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.     

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.     

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.     

Stability Charts for Uniform Slopes

While computational tools have made most graphical methods and charts obsolete, stability charts for slopes are still routinely used in practice. The charts presented here are based on the kinematic approach of limit analysis that leads to a strict lower bound on stability number c/γH or an upper bound on the safety factor. An earlier suggestion is employed in this paper to produce charts that eliminate the necessity for iterations. Charts are presented for slopes subjected to pore water pressure and also for those exposed to seismic forces.     

Critical Pool Level and Stability of Slopes in Granular Soils

The influence of pore-water pressure and the pool water pressure on stability of submerged slopes was investigated using the kinematic approach of limit analysis. For soils with some cohesive component of strength, the critical pool level is slightly below half of the slope height, whereas for slopes built of purely granular soils the critical pool level is not well defined. The most critical mechanism of failure for submerged granular slopes was found to have the failure surface intersecting the face of the slope, with one intersection point above, and the other one below the pool level. The solution to the stability problem was found to be independent of the length scale (slope height), and equally critical mechanisms of failure can be triggered “locally” with any water level in the pool. The safety factor associated with these mechanisms is lower than the well-known factor defined by a planar failure surface approaching the slope face.     

Lateral Resistance of Short Single Piles and Pile Groups Located Near Slopes

Many transmission towers, high-rise buildings, and bridges are constructed near steep slopes and are supported by large-diameter piles. These structures may be subjected to large lateral loads, such as violent winds and earthquakes. Widely used types of foundations for these structures are pier foundations, which have large diameter with high stiffness. The behavior of a pier foundation subjected to lateral loads is similar to that of a short rigid pile, because both elements seem to fail by rotation developing passive resistance on opposite faces above and below the rotation point, unlike the behavior of a long flexible pile. This paper describes the results of several numerical studies performed with a three-dimensional finite-element method (FEM) of model tests and a prototype test of a laterally loaded short pile and pier foundation located near slopes, respectively. Initially, in this paper, the results of model tests of single piles and pile groups subjected to lateral loading, in homogeneous sand with 30° slopes and horizontal ground were analyzed by the three- dimensional (3D) finite-element (FE) analyses. Furthermore, field tests of a prototype pier foundation subjected to lateral loading on a 30° slope was reported. The FE analyses were conducted to simulate these results. The main purpose of this paper is the validation of the 3D elasto–plastic FEM by comparisons with the experimental data.     

Stability Charts for 3D Failures of Steep Slopes Subjected to Seismic Excitation

Design of slopes and analysis of existing slopes subjected to seismic shaking are carried out routinely using approximations of plane strain and substitution of a quasi-static load for the seismic excitation. A three-dimensional (3D) analysis of slopes is carried out, based on the kinematic theorem of limit analysis. A rotational failure mechanism is used with the failure surface in the shape of a curvilinear cone sector passing through the slope toe, typical of steep slopes. A quasi-static approach is used to develop stability charts allowing assessment of the factor of safety of slopes without the need for an iterative procedure. The charts are of practical importance in cases of excavation slopes and whenever a slope is physically constrained, preventing a plane failure.     

GIS-Based Implementation of Three-Dimensional Limit Equilibrium Approach of Slope Stability

The growing popularity of the geographical information system (GIS), with capacities ranging from conventional data storage to complex spatial analysis and graphical presentation, means it is also becoming a powerful tool for geotechnical engineers. In this technical note, integrating the GIS grid-based data with four proposed column-based limit equilibrium models of three-dimensional (3D) slope stability analysis, new correspondent GIS grid-based 3D deterministic models have been devised in order to calculate the safety factor of the slope. Based on four GIS-based 3D slope stability analysis models, a GIS-based program, 3DSlopeGIS, has been developed to implement the algorithm where the whole of the input data is in the same form as the GIS dataset. Certain widely addressed examples have been evaluated using 3DSlopeGIS and the results show the correction and potential of this GIS-based tool as a means of assessing the 3D stability of a slope. A practical slope problem has also been evaluated using the 3DSlopeGIS system, and the results have illustrated the convenience of data management.     

Yield Strength Ratio and Liquefaction Analysis of Slopes and Embankments

A procedure is proposed to evaluate the triggering of liquefaction in ground subjected to a static shear stress, i.e., sloping ground, using the yield strength ratio, su(yield)/σv0′. Thirty liquefaction flow failures were back analyzed to evaluate shear strengths and strength ratios mobilized at the triggering of liquefaction. Strength ratios mobilized during the static liquefaction flow failures ranged from approximately 0.24 to 0.30 and are correlated to corrected cone and standard penetration resistances. These yield strength ratios and previously published liquefied strength ratios are used to develop a comprehensive liquefaction analysis for ground subjected to a static shear stress. This analysis addresses: (1) liquefaction susceptibility; (2) liquefaction triggering; and (3) post-triggering/flow failure stability. In particular, step (2) uses the yield strength ratio back-calculated from flow failure case histories and the cyclic stress method to incorporate seismic loading.     

