Computation of bounds on chosen measures of real plastic deformation for beams

Abstract

A bounding technique for producing plastic deformation in elastic-plastic beam structures subjected to the contemporaneous action of a steady load and a cyclically variable load, such as the combined load results slightly above the shakedown limit, is studied. The technique is based on the proportionality between the kinematical part of the solution to the Euler-Lagrange equations relative to the shakedown cyclic load factor problem (for the structure subjected to a fixed steady load and to an amplified cyclic load), and the gradient, with respect to the cyclic load multiplier, of the kinematical part of the elastic-plastic steady-state response of the structure to loads at the shakedown limit. A suitable bounding principle allows us to evaluate a bound on the proportionality factor between the solution to the above mentioned problems for loads at the shakedown limit. Knowing such a bound and the kinematical part of the solution to the shakedown load factor problem, it is possible then to compute other bounds on any measure of real plastic deformation produced by loads slightly above the shakedown limit. A numerical example concludes the paper.