Application of the Improved Entry and Exit Method in Slope Reliability Analysis

Abstract

The entry and exit method is a simple and practical method to decide the critical slip surface of slope. Nevertheless, it has the drawback of sacrificing computational efficiency to improve search accuracy. To solve this problem, this paper proposes an improved entry and exit approach to search for the critical slip surface. On basis of the random fields produced by applying the Karhunen–Loève expansion approach, the simplified Bishop’s method combined with the improved entry and exit method is used to decide the critical slip surface and its relevant minimum factor of security. Then, the failure probability is calculated by conducting Monte Carlo simulation. Two instances are reanalyzed to validate the precision and efficiency of the method. Meaningful comparisons are made to show the calculating precision and calculating efficiency of the improved entry and exit method in searching for the minimum security factor of slope, based on which the effect of the reduced searching range on slope reliability was explored. The outcomes suggest that the approach offers a practical device for assessing the reliability of slopes in spatially variable soils. It can significantly enhance the computational efficiency in relatively high-computational precision of slope reliability analysis.

1. Introduction

With the exploration and exploitation of offshore oil and gas resources, underwater engineering has become increasingly diverse. Submarine landslides caused by earthquakes and other reasons have seriously threatened the safety of deep-sea oil and gas drilling platforms, submarine pipelines, and other underwater oil production facilities. Evaluating the stability of submarine slopes is one of the important topics in marine geotechnical engineering. When analyzing the stability of marine slopes, the soil is usually regarded as a homogeneous material, and the strength coefficients of the soil are taken as the average of the statistical data. The stability of the slope is evaluated through a safety factor. Due to the factors such as material composition, sedimentation history, and consolidation pressure, soil strength parameters exhibit spatial variability [1,2,3,4], which significantly affects the slope stability [5,6,7,8,9,10]. It is difficult to comprehensively reflect the reliability of submarine slopes by using safety factors to evaluate the stability of marine slopes. Nguyen et al. [11] studied the role of spatial variability of the soil nature during rainfall infiltration on sandy slope steadiness; they discovered that the uncertainty in the effective friction angle parameter had a significant influence on the obtained safety factor and failure probability. Wang et al. [12] proposed a reliability exploration method for assessing the earth dam slope failure possibility based on extreme gradient boosting (XGBoost), and by conducting parameter sensitivity analysis on case studies, they found that the parameter of variation and fluctuation scale of spatially variable soil properties significantly influenced the failure probability of earth dam slopes. Furthermore, strain rate or loading rate [13] is also one of the key coefficients that affect the shear strength and deformation response of soil particles. Deng et al. [14] revealed the mechanism of toe loading through triaxial compression tests, verifying the important relationship between strain rate effect and slope sliding. However, in practice, it is unrealistic to conduct slope reliability analysis considering all influencing parameters. In such cases, the emerging methods based on model and field testing have been widely adopted for slope failure prediction in recent years. For example, Liu et al. [15] proposed an improved Markov chain model that can accurately simulate soil boundary uncertainty and obtain more accurate reliability results based on limited site drilling information. Xie et al. [16] conducted a set of model experiments and field tests to examine the tilting actions of landslides, and then they proposed a novel method that can effectively predict the remaining time before slope failure occurs. He et al. [17] introduced a machine-learning-aided approach for the stochastic reliability analysis of spatially variable soil slopes; the model generated based on machine learning in this method not only accurately predicted the safety factor, but also greatly reduced the computational workload.

