Nonlinear analysis and buckling of shallow arches with unequal rotational end restraints

Arches are often supported by other structural elements which provide elastic types of restraint at both of their ends and these elastic restraints usually have different stiffnesses. This paper presents a nonlinear elastic in-plane buckling and postbuckling analysis of pin-ended shallow circular arches having elastic rotational end restraints of unequal stiffness when subjected to a uniform radial load. An energy method is used to derive the differential equations of equilibrium, and the differential equations of buckling equilibrium. The nonlinear equilibrium equation between the external load and the axial force is also established. Analytical solutions for the nonlinear in-plane behaviour and buckling loads of the arches are derived. It is found that the effects of the unequal stiffness of the rotational end restraints on the nonlinear behaviour and buckling of the arches are significant. The nonlinear buckling loads increase with an increase of the stiffness of rotational end restraints. It is also found that the pin-ended arch with elastic rotational end restraints of unequal stiffness may buckle in a limit point instability mode, but they cannot buckle in a bifurcation mode. This is different from pin-ended arches with equal rotational end restraints, which can buckle in either a limit point instability or a bifurcation mode. It is demonstrated that there exists a specific geometric parameter (as defined in the paper) that defines a switch between arches and beams curved in elevation. Comparisons with finite element results show that the analytical solutions can accurately predict the nonlinear behaviour and buckling loads of pin-ended shallow circular arches with elastic rotational end restraints of unequal stiffness. The analytical solutions are shown to reduce to those of pin-ended arches with equal rotational end restraints or without rotational end restraints, fixed arches, and pinned-fixed arches.

► Nonlinear analysis of arches with unequal rotational end restraints.
► Analytical solution for nonlinear equilibrium and buckling of these arches.
► Arches with unequal rotational end restraints can buckle in a limit point mode.
► These arches cannot buckle in a bifurcation mode.
► Significant effects of rotational end restraints on nonlinear behaviour of arches.