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.     

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 Stability Analysis of Convex Slopes in Plan View

A method of three-dimensional slope stability analysis for convex slopes in plan view is presented here based on the upper-bound theorem of the limit analysis approach. The method can also be used to determine the bearing capacity of foundations adjacent to such slopes. A rigid-block translational collapse mechanism is considered in which energy dissipation takes place along planar velocity discontinuities. Comparing the bearing capacity of foundations, numerical results indicate that the one located near convex slopes has less capacity than the one located near straight slopes. Inversely, slopes not subjected to surcharge loads are more stable when they are convex. Concerning the bearing capacity analyses, consideration of the curvature effect of convex slopes is more significant in frictional soils whereas for slope stability analyses, mentioned effect is of prime importance in cohesive soils. Numerical results of proposed algorithm are presented in the form of nondimensional graphs.     

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.     

Calculations of Bearing Capacity Factor Nγ Using Numerical Limit Analyses

This paper presents numerical upper- and lower-bound solutions for the well-known bearing capacity factor Nγ of a surface strip footing on a frictional soil. The analyses use linear programming and finite-element spatial discretization to solve limit analysis of perfect plasticity, assuming a linear Mohr-Coulomb failure envelope with associated flow within the soil and along the soil-footing interface. The current analyses are to bound the exact value of Nγ within ±5% increasing to ±30% as the internal friction angle increases from 5° to 45° for both smooth and rough interface conditions. Previous solutions by Hansen and Christensen in 1969 and Booker in 1969 are in excellent agreement with the best estimate of Nγ (average of upper and lower bounds) obtained from the current numerical limit analyses. Other well-known analytical solutions and numerical calculations appear to overestimate Nγ for rough footings. Comparisons of predicted upper-bound failure mechanisms and lower-bound contact pressures help to explain similarities and differences among these solutions.     

Eigenvalue Problem from the Stability Analysis of Slopes

Of the existing methods for the three-dimensional (3D) limit equilibrium analysis of slopes, none can simultaneously satisfy all six equilibrium equations. Except for Fellenius’ method that satisfies only one condition of moment equilibrium, all these methods could encounter numerical problems in their applications. Based on the global analysis procedure that considers the whole sliding body instead of individual columns as the loaded body, it is shown that the 3D limit equilibrium analysis of slopes simply reduces to the solution of a generalized eigenvalue problem in which the largest real eigenvalue is just the factor of safety (FOS). The proposed solution is rigorous and can accommodate any shape of slip surfaces. Under undrained conditions, the problem has a unique solution and the FOS has an explicit expression. In addition, through transforming the volume integrals over the sliding body into the boundary integrals, the proposed method does not need to partition the sliding body into columns.     

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.     

Upper-Bound Solutions for Rigid-Plastic Beams and Plates of Large Deflections by Variation Principles

This paper demonstrates deriving upper-bound solutions of geometrically nonlinear problems for beams and plates from rigid perfectly plastic material by the principles of virtual work in general form and stationary of total energy. Presented noncomplicated examples justify that the first is more appropriate when a kinematically admissible displacement field is defined by several generalized displacements. The second can serve as effective means for comparison in accuracy solutions corresponding to different displacement fields playing the same role as the upper-bound theorem in the limit analysis. Procedures of the latter for obtaining upper-bound solutions mainly remain valid. Solutions for a beam and rectangular plate subjected to uniformly distributed load illustrate importance of taking into account transformation forms of displacements in loading process.     

Three-Dimensional Passive Earth Pressures by Kinematical Approach

The 3D passive earth pressure problem is investigated by the upper-bound method in limit analysis. Three kinematically admissible failure mechanisms referred to as M1, Mn, and Mnt are considered for the calculation schemes. The M1 mechanism is an extension into three dimensions of the classical 2D Coulomb mechanism. The Mn mechanism is a generalization of the M1 mechanism and is composed of a sequence of rigid blocks. Finally, the Mnt mechanism is a more elaborate mechanism in which the final block of the Mn mechanism is truncated by two portions of right circular cones. The lowest upper-bound solutions given by the present analysis are compared with other authors' results and presented in a form of design tables relating the geometrical parameters, soil properties, and the 3D passive earth pressure coefficients.     

