Upper bound limit analysis using discontinuous quadratic displacement fields

Abstract

The 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.