An efficient method for computing slope reliability calculation based on rigorous limit equilibrium

Juxiang Chen; Dayong Zhu; Yalin Zhu

doi:10.1515/arh-2022-0147

Open Access Published by De Gruyter Open Access April 13, 2023

An efficient method for computing slope reliability calculation based on rigorous limit equilibrium

Juxiang Chen , Dayong Zhu and Yalin Zhu

From the journal Applied Rheology

https://doi.org/10.1515/arh-2022-0147

Abstract

Traditional rigorous limit equilibrium methods satisfy all equilibrium conditions and usually have high accuracy, however, which are less efficient for slope reliability analysis. The main reason is that the limit state functions are highly nonlinear implicit functions of safety factor. Complex numerical iterations are required, which may sometimes lead to computational convergence problems. A new method for computing slope reliability calculation with high efficiency and accuracy was proposed. This method was based on the rigorous limit equilibrium method by modifying normal stresses over the slip surface. The critical horizontal acceleration factor K c , which can be expressed explicitly, was used to replace the implicit safety factor as a representation of slope stability. The difference between K c and the known value K c 0 was used as the limit state function. Two slope examples were analyzed. The results showed that the calculation results of this method were in good agreement with those of the traditional Morgenstern–Price limit equilibrium method, but the computational efficiency was significantly improved. When this method was combined with the subset simulation method, the calculation time was only a few seconds. Therefore, this method can be used for rapid calculation of slope reliability.

Keywords: slope reliability; normal stresses over the slip surface; rigorous limit equilibrium method; critical horizontal acceleration factor; limit state function

1 Introduction

The value of safety factor cannot truly reflect the safety state of the slope, due to the inherent variability of geotechnical materials and the uncertainty of loads, as well as the additional assumptions of inter-slice forces required by the limit equilibrium methods. The slope with safety factor greater than one is not necessarily safe [1]. As a matter of fact, some slopes with high safety factors have been damaged [2,3]. Slope reliability is another index that can reflect slope stability more effectively than safety factor. The purpose of slope reliability analysis is to solve the problem of stability uncertainty caused by the randomness of geotechnical parameters. The calculations of failure probability and reliability index must adopt a calculation method with sufficient accuracy. Otherwise, the reliability calculation results will be less objective than that of the safety factor. Therefore, it is necessary to develop a slope reliability analysis method with simple algorithm, high calculation accuracy, and high efficiency.

In practical engineering, the reliability analysis of slope stability is often based on the limit equilibrium method. Generally, the difference between the safety factor and one is used as the limit state function, which is combined with the reliability calculation method to determine the slope reliability. The accuracy and efficiency of slope reliability calculation are closely related to the complexity of safety factor expression and the effectiveness of reliability algorithm. At present, there are two kinds of limit equilibrium methods. One is the “simplified” method that does not rigorously meet the static equilibrium conditions, such as Fellenius method [4], simplified Bishop method [5], and simplified Janbu method [6]. The other is “rigorous” method that fully satisfies all the force and moment equilibrium conditions, such as the rigorous Janbu method [7], Spencer method [8], and Morgenstern–Price method [9,10]. Many studies have shown that only the safety factor that satisfies all equilibrium conditions is reliable for complex slopes with noncircular slip surfaces [11,12]. The calculation accuracy of safety factors of “rigorous” methods is generally higher than that of “simplified” methods. However, the expression of safety factor of the “rigorous” method is usually not explicit. Multiple complex numerical iterations are required to obtain the results. Sometimes, the calculation results do not converge [13].

Some scholars have proposed many efficient slope reliability analysis methods by constructing the approximate explicit expression of performance function and/or reducing the number of samplings [14]. The surrogate model method simplifies the expression of performance function from a complex nonlinear implicit format to an approximately equivalent explicit format. Nonintrusive stochastic finite element method (NISFEM) and response surface method (RSM) are representative methods [15,16]. These methods can improve the accuracy of the complex slope reliability calculation by combining with machine learning methods such as artificial neural network [17] and support vector machine [18]. However, a large number of data samples are needed to obtain the surrogate models. The calculation accuracy and efficiency mainly depend on the selection of parameters and the location of sample points. For the slope reliability problem with highly nonlinear implicit performance function and small failure probability, the number of sample points required to fit the surrogate model is huge. At present, there is no consensus about the effective point selection method.