Limit Analysis and Stability Charts for 3D Slope Failures

The kinematic approach of limit analysis is explored in three-dimensional (3D) stability analysis of slopes. A formal derivation is first shown indicating that, in a general case, the approach yields an upper bound to the critical height of the slope or an upper bound on the safety factor. A 3D failure mechanism is used to produce stability charts for slopes. The slope safety factor can be read from the charts without the need for iterations. While two-dimensional (2D) analyses of uniform slopes lead to lower safety factors than 3D analyses do, a 3D calculation is justified in cases where the width of the collapse mechanism has physical limitations, for instance, in the case of excavation slopes, or when the analysis is carried out to back-calculate the properties of the soil from 3D failure case histories. Also, a 3D failure can be triggered by a load on a portion of the surface area of the slope. Calculations indicate that for the 3D safety factor of the loaded slope to become lower than the 2D factor for the same slope (but with a load-free surface), the load has to be very significant and equal to the weight of a soil column of the order 10?1 of the slope height.     

Evaluation of Interslice Force Function and Discussion on Convergence in Slope Stability Analysis by the Lower Bound Method

The interslice force function f(x) is a major assumption of the limit equilibrium method, which is important but has not been adequately considered in the past. In this paper, f(x) is taken as the control variable, and the upper and lower limits of the factor of safety for a general slope will be determined by a global optimization analysis. Based on this approach, f(x) will be determined and investigated. We demonstrate that f(x) cannot be arbitrarily assigned if a set of acceptable internal forces is required. The present approach can be presented practically as a lower bound approach with the advantage that failure to converge is virtually eliminated, which is not possible with all other existing “rigorous” methods. The “present proposal” attempts to answer several important questions in the basic theory of slope stability analysis, and provides a f(x) based on the lower bound approach statically admissible forces throughout the whole failure zone. Currently, different assumptions will give different factors of safety to the same problem, and this situation will be overcome by the use of the present proposal. The present proposal is also proven to give a result equal to the slip line solution for a simple footing on clay which is not possible for other classical slope stability methods, which has demonstrated that the applicability of the “present proposal” for general difficult problems.     

Elastoplastic 3D Deformation and Stress Analysis of Strip Rolling

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Significance of Soil Dilatancy in Slope Stability Analysis

A finite-element approach is used to analyze the slope stability problem and to examine the effect of soil dilatancy on the stability of slopes. It is found that soil dilatancy has a significant effect on the stability of slopes, and that higher values of dilation angle lead to larger stability numbers. Therefore, the stability numbers obtained from limit analyses (lower∕upper bound solutions) are not conservative for granular soils that exhibit a dilation angle smaller than a soil's friction angle.     

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.     

Quasi-Three-Dimensional Slope Stability Analysis Method for General Sliding Bodies

A quasi-three-dimensional procedure has been developed for computing the stability of earth slopes and waste containment facilities along general slip surfaces. The procedure, termed the Resistance-Weighted procedure, is an extension of existing quasi-three-dimensional procedures, which utilize results from two-dimensional slope stability analyses to estimate three-dimensional stability. As a part of this procedure, a scheme has been developed for modeling realistic three-dimensional sites and efficiently generating input data for two-dimensional analyses. This scheme enables Resistance-Weighted calculations to be performed using existing commercial spreadsheet and two-dimensional slope stability software. The method can also be incorporated into existing two-dimensional slope stability software with relatively little development effort. The Resistance-Weighted procedure and geometric modeling scheme are presented in this paper. Application of the procedure to a case history and results of a series of analyses to validate the procedure are described. The results of the analyses show that the Resistance-Weighted procedure produces results that compare favorably with more rigorous three-dimensional procedures. Although the Resistance-Weighted procedure is approximate, it serves as a simple means for estimating the magnitude of three-dimensional effects when a more rigorous three-dimensional procedure is not available.     

Analysis of Cavity Expansion in Sand

Cavity expansion analysis plays a significant role in modern soil mechanics. The analysis of many of the most important problems in the practice of geotechnical engineering (such as cone penetration testing, pile loading, or pressuremeter testing) rely to a large extent on cavity expansion analyses. Cavity expansion processes are of two basic types: expansion from a finite radius and expansion from zero initial radius. It is usual to use a different type of analysis for each of these problems. Analysis of the cavity creation problem yields only the limit pressure, but not necessarily information on the pressure‐strain relationship during expansion. Analysis of expansion from an initially finite cavity radius gives a pressure‐strain curve, but no information on the limit pressure. In this article, we present a simple numerical analysis that provides the solution to both problems simultaneously. The analysis takes full account of the flow rule and dependence of the friction angle on stress state, providing a rigorous solution for the cavity expansion problem throughout the plastic zone. The analysis can be used for both spherical and cyclindrical cavities. As illustration of the versatility of the analysis, plots of limit pressure versus soil state, cavity pressure versus strain for various soil states, and evolution of soil state within the plastic zone are provided.     

