Nonlinear Dynamic Buckling of Fixed Shallow Arches under an Arbitrary Step Radial Point Load

Abstract

This paper investigates the nonlinear elastic in-plane dynamic buckling of a fixed shallow circular arch that was subjected to an arbitrarily located step radial point load, which has not been previously reported in the literature. The arbitrary step load produced unsymmetrical deformations of the arch and unsymmetrical internal forces in the arch and caused the arch to oscillate unsymmetrically. Because the unsymmetrical equilibrium path was nonlinear and consisted of a primary stable branch, an unstable branch, and a remote stable branch, when the step load reached a critical value, the arch could oscillate to reach the unstable equilibrium branch and subsequently lose its in-plane stability in a dynamic buckling mode. Hence, the first condition for dynamic buckling of the arch was that the load could induce oscillation of the arch sufficiently large to reach an unstable equilibrium position. The second condition was derived on the basis of the principle of the conservation of energy: the total potential energy of the system needed to vanish for its possible dynamic buckling. This condition could be used in conjunction with the nonlinear equilibrium path to derive the analytical solution for the dynamic buckling load of shallow arches under an arbitrary step radial point load. The unsymmetrical effects of the load position on the dynamic buckling load were found to be significant. First, if the load was not applied at the crown of the arch, there was only one dynamic buckling load. This was different from the case of a fixed arch subjected to a step point load at its crown, in which an upper and a lower dynamic buckling load exist. Second, the dynamic buckling load first decreased and then increased as the point of the application of the load moved away from the crown. Third, when the load position exceeded a specific value from the crown, the arch did not experience dynamic buckling but oscillated around a stable equilibrium point under the step load. The effects of the modified slenderness on the dynamic buckling were also investigated, and it was found that the dynamic buckling load increased with an increase of the modified slenderness. Finally, the comparisons with the nonlinear elastic static in-plane buckling showed that the nonlinear elastic dynamic buckling load of an arch was lower than its static counterpart.