In-plane nonlinear multiple equilibria and switches of equilibria of pinned–fixed arches under an arbitrary radial concentrated load

Nonlinear in-plane multiple equilibria and buckling of pinned–fixed shallow circular arches under an arbitrary radial concentrated load are investigated. Analytical solutions for the multiple nonlinear equilibria, buckling and limit points are derived. New findings are: (1) pinned–fixed shallow arches under the arbitrary concentrated load have multiple stable and unstable equilibria; (2) the position of the arbitrary concentrated load and the modified slenderness of the arch influence the number of multiple equilibria and limit points as well as the first buckling load significantly; (3) a pinned–fixed arch under the arbitrary concentrated load can buckle in a limit point instability mode, but not in a bifurcation mode; (4) when the load is located between the crown and the pinned end, the buckling load is lower than that when the load is located between the crown and the fixed end, and (5) in addition to limit points, the nonlinear equilibria of pinned–fixed arches under the arbitrary concentrated load have inflexion points, which corresponds to specific modified slenderness switching the number of equilibria and limit points or switching buckling and no buckling behaviour. The analytical solutions for inflexion points and specific modified slenderness, and for the corresponding load, axial force and displacement are also derived for the first time in the literature. Comparisons with the finite element results have shown that the analytical solutions can accurately predict the multiple equilibria, limit points, inflexion points, and buckling load of shallow pinned–fixed arches under the arbitrary concentrated load.