Constrained Euler buckling: The von Kármán approximation

Abstract

We consider the classical problem of the buckling of a planar elastica inside a rectangular cavity. We compute the equilibrium solutions analytically in the (von Kármán) small deflection approximation. We list the different equilibrium states and their domain of validity in terms of the imposed horizontal $\Delta$ and vertical $H$ displacements. We compute the horizontal $P$ and the vertical $F$ applied forces and show how they increase and scale when the compaction ratio $\sqrt{\Delta }/H$ is increased. Finally, we introduce an approximate response state, where the system adopts a periodic configuration with a noninteger number of repeated folds. This solution represents an average response of the structure and brings information on its global behavior.