Response of an elastic, plastic tubular cantilever beam subjected to a force pulse at its tip-Small deflection analysis

According to experiments, the bending behaviour of a pipe is characterized by its global elastic-plastic(hardening-softening) bending moment-curvature constitutive equation which depends on the interaction between its material properties and the ovalization of the cross-section when it is subjected to large plastic deformation. In the context of dynamic structural plasticity, this behaviour should have an important role in problems such as pipe-whip. This paper presents the results of a study of the dynamic response of a tubular cantilever beam which possesses such hardening-softening behaviour in the plastic range. With the help of a numerical approach based on a small deflection formulation, three examples are given. The first example which utilizes parameters selected from a typical pipe-whip test is compared with experimental data and gives a good prediction of the degree of ovalization of the cross-section, the distribution of plastic work in the pipe and the instantaneous deformation of the pipe. Additional examples consider various pulse loadings and constitutive relations to demonstrate that the evolution of softening can undergo three stages, viz. initial softening of one cross-section, growth of the softening region and finally shrinking of the softening region to a particular cross-section which exhibits a sharp localization of the bending deformation.