Effect of Normal Strains in Buckling of Thick Orthotropic Shells

An improved elasticity solution to the problem of buckling of orthotropic cylindrical shells subjected to external pressure is presented. The 2D axisymmetric cylindrical shell is studied (ring approximation). Specifically, in the development of the governing equations and boundary conditions for the buckling state, the solution includes the terms with the prebuckling normal strains and stresses as coefficients (i.e., the terms ekk0σij′ and σkk0eij′, which were neglected in the earlier work as being too small compared to the terms σij′ and σkk0ωj′, respectively). The formulation results in a two-point boundary eigenvalue problem for ordinary differential equations in r, with the external pressure p as the parameter. The results show that the effect of including the normal strains and stresses is to further decrease the critical load. This decrease (versus the earlier elasticity solution without these terms) depends on the shell thickness and is generally moderate, and in no event comparable with the (quite large) decrease of the elasticity versus the shell theory prediction. This decrease depends also on the degree of orthotropy, and it is smaller for the isotropic case. Finally, a formula is derived for the critical pressure based on a first-order shear deformation formulation, and the comparison shows an improvement versus the classical shell for thick shells, but still the elasticity solution is noticeably lower than the first-order shear deformation prediction.