Probabilistic assessment of tunnel convergence considering spatial variability in rock mass properties using interpolated autocorrelation and response surface method

This study aims at the probabilistic assessment of tunnel convergence considering the spatial variability in rock mass properties. The method of interpolated autocorrelation combined with finite difference analysis is adopted to model the spatial variability of rock mass properties. An iterative procedure using the first-order reliability method(FORM) and response surface method(RSM) is employed to compute the reliability index and its corresponding design point. The results indicate that the spatial variability considerably affects the computed reliability index. The probability of failure could be noticeably overestimated in the case where the spatial variability is neglected. The vertical scale of fluctuation has a much higher effect on the probabilistic result with respect to the tunnel convergence than the horizontal scale of fluctuation. And the influence of different spacing of control points on the computational accuracy is investigated.     

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.     

Reliability Analysis of Quick Sand Condition

First order reliability method (FORM) is generally used for reliability analysis in geotechnical engineering. This article adopts generalized regression neural network (GRNN) based FORM, Gaussian process regression (GPR) based FORM and multivariate adaptive regression spline (MARS) based FORM for reliability analysis of quick sand condition. GRNN is related to the radial basis function (RBF) network. GPR is developed based on probabilistic framework. MARS is a nonparametric regression technique. A comparative study has been carried out between the developed models. The performance of GPR based FORM and MARS based FORM match well with the FORM. This article gives the alternative methods for reliability analysis of quick sand condition.     

Long embankment failure accounting for longitudinal spatial variation – A probabilistic study

For long earth embankments or levees, it is of interest to investigate the slope failure mode in the longitudinal direction. However, this is less commonly discussed in comparison to the plane-strain failure mode. In this paper, the longitudinal failure mode of a long embankment consisting of homogeneous soils is examined. A probabilistic approach using the first-order reliability method (FORM) is adopted to consider the uncertainty of soil properties. In particular, the spatial variability of the undrained shear strength of the soil is modelled in the probabilistic analysis. Parametric studies are subsequently conducted to examine the influence of this soil characteristic on the failure mode of the long embankment.     

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.     

Reliability-based design and its complementary role to Eurocode 7 design approach

Reliability-based design (RBD) can play a useful complementary role to overcome some limitations in the Eurocode 7 (EC7) design approach, for example in situations with parameters not covered in EC7, different parametric sensitivities across different problems, cross-correlated or spatially correlated parameters, design aiming at a target reliability or failure probability, or when uncertainty in unit weight of soil is modeled. The complementary role played by RBD under these circumstances is illustrated and discussed for a shallow foundation, a reinforced rock slope, a Norwegian clay slope with spatial variability, a laterally loaded pile requiring implicit numerical analysis, and an anchored sheet pile wall. A pragmatic RBD approach involving first-order reliability method (FORM) only and a more rigorous RBD approach involving both first-order and second-order reliability method (SORM) are offered. Both approaches are implementable using either spreadsheet-based FORM and SORM procedures, or using various commercially available FORM/SORM packages.     

Seismic reliability analysis of earth slopes under short term stability conditions

Different models were developed for evaluating the probabilistic three-dimensional (3-D) stability analysis of earth slopes and embankments under earthquake loading. The 3-D slope stability model assumed is that of a simple cylindrical failure surface. The probabilistic models evaluate the probability of failure under seismic loading considering the randomness of earthquake occurrence, and earthquake induced acceleration and uncertainties stemming from the discrepancies between laboratory-measured and in-situ values of shear strength parameters. The models also takes into consideration the spatial variabilities and correlations of soil properties. The probabilistic analysis and design approach is capable of obtaining the 2-D and 3-D static and dynamic safety factors, the probability of slope failure, the earthquake induced acceleration coefficient, the yield acceleration coefficient, the earthquake induced displacement, and the probability of allowable displacement exceedance taking into account the local site effect. The approach is applied to a well known landslide case: Congress Street Landslide in Chicago. A sensitivity analysis was conducted on the different parameters involved in the models by applying those models to the Congress Street landslide considering different levels of seismic hazard. Also, a sensitivity analysis was carried out to study the sensitivity of computed results to input parameters of undrained shear strength, and corrective factors. A comparison was made between the different models of failure. The parametric study revealed that the hypocentral distance and earthquake magnitude have major influence on the earthquake induced displacement, probability of failure and dynamic 2-D and 3-D safety factors.     

