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。      

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.     

Probabilistic Stability Analyses of Embankments Based on Finite-Element Method

Conventional limit equilibrium methods are commonly used to assess the stability of embankments. The finite-element method, as an alternative to limit equilibrium methods, is being increasingly used in the deterministic stability analysis of slopes or embankments. In this paper, a practical procedure for integrating the finite-element method and the limit equilibrium methods into probabilistic stability analysis for embankments is presented. The response surface method is adopted to approximate the performance function for the stability problems and the first-order reliability method is used to calculate the reliability index based on an intuitive expanding ellipsoid perspective. The advantages of the response surface method as a bridge between stand-alone numerical packages and spreadsheet-based reliability analysis via automatic constrained optimization are demonstrated and discussed through a hypothetical two-layer slope and an actual case of the James Bay Dykes. The results show the ease and successful implementation of the proposed procedure for reliability analysis of embankments.     

石宝露天矿采场最终边坡稳定性分析

针对石宝露天矿3个采坑采场最终边坡出现的不稳定现象，根据其工程地质和水文地质条件，并结合现场工程地质勘察和岩石力学试验，运用极限平衡分析法对其整个采场的最终边坡进行了详细的分析和研究。分析结果表明，石宝露天矿采场边坡均处于稳定状态，但是A、B、C、E、F、I、K、L8个分区边坡从稳定性考虑，需要减缓边坡角，D、G、H、J4个分区边坡角可提高，加陡时应考虑现场工程地质状况。     

Displacements of Reinforced Slopes Subjected to Seismic Loads

Traditional analyses of stability of slopes subjected to seismic loads entail global equilibrium considerations with seismic influence included as a quasi-static force. Such an analysis does not reflect the earthquake shaking process, and it does not provide any information about permanent displacements that may have occurred as a result of that process. Earthquake events in recent years have brought about renewed interest in analyses of slopes subjected to seismic loads. This paper focuses on displacement calculations of reinforced slopes. Design of reinforced slopes using the quasi-static approach may lead to an unrealistically long reinforcement for large ground accelerations. If slopes are allowed to move by even a small displacement, then the reinforcement length can be reduced significantly. Two mechanisms of failure of reinforced slopes subjected to seismic conditions are considered: (1) Rotational collapse; and (2) sliding directly over the bottom layer of reinforcement. Yield accelerations and integrals of seismic records are presented in charts for easy use in practical applications. An example is shown to illustrate the method.     

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.     

Slope Stability Analysis with Nonlinear Failure Criterion

A linear failure criterion is widely used in slope stability analyses. However, the strength envelope of almost all geomaterials has the nature of nonlinearity. This paper computes rigorous upper bounds on slope stability factors under the condition of plane strain with a nonlinear yield criterion by employing the upper bound theorem of plasticity. A stability factor (or a limit load) computed using a linear Mohr-Coulomb (MC) failure criterion which circumscribes the actual nonlinear failure criterion is an upper bound value of the actual stability factor (or limit load). In this paper, an improved method using a “generalized tangential” technique to approximate a nonlinear failure criterion is proposed to estimate the stability factor of a slope on the basis of the upper bound theorem of plasticity. Using the “generalized tangential” technique, the curve of the nonlinear failure criterion is simplified as a set of straight lines according to the linear MC failure criterion. The straight line is tangential to the curve of the nonlinear failure criterion. The set of straight lines of the linear MC failure criterion is employed to formulate the slope stability problem as a classical optimization problem. The objective function formulated in this way is minimized with respect to the location of sliding body center and the location of tangency point. Two typical slope stability problems (a homogeneous soil slope with two slope angles and a vertical cut slope with a tension crack) are analyzed using the proposed method. For the soil slope with two slope angles, the computed results are compared with published solutions by others. The comparison shows that the proposed method gives reasonable and consistent values of the stability factor of the slope. For the vertical cut slope with a tension crack, a statically admissible stress field is constructed for the slope. The stress field does not violate the nonlinear failure criterion. Lower bound solutions are obtained by satisfying stress equilibrium conditions. The upper bound solutions obtained from the proposed method are equal to the lower bound solutions for the vertical cut slope. The agreement further supports the validation of the proposed approach. The influences of the strength parameters in the nonlinear criterion on the stability of slopes are also studied and discussed in this paper.     

