Non-linear stability and remote unconnected equilibria of shallow arches with asymmetric geometric imperfections

• Closed-form expressions of the equilibria and critical loads are derived.
• These expressions can be applied to shallow arches with arbitrary shapes.
• Asymmetric geometric imperfections make equilibria split at bifurcation points.
• Remote unconnected equilibria exist for imperfect shallow arches.
• Unable to obtain remote unconnected equilibria by FEA without initial perturbation.

This paper presents an analytical method to study the non-linear stability and remote unconnected equilibria of shallow arches with non-symmetric geometric imperfections. The exact solutions of the equilibria and critical loads are obtained. Unlike many previous studies, these solutions can be applied to arbitrary shallow arches with arbitrary geometric imperfections. It is found that slightly imperfect arches have multiple remote unconnected equilibria that cannot be obtained in experiments or using finite element simulations if a proper perturbation is not performed. The formulas to directly calculate the critical loads, including those of the remote unconnected equilibria, are also derived. The effect of asymmetric geometric imperfections on the equilibria and critical loads is revealed by applying the derived formulas to half-sine arches with different geometric imperfections.