Sarma proposed a new method to characterize the slope stability by using critical horizontal acceleration factor K c instead of safety factor [19]. The solution of K c satisfies the physical acceptable conditions. The calculation does not require numerical iteration, which can greatly save the time cost and will not lead to the problem of calculation convergence. However, the equations of Sarma method are complex and inconvenient to calculate. Zhu and Lee [13] proposed a rigorous limit equilibrium method by modifying normal stresses over the slip surface, in which the slope safety factor can be solved explicitly. After one or two iterations, the accuracy of safety factor calculation results is equivalent to that of the Morgenstern–Price method [20]. Based on the rigorous limit equilibrium method proposed by Zhu and Li, combined with the idea of Sarma using critical horizontal acceleration factor to characterize slope stability, a new method for calculating slope reliability was proposed in this article.

Even if the performance functions are implicit, as long as there are enough samples, the results of slope reliability calculation by Monte Carlo simulation (MCS) method are accurate. However, for complex slopes with failure probability less than 10 − 4 , more than 10 6 times of sampling are needed to achieve sufficient accuracy and the calculation processes are extremely time-consuming. Some scholars have tried to use importance sampling (IS) method [21], subset simulation (SS) [22] method, and other methods to reduce unnecessary sampling times and improve the computational efficiency of MCS method [23]. IS method improves the computational efficiency by changing the sampling center and sampling direction, increasing the probability of samples falling into failure domain and the weight of important variables [21,24]. However, “curse of dimensionality” problem will be encountered in complex slopes with multiple random variables [22]. SS method equivalently transforms the solution of the small failure probability problem into the product of a series of large conditional probabilities by introducing reasonable intermediate failure events. It is very suitable for the reliability analysis of high-dimensional complex slopes. In this article, SS method was used to calculate the slope failure probability.

An efficient calculation method of slope reliability based on the rigorous limit equilibrium method was proposed in this article. First, based on the rigorous limit equilibrium method by modifying the normal stress on the sliding surface, the explicit expression of the critical horizontal acceleration factor K c was derived. Then, the limit state function is established with K c characterizing the slope stability instead of the safety factor. Finally, the failure probability of slope was calculated by SS method. The reliability calculation results of example slopes showed the effectiveness and reliability of the proposed method. The calculation accuracy was close to that of the Morgenstern–Price method, but the efficiency was greatly improved.

2 Slope reliability calculation method based on rigorous limit equilibrium

2.1 Limit equilibrium equations

A two-dimensional slope with a general shape slip surface is shown in Figure 1. The sliding body boundary consists of the ground surface y = g ( x ) and the sliding surface y = s ( x ) . The horizontal coordinates at both ends of the sliding body are a and b . The horizontal and vertical forces acting on the left and right ends of the sliding body are E a , T a , E b , and T b , respectively. q x ( x ) and q y ( x ) are defined as the vertical and horizontal loads acting on the ground per unit width of the sliding body, respectively. The slope is divided into several vertical slices. The combined forces acting on the soil slice with a width of d x are gravity w d x (where w is the weight of a slice per unit width), horizontal seismic force K w d x (where K is the horizontal acceleration factor), pore water pressure u ( x ) , normal stress σ ( x ) , shear stress τ ( x ) , horizontal internal force E ( x ) , and vertical internal force T ( x ) . The line connecting the action points of E ( x ) is the thrust line y t ( x ) .

$Figure 1 Forces acting on sliding mass and the soil slice with a width of d x \text{d}x .$

Figure 1

Forces acting on sliding mass and the soil slice with a width of d x .

Assuming that the sliding body is in a limiting state, the safety factor F s of the entire sliding surface is a constant. The normal and shear stresses on the slip surface should satisfy the Mohr–Coulomb failure criterion,

(1) τ ( x ) = 1 F s { [ σ ( x ) − u ( x ) ] tan ϕ ( x ) + c ( x ) } ,

where ϕ ( x ) and c ( x ) are the effective internal friction angle and cohesion on the slip surface. By omitting “ ( x ) ,” equation (1) can be simplified as:

(2) τ = 1 F s [ ( σ − u ) tan ϕ + c ] .

