In-Plane Stability of Parabolic Arches with Horizontal Spring Supports. I: Theory

This paper investigates the in-plane elastic stability of a shallow parabolic arch with horizontal spring supports subjected to a uniformly distributed vertical load. A virtual work formulation is used to establish both the nonlinear equilibrium equations and the buckling equilibrium equation for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained, and a closed-form solution for the in-plane antisymmetric bifurcation buckling load and an approximation to the symmetric snap-through buckling load of shallow arches are proposed. Comparisons with the finite-element predictions of these buckling loads, together with the measured buckling loads of arches tested in the laboratory and reported in the companion paper, demonstrate that the analytical solutions are accurate. It is found that the effects of the stiffness of the horizontal springs on the buckling load and buckling behavior of arches are significant, and that the buckling load of an arch increases with an increase in the flexibility of the horizontal springs at the supports. Values of the slenderness that delineate the buckling modes increase with an increase of the flexibility of the horizontal springs. Values of the slenderness that distinguish between an arch (for which buckling is possible) and a beam curved in elevation (that cannot buckle in plane) also increase with an increase of the flexibility of the horizontal springs.