The stability of slope is analyzed by extensively applying the limit equilibrium approach because of its simple principle and reliable results. Combining the limit equilibrium approach with the random field theory, Cho [18] proposed the random limit equilibrium approach to analyze the reliability of slopes. The random limit equilibrium approach has been used by many scholars to analyze the reliability of slopes [19,20,21,22,23]. When combining the random limit equilibrium method with Monte Carlo simulation to calculate the failure probability of slopes, the number of Monte Carlo simulations and the time required for each Monte Carlo simulation are key factors that affect the efficiency of slope reliability calculation. The number of critical slip surfaces analyzed during each Monte Carlo simulation is a key factor that affects the efficiency of the calculation. The traditional method of grid search [24] is a type of violent search method that can quickly search for a limited number of slip surfaces in grid points with a rough division form. But it is difficult to determine the true critical slip surface using this method. In order to obtain more accurate search results, it is usually necessary to exponentially increase the number of trial slip surfaces by reducing the spacing between grid points. However, this significantly increases computational costs. In order to improve search efficiency, Jiang et al. [25] proposed the binary method based on the grid search method, which uses the previously searched objective function value information to find the possible positions of extreme points to improve the descent speed of the function. Mo et al. [26] used a similar approach and predetected each search direction from the central point to determine the next center point movement direction, which overcame the limitations of the search area and improved the reliability of slope stability analysis. After using traditional approaches to find the circular slip surface, Malkawi et al. [27] simplified the second search process by changing the position of some points on the slip surface and applying the random jumping method and two-point random walking method. This achieved an optimized search for the key slip surface, improving the accuracy of the safety factor. The traditional grid search method is independent of each trial slip surface [28] and has strong adaptability. Zhang et al. [29] used the grid search method to determine the critical sliding surface of expansive soil slopes and established a reliability calculation model for expansive soil slopes. Wu et al. [30] combined the grid search method with the pattern search method to search for noncircular slip surfaces of complex soil slopes. Kostić et al. [31] expanded the grid-based search approach for identifying critical failure surfaces and predicting the safety factor of slopes by developing supplementary analytical expressions for the slip center grid. These traditional methods are extensively applied to determine the critical slip surface in the stability analysis of slopes because of their simple principles and reliable results. However, if a high accuracy of the safety factor is necessary, these methods can be very time-consuming. The contradiction between the precision and efficiency of traditional approaches in searching for the minimum safety factor limits their application in slope reliability analysis. To settle this problem, this paper proposes an improved entry and exit approach to determine the critical slip surface. Meaningful comparisons are made to verify the performance of the approach.

2. Random Field Modeling of Spatially Variable Soil Properties

To conduct slope reliability analysis, it is essential to characterize soil spatial variability, often modeled applying random field theory. The autocorrelation function adopted in this research is as follows [6]:

$$\rho (x,y)=\exp \left(-\left[{\left(\frac{x-x^{\prime }}{l_x}\right)}^2+{\left(\frac{y-y^{\prime }}{l_y}\right)}^2\right]\right)$$

(1)

where (x, y) and (x′, y′) are two random points, and the autocorrelation distances are represented as l_x in the horizontal direction and l_y in the vertical direction.

In this research, under Karhunen–Loève (KL) expansion [9,32,33], the lognormal random field is shown as

$$H\left(\mathit{x},\theta \right)=\exp \left({\mu }_{\ln }+{\displaystyle \sum _{i=1}^M{\sigma }_{\ln }\sqrt{{\lambda }_i}{\phi }_i\left(\mathit{x}\right){\chi }_i\left(\theta \right)}\right)$$

(2)

where μ_ln = lnμ − (σ_ln)²/2 is the mean of ln(H) and σ_ln = (ln(1 + (σ/μ)²))^0.5 is the standard deviation of ln(H), χ_i(θ) is a set of orthogonal random coefficients (uncorrelated random variables with zero mean and unit variance), and φ_i(x) and λ_i are the characteristic values and characteristic functions of the autocorrelation function. The value of M, which represents the number of truncated terms, is closely tied to the desired precision level. However, the level of accuracy is achieved through ε to quantify ε, defined as

$$\epsilon ={\displaystyle \sum _{i=1}^M{\lambda }_i/{\displaystyle \sum _{i=1}^{\infty }{\lambda }_i}}$$

(3)

In slope reliability analysis, cohesion (c) and friction angle (φ) are commonly regarded as uncertain geotechnical coefficients. The random fields (regarding c and φ) produced applying the KL expansion method can be represented as follows [6]:

$$H_c\left(\mathit{x},\theta \right)=\exp \left({\mu }_{\ln c}+{\displaystyle \sum _{i=1}^M{\sigma }_{\ln c}\sqrt{{\lambda }_i}{\phi }_i\left(x\right){\chi }_{ci}\left(\theta \right)}\right)$$

(4)

$$H_{\phi }\left(\mathit{x},\theta \right)=\exp \left[{\mu }_{\ln \phi }+{\displaystyle \sum _{i=1}^M{\sigma }_{\ln \phi }\sqrt{{\lambda }_i}{\phi }_i\left(\mathit{x}\right)\left({\chi }_{ci}\left(\theta \right)\cdot {\rho }_{c,\phi }+{\chi }_{\phi i}\left(\theta \right)\cdot \sqrt{1-{\rho }^2{}_{c,\phi }}\right)}\right]$$

(5)

where χ_ci(θ) and χ_φi(θ) are two independent vectors, and ρ_c,φ means the cross-association parameter between c and φ.