The centroid coordinate of the sliding body is ( x c , y c ) . Suppose the values of E a , T a , E b , and T b are all zero. With reference to Figure 1, taking the whole sliding body as the object, the three overall limit equilibrium equations (horizontal and vertical force equilibrium and moment equilibrium) can be written in integral forms:

(3a) ∫ a b ( − σ ⋅ s ′ + τ ) d x − ∫ a b ( K w − q x ) d x = 0 ,

(3b) ∫ a b ( σ + τ ⋅ s ′ ) d x − ∫ a b ( w + q y ) d x = 0 ,

(3c) ∫ a b ( w + q y ) ( x − x c ) + K w ( y c − s + g 2 ) − q x ( y c − g ) d x − ∫ a b [ ( − σ ⋅ s ′ + τ ) ( y c − s ) + ( σ + τ ⋅ s ′ ) ( x − x c ) ] d x = 0 ,

where s ′ = d s d x = tan α , in which α is the inclination of the slip surface at x .

By substituting equation (2) into equations (3a)–(3c), the equations thus become:

(4a) ∫ a b σ ( − s ′ + tan ϕ F s ) d x = 1 F s ∫ a b ( u tan ϕ − c ) d x + ∫ a b K w d x − ∫ a b q x d x ,

(4b) ∫ a b σ ( 1 + tan ϕ F s s ′ ) d x = 1 F s ∫ a b ( u tan ϕ − c ) s ′ d x + ∫ a b ( w + q y ) d x ,

(4c) ∫ a b σ r σ ( x ) + tan ϕ F s r τ ( x ) d x = 1 F s ∫ a b ( u tan ϕ − c ) r τ ( x ) d x + ∫ a b ( w + q y ) ( x − x c ) + K w y c − s + g 2 − q x ( y c − g ) d x ,

where

(5a) r σ ( x ) = − s ′ ⋅ ( y c − s ) + x − x c ,

(5b) r τ ( x ) = y c − s + s ′ ( x − x c ) .

2.2 Modified normal stresses over the slip surface

It can be seen from equations (4a)–(4c) that if the distribution of normal stress σ ( x ) over the slip surface can be determined, the safety factor F s can be obtained. Zhu and Lee [13] proposed that the initial normal stress distribution σ 0 ( x ) was multiplied by the correction function ξ ( x ) as the normal stress σ ( x ) acting on the sliding surface, that is,

(6) σ ( x ) = ξ ( x ) σ 0 ( x ) .

To satisfy the three equilibrium equations, the correction function can be taken as a linear function with two auxiliary variables:

(7) ξ ( x ) = λ 1 ξ 1 ( x ) + λ 2 ξ 2 ( x ) ,

where λ 1 and λ 2 are the undetermined coefficients. Initial normal stress distribution of the slip surface σ 0 ( x ) can take many forms. In order to facilitate the calculation, σ 0 ( x ) , ξ 1 ( x ) , and ξ 2 ( x ) are calculated by the formula in the study by Zhu et al. [12]:

(8) σ 0 ( x ) = w 1 + s ′ 2 ,

(9a) ξ 1 ( x ) = x − b a − b ,

(9b) ξ 2 ( x ) = x − a b − a .

2.3 Rigorous limit equilibrium equations by modifying normal stresses over the slip surface

By substituting equations (6), (7), (8), (9a), and (9b) into equations (4a)–(4c), we can obtain:

(10a) λ 1 A 1 + 1 F s A 1 ′ + λ 2 A 2 + 1 F s A 2 ′ − K A 3 = A 4 + 1 F s A 4 ′ ,

