Large deformations of a heavy cantilever

A cantilever of uniform cross-section and density is held at an angle $\alpha$ at one end. The shape of the cantilever depends heavily on $\alpha$ and a nondimensional parameter $K$ which represents the relative importance of density and length so that of flexural rigidity. Perturbations on the elastica equations for small and large $K$ show good agreement with exact numerical integration. It is found that whenever $K$ reaches a critical value, bifurcations of the solutions occur. This nonuniqueness can be observed by the flipping phenomena as $\alpha$ is increased.