Investigating the Impact of Random Field Element Size on Soil Slope Reliability Analysis

Abstract

The determination of the optimal random field element (RFE) size is crucial in soil slope reliability analysis as it governs the trade-off between precision in failure probability calculations and computational efficiency. Given the substantial computational burden associated with smaller RFE sizes, studies on their impact on slope failure probability are scarce. This research examines the influence of RFE size on failure probability and safety factor, employing the Karhunen–Loève expansion to generate random fields and integrating the simplified Bishop method with particle swarm optimization (PSO) to assess slope stability. Through Monte Carlo Simulation (MCS), this study investigates the effects of the ratio of slope height to RFE size (H/D_e) on slope reliability metrics across two illustrative cases. Results reveal a notable influence of H/D_e on the distribution of safety factors (F_s) and failure probability (PF), with overestimation observed at smaller H/D_e ratios. When H/D_e exceeds 10 for Example 1 and 15 for Example 2, the F_s distribution patterns in both scenarios stabilize significantly, displaying minimal variability. The PF of Example 1 and Example 2 decreases with the increase of H/D_e and remains basically unchanged when H/D_e exceeds 10 and 15, respectively. Consequently, a recommended H/D_e ratio of 20 is proposed based on the analyzed cases, facilitating accurate calculations while mitigating computational overhead.

1. Introduction

Slope stability analysis is a key research area in geotechnical engineering. Large-scale landslides can bury houses and factories, disrupt traffic, and block rivers, posing severe threats to people’s lives and property safety [1]. Assessing slope stability is vital for ensuring the safety and reliability of civil engineering projects. The deterministic approaches employed in assessing slope stability encompass various methodologies, notably including the finite element analysis method (FEAM) [2,3,4], limit analysis method (LAM) [5,6], and limit equilibrium method (LEM) [7,8,9]. The LEM is simple in principle and calculation and can obtain reliable results. Therefore, the LEM has garnered widespread adoption in the assessment of soil slope stability.

Nonetheless, the uncertainties inherent in geotechnical engineering exert a profound influence on the reliability of slope stability, consequently leading to potential unreliability in the deterministic analyses [10]. Among these uncertainties, the spatial variability of soil properties stands out as a pivotal factor, constituting one of the key uncertainties affecting slope stability [11,12,13]. Although soil parameters can be determined through geotechnical testing, achieving accurate assessments for each specific location remains infeasible. Therefore, in the context of slope reliability analysis, random field theory assumes considerable significance as an effective means for characterizing the spatial variability of soil parameters [13,14]. The Karhunen–Loève (KL) expansion is recognized as one of the methodologies for generating random fields. This technique is widely utilized by numerous researchers [14,15] due to its ability to achieve the same accuracy requirements with a lesser number of expansion terms.

Over the past few years, increasing attention has been paid to the reliability analysis of slopes considering the uncertainties of soil parameters [15,16,17]. Cho [14] presented the Random Limit Equilibrium Method (RLEM), an innovative approach that harmonizes random field theory with the well-established LEM. This integration results in a robust methodology particularly suited to assessing the reliability of soil slope designs. Then, the RLEM was used by various researchers to study the probability of the saturated slope [18], the slope considering the nonlinear failure criterion [16], the slope reinforced permeable polymer [19], and the three dimensional slope [15]. Taking into account the spatial variability of soil parameters, slope reliability analysis can become exceedingly time-consuming [20]. When employing RLEM for slope reliability analysis, a substantial number of numerical simulations must be conducted, with each simulation requiring the determination of the critical slip surface (CSS). Hence, one of the keys to increasing the efficiency of reliability analysis lies in effectively reducing the time needed to determine the CSS of the slope [21]. The Particle Swarm Optimization (PSO) method has been employed by numerous researchers for identifying the CSS of slope [22] owing to its advantages of simple operation, high computational efficiency, and robust stability [23,24,25]. When applying the RLEM for slope reliability analysis, the random field is discretized into elements of specific dimensions to characterize the spatial variability of soil. The random field element (RFE) size plays a crucial role in the slope reliability analysis [26]. A larger RFE size may inadequately describe the spatial variability of soil parameters, negatively impacting slope reliability analysis results. Conversely, a smaller RFE size can better describe this variability but may also introduce significant computational demands [26]. It is very important to determine the appropriate RFE size in slope reliability analysis, yet it remains a topic that has garnered limited attention from scholars.

The primary objective of this study is to conduct a thorough analysis of the influence of RFE size on slope reliability. To achieve this, the K-L expansion method is employed to generate random fields for soil parameters, and an integration of RLEM with the PSO algorithm is utilized to ascertain the CSS of slopes precisely. The Monte Carlo Simulation (MCS) method is applied to compute the failure probability of the slopes. Ultimately, this study will focus on analyzing the crucial role of the H/D_e ratio in slope reliability analysis, providing valuable insights for related fields.