(10b) λ 1 B 1 + 1 F s B 1 ′ + λ 2 B 2 + 1 F s B 2 ′ = A 3 + B 3 + 1 F s B 3 ′ ,

(10c) λ 1 E 1 + 1 F s D 1 + λ 2 E 2 + 1 F s D 2 − K E 4 = E 3 + 1 F s D 3 ,

where

(11a) A 1 = − ∫ a b σ 0 ( x ) ξ 1 ( x ) s ′ d x ,

(11b) A 2 = − ∫ a b σ 0 ( x ) ξ 2 ( x ) s ′ d x ,

(11c) A 3 = ∫ a b w d x ,

(11d) A 4 = − ∫ a b q x d x ,

(11e) A 1 ′ = ∫ a b σ 0 ( x ) ξ 1 ( x ) tan ϕ d x ,

(11f) A 2 ′ = ∫ a b σ 0 ( x ) ξ 2 ( x ) tan ϕ d x ,

(11g) A 4 ′ = ∫ a b ( u tan ϕ − c ) d x ,

(11h) B 1 = ∫ a b σ 0 ( x ) ξ 1 ( x ) d x ,

(11i) B 2 = ∫ a b σ 0 ( x ) ξ 2 ( x ) d x ,

(11j) B 3 = ∫ a b q y d x ,

(11k) B 1 ′ = ∫ a b σ 0 ( x ) ξ 1 ( x ) s ′ tan ϕ d x ,

(11l) B 2 ′ = ∫ a b σ 0 ( x ) ξ 2 ( x ) s ′ tan ϕ d x ,

(11m) B 3 ′ = ∫ a b ( u tan ϕ − c ) s ′ d x ,

(11n) D 1 = ∫ a b σ 0 ( x ) ξ 1 ( x ) r τ ( x ) tan ϕ d x ,

(11o) D 2 = ∫ a b σ 0 ( x ) ξ 2 ( x ) r τ ( x ) tan ϕ d x ,

(11p) D 3 = ∫ a b ( u tan ϕ − c ) r τ ( x ) d x ,

(11q) E 1 = ∫ a b σ 0 ( x ) ξ 1 ( x ) r σ ( x ) d x ,

(11r) E 2 = ∫ a b σ 0 ( x ) ξ 2 ( x ) r σ ( x ) d x ,

(11s) E 3 = ∫ a b [ ( w + q y ) ( x − x c ) − q x ( y c − g ) ] d x ,

(11t) E 4 = ∫ a b w y c − s + g 2 d x .

When the initial normal stresses distribution σ 0 ( x ) acting upon the slip surface is determined, these 20 parameters in equation (11a)–(11t), namely, A 1 – A 4 , A 1 ′ – A 2 ′ , A 4 ′ , B 1 − B 3 , B 1 ′ − B 3 ′ , D 1 − D 3 , and E 1 − E 4 , can be calculated by the definite integral. By substituting the values of these parameters into equations (10a)–(10c), rigorous limit equilibrium equations with four unknowns of λ 1 , λ 2 , F s , and K are obtained.

If F s is assumed to be a given value (e.g., F s = 1 ), then the horizontal acceleration factor K can be obtained by solving the equation (10a)–(10c):

(12) K = λ 1 A 1 + 1 F s A 1 ′ + λ 2 A 2 + 1 F s A 2 ′ − A 4 − 1 F s A 4 ′ λ 1 B 1 + 1 F s B 1 ′ + λ 2 B 2 + 1 F s B 2 ′ − B 3 − 1 F s B 3 ′ ,

where

(13) λ 1 = G 3 B 2 + 1 F s B 2 ′ − G 2 A 3 + B 3 + 1 F s B 3 ′ G 1 B 2 + 1 F s B 2 ′ − G 2 B 1 + 1 F s B 1 ′ ,

(14) λ 2 = G 1 A 3 + B 3 + 1 F s B 3 ′ − G 3 B 1 + 1 F s B 1 ′ G 1 B 2 + 1 F s B 2 ′ − G 2 B 1 + 1 F s B 1 ′ ,

(15) G 1 = E 4 A 1 + 1 F s A 1 ′ − A 3 E 1 + 1 F s D 1 ,

(16) G 2 = E 4 A 2 + 1 F s A 2 ′ − A 3 E 2 − 1 F s D 2 ,

(17) G 3 = A 4 E 4 − A 3 E 3 + 1 F s ( A 4 ′ E 4 − A 3 D 3 ) .

2.4 Limit state performance function of slope stability characterized by K c

The accuracy and efficiency of the slope reliability calculation depend on the complexity of the limit state performance function expression. Sarma [19] proposed a method for calculating the critical horizontal acceleration factor K c required to make the sliding mass reach the limit equilibrium state. That is to say, K c is the coefficient of the horizontal seismic force required to bring the sliding body to the limit equilibrium state when the safety factor is equal to one. Sarma [25] pointed out that the safety factor F s is defined as that factor by which the available shear strength should be reduced to bring it into equilibrium with the mobilized shear stress. Within the same principle, the critical horizontal acceleration factor, K c , gives the horizontal load as a fraction of the total weight of the free body, which, when applied to the free body, brings the stresses along the slip surface into equilibrium with the available strength. The critical horizontal acceleration factor can be used as an index to measure the safety factor.

After calculations, Sarma found that there was a monotonically decreasing function relationship between the horizontal acceleration coefficient K and the safety factor F s . As shown in Figure 2, the lower the safety factor F s , the larger the horizontal acceleration coefficient K . When F s is equal to one, K is equal to K c and K c is greater than zero. When K is equal to zero, F s is greater than one.

$Figure 2 Variation in factor of safety F s {F}_{\text{s}} with horizontal acceleration factor K.$

Figure 2

Variation in factor of safety F s with horizontal acceleration factor K.

