基于随机响应面法的边坡系统可靠度分析

提出了一套基于随机响应面法的边坡系统可靠度分析方法。该方法首先从大量潜在滑动面中筛选出代表性滑动面。针对每条代表性滑动面,采用Hermite多项式展开建立其安全系数与土体参数间的非线性显式函数关系（即随机响应面）。然后,采用直接蒙特卡洛模拟计算边坡系统失效概率。在蒙特卡罗模拟中,采用所有代表性滑动面的随机响应面计算每一组样本所对应的边坡最小安全系数。最后,以两个典型多层边坡系统可靠度问题为例验证了该方法的有效性。结果表明：文中提出的边坡系统可靠度分析方法能够有效地识别边坡代表性滑动面,具有较高的计算精度和效率,并且确定代表性滑动面时无需计算滑动面间的相关系数。同时该方法可以有效地计算低失效概率水平的边坡系统可靠度,为含相关非正态参数的边坡系统可靠度问题提供了一条有效的分析途径。此外,多层边坡可能同时存在多条潜在滑动面,基于单一滑动面（如临界确定性滑动面）或者部分代表性滑动面进行边坡系统可靠度分析均会低估边坡失效概率。     

Risk de-aggregation and system reliability analysis of slope stability using representative slip surfaces

This paper develops a risk de-aggregation and system reliability approach to evaluate the slope failure probability, p_f, using representative slip surfaces together with MCS. An efficient procedure is developed to strategically select the candidate representative slip surfaces, and a risk de-aggregation approach is proposed to quantify contribution of each candidate representative slip surface to the p_f, identify the representative slip surfaces, and determine how many representative slip surfaces are needed for estimating the p_f with reasonable accuracy. Risk de-aggregation is performed by collecting the failure samples generated in MCS and analyzing them statistically. The proposed methodology is illustrated through a cohesive soil slope example and validated against results from previous studies. When compared with the previous studies, the proposed approach substantially improves the computational efficiency in probabilistic slope stability analysis. The proposed approach is used to explore the effect of spatial variability on the p_f. It is found that, when spatial variability is ignored or perfect correlation assumed, the p_f of the whole slope system can be solely attributed to a single representative slip surface. In this case, it is theoretically appropriate to use only one slip surface in the reliability analysis. As the spatial variability becomes growingly significant, the number of representative slip surfaces increases, and all representative slip surfaces (i.e., failure modes) contribute more equally to the overall system risk. The variation of failure modes has substantial effect on the p_f, and all representative surfaces have to be incorporated properly in the reliability analysis. The risk de-aggregation and system reliability approach developed in this paper provides a practical and efficient means to incorporate such a variation of failure modes in probabilistic slope stability analysis.     

基于广义子集模拟的土坡系统可靠度分析

如何有效地评价边坡的系统可靠度并识别出对边坡稳定性具有重要影响的关键滑面一直是边坡稳定性分析的关键问题。提出了基于广义子集模拟的边坡系统可靠度分析方法及代表性滑面识别方法,并推导了基于广义子集模拟的边坡系统可靠度计算公式及边坡中滑面对边坡系统失效的相对贡献量化公式。基于广义子集模拟计算结果,采用概率网络评价方法识别边坡代表性滑面。以一个双层黏性土坡和芝加哥国会切坡算例验证了所提方法的有效性。结果表明：提出的基于广义子集模拟的边坡系统可靠度分析方法可有效地估计边坡系统及其单一滑面的失效概率,对于具有低失效概率水平边坡可靠度的求解,其计算效率明显优于传统蒙特卡洛模拟方法。此外,对于单个失效模式而言,广义子集模拟与子集模拟计算效率相当。对于多个失效模式的失效概率计算问题,广义子集模拟不需要重复对每个失效模式失效概率进行计算,计算效率明显优于子集模拟。提出的代表性滑面选择方法是在系统失效概率及单滑面失效概率的高效计算基础上实现的,代表性滑动面能够较好地代表边坡系统失效,从而有效地降低了边坡系统失效概率对代表性滑面数目及代表性滑面失效概率估计准确性的依赖性。     

