Optimality conditions for shakedown design of trusses

Abstract

This paper deals with optimal shakedown design of truss structures constituted by elastic perfectly plastic material. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problem are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the maximum shakedown limit load design whose volume is assigned. The Kuhn-Tucker equations of the four problems here above mentioned are found by utilizing a variational approach; these equations prove the equivalence of the two types of design problem and provide useful information on the structure behaviour in optimality conditions. A suitable computational procedure of iterative type devoted to the reaching of the minimum volume design is presented. It is shown that the design obtained by this technique is the optimal one, since it satisfies the optimality conditions of the relevant search problem. In the typical step of this technique the dependency of the elastic response on the design variables is approximately taken into account. In the application stage a numerical example, aimed at utilizing this special technique, is presented.