Analytical Study of the Snap-Through and Bistability of Beams With Arbitrarily Initial Shape

We derive the snap-through solution and the governing snapping force equations for an arbitrarily pre-shaped beam deflected under a mid-length lateral point force. The exact solution is obtained based on the classical theory of elastic beams as a superposition of the initial shape and the modes of buckling. Two kinds of solution are identified depending on the axial force level. The two solutions, bifurcation conditions, bistability conditions, and the snapping force equations are derived and discussed. The snap-through and snapping force solutions are then calculated for two common beam initial shapes, the curved (first buckling shape) and the inclined one (V-shape). In both cases, explicit expressions are obtained describing the snap-through behavior. The analytical modeling results show excellent agreement with the finite element simulations. The comparison between the two cases shows a similar snap-through behavior qualitatively, while several differences and similarities are noticed quantitatively.