Reliability Analysis of Three-dimensional Soil Slopes Considering Spatial Variability of Soil Parameters - Wan Yukuai; Zhou Yuqi; Shao Linlan; Wang Yuke; Zhang Fei (工程科学与技术)
ObjectiveThree-dimensional (3D) slope reliability analysis encounters two primary challenges: (1) the stochastic modeling of soil properties involves intricate spatial variability, necessitating extensive data processing and imposing prohibitive computational burdens; and (2) brute-force trial-and-error methods for locating critical failure surfaces are computationally inefficient, impeding real-world implementation despite the imperative for precise stability evaluations in civil infrastructure. To address these limitations, this study employs the covariance matrix decomposition method to generate 3D lognormal random fields for soil parameters, enabling efficient modeling of spatial variability. The particle swarm optimization (PSO) algorithm is refined with enhanced termination criteria and integrated with the 3D Bishop method to search for the minimum factor of safety (F). This approach significantly improves the accuracy and computational efficiency of slope reliability analysis. Methods The core of this framework lies in employing a stepwise covariance matrix decomposition method to generate 3D lognormal random fields for slope soil parameters. By systematically decomposing the covariance matrix, the stepwise covariance matrix decomposition efficiently partitions the problem into tractable components, drastically reducing computational demands while enabling high-resolution random field generation with limited resources. This decomposition facilitates rapid simulation of spatially variable soil parameters without excessive overhead. To complement the random field modeling, an enhanced PSO algorithm is proposed for 3D slope stability analysis. The algorithm integrates the 3D Bishop method—a well-established limit equilibrium technique—into PSO's global search mechanism, combining PSO's exploratory power with the Bishop method's precise stability calculations. This synergy enhances the identification of critical slip surfaces. Refined iteration termination criteria are incorporated to expedite convergence toward the minimum factor of safety (F) and corresponding critical surface, thereby improving computational efficiency for reliability analysis. The failure probability of 3D slopes is subsequently assessed via Monte Carlo simulation, which accounts for inherent soil property uncertainties. The method's accuracy and efficacy are validated through numerical examples systematically analyzing diverse slip surface geometries (e.g., cylindrical and cylindrical-ellipsoidal combinations). Results and Discussions The proposed method was applied to analyze Example 1. Excluding spatial variability of soil parameters and assuming a cylindrical sliding surface, a deterministic analysis yielded a F of 1.3524, closely matching Wan et al.'s (2023) result of 1.3525. For a sliding surface combining cylindrical and ellipsoidal shapes, F decreased as the failure surface width (B) increased, gradually approaching the two dimensional analysis outcome. When incorporating spatial variability of soil parameters, the calculated failure probability (Pf) range of 0.0905–0.0916 (average 9.11%) for Example 1, closely aligning with Wan et al.'s (2023) finding of 9.25%, with a relative error of approximately 1.5%. The comparative analysis of safety factors and failure probabilities confirms the accuracy of the proposed method and computational procedure. Systematic studies are conducted for various slip surface forms, encompassing combinations of "cylindrical" and "cylindrical + ellipsoidal" shapes, as well as different correlation lengths and coefficients of variation. When the sliding surface is cylindrical and the characteristic length scale in the out-of-plane direction (ly) is set to 20 m, the Pf gradually decreases as B increases. This observation suggests that the spatial variability of soil parameters is one of the pivotal factors contributing to the enhanced stability assessment accuracy of 3D slope analysis compared to two-dimensional counterparts. When ly approaches infinity (i.e., ly =+∞), as B increases, Pf gradually rises while the average safety factor (Fav ) decreases. These findings indicate that B significantly influences the stability of 3D slopes. When the coefficient of variation for cohesion (COVc) increases from 0.1 to 0.6, Pf rises from 0.0 to 0.25, reflecting an approximate increment of 0.25. When the coefficient of variation for the internal friction angle (COVφ) increases from 0.1 to 0.4, Pf increases from 0.04 to 0.14, corresponding to an approximate increment of 0.10. When the correlation coefficient between cohesion and the internal friction angle (ρc,φ) increases from -0.5 to 0.5, Pf rises from 0.055 to 0.070, indicating an approximate increment of 0.015. These findings demonstrate that COVc, COVφ , and ρc,φ all exert significant influences on the failure probability. When the characteristic length scale in the x-direction (lx) increases from 10 m to 40 m, Pf increases by approximately 0.03. Conversely, when ly increases from 10 m to 40 m, Pf increases by approximately 0.075. This result suggests that ly has a more pronounced impact on slope failure probability compared to lx. Additionally, when the characteristic length scale in the z-direction (lz) increases from 1 m to 4 m, Pf increases by approximately 0.05, indicating a notable influence on failure probability. ConclusionsThe results demonstrate that the enhanced PSO algorithm, when integrated with the 3D Bishop method, markedly improves the computational efficiency of 3D slope reliability analysis without compromising accuracy. When only spatial variability is considered, the failure probability decreases as the width of the failure surface increases. Conversely, when only the 3D effect is considered, the failure probability increases with the width of the failure surface. When both factors are considered simultaneously, their effects on the failure probability tend to offset each other, resulting in a significantly lower failure probability for 3D slopes compared to 2D slopes. When accounting for the variability of soil parameters and 3D effects, the failure probability of slopes initially increases and then decreases with the widening of the failure surface. This pattern indicates that 3D effects dominate when the failure surface is narrow, as geometric constraints and stress redistribution play a critical role. In contrast, spatial variability of soil properties becomes the primary factor when the failure surface exceeds a critical width, as material heterogeneity governs the failure process. The analysis reveals a critical failure surface width, marking the transition from geometry-controlled to material-controlled failure mechanisms in slope stability assessment.