3. Limit Equilibrium Method

3.1. Entry and Exit Method

The critical slip surface is determined by analyzing a certain number of trial slip surfaces, and the one with the minimum safety factor is selected as the key slip surface for the slope. The number of the trial slip surfaces has a significant effect on the calculation accuracy. The entry and exit approach generates a circular slip surface based on three parameters: entry point, exit point, and tangent line (Figure 1). By dividing the entry range L₁ into n₁ points, the exit range L₂ into n₂ points, and the vertical distance range between the line A and B into n₃ points, n₁ × n₂ × n₃ trial slip surfaces is generated.

3.2. Improved Entry and Exit Method

The essence of the improved entry and exit method is to adjust the searching range according to the previous search results and then perform a second search to decide the critical slip surface of slope. The general steps are as follows:

• The entry and exit approach is employed to search for the potential key slip surface. The distances of adjacent two entry points, adjacent two exit points, and adjacent two tangent lines are D₁ (D₁ = L₁/20), D₂ (D₂ = L₂/20), and D₃ (D₃ = H/10), respectively. The simplified Bishop approach is used to calculate the security factors of all trial slip surfaces. The slip surface with the minimum factor of security is taken as “the potential critical slip surface (Css′)”, as shown in Figure 2.

2.

The key points (key entry point, key exit point, and key tangent line) are determined according to the location of the potential critical slip surface (Css′), and the searching range is reduced. Based on the reduced searching range, the entry and exit approach is adopted again to determine the critical slip surface (Css) of the slope. The reduced searching range is set to unilateral 4D₁, 4D₂, and 4D₃, as shown in Figure 3.

3.

On basis of the reduced searching range, we increase the multiple of points within the reduced searching range according to the requirements of target search accuracy, as shown in Figure 4. In this paper, the factor of safety obtained by analyzing 81 × 81 × 41 = 269,001 trial slip surfaces is used as the target accuracy. The distances of adjacent two entry points, adjacent two exit points, and adjacent two tangent lines are L₁/80, L₂/80, and H/40, respectively.

It can be seen that when using traditional methods to achieve target search accuracy, it is necessary to analyze 81 × 81 × 41 = 269,001 trial slip surfaces. When using the improved entry and exit method to achieve target search accuracy, {21 × 21 × 11 (first search) + 33 × 33 × 33 (second search)} = 40,788 trial slip surfaces need to be analyzed (reduced searching range equal to unilateral 4D). The number of trial slip surfaces calculated by the improved entry and exit method is 15% of the traditional entry and exit method, and its calculation efficiency is significantly better than the traditional entry and exit method.

3.3. Simplified Bishop Method and Failure Probability

The simplified Bishop approach proposed by [34] is considered one of the best limit equilibrium approaches for calculating the safety factor of circular slip surfaces in slope stability analysis [35]. The factor of security F_s can be shown as follows:

$$F_s=\frac{{\displaystyle \sum \frac{1}{m_{\alpha i}}\left[\left(W_i-{\mu }_{i}b\right)\tan {\phi }^{\prime }+c^{\prime }b\right]}}{{\displaystyle \sum W_i\sin {\alpha }_i}}$$

(6)

where W_i is gravity force; c′ is the cohesion; μ_i is the pore water pressure; φ′ means the internal friction angle; b represents the soil strip width; α_i refers to the inclination of the bottom of the ith soil strip; m_αi = cosα_i + tanφsinα_i/F_s.

In this paper, the Karhunen–Loève (KL) expansion method [33] is adopted to generate two-dimensional random fields. The improved entry and exit approach is employed to recognize the critical slip surface associated with the minimum safety factor. Monte Carlo simulations are utilized to derive the failure possibility of the slope, and the failure probability (Nf/N), defined as the ratio of the number of failure samples (Nf) to the sample size of Monte Carlo simulations (N). The process for computing the failure probability can be outlined in Figure 5 as follows:

4. Example Analysis

4.1. Illustrative Example 1: Cohesive Slope

Example 1 is a cohesive soil slope, which has been used by several scholars to verify the application of their methods [18,36,37,38]. The same example is reanalyzed in this paper to verify the feasibility of the proposed method. The soil parameters of Example 1 are shown in

Table 1. Calculation parameters of Example 1.

