Variational Stability Analysis of Cohesive Slope by Applying Boundary Integral Equation Method

A GIS-based method for locating the critical 3D slip surface in a slope

The various methods of optimization or random search have been developed for locating the critical slip surface of a slope and the related minimum safety factor in the limit equilibrium stability analysis of slope. However, all these methods are based on a two-dimensional (2D) method and not one had been adapted for a search of the three-dimensional (3D) critical slip surface. In this paper, a new Monte Carlo random simulation method has been proposed to identify the 3D critical slip surface, in which assuming the initial slip to be the lower part of an ellipsoid, the 3D critical slip surface in the 3D slope stability analysis is located by minimizing the 3D safety factor of limit equilibrium approach. Based on the column-based 3D limit equilibrium slope stability analysis models, new geographic information systems (GIS) grid-based 3D deterministic limit equilibrium models are developed to calculate the 3D safety factors. Several practical examples, of obtained minimum safety factor and its critical slip surface by a 2D optimization or random technique, are extended to 3D slope problems to locate the 3D critical slip surface and to compare with the 2D results. The results shows that, comparing with the 2D results, the resulted 3D critical slip surface has no apparent difference only from a cross-section, but the associated 3D safety factor is definitely higher.

The Various methods of optimization or random search have been developed for locating the critical slip surface of a slope and the related minimum safety factor in the limit equilibrium stability analysis of slope. But all these methods are based on a two-dimensional (2D) method and no one had been adapted for a search of the three-dimensional (3D) critical slip surface. In this paper, a new Monte Carlo random simulating method has been proposed to identify the 3D critical slip surface, in which assuming the initial slip to be the lower part of an ellipsoid, the 3D critical slip surface in the 3D slope stability analysis is located by minimizing the 3D safety factor of limit equilibrium approach. Based on the column-based three-dimensional limit equilibrium slope stability analysis models, new Geographic Information Systems (GIS) grid-based 3D deterministic limit equilibrium models are developed to calculate the 3D safety factors. Several practical examples, of obtained minimum safety factor and its critical slip surface by a 2D optimization or random technique, are extended to 3D slope problems to locate the 3D critical slip surface and to compare with the 2D results. The results shows that, comparing with the 2D results, the resulting 3D critical slip surface has no apparent difference only from a cross section, but the associated 3D safety factor is definitely higher.

This paper develops a new numerical procedure for the slope stability analysis, by combining the slice method in the limit equilibrium method with the lower-bound approach in the limit analysis method. The procedure provides an appropriate lower-bound solution subject to the collapse mechanism represented by a potential slip surface. First, the safety factor for the shear strength is calculated for the case in which the position of the slip surface is known beforehand. The problem is formulated as an optimization problem for finding a stress field which minimizes the safety factor within the limitations of satisfying the lower-bound conditions. The formulated problem is solved numerically by the mathematical programming technique. The obtained stress field appears physically reasonable. Several case studies have shown that the simplified Janbu method based on the critical slip surface tends to underestimate the safety factor on the safety side, compared with the lower-bound solution by the proposed procedure. Subsequently, the assumption used in each of the representative slice methods, which are necessary to render the stability problem to a statically determinate, is verified by the use of the stress field determined by the proposed procedure. Finally, a procedure for locating the critical noncircular slip surface which minimizes the safety factor within the lower-bound conditions is developed. Many important properties in the slope stability analysis are clarified through the development of this procedure.

The design of slopes in open pit mines requires an in-depth understanding of the ground behavior to predict the potential failure mechanism and to better determine the stabilization measures. This study compares the critical slip surface defined by the limit equilibrium method (LEM), the limit analysis (LA), and the finite elements method (FEM) for the stability analysis to better approach the stability on a multi-layered slope. The safety factor, the size, and the location of the critical slip surface obtained from the applied methods are considered in the comparisons. This study highlights some features that affect the slope stability and presents a procedure for addressing the evaluation challenges in a multi-layered formation. Moreover, it presents some aspects of the upper-bound computation on the safety factor of a layered slope subjected to the effects of pore water pressures. Based on the obtained results, the critical slip surface defined by the limit equilibrium method compares well with the slip mechanism suggested by the limit analysis and the finite element approach. In view of the differences in the shape and location of the critical slip surface, as well as the values obtained for the safety factor, it is recommended that an engineer should analyze critical slopes using the finite element method in combination with the limit equilibrium or limit analysis method as a cross reference. The authors propose that in defining the potential failure mass, consideration must be given to the conducted field research and monitoring.

