On the Equations of the Linear Theory of Elastic Conical Shells

Abstract

In a recent article [19], it was shown that the equations of the linear theory of elastic shells of revolution under arbitrary loads, after a harmonic analysis in the polar angle of the base plane of the shell, can be reduced to two simultaneous fourth order ordinary differential equations for a stress function and a displacement function. The exact reduction procedure of [19] was applied to a circular cylindrical shell. Also mentioned was the possibility of a simplified procedure based on the fact that there is an inherent error in shell theory of the order h/R, where h is the shell thickness and R is a representative magnitude of the two principal radii of curvature. In the present paper, the exact and the simplified procedure of [19] will be applied to a conical shell frustrum. The simplified procedure leads to two differential equations as well as to auxiliary equations for the calculation of stress resultants and couples which differ from those of the exact procedure only in terms which are O(h/R). The differential equations obtained here are very similar in form to those of shallow shell theory [6,8] as well as to those of the Donnell type theory [15]. They do in fact reduce to the shallow shell equations when the slope angle /!? between a meridian and the base plane is very small. On the other hand, for /3 = z/2, they reduce to the “correct” equations for circular cylindrical shells of [16,19] while the results of [15] reduce only to the Donnell equation. Aside from their usefulness in obtaining the solution of specific problems, our results also delimit the range of applicability of the Donnell type equations for conical shells previously obtained in [5,15].