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.     

简单边坡的稳定性分析

提出了一种新方法用于分析边坡的稳定性,对土条间作用力作了如下假定：认为土条侧面作用力平行于条底,但其大小并不相等。依据平衡条件,结合强度破坏准则,求出边坡的安全系数。算例表明,该方法不失为一种合理、有效的边坡稳定分析方法。     

Probabilistic analysis of simple slopes with cohesive soil strength using RLEM and RFEM

The 2D random finite element method and the one-dimensional and 2D random limit equilibrium method are used to investigate the influence of spatial variability of soil strength parameters on the probability of failure of simple soil slopes with cohesive undrained shear strength. The combined influence of spatial variability of soil properties and cross-correlation between undrained soil strength and unit weight on the computed probability of failure is explored. The paper identifies conditions where numerical outcomes are similar and where they are not. The limitations of each analysis method are described and implications to analysis and design are identified.

Abbreviations: FEM: finite element method; LEM: limit equilibrium method; RFEM: random finite element method; RLEM: random limit equilibrium method     

Improved Bell's method for the stability analysis of slopes

A new procedure based on the approximation to the total normal pressure along the slip surface is developed to compute the factor of safety of slopes for slip surfaces of all shapes. By taking the whole sliding body, instead of an individual slice, as the loaded object, all the equilibrium equations are formulated according to the three‐moment equilibrium conditions rather than the two force equilibrium conditions and one‐moment equilibrium condition. The system of nonlinear equations deduced in this way is well‐scaled and enjoys excellent numerical properties such as the existence of solution with a positive factor of safety, a nearly unlimited scope of convergence and a rapid convergence rate associated with the Newton method. In the case of ?_u =0—the situation where no drainage and no consolidation are involved, furthermore, the system has a unique solution and the factor of safety has an explicit expression. Some typical examples are analyzed to demonstrate the numerical properties of the proposed procedure. Copyright © 2009 John Wiley & Sons, Ltd.     

Neural network prediction of the reliability of heterogeneous cohesive slopes

The reliability of heterogeneous slopes can be evaluated using a wide range of available probabilistic methods. One of these methods is the random finite element method (RFEM), which combines random field theory with the non‐linear elasto‐plastic finite element slope stability analysis method. The RFEM computes the probability of failure of a slope using the Monte Carlo simulation process. The major drawback of this approach is the intensive computational time required, mainly due to the finite element analysis and the Monte Carlo simulation process. Therefore, a simplified model or solution, which can bypass the computationally intensive and time‐consuming numerical analyses, is desirable. The present study investigates the feasibility of using artificial neural networks (ANNs) to develop such a simplified model. ANNs are well known for their strong capability in mapping the input and output relationship of complex non‐linear systems. The RFEM is used to generate possible solutions and to establish a large database that is used to develop and verify the ANN model. In this paper, multi‐layer perceptrons, which are trained with the back‐propagation algorithm, are used. The results of various performance measures indicate that the developed ANN model has a high degree of accuracy in predicting the reliability of heterogeneous slopes. The developed ANN model is then transformed into relatively simple formulae for direct application in practice. Copyright © 2016 John Wiley & Sons, Ltd.     

边坡稳定性分析的条块稳定系数法

鉴于传统条分法对边坡安全系数定义和使用的局限性,引入了条块稳定系数的概念。分析了条间力推力线高度的影响因素,假定条间力传递时都具有相同的条间力稳定系数 ,并考虑了条块重力偏心矩的影响。根据条块的力和力矩平衡关系以及条间力假定,建立了求解条块稳定系数的四元方程组。在此基础上,求得条块稳定系数随滑动面的分布状况。根据条块稳定系数随滑动面的分布规律划分了滑体的主滑段、抗滑段和牵引段,大致判断滑坡的滑动模式。根据滑面上抗滑力和下滑力之比的定义,求解边坡整体稳定系数。采用传统条分法和所建立的条块分析法分别以圆弧滑动和折线形滑动的两个实例进行了验算,说明了条块分析法的优越性和可靠性。     

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°.     

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.     