When there is no seismic force (i.e., the known horizontal seismic coefficient K c 0 is equal to zero) on the slope and the slope mass is in a stable state, F s is usually a value greater than one. Let F s = 1 , when the sliding body reaches the limit equilibrium state, K c is expected to be greater than zero. Therefore, the condition for slope stability in the absence of a seismic force is F s > 1 at K c = 0 . This condition can be represented as K c > 0 at F s = 1 . Similarly, when there is a seismic force (the magnitude is expressed by the known horizontal seismic coefficient K c 0 ) on the slope, the condition for slope stability is K c > K c 0 at F s = 1 [26].

According to the aforementioned idea, the critical horizontal acceleration factor K c can be calculated directly by substituting F s = 1 into equation (12).

(18) K c = λ 1 ( A 1 + A 1 ′ ) + λ 2 ( A 2 + A 2 ′ ) − A 4 − A 4 ′ λ 1 ( B 1 + B 1 ′ ) + λ 2 ( B 2 + B 2 ′ ) − B 3 − B 3 ′ ,

where

(19) λ 1 = G 3 ( B 2 + B 2 ′ ) − G 2 ( A 3 + B 3 + B 3 ′ ) G 1 ( B 2 + B 2 ′ ) − G 2 ( B 1 + B 1 ′ ) ,

(20) λ 2 = G 1 ( A 3 + B 3 + B 3 ′ ) − G 3 ( B 1 + B 1 ′ ) G 1 ( B 2 + B 2 ′ ) − G 2 ( B 1 + B 1 ′ ) ,

(21) G 1 = E 4 ( A 1 + A 1 ′ ) − A 3 ( E 1 + D 1 ) ,

(22) G 2 = E 4 ( A 2 + A 2 ′ ) − A 3 ( E 2 − D 2 ) ,

(23) G 3 = A 4 E 4 − A 3 E 3 + A 4 ′ E 4 − A 3 D 3 .

It can be seen from equation (18) that K c is only an explicit function of the geotechnical strength parameters c i and ϕ i . Therefore, using the difference between K c and the known seismic coefficient K c 0 , that is, equation (24) as the limit state function, the calculation time of slope reliability analysis will be greatly reduced. When there is seismic load on the slope, if the calculation result of K c is less than the known seismic coefficient K c 0 , the slope will be unstable. Similarly, if there is no seismic load on the slope (i.e., K c 0 is zero), K c less than zero indicates the slope instability.

(24) Z = K c − K c 0 .

The method of calculating K c is simple and easy to understand. It can be applied to sliding surfaces of any shape, does not require iterative calculation, and will not lead to convergence problems [27]. It is very simple to use K c to evaluate the slope stability. Therefore, using the expression containing K c as the performance function to analyze the slope reliability is bound to significantly improve the computational efficiency.

It should be noted that the calculation results must be checked to satisfy the conditions that the horizontal thrust of the entire sliding body is zero, there is no tension between the soil slices, and the normal stresses over the sliding surface are positive.

2.5 Slope failure probability calculation based on SS method

It is assumed that the random variables affecting the slope stability are { x : X 1 , X 2 , ⋯ X n } , and equation (24) is used as the limit state performance function. When Z < 0, the failure probability of the slope is as follows:

(25) P F = P ( K c ( x ) < K c 0 ) = ∫ K c ( X i ) < K c 0 f X ( x ) d x ,

where f X ( x ) is the probability density function.

Among many slope reliability calculation methods, MCS method is recognized as an accurate method. Based on the law of large numbers, MCS method uses the frequency of failure events to approximate the probability of failure events. When the number of samples is large enough, the estimated failure probability can converge to the true value. Let the sampling number be N, and the unbiased estimation of slope failure probability can be obtained:

(26) P ⌢ F = 1 N ∑ i = 1 N I F ( x ) ,

where I F ( x ) is the indicator function, which is given by:

I F ( x ) = 1 , K c ( x ) < 0 0 , K c ( x ) ≥ 0 .

The concept of MSC method is simple, but its computational efficiency is very low for small failure probability problems. Au and Wang [22] improved the MCS method and proposed an SS method that can effectively solve small failure probability problems. SS method divides the probability space into a series of subsets with a sequence inclusion relationship by introducing reasonable intermediate failure events. Therefore, the small failure probability can be expressed as the product of a series of large conditional probabilities that can be efficiently obtained.