This paper presents a new approach to determine the minimum factor of safety and the associated critical line segment pseudostatic slip surface. The new technique is called Alternating Variable Local Gradient (AVLG) that it is based on Univariat method. The slip surface shape is line segment (or non-circular). The safety factor is calculated using Janbu function. This procedure can truly convert the initial slip surface to critical slip surface with the different conditions of slope and soil properties. The critical slip surface obtained in this method provides both minimum factor of safety and more accordance with reality in nature. Resultant critical slip surface is reasonable also comparative study of numerical examples indicate the efficiency and accuracy of proposed method.

Over the last few years, particle swarm optimization (PSO) has been extensively applied in various geotechnical engineering including slope stability analysis. However, this contribution was limited to two-dimensional (2D) slope stability analysis. This paper applied PSO in three-dimensional (3D) slope stability problem to determine the critical slip surface (CSS) of soil slopes. A detailed description of adopted PSO was presented to provide a good basis for more contribution of this technique to the field of 3D slope stability problems. A general rotating ellipsoid shape was introduced as the specific particle for 3D slope stability analysis. A detailed sensitivity analysis was designed and performed to find the optimum values of parameters of PSO. Example problems were used to evaluate the applicability of PSO in determining the CSS of 3D slopes. The first example presented a comparison between the results of PSO and PLAXI-3D finite element software and the second example compared the ability of PSO to determine the CSS of 3D slopes with other optimization methods from the literature. The results demonstrated the efficiency and effectiveness of PSO in determining the CSS of 3D soil slopes.

A hybrid formulation of the 2.5D elastodynamic scattering problem combining the finite-element method and boundary integral equation method is presented and validated. The formulation of the 2.5D boundary integral equation method that is used was presented in detail by Papageorgiou and Pei (1998) and is an extension of the discrete wavenumber boundary integral equation method originally proposed by Kawase (1988) for 2D scattering problems. Modeling of the wave field in the domain of the scatterer is based on the variational principle of virtual displacements, and discretization of the domain is accomplished using the finite-element method.The formulation may be used to study the wave field in models of sedimentary deposits (e.g., valleys) or topography (e.g., canyons or ridges) with a 2D variation in structure but obliquely incident plane waves. The hybrid method exploits the versatility of the finite element method for modeling the scatterer and the effectiveness of the boundary integral equation method for taking care of the radiation condition in the half-space. The advantage of the 2.5D formulation is that it provides the means for calculations of 3D wave fields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations.

The principles and development of the boundary integral equation (BIE) method for solving engineering problems are reviewed, with particular emphasis on applications in three-dimensional stress analysis. Its use in problems of this type is illustrated with the aid of three examples, one a classical stress concentration problem, the other two involving cracked components. Compared with a finite-element method, the BIE approach is shown to be capable of being both more accurate and more economical to use in terms of competing resources required and the cost of data preparation. These advantages may, however, be less clear-cut when predictions of stresses and displacements at a large number of points within a component are required, or when the component concerned has a relatively high surface-to-volume ratio.

Acoustic wave scattering by scatterers in layered media is studied using a new method that combines the hybrid ray–mode and boundary integral equation methods. The latter is used to formulate the scattering process and the former is used to provide the Green’s function of the layered environment. Following standard procedures of the boundary integral equation method, integral equations are formulated first for field distributions along the boundaries of the obstacles. By expressing these unknown distributions in terms of appropriate basis functions, the integral equations are then reduced to algebraic equations that are solved numerically. In the integral equations, it is necessary to compute the Green’s function of the layered environment for various arrangements of locations of source and receiver. None of these conventional approaches (rays, modes, and spectral integration) to evaluate the Green’s function are satisfactory for all possible arrangements. Only the newly developed hybrid ray–mode method is best suited for this purpose because it combines rays and modes self-consistently within a single framework and optimizes the advantages of each. Numerical implementation illustrates these aspects.