基于数值应力场的边坡临界滑动场

数值分析方法（有限元、有限差分等）可计算边坡应力、变形和破坏,但直接计算边坡安全系数比较困难;而极限平衡法需对边坡体内力作出假设,计算精度受到限制。利用数值分析方法计算得到的应力场,重新定义推力最大原理中推力的含义,将常规边坡临界滑动场数值模拟方法作进一步的改进,提出基于数值应力场的边坡临界滑动场方法。该方法可充分发挥两大方法的优势,能够准确、方便地搜索出边坡临界滑动面位置,并给出更合理的边坡安全系数。通过两个算例的稳定性分析,并与其他方法进行比较,验证了该方法的准确性和合理性。     

土坡稳定极限分析上限法中的多解性分析

土坡稳定极限分析上限法中采用了完全塑性区假定,即假定土坡内存在一个处处位于屈服面上的滑动体。将滑动体内部条块界面上的强度发挥系数视为优化变量,利用功能平衡方程得到了对于给定滑动面的多个安全系数,基于和声算法和潘家铮极值原理选择其中最大的安全系数作为该滑动面的抗滑稳定安全系数,利用该方法对于两个典型土坡算例进行了分析,将其结果与常规极限分析上限法结果进行了比较,给出了完全塑性区假定对计算结果的影响。     

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。     

地震作用下边坡安全系数时程计算及参数研究

提出了一种计算地震作用下边坡动力安全系数时程的方法,该方法以动力有限元时程分析得到应力场,再使用遗传算法搜索每个时刻下的临界滑裂面,进而确定其安全系数,从而得到边坡在动力作用下的安全系数时程,物理意义明确。通过在基岩输入不同周期的正弦波加速度,研究了动力作用周期对安全系数时程的影响,发现相同加速度幅值下周期越大安全系数越小,安全性越低。与边坡自振周期和振型比对后发现,地震波周期与边坡自振周期耦合引起的动力响应增大,不仅与周期耦合相关,还与相应的自振振型相关,只有与以坡体本身振动为主的振型耦合时,才会发生明显的动力响应增大,相应的安全系数明显降低。     

Probabilistic stability analyses of undrained slopes by 3D random fields and finite element methods

A long slope consisting of spatially random soils is a common geographical feature. This paper examined the necessity of three-dimensional(3 D) analysis when dealing with slope with full randomness in soil properties. Although 3 D random finite element analysis can well reflect the spatial variability of soil properties, it is often time-consuming for probabilistic stability analysis. For this reason, we also examined the least advantageous(or most pessimistic) cross-section of the studied slope. The concept of"most pessimistic" refers to the minimal cross-sectional average of undrained shear strength. The selection of the most pessimistic section is achievable by simulating the undrained shear strength as a 3 D random field. Random finite element analysis results suggest that two-dimensional(2 D) plane strain analysis based the most pessimistic cross-section generally provides a more conservative result than the corresponding full 3 D analysis. The level of conservativeness is around 15% on average. This result may have engineering implications for slope design where computationally tractable 2 D analyses based on the procedure proposed in this study could ensure conservative results.     

土质边坡三维可靠度分析

基于空间随机场理论,利用局部平均法研究空间各向异性的土坡的可靠度问题,推导出相应计算公式,并通过计算边坡的最大破坏概率来进行滑坡的空间预测。     

非均质土坡的有限元塑性极限分析

在评价现有边坡稳定分析方法的基础上,提出了边坡稳定的有限元塑性极限分析方法。借助有限单元思想和线性规划,建立了边坡稳定的数学规划模型,由此可以求出安全系数的上下限解,从而,界定了边坡的安全范围,同时,可以给出下限状态下的应力场和上限状态下的速度场。以一个经典非均质土坡的边坡稳定作为算例,比较了多种方法的分析结果,论证了本方法的正确性。     

