In-plane buckling of circular arches and rings with shear deformations

A finite strain formulation is developed for elastic circular arches and rings in which the effects of shear deformations are included. Timoshenko beam hypothesis is adopted for incorporating shear. Finite strains are defined in terms of the normal and shear component of the longitudinal stretch. The constitutive relations for stress and finite strain are based on a hyperelastic constitutive model. Virtual work and equilibrium equations are derived. Closed-form in-plane buckling solutions are developed for circular rings and high arches under hydrostatic pressure. The effects of axial deformation prior to buckling as well as shear deformations are included in the buckling analysis. The formulation developed is compared with solutions in the literature and to the predictions of the finite element package ANSYS. The importance of including the effects of shear deformations for deep arches is investigated.