The authors present a procedure for deriving an upper-bound solution of 3D slope stability problems based on a rigid finite element method (RFEM) previously proposed by them. Unfortunately, however, their paper includes serious omissions and non-accurate statements, and their results are not consistent. The following are examples of problematic points in this paper.
In their introduction the authors state: ‘Two common types of method have been proposed that attempt to perform a quantitative assessment of the stability of slopes: limit equilibrium [LE] methods and limit analysis [LA] methods.’ This statement ignores a third class of methods commonly known as ‘stress reduction methods’ (SRM). In principle these methods involve solving a series of complete continuum mechanics elasto-plastic boundary value problems (using FE or FD techniques), for materials in which the strength function is reduced by a safety factor F. An F value for which the solution does not exist (the program does not converge) represents a state of failure, and it is taken as the safety factor of the slope. This approach was originally introduced by Zienkiewicz et al. (1975), and it has since been extensively utilised (e.g. Dawson et al., 1999; Griffiths & Lane, 1999; and many others), to the extent that it is now incorporated in many commercial slope stability programs. It is not the intention to discuss here the relative merits of SRM as against LE and LA methods, except to point out that, as the authors' procedure involves repeated application of FE analysis in order to minimise their objective function, would it not be more beneficial to use the FE program in order to obtain the complete solution of the elasto-plastic problem (as is done in SRM) rather than only an upper bound on this solution?
The authors continue the above statement, saying: ‘These methods [i.e. LE and LA] are based mainly on the assumption of rigid-perfectly plastic material behaviour.’ This statement is obviously wrong, at least for LE methods. These methods deal only with forces and stress, with no consideration of deformation. Consequently the notion of rigid behaviour is not even defined in the LE framework. It is frequently stated that LE deals with rigid bodies, but such statements are wrong in principle, being similar to a statement that Euclidian geometry deals with weightless bodies. (Weight is not defined in Euclidian geometry, so it cannot deal with weightless bodies. Similarly, strains are not defined in LE, and this approach does not make a distinction between deformable and rigid bodies; it simply has no implications with respect to deformation.)
The first two verification problems considered by the authors (Figs 4 and 7) have a 2D plane strain character (being homogeneous in the third direction, x). A two-dimensional problem must have a 2D solution, yet the solutions presented by the authors (Figs 6, 8 and 9) have a 3D character. This is clearly unacceptable. At what location x is this 3D slide being located? The third problem considered by the authors is not strictly plane strain owing to some (relatively small) undulations of the soil surface (see Fig. 12(a)). However, it is rather doubtful that these undulations can be responsible for the localised failure surface shown in Fig. 12(b). It is realised that all actual slope failures have a finite extent in the third direction, but in all probability this extent is governed by local strength non-homogeneities (which are not considered by the authors), rather than by the numerical application of the upper-bound theorem to an approximately plane strain problem.
The authors refer to the works of Leshchinsky et al. (1985) and Leshchinsky & Baker (1986), which deal with the variational approach to 3D slope stability problems. However, they appear to miss an essential point derived in these works. Leshchinsky et al. (1985) showed that the 3D variational problem admits two types of solution: (i) a 2D solution, which has a cylindrical character with a log-spiral cross-section; and (ii) a 3D solution, which closes up on itself. Leshchinsky & Baker (1986) showed that these two solutions can be combined into a composite failure mechanism consisting of a central cylindrical part having a length l, and two end cups consisting of the derived 3D solutions. They then minimised the safety factor with respect to l, and found that for a plane strain problem the minimal safety factor was obtained where l → ∞ , and the 3D safety factor becomes identical with the 2D variational solution of Baker (1981). Similar results were obtained by Baker & Leshchinsky (1987) when studying truncated conical heaps in the limit when the ratio between the upper truncated radius and the height of the heap approaches infinity. These results show that a consistent analysis of a plane strain problem yields (as it should) a 2D solution. It is noted in passing that Leshchinsky et al. (1985) verified that the 3D variational LE problem is entirely equivalent to the corresponding upper-bound problem, so the authors should have obtained similar results.