2. Methods

2.1. Karhunen–Loève (KL) Method

The characteristics of a random field are defined by its mean μ, standard deviation σ, and autocorrelation function [14]. In this study, the adopted autocorrelation function expression is formulated as follows:

$$\rho \left({\tau }_x,{\tau }_y\right)=\exp \left(-{\left({\tau }_x/{\delta }_x\right)}^2-{\left({\tau }_y/{\delta }_y\right)}^2\right),$$

(1)

where τ_x and τ_y are the relative separation distances along the x-axis and y-axis, respectively, while δ_x and δ_y denote the corresponding fluctuation ranges along these axes.

The cohesion (c) and friction angle (φ) are considered to be the uncertain characteristics of soil, exhibiting a close and inseparable interdependence that cannot be ignored. The cross-correlated lognormal random fields can be represented as follows [21]:

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

(2)

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

(3)

where χ_i(θ) is random variable; λ_i and φ_i(x) are the eigenvalues and eigenfunctions of the autocorrelation function, respectively; M is the truncation term of the series expansion; μ_ln = lnμ − (σ_ln)²/2 and σ_ln = (ln(1 + (σ/μ)²))^0.5 are the mean and standard deviation of lognormal random field, respectively; and ρ_c,φ is the cross correlation coefficient between c and φ.

2.2. Bishop’s Method

Compared to rigorous analytical methods, Bishop’s method can yield very accurate results [27]. Therefore, this study employs this method, and the factor of safety F_s can be expressed as

$$F_s={\displaystyle \sum \frac{1}{m_{\alpha i}}\left[\left(W_i-u_i{b}_i\right)\tan {\varphi }_i+c_i{b}_i\right]},$$

(4)

in which

$$m_{\alpha i}=\cos {\alpha }_i+\tan {\varphi }_i\cdot \sin {\alpha }_i/F_s,$$

(5)

where W_i is the weight of soil slice, μ_i is the pore water pressure, and α_i is the inclination of the bottom.

Wan et al. [21] optimized the PSO algorithm to enhance the computational efficiency in determining the CSS of slope. This study adopts the improved PSO method to determine the CSS of slope.

2.3. Estimation of Failure Probability of Slope

To assess the failure probability (PF) of a slope, an MCS is performed. The algorithmic approach employed for computing PF is shown in Figure 1.

3. Results and Discussion

In this study, an approximate relative error (RE) is used to study the impact of RFE size on the PF. Considering the slope discretized into square elements, each with a width denoted as D_e = w₁, w₂, w₃…w_n, where w₁ represents the minimum value among w₁, w₂, w₃…w_n and is deemed sufficiently small. Thus, the RE pertaining to PF can be formulated as follows:

$$\mathrm{RE}={\displaystyle \frac{P{F}^i-P{F}^1}{P{F}^1}}$$

(6)

where PF¹ is the PF with D_e = w₁, and PFⁱ is the PF with D_e = w_i. The critical dimension of RFE corresponds to the minimum size at which any further decrement in D_e fails to elicit a notable reduction in the absolute RE of PF, thereby indicating the optimal balance between accuracy and computational efficiency.

This study utilizes an analysis program that was developed by the author using the MATLAB programming language (R2020a, 9.8.0).

Example 1:

Example 1 proposed by Cho [14] is typical slope with single layer undrained cohesive.

Table 1. Soil parameters in two examples.

Examples	μc/(kPa)	COVc	γ/(kN/m3)	μφ	COVφ
Example 1	23	0.3	20	0	0.0
Example 2	10	0.3	20	30	0.2

displays the soil parameters for Example 1, including the mean (μ_c) and covariance (COV_c) of c and the soil weight (γ), mean (μ_φ), and covariance (COV_φ) of φ. This study considered the scenarios with constant δ_h (δ_h = 40 m) and different δ_v (δ_v = 3 m, 5 m, 7 m). In this example, we modeled the shear strength c as a random field using the K-L expansion method with different RFE sizes; D_e = 10 m, 5.0 m, 2.5 m, 1.0 m, 0.5 m, 0.25 m, 0.2 m, 0.15 m, 0.1 m, 0.09 m, and 0.08 m. And the corresponding ratios of slope height and RFE size (H/D_e) are 0.5, 1, 2, 5, 10, 20, 25, 33, 50, 55, and 62.5, respectively. When employing the MCS for slope reliability analysis, the accuracy of the results improves with the increasing number of simulations. Typically, a relatively accurate result can be obtained after 10,000 simulations. To ensure the precision of the computational outcomes in this study, we generated 50,000 random field realizations using MCS. Figure 2 shows the typical random field realization by the RFE size of 5 m, 1 m, 0.25 m, and 0.1 m, respectively.

