Limit analysis

The limit analysis adopts the concept of an idealized stress–strain relation, that is, the soil is assumed as a rigid, perfectly plastic material with an associated flow rule. Without carrying out the step-by-step elasto-plastic analysis, the limit analysis can provide solutions to many problems. Limit analysis is based on the bound theorems of classical plasticity theory (Drucker et al., 1951; Drucker and Prager, 1952). The general procedure of limit analysis is to assume a kinemati-cally admissible failure mechanism for an upper bound solution or a statikinemati-cally admissible stress field for a lower bound solution, and the objective function will be optimized with respect to the control variables. Early efforts of limit analysis were mainly made on using the direct algebraic method or analytical method to obtain the solutions for slope stability problems with simple geome-try and soil profile (Chen, 1975). Since closed form solutions for most practical problems are not available, later attention has been shifted to employing the slice techniques in traditional limit equilibrium to the upper bound limit analysis (Michalowski, 1995; Donald and Chen, 1997).

Limit analysis is based on two theorems: (a) the lower bound theorem, which states that any statically admissible stress field will provide a lower bound estimate of the true collapse load, and (b) the upper bound theorem, which states that when the power dissipated by any kinematically admissible velocity field is equated with the power dissipated by the external loads, then the external loads are upper bounds on the true collapse load (Drucker and Prager, 1952).

A statically admissible stress field is one that satisfies the equilibrium equations, stress boundary conditions, and yield criterion. A kinematically admissible velocity field is one that satisfies strain and velocity compatibil-ity equations, veloccompatibil-ity boundary conditions and the flow rule. When combined, the two theorems provide a rigorous bound on the true collapse load. Application of the lower bound theorem usually proceeds as stated in the following. (a) First, a statically admissible stress field is constructed.

Often it will be a discontinuous field in the sense that we have a patchwork of regions of constant stress that together cover the whole soil mass. There will be one or more particular value of stress that is not fully specified by

the conditions of equilibrium. (b) These unknown stresses are then adjusted so that the load on the soil is maximized but the soil remains unyielded. The resulting load becomes the lower bound estimate for the actual collapse load.

Stress fields used in lower bound approaches are often constructed with-out a clear relation to the real stress fields. Thus, the lower bound solutions for practical geotechnical problems are often difficult to find. Collapse mechanisms used in the upper bound calculations, however, have a distinct physical interpretation associated with actual failure patterns and thus have been extensively used in practice.

2.4.1 Lower bound approach

The application of the conventional analytical limit analysis was usually limited to simple problems. Numerical methods therefore have been employed to compute the lower and upper bound solutions for the more complex problems.

The first lower bound formulation based on the finite element method was proposed by Lysmer (1970) for plain strain problems. The approach used the concept of finite element discretization and linear programming. The soil mass is subdivided into simple three-node triangular elements where the nodal normal and shear stresses were taken as the unknown variables. The stresses were assumed to vary linearly within an element, while stress discontinuities were permitted to occur at the interface between adjacent triangles. The statically admissible stress field was defined by the constraints of the equilibrium equations, stress boundary conditions and the linearized yield criterion. Each non-linear yield criterion was approximated by a set of linear constraints on the stresses that lie inside the parent yield surface, thus ensuring that the solutions are a strict lower bound. This led to an expression for the collapse load which was maximized, subjected to a set of linear constraints on the stresses. The lower bound load could be solved by optimization, using the techniques of linear programming. Other investigations have worked on similar algorithms (Anderheggen and Knopfel, 1972; Bottero et al., 1980). The major disadvantage of these formulations was the linearization of the yield criterion which generated a large system of linear equations, and required excessive computational times, especially if the traditional simplex or revised simplex algorithms were used (Sloan, 1988a). Therefore, the scope of the early investi-gations was mainly limited to small-scale problems.

Efficient analyses for solving numerical lower bounds by the finite element method and linear programming method have been developed recently (Bottero et al., 1980; Sloan, 1988a,b). The key concept of these analyses was the introduction of an active set algorithm (Sloan, 1988b) to solve the linear programming problem where the constraint matrix was sparse. Sloan (1988b) has shown that the active set algorithm was ideally suited to the numerical lower bound formulation and could solve a large-scale linear programming problem efficiently. A second problem associated with the numerical lower bound solutions occurred when dealing with statically admissible conditions for an infinite-half space. Assdi and Sloan (1990) have

solved this problem by adopting the concept of infinite elements, and hence obtained rigorous lower bound solutions for general problems.

