A consistent first-order nonlinear shell theory

Abstract

A general nonlinear theory of shells is derived. The theory is valid for large extensions, shear and rotations. The large rotations of the shell are described in detail. The measures of bending deformation then are written explicitly in terms of these large rotations. The general nonlinear theory contains all of the existing approximate theories as special cases including a new first-order shell theory. The remaining portion of the dissertation consists of deriving the consistent first-order nonlinear shell theory from the general theory. The first-order approximation is obtained by consistently retaining only terms of the same order of magnitude as the squares and products of the shell reference surface displacements and their derivatives while discarding higher-order terms. Comparisons are made between the first-order equilibrium equations, the Mushtari-Calimov equations and the equations obtained by making the assumptions in Sanders' small strain-small rotation theory. The comparions are made for shells of arbitrary geometry and a cylindrical shell.