A new approach to the asymptotic integration of the equations of shallow convex shell theory in the post-critical stage

Abstract

A method is proposed for the asymptotic integration of the non-linear equations of shallow elastic shell theory on the basis of a new definition of the small parameter that is selected to be proportional to the ratio between the shell thickness and the amplitude of its deflection. This parameter is actually small if the shell is in the post-critical stage, i.e., its deflections are large. An asymptotic expansion of the solution of the shell equilibrium equations in the parameter mentioned is carried out. It is established that the first two approximations result in the geometric theory of shell stability formulated by Pogorelov /1/. By comparing the asymptotic and numerical solutions /2/ found for a spherical shell under axisymmetric deformation, satisfactory accuracy of the proposed method is obtained for fairly large deflection. The well-known Koiter approach is used in the small-deflection domain. The two asymptotic expansions, one of which is suitable for small deflections and the other for large, are merged using the Pade approximation.