Free periodic vibrations of beams with large displacements and initial plastic strains

This paper intends to analyse free vibrations of beams in the geometrically non-linear regime and with plastic strains. The specific goal is to find how plastic strains combined with large displacements influence the non-linear modes of vibration, by analysing the influence of the former two factors in mode shapes and natural frequencies of vibration. The geometrical non-linearity is represented by the Von Kármán type strain–displacement relations. A stress–strain relation of the bilinear type, with isotropic strain hardening, is assumed, the Von Mises yield criterion is employed and the flow theory of plasticity applied. To obtain the time domain ordinary differential equations of motion the principle of virtual work is used and a Timoshenko p-version finite element model with hierarchical basis functions is adopted. The equations of motion are naturally different from the usual large displacement equations, due to the appearance of matrices and vectors related with plastic terms. In the cases studied, plastic strains are imposed on the beam by equally distributed static forces; the forces are then removed and a study on the free vibrations is carried out. It is assumed that, once defined, the plastic strain field does not change. The time domain equations are transformed to the frequency domain by the harmonic balance method and these frequency domain equations are solved by an arc-length continuation method. The variations of mode shapes of vibration and of natural frequencies with vibration amplitude are investigated. It is found that the plastic strain distribution defines if and how much softening spring effect occurs. Hardening spring effect always appears, but with some plastic strain fields hardening spring takes place only at large vibration amplitudes. Plastic deformations also have an important effect in the vibration shapes.