Efficient system reliability analysis of soil slopes using multivariate adaptive regression splines-based Monte Carlo simulation

System effects should be considered in the probabilistic analysis of a layered soil slope due to the potential existence of multiple failure modes. This paper presents a system reliability analysis approach for layered soil slopes based on multivariate adaptive regression splines (MARS) and Monte Carlo simulation (MCS). The proposed approach is achieved in a two-phase process. First, MARS is constructed based on a group of training samples that are generated by Latin hypercube sampling (LHS). MARS is validated by a specific number of testing samples which are randomly generated per the underlying distributions. Second, the established MARS is integrated with MCS to estimate the system failure probability of slopes. Two types of multi-layered soil slopes (cohesive slope and c–φ slope) are examined to assess the capability and validity of the proposed approach. Each type of slope includes two examples with different statistics and system failure probability levels. The proposed approach can provide an accurate estimation of the system failure probability of a soil slope. In addition, the proposed approach is more accurate than the quadratic response surface method (QRSM) and the second-order stochastic response surface method (SRSM) for slopes with highly nonlinear limit state functions (LSFs). The results show that the proposed MARS-based MCS is a favorable and useful tool for the system reliability analysis of soil slopes.     

边坡可靠度分析的一种新的优化求解方法

介绍了Low & Tang提出的一种新的可靠度优化求解方法,并将之用于边坡可靠度分析中：该方法适用于任何概率分布的相关变量,不必计算当量正态均值和方差、相关变量独立变换,直接在变量的原始空间内搜索边坡的最小可靠指标和概率临界滑面,可采用任何合适的约束优化方法进行求解,方法清晰简洁。边坡可靠度分析常用的滑面有2个：最小安全系数（变量均值处）对应的确定性临界滑面和最小可靠指标对应的概率临界滑面,但这2个滑面在有些情况下差别较大,Hassan & Wolff提出了一种简化方法可以方便地获得概率临界滑面,但由于方法简单,受到质疑。通过一系列算例分析,优化求解方法得到的概率临界滑面和Hassan & Wolff的简化方法滑面非常接近,显示了简化方法的有效性,值得在工程实践中推广。     

Multiple response surfaces for slope reliability analysis

This paper develops a multiple response surfaces approach to approximate the limit state function for slope failure by second‐order polynomial functions, to incorporate the variation of the most probable slip surfaces, and to evaluate the slope failure probability p_f. The proposed methodology was illustrated through a cohesive soil slope example. It is shown that the p_f values estimated from multiple response surfaces agree well with those p_f values that have been obtained by searching a large number of potential slip surfaces in each Monte Carlo simulation (MCS) sample. The variation of number of the most probable slip surfaces is studied at different scale of fluctuation (λ) values. It is found that when full correlation assumed for each of random fields (i.e., spatial variability is ignored), the number of the most probable slip surfaces is equal to the number of random fields (in this study, it is 3). When the spatial variability grows significantly, the number of the most probable slip surfaces or number of multiple response surfaces firstly increases evidently to a higher value and then varies slightly. In addition, the contribution of a specific most probable slip surface varies dramatically at different spatial variability level, and therefore, the variation of the most probable slip surfaces should be accounted for in the reliability analysis. The multiple response surfaces approach developed in this paper provides a limit equilibrium method and MCS‐based means to incorporate such a variation of the most probable slip surfaces in slope reliability analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient system reliability analysis of rock slopes based on Subset simulation

How to efficiently assess the system reliability of rock slopes is still challenging. This is because when the probability of failure is low, a large number of deterministic slope stability analyses are required. Based on Subset simulation, this paper proposes an efficient approach for the system reliability analysis of rock slopes. The correlations among multiple potential failure modes are properly accounted for with the aid of the “max” and “min” functions. A benchmark rock slope and a real engineered rock slope with multiple correlated failure modes are used to demonstrate the effectiveness of the proposed approach.     

