Snapping of an elastica under various loading mechanisms

This paper studies the snap-through buckling of a hinged elastica subject to a midpoint force. The focus is placed on how different load models affect the deformation and snapping load. Three different load models are considered. In the first model, the point force is fixed onto a midpoint of the elastica. In the second model, the point force is fixed on a central line in space. In the third model, the external force is applied through a rigid bar, which is allowed to slide along the central line in space. When the loaded elastica deforms symmetrically, as in the case when the magnitude of the point force is small, there is no difference between these three load models. However, when the elastica deforms unsymmetrically, the three load models produce different results. Vibration method is used to determine the stability of the equilibrium configurations. It is found that the elastica may snap via either a sub-critical pitchfork bifurcation or a limit-point bifurcation. If the elastica snaps via a sub-critical pitchfork bifurcation, the pre-snapping deformation is symmetric. If the elastica snaps via a limit-point bifurcation, on the other hand, the pre-snapping deformation is unsymmetric. For a specified initial shape of the elastica, different load models predict different snapping loads and pre-snapping deformations. The theoretical predictions are confirmed by experimental observations.