Reliable measures of plastic deformations for elastic plastic structures in shakedown conditions

A new formulation for evaluating reliable measures of the plastic deformations occurring in the transient phase of a structure in shakedown conditions is proposed. The structure is thought as constituted by elastic perfectly plastic material and subjected to a combination of fixed and cyclic loads. The proposed formulation consists in the search for the optimal plastic strain field that minimize a suitable objective function defining a strain energy measure related to the plastic strains at the shakedown limit. The typical self-stress field can be obtained as the elastic structural response to an assigned plastic strain field respecting appropriate ductility limits for the material. Without loss of generality, reference is made to plane steel frames and the limit behaviour of the typical cross section is evaluated just in terms of bending moment. Once the optimization problem, formulated under appropriate constraints related to the structure plastic admissibility and to the ductility limit behaviour of the material, is solved, the chosen measure of the plastic strains occurring during the transient phase is evaluated solving a simple linear elastic analysis problem. The design variables reduce just to the plastic strains evaluated in correspondence of suitably chosen cross sections of the structure (Gauss points).