Quasi-Bifurcation of Discrete Systems with Unstable Post-Critical Behavior under Impulsive Loads

Abstract

The huge economic and environmental consequences due to explosions in the oil industry have recently motivated the study of thin-walled cylindrical shells subjected to impulsive waves due to an explosion. One of the main challenges in this field is the identification of suitable dynamic buckling criteria for this class of loads. This paper focuses on a simple geometrically nonlinear two degree-of-freedom system subjected to impulsive loads of very short duration with respect to their fundamental periods. Similar to what occurs in shell structures, the model includes both membrane and bending stiffness components and displays an unstable static behavior at the critical state. The description of the motion is formulated using Lagrange equations and they are integrated numerically by means of an implicit code. By analogy with the static case, the dynamic path that follows the motion of a degree-of-freedom at a given load level is defined as the fundamental motion; any perturbed path that departs from the fundamental path is interpreted as a bifurcation of the original motion and has been given the name of quasi-bifurcation by Lee. Following the original criterion due to Lee, a coefficient calculated as the projection of the perturbed motion on the corresponding perturbed acceleration is investigated, and instability is said to occur for positive values of the coefficient. The present results show that the quasi-bifurcation criterion, as originally presented by Lee, is a necessary condition but may yield low values of instability loads because a positive projection may occur for a short time before the structure returns to a stable condition with a negative coefficient. A modification is proposed in which the time integration of the inner product due to Lee is carried out, and the analysis shows that this allows a clearer identification of bounds in the response that are associated with dynamic buckling.