Non-linear buckling and postbuckling analysis of arches with unequal rotational end restraints under a central concentrated load

Abstract

This paper presents an analytical study of the non-linear elastic in-plane buckling and postbuckling behaviour of pin-ended shallow circular arches having unequal elastic rotational end restraints under a central concentrated radial load. The principle of stationary potential energy is used to derive the differential equations of equilibrium, based on which the analytical solution for the non-linear equilibrium path is derived. It is found that the non-linear behaviour of an arch having unequal rotational end restraints is much more complicated than that of an arch with equal rotational end restraints. The arch may have a non-linear equilibrium path that consists of one or two unstable equilibrium paths and two or four limit points, and it may even have a non-linear looped equilibrium path in some cases. The number of limit points on the non-linear equilibrium path of an arch depends on its slenderness ratio and included angle, and on the stiffnesses of the unequal rotational end restraints. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points, as well as for distinguishing between arches and beams curved in-elevation. The principle of conservation of energy at neutral equilibrium is used to derive the differential equations of buckling equilibrium, which are then used to investigate the buckling behaviour. It is found that an arch with unequal rotational end restraints cannot buckle in a bifurcation mode. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour.