrm{s}}}(L\cos \alpha -H\sin \alpha ),\end{array}

(30)\begin{array}{rl}\frac{\hat{W}\left({\psi }_s,\theta ,l\right)}{\kappa \gamma }=& \frac{\kappa k_0}{m+1}[{\left(\frac{n_0}{\gamma \left(1-r_{\mathrm{u}}\right)\cos \alpha }-\kappa L\cos \alpha \right)}^{m+1}\\ & -{\left(\frac{n_0}{\gamma \left(1-r_{\mathrm{u}}\right)\cos \alpha }-\kappa H\sin \alpha \right)}^{m+1}]\\ & +n_1(L\cos \alpha -H\sin \alpha )+H\cos \alpha (L\cos \alpha \\ & -H\sin \alpha )-0.5l^2\sin (\theta +\alpha )\cos (\theta +\alpha ).\end{array}

The optimization can be implemented by MATLAB’s built-in multi-parameter optimization toolbox.

4 Probabilistic stability study of optimally designed retaining structures

This research focuses on conducting a reliability analysis for retaining structures using an optimization approach. Based on the proposed passive and active failure mechanisms, it is evident that the failure probability is influenced by variations in earth pressures and geotechnical parameters. Therefore, the failure probability against the optimally designed retaining structures for a given earth pressure was proposed. The performance function used in this work can be expressed as

(31)\begin{array}{rl}& G_1=F_{\mathrm{t}}-F_{\mathrm{A}},\\ & G_2=F_{\mathrm{P}}-F_{\mathrm{t}},\end{array}

where F_{\mathrm{t}} stands for the lateral force supporting the retaining structures, F_{\mathrm{A}} and F_{\mathrm{P}} are the optimal active and passive earth pressure, respectively. The soil parameters for the two mechanisms are shown in Tab.1. The height of the retaining wall for passive and active modes is 4 and 5 m, respectively. And the deterministic earth pressure can be obtained by the developed optimization procedure in this study, and the variation earth pressure F with {\psi }_{\mathrm{s}}\mathrm{a}\mathrm{n}\mathrm{d} \theta , can be visualized in Fig.4. In this study, the effect of the uncertainties related to the soil properties on the retaining structure stability is investigated and five uncertain parameters were taken into consideration.

4.1 Uncertain input data

Since the input random variables are unlikely to be accurately estimated in practical engineering, statistical information is adopted to model the probabilistic failure in this work. To calculate the failure probability against the retaining structures for a given earth pressure in passive failure mechanism, the corresponding statistical information of random variables is shown in Tab.2 with the assumption that variables are following normal and lognormal distribution. And for the active failure mechanism, the corresponding statistical information of random variables is shown in Tab.3. The coefficients of variation (COVs) are influenced by the inherent uncertainty in soil characteristics as well as potential data collection inaccuracies. It is significant to note that soil properties can be quantified through various testing methods, contributing to the relatively low COV values of these variables.

Each random input variable is represented by two distinct probability density functions (PDFs). Specifically, for those cases exhibiting non-normal distributions, the five variables are assumed to adhere to a lognormal distribution. This selection is warranted due to the non-negativity of the random variables involved, ensuring that the modeling accurately reflects their inherent characteristics.

4.2 Failure probability analysis

Normally the performance function is based on the resistance-load effect model. Most of the methods used for uncertainty propagation are applicable for the expression of performance function. However, in some complicated uncertainty system, the relationship between the input variables and model outputs tends to be implicit. Therefore, this study adopted improved RSM procedures to conduct reliability analysis to improve the accuracy and efficiency. The error analysis for RSM was first evaluated by calculating its quantitative error metrics in comparison with the limit analysis solutions. The mean absolute error (MAE), root mean square error (RMSE) of RSM technique was calculated considering the effect of sample size. As shown in Fig.5, the MAE and RMSE values of RSM are low in both active and passive mechanisms, and these values decreased with the increase of sample size. When a sample size exceeding 10000 was employed, the MAE and RMSE values are lower than 0.1, with the R² values larger than 0.99, confirming the accuracy of the proposed RSM technique.

The overall failure probability for both passive and active case relates to a specific scenario. By selecting a particular probabilistic scenario and type of random variables (normal or nonnormal), it is feasible to perform MCS. For Scenario 1, five variables listed in Tab.2 and Tab.3 are assumed to follow a normal distribution. For Scenario 2, five variables listed in Tab.2 and Tab.3 are assumed to follow a lognormal distribution. Additionally, a scenario (Scenario 3) was considered in which the nonlinear parameter and shear strength coefficient follow a lognormal distribution, while the initial cohesion, axial tensile stress, and unit weight follow a normal distribution. A convergence analysis of MCS was performed, and the results showed that ten independent simulation trials, each with a sample size of one million, yielded a standard deviation below 0.00031 for the obtained failure probability.

To verify the accuracy of the proposed method for probability analysis, the comparison in failure probability between the proposed method and MC simulation are made with a constant earth pressure F_{\mathrm{t}} (F_{\mathrm{t}}= 390 kN/m for passive case and F_{\mathrm{t}}= 80 kN/m for active case), shown in Tab.4. The pore pressure coefficient r_{\mathrm{u}} = 0.1 is used in this study. Approximating failure probability was obtained by improved RSM and MC technique, despite their significant computational efficacy. Take the scenario of active mechanism and normal distribution as an example, the improved RSM completed the calculation in 2.2 s, while MCS required ~40 h to perform 1 million simulations on the same computation device (central processing unit: i9-13900H).

