Stability Analysis of Complex Soil Slopes using Limit Analysis

The limit equilibrium method is commonly used for slope stability analysis. Limit equilibrium solutions, however, are not rigorous because neither static nor kinematic admissibility conditions are satisfied. Limit analysis takes advantage of the lower- and upper-bound theorems of plasticity theory to provide rigorous bounds on the true solution of a stability problem. In this study, finite-element models are used to construct both statically admissible stress fields for lower-bound analysis and kinematically admissible velocity fields for upper-bound analysis of soil slopes. While limit analysis of relatively simple slopes, typically homogeneous and of simple geometry, has been done previously, limit analysis of slopes with complex geometries, soil profiles, and groundwater patterns could not be effectively done in the past. In this paper, the theoretical basis and procedure for limit analysis of such slopes is presented. Various examples of slopes are selected from the literature and analyzed using both limit equilibrium and limit analysis. Factors of safety from limit equilibrium and limit analysis are compared. A comparison is also made, for each example, between the critical slip surfaces from limit equilibrium with the velocity field and plastic zone from the upper-bound solution and with the stress field from the lower-bound solution.     

Observations on Limit Equilibrium–Based Slope Reliability Problems with Inclined Weak Seams

This study addresses the complexity of slope reliability problems based on limit equilibrium methods (LEMs). The main focus is on the existence of multiple failure modes that poses difficulty to many LEM-based slope reliability methods. In particular, when weak seams are present, the failure modes associated with those seams may be difficult to detect. A systematic way of searching the failure modes is proposed, and its robustness over slopes with or without weak seams is demonstrated. It is found that in the presence of weak seams, assuming circular slip surfaces may cause underestimation of slope failure probability. The conclusion of the study promotes the use of finite elements as the stability method for reliability evaluation because it is not necessary to search for failure surfaces in finite-element stability analysis.     

Probabilistic Assessment of Slope Stability That Considers the Spatial Variability of Soil Properties

In this paper, a numerical procedure for probabilistic slope stability analysis is presented. This procedure extends the traditional limit equilibrium method of slices to a probabilistic approach that accounts for the uncertainties and spatial variation of the soil strength parameters. In this study, two-dimensional random fields were generated based on a Karhunen-Loève expansion in a fashion consistent with a specified marginal distribution function and an autocorrelation function. A Monte Carlo simulation was then used to determine the statistical response based on the generated random fields. This approach makes no assumption about the critical failure surface. Rather, the critical failure surface corresponding to the input random fields of soil properties is searched during the process of analysis. A series of analyses was performed to verify the application potential of the proposed method and to study the effects of uncertainty due to the spatial heterogeneity on the stability of slope. The results show that the proposed method can efficiently consider the various failure mechanisms caused by the spatial variability of soil property in the probabilistic slope stability assessment.     

Influence of Spatial Variability on Slope Reliability Using 2-D Random Fields

The paper investigates the probability of failure of slopes using both traditional and more advanced probabilistic analysis tools. The advanced method, called the random finite-element method, uses elastoplasticity in a finite-element model combined with random field theory in a Monte-Carlo framework. The traditional method, called the first-order reliability method, computes a reliability index which is the shortest distance (in units of directional equivalent standard deviations) from the equivalent mean-value point to the limit state surface and estimates the probability of failure from the reliability index. Numerical results show that simplified probabilistic analyses in which spatial variability of soil properties is not properly accounted for, can lead to unconservative estimates of the probability of failure if the coefficient of variation of the shear strength parameters exceeds a critical value. The influences of slope inclination, factor of safety (based on mean strength values), and cross correlation between strength parameters on this critical value have been investigated by parametric studies in this paper. The results indicate when probabilistic approaches, which do not model spatial variation, may lead to unconservative estimates of slope failure probability and when more advanced probabilistic methods are warranted.     

Reliability Analysis of Earth Dams

In the present study, results of reliability analyses of four selected rehabilitated earth dam sections, i.e., Chang, Tapar, Rudramata, and Kaswati, under pseudostatic loading conditions, are presented. Using the response surface methodology, in combination with first order reliability method and numerical analysis, the reliability index (β) values are obtained and results are interpreted in conjunction with conventional factor of safety values. The influence of considering variability in the input soil shear strength parameters, horizontal seismic coefficient (αh), and location of reservoir full level on the stability assessment of the earth dam sections is discussed in the probabilistic framework. A comparison of results with those obtained from other method of reliability analysis, viz., Monte Carlo simulations combined with limit equilibrium approach, provided a basis for discussing the stability of earth dams in probabilistic terms, and the results of the analysis suggest that the considered earth dam sections are reliable and are expected to perform satisfactorily.     