γ (kN/m3)	Cu (kPa)	φ (°)	COVCμ	lh (m)	lv (m)
20	23	0	0.3	20	2

, and Figure 6 shows the geometry of Example 1.

In Example 1, the number of random field samples adopted to perform Monte Carlo simulation is 50,000. In

Table 2. Failure probabilities of Example 1.

Method	Failure Probability Pf	Source
MCS (100,000)	7.60 × 10−2	[18]
MRSM (500,000)	7.90 × 10−2	[36]
MCS (10,000)	7.73 × 10−2	[37]
RSSs+MCS (10,000)	7.50 × 10−2	[38]
MCS (50,000)	7.84 × 10−2	This article (21 × 21 × 11)

, the failure probability acquired by applying the traditional entry and exit approach is 7.84 × 10⁻², which is reasonably close to the calculation results of other scholars [18,36,37,38]. The minimum relative difference between the calculated failure probability in this article and those from other scholars is about 0.76%.

The failure probability of Example 1 is calculated by applying the traditional entry and exit approach and the improved entry and exit approach with the same search accuracy.

Table 3. Effects of reduced searching range on the calculated failure probabilities.

Calculation Results	Reduced Searching Range (Unilateral)						Traditional Method 81 × 81 × 41 (Target Value)
	1D	2D	3D	4D	5D	6D
Average Fs	1.2512	1.2495	1.2491	1.2488	1.2487	1.2486	1.2482
Pf	0.0894	0.0907	0.0912	0.0914	0.0914	0.0915	0.0917
Number of slip surfaces (simulation once)	5580	9764	20,476	40,788	73,772	122,500	269,001
Calculation time (h)	0.54	0.97	2.08	4.09	7.23	11.67	25.29

shows the calculated results. According to Table 3, failure probability increases with the increasing reduced searching range. When the reduced searching range is equal to unilateral 2D, the calculated failure probability by applying the improved entry and exit method is close to the target value. The relative difference between the failure probabilities obtained by applying the traditional entry and exit approach (81 × 81 × 41 trial slip surfaces is analyzed) and the improved entry and exit method (the reduced searching range is equal to unilateral 2D) is 1.1%. When the reduced searching range is equal to unilateral 4D, the failure probability obtained by using the improved entry and exit approach is slightly smaller (0.33%) than the target value. Furthermore, when applying the improved entry and exit approach (the reduced searching range is equal to unilateral 2D) to decide the critical slip surfaces, the number of slip surfaces (9764 trial slip surfaces) is about 3.6% of the traditional entry and exit method (269,001 trial slip surfaces) in each simulation. When using the improved entry and exit method (the reduced searching range is equal to unilateral 4D) to decide the critical slip surfaces, the number of slip surfaces (40,788 trial slip surfaces) is about 15% of the traditional entry and exit method (269,001 trial slip surfaces) in each simulation. The calculation efficiency of the improved entry and exit method is significantly better than traditional entry and exit method.

4.2. Illustrative Example 2: c-φ Slope

Example 2 is a c-φ slope used by several scholars to validate the application of their method [18,37,38,39]. The same example is reanalyzed herein to verify the feasibility of the approach. The soil parameters of Example 2 are shown in

Table 4. Parameters of Example 2.

γ (kN/m3)	Shear Strength		Coefficient of Variation		lh (m)	lv (m)	ρ (c,φ)
	c (kPa)	φ (°)	COVc	COVφ
20	10	30	0.3	0.2	20	2	−0.5

, and Figure 7 shows the geometry of Example 2.

The number of random field samples adopted to perform the Monte Carlo simulation in Example 2 is 50,000. In

Table 5. Reliability analysis outcomes in Example 2.

Method	Failure Probability Pf	Source
MCS (100,000)	1.71 × 10−2	[18]
MRSM (500,000)	1.87 × 10−2	[39]
MCS (10,000)	1.6 × 10−2	[37]
RSSs+MCS (10,000)	1.7 × 10−2	[38]
MCS (50,000)	1.81 × 10−2	This article (21 × 21 × 11)

, the failure probability acquired by applying the traditional entry and exit approach is 1.81 × 10⁻², which is reasonably close to the calculation results of other scholars [18,37,38,39]. The minimum relative difference between the calculated results in this article and those from other scholars is about 3.2%.

