Abstract
In geomechanics, limit analysis provides a useful method for assessing the capacity of struc-
tures 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 kinematically admissible discontinuities. It is well known (and confirmed here)
that the accuracy of the latter approach is highly dependent on the alignment of the discon-
tinuities, such that it can perform poorly when an unstructured mesh is employed. Second,
the optimization of the displacement field is formulated as a standard second-order cone pro-
gramming (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.
Key words: limit analysis; upper bound; cohesive-frictional; finite element; optimization;
conic programming