Conditional random field reliability analysis of a cohesion-frictional slope

Discarding known data from cored samples in the reliability analysis of a slope in spatially variable soils is a waste of site investigation effort. The traditional unconditional random field simulation, which neglects these known data, may overestimate the simulation variance of the underlying random fields of the soil properties. This paper attempts to evaluate the reliability of a slope in spatially variable soils while considering the known data at particular locations. Conditional random fields are simulated based on the Kriging method and the Cholesky decomposition technique to match the known data at measured locations. Subset simulation (SS) is then performed to calculate the probability of slope failure. A hypothetical homogeneous cohesion-frictional slope is taken as an example to investigate its reliability conditioned on several virtual samples. Various parametric studies are performed to explore the effect of different layouts of the virtual samples on the factor of safety (FS), the spatial variation of the critical slip surface and the probability of slope failure. The results suggest that whether the conditional random fields can be accurately simulated depends highly on the ratio of the sample distance and the autocorrelation distance. Better simulation results are obtained with smaller ratios. Additionally, compared with unconditional random field simulations, conditional random field simulations can significantly reduce the simulation variance, which leads to a narrower variation range of the FS and its location and a much lower probability of failure. The results also highlight the great significance of the conditional random field simulation at relatively large autocorrelation distances.     

考虑空间变异性条件下的边坡稳定可靠度高效敏感性分析

在有限数据条件下,可靠度敏感性分析是研究各种不确定性因素对边坡失稳概率影响规律的重要途径。基于直接蒙特卡洛模拟和概率密度加权分析方法提出了一种高效边坡稳定可靠度敏感性分析方法。所提出的方法通过随机场表征岩土体参数的空间变异性,并采用局部平均理论建立岩土体参数的缩维概率密度函数,用于概率密度加权分析中高效、准确地计算不同敏感性分析方案对应的边坡失稳概率。最后,通过一个工程案例--詹姆斯湾堤坝说明了所提出方法的有效性和准确性。结果表明：在敏感性分析过程中,所提出的方法只需要执行一次直接蒙特卡洛模拟,避免了针对不同敏感性分析方案重新产生随机样本和执行边坡稳定分析,节约了大量的计算时间和计算资源,显著提高了基于蒙特卡洛模拟的敏感性分析计算效率;在概率密度加权分析中采用岩土体参数的缩维概率密度函数能够准确地计算边坡失稳概率,避免了有偏估计,使概率密度加权分析方法适用于考虑空间变异性条件下的边坡稳定可靠度敏感性分析问题。     

基于高效随机有限元法的边坡风险评估

提出了基于子集模拟的边坡风险评估的高效随机有限元法（RFEM）,推导了基于子集模拟的边坡失效概率和失效风险的计算公式,并给出了基于高效RFEM的边坡可靠度分析和风险评估流程图。采用一个边坡算例验证了所提方法的有效性。结果表明,基于子集模拟的高效RFEM可以视为是对基于蒙特卡洛模拟的传统RFEM的改进,显著地提高了失效概率和失效风险的计算效率以及失效样本的产生能力,非常适用于分析小失效概率的可靠度问题,极大地增强了RFEM在边坡可靠度分析和风险评估中的实用性。高效RFEM将边坡的整体失效风险分解为对应不同概率水平的边坡失效风险,并量化了它们对整体风险的相对贡献度。在该方法中,边坡可靠度分析和风险评估与确定性边坡有限元分析互不耦合,极大地简化了它们的计算过程。此外,土体不排水抗剪强度的竖向空间变异性对边坡失效风险具有显著的影响。     