In this section, the effects of the REF size on F_s is analyzed. Figure 3 illustrates the distribution of F_s for various H/D_e of Example 1. The F_s corresponding the maximum probability density function value is around 1.25. It can be seen that when H/D_e is smaller than 5, the probability density of F_s becomes higher, and the distribution of F_s becomes more and more concentrated at 1.25 with the increase in H/D_e. Nevertheless, when the element has a relatively smaller size like H/D_e > 10, the distribution of F_s tends to be stable without much change.

The effect of the RFE size on the PF of slopes was analyzed in this section. Figure 4 and Figure 5 show the PF values for various H/D_e. As shown in Figure 4, when the RFE size D_e is large and H/D_e is small, the value of PF was obviously overestimated, and as D_e decreases and H/D_e increases, it shows a downward trend and gradually becomes a steady state. This is because when the RFE size is relatively large, the number of elements into which the slope is divided is smaller, making the slope appear relatively homogeneous. This means that the c values at different locations are very similar. Therefore, the average c at the slip surface has a relatively high probability to take a very low value. But when the D_e value is large, this phenomenon will be reversed due to the influence of spatial averaging. When RFE size is small, the number of elements would be large, and the realization of the c field would be heterogeneous. It means that the value of c at different locations are likely to be different, and the smaller c at one point could be offset by a larger value at another point. Thus, the failure probability would be overestimated by a large RFE size and a small H/D_e. Figure 5 illustrates the RE of PF, as the RFE size decreases, the relative error of PF also decreases and gradually stabilizes, and finally no longer decreases and approaches 0.

Example 2:

Example 2, proposed by Cho [14], is an c-φ slope. The property parameters associated with this slope are presented in Table 1, and δ_v and δ_h are the same as in Example 1. D_e is set to 10 m, 5.0 m, 2.5 m, 1.0 m, 0.5 m, 0.25 m, 0.2 m, 0.15 m, and 0.1 m, and the corresponding H/D_e is equal to 1.5, 3, 6, 15, 30, 60, 75, 100, and 150, respectively. Figure 6 shows the typical random field realization with D_e = 5 m, 1 m, 0.25 m, and 0.1 m.

The distribution of F_s was similar to Example 1, as shown in Figure 7. When H/D_e is larger than 15, the distribution of FS becomes lightly flat. And when H/D_e is less than 15, with the increase in H/D_e, the distribution of Fs is more and more concentrated at 1.3. When H/D_e = 15, the distribution of F_s reaches the most concentrated state. As H/D_e continues to increase, the improvement in the distribution concentration state is no longer obvious.

Figure 8 plots the failure probability for various H/D_e and δ_v. The PF decreases rapidly as H/D_e increases when H/D_e < 15 and maintains a stable state of an approximately straight line after H/D_e > 15. And when H/D_e is the same, different δ_v will also have a significant impact on F_s. It can be found that the PF becomes smaller with the decrease in δ_v. Figure 9 illustrates the RE of PF; as the RFE size decreases, the relative error of PF also decreases and gradually stabilizes and finally no longer decreases and approaches 0. The results of Example 1 and Example 2 indicate that a critical ratio H/D_e = 20 emerges as pivotal in accurately assessing the probability of slope failure. This critical ratio signifies the optimal balance between precision and efficiency in slope stability analysis.

4. Conclusions

In this study, a reliability calculation program taking into account the spatial variability of soil parameters was presented. Based on the program, the impact of H/D_e on PF and F_s is analyzed. The following conclusions can be drawn:

The H/D_e has significant influence on the distribution of F_s. When H/De is less than 10, the distribution of F_s becomes more concentrated as H/D_e increases; when H/D_e is greater than 10, the distribution of F_s becomes very close as the grid size increases.

The H/De has significant influence on PF. The PF decreases with the increase in H/D_e and gradually tends to remain unchanged. When the value of H/D_e is small, it may lead to an overestimation of PF. This phenomenon underscores the importance of selecting an appropriate value for H/D_e to ensure accurate assessment of slope stability and avoid misleading results.

The results show that when the H/D_e exceeds 10 and 15 in Examples 1 and 2, respectively, PF remains essentially unchanged, which indicates that setting H/D_e to 20 can not only ensure the accuracy of calculations but also prevent the unnecessary consumption of computing resources.

This study focuses on the influence of H/D_e on computational accuracy. In order to obtain relatively stable computation results, this study has adopted more conservative parameter settings for both the number of MCS and the iterations of PSO. Reducing the number of MCS while maintaining computational accuracy, as well as rapidly searching for the CSS, are two crucial research directions for optimizing the computational strategy of slope reliability, warranting further investigation.