Probabilistic analysis of reinforced slopes using RFEM and considering spatial variability of frictional soil properties due to compaction

Probabilistic stability analyses of constructed wrapped-face reinforced slopes (or embankments) using frictional soils were carried out using the random finite element method (RFEM). Soil properties reported in the literature for unsaturated frictional fills compacted to different densities were used in the simulations. Bar elements were added to the RFEM code to simulate extensible geosynthetic reinforcement layers and the Davis approach was used to improve numerical stability for purely frictional soil slopes at collapse. The influence of isotropic and anisotropic spatially variable soil strength was investigated and shown to have a large influence on the variation of maximum mobilised tensile forces in reinforcement layers for the steep 5 m-high slopes in the study. The influence of fill placed at different layer thickness and compacted to different levels was simulated by adjusting the soil strength and unit weight, and the vertical strength correlation length in the anisotropic spatially variable strength field used in each slope realisation. Numerical results showed that vertical strength correlation lengths approaching the magnitude of fill lift heights can control the probability of failure for reinforced slopes constructed with weak fills placed in lift heights close to but less than the wrapped reinforcement spacing used in the study.     

Probabilistic assessment of soil liquefaction considering spatial variability of CPT measurements

The conventional liquefaction potential assessment methods (also known as simplified methods) profoundly rely on empirical correlations based on observations from case histories. A probabilistic framework is developed to incorporate uncertainties in the earthquake ground motion prediction, the cyclic resistance prediction, and the cyclic demand prediction. The results of a probabilistic seismic hazard assessment, site response analyses, and liquefaction potential analyses are convolved to derive a relationship for the annual probability and return period of liquefaction. The random field spatial model is employed to quantify the spatial uncertainty associated with the in-situ measurements of geotechnical material.     

Effect of spatial variability of cross‐correlated soil properties on bearing capacity of strip footing

Geotechnical engineering problems are characterized by many sources of uncertainty. Some of these sources are connected to the uncertainties of soil properties involved in the analysis. In this paper, a numerical procedure for a probabilistic analysis that considers the spatial variability of cross‐correlated soil properties is presented and applied to study the bearing capacity of spatially random soil with different autocorrelation distances in the vertical and horizontal directions. The approach integrates a commercial finite difference method and random field theory into the framework of a probabilistic analysis. Two‐dimensional cross‐correlated non‐Gaussian random fields are generated based on a Karhunen–Loève expansion in a manner consistent with a specified marginal distribution function, an autocorrelation function, and cross‐correlation coefficients. A Monte Carlo simulation is then used to determine the statistical response based on the random fields. A series of analyses was performed to study the effects of uncertainty due to the spatial heterogeneity on the bearing capacity of a rough strip footing. The simulations provide insight into the application of uncertainty treatment to geotechnical problems and show the importance of the spatial variability of soil properties with regard to the outcome of a probabilistic assessment. Copyright © 2009 John Wiley & Sons, Ltd.     

表层土渗透系数空间变异与随机模拟研究

在汉江遥堤典型堤段对堤后200 m?100 m范围内表层土取样进行渗透性试验,基于试验结果分析了土体渗透系数的随机性和结构性的空间变异特征,结果表明细砂渗透系数在水平方向具有各向同性,空间结构可用指数模型描述。采用转向带法进行渗透系数随机场模拟,从基本统计量、空间结构等方面进行的分析表明,转向带法模拟结果保持了原随机场的统计特性,具有较好的收敛性。     

考虑参数空间变异性多层土坡系统可靠度分析

多层土坡在岩土工程实际中十分常见,不仅土体参数存在一定的空间变异性,而且土体框架呈现明显的层状分布特征,然而目前对考虑土体参数空间变异性的多层土坡稳定可靠度研究的远远不够。提出了基于多重响应面边坡系统可靠度分析的蒙特卡洛模拟（MCS）方法,给出了计算流程图,系统地研究了考虑土体参数空间变异性的多层土坡系统可靠度问题。结果表明,提出方法能够有效地分析考虑参数空间变异性低失效概率水平的多层土坡系统可靠度问题,并且具有较高的参数敏感性分析计算效率。     

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.     

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.     

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.     

