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.     

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.     

The evaluation of the probability of rock wedge failure using the point estimate method

In a probabilistic analysis of rock slope stability, the Monte Carlo simulation technique has been widely used to evaluate the probability of slope failure. While the Monte Carlo simulation technique has many advantages, the technique requires complete information of the random variables in stability analysis; however, in practice, it is difficult to obtain complete information from a field investigation. The information on random variables is usually limited due to the restraints of sampling numbers. This is why approximation methods have been proposed for reliability analyses. Approximation methods, such as the first-order second-moment method and the point estimate method, require only the mean and standard deviation of the random variable; therefore, it is easy to utilize when the information is limited. Usually, a single closed form of the formula for the evaluation of the factor of safety is needed for an approximation method. However, the commonly used stability analysis method of wedge failure is complicated and cumbersome and does not provide a simple equation for the evaluation of the factor of safety. Consequently, the approximation method is not appropriate for wedge failure. In order to overcome this limitation, a simple equation, which is obtained from the maximum likelihood estimation method for wedge failure, is utilized to calculate the probability of failure. A simple equation for the direct estimation of the safety factors for wedge failure has been empirically derived from failed and stable cases of slope, using the maximum likelihood estimation method. The developed technique has been applied to a practical example, and the results from the developed technique were compared to the results from the Monte Carlo simulation technique.     

System Reliability Assessment for a Rock Tunnel with Multiple Failure Modes

This paper presents a practical procedure for assessing the system reliability of a rock tunnel. Three failure modes, namely, inadequate support capacity, excessive tunnel convergence, and insufficient rockbolt length, are considered and investigated using a deterministic model of ground-support interaction analysis based on the convergence–confinement method (CCM). The failure probability of each failure mode is evaluated from the first-order reliability method (FORM) and the response surface method (RSM) via an iterative procedure. The system failure probability bounds are estimated using the bimodal bounds approach suggested by Ditlevsen (1979), based on the reliability index and design point inferred from the FORM. The proposed approach is illustrated with an example of a circular rock tunnel. The computed system failure probability bounds compare favorably with those generated from Monte Carlo simulations. The results show that the relative importance of different failure modes to the system reliability of the tunnel mainly depends on the timing of support installation relative to the advancing tunnel face. It is also shown that reliability indices based on the second-order reliability method (SORM) can be used to achieve more accurate bounds on the system failure probability for nonlinear limit state surfaces. The system reliability-based design for shotcrete thickness is also demonstrated.     

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。     

应变软化土坡渐进破坏的可靠度分析

在考虑土性参数的不确定性和土体应变软化特性的基础上,以可靠度方法建立了边坡渐进破坏的极限状态方程。通过计算边坡局部破坏概率,确定边坡中最危险的土条,并推导出破坏起始于该土条,逐渐向两侧扩展的渐进破坏概率计算公式。该方法能很好地反映边坡局部破坏的产生、扩展及对边坡整体稳定性的影响,找到最贴近实际情况的破坏传播路径,为边坡稳定分析提供了理论依据。通过算例分析可知,滑动面上各个土条的局部破坏概率均不相等,破坏起始于中部第8土条,破坏先向上传递,后向下传递至坡脚,并非从坡脚向坡顶传递的牵引式破坏,或从坡顶向坡脚传递的推移式破坏。     

Study of the Engineering Geological Problems of the Havasan Dam,with Emphasis on Clay-Filled Joints in the Right Abutment

Havasan dam site is located in northwest of Iran. The planned concrete dam is to be built on Cretaceous limestone. Faulted and fractured limestone is exposed at the dam abutments and in the reservoir area. Rock mass properties including the deformation modulus and uniaxial compressive strength were calculated using different rock mass classification systems (RMR, Q, GSI and DMR). Laboratory tests indicate that joint filling materials contain clay with low to high plasticity (CL to CH) and low to medium potential swelling pressures. X-ray diffraction analysis confirms that the reason for potential swelling of joint fillings is the existence of clay minerals (such as illite and montmorillonite). The study results about the shear strength of clay-filled joints show that under JRC–JCS condition (laboratory scale), JRC _n –JCS _n (large scale) and normal stress equal to 0.25–4 MPa, the range of shear strength of clay-filled joints will be equal to 0.2–2.17 and 0.14–1.72 MPa. In some areas dissolution along the joints results in high permeability, especially in the right abutment. Three dominant joint sets occur in the exploration galleries which have been excavated in the right abutment. The maximum aperture of these joints varies from 7 to 9 cm, and the joints are typically filled with clay. Preliminary analysis shows that the presence of open joints which will cause seepage of water, combined with the impact of the clay-filled joints and forces acting on the slopes, could lead to slope failures and rock falls. In addition, the assessment of slope stability results in abutments using limited equilibrium method and Swedge software under dynamic and static conditions shows that two wedges formed on the slopes of the abutment by the natural joints are potentially unstable. The rock wedge on the left abutment is smaller but presents higher sliding potential. In addition, there is no probability of planar failure due to the geological condition of the dam abutments. This paper summarizes the site investigation and subsequent analysis, which resulted in a recommendation not to construct this site. We offer some potential mitigation plans to consider if a dam were to be built at this site.     