New Method for 3D and Asymmetrical Slope Stability Analysis

A new three-dimensional (3D) slope stability analysis method is developed based on two-directional moment equilibrium. This method calculates not only the safety factor but also the possible direction of sliding for semispherical and composite failure surfaces. As a result, the possible errors associated with assuming a plane of symmetry in 3D stability analyses are eliminated. Another advantage of the new method is to eliminate the tedious work on the coordinate transformation prior to the analysis. Two examples of symmetrical failure surfaces are used to verify the basic formulation in the present study. Three additional examples further demonstrate the applicability of the proposed method in analyzing 3D asymmetrical failure surfaces. An analysis on a slope, subject to asymmetrical excavation unloading and geological conditions, shows that using the method of one-directional moment equilibrium may give an overestimated safety factor of the slope.     

Keyblock Stability in Seismically Active Rock Slopes—Snake Path Cliff, Masada

Keyblock stability in the “Snake Path” cliff of the Masada monument, situated on the western margin of the seismically active Dead Sea transform, is studied using field mapping, mechanical analysis, and monitoring of displacement, pressure, temperature, and relative humidity, over a period of 11 months. A linear nonreversible displacement trend is interpreted as the block response to regional microseismicity. A more pronounced cyclic displacement trend however is shown to be a response to climatic changes on the cliff face. This finding introduces a new, time-dependent, failure process in jointed rock slopes—the degradation of shear and/or cohesive strength of joints due climatic effects. Using two-dimensional (2-D) and three-dimensional (3-D) limit equilibrium analyses it is demonstrated that the 2-D solution overestimates the factor of safety against sliding by as much as 15% if water pressures in the boundary joints are considered. Application of a 2-D solution for a truly 3-D case where prismatic blocks are considered proves therefore to not be conservative.     

Classic Buckling of Three-Dimensional Multicolumn Systems under Gravity Loads

The elastic stability of three-dimensional (3D) multicolumn systems under gravity loads is analyzed in a condensed manner using the classical Timoshenko stability functions. The characteristic equations corresponding to multicolumn systems with sidesway uninhibited, partially inhibited, and totally inhibited are derived. Using the transcendental equations of the proposed method, the effective length K factor for each column and the total critical axial load of an entire story can be determined directly. The proposed method is applicable to 3D framed structures with rigid, semirigid, and simple connections. It is shown that the elastic stability of framed structures depends on: (1) the axial load pattern on the columns; (2) the variation in size and height among the columns; (3) the plan layout of the columns; (4) the overall floor-torsional sway caused by any asymmetries in the loading pattern, column layout, and column sizes and heights (all of which reduce the flexural-buckling capacity of multicolumn systems); (5) the end restraints of the columns; and (6) the bracings along the two horizontal and rotational directions of the floor plane. The proposed method solves the classical bifurcation stability of 3D frames directly without complex matrix solutions, however, it is limited to frames made up of columns of doubly symmetrical cross section with their principal axes parallel to the global axes. Examples are presented that show the effectiveness of the proposed method and the results compared with those obtained by complex matrix methods.     

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.     

Evaluation of Surficial Stability for Homogeneous Slopes Considering Rainfall Characteristics

Shallow slope failures in residual soil during periods of prolonged infiltration are commonly occurring in the world. This study examines an infinite slope analysis to estimate the influence of infiltration on surficial stability of slopes by the limit equilibrium method. An approximate method that accommodates the boundary condition of a uniform rainfall has been proposed to evaluate the likelihood of shallow slope failure that is induced by a particular rainfall event. The method based on the Mein and Larson model, which provides an explicit solution for preponding infiltration, has been applied to various types of soil having measured unsaturated hydraulic properties. To compare results with those obtained from the approximate method, a series of numerical analyses were carried out. According to the results, with the use of properly estimated input parameters, the approximate method was found to give results that compare reasonably well with those of more rigorous finite element analyses.     

