Numerical Analysis of Combined Bending and Torsion of a Work-Hardening Plastic Square Bar

Abstract

An iterative scheme using a backward difference in the timelike variable is developed, which makes possible a numerical-analysis solution for the incremental plasticity theory formulation with hardly any extra computational effort than that required for total-strain theory; this permits easy comparison between the predictions of the two theories for a variety of bending-twist histories. Isotropic hardening is assumed, using comparable generalizations of the Ramberg-Osgood stress-strain law to combined-stress incremental and total-strain theories, for continuous loading with no unloading, and for a material assumed to have no initial elastic range, so that the entire cross section is plastic. Analysis, of a uniform prismatic square bar, continuously loaded from a stress-free state, by a bending couple in a plane of symmetry, and a twisting couple, shows that the predictions of the two theories agree when the ratio of curvature to angle of twist per unit length is held constant during the deformation, even though the stress path in stress space is nonradial. When the ratio of curvature to unit angle of twist varies during the deformation, the two theories predict different results.