On the efficient estimation of small failure probability in slopes

The random finite element method (RFEM) combines the random field theory and finite element method in the framework of Monte Carlo simulation. It has been applied to a wide range of geotechnical problems such as slope stability, bearing capacity and the consolidation of soft soils. When the RFEM was first developed, direct Monte Carlo simulation was used. If the probability of failure (p _f ) is small, the direct Monte Carlo simulation requires a large number of simulations. Subset simulation is one of most efficient variance reduction techniques for the simulation of small p _f . It has been recently proposed to use subset simulation instead of direct Monte Carlo simulation in RFEM. It is noted, however, that subset simulation requires calculation of the factor of safety (FS), while direct Monte Carlo requires only the examination of failure or non-failure. The search for the FS in RFEM could be a tedious task. For example, the search for the FS of slope stability by the strength reduction method (SRM) usually requires much more computational time than a failure or non-failure checking. In this paper, the subset simulation is combined with RFEM, but the need for the search of FS is eliminated. The value of yield function in an elastoplastic finite element analysis is used to measure the safety margin instead of the FS. Numerical experiments show that the proposed approach gives the same level of accuracy as the traditional subset simulation based on FS, but the computational time is significantly reduced. Although only examples of slope stability are given, the proposed approach will generally work for other types of geotechnical applications.     

Evaluating slope stability uncertainty using coupled Markov chain

Geological uncertainty appears in the form of one soil layer embedded in another or the inclusion of pockets of different soil type within a more uniform soil mass. Uncertainty in factor of safety (FS) and probability of failure (Pf) of slope induced by the geological uncertainty is not well studied in the past. This paper presents one approach to evaluate the uncertainty in FS and Pf of slope in the presence of geological uncertainty using borehole data. The geological uncertainty is simulated by an efficient coupled Markov chain (CMC) model. Slope stability analysis is then conducted based on the simulated heterogeneous soils. Effect of borehole layout schemes on uncertainty evaluation of FS and Pf is investigated. The results show that borehole within influence zone of the slope is essential for a precise evaluation of FS statistics and Pf. The mean of FS will converge to the correct answer as the borehole number increases.     

Efficient Monte Carlo Simulation of parameter sensitivity in probabilistic slope stability analysis

Monte Carlo Simulation (MCS) method has been widely used in probabilistic analysis of slope stability, and it provides a robust and simple way to assess failure probability. However, MCS method does not offer insight into the relative contributions of various uncertainties (e.g., inherent spatial variability of soil properties and subsurface stratigraphy) to the failure probability and suffers from a lack of resolution and efficiency at small probability levels. This paper develop a probabilistic failure analysis approach that makes use of the failure samples generated in the MCS and analyzes these failure samples to assess the effects of various uncertainties on slope failure probability. The approach contains two major components: hypothesis tests for prioritizing effects of various uncertainties and Bayesian analysis for further quantifying their effects. Equations are derived for the hypothesis tests and Bayesian analysis. The probabilistic failure analysis requires a large number of failure samples in MCS, and an advanced Monte Carlo Simulation called Subset Simulation is employed to improve efficiency of generating failure samples in MCS. As an illustration, the proposed probabilistic failure analysis approach is applied to study a design scenario of James Bay Dyke. The hypothesis tests show that the uncertainty of undrained shear strength of lacustrine clay has the most significant effect on the slope failure probability, while the uncertainty of the clay crust thickness contributes the least. The effect of the former is then further quantified by a Bayesian analysis. Both hypothesis test results and Bayesian analysis results are validated against independent sensitivity studies. It is shown that probabilistic failure analysis provides results that are equivalent to those from additional sensitivity studies, but it has the advantage of avoiding additional computational times and efforts for repeated runs of MCS in sensitivity studies.     