The failure probability of Example 2 is calculated by applying the traditional entry and exit approach and the improved entry and exit approach with the same search accuracy. The computation outcomes are shown in

Table 6. Effects of reduced searching range on the calculated failure probabilities.

Calculation Results	Reduced Searching Range (Unilateral)						Traditional Method 81 × 81 × 41 (Target Value)
	1D	2D	3D	4D	5D	6D
Average Fs	1.1934	1.1929	1.1926	1.1925	1.1925	1.1925	1.1924
Pf	0.0195	0.0197	0.0200	0.0201	0.0201	0.0201	0.0201
Number of slip surfaces (simulation once)	5580	9764	20,476	40,788	73,772	122,500	269,001
Calculation time (h)	0.52	0.96	2.09	4.16	7.36	12.09	24.31

. According to Table 6, failure probability increases with the increasing reduced searching range. When the reduced searching range is equal to unilateral 2D, the calculated failure probability by applying the improved entry and exit method is close to the target value. The relative difference between the failure probabilities obtained by applying the traditional entry and exit method (81 × 81 × 41 trial slip surfaces is analyzed) and the improved entry and exit method (the reduced searching range is equal to unilateral 2D) is 2.0%. When the reduced searching range is equal to unilateral 4D, the failure probability acquired by applying the improved entry and exit method is almost equal to that obtained by applying the traditional entry and exit approach. Furthermore, when applying the improved entry and exit approach (the reduced searching range is equal to unilateral 2D) to decide the critical slip surfaces, the number of slip surfaces (9764 trial slip surfaces) is about 3.6% of the traditional entry and exit method (269,001 trial slip surfaces) in each simulation. When using the improved entry and exit method (the reduced searching range is equal to unilateral 4D) to decide the critical slip surfaces, the number of slip surfaces (40,788 trial slip surfaces) is about 15% of the traditional entry and exit method (269,001 trial slip surfaces) in each simulation.

These results indicate that when using the improved entry and exit method to search for critical slip surfaces, even with a smaller reduced searching range (unilateral 2D or unilateral 4D), it is possible to calculate failure probabilities that are very close to the target value. Moreover, this method significantly enhances computational efficiency while providing more accurate estimates of failure probabilities. Therefore, the safety factor search method for improved entry and exit is an effective and suitable approach for addressing the contradiction between computational efficiency and search accuracy in traditional methods for slope reliability analysis.

4.3. Discussion on Reduced Searching Range

This section provides the number of potential critical sliding surfaces (with the global minimum safety factor) that were omitted due to the reduced searching range during the second search. Its primary purpose is to discuss the fluctuation degree of the difference value in positions between potential critical sliding surfaces and critical sliding surfaces under two different search accuracies. This also fundamentally validates the result in the example analysis that a larger reduced searching range leads to a more accurate estimation of failure probability.

4.3.1. The Difference of the Location between Critical Slip Surface and Potential Critical Slip Surface

The traditional entry and exit method is used to analyze Example 1 and Example 2. The slip surface with the lowest safety factor calculated by analyzing 81 × 81 × 41 = 269,001 trial slip surfaces is the critical slip surface. The slip surface with the lowest safety factor calculated by analyzing 21 × 21 × 11 = 4851 trial slip surfaces is illustrated by using the term “potential critical slip surface”. The number of random field samples adopted to perform the Monte Carlo simulation in the two examples is 1000. The potential critical slip surface key point (key entry point, key exit point, and key tangent line) positions are subtracted from the critical slip surface key points (key entry point, key exit point and key tangent line) positions, and the difference value is obtained, as shown in Figure 8.

In the calculation results of Example 1 in Figure 8, there are 123 exit points, 135 entry points, and 36 tangent lines whose difference values exceeded the length of 2D. There are 39 exit points, 42 entry points, and 22 tangent lines whose difference values exceeded the length of 4D. In the calculation results of Example 2 in Figure 8, there are 163 exit points, 1 entry point, and 3 tangent lines whose difference values exceeded the length of 2D. There are only 32 exit points whose difference values exceeded the length of 4D. The results indicate that most difference values do not exceed the range of 2D, and fewer exceed 4D. This means that when using the improved entry and exit method, compared with the reduced searching range based on unilateral 2D, the critical slip surface obtained for unilateral 4D is closer to the accurate results of the global range search situation. However, the search results for the critical slip surface on the basis of unilateral 2D are also good.