Methods and Object-Oriented Software for FE Reliability and Sensitivity Analysis with Application to a Bridge Structure

This paper addresses the growing demand for finite-element software with capabilities to incorporate uncertainty in the input parameters. Reliability and response sensitivity algorithms are implemented in the general-purpose finite-element software OpenSees, which employs an object-oriented programming approach to achieve a sustainable software with focus on maintainability and extensibility. The product is a comprehensive and freely available library of software tools for finite-element reliability and response sensitivity analysis. A numerical example involving a detailed model of a highway bridge with inelastic material behavior and 320 random variables is presented to demonstrate features of the methodology and the software. Importance vectors are employed to rank the input parameters according to their relative influence on the structural reliability. The required response sensitivities are obtained by an extensive implementation of the direct differentiation method.     

Search for Critical Slip Surface in Slope Stability Analysis by Spline-Based GA Method

The stability of a soil slope is usually analyzed by limit equilibrium methods, in which the identification of the critical slip surface is of principal importance. In this study the spline curve in conjunction with a genetic algorithm is used to search the critical slip surface, and Spencer’s method is employed to calculate the factor of safety. Three examples are presented to illustrate the reliability and efficiency of the method. Slip surfaces defined by a series of straight lines are compared with those defined by spline curves, and the results indicate that use of spline curves renders better results for a given number of slip surface nodal points comparing with the approximation using straight line segments.     

Handling of Constraints in Finite-Element Response Sensitivity Analysis

In this paper, the direct differentiation method (DDM) for finite-element (FE) response sensitivity analysis is extended to linear and nonlinear FE models with multi-point constraints (MPCs). The analytical developments are provided for three different constraint handling methods, namely: (1) the transformation equation method; (2) the Lagrange multiplier method; and (3) the penalty function method. Two nonlinear benchmark applications are presented: (1) a two-dimensional soil-foundation-structure interaction system and (2) a three-dimensional, one-bay by one-bay, three-story reinforced concrete building with floor slabs modeled as rigid diaphragms, both subjected to seismic excitation. Time histories of response parameters and their sensitivities to material constitutive parameters are computed and discussed, with emphasis on the relative importance of these parameters in affecting the structural response. The DDM-based response sensitivity results are compared with corresponding forward finite difference analysis results, thus validating the formulation presented and its computer implementation. The developments presented in this paper close an important gap between FE response-only analysis and FE response sensitivity analysis through the DDM, extending the latter to applications requiring response sensitivities of FE models with MPCs. These applications include structural optimization, structural reliability analysis, and finite-element model updating.     

Load-Factor Stability Analysis of Embankments on Saturated Soil Deposits

A continuum-based finite-element methodology is established for quantifying the stability of earthen embankments built on saturated soil deposits. Within the methodology the soil is treated as a fluid-solid porous medium, in which the soil skeleton's constitutive behavior is modeled using a smooth elastoplastic cap model that features continuous coupling between deviatoric and volumetric plasticity. In the stability analysis procedure, self-weight of the embankment soils is monotonically increased at rates characteristic of the embankment construction time, until instability mechanisms develop. The transient effects of excess pore pressures and their impact on soil strength are explicitly modeled, allowing for computation of embankment safety factors against instability as a function of construction rate. Details on the proposed method are presented and discussed, including (1) how the construction rate of an embankment can be modeled; (2) how load-based safety factors can differ from resistance-based safety factors; and (3) solved example problems corresponding to a case history of an embankment failure.     

基于极限平衡法和强度折减法的边坡稳定性分析

考虑传统的极限平衡法和FLAC二维非线性数值分析方法的优缺点,将边坡稳定性分析的Slide极限平衡法和FLAC2D强度折减法相结合,以某排土场为工程实际,进行了边坡在天然、天然地震、降雨和降雨地震四种不同工况下的稳定性验算比较;鉴于影响边坡稳定因素的不确定性,采用基于Spencer法的Mont-Carlo法进行了边坡可靠度指标的计算,并对强度参数C、（φ）和容重γ对边坡稳定的影响进行了分析,得出边坡稳定对内摩擦角（φ）的敏感性高于内聚力C的敏感性,而容重γ的影响较C、（φ）可忽略不计.该方法通过两者的相互校核,不仅给出边坡非线性应力-应变关系和安全系数来共同评价边坡的稳定性,而且结合排土场边坡安全系数的概率分析提供边坡潜在的失效概率使边坡稳定性评价更加科学、精确.     