Development and application of a quantitative risk assessment to a very slow moving rock slope and potential sudden acceleration

The benefits of quantitative risk assessments for landslide management have been discussed and illustrated in several publications. However, there still are some challenges in its application for low-probability, high-magnitude events. These challenges are associated with the difficulties in populating our models for risk calculations, which largely require the input of expert opinion. This paper presents a quantitative risk assessment to a very slow moving rock slope within a dam reservoir in the Province of British Columbia, Canada. The assessment is focused on the risk to the population in the vicinity of the dam and the populated areas downstream. Expert opinions quantified the slope failure probabilities in the order of 10^?3 to 10^?1 per year for the smallest failure scenario considered and less than 10^?6 for a failure of the entire slope. However, these estimations are associated with high levels of uncertainty. Our approach starts with the calculation and assessment of the magnitude and probability of the potential slope failure consequences, minimizing the uncertainties associated with estimated slope failure probabilities. Then, these consequences and failure probabilities are combined to obtain a measure of risk. The uncertainty associated with the slope failure probabilities is managed by the estimation of plausible ranges for these. The calculated risk levels are then presented as ranges of values and assessed against adopted evaluation criteria. The consequence and risk assessment of the rock slope suggest that the risk to the population exposed in the vicinity of the dam and populated areas downstream is under adequate control. The probability of large consequence scenarios is extremely low, in the order of 10^?7 chance of an event causing more than 100 fatalities. We propose an observational technique to assess changes in risk levels and decide when to update the risk management approach or deploy emergency measures. The technique is focused on the detection of changes in the slope deformation patterns that would indicate an increase in the potential failure volumes or an imminent failure. It can be considered an extension to the current early warning system in place, easy to implement and enhanced with the strength of the comprehensive analysis required for a quantitative risk assessment.     

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。     

节理岩体关键块体稳定的概率分析

基于地下岩体受节理面的控制,节理面的几何和力学参数随机分布,从而导致岩体系统具有高度不确定性,提出以关键块体理论为基础,考虑节理几何和力学参数随机性的岩体开挖可靠度分析方法,并给出了块体稳定的总失效概率评价模型。以澳大利亚阿德莱德地区一铜矿地质条件为例,以节理面倾角、倾向、摩擦系数和黏聚力为随机变量,通过Monte Carlo模拟和概率图方法,进行了岩体可靠度和失效概率的计算。最后,采用条件概率的分析方法,计算了单面滑动块体的总失效概率。计算结果表明,块体沿单面滑动并且出现的概率为11.0%,总的失效概率为3.85%,超过一般岩体工程可允许的风险水平,认为该方法可以作为评价块体可靠性的依据。     

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.     

基于广义Hoek-Brown强度准则的PEM节理边坡可靠性分析

广义Hoek-Brown强度准则(简称GH-B准则)作为估计完整岩石或节理岩体剪切强度的半经验准则,已成为岩体强度预测及数值计算领域中应用最广的准则之一。经验公式中涉及3个与岩体特性有关的材料参数m_b、s和a,实际应用中还不能直接对这3个参数进行精确取值,只能通过中间变量推导或结合地质经验来确定。为尽可能大地消除因用Hoek-Brown经验公式(简称H-B经验公式)的中间变量推导带来的累积误差,消除边坡稳定性评价结果的偏差,提出了从基本随机变量着手,考虑基本变量的可变性与不确定性,借助可靠性理论来综合分析评价边坡潜在破坏概率和风险程度的方法。研究基于GH-B准则的节理岩质边坡的可靠性问题,首先从决定H-B经验公式的4个基本参数GSI、σ_ci、mi和D出发,研究变量的分布概型及取值范围,然后引入到H-B经验公式中去,利用有效地可靠性计算方法求解边坡的失稳概率,将统计矩点估计法(PEM法)引入到节理岩质边坡失稳概率的求解中,通过对一开挖桥墩承台边坡的可靠性分析表明,基于GH-B准则的PEM方法提高了稳定性评价的客观性,结果更加真实可靠。      