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.     

Three-Dimensional Lower Bound Solutions for Stability of Plate Anchors in Clay

Soil anchors are commonly used as foundation systems for structures that require uplift or lateral resistance. These types of structures include transmission towers, sheet pile walls, and buried pipelines. Although anchors are typically complex in shape (e.g., drag or helical anchors), many previous analyses idealize the anchor as a continuous strip under plane strain conditions. This assumption provides numerical advantages and the problem can be solved in two dimensions. In contrast to recent numerical studies, this paper applies three-dimensional numerical limit analysis to evaluate the effect of anchor shape on the pullout capacity of horizontal anchors in undrained clay. The anchor is idealized as either square, circular, or rectangular in shape. Estimates of the ultimate pullout load are obtained by using a newly developed three-dimensional numerical procedure based on a finite-element formulation of the lower bound theorem of limit analysis. This formulation assumes a perfectly plastic soil model with a Tresca yield criterion. Results are presented in the familiar form of break-out factors based on various anchor shapes and embedment depths, and are also compared with existing numerical and empirical solutions.     

Evaluation of Microirrigation Uniformity on Laterals Considering Field Slopes

Water application uniformity is an important performance criterion that must be considered during the design and evaluation of the microirrigation systems. This study was conducted to evaluate the water application uniformity considering field slopes. The uniformity parameters including coefficient of variation (CV), Christiansen’s uniformity coefficient (UCC), distribution uniformity (DU), emitter discharge variation qv(q?max) expressed by the difference between the maximum emitter flow rate qmax and the minimum emitter flow rate qmin to qmax, emitter discharge variation qv(qd) expressed by the difference between qmax and qmin to design emitter flow rate qd, emitter discharge variation qv(qd) expressed by the difference between qmax and qmin to average emitter flow rate q(avg), emission uniformity (EU(q?avg)) expressed by qmin to qavg, and EU(qd) expressed by qmin to qd. Furthermore, the relationships of CV versus UCC, CV versus DU, CV versus qv(qd), CV versus qv(q?max), CV versus qv(q?avg), CV versus EU(q?avg), and CV versus EU(qd) were also discussed. The results of the study revealed that the field slope does not obviously affect the relationships of CV versus UCC, CV versus qv(q?max), and CV versus qv(q?avg) for any of the slopes evaluated, which include uphill, zero slope, and downhill. Furthermore, the effect of slope on the relationships of CV versus EU(q?avg), CV versus EU(qd), and CV versus qv(qd) was obvious, especially when large downhill slopes with a CV>0.05 were considered. However, the effect of slope on the relationships of CV versus DU was intermediate. Taken together, the results of this study indicated that the UCC, qv(q?max), and qv(q?avg) should be recommend in addition to CV when evaluating the water application uniformity of the microirrigation systems, regardless of slope, and that qv(qd) and EU(qd) should be recommend when evaluating systems placed in fields with a slope of zero or an uphill slope.     

Strain Distribution within Geosynthetic-Reinforced Slopes

Geosynthetic-reinforced slopes are conventionally designed using methods based on limit equilibrium. In order to estimate the factor of safety against internal stability using these methods, the distribution of the reinforcement peak tensile forces with height must be assumed. A linear distribution of reinforcement peak tension with height, with zero tension at the crest and maximum peak tension at the toe of the structure, has often been assumed. Although this assumption may be appropriate for the design of vertical geosynthetic-reinforced walls, little evidence has been collected so far justifying this distribution for the design of geosynthetic-reinforced slopes. A combination of centrifuge testing and digital image analysis is undertaken in order to obtain the strain distribution within geosynthetic-reinforced slopes under prefailure conditions. Specifically, digital image analysis techniques are used to determine the displacement distribution along reinforcement layers in reduced-scale models subjected to increasing g levels. A sigmoid function was useful to fit raw displacement data and estimate the strain distribution along reinforcement layers. Analysis of reinforcement strain results shows that the location of the reinforcement maximum peak strain does not occur near the toe of the structure, but was located approximately at midheight of the reinforced slopes, at the point along the critical failure surface directly below the crest of the slope. The pattern of reinforcement peak strain with height obtained for prefailure conditions is similar to that obtained for failure conditions. The estimated factor of safety is found to be a good indicator of the magnitude of the reinforcement maximum peak strain for geosynthetic-reinforced slopes built with different configurations.     