This point ties up points (a) and (d). The authors report running times of 117·3 h and 142·5 h for their problems 1 and 2 respectively. Studying the stability of a vertical trench of finite length, Ugai & Leshchinsky (1995) compared the numerical performance of 3D SRM and LE variational analysis. The FE program implementing the SRM was run on a mainframe computer, whereas the LE procedure was implemented on a conventional laptop. They reported a good correspondence between results obtained by these two computation procedures. Leshchinsky (private communication) reports that a typical running time for the 3D SRM did not exceeded 5 h, whereas in their paper it is written that ‘The computational time required for limit equilibrium analysis, however, is less by orders of magnitude than that required for the 3-D finite element analysis’: that is, it took only a few minutes. It is realised that we are comparing here performance of different computers, and possibly different discretisation levels. However, the 25-fold reduction in running times between the authors' procedure and SRM appears to indicate that their formulation is not efficient compared with conventional SRM. This inefficiency (coupled with the fact that SRM yields a complete elasto-plastic solution, rather than only an upper bound on this solution), raises serious doubts about the practical utility of the procedure proposed by the authors.
Authors' reply
The authors thank the discusser for his interest in the paper. The following are responses to the comments made in the discussion.
The attention of the paper is focused on the upper- bound limit analysis of slope stability by using rigid finite elements. An upper-bound solution is obtained using this approach. It is known that a complete solution of the elasto-plastic problem can be obtained using the stress reduction method (SRM) approach. In our view, we should appreciate the power of the limit analysis, in which a solution to a plasticity problem is approached with a set of ‘bounds’ without ever solving the complete set of partial differential governing equations of the problem. It is noted that, in the authors' approach, the rigid finite elements are used only to discretise the soil mass and construct a kinematical velocity field. Of course, there is no doubt that SRM is a positive development. But SRM should be used carefully and safely, because the non-convergence phenomenon is taken as an indicator of collapse. It is well known, however, that non-convergence can be caused by many factors, and does not necessarily correspond to collapse (Crisfield, 1991).
The discusser states that the assumption of rigid- perfectly plastic material behaviour for limit equilibrium (LE) methods is wrong in principle. The reason is that ‘These methods deal only with forces and stress, with no consideration for deformation. Consequently the notion of rigid behaviour is not even defined in the LE framework.’ This seems reasonable, but is actually a misconception. The discusser appears to miss another notion of strength that is employed in the LE framework to mark collapse of the structure. In the case of perfectly plastic materials, the stress state experiences no change upon attainment of limit load. However, in nature, the strength of real soils is dependent on the level of deformation: for example, many soils experience post-peak reductions in shearing. If we do not make the assumption of perfectly plastic behaviour, it will raise the question of which point along the stress–strain curve should be used as the shear strength in an LE model. In essence, the LE methods work well for perfectly plastic materials because the perfect plasticity implies uncoupling of forces and deformations.
Simulation of a slope with infinite distance or very long length across the width (the third direction, x) of the slope can be viewed as a 2D plane strain problem. The failure surface in such cases will be fairly long in the width direction, and the stability analysis should be based on a 2D mechanics. However, this perfect 2D case is rarely encountered in reality. Real slopes—cut slopes, for example—have a finite extent in both geometry and possible failure surface. The models adopted by the authors in the first two verification problems have been set having a finite width accordingly, as illustrated in Fig. 8(a). This is close to the real phenomenon. These three problems all have 3D characteristics and thus a 3D failure mechanism should be adopted. The 2D analysis for these problems will cause a large deviation between the reality and the assumed failure mechanism. In these cases, 3D analysis based on a 3D model will be the only right way to simulate reality.
In the works of Leshchinsky et al. (1985), Leshchinsky & Baker (1986) and Baker & Leshchinsky (1987), the failure surface consists of two portions: a cylinder with length l and 3D end cup surfaces. In plane strain cases, l → ∞, the 3D safety factor will become identical with the 2D variational solution. However, in the verification problems adopted by us, our models are set up within a certain finite extent in accordance with reality. Moreover, it is noted that the boundary elements in these models have been fixed during calculations. This approach in fact imposes a constraint to prevent the end cup portions from extending beyond the lateral sides in our models. Thus the cylinder length l has a length limit, or the failure surface has 3D end cup portions without the cylindrical portion, which corresponds to l → 0. In fact, our approach is closer to the real situation.
The authors admit that the solution procedure of the proposed approach is quite time consuming, because the task of minimising the factor of safety is stated as a non-linear programming problem subject to a highly sparse set of constraints. It is necessary to explore and develop more powerful and robust non-linear optimisation algorithms for solving a system of non-linear equations for determining the velocity fields and the minimum factor of safety. Much more work is needed to be done before the full potential of the approach can be realised.
