Snapping of a buckled beam on elastic foundation under a midpoint force

Abstract

Abstract This paper investigates the deformation and buckling behavior of a hinged buckled beam resting on an elastic foundation and under a point force at the midpoint. The initial stable configuration before the point force is applied may be symmetric or anti-symmetric, depending on the stiffness of the elastic foundation and the end shortening of the buckled beam. In the case when the initial stable configuration is symmetric, the buckled beam may snaps to the other symmetric position via either a sub-critical bifurcation or a limit-point bifurcation when the point force is applied quasi-statically. In the case when the initial stable configuration is anti-symmetric, the buckled beam may branch to a symmetric shape via a super-critical bifurcation. The sub-critical snapping load and the super-critical branching load can be written in closed-form expressions. In the case when the point force is applied suddenly, the buckled beam with symmetric initial shape may snap either unsymmetrically or symmetrically. It is guaranteed that the buckled beam will not snap dynamically as long as the magnitude of the step force is smaller than a conservative dynamic snapping load, which can be written in a closed-form expression in case the buckled beam snaps unsymmetrically.