Nonlinear in-plane buckling of shallow parabolic arches with tension cables under step loads

Shallow parabolic arches have been commonly applied in engineering structures, whereas their structural behavior has not been fully investigated due to significant geometric nonlinearity. In general, a shallow parabolic arch has a tendency to suddenly buckle when it is subjected to a step load. An analytical study and a numerical simulation on the nonlinear in-plane buckling behavior of shallow parabolic arches with tension cables and pin-joints are presented. To establish nonlinear buckling equilibrium equations for shallow parabolic arches, motion equations are evaluated using Hamilton’s principle. Then, the law of conservation of energy is adopted for unstable equilibrium paths, to establish the criteria for the nonlinear buckling of these shallow arches. Closed-form solutions for the lower and upper nonlinear buckling loads of a shallow arch under step loads are obtained. The analytical solutions show that the modified slenderness ratio is an important parameter affecting buckling. Nonlinear buckling under step loads or static snap-through buckling with symmetry is impossible when the slenderness ratio is out of the given interval. Moreover, nonlinear buckling under step loads is sensitive to the stiffness of the tension cables. In comparison with an equivalent distributed step load, a central concentrated load is much more likely to induce nonlinear buckling.