Application of an Anisotropic Constitutive Model for Structured Clay to Seismic Slope Stability

The anisotropic nature of response and degradation of shear strength from the undisturbed condition to the remolded state are two fundamental and challenging aspects of response in some clay deposits. This paper presents a comprehensive, yet flexible and practical, version of the SANICLAY model and its application to a seismic slope-stability problem. The model is based on the well-known isotropic modified Cam-Clay model with two additional mechanisms to account for anisotropy and destructuration. The model has been efficiently implemented in a three-dimensional (3D) continuum, coupled, dynamic, finite-difference program. The program has been used to analyze the seismic response of clay slopes to gain better insight into the role of the previously mentioned parameters in real applications. Different aspects of the model, including anisotropy and destructuration, and their effects on the earthquake-induced strains and deformations in the slope have then been explored and presented. By providing a link between the model parameters and the soil’s undrained shear strength, which is a well-known engineering parameter, a benchmark comparison has been made between the results of the present advanced model and those of an engineering approach. To this end, a modified Newmark sliding-block analysis has been used, in which the yield acceleration is gradually reduced as block sliding progresses during the earthquake. It is observed that although the two analyses display the same trends, the modified Newmark sliding-block method provides conservative results compared with those obtained from the developed simulation model.     

Three-Dimensional Analysis of Nonhomogeneous Slopes

Presented is a method of three-dimensional slope stability analysis for homogeneous and nonhomogeneous symmetrical slopes based on the upper-bound theorem of the limit analysis approach. A rigid-block translational toe, above-the-toe or below-the-toe collapse mechanism is considered, with energy dissipation taking place along planar velocity discontinuities. The approach can be considered as a modification and extension of the procedure proposed by Michalowski in 1989. An effective iterative algorithm is applied to find the optimum (least) upper bound in constrained or unconstrained problems. The present procedure removes some essential limitations of limit analysis methods in two- and three-dimensional stability analysis of nonhomogeneous slopes.     

Three-Dimensional Asymmetrical Slope Stability Analysis Extension of Bishop’s, Janbu’s, and Morgenstern–Price’s Techniques

Most existing three-dimensional (3D) slope stability analysis methods are based on simple extensions of corresponding two-dimensional (2D) methods of analysis and a plane of symmetry or direction of slide is implicitly assumed. In this paper, 3D asymmetric slope stability models based on extensions of Bishop’s simplified, Janbu’s simplified, and Morgenstern–Price’s methods are developed. Under these new formulations, the direction of slide is unique and is determined from 3D force/moment equilibrium. Results from the new formulations are similar to the classical methods in normal cases but are numerically stable under transverse load. Further, the writers demonstrate that the present formulation is actually equivalent to the axes rotation formulation by Jiang and Yamagami but is much more convenient to be used for general problems. The writers have also discovered some inherent limitations of 3D limit equilibrium analysis which are absent in the corresponding 2D analysis.     

Required Unfactored Strength of Geosynthetic in Reinforced Earth Structures

Current reinforced earth structure designs arbitrarily distinguish between reinforced walls and slopes, that is, the batter of walls is 20° or less while in slopes it is larger than 20°. This has led to disjointed design methodologies where walls employ a lateral earth pressure approach and slopes utilize limit equilibrium analyses. The earth pressure approach used is either simplified (e.g., ignoring facing effects), approximated (e.g., considering facing effects only partially), or purely empirical. It results in selection of a geosynthetic with a long-term strength that is potentially overly conservative or, by virtue of ignoring statics, potentially unconservative. The limit equilibrium approach used in slopes deals explicitly with global equilibrium only; it is ambiguous about the load in individual layers. Presented is a simple limit equilibrium methodology to determine the unfactored global geosynthetic strength required to ensure sufficient internal stability in reinforced earth structures. This approach allows for seamless integration of the design methodologies for reinforced earth walls and slopes. The methodology that is developed accounts for the sliding resistance of the facing. The results are displayed in the form of dimensionless stability charts. Given the slope angle, the design frictional strength of the soil, and the toe resistance, the required global unfactored strength of the reinforcement can be determined using these charts. The global strength is then distributed among individual layers using three different assumed distribution functions. It is observed that, generally, the assumed distribution functions have secondary effects on the trace of the critical slip surface. The impact of the distribution function on the required global strength of reinforcement is minor and exists only when there is no toe resistance, when the slope tends to be vertical, or when the soil has low strength. Conversely, the impact of the distribution function on the maximum unfactored load in individual layers, a value which is typically used to select the geosynthetics, can result in doubling its required long-term strength.     