运动学和极限平衡法相结合边坡稳定性分析

本文根据非洲如瓦西矿西南边坡为案例,分别采用运动学分析方法和极限平衡法,对边坡稳定性进行分析研究。研究结果表明:西南边坡发生平面滑坡和楔形体滑坡的概率较小,但由于边坡部分岩体较破碎,结构面较发育,不排除小规模楔形体滑坡的可能。极限平衡法计算的最小安全系数均高于规范要求值,其边坡为稳定边坡。     

Spatial Distribution of Safety Factors

Classical limiting equilibrium analysis seeks the minimum factor of safety and its associated critical slip surface. This objective is mathematically convenient; however, it limits the analysis' practical usefulness. Introduced is a general framework for safety maps that allow for a physically meaningful extension of classical slope stability analysis. Safety maps are represented by a series of contour lines along which minimal safety factors are constant. Each contour line is determined using limit equilibrium analysis and thus represents a value of global safety factor. Since most problems of slope stability possess a flat minimum involving a large zone within which safety factors are practically the same, representation of the results as a safety map provides an instant diagnostic tool about the state of the stability of the slope. Such maps provide at a glance the spatial scope of remedial measures if such measures are required. That is, unlike the classical slope stability approach that identifies a single surface having the lowest factor of safety, the safety map displays zones within which safety factors may be smaller than an acceptable design value. The approach introduced results in a more meaningful application of limiting equilibrium concepts while preserving the simplicity and tangibility of limit equilibrium analysis. Culmann's method is used to demonstrate the principles and usefulness of the proposed approach because of its simplicity and ease of application. To further illustrate the practical implications of safety maps, a rather complex stability problem of a dam structure is analyzed. Spencer's method using generalized slip surfaces and an efficient search routine are used to yield the regions within the scope where the safety factors are below a certain value.     

Aerostatic Reliability Analysis of Long-Span Bridges

The response surface Monte Carlo method (RSMCM) is proposed for reliability analysis of aerostatic response and aerostatic stability for different types of long-span bridges, in which the nonlinear effects due to geometric nonlinearity and deformation-dependent aerostatic loads are taken into consideration. The geometric parameters, the material parameters, and the aerostatic coefficients of the bridge girder are regarded as random variables in the proposed method. RSMCM has higher accuracy in comparison with the traditional response surface method and requires much less computational cost than the conventional Monte Carlo method. The proposed method is applied to reliability analysis of aerostatic response and aerostatic stability of the Hong Kong Ting Kau Bridge, and reasonable results illustrating effectiveness of the method are obtained.     

Reliability Analysis of Suspension Bridges against Fatigue Failure from the Gusting of Wind

A fatigue reliability analysis of suspension bridges due to the gustiness of the wind velocity is presented by combining overall concepts of bridge aerodynamics, fatigue analysis, and reliability analysis. For this purpose, the fluctuating response of the bridge deck is obtained for buffeting force using a finite-element method and a spectral analysis in frequency domain. Annual cumulative fatigue damage is calculated using Palmgren–Miner’s rule, stress-fatigue curve approach and different forms of distribution for stress range. In order to evaluate the reliability, both first-order second-moment (FOSM) method and full distribution procedure (assuming Weibull distribution for fatigue life) are used to evaluate the fatigue reliability. Probabilities of fatigue failure of the Thomas Bridge and the Golden Gate Bridge for a number of important parametric variations are obtained in order to make some general observations on the fatigue reliability of suspension bridges. The results of the study show that the FOSM method predicts a higher value of the probability of fatigue failure as compared to the full distribution method. Further, the distribution of stress range used in the analysis has a significant effect on the calculated probability of fatigue failure in suspension bridges.     

Optimum Design for External Seismic Stability of Geosynthetic Reinforced Soil Walls: Reliability Based Approach

In this paper, an analytical study considering the effect of uncertainties in the seismic analysis of geosynthetic-reinforced soil (GRS) walls is presented. Using limit equilibrium method and assuming sliding wedge failure mechanism, analysis is conducted to evaluate the external stability of GRS walls when subjected to earthquake loads. Target reliability based approach is used to estimate the probability of failure in three modes of failure, viz., sliding, bearing, and eccentricity failure. The properties of reinforced backfill, retained backfill, foundation soil, and geosynthetic reinforcement are treated as random variables. In addition, the uncertainties associated with horizontal seismic acceleration and surcharge load acting on the wall are considered. The optimum length of reinforcement needed to maintain the stability against three modes of failure by targeting various component and system reliability indices is obtained. Studies have also been made to study the influence of various parameters on the seismic stability in three failure modes. The results are compared with those given by first-order second moment method and Monte Carlo simulation methods. In the illustrative example, external stability of the two walls, Gould and Valencia walls, subjected to Northridge earthquake is reexamined.     

