Improved bounds for displacements at shakedown

Abstract

A method is proposed for obtaining an upper bound for the displacement which can occur in a structure adapted to a given variable-repeated load. First, the exact formulation of the problem is presented which leads to a problem of optimal control. Then the differential constraints are neglected or replaced by algebraic conditions. The solution of the resulting linear programming problem supplies an upper bound to the exact elastic-plastic deflection. Further approximation can be introduced, if desired, which allows the decomposition of the problem into separate bounds to elastic and plastic parts of displacement. Numerical examples include a two-span continuous beam and a two-storey frame. The first was solved analytically, the second by means of a specially developed computer routine capable of evaluating the displacements at shakedown of arbitrary planar frames or trusses. The results obtained confirm the validity of the small displacement approach.