Extension of Preissmann Scheme to Two-Dimensional Flows

A numerical solution to the finite difference of two-dimensional (2D) depth-averaged equations on nonstaggered grid points is proposed in this technical note. Following a locally one-dimensional procedure, the basic equations are split into a pair of one-dimensional equations. Therefore, the solution of a 2D problem is reduced to the solution of a sequence of two one-dimensional problems. The discretization of the split one-dimensional equations is obtained with the use of the original Preissmann operator. Using Fourier’s classic linear analysis, stability, dissipation and dispersion with frictional resistance are investigated for the variations of the Courant number and weighting time factor.     

Generalized Method for Three-Dimensional Slope Stability Analysis

This paper describes an extension of a new three-dimensional (3D) stability analysis method, including formulation, comparative studies, and examples of new application. The new method uses “two-directional force and moment equilibrium” in the stability analysis of 3D potential failure mass with arbitrary shapes. The use of this new method has resulted in a novel situation wherein the direction of the resultant shear force (or direction of sliding) generated on the potential failure surface can now be calculated instead of the guesswork assumptions that were formerly made. It is also demonstrated that this new method eliminates the labor-intensive work for establishing local coordinate systems performed in conventional 3D analysis. Consequently, this new method facilitates a computer-aided 3D search for the critical failure surfaces in slope areas.     

Reservoir Bank Deformation Modeling: Application to Grangent Reservoir

Within inshore or fluvial environments, submerged fine matter mud banks are characterized by a high water content, a great spatial variability, and a strong deformability. The study of their instabilities induced by the variation of hydraulic stress requires a coupled modeling of sliding, erosion, and deposition mechanisms. In order to predict the impact of dam reservoir emptying on the stability of immersed upstream slopes, the method of approach to the problem proposed here combines theoretical developments, numerical modeling, site observations, and measurements. First, the theoretically achieved sliding criterion is compared with unstable mud height measurements. For more accuracy in the representation of the natural events, the sliding criterion is then integrated within a numerical code which couples the computation of hydrodynamic conditions, the erosion, and deposition of mud and the banks sliding. Finally, the results of the combination of all these mechanisms are compared with the variations in the bathymetric profiles obtained on the experimental site.     

Three-Dimensional Elastic Catenary Cable Element Considering Sliding Effect

The nonlinear behavior of cable-supported bridges is governed by the geometric nonlinearity of cables, which is attributable to sag and sliding effects at the saddle. In a cable-stayed bridge with a midspan saddle, and in all suspension bridges, cable sliding can occur at the saddle under extreme forces, such as those caused by an earthquake or typhoon. However, the conventional method of analysis of cable-supported bridges does not consider the effect of cable sliding at the saddle; instead it regards those cables as fixed. This assumption might lead to a misunderstanding of the global structure system. The goal of this study is to develop a three-dimensional (3D) elastic cable finite element that considers the sliding effect and uses a geometric nonlinear cable finite element based on elastic catenary theory. In this study, two types of sliding were considered: the roller sliding condition without friction and the frictional sliding condition. These were formulated to derive the nodal force vectors and tangential stiffness matrices. To validate the proposed 3D cable sliding element, experiments were conducted for both sliding conditions, and results were compared with calculations of the amount of sliding and displacement at the loading point. In addition, a cable-supported structural system was analyzed to investigate the characteristics of a realistic structure with cable sliding. Overall calculations using the 3D cable sliding model were in very good agreement with the measured values.