Slope stability analysis is a crucial task in the design of geotechnical structures. In many computational methods, such as those combining displacement-based finite element methods and strength reduction techniques, no assumptions are made a priori about the shape or position of the failure mechanism. This is usually considered to be a positive feature but there are situations where the ability to control the search domain for the critical slip surface might be of interest. These are the cases where the critical slip surface converges towards a local (unintended) one, or when it is necessary to determine the factor of safety for slopes with varying inclinations such as the upstream and downstream slopes of embankment dams. This article presents the Finite Element Limit Equilibrium Method (FELEM), which integrates stress states computed by the finite element method along a predefined trial slip surface to determine the factor of safety. This method is combined with a swarm-based metaheuristic optimization algorithm to identify the critical slip surface with the lowest factor of safety. The combined approach is tested on two numerical examples, addressing issues such as discretization error, optimization procedure settings, and non-associated plastic flow. Finally, the combination of FELEM and the optimization algorithm is applied to a boundary value problem of a newly designed embankment dam. Slope stability is evaluated independently for both the upstream and downstream slope, each with a different inclination. Computations are performed for the hydrostatic and steady-state seepage conditions.

The applicability of the dynamic programming method to two-dimensional slope stability analyses is studied. The critical slip surface is defined as the slip surface that yields the minimum value of an optimal function. The only assumption regarding the shape of the critical slip surface is that the surface is an assemblage of linear segments. Stresses acting along the critical slip surface are computed using a finite element stress analysis. Assumptions associated with limit equilibrium methods of slices related to the shape of the critical slip surface and the relationship between interslice forces are no longer required. A computer program named DYNPROG was developed based on the proposed analytical procedure, and numerous example problems have been analyzed. Results obtained when using DYNPROG were compared with those obtained when using several well-known limit equilibrium methods. The comparisons demonstrate that the dynamic programming method provides a superior solution when compared with conventional limit equilibrium methods. Analyses conducted also show that factors of safety computed when using the dynamic programming method are generally slightly lower than those computed using conventional limit equilibrium methods of slices; however, as Poisson's ratio approaches 0.5, the computed factors of safety from the dynamic programming method and the limit equilibrium method appear to become similar.Key words: dynamic programming, slope stability, stress analysis, optimization theory, limit equilibrium methods of slices.

In past few years, we have resorted to the methodology of special-purpose computer in order to accelerate the Boundary Integral Equation Method (BIEM). Here we applied a special-purpose computer MDGRAPE-2 to the acceleration of the BIEM for three-dimensional elastostatics. The formulation was developed so that MDGRAPE-2 can evaluate the layer-potentials of the BIEM. The implemented BIEM code using MDGRAPE-2 and the iterative solver GMRES was nearly 70 or more times faster than the naive BIEM code in solving a boundary value problem with 3.9 million unknowns.

Micropiles are widely used to reinforce slopes due to their successful performance and fast construction. In this study, a simple nonlinear method is proposed to analyze the stability of a homogeneous slope reinforced with micropiles. This method is based on shear strength reduction technique, in which the soil behavior is described using the nonassociated Mohr‐Coulomb criterion and micropiles are modeled as 3D pile elements. A series of slope stability analyses is performed to investigate the coupled mechanism of micropile system, and the optimum of pile position, depth of embedment, and length of truncation are analyzed. Results show that the position of micropile system plays an important role not only in the calculation of the safety factor, but also in locating the failure surface, which demonstrates the dominating coupled effect exists between micropiles and slope. The critical embedment depth of the micropile is about 2 times the length of micropile above the critical slip surface, and the micropiles flexure rather than rotation becomes increasingly prevalent as the depth of micropiles embedment increases. Truncation of micropiles may improve the capacity of the micropile system, and the largest truncation length of micropile is about 1/4 depth of critical slip surface in this study.

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary‐value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous boundary potential, which we shall refer to as S1, and those due to a sudden change from the specification of boundary potential flux, an S2 singularity. In this paper the analysis is presented for a third type of boundary singularity, namely an S3 singularity: that arising from a discontinuous boundary flux. Such a condition is frequently encountered in the field of heat transfer where, for example, a system or pipe has a change of lagging material.In general, it is found that the BIE method is superior, with regards to the computational time required to achieve a certain degree of accuracy, over standard FD methods even when there is a boundary singularity. Further, the BIE method determines the solution near the singularity much more accurately than the FD method. The FD method does, however, have advantages for a very restrictive class of problems; for example, when the boundary conditions are of the Dirichlet type and the boundary geometry is rectangular. In this case an optimum relaxation parameter can easily be obtained. A soon as Neumann conditions are prescribed, the BIE is far more efficient than the FD, whatever the boundary geometry.It is concluded that, for fast, accurate solutions of general Laplacian boundary‐value problems, the BIE is appreciably superior to the FD and this is even more pronounced when there is a boundary singularity.