边坡可靠度分析的支持向量机法

提出了基于支持向量机（SVM）的边坡可靠度分析新算法。该方法采用均匀设计确定样本点,通过一定数量的确定性计算来训练SVM,拟合边坡的功能函数;采用一阶可靠度方法（FORM）和迭代算法优化SVM模型,获得可靠度指标和验算点信息;在SVM模型基础上进一步通过二阶可靠度方法（SORM）和蒙特卡罗模拟（MCS）计算边坡的失稳概率。以两个典型边坡为例,通过与其他方法比较,证明了该方法的准确性和高效性。结果表明：提出的在标准正态空间（U空间）中取样并构建SVM,在原始空间（X空间）中计算功能函数的算法,有效地解决了具有相关非正态分布变量的可靠度分析问题,并且可很容易扩展到SORM的计算。算例结果证明,该方法的精度高于FORM;而效率优于MCS。分析过程中,边坡安全系数计算和可靠度分析相互独立。因此,该方法既适用于具有显式功能函数的简单问题,也适用于需要软件计算安全系数的实际边坡问题。     

Full probabilistic design of slopes in spatially variable soils using simplified reliability analysis method

ABSTRACT

A simplified reliability analysis method is proposed for efficient full probabilistic design of soil slopes in spatially variable soils. The soil slope is viewed as a series system comprised of numerous potential slip surfaces and the spatial variability of soil properties is modelled by the spatial averaging technique along potential slip surfaces. The proposed approach not only provides sufficiently accurate reliability estimates of slope stability, but also significantly improves the computational efficiency of soil slope design in comparison with simulation-based full probabilistic design. It is found that the spatial variability has considerable effects on the optimal slope design.     

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.     

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).     

Effect of 2D spatial variability on slope reliability: A simplified FORM analysis

To meet the high demand for reliability based design of slopes, we present in this paper a simplified HLRF(Hasofere Linde Rackwitze Fiessler) iterative algorithm for first-order reliability method(FORM). It is simply formulated in x-space and requires neither transformation of correlated random variables nor optimization tools. The solution can be easily improved by iteratively adjusting the step length. The algorithm is particularly useful to practicing engineers for geotechnical reliability analysis where standalone(deterministic) numerical packages are used. Based on the proposed algorithm and through direct perturbation analysis of random variables, we conducted a case study of earth slope reliability with complete consideration of soil uncertainty and spatial variability.     

Reliability Analysis of Earth Slopes Considering Spatial Variability

The paper presents a computational procedure for reliability analysis of earth slopes considering spatial variability of soils under the framework of the Limit Equilibrium Method. In the reliability analysis of earth slopes, the effect of spatial variability of soil properties is generally included indirectly by assuming that the probabilistic critical slip surface is the same as that determined without considering spatial variability. In contrast to this indirect approach, in the direct approach, the effect of spatial variability is included in the process of determination of the probabilistic critical surface itself. While the indirect approach requires much less computational effort, the direct approach is definitely more rigorous. In this context this paper attempts to investigate, with the help of numerical examples, how far away are the results obtained from the indirect approach from that obtained from the direct approach. In both the approaches, it is required to use a model of discretization of random fields into finite random variables. A few such models are available in the literature for one-dimensional (1D) as well as two-dimensional (2D) spatial variability. The developed computational scheme is based on the First Order Reliability Method (FORM) coupled with the Spencer Method of Slices valid for limit equilibrium analysis of general slip surfaces. The study includes bringing out the computational advantages and disadvantages of the three commonly used discretization models. The sensitivity of the reliability index to the magnitudes of the scales of fluctuation has also been studied.     

Effects of anisotropy in correlation structure on the stability of an undrained clay slope

Slopes are mainly naturally occurred deposits, so slope stability is highly affected by inherent uncertainty. In this paper, the influence of heterogeneity of undrained shear strength on the performance of a clay slope is investigated. A numerical procedure for a probabilistic slope stability analysis based on a Monte Carlo simulation that considers the spatial variability of the soil properties is presented to assess the influence of randomly distributed undrained shear strength and to compute reliability as a function of safety factor. In the proposed method, commercially available finite difference numerical code FLAC 5.0 is merged with random field theory. The results obtained in this study are useful to understand the effect of undrained shear strength variations in slope stability analysis under different slope conditions and material properties. Coefficient of variation and heterogeneity anisotropy of undrained shear strength were proven to have significant effect on the reliability of safety factor calculations. However, it is shown that anisotropy of the heterogeneity has a dual effect on reliability index depending on the level of safety factor adopted.     