Fig.6 demonstrates the variation of failure probability for different single failure mechanisms with different pore pressure coefficients. For passive failure mechanism, the failure probability increases with the increase of pore pressure coefficients, while the active case presents reverse trend. Tab.5 clearly indicates the variation of failure probability according to different magnitudes of the applied earth pressure. For active failure mechanism, the failure probability decreases with the increment of earth pressure, while the passive pressure presents reverse trend. Therefore, the range of applied earth pressure can be obtained when an acceptable failure probability is given. It can also be concluded that the distribution types of random variables have limited influence on failure probability, which is consistent with the previous findings [27].

4.3 Sensitivity analysis

Sensitivity analysis was employed to quantitatively evaluate the influence of parameters, involved in the limit state function, on failure probability. This information can subsequently be utilized to identify the input variables that most significantly influence system failure, thereby offering valuable guidance in the design process. In this research, the design point sensitivities were first determined through improved RSM, aiming to prioritize variables for risk mitigation.

The importance factors of variables involved in both active and passive mechanism were plotted using their absolute value. As shown in Fig.7, distribution type poses minor effect on the importance factors, which is consistent with the failure probability result. Nonetheless, the ranking of importance factors changes with the failure mechanism. In active mechanism, unit weight of the soil, γ, is the variable with the greatest impact on failure probability, followed by initial cohesion and shear strength coefficient. In this case, the nonlinear parameter m is suggested to have the least influence on failure probability. For comparison, initial cohesion poses the most significant influence on the failure probability concerning passive mechanism, and shear strength coefficient and nonlinear parameter are the two variables with the least influence on failure probability.

The first-order and total-order Sobol indices of these parameters were also calculated to explore the parameter interactions and global sensitivity of these variables. As shown in Fig.8, Sobol indices show that initial cohesion and unit weight are the most two influencing factors in both active and passive mechanism, which is consistent with the importance factor results. In addition, the first-order Sobol indices of all parameters are quite close to their total-order Sobol indices, indicating that these parameters have almost no interactions with each other.

Sensitivity analysis has been performed to evaluate how variations in these variables impact the failure probability. For both active and passive mechanisms, the COV of the shear strength coefficient and nonlinear parameter is reduced by 40%, i.e., from 0.05 to 0.03, and the corresponding failure probability was evaluated. As shown in Fig.9, reducing the COV of the shear strength coefficient and nonlinear parameter can significantly lowered the failure probability for both active and passive mechanisms. For instance, the failure probability decreased from 7% to 1.5% when both the COVs of shear strength coefficient and nonlinear parameter reduced from 0.05 to 0.03. Fig.9 can also be used for the design of retaining structures, as the required lateral supporting force can be determined with a prefixed failure probability when the mean value and COV of variables are obtained.

5 Conclusions

This study presents a comprehensive reliability-based approach for analyzing and designing safe earth pressure ranges against retaining walls, focusing on the impact of soil property uncertainties on retaining structure stability. The kinematic framework of upper bound theory has been applied to the nonlinear optimization of the limit lateral force supporting the retaining structure. A novel variational approach is extended to the analysis of active and passive earth pressure, subjected to seepage forces, in nonlinear Power-type soils. By selecting different coordinate systems, the rigid one-wedge failure mechanism is applied to evaluate the limit thrust by optimization. This study introduces an innovative method to incorporate seepage effects from the perspective of energy balance, with potential applications to a broader range of geotechnical challenges. By employing an improved RSM, failure probabilities across various probabilistic scenarios and distribution types can be accurately determined, as examined by MCS. Some conclusions can be drawn.

1) The failure probability of retaining structures varies with changes in earth pressure and geotechnical parameters. For both passive and active failure modes, the proposed optimization procedure successfully identified the range of applied earth pressure that results in an acceptable failure probability. The analysis also revealed that increasing pore pressure coefficients leads to higher failure probabilities in passive modes, whereas the active mode shows the opposite trend.

2) The improved RSM employed in this study proved to be an efficient and accurate approach for assessing failure probabilities, particularly in complex nonlinear systems. Comparison with MCS (despite its computational intensity) confirmed the robustness of the RSM, showing minimal error with significantly higher computational efficiency. Additionally, the choice of probability distribution (normal vs. non-normal) had a limited impact on failure probabilities, aligning with previous studies. These insights underscore the necessity for meticulous site investigation, soil characterization, and continuous monitoring during excavation to mitigate uncertainties and enhance the reliability of retaining wall designs.

3) Sensitivity analysis highlighted the most influential variables contributing to the failure probability for both active and passive mechanisms. Soil unit weight and initial cohesion emerged as critical factors. Furthermore, reducing the COVs for shear strength and nonlinear parameters resulted in a decrease in failure probability. This insight offers practical guidance for optimizing the design of retaining structures by focusing on controlling key parameters to minimize failure risk.

Overall, this research advances the understanding of RBD in geotechnical engineering, offering practical methodologies for incorporating uncertainties into the stability assessment of retaining structures. The proposed approach not only improves safety and performance but also provides a valuable tool for engineers to achieve optimal design solutions under various probabilistic conditions. However, the adopted input variables may affect the convergence stability of the proposed method to some extent. Future work will focus on the rough retaining wall with considering the effects of the friction on the retaining wall interface.

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