Snap-through of an elastica under bilateral displacement control at a material point

Snap-through phenomenon widely occurs for elastic systems, where the systems lose stability at critical points. Here snap-through of an elastica under bilateral displacement control at a material point is studied, by regarding the whole elastica as two components, i.e., pinned-clamped elasticas. Specifically, stiffness−curvature curves of two pinned-clamped elasticas are firstly efficiently located based on the second-order mode, which are used to determine the shapes of two components. Similar transformations are used to assemble two components together to form the whole elastica, which reveals four kinds of shapes. One advantage of this way compared with other methods such as the shooting method is that multiple coexisting solutions can be located accurately. On the load−deflection curves, four branches correspond to four kinds of shapes and first two branches are symmetrical to the last two branches relative to the original point. For the bilateral displacement control, the critical points can only appear at saddle-node bifurcations, which is different to that for the unilateral displacement control. Specifically, one critical point is found on the first branch and two critical points are found on the secondary branch. In addition, the snap-through among different branches can be well explained with these critical points.