MULTILEVEL FORMULATIONS IN THE LIMIT ANALYSIS AND DESIGN OF STRUCTURES WITH BILATERAL CONTACT CONSTRAINTS

Abstract

In this paper, we study the rich class of formulations that arise in the limit analysis and design of elastic/plastic structures in the presence of contact constraints. It is well-known that in the absence of contacts, both the limit analysis and limit design problems can be written as linear programs. However, when contact constraints are present, the structure effectively exhibits both softening and stiffening behaviour under monotonically increasing loading. The resulting limit analysis and limit design problems are non-convex and are difficult to solve due to the presence of complementary type of equality constraints. We show that by using a mixed form of the minimum principle, we can restate the limit analysis and limit design problems as two- and three-level formulations, respectively. Further, under a strong assumption on the problem and solution data, we can take advantage of the underlying convexity to reduce both these multilevel formulations to equivalent linear programs. While it may not be possible to always verify this assumption in practice, we show that a two-step iterative procedure is effective in reaching a solution to the limit design problem.