Non-linear inplane deformation and buckling of rings and high arches

Abstract

A non-linear theory is presented for stretching and inplane-bending of isotropic beams which have constant initial curvature and lie in their plane of symmetry. For the kinematics, the geometrically exact one-dimensional (1-D) measures of deformation are specialized for small strain. The 1-D constitutive law is developed in terms of these measures via an asymptotically correct dimensional reduction of the geometrically non-linear 3-D elasticity under the assumptions of comparable magnitudes of initial radius of curvature and wavelength of deformation, small strain, and small ratio of cross-sectional diameter to initial radius of curvature ( h/R). The 1-D constitutive law contains an asymptotically correct refinement of O(h/R) beyond the usual stretching and bending strain energies which, for doubly symmetric cross sections, reduces to a stretch–bending elastic coupling term that depends on the initial radius of curvature and Poisson’s ratio. As illustrations, the theory is applied to inplane deformation and buckling of rings and high arches. In spite of a very simple final expression for the second variation of the total potential, it is shown that the only restriction on the validity of the buckling analysis is that the prebuckling strain remains small. Although the term added in the refined theory does not affect the buckling loads, it is shown that non-trivial prebuckling displacements, curvature, and bending moment of high arches are impossible to calculate accurately without this term.