考虑能量-时间分布的边坡动力可靠性分析新方法

提出一种考虑能量-时间分布的边坡动力可靠性分析方法。该方法将动态最危险滑动面及其稳定系数以时间序列加以刻画,并根据边坡动力反应的能量分布特征,提取持续时间统计窗,用于对上述时间序列的统计分析,以获取边坡动力模糊失效概率、边坡动力可靠度指标和基于保证率的边坡动力稳定系数。以澳大利亚计算机应用协会边坡稳定考核题为例,应用上述新方法考察其在芦山7.0级主震波形条件下的稳定性,研究了在不同统计窗下的边坡动力可靠性。案例分析表明： （1）新方法能够抓住影响边坡动力稳定的主要时间段,使分析结果更为凝练、可信。（2）通过引入边坡失效状态的模糊判别,使得可靠性评价中能够考虑模糊性,解决了以往常规方法区分度不够的问题。（3）基于保证率的边坡动力稳定系数具有很好的应用前景,它在内涵上体现了可靠性分析,在形式上与静力稳定系数的定义兼容,在数值上反映了边坡瞬时动力稳定系数的保守估计值,在实践上与现行规范的拟静力法具有良好的可比性,因而具有多方面的优势。（4）就本案例而言,动力条件下最危险滑动面的发育位置趋向于静力条件下的最危险滑动面,体现了依据静力和拟静力理论框架所进行的防护工程设计,在动力条件下仍然具有积极的意义。（5）新方法对定量研究现行边坡规范的抗震设计冗余提供了一条途径。提出的新方法为边坡抗震研究提供了新的思路、方法和可供参考的实例。     

Three-dimensional slope reliability and risk assessment using auxiliary random finite element method

This paper aims to propose an auxiliary random finite element method (ARFEM) for efficient three-dimensional (3-D) slope reliability analysis and risk assessment considering spatial variability of soil properties. The ARFEM mainly consists of two steps: (1) preliminary analysis using a relatively coarse finite-element model and Subset Simulation, and (2) target analysis using a detailed finite-element model and response conditioning method. The 3-D spatial variability of soil properties is explicitly modeled using the expansion optimal linear estimation approach. A 3-D soil slope example is presented to demonstrate the validity of ARFEM. Finally, a sensitivity study is carried out to explore the effect of horizontal spatial variability. The results indicate that the proposed ARFEM not only provides reasonably accurate estimates of slope failure probability and risk, but also significantly reduces the computational effort at small probability levels. 3-D slope probabilistic analysis (including both 3-D slope stability analysis and 3-D spatial variability modeling) can reflect slope failure mechanism more realistically in terms of the shape, location and length of slip surface. Horizontal spatial variability can significantly influence the failure mode, reliability and risk of 3-D slopes, especially for long slopes with relatively strong horizontal spatial variability. These effects can be properly incorporated into 3-D slope reliability analysis and risk assessment using ARFEM.     

一种基于位移场分析的临界滑动面确定方法研究

安全系数的计算和临界滑动面的确定是边坡稳定性分析的两个组成部分,强度折减法可以得到合理的安全系数,但却不能准确刻画临界滑动面的位置。基于对滑动现象的认识,提出了一种基于位移场分析的临界滑动面确定方法,认为当边坡处于临界状态时,其潜在滑动面附近的位移等值线最为密集,潜在滑动面上的点往往是在深度方向上沿垂直滑面的位移变化率达到最大值的位置。将本文方法与Spencer法、岩质边坡模型试验进行了对比,验证了该方法在曲线和折线滑动面搜索方面的适用性和可靠性。同时,探讨了单元形状、疏密程度、离散点间距等因素对滑动面的影响。     

Identification of representative slip surfaces for reliability analysis of soil slopes based on shear strength reduction

Although a slope may have numerous potential slip surfaces, its failure probability is often governed by several representative slip surfaces (RSSs). Previous efforts mainly focus on the identification of circular RSSs based on limit equilibrium methods. In this paper, a method is suggested to identify RSSs of arbitrary shape based on the shear strength reduction method. Monte Carlo simulation is used to generate a large number potential slip surfaces. The RSSs are identified through analyzing the failure domains represented by these samples. A kriging-based response surface model is employed to enhance the computational efficiency. These examples shows that the RSSs may not always be circular, and that the suggested method can effectively locate the RSSs without making prior assumptions about the shape of the slip surfaces. For the examples investigated, the system failure probabilities computed based on the shear strength reduction method are comparable to, but not the same as those computed based on the limit equilibrium methods. The suggested method significantly extends our capability for identifying non-circular RSSs and hence probabilistic slope stability analysis involving non-circular slip surfaces.     

