Non-linear in-plane multiple equilibria and buckling of pin-ended shallow circular arches under an arbitrary radial point load

Abstract

• Load location influences multiple limit points and non-linear equilibria of pin-ended arch. • Criteria distinguishing limit points and multiple equilibria are established. • Special modified slenderness corresponding to point of inflexion or cusp for switching multiple limit points and equilibria. • Load, axial force and displacement at limit points, points of inflexion and cusps are derived. • In some cases, mathematical solutions of nonlinear equilibria do not represent mechanical equilibrium paths. This paper analytically investigates effects of the load location on the non-linear in-plane multiple equilibria, limit points, stationary points of inflexion, cusp, and buckling behavior of a pin-ended elastic shallow circular arch under a radial point load at an arbitrary location along the arch length. Theoretical solutions for the non-linear response of the arch to the arbitrary radial point load including the limit points, stationary points of inflexion, cusp and multiple equilibria are derived. The major findings are: (1) there exists special modified slenderness corresponding to an arch, whose non-linear equilibrium path has stationary point of inflexion or a cusp and which can distinguish the number of multiple limit points and equilibria; (2) criteria distinguishing multiple limit points and equilibria are developed by relating the special modified slenderness to the load location; (3) theoretical solutions for the load, axial force and displacement at stationary points of inflexion and at cusps are also deduced; (4) the load location and the modified slenderness of an arch significantly influence the non-linear multiple equilibria of the arch; and (5) the load location has significant influence on the buckling pattern of an arch, and when the point load is applied at a location away from the apex of the arch, the arch can buckle only in a limit point instability pattern, but not in a bifurcation pattern.