双滑块边坡锚固系统时变可靠性分析

考虑锚杆锚固段从岩体中拔出、拉杆拉断、拉杆从注浆体中拔出等失效模式,利用系统可靠性原理和极限平衡分析方法,并基于Monte-Carlo抽样原理提出双滑块岩质边坡锚固系统破坏概率的直接求解方法。同时考虑锚杆钢筋的腐蚀与软弱滑动面抗剪强度c、? 的时变性,建立了考虑锚杆多失效模式双滑块岩质边坡锚固系统的时变可靠性模型。算例计算结果表明：软弱滑动面上的强度参数c、? 的时变性和注浆体与围岩之间的抗力时变性对锚固系统的破坏概率的影响较大,而锚杆的腐蚀对锚固系统破坏概率的影响不明显。     

基于克里金代理模型与子集模拟的边坡高效可靠度分析

作为一种高效且准确的代理模型,克里金方法近年来被广泛用于边坡高效可靠度分析。然而,传统方法一般直接将克里金模型与蒙特卡洛模拟耦合进行可靠度分析,导致其在高维小失效概率的边坡可靠度计算中容易出现内存占用过大甚至溢出而无法求解的问题。为此,提出一种基于克里金代理模型的子集模拟方法,以高效解决小概率水平的边坡可靠度分析问题。该方法首先采用一定数量的样本校准克里金模型并进行精度验证,然后基于构建的模型开展子集模拟边坡可靠度计算。最后,采用一个单层粘性土坡与一个工程实例土坡验证所提方法的有效性,并研究回归模型、相关函数模型以及训练样本对该方法精度的影响。结果表明：（1）该方法可以有效计算边坡的失效概率,并且比传统方法更高效;（2）构建克里金模型时,采用10倍随机变量数的训练样本即可得到满足计算精度需求的模型,而额外增加训练样本对计算结果影响较小。     

A system reliability approach for evaluating stability of rock wedges with correlated failure modes

This paper proposes a system reliability approach for evaluating the stabilities of rock wedges considering multiple correlated failure modes. A probabilistic fault tree is employed to model the system aspects of the problem. The system reliability analysis is performed using an N-dimensional equivalent method taking into account correlations between different failure modes. Reliability sensitivity analyses at three different levels, namely, single limit state function level, single failure mode level, and system reliability level, were carried out to study the effect of changes in variables on the stability of the wedge. An example case was analysed to illustrate the proposed approach. The stability of the wedge can be evaluated efficiently using the proposed system reliability approach in a more systematic and quantitative way. The probabilities of failure of the wedge from the N-dimensional equivalent method are fairly consistent with those from the Monte Carlo simulation method. The results demonstrate that the probability of failure will be overestimated if the correlations between different failure modes of the wedge are not taken into account. They also demonstrate that the relative importance of different failure modes to the system reliability of the wedge can differ considerably and be treated systematically and quantitatively by the proposed approach. The sensitivity results are highly dependent on the selected sensitivity analysis level.     

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.     

Efficient system reliability analysis illustrated for a retaining wall and a soil slope

Although first-order reliability method is a common procedure for estimating failure probability, the formulas derived for bimodal bounds of system failure probability have not been widely used as expected in present reliability analyses. The reluctance for applying these formulas in practice may be partly due to the impression that the procedures to implement the system reliability theory are tedious. Among the methods for system reliability analysis, the approach suggested in Ditlevsen 1979 is considered here because it is a natural extension of the first-order reliability method commonly used for failure probability estimation corresponding to a single failure mode, and it can often provide reasonably narrow failure probability bounds. To facilitate wider practical application, this paper provides a short program code in the ubiquitous Excel spreadsheet platform for efficiently calculating the bounds for system failure probability. The procedure is illustrated for a semi-gravity retaining wall with two failure modes, a soil slope with two and eight failure modes, and a loaded beam with three failure modes. In addition, simple equations are provided to relate the correlated but unrotated equivalent standard normals of the Low and Tang 2007 FORM procedure with the uncorrelated but rotated equivalent standard normals of the classical FORM procedure. Also demonstrated are the need for investigating different permutations of failure modes in order to get the narrowest bounds for system failure probability, and the use of SORM reliability index for system reliability bounds in a case where the curvature of the limit state surface cannot be neglected.     

甘肃黄土高边坡可靠度研究

以甘肃省266个黄土自然极限状态坡为研究对象,将其按地形地貌、地层岩性等特征划分为4个亚区,对每个亚区的极限状态坡高与坡宽进行双对数线性模型回归和不同置信度下的统计分析,给出4个亚区不同置信度下坡高与坡宽的相关关系。以1 024组黄土物理力学指标统计结果作为计算参数,用Bishop法的稳定系数计算公式建立极限状态方程,利用 Monte Carlo法对各亚区不同置信度下的边坡进行可靠度模拟,系统分析了坡高、坡度、参数变异性对失效概率的影响,以及失效概率与稳定系数的关系。结果表明,临洮-永靖亚区边坡坡度低于31.5?,失效概率可控制在10%以下;陇东亚区、兰州-定西-会宁亚区和天水-秦安-通渭亚区在强度参数变异系数取当地平均水平,稳定系数为1.3情况下失效概率在10%以内,稳定系数为1.2情况下,失效概率多在20%以内。     