The effects of method of generating circular slip surfaces on determining the critical slip surface by particle swarm optimization

The main objectives of slope stability analysis are evaluating factor of safety for a given slip surface and determining the critical slip surface for a given slope. Factor of safety is usually calculated by limit equilibrium method. The main steps to determine the critical slip surface are generating trial slip surfaces as probable solutions and searching among them to determine the one with the lowest factor of safety. Although the process of searching the critical slip surface received much attention between researchers, the significance of method of generating slip surfaces is seldom addressed in the literature. The authors believe that this ignorance can affect the accuracy of the results of slope stability analysis even in the simplest problems with circular slip surfaces. Consequently, this paper focused on the method of generating circular trial slip surfaces as the simplest mechanism of sliding and considered its effect on determining the critical slip surface. A new method of generating circular slip surface was presented, which is more efficient and less restricted than the conventional method. A computer program was also developed to determine the critical slip surface of slopes by using particle swarm optimization. The performances of the proposed method and developed computer program were verified during comparative studies and sensitivity analysis. Based on the results, the effect of method of generating circular slip surfaces on determining the critical slip surface was confirmed successfully. In all considered problems, the proposed method of generating circular slip surfaces led to the lower values of factor of safety compare with the conventional method.     

Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D validation

The objective is to capture the 3D spatial variation in the failure mode occurring in accretionary wedges and their analog experiments in the laboratory from the sole knowledge of the material strength and the structure geometry. The proposed methodology relies on the maximum strength theorem which is inherited from the kinematic approach of the classical limit analysis. It selects the optimum virtual velocity field which minimizes the tectonic force. These velocity fields are constructed by interpolation thanks to the spatial discretization conducted with ten-noded tetrahedra in 3D and six-noded triangles in 2D. The resulting, discrete optimization problem is first presented emphasizing the dual formalism found most appropriate in the presence of nonlinear strength criteria, such as the Drucker–Prager criterion used in all reported examples. The numerical scheme is first applied to a perfectly triangular 2D wedge. It is known that failure occurs to the back for topographic slope smaller than and to the front for slope larger than a critical slope, defining subcritical and supercritical slope stability conditions, respectively. The failure mode is characterized by the activation of a ramp, its conjugate back thrust, and the partial or complete activation of the décollement. It is shown that the critical slope is captured precisely by the proposed numerical scheme, the ramp, and the back thrust corresponding to regions of localized virtual strain. The influence of the back-wall friction on this critical slope is explored. It is found that the failure mechanism reduces to a thrust rooting at the base of the back wall and the absence of back thrust, for small enough values of the friction angle. This influence is well explained by the Mohr construction and further validated with experimental results with sand, considered as an analog material. 3D applications of the same methodology are presented in a companion paper.     

路堤边坡稳定可靠度计算中的模型不确定性分析

基于均质路堤边坡Monte Carlo法的稳定可靠度计算,分析了临界滑面搜索策略和稳定分析方法两类模型不确定性对边坡可靠度的影响特性,讨论了边坡失效概率随土工参数变异性的变化规律。研究表明,选用不同的临界滑面搜索策略所得可靠度结果差异不大,参数滑面法（overall slope）的失效概率略大于均值滑面法（global minimum）,但差别对边坡稳定性分析没有实质性影响;土性参数变异水平是影响边坡可靠度的最重要因素,边坡在相同设计参数安全系数下的可靠度指标随参数变异性增大而急剧降低;不同稳定性分析方法对应的安全系数概率密度函数曲线形态基本一致,但失效概率差异明显,因此目标可靠度指标取值应与稳定性分析方法相适应。提出的考虑土工参数变异水平的安全系数取值修正原则,对改进确定性设计的边坡稳定分析技术有积极意义。