On collapse of non-uniform shallow arch under uniform radial pressure

Analytical investigation of collapse of a pinned non-uniform shallow circular arch under uniform radial pressure is presented in this paper. The non-uniformity is introduced by dividing the arch into three regions of constant stiffness. The equilibrium equations are obtained by using least potential energy principle. By proper nondimensionalization, the presented solution is independent of total length of the arch. And two modified slenderness parameters are identified which enables the result to be valid for any symmetric cross-section shape. Parametric study on different geometric parameters has been carried out on the snap-through buckling load and anti-symmetric bifurcation load. The validity of the analytical result is verified by comparison with FEA results. Criteria concerning four deformation modes are identified. We show there is an equal potential energy load for some non-uniform shallow arches by a straightforward deduction, the existence of which load is a necessary condition for the buckle propagation of a corresponding long shallow panel. Finally two limiting cases (rigid center case and rigid end case) with extreme non-uniformity are analytically studied by using augmented potential energy with Lagrangian multipliers. For rigid center case, closed-form condition for the possible occurrence of symmetric snap-through is presented. For rigid end case, an asymptotical analysis leads to a somewhat surprising critical value of slenderness which makes the snap-through buckling just possible. This paper intends to improve the understanding of the effect of non-uniformity on collapse of shallow arch under radial uniform pressure.