岩质边坡渐进破坏的三维随机分析

层间错动带是岩体结构中普遍存在的一种地质现象,含错动带的岩质边坡三维分析更是岩土工程关注的难点问题。结合国内某高拱坝工程,根据岩体层间错动带的分布状况,利用等厚度节理单元模拟了坝区22层错动带和软弱夹层,利用Mohr-Coulomb准则确定节理单元的破环方式,提出了岩质边坡三维滑动面的确定方法,该方法考虑了岩石和错动带（软弱夹层）的随机性。利用弹塑性理论、可靠度理论和破坏追踪相结合的方法,通过3 000多次有限元的计算,得到了边坡滑动面的位置、形态、破坏的先后顺序、滑动的方向以及滑动到任何程度的概率度量。含层间错动带的岩质边坡渐进破坏模式可概括为边坡的破坏起始于层间错动带的剪切破坏。随着各层间错动带的破坏、边坡结构性能恶化,部分较薄的岩层变成不同支撑形式的板,然后部分岩层开始破坏,边坡开始滑动,形成一个或几个破坏台阶,直至失稳。     

Efficient subset simulation for evaluating the modes of improbable slope failure

For analyzing low probability slope failures, a modified version of subset simulation, based on performance-based subset selection rather than the usual probability-based subset selection, is combined with the random finite element method. The application to an idealized slope is used to study the efficiency and consistency of the proposed method compared to classical Monte Carlo simulations and the shear strength reduction (SSR) method. Results demonstrate that failure events taking place without strength reduction have different modes of failure than stable slopes brought to failure by SSR. The correlation between sliding volume and factor of safety is also demonstrated.     

Plane failure analysis of rock slopes

Summary Hoek and Bray (1981) gave an analytical approach for plane failure analysis for rock slopes that is limited to those slopes in which the upper slope surface is horizontal and the tension crack is vertical. An analysis is presented here which can take these factors into account. It is found that varying the angle of the upper slope from 0° to 30° causes a significant reduction in the factor of safety. Varying the tension crack from vertical to 70° only has an effect when the upper slope angle is less than 20°.     

基坑体系可靠度的条件概率计算方法

基坑存在多种失效模式,考虑失效模式之间的相关性,双界限法计算体系失效概率存在计算结果区间范围较大的弊端。利用均匀试验和非参数回归方法建立响应面,在响应面的基础上,对Monte Carlo模拟生成的随机参数进行插值,得到各个失效模式指标,结合Pearson相关系数检验两两失效模式之间的相关性,用条件概率方法计算基坑体系失效概率,提出了基于条件概率考虑多失效模式相关的基坑体系可靠度分析方法。在此基础上,通过1个典型算例进行对比分析,计算结果表明,该方法不仅计算简便,而且结果可靠,其结果可为基坑体系可靠度分析理论提供一条新的途径。     

System probabilistic model of rock slope stability considering correlated failure modes

In this paper, a system reliability analysis of rock slope stability with considering all input parameters as stochastic parameter is presented. To perform reliability analysis a cut-set system has been used. For this purpose, Sequential Compounding Method (SCM) as a powerful method for reducing the computational time and accurate evaluation is employed to determine the reliability indices with considering correlations between failure modes which are calculated by defining equivalent linear safety margin for each failure mode. Furthermore, the 3-D system probability of failure surface is presented and the probabilistic model is developed to evaluate the rock slope probability of failure.     

Influence of rock property correlation on reliability analysis of rock slope stability: From property characterization to reliability analysis

Cohesion(c) and friction angle(φ) of rock are important parameters required for reliability analysis of rock slope stability. There is correlation between c and φ which affects results of reliability analysis of rock slope stability. However, the characterization of joint probability distribution of c and φ through which their correlation can be estimated requires a large amount of rock property data, which are often not available for most rock engineering projects. As a result, the correlation between c and φ is often ignored or simply assumed during reliability studies, which may lead to bias estimation of failure probability. In probabilistic rock slope stability analysis, the influence of ignoring or simply assuming the correlation of the rock strength parameters(i.e., c and φ) on the reliability of rock slopes has not been fully investigated. In this study, a Bayesian approach is developed to characterize the correlation between c and φ, and an expanded reliability-based design(RBD) approach is developed to assess the influence of correlation between c and φ on reliability of a rock slope. The Bayesian approach characterizes the sitespecific joint probability distribution of c and φ, and quantifies the correlation between c and φ using available limited data pairs of c and φ from a rock project. The expanded RBD approach uses the joint probability distribution of c and φ obtained through the Bayesian approach as inputs, to determine the reliability of a rock slope. The approach gives insight into the propagation of the correlation between c and φ through their joint probability into the reliability analysis, and their influence on the calculated reliability of the rock slope. The approaches may be applied in practice with little additional effort from a conventional analysis. The proposed approaches are illustrated using real c and φ data pairs obtained from laboratory tests of fractured rock at Forsmark, Sweden.     

