An elastic–plastic hardening–softening cantilever beam subjected to a force pulse at its tip: a model for pipe whip

Abstract

A theoretical model is proposed to simulate the large deflection dynamic behaviour of an elastic–plastic, hardening–softening cantilever beam with an attached end mass subjected to a suddenly applied force pulse at its free end. The model is a development of a previous one by the present authors which was based on a small deflection formulation. Numerical examples are given to illustrate the effects of the softening parameter, α , the ratio of the end mass to the mass of the beam and the intensity of the force pulse on the dynamic reponse of a beam. The changes in shape of the beams, the growth of the softening region and the possible onset of local collapse are examined. The model, though applicable generally to beams with elastic–plastic hardening–softening characteristics, was developed primarily to describe the phenomenon of pipe whip, the large deflection behaviour of a high–pressure piping system following a sudden rupture of the pipe. During the motion and deformation of a whipping pipe, softening can occur in the local bending characteristic and this in turn can lead to severe localization of the deformation, i.e. the formation of a ‘kink’. Following initial elastic behaviour, the local bending behaviour of the pipe involves sequential hardening and softening phases as the pipe deforms plastically. This results from the interaction between the material properties of the pipe and the ovalization of its cross–sections as it undergoes large plastic deformations. It has been demonstrated recently that the model can accurately predict the transient deformation of the centre lines of freely whipping pipes (Reid et al . 1996). The practical issues associated with applying the theory to tubular beams (pipes) and the mesh sensitivity of the numerical solution of the governing equations are discussed herein. However, the main aim of the paper is to add to the range of beam models available for the solution of dynamic structural plasticity problems.