An analytical solution for in-plane free vibration and stability of loaded elliptic arches

In-plane free vibration and stability analyses of elliptic arches subjected to a uniformly distributed vertical static loading are performed here. A variational principle is applied to derive the governing equations for free vibration and stability of preloaded arches, considering the effect of the extensibility of the arch centerline but neglecting the effect of shear deformation. Particular attention is given to present a general procedure for combining series solutions with stiffness matrixes to construct an analytical solution for free vibration and stability of loaded arches with varying curvature. The correctness of the proposed solution is verified through a convergence study on the vibration frequencies of a loaded circular arch and by comparing the results with published data. The solution is further applied to investigate the behaviors of clamped or fixed–free elliptic arches.