Reliability-Based Analysis and Design of Strip Footings against Bearing Capacity Failure

This paper presents a reliability-based approach for the analysis and design of a shallow strip footing subjected to a vertical load with or without pseudostatic seismic loading. Only the punching failure mode of the ultimate limit state is studied. The deterministic models are based on the upper-bound method of the limit analysis theory. The random variables used are the soil shear strength parameters and the horizontal seismic coefficient. The Hasofer-Lind reliability index and the failure probability are determined. A sensitivity analysis is also performed. The influence of the applied footing load on the reliability index and the corresponding design point is presented and discussed. It was shown that the negative correlation between the soil shear strength parameters highly increases the reliability of the foundation and that the failure probability is highly influenced by the coefficient of variation of the angle of internal friction of the soil and the horizontal seismic coefficient. For design, an iterative procedure is performed to determine the breadth of the footing for a target failure probability.     

Meta-Analysis of 301 Slope Failure Calculations. I: Database Description

Since the early part of the twentieth century, two-dimensional limit equilibrium (2DLE) analysis has been the scientific community’s primary means of slope stability calculation. However, it is well established that the input parameters to 2DLE, namely, soil strength and anisotropy, slope geometry, pore water pressures, failure surface geometry, applicable correction factors, and loading conditions are all inherently uncertain. Effective modeling must account for these uncertainties statistically. Unfortunately, most of the key statistical parameters, such as the safety factor statistical distribution and standard deviation (sd), are unknown and must be estimated by the analyst. In response to this growing need for statistical information, a database was established from the literature of 157 different failed slopes and the corresponding published 301 safety factor (SF) calculations. The database, which covered more than five decades of slope stability research, also included a number of the slope stability factors, including analytical method used, stress approach (effective versus total), assumed slip surface geometry, slope type, applied correction factors, and soil Atterberg limits. A temporal analysis found no evidence that SF prediction or deviation had significantly changed. A log (base 10) normal distribution was found to adequately describe the SF data, with a (nontransformed) mean of 1.03 and a (transformed) sd of 0.087, but the pronounced curvature of the residuals indicated significant, unresolved slope factors, further investigated in the companion paper.     

Meta-Analysis of 301 Slope Failure Calculations. II: Database Analysis

In response to the growing need for statistical information regarding slope stability risk analysis, this work applies inferential analysis to a compiled database of 157 failed slopes and corresponding 301 safety factor (SF) calculations. As presented in the companion paper, this database also includes a number of slope stability factors, including analytical method used, stress approach (effective versus total), assumed slip surface geometry, slope type, applied correction factors, and soil Atterburg limits. Although the SF data were found to be fairly well fit by a lognormal distribution, pronounced curvature of the residuals was observed, likely related to various unaccounted slope factors. In response, inferential statistics are used in this paper to analyze the effects of analytical method, slope type, soil plasticity, and effective versus total stress analysis. ANOVA hypothesis testing indicated significant differences between analytical methods and significant interactions between slope types and pore-water stress approaches. Direct SF calculation methods, such as infinite slope, wedge, and the ordinary method of slices were found to produce SF near 1 as expected, but higher order methods in general, and force methods in particular, predicted safety factors significantly greater than 1. Clay content alone was not a discernible influence on SF calculations. A reduced factor ANOVA model was developed to predict SF, given analytical method (a main effect) and the interactions between analytical method with both slope type and pore-water pressure approach.     

Stability of Steep Slopes in Cemented Sands

The analysis of steep slope and cliff stability in variably cemented sands poses a significant practical challenge as routine analyses tend to underestimate the actually observed stability of existing slopes. The presented research evaluates how the degree of cementation controls the evolution of steep sand slopes and shows that the detailed slope geometry is important in determining the characteristics of the failure mode, which in turn, guide the selection of an appropriate stability analysis method. Detailed slope-profile cross sections derived from terrestrial lidar surveying of otherwise inaccessible cemented sand cliffs are used to investigate failure modes in weakly cemented [unconfined compressive strength (UCS)<30?kPa] and moderately cemented (30相似文献