Three-Dimensional Multiscale Bifurcation Analysis of Granular Media

This paper deals with Hill’s bifurcation criterion, which is very well suited to describe various failure modes in granular media. The first part of this paper is dedicated to the analytical and numerical investigation of this criterion by considering phenomenological constitutive relations: the incrementally piece-wise linear and nonlinear relations proposed by Darve. The 3D bifurcation domain and 3D cones of unstable directions are given for these two relations. A similar analysis is performed with a micromechanical model in the second part of the article. Finally, a qualitative comparison of the results obtained with these two different approaches leads us to some key conclusions about this material instability criterion, which is studied for the first time for general 3D conditions, by considering these nonconventional constitutive relations.     

Block Element Method for the Seismic Stability of Rock Slopes

The seismic stability analysis of rock slope is implemented using a block element method (BEM) in this paper. Based on the formulations of the matrices of stiffness, mass, and damping, the dynamic governing equation for the rock block system is established. The Wilson method is used to solve the dynamic governing equation, and the viscoelastic artificial boundary condition is introduced to treat the unbound domain problem. The proposed method is applied to the seismic stability analysis of the intake slope in a hydropower project, from which the dynamic safety factors of key block element combinations during earthquake and their dynamic amplification factors of acceleration are evaluated.     

Two-Stepped Evolutionary Algorithm and Its Application to Stability Analysis of Slopes

Based on genetic algorithm and genetic programming, a new evolutionary algorithm is developed to evolve mathematical models for predicting the behavior of complex systems. The input variables of the models are the property parameters of the systems, which include the geometry, the deformation, the strength parameters, etc. On the other hand, the output variables are the system responses, such as displacement, stress, factor of safety, etc. To improve the efficiency of the evolution process, a two-stepped approach is adopted; the two steps are the structure evolution and parameter optimization steps. In the structure evolution step, a family of model structures is generated by genetic programming. Each model structure is a polynomial function of the input variables. An interpreter is then used to construct the mathematical expression for the model through simplification, regularization, and rationalization. Furthermore, necessary internal model parameters are added to the model structures automatically. For each model structure, a genetic algorithm is then used to search for the best values of the internal model parameters in the parameter optimization step. The two steps are repeated until the best model is evolved. The slope stability problem is used to demonstrate that the present method can efficiently generate mathematical models for predicting the behavior of complex engineering systems.     

Three-Dimensional Discrete Element Method of Analysis of Clays

Particles of cohesive soils such as clays are platelike and very small in size (e.g., cuboids of dimensions 1.0?μm×1?μm×0.06?μm). There are not only mechanical interactions between two clay particles, but also physico–chemical interactions. Properties such as the stress–strain behavior and shear strength of such a material result from a complex microscopic interactions of particles and interparticle forces. Development of physically meaningful mathematical models requires a microlevel understanding of these interactions. These interactions are difficult to observe experimentally, owing primarily to the minute size of particles. Numerical simulation studies have been conducted in the past, but using two-dimensional idealizations of particles. In the present study, a three-dimensional discrete element method is developed and implemented into a computer program. The method is used to conduct one-dimensional compression of an assembly of particles, and the macroscopic and microscopic results are presented and discussed.     

Three-Dimensional Finite Element Analysis of Underground Caverns

Detailed performance monitoring studies have been carried out for determining the deformations and stress distribution around underground powerhouse caverns in nonhomogeneous rock mass, using three-dimensional finite element method. The behavior of rock mass has been analyzed by elasto-plastic model. The effect of weak zones and creation of multiple caverns in the rock mass has been investigated. The deformations predicted in the surrounding area of the caverns are compared with those obtained by multi-point borehole extensometers (MPBX) measurements. It is observed that the computed deformations compare reasonably well with the MPBX measurements.     

Bed-Load Transport Flume Experiments on Steep Slopes

Experiments were conducted over uniform gravel bed materials to obtain 143 friction factor values under bed-load equilibrium flow conditions in an attempt to add to the scarce data available on slopes between 1 and 9% for Shields numbers between 0.08 and 0.29. Analyses showed that when only flows over flat beds are considered, a distinction must be made between flows with and without bed load. More particularly, fitting flow resistance equations indicated that the roughness parameter increases by a factor of 2.5 from clear water flow to intense bed-load transport. Between these two states, the flow resistance can be approximated by a constant for a given slope.     

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.