考虑参数空间变异性的非饱和土坡可靠度分析

在考虑多个土体参数空间变异性的基础上,提出了基于拉丁超立方抽样的非饱和土坡稳定可靠度分析的非侵入式随机有限元法。利用Hermite随机多项式展开拟合边坡安全系数与输入参数间的隐式函数关系,采用拉丁超立方抽样技术产生输入参数样本点,通过Karhunen-Loève展开方法离散土体渗透系数、有效黏聚力和内摩擦角随机场,并编写了计算程序NISFEM-KL-LHS。研究了该方法在稳定渗流条件下非饱和土坡可靠度分析中的应用。结果表明：非侵入式随机有限元法为考虑多个土体参数空间变异性的非饱和土坡可靠度问题提供了一种有效的分析工具。土体渗透系数空间变异性和坡面降雨强度对边坡地下水位和最危险滑动面位置均有明显的影响。当降雨强度与饱和渗透系数的比值大于0.01时,边坡失效概率急剧增加。当土体参数变异性或者参数间负相关性较大时,忽略土体参数空间变异性会明显高估边坡失效概率。     

Reliability analysis of piles in spatially varying soils considering multiple failure modes

In most limit state design codes, the serviceability limit checks for drilled shafts still use deterministic approaches. Moreover, different limit states are usually considered separately. This paper develops a probabilistic framework to assess the serviceability performance with the consideration of soil spatial variability in reliability analysis. Specifically, the performance of a drilled shaft is defined in terms of the vertical settlement, lateral deflection, and angular distortion at the top of the shaft, corresponding to three limit states in the reliability analysis. Failure is defined as the event that the displacements exceed the corresponding tolerable displacements. The spatial variability of soil properties is considered using random field modeling. To illustrate the proposed framework, this study assesses the reliability of each limit state and the system reliability of a numerical example of a drilled shaft. The results show the system reliability should be considered for the serviceability performance. The importance measures of the random variables indicate that the external loads, the performance criteria, the model errors of load transfer curves and soil strength parameter are the most important factors in reliability analysis. Moreover, it is shown that the correlation length and coefficient of variation of soil strength can exert significant impacts on the calculated failure probability.     

降雨条件下考虑饱和渗透系数变异性的边坡可靠度分析

土体饱和渗透系数表现为天然的变异性,为此基于Green-Ampt模型建立了考虑饱和渗透系数变异性的降雨入渗物理模型,并藉此模型确定了坡体湿润锋深度和含水率分布。然后结合无限长非饱和土边坡稳定模型得到解析形式的反映边坡稳定性的极限状态函数。采用Monte Carlo法对饱和渗透系数进行随机抽样并最终建立降雨条件下考虑饱和渗透系数变异性的边坡概率分析框架。针对一假想边坡,探讨了饱和渗透系数的变异系数、降雨持时和降雨强度对边坡破坏概率以及破坏发生时间概率分布的影响,结果表明：在降雨初期,边坡的破坏概率随饱和渗透系数变异性的增强而逐渐增加,但随着降雨的持续,破坏概率开始随变异性的增强而显著降低;滑坡最可能发生时间的大小并不受饱和渗透系数变异性的影响,而是直接取决于降雨强度;滑坡最可能发生时间所对应的概率却随变异性的增强而逐渐减小。     

Spatial probabilistic modeling of slope failure using an integrated GIS Monte Carlo simulation approach

Spatial probabilistic modeling of slope failure using a combined Geographic Information System (GIS), infinite-slope stability model and Monte Carlo simulation approach is proposed and applied in the landslide-prone area of Sasebo city, southern Japan. A digital elevation model (DEM) for the study area has been created at a scale of 1/2500. Calculated results of slope angle and slope aspect derived from the DEM are discussed. Through the spatial interpolation of the identified stream network, the thickness distribution of the colluvium above Tertiary strata is determined with precision. Finally, by integrating an infinite-slope stability model and Monte Carlo simulation with GIS, and applying spatial processing, a slope failure probability distribution map is obtained for the case of both low and high water levels.     

Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this article, it is employed in the case where the uncertain parameters are modelled by random fields. This is illustrated through the probabilistic analysis at the serviceability limit state (SLS) of a strip footing resting on a soil with a spatially varying Young's modulus. The probabilistic numerical results have shown that the probability of exceeding a tolerable vertical displacement (P _e) calculated by subset simulation is very close to that computed by MCS methodology but with a significant reduction in the number of realisations. A parametric study to investigate the effect of the soil variability (coefficient of variation and the horizontal and vertical autocorrelation lengths of the Young's modulus) on P _e was presented and discussed. Finally, a reliability-based design of strip footings was presented. It allows one to obtain the probabilistic footing breadth for a given soil variability.     

含水层渗透系数空间变异性对地下水数值模拟的影响

地下水数值模拟是目前定量研究地下水水量和水质的重要手段。使用基于随机理论的MonteCarlo方法来进行地下水数值模拟。这种方法能较好地考虑水文地质参数的空间变异性。主要将MonteCarlo方法和确定性模型模拟方法的模拟结果在渗透系数场、水头场、速度场和浓度场等方面进行了比较。结果表明:在模拟三维非均质含水层中的溶质运移问题时,充分考虑了含水层渗透系数空间变异性的MonteCarlo法比确定性方法更为有效,模拟精度提高了很多,且对模拟误差及误差来源有合理的数学解释。     

Reliability-based semi-analytical solution for ground improvement by PVDs incorporating inherent (spatial) variability of soil

The design of soil consolidation via prefabricated vertical drains (PVDs) has been traditionally carried out deterministically and thus can be misleading due to the ignorance of the uncertainty associated with the inherent (spatial) variation of soil properties. To treat such uncertainty in the design process of soil consolidation by PVDs, stochastic approaches that combine the finite element method with the Monte Carlo technique (FEMC) have been usually used. However, such approaches are complex, computationally intensive and time consuming. In this paper, a simpler reliability-based semi-analytical (RBSA) method is proposed as an alternative tool to the complex FEMC approach for soil consolidation by PVDs, considering soil spatial variability. The RBSA method is found to give similar results to those obtained from the FEMC approach and can thus be used with confidence in practice.     

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.     

考虑土参数自相关距离影响的单桩可靠性设计方法

在岩土工程设计中,准确地衡量土层参数的空间变异性决定了设计结构的可靠性和经济性,土参数的自相关距离是反映其空间变异性的一个重要指标。目前对于自相关距离与结构可靠性之间的关系的研究还很不完善。本文以单桩可靠性设计为例,选取工程场地的静力触探试验数据,引入随机场理论描述侧摩阻力和锥尖阻力的分布特征,利用递推平均法计算各地基土层侧摩阻力和锥尖阻力的自相关距离,在描述土层侧摩阻力和锥尖阻力的分布特征时考虑了自相关距离的影响,进一步结合蒙特卡洛法计算单桩的可靠指标。结果表明土参数取值考虑自相关距离的影响后,降低了土参数分布的离散程度,计算的单桩可靠指标更高。在保证桩基安全度的前提下,设计结果能够降低工程成本。     

Uncertainty of rainfall-induced landslides considering spatial variability of parameters

A cross-correlation analysis is conducted to determine the impacts of the heterogeneity of hydraulic conductivity K_s, soil cohesion c′ and soil friction angle (tan φ′) on the uncertainty of slope stability in time and space during rainfall. We find the relative importance of tan φ′ and c′ depends on the effective stress. While the sensitivity of the stability to the variability of K_s is small, the large coefficient of variation of K_s may exacerbate the variability of pore-water pressure. Therefore, characterizing the heterogeneity of hydraulic properties and pore-water distribution in the field is critical to the stability analysis.     

Reliability bearing capacity analysis of footings on cohesive soil slopes using RFEM

Reliability analysis of bearing capacity of a strip footing at the crest of a simple slope with cohesive soil was carried out using the random finite element method (RFEM). Analyses showed that the coefficient of variation and the spatial correlation length of soil cohesion can have a large influence on footing bearing capacity, particularly for slopes with large height to footing width ratios. The paper demonstrates cases where a footing satisfies a deterministic design factor of safety of 3 but the probability of design failure is unacceptably high. Isotropic and anisotropic spatial variability of the soil strength was also considered.