Lyamin and Sloan (1997) proposed a new lower bound formulation which used linear stress finite elements, incorporating non-linear yield conditions, and exploiting the underlying convexity of the corresponding optimization problem. They showed that the lower bound solution could be obtained efficiently by solving the system of non-linear equations that define the Kuhn–

Tucker optimality conditions directly.

Recently, Zhang (1999) presented a lower bound limit analysis in con-junction with another numerical method – the rigid finite element method (RFEM) to assess the stability of slopes. The formulation presented satisfies both static and kinematical admissibility of a discretized soil mass without requiring any assumption. The non-linear programming method is employed to search for the critical slip surface.

2.4.2 Upper bound approach

Implementation of the upper bound theorem is generally carried out as fol-lows. (a) First, a kinematically admissible velocity field is constructed. No separations or overlaps should occur anywhere in the soil mass. (b) Second, two rates are then calculated: the rate of internal energy dissipation along the slip surface and discontinuities that separate the various velocity regions, and the rate of work done by all the external forces, including grav-ity forces, surface tractions and pore water pressures. (c) Third, the above two rates are set to be equal. The resulting equation, called the energy–

work balance equation, is solved for the applied load on the soil mass. This load would be equal to or greater than the true collapse load.

The first application of the upper bound limit analysis to the slope stability problem was by Drucker and Prager (1952) in finding the critical height of a slope. A failure plane was assumed, and analyses were performed for isotropic and homogeneous slopes with various angles. In the case of a vertical slope, it was found that the critical height obtained by the upper bound theorem was identical with that obtained by the limit equilibrium method. Similar studies have been done by Chen and Giger (1971) and Chen (1975). However, their attention was mainly limited to a rigid body sliding along a circular or log-spiral slip surface passing both through the toe and below the toe in cohesive materials. The stability of slopes was evaluated by the stability factor, which could be minimized using an analytical technique.

Karel (1977a,b) presented an energy method for soil stability analysis.

The failure mechanisms used in the method included: (a) a rigid zone with a planar or a log-spiral transition layer; (b) a soft zone confined by plane or log-spiral surfaces; and (c) a composed failure mechanism consisting of rigid and soft zones. The internal dissipation of energy occurred along the transition layer for the rigid zone, and within the zone and along the transition layer for the soft zone. However, no numerical technique was proposed to determine the least upper bound of the factor of safety.

Izbicki (1981) presented an upper bound approach to slope stability analysis. A translational failure mechanism, which was confined by a cir-cular slip surface in the form of rigid blocks similar to the traditional slice method, was used. The factor of safety was determined by an energy bal-ance equation and the equilibrium conditions of the field of force associated with the assumed kinematically admissible failure mechanism. However, no numerical technique was provided to search for the least upper bound of the factor of safety in the approach.

Michalowski (1995) presented an upper bound (kinematical) approach of limit analysis in which the factor of safety for slopes derived is associated with a failure mechanism in the form of rigid blocks analogous to the vertical slices used in traditional limit equilibrium methods. A convenient way to include pore water pressure has also been presented and implemented in the analysis of both translational and rotational slope collapse. The strength of the soil between blocks was assumed explicitly that it was taken as zero or its maxi-mum value set by the Mohr–Coulomb yield criterion.

Donald and Chen (1997) proposed another upper bound approach to eval-uate the stability of slopes based on a multi-wedge failure mechanism. The slid-ing mass was divided into a small number of discrete blocks, with linear interfaces between the blocks and with either linear or curved bases to individ-ual blocks. The factor of safety was iteratively calculated by equating the work done by external loads and body forces to the energy dissipated along the bases and interfaces of the blocks. Powerful optimization routines were used to search for the lowest factor of safety and the corresponding critical failure mechanism.

