Symmetric and unsymmetric buckling of circular arches

A nonlinear geometrically exact inextensible elastica theory is used to derive a mathematical system which models a clamped circular arch of central angle $2\alpha$ under the action of a vertical force field of amplitude $P$ (e.g., gravity). The equilibria of the arch are studied for various values of $\alpha , 0 < \alpha < \pi$. The existence of a solution of symmetric form for all fixed values of $P$ and $\alpha$ is proved analytically by arguments based on variational principles. Numerical solutions are calculated for a variety of choices of $\alpha$, and in each case buckling (nonuniqueness) is shown to occur when $P$ is sufficiently large. In some cases, both symmetric and unsymmetric configurations are found, but each unsymmetric configuration obtained is found to be an unstable equilibrium, having energy greater than that of the symmetric configuration. Implications concerning the relative strengths and weaknesses of the various arches are discussed.