Multiple unstable equilibrium branches and non-linear dynamic buckling of shallow arches

• Arch with unequal rotational end restraints has multiple unstable branches.
• Multiple unstable branches influence dynamic buckling of arches significantly.
• The arch under central radial may have up to three dynamic buckling loads.
• Analytical solution for dynamic buckling load is derived using an energy method.
• Effects of arch geometry and stiffness of end restraints are significant.

An arch under a suddenly-applied load will oscillate about its equilibrium position. If the suddenly-applied load is sufficiently large, the oscillation may reach a position on the unstable equilibrium branch of the arch, triggering its dynamic buckling. In many cases, arches are supported by other structural members or by elastic foundations which provide elastic types of rotational restraints to the ends of the arch. When the rotational end restraints of an arch are not equal to each other, the in-plane non-linear equilibrium path of the arch may have multiple unstable branches, which will influence the dynamic buckling of the arch significantly. This paper investigates effects of multiple unstable equilibrium branches on the non-linear in-plane dynamic buckling of a shallow circular arch under a suddenly-applied central concentrated load. Two methods based on the energy approach are used to derive the dynamic buckling loads. It is found that the number and magnitude of dynamic buckling loads are influenced significantly by the number of unstable equilibrium branches, by the stiffness of the unequal rotational end restraints, and by the included angle and slenderness ratio of the arch.