Other efforts have been made in solving the limit analysis problems by the finite element method, which represents an attempt to obtain the upper bound solution by numerical methods on a theoretically rigorous foundation of plas-ticity. Anderheggen and Knopfel (1972) appeared, having developed the first formulation based on the upper bound theorem, which used constant-strain triangular finite elements and linear programming for plate problems. Bottero et al. (1980) later presented the formulation for plain strain problems. In the formulation, the soil mass is discretized into three-node triangular elements whose nodal velocities were the unknown variables. Each element was associated with a specific number of unknown plastic multiplier rates. Velocity discontinuities were permitted along pre-specified interfaces of adjacent triangles. Plastic deformation could occur within the triangular element and at the velocity discontinuities. Kinematically admissible velocity fields were defined by the constraints of compatibility equations, flow rule of the yield criterion and velocity boundary conditions. The yield criterion was linearized using a polygonal approximation. Thus, the finite element formulation of the upper bound theorem led to a linear programming problem whose objective function was the minimization of the collapse load and was expressed in terms of the unknown velocities and plastic multipliers. The upper bound loads were obtained using the revised simplex algorithm. Sloan (1988b, 1989) adopted the same basic formulation as Bottero et al. (1980) but solved the linear programming problem using an active set algorithm. The major problem

encountered by Bottero et al. (1980) and Sloan (1988b, 1989) was caused by the incompressibility condition of the perfectly plastic deformation. The discretization using linear triangular elements must be arranged such that four triangles form a quadrilateral with the central nodes lying at its centroid. Yu et al. (1994) have shown that this constraint can be removed using higher order (quadratic) interpolation of the nodal velocities.

Another problem of the formulation used by Bottero et al. (1980) and Sloan (1988b, 1989) was that it could only handle a limited number of velocity discontinuities with pre-specified directions of shearing. Sloan and Kleeman (1995) have made significant progress in developing a more general numerical upper bound formulation in which the direction of shearing was solved automatically during the optimization solution. Yu et al. (1998) compare rigorous lower and upper bound solutions with con-ventional limit equilibrium results for the stability of simple earth slopes.

Many researchers (Mroz and Drescher, 1969; Collins, 1974; Chen, 1975;

Michalowski, 1989; Drescher and Detournay, 1993; Donald and Chen, 1997;

Yu et al., 1998) pointed out that an upper bound limit analysis solution may be regarded as a special limit equilibrium solution but not vice versa. The equiva-lence of the two approaches plays a key role in the derivations of the limit load or factor of safety for materials following the non-associated flow rule.

Classically, algebraic expressions for the upper bound method are deter-mined for the simple problems. Assuming a log-spiral failure mechanism for failure surface A shown in Figure 2.10, the work done by the weight of the soil is dissipated along the failure surface based on the upper bound approach by Chen (1975) using an associated flow rule, and the height of the slope can be expressed as

(2.29)

The critical height of the slope is obtained by minimizing eq. (2.29) with respect to θ0 and θh which has been obtained by Chen (1975). Chen has also found that failure surface A is the most critical log-spiral failure surface unless βis small. When βand φ′are small, a deep-seated failure shown by failure surface B in Figure 2.10 may be more critical. The basic solution as given by eq. (2.29) can however be modified slightly for this case. The critical result of f(φ′,α,β) as given by eq. (2.29) can be expressed as a dimen-sionless stability number N_s which is given by Chen (1975). In general, the stability numbers by Chen (1975) are very close to that by Taylor (1948).

Within the strict framework of limit analysis, 2D slice-based upper bound approaches have also been extended to solve 3D slope stability prob-lems (Michalowski, 1989; Chen et al. 2001a,b). The common features for these approaches are that they all employ the column techniques in 3D limit equilibrium methods to construct the kinematically admissible velocity field, and have exactly the same theoretical background and numerical algorithm which involves a process of minimizing the factor of safety. More recently, a promising 2D and 3D upper bound limit analysis approach by means of linear finite elements and non-linear programming (Lyamin and Sloan, 2002b) has emerged. The approach obviates the need to linearize the yield surface as adopted in the 2D approach using linear programming (Sloan, 1989; Sloan and Kleeman, 1995). However, the approach nonethe-less has stress involvement in performing the upper bound calculations.