In this method, the target failure event is F = { K c ( x ) < K c 0 } . The intermediate failure events can be defined as F i = { K c ( x ) < K c i , i = 1 , 2 , ⋯ , m } and F 1 ⊃ F 2 ⊃ ⋯ ⊃ F m = F . That is,

F k = ⋂ i = 1 k F i ( k = 1 , 2 , ⋯ , m ) ,

where K c 1 > K c 2 > ⋯ > K c m − 1 > K c m = K c 0 is a decreasing intermediate threshold sequence. According to the multiplication theorem and the relationship between events, the failure probability can be expressed as:

(27) P F = P ( F ) = P ( F m ) = P ( F 1 ) ∏ i = 2 m P ( F i ∣ F i − 1 ) .

Let P ⌢ 1 = P ( F 1 ) , P ⌢ i = P ( F i ∣ F i − 1 ) = P ( K c ( x ) < K c i ∣ K c ( x ) < K c i − 1 ) ( i = 1 , 2 , ⋯ , m ) , then the estimation of failure probability can be rewritten as:

(28) P ⌢ F = ∏ i = 1 m P ⌢ i .

The outline of calculating failure probability by the SS method is as follows:

Step 1. MCS method is used to generate N 1 independent and identically distributed samples { x k ( 0 ) : k = 1 , 2 , ⋯ , N 1 } of multidimensional random variables affecting slope stability. The subscript “0” indicates that these samples correspond to the “zeroth conditional level.”

Step 2. The performance function values { K c ( x k ( 0 ) ) : k = 1 , 2 , ⋯ , N 1 } corresponding to the N 1 samples are calculated by equation (18) and sorted in a descending order. p 0 is the intermediate failure probability value, which is generally taken as 0.1. Taking the ( 1 − p 0 ) N 1 th function value as the critical value K c 1 of the intermediate event F 1 = { x : K c ( x ) < K c 1 } , then the sample estimation of P ( F 1 ) is equal to p 0 .

P ⌢ 1 = P ( F 1 ) = P [ K c ( x ) < K c 1 ] = p 0 .

Step 3. The p 0 N 1 samples corresponding to the performance function values in the range of F i − 1 ( i ≥ 2 ) in the i − 1 layer are taken as seed samples. Additional ( 1 − p 0 ) N 1 samples are generated using Markov chain Monte Carlo simulation (MCMCS). At this time, the total number of samples falling in F i is still N 1 .

Step 4. The performance function values corresponding to this N 1 sample are sorted in a descending order again. The ( 1 − p 0 ) N 1 th function value is taken as the critical value K c i ( i ≥ 2 ) of the intermediate event F i = { x : K c ( x ) < K c i } . The estimated value of P i is obtained as follows:

P ⌢ i = P ( F i ∣ F i − 1 ) = P [ K c ( x ) < K c i ∣ K c ( x ) < K c i − 1 ] = p 0 .

Step 5. Repeat Steps 3–4 until the failure domain F m = { K c ( x ) < K c m = K c 0 } is reached. Counting the number of samples N m satisfying K c < K c m in the current simulation layer, then

P ( F m ∣ F m − 1 ) = N m N 1 .

Step 6. The failure probability of slope is estimated by equation (28).

The improved Metropolis algorithm is used to perform MCMCS to generate conditional samples of intermediate failure events. The detailed algorithm and calculation steps can be seen in previous studies [28–30]. The number of samples produced in the whole process is as follows: N = N 1 + ( m − 1 ) ( 1 − p 0 ) N 1 .

The reliability of the structure can be expressed by failure probability or reliability index. There is a one-to-one correspondence between failure probability and reliability index. When the failure probability is obtained, the reliability index β can be calculated by equation (29) according to the inverse function of the standard normal distribution function.

(29) β = Φ − 1 ( 1 − P ⌢ F ) ,

where Φ − 1 ( x ) represents the inverse function of the standard normal distribution function.

A flowchart of coupling the proposed method with the SS method to calculate the slope failure probability is shown in Figure 3.

Figure 3

Flowchart of the coupling method of the proposed method and the SS method.

3 Examples

In order to verify the effectiveness of the proposed method, the reliability indexes, failure probability, and computer CPU time of two slope examples were compared. The explicit expression of K c (hereinafter referred to as K c method) and the implicit expression of F s based on the Morgenstern–Price method [31] (hereinafter referred to as F s method) were used as the limit state performance function, respectively. CPU time was recorded by a computer with a CPU frequency of 3.60 GHz and a memory of 16 GB. It must be pointed out that the computing time on computers with better hardware configuration will certainly be significantly reduced, but the relative law of time cost should be consistent.

