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

A non-linear theory is presented for stretching and inplane-bending of isot ropic 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 constit utive law is developed in terms of these measures via an asymptotically cor rect dimensional reduction of the geometrically non-linear 3-D elasticity u nder the assumptions of comparable magnitudes of initial radius of curvatur e and wavelength of deformation, small strain, and small ratio of cross-sec tional diameter to initial radius of curvature (h/R). The 1-D constitutive law contains an asymptotically correct refinement of O(h/R) beyond the usua l stretching and bending strain energies which, for doubly symmetric cross sections, reduces to a stretch-bending elastic coupling term that depends o n the initial radius of curvature and Poisson's ratio. As illustrations, th e theory is applied to inplane deformation and buckling of rings and high a rches. 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 validi ty of the buckling analysis is that the prebuckling strain remains small. A lthough the term added in the refined theory does not affect the buckling l oads, it is shown that non-trivial prebuckling displacements, curvature, an d bending moment of high arches are impossible to calculate accurately with out this term. (C) 1999 Elsevier Science Ltd. All rights reserved.