二维随机场下边坡稳定性的径向基函数神经网络分析法

岩土参数的随机性会直接影响边坡稳定性评价结果的精度。首先,依据边坡参数的常用分布特征,利用拉丁超立方抽样法生成若干组边坡土性参数和几何参数的随机样本,用有限元强度折减法求解各组样本对应的边坡安全系数。再考虑土性参数的空间变异性,在二维随机场模型下将蒙特卡罗模拟和有限元强度折减法相结合求解各组样本对应的边坡失效概率。然后,利用样本数据及其安全系数和失效概率对径向基函数（RBF）神经网络进行训练和测试,从而建立边坡安全系数和失效概率的预测模型。算例表明,二维随机场模型能相对精确地考虑参数的空间变异性;在此基础上建立的神经网络模型对边坡的安全系数和失效概率具有较高的预测精度,且能极大地节省边坡稳定性分析的时间。     

再论岩质高边坡稳定性分析方法

岩质高边坡稳定性分析是岩土工程中经典的研究课题,其假定不同,分析方法就不同。从现有稳定性分析方法中存在的问题入手,指出其存在的优劣性,在刚体极限平衡和有限元等传统边坡稳定性分析方法的基础上,提出了用有限单元法直接求解边坡稳定性。该方法的思路是用潜在滑动面上在开挖、支护完成后尚未发生滑动的所有滑面单元模拟边坡在各工况下的真实应力场和变形场,根据这个真实应力场直接计算边坡安全系数。结合现场即时监测手段和计算机仿真技术可逐步模拟边坡在分步开挖期间的稳定性。算例分析表明,该方法完善了刚体极限平衡法的不足,克服了强度折减法强度参数c、?值折减比例的难题,弥补了超载法力学方向不明确的缺陷。计算所得的边坡安全系数和稳定状态与工程实际相当接近,表明提出的方法是可靠有效的新方法,为复杂岩质边坡稳定性分析提供了新的思路和途径。     

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.     

Probabilistic failure analysis of infinite slopes under random rainfall processes and spatially variable soil

Rainfall-induced landslides occur during or immediately after rainfall events in which the pore water pressure builds up, leading to shallow slope failure. Thereby, low permeability layers result in high gradients in pore water pressure. The spatial variability of the soil permeability influences the probability such low permeability layers, and hence the probability of slope failure. In this paper, we investigate the influence of the vertical variability of soil permeability on the slope reliability, accounting for the randomness of rainfall processes. We model the saturated hydraulic conductivity of the soil with a one-dimensional random field. The random rainfall events are characterised by their duration and intensity and are modelled through self-similar random processes. The transient infiltration process is represented by Richards equation, which is evaluated numerically. The reliability analysis of the infinite slope is based on the factor of safety concept for evaluating slope stability. To cope with the large number of random variables arising from the discretization of the random field and the rainfall process, we evaluate the slope reliability through Subset Simulation, which is an adaptive Monte Carlo method known to be especially efficient for reliability analysis of such high-dimensional problems. Numerical investigations show higher probability of slope failure with increased spatial variability of the saturated hydraulic conductivity and with more uniform rainfall patterns.     

Evaluation of factor of safety for vegetated and barren soil slopes with limit equilibrium computations

Limit equilibrium method (LEM) is the most conventional and popular technique in slope stability analysis. Location of the unique failure surface and the determination of corresponding critical factor of safety (critical-FOS) of the soil slope requires a lot of trial and research. With this amount of trial and research results, we evaluate the field of factor of safety (FOS-field), which represents the critical-FOS at a point within the domain of all possible failure surfaces. Such a field can be very important for the precise judgment of the most critical-FOS of the slope and its perturbation. This paper also presents the evaluation of the FOS-field of vegetated slopes, thus providing an analytical way to examine the effect of vegetation on the soil slope stability. We evaluated the stability of vegetated and barren soil slopes under dry and fully saturated conditions. With dry and fully saturated conditions, the behaviour of slopes in most favourable and worst conditions can be simulated. Evaluation of FOS-field for various slope geometries and conditions show that the more unstable the slope becomes, the more difficult it will be to pinpoint the location of the critical failure surface with the unique least FOS. We verified the LEM codes with finite element method (FEM) and spectral element method (SEM) codes considering a sample problem from Smith and Griffiths’ book (Smith and Griffiths 2003).