Probabilistic reliability analysis of multiple slopes with genetic algorithms

Uncertainty estimation and consideration in engineering is an important practice to design reliable structures especially in geotechnics since the level of control with regards to the material parameters is relatively low. The definition of reliability indices to approximate the probability of failure allows for a better assessment of stability with fewer computations than using alternative methods. Nonetheless, yet an optimisation problem needs to be solved. In this work, a genetic algorithm is developed to solve this optimisation problem considering the limit equilibrium method to search for multiple critical failures. Study cases are presented to illustrate the capabilities of the method.     

基于可靠度理论的滑坡稳定性及其影响因素分析

基于可靠度理论,利用Morgenstern-Price法、Ordinary法、Bishop法和Janbu法等4类极限平衡方法对台州市北水南调工程朱溪引水段Ⅷ号滑坡的稳定性进行了分析,设计了最不利组合、最有利组合、平均值组合的确定性分析方案,目的是为可靠度计算结果提供对比基准;设计了Monte-Carlo循环次数为5 000,50 000,500 000,5 000 000次等4种计算工况,岩土参数的概率分布函数为正态分布、三角形分布和均匀分布等3种情况,方差等于0.5,1.0,2.0,3.0,4.0,5.0倍初始实验方差等6种计算方案,岩土参数间相关性在-1.0～1.0间按0.25等幅变化的9种情况。研究发现,循环次数仅对安全系极值有较大影响,对可靠度指标影响不大;分布函数和数据离散性均会对结果精度造成影响;参数相关性仅对失效概率造成较大影响。     

基于不平衡推力法的边坡可靠度分析

Rosenblueth方法是一种简单实用的可靠度指标计算方法,不需要了解各状态变量分布函数,只要利用它们的均值和方差,就可求得边坡的可靠指标。推导了下滑力超载储备安全系数的显式表达式,以此式作为边坡可靠度分析的状态函数,结合Rosenblueth方法的优点,给出了一种边坡稳定可靠度分析的实用方法。     

隧道结构系统可靠度研究

在结构系统可靠度研究的基础上,根据隧道结构的特点,应用概率理论及工程结构系统可靠度分析方法,对整个隧道结构系统可靠度进行了探讨。隧道结构系统可靠度研究包括衬砌断面可靠度、在各种不同围岩压力作用下衬砌断面可靠度、同种围岩地段衬砌可靠度以及整座隧道可靠度的研究。整座隧道可看作由两端洞门和洞内不同围岩地段隧道衬砌所组成的串联系统,任一围岩地段衬砌和任一洞门结构的破损,都认为该隧道破损,则运用“概率网络估算技术”（又称PNET法）可求得整座隧道系统的失效概率与可靠指标。通过实例计算得到,整座隧道系统的总体可靠指标比所有单段结构的可靠指标都要低。     

Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables

This paper proposes a stochastic response surface method for reliability analysis involving correlated non-normal random variables, in which the Nataf transformation is adopted to effectively transform the correlated non-normal variables into independent standard normal variables. Transformations of random variables that are often used in reliability analyses in terms of standard normal variables are summarized. The closed-form expressions for fourth to sixth order Hermite polynomial chaos expansions involving any number of random variables are formulated. The proposed method will substantially extend the application of stochastic response surface method for reliability problems. An example of reliability analysis of rock slope stability with plane failure is presented to demonstrate the validity and capability of the proposed stochastic response surface method. The results indicate that the proposed stochastic response surface method can evaluate the reliability of rock slope stability involving correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that for the first-order reliability method, and it is much more efficient than direct Monte-Carlo simulation. The results also show that the number of collocation points selected should ensure that the Hermite polynomial matrix has a full rank so that different order SRSMs can produce a robust estimation of probability of failure for a specified performance function. Generally, the accuracy of SRSM increases as the order of SRSM increases.     

双临空面岩质边坡滑动与倾倒破坏的运动学分析

临空面的几何形状在边坡破坏模式与稳定性分析中起着举足轻重的作用。运动学分析是确定边坡破坏模式与评价边坡稳定性的一种有效方法。目前基于运动学分析的边坡稳定性研究主要集中于单临空面边坡的破坏模式与最大安全开挖边坡角的确定。本文将此项研究扩展至双临空面边坡,将其破坏模式细分为4种,分别为沿结构面发生单平面滑动、沿结构面发生楔形体滑动、沿两个结构面的交线发生楔形体滑动以及倾倒破坏。在立体投影中得出,平滑滑动与楔形体滑动的滑动区为双临空面真倾向线与摩擦圆所组成的区域,单个结构面倾角矢量与两个结构面交线矢量位于该区城内;倾倒破坏区为双临空面真倾向线、摩擦圆与基圆所组成的区域,结构面的法向矢量位于该区城内。提出了双临空面边坡最大安全开挖边坡角的确定方法及边坡设计原则。最后将上述方法应用于三峡库区湖北省秭归县郭家坝镇郭家坝村生基坡高边坡,研究了该双临空面边坡的破坏模式并给出了最大安全边坡角的建议值     

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.