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The equations of linear elasticity for rotationally symmetric deformations are expanded using a small parameter related to the thickness to radius of curvature ratio of the shell to obtain the classical thin shell equations of conical shells as a first approximation. These classical equations with variable coefficients permit further asymptotic expansions in the cases of steep as well as shallow cones, yielding systems of equations with constant coefficients. Solutions of these equations are used to compute the influence coefficients relating edge loads and edge displacements.

Introduction

In the case of circular cylindrical shells, asymptotic expansions of the equations of elasticity resulting in a sequence of systems of equations, the first of which is the classical theory, have been obtained by Johnson and Reissner [1]. Further results for cylindrical shells using these expansions have been reported by Reissner and coworkers [2-5]. A study of conical shells along the lines of [1] has not been carried out so far. In the present paper first we show that such a derivation is indeed possible. We list two sets of differential equations, the first being the classical conical shell theory and the second representing the effects due to finite thickness. We also derive a characteristic length for the edge bending effect, which varies from the entire domain in the case of a shallow conical shell to the characteristic length associated with the cylindrical shell for high conicity. The classical theory of conical shells is embodied in a system of differential equations which can be reduced to a single fourth order differential equation with variable coefficients. The solutions of these equations are in terms of Kelvin functions [6], which are related to Bessel functions. Here we show that the introduction of a second small parameter related to the conicity of the shell can be used to perturb the classical equations to obtain an asymptotic sequence, the first of which represents a cylindrical shell and the effects of conicity appearing in the higher order systems. This small parameter is in fact the ratio of the characteristic length to the length of the shell. Interestingly, a one term correction gives sufficient accuracy even for apparently shallow cones with serniapex angles as high as 70°.

The results for steep conical shells are supplemented with asymptotic results for shallow shells using a second small parameter. Effectively, we consider shallow shells as obtained by perturbing a circular plate. Finally, for the sake of comparison the exact influence coefficients are computed using the Kelvin functions. Numerical results are given in tabular form.

Formulation of the problem

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