4.3.2. Further Verification of the Reliability of the Reduced Searching Range

The traditional entry and exit method with different trial slip surfaces is adopted for calculating the failure probabilities of Example 1 and Example 2.

Table 7. The probability of failure when applying the traditional entry and exit approach in this paper.

Number of Trial Slip Surfaces	21 × 21 × 11	41 × 41 × 21	61 × 61 × 31	81 × 81 × 41	101 × 101 × 51
Example 1 (Pf)	0.0784	0.0905	0.0877	0.0917	0.0902
Example 2 (Pf)	0.0181	0.0193	0.0199	0.0201	0.0203

shows the calculated results. As can be seen from Table 7, the failure probabilities increase with the number of increasing trial slip surfaces. However, the calculated failure probabilities obtained with 61 × 61 × 31 trial slip surfaces and 101 × 101 × 51 trial slip surfaces is anomalous. This is because the trial slip surfaces used to decide the critical slip surface are different. For example, the P_f,_{61 × 61 × 31} is larger than P_f,_{21 × 21 × 11} but smaller than P_f,_{41 × 41 × 21}, because the set of the trial slip surfaces with 61 × 61 × 31 trial slip surfaces contains the set of the trial slip surfaces with 21 × 21 × 11 trial slip surfaces but intersects with the set of the trial slip surfaces with 41 × 41 × 21 trial slip surfaces. In addition, failure probabilities of Example 1 obtained by analyzing 81 × 81 × 41 trial slip surfaces is 0.0917, which is 16.9% larger than that obtained by analyzing 21 × 21 × 11 trial slip surfaces. The failure probabilities of Example 2 obtained by analyzing 81 × 81 × 41 trial slip surfaces is 0.0201, which is 11.0% larger than that obtained by analyzing 21 × 21 × 11 trial slip surfaces. This is because the more trial sliding surfaces analyzed under the higher calculation accuracy, the smaller the safety factors will be. These results indicate that the calculation accuracy of the safety factor has an effect on the failure probability of slopes.

The failure probabilities obtained by using the improved entry and exit method with a fixed reduced searching range (unilateral 2D and unilateral 4D) are compared with failure probabilities obtained by using the traditional entry and exit method, as shown in Figure 9. It can be seen that when reduced searching range is equal to unilateral 2D and unilateral 4D, the failure probabilities are reasonably close to those obtained by using the traditional entry and exit method. The relative differences for unilateral 2D between the results of Example 1 and Example 2 are about below 1.1% and 2.0%, respectively. The relative differences for unilateral 4D between the results of Example 1 and Example 2 are about below 0.3% and 0.5%. These outcomes further show the rationality of the given reduced searching range.

Regarding the research findings obtained in this discussion section, it is evident that the number of trial slip surfaces significantly affects the accuracy of computed failure probability results. The proposed improved entry and exit method can significantly reduce the computational cost while obtaining a more accurate result in calculating failure probability. This method is applicable for realizing rapid and practical reliability analysis assessments on a large number of slopes—both at the design stage and at the stage of planning the technology for the execution of geotechnical works. Note that by the means of the enhanced computational efficiency provided by the method, in the practice of slope reliability analysis, the reduced searching range for the second search calculation can be reasonably selected based on the requirements of calculation accuracy.

5. Conclusions

This paper proposes an improved entry and exit approach to decide the critical sliding surface and the minimum factor of security. The improved entry and exit method is combined with the Monte Carlo simulation to calculate the failure probability of slope considering spatially variable soils. The outcomes of the analysis lead to the following conclusions:

The improved entry and exit method, compared with the traditional entry and exit method, can achieve high calculation efficiency because the searching range is significantly reduced according to the location of the potential critical slip surface.

The failure probability increases with the increasing reduced searching range. When reduced searching range is equal to unilateral 2D, the improved entry and exit method can obtain reasonable results in the slope reliability analysis.

The calculation precision of the safety factor has an effect on the failure probability of slopes. The failure probabilities of Example 1 and Example 2 obtained by analyzing 81 × 81 × 41 trial slip surfaces are 16.9% and 11.0% higher than those obtained by analyzing 21 × 21 × 11 trial slip surfaces, respectively.