3.1 Example 1

The first example slope is taken from a test question of the Australian CAD Association [32]. The slope has three soil layers and a known circular sliding surface, as shown in Figure 4. The unit weight of each soil layer is constant, and cohesion and friction coefficients are normally distributed. The statistical parameters are given in Table 1. There is no seismic force on the slope, so K c 0 = 0 . The K c method is coupled with the MCS method and the SS method, respectively (hereinafter referred to as the K c -MCS method and the K c -SS method). The results of these two methods were compared with that of the method coupled with F s method and the MCS method (hereinafter referred to as the F s -MCS method), as given in Table 2. Relationship curves of slope failure probability, reliability indexes, CPU time, and sampling times calculated by the three methods are shown in Figure 5.

Figure 4

Cross-section of heterogeneous slope for example 1.

Table 1

Soil parameters

Soil layer number	c (kPa)		f = tan φ		γ (kN/m³)
Soil layer number	μ _c	σ _c	μ _f	σ _f	γ (kN/m³)
①	0	0	0.781	0.1	19.5
②	5.3	0.7	0.424	0.05	19.5
③	7.2	0.2	0.364	0.05	19.5

Note: In soil layer ①, μ _c = 0, σ _c = 0 means that the soil is cohesionless soil, and its cohesion is minimal and can be ignored.

Table 2

Results of reliability analysis of example 1 by different methods

Performance functions	Reliability methods	Sampling number N	β	P _f	Computation time t (s)
F s method	MCS	1,000,000	3.183	0.000730	1557.60
F s method	MCS	512,000	3.185	0.000725	788.39
K c method	MCS	1,000,000	3.172	0.000756	33.36
K c method	MCS	512,000	3.169	0.000761	16.48
K c method	SS	1,000,000	3.183	0.000730	31.98
K c method	SS	64,000	3.184	0.000727	3.78
Bishop			3.281 [33]

Note: MCS: Monte Carlo simulation; SS: subset simulation.

Figure 5

Relationship curves of reliability results, computation time, and sampling number of example 1: (a) relationship curves between reliability results and sampling number of example 1 and (b) relationship curves between computation time and sampling number of example 1.

After one million samplings, reliability indexes and failure probability of three methods were all very close, and their reliability index errors with the simplified Bishop method [33] were all less than 4%. The CPU time of the K c -SS method was slightly less than that of the K c -MCS method. But they were both about one-fiftieth of the calculation time of the F s -MCS method.

According to Figure 5, when the calculation results tended to be stable, the reliability index and failure probability calculation results of the K c -SS and K c -MCS method were close to those of the F s -MCS method. The reliability index errors of the two methods and the F s -MCS method were −0.50% and −0.03%, and the errors of the failure probability were 4.97% and 0.28%, respectively. The K c -SS method performed about 64,000 samplings, which was one-eighth of the other two methods.

Although the sampling times of the K c -MCS method were the same as those of the F s -MCS method, the calculation time was about one-fiftieth of that of the latter. The computation time of the K c -SS method was less than five-thousandths of that of the F s -MCS method.

3.2 Example 2

Another slope is James Bay Dam in northern Quebec, Canada [34]. As shown in Figure 6, the slope contains four soil layers, and the slip surface is noncircular. The statistical values of soil parameters of each layer are listed in Table 3. The calculation results of slope reliability are presented in Table 4. The comparison curves of reliability indexes, CPU time, and sampling times are shown in Figure 7.

Figure 6

Cross-section of heterogeneous slope for example 2.

Table 3

Soil parameters of example 2

Soil layer number	c (kPa)		f = tan φ		γ (kN/m³)
Soil layer number	μ _c	σ _c	μ _f	σ _f	γ	σ _γ
①	0	0	0.577	0.031	20	1.1
②	41	0	0	0	19	0
③	34.5	3.95	0	0	19	0
④	31.2	6.31	0	0	20.5	0

Note: In soil layer ①, μ _c = 0, σ _c = 0 means that the soil is cohesionless soil, and its cohesion is minimal and can be ignored.

Table 4

Results of reliability analysis of example 2 by different methods

Performance functions	Reliability methods	Sampling number N	β	P _f	Computation time t (s)
F s method	MCS	1,000,000	1.477	0.0698	37441.00
F s method	MCS	32,000	1.476	0.0700	1233.70
K c method	MCS	1,000,000	1.463	0.0718	166.25
K c method	MCS	64,000	1.476	0.0700	5.89
K c method	SS	1,000,000	1.464	0.0716	122.55
K c method	SS	32,000	1.464	0.0716	3.53
FEM	RSM		1.49 [34]

Note: FEM: finite element method; RSM: response surface method.

