On the Final States of Shallow Arches on Elastic Foundations Subjected to Dynamical Loads

Abstract

Given in this paper is a nonlinear analysis of snap-through problems of shallow arches on linear elastic foundations subjected to time-varying loads which have a time-independent character as τ → ∞. Specific problems studied in detail are simply supported sinusoidal arches under impulsive loads, under time-varying loads with asymptotic spatial distributions of sinusoidal shape, and under time-varying loads with uniform asymptotic spatial distributions. It is found that for a very wide range of the foundation modulus a necessary and sufficient condition for stability against snap-through can be established and the final state of the arch predicted. Outside this range and when the loads are timewise step loads, useful sufficient conditions for instability and sufficient conditions for stability can be found separately. The nonlinear treatment presented here is exact in the sense that no approximation is made or is required in the mathematical analysis.