Elasto-plastic and geometrically nonlinear vibrations of beams by the p-version finite element method

A model based on a Timoshenko beam p-version finite element is developed to analyse oscillations that are, simultaneously, elasto-plastic and geometrically nonlinear. The geometrical nonlinearity is represented by Von Kármán type strain–displacement relations and the stress–strain relation is of the bilinear type, with mixed strain hardening. The equations of motion are obtained using the principle of virtual work and are solved in the time domain by an implicit Newmark method. The Von Mises yield criterion is employed and the flow theory of plasticity applied; if plastic flow is found at a point of the domain, the total plastic strain is determined by summation. Numerical examples are carried out in order to demonstrate that the p-version element here advocated has a number of advantages and to show the influence of the plastic and geometrically nonlinear terms on the oscillations of beams.