Variational Methods and Upper Bound Theorem

Abstract

Recent studies on the validity of some variational methods applied to stability problems in soil mechanics point out the existence of some important errors in their statement. These errors seem to be due to the lack of a rigorous mathematical analysis that assures the existence of an absolute minimum of the suggested functionals. Such analysis will usually involve considerations of the second variation of these functionals. Therefore, it will frequently lead to serious difficulties; indeed, it may be unfeasible in many cases. In this work, the limit analysis theorems are proposed to define “security” or “load” functionals to be optimized. This methodology assures the existence of a lower or upper bound of the suggested functional depending on which theorem is applied, provided that the mathematical conditions assumed for the validity of limit theorems are satisfied. Moreover, both the functional and the solution based on it have a coherent physical meaning. As a consequence, it is possible to reduce the problem to the analysis of the first variation of the functional. This methodology seems to be applicable to any stability problem in soil mechanics. In particular, it is applied to a slope stability problem by defining a security functional, based on the upper bound theorem. A system of equations, therefore, is obtained from the vanishing of its first variation.