Figure 7

Relationship curves of reliability results, computation time, and sampling number of example 2: (a) relationship curves between reliability results and sampling number of example 2 and (b) relationship curves between computation time and sampling number of example 2.

The results of the three methods after one million samplings were compared with that of the combination method of FEM and RSM. The reliability index errors were all less than 2%. Among them, the error of the F s -MCS method was the smallest, but the calculation time was the most, which was about 305 times that of the K c -SS method.

It can be seen from Figure 7 that the sampling times of the F s -MCS method were 32,000 when the calculation results of reliability indexes and failure probability were gradually stable. It was the same as that of the K c -SS method and was half of that of the K c -MCS method. However, the CPU time of the F s -MCS method was about 209 times and 349 times that of the latter two methods, respectively. The reliability results of the K c -SS and K c -MCS method were very close to that of the F s method, and the errors were still not more than 2%.

According to the analyses of the aforementioned two examples, when the reliability calculation method was the same, the reliability results of performance function using K c expression and F s expression were very close. However, the calculation time of two methods varied greatly, from a few hours to several seconds. When the performance function is expressed by K c , the reliability results and calculation time of the SS method were not much different from those of the MCS method.

In summary, the accuracy of slope reliability calculation results using the K c expression in this article is equivalent to that of F s expression, but the calculation efficiency is greatly improved. It shows that the proposed method is an efficient and high-precision method for calculating the slope reliability.

4 Conclusion

For slopes with general shape slip surfaces, based on the limit state performance functions of safety factor expressions of traditional rigorous limit equilibrium methods, the accuracy of the reliability analysis results is high, but the calculation efficiency is very low. Based on the rigorous limit equilibrium method by modifying normal stresses over the slip surface, a slope reliability calculation method using the explicit expression containing the critical horizontal acceleration factor as the limit state performance function was proposed in this article. Through the case studies, the following conclusions were drawn:

This method is based on the limit equilibrium method by modifying the normal stress over the slip surface, which strictly satisfies all the equilibrium conditions and is suitable for the reliability analysis of two-dimensional slope with arbitrary shape slip surface.
The accuracy of calculation results of this method was very close to that of the limit state performance function using the safety factor implicit expression of the Morgenstern–Price method, but the computational efficiency was significantly improved.
This method is especially suitable for the reliability calculation of small failure probability slope with complex soil layer and a large number of random variables. It can be used as an effective tool for engineering designers to carry out a rapid slope reliability analysis.

In this article, only the reliability analyses of slopes with known slip surfaces were carried out. In practical engineering applications, this method can be easily combined with the optimization method to calculate the slope system reliability, which will be the focus of the next research work of this method.

Funding information: This work was supported by the National Natural Science Foundation of China (NSFC Grant No. 52079121).
Author contributions: Conceptualization, methodology, software, formal analysis, investigation, data curation, writing – original draft preparation, writing – review and editing, visualization. Conceptualization: Juxiang Chen and Dayong Zhu; methodology: Juxiang Chen and Dayong Zhu; software: Juxiang Chen; formal analysis: Juxiang Chen, Dayong Zhu, and Yalin Zhu; investigation: Juxiang Chen; data curation: Juxiang Chen; writing – original draft preparation: Juxiang Chen; writing – review and editing: Juxiang Chen, Dayong Zhu, and Yalin Zhu; visualization: Juxiang Chen; all authors have read and agreed to the published version of the manuscript.
Conflict of interest: 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 article.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: All data generated or analyzed during this study are included in this published article (and its supplementary information files).

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Received: 2023-01-29

Revised: 2023-02-28

Accepted: 2023-03-16

Published Online: 2023-04-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

An efficient method for computing slope reliability calculation based on rigorous limit equilibrium

Abstract

1 Introduction

2 Slope reliability calculation method based on rigorous limit equilibrium

2.1 Limit equilibrium equations

2.2 Modified normal stresses over the slip surface

2.3 Rigorous limit equilibrium equations by modifying normal stresses over the slip surface

2.4 Limit state performance function of slope stability characterized by K c

2.5 Slope failure probability calculation based on SS method

3 Examples

3.1 Example 1

3.2 Example 2

4 Conclusion

References

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