锚杆多种失效模式与双滑块边坡锚固系统可靠性分析

考虑锚杆拉杆拉断、拉杆从注浆体中拔出、锚固段注浆体从岩体中拔出、外锚头破坏以及垫墩底岩体的压坏等失效模式,利用系统可靠性原理和极限平衡分析方法,建立了双滑块边坡多锚杆锚固系统可靠性分析模型。基于蒙特卡罗随机抽样原理提出了该类边坡锚固系统破坏概率的直接求解方法。最后结合算例,分别基于中值安全系数和破坏概率指标分析了各计算参数对计算结果的影响,并讨论了锚杆锚固角和被动滑块可能滑裂面倾角对锚固系统稳定性的影响。     

Probabilistic Stability Evaluation of Oppstadhornet Rock Slope,Norway

Probabilistic analyses provide rational means to treat the uncertainties associated with underlying parameters in a systematic manner. The stability of a 734-m-high jointed rock slope in the west of Norway, the Oppstadhornet rock slope, is investigated by using a probabilistic method. The first-order reliability method (FORM) is used for probabilistic modeling of the plane failure problem in the rock slope. The Barton–Bandis (BB) shear strength criterion is used for the limit state equation. The statistical distributions of the BB criterion parameters, for which comprehensive data were collected and statistically analyzed, are determined by using distribution fitting algorithms. The sensitivity of the FORM model for the BB criterion is also investigated. It is found that the model is most sensitive to the mean value of the residual friction angle (ϕ _r) and least sensitive to the mean value of the slope angle (β _f). It is also found that the standard deviation of joint compressive strength (JCS) causes the greatest difference in the reliability index, which has the least sensitivity to the change in the mean and standard deviation of joint roughness coefficient (JRC).     

边坡渐进破坏的模糊随机可靠性研究

研究边坡的渐进破坏过程对边坡稳定性的影响,认为边坡破坏是由局部向整体扩展的渐进过程,它具有随机性和模糊性双重特性.渐进破坏模糊随机法能很好地反映边坡体内局部破坏的产生、扩展及对边坡整体可靠性的影响.基于岩土体的空间变异性,考虑了随机变量和极限状态函数的模糊性,以模糊随机变量为基本变量建立了边坡渐进破坏的模糊极限状态方程...     

Influence of heterogeneity on 3D slope reliability and failure consequence

This paper investigates the influence of heterogeneity of undrained shear strength on the reliability of, and risk posed by, a long slope cut in clay, for different depths of foundation layer. The clay has been idealised as a linear elastic, perfectly plastic Von Mises material and its spatial variability has been modelled using random field theory, whereas slope performance has been computed using a parallel 3D finite element program. The results of Monte Carlo simulations confirm previous findings that three categories of failure mode are possible and that these are significantly influenced by the horizontal scale of fluctuation relative to the slope geometry. In particular, discrete 3D failures are likely for intermediate scales of fluctuation and, in this case, reliability is a function of slope length. The risk posed by potential slides has been quantified in terms of slide volumes and slide lengths, which have been estimated by considering the computed out-of-face displacements. The results show that, for a given horizontal scale of fluctuation relative to the slope geometry, there is a wide range of possible slide volumes and slide geometries. Indeed, the results highlight just how difficult it is to compute a 2D slope failure in a heterogeneous soil. However, they also indicate that, for low probabilities of failure, the volumes of potential slides can be small. This suggests that, for some problems, it may not be necessary to design to very small probabilities of failure, due to the reduced consequence of failure in this case. The techniques developed in this paper will be important in benchmarking simpler 2D and 3D solutions used in design, as there is a need to quantify slide geometries when benchmarking simpler methods based on predefined failure mechanisms.     

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。     

基于节理不确定性的可靠度分析

边坡稳定性一直是边坡安全的重点研究对象,针对边坡评价中常见的不确定性因素,可靠度分析是值得利用的方法。为评价某节理发育的岩质岸坡稳定性,通过有限元计算软件,结合现场勘探测绘数据,建立以边坡节理强度参数c、φ为输入变量,安全系数为输出变量的点估计（PEM）计算概率模型,计算结果表明:节理发育对该边坡变形具有明显控制作用;边坡整体可靠性较好,破坏概率极低。最后,通过蒙托卡罗法对可靠度结果进行验证,结果表明两种方法的计算结果不存在显著性差异。研究结果表明节理对岩质边坡稳定具有良好的敏感性,基于节理不确定性的点估计法分析边坡可靠度是一种有效的方法。     

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.     

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。