Influence of Route Direction on Thermal Field of Embankments Constructed in High-Altitude Permafrost Area

This paper discusses the effect of route direction, embankment height, and pavement type on the thermal field of embankments built in permafrost regions. A finite-element model (FEM) is adopted to simulate diverse conditions of the embankment. The 30-year meteorological data including the solar radiation, air temperature, and wind velocity are used as the boundary conditions for the Qinghai-Tibet Highway. The results obtained from the FEM calculations are found to be in good agreement with the actual measurements on the thermal field. Further, the results show that route direction has great impact on the equilibrium of the thermal field within embankments in permafrost regions. The thermal imbalance is more obvious for embankments in the east–west direction and less in the north–south direction. In addition, the thermal asymmetry is closely related to seasonal variation and it is more pronounced in winter and less in summer.     

基于有限元极限平衡法的三维边坡稳定性

提出了一种基于有限元弹塑性应力场和极限平衡状态的三维边坡稳定分析方法——三维有限元极限平衡法。首先,考虑三维空间中滑动方向,提出滑动面上一点在滑动方向上的极限平衡条件,并证明滑动面上土体整体达到极限平衡状态与滑动面上土体各处在滑动方向上处于极限平衡状态等价。再通过刚体极限平衡假定计算主滑方向和滑动面上各点滑动方向。最后,定义局部安全系数为抗剪强度与滑动方向上剪应力投影的比值,基于三维边坡整体极限平衡条件将局部安全系数通过积分中值定理转变为整体安全系数。该方法计算简单,消除了剪应力比形式定义安全系数滑动面形状限制,具备合理性与有效性。算例验证结果表明,该方法滑动方向假设合理,安全系数与严格三维极限平衡法结果一致。      

Numerical Method for Lower-Bound Solution of the Rigid-Plastic Limit Analysis Problem

This paper describes a numerical method to determine the lower-bound solution of limit load of a rigid–perfectly plastic body obeying the von Mises yield criterion. The idea of this method is to construct a smoothed linear stress field that satisfies the yield criterion everywhere in the body. Applying the similar stress recovery techniques as superconvergent patch recovery and recovery by equilibrium in patch in the elastic finite-element analysis, the nodal stresses are obtained from those stresses at the integration points from an iterative process of upper-bound limit analysis. Then, the improved stress fields and lower-bound solutions can be derived by ensuring all the nodal stresses within the yield surface. The convergence of this method is guaranteed. The validity of the proposed method is demonstrated with some numerical examples. The computational results show that more reliable lower-bound solutions can be obtained by using this method, especially for problems with strain singularity.     

基于Slide程序对云南某矿排土场边坡稳定性与不确定性分析

露天矿排土场边坡的稳定性对露天矿山安全有序的生产与运行起着至关重要的作用。该文采用极限平衡法对排土场边坡进行了稳定性分析,通过排土场边坡的各种不确定影响因素及其相互关系,对排土场边坡进行了不确定性分析。基于Slide程序对云南某露天矿排土场进行了简要分析与模拟,其模拟结果与该排土场现状基本吻合,同时分析了该排土场的不确定性,为今后如何进行堆排提出了建议。     

Reliability Analysis of Laterally Loaded Piles Involving Nonlinear Soil and Pile Behavior

An alternative approach of analyzing laterally loaded piles in the ubiquitous spreadsheet platform is presented. The numerical procedure couples nonlinear pile flexural rigidity (EpIp) with nonlinear p-y analysis. The deterministic study is then extended to carry out reliability analysis, which reflects the uncertainties and correlation structure of the underlying parameters. The reliability index is evaluated based on the alternative intuitive perspective of an expanding equivalent ellipsoid in the original space of the random variables. This paper investigates two modes of failure: deflection and bending moment, and considers non-normal random variables. Spatial variability of the soil medium is accounted for by incorporating an autocorrelation model. The spreadsheet-based reliability approach can also be coupled with stand-alone programs via the response surface method. The probabilities of failure inferred from reliability indices agree well with Monte Carlo simulations. Simple reliability-based design is demonstrated, in which the appropriate pile section or length that satisfies target reliability in one or more limit states is sought.