Two Methods for Predicting Reliability Index and Critical Probabilistic Slip Surface of Soil Slopes

Two methods of reliability analysis of soil slopes are studied, and the representative flow charts of both methods are illustrated. Method 1 can predict the reliability index and the critical probabilistic slip surface directly and it is computational efficient, but it needs the development of new codes for integrating the reliability analysis code and the slope stability code. Method 2 makes the reliability analysis code call the slope stability analysis code directly, and each code can be considered as an intact part. The main result of Method 2 is the reliability index of soil slope. Combined with the proposed method for locating the critical slip surface, Method 2 can also predict the probabilistic slip surface. Although Method 2 needs much more callings of the subprogram of slope stability analysis code, it needs not the developing of new computer program. Thus, Method 2 is easy to use and can be applied to different reliability analysis methods and slope stability analysis methods.     

基于多源试验数据空间变异土体参数概率反演及边坡可靠度更新

岩土工程现场勘察试验通常只能获得有限的试验数据,据此难以真实地量化土体参数的空间变异性。提出了考虑土体参数空间变异性的概率反演和边坡可靠度更新方法,基于室内和现场两种不同来源的试验数据概率反演空间变异参数统计特征和更新边坡可靠度水平,并给出了计算流程。此外为合理地描述土体参数先验信息,发展了不排水抗剪强度非平稳随机场模型。最后通过不排水饱和黏土边坡算例验证了提出方法的有效性,并探讨了试验数据和钻孔位置对边坡后验失效概率的影响。结果表明：提出方法实现了空间变异土体参数概率反演与边坡可靠度更新的一体化,基于有限的多源试验数据概率反演得到的土体参数均值与试验数据非常吻合,明显降低了对参数不确定性的估计,更新的边坡可靠度水平显著增加。受土体参数空间自相关性的影响,试验数据对钻孔取样点附近区域土体参数统计特征更新的影响明显大于距离取样点较远区域。     

基于响应面法的边坡稳定二阶可靠度分析

提出了基于响应面法的边坡稳定二阶可靠度分析的实用算法。选择U空间中的随机变量,通过空间变换和相关矩阵分解,计算试验点的功能函数;通过迭代算法构造响应面、以确保通过最小的计算量获得最优精度,并在此基础上进行FORM/SORM计算。以一岩石边坡的平面滑动问题为例,通过与蒙特卡洛模拟、FORM及随机响应面法的比较,证明了该方法的准确性和高效性。分析了参数的相关性及试验点取值范围对计算结果的影响,讨论了可靠度分析结果中参数敏感性和物理属性问题。该方法可为实际边坡问题的可靠度分析提供参考,并可以用来进行基于可靠度分析的加固设计。     

基于模糊随机理论的广义可靠度在边坡稳定性分析中的应用

在常规的概率极限状态理论基础上 ,分析了边坡体安全状态的模糊性 ,在此基础上构造了基于MCS以及FORM算法的模糊随机可靠度算法 ,分别就边坡体的滑动失效、渗透破坏两种失效模式作了参数敏感性分析 ,并进一步对边坡体的系统失效模式作了探讨     

Uncertainty and Reliability Analysis Applied to Slope Stability: A Case Study From Sungun Copper Mine

Nowadays, there are many new methods for slope stability analysis; including probabilistic methods assessing geotechnical uncertainties to develop safety factors. In this paper, a reliability index analysis for the Sungun copper mine slope stability is evaluated based on three methods of uncertainties consisting Taylor series method, Rosenblueth point estimate method and Monte-Carlo simulation method. Sungun copper mine will be one of the Iran’s biggest mines with final pit’s height of 700 meters. For this study two of its main slopes were assessed, one dipping to the NE (030) and the other to the SE (140). Probability density function of cohesion and angle of friction for the slopes were developed using limit equilibrium methods. These shear strengths were then used to determine the probability density function of safety factor and reliability index using the probabilistic methods. Results of the probabilistic analysis indicate that with ascending values of the uncertainties the reliability index decreases. Furthermore, it was determined that with the Monte Carlo simulation the seed number used has little effect on the reliability index of the safety factor especially with seed numbers in excess of 1200. Variations in the overall reliability index of safety factor were observed between the two slopes and this difference is explained by the differences in complexities of the geology within the cross-section.     

A Bayesian framework for updating model parameters while considering spatial variability

The study presents a recent slope failure in India which resulted in the burial of a village and claimed large number of lives. Current methods of probabilistic back analysis incorporate uncertainty in the analysis but do not consider spatial variability. In this study, back analysis is performed using Bayesian analysis in conjunction with random field theory. The probabilistic method is shown to be efficient in back-analysing a slope failure. It also provides confidence in parameter values to be used for post-failure slope design. The back analysis method which does not consider spatial variability overestimates the uncertainty in analysis, which can lead to uneconomical slope remediation design and measures.