A $\Gamma$-Convergence Result for the Upper Bound Limit Analysis of Plates

TL;DR

This paper proves the mathematical convergence of finite element methods, including those with discontinuous derivatives, for upper bound limit analysis of plates, ensuring reliable numerical evaluation of ultimate loads in structural engineering.

Contribution

It establishes the $\Gamma$-convergence of finite element discretizations, including quadratic and cubic elements, for the limit analysis of plates, enhancing theoretical foundations.

Findings

Finite element methods with discontinuous derivatives converge to the continuous problem.

Numerical results validate the theoretical convergence for various materials.

The analysis improves the reliability of yield design calculations.

Abstract

Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have proposed to use various finite elements discretizations. We provide in this paper a mathematical analysis which ensures the convergence of the finite element method, even with finite elements with discontinuous derivatives such as the quadratic 6 node Lagrange triangles and the cubic Hermite triangles. More precisely, we prove the $\Gamma$-convergence of the discretized problems towards the continuous limit analysis problem. Numerical results illustrate the relevance of this analysis for the yield design of both homogeneous and non-homogeneous materials.