SUMMARYIn geomechanics, limit analysis provides a useful method for assessing the capacity of structures such as footings and retaining walls, and the stability of slopes and excavations. This paper presents a finite element implementation of the kinematic (or upper bound) theorem that is novel in two main respects. First, it is shown that conventional linear strain elements (6-node triangle, 10-node tetrahedron) are suitable for obtaining strict upper bounds even in the case of cohesive-frictional materials, provided that the element sides are straight (or the faces planar) such that the strain field varies as a simplex. This is important because until now, the only way to obtain rigorous upper bounds has been to use constant strain elements combined with a discontinuous displacement field. It is well known (and confirmed here) that the accuracy of the latter approach is highly dependent on the alignment of the discontinuities, such that it can perform poorly if an unstructured mesh is employed. Second, the optimization of the displacement field is formulated as a standard second-order cone programming (SOCP) problem. Using a state-of-the-art SOCP code developed by researchers in mathematical programming, very large example problems are solved with outstanding speed. The examples concern plane strain and the Mohr-Coulomb criterion, but the same approach can be used in 3D with the Drucker-Prager criterion, and can readily be extended to other yield criteria having a similar conic quadratic form.

SUMMARYThe formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second-order cone programming (SOCP), for which efficient primaldual interior-point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr-Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker-Prager criterion in 2D or 3D, and Nielsen's criterion for plates).

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SUMMARYThe classical lower and upper bound theorems allow the exact limit load for a perfectly plastic structure to be bracketed in a rigorous manner. When the bound theorems are implemented numerically in combination with the finite element method, the ability to obtain tight bracketing depends not only on the efficient solution of the arising optimization problem, but also on the effectiveness of the elements employed. Elements for (strict) upper bound analysis pose a particular difficulty since the flow rule is required to hold throughout each element, yet it can only be enforced at a finite number of points. For over 30 years, the standard choice for this type of analysis has been the constant strain element combined with discontinuities in the displacement field. Here we show that, provided certain conditions are observed, conventional linear strain triangles and tetrahedra can also be used to obtain strict upper bounds for a general convex yield function, even when the displacement field is discontinuous. A specific formulation for the Mohr-Coulomb criterion in plane strain is given in terms of second-order cone programming, and example problems are solved using both continuous and discontinuous quadratic displacement fields.

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SUMMARYA major difficulty when applying the kinematic theorem in limit analysis is the derivation of expressions of the dissipation functions and the set of plastically admissible strains. At present, no standard methodology exists. Here, it is shown that they can be readily obtained, provided that the yield restriction can be rewritten as an intersection of cones, and that the expression defining the dual cones is available. This is always possible for the case of self-dual cones and some other classes, and covers many of the well-known criteria. Therefore, a difficult obstacle with respect to the use of the kinematic theorem in conjunction with any numerical method can be overcome. The methodology is illustrated by giving the expressions of the dissipation functions for various conic yield restrictions. A special emphasis is given on upper bound finite element limit analysis. Taking advantage of duality in conic programming, we can obtain the dual problem, where knowledge of the dual cone is not necessary. Therefore, this formulation is feasible for any cone. Finally, it is interesting that the form of the dual problem, for varying yield strength within the finite element, differs from that presented in other papers.

There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.