Stiffness of thin pressurized shells of revolution

Abstract

A quasi-linear approach is developed. The strains and rotations are assumed to be small, but the equilibrium conditions are fulfilled on the deformed shell. In the resulting differential equation, bending and appear side by side. The asymptotic form of this equation is discussed for the case of the circular torus. Considered separately, the two effects are fairly similar. Suitable superposition of the two limit solutions yields a close approximation to the composite solution. As a sample application, the transverse of torus shells is derived and is presented as a function of internal p and shell thickness h. It is found in particular that membrane theory remains correct for h small but finite. Linear shell theory, on the other hand, involves an error that is proportional to p if p is small. I. Introduction T HIS paper is concerned with the rotationally symmetric deformations that pressurized, thin elastic shells of revolution experience under axisymmetric external loads. The approach to this problem presented here is a quasilinear approach: the shell strains and rotations are assumed to be small, but, contrary to the approach of linear shell theory, the shell equilibrium conditions will be fulfilled on the deformed shell. Thereby, the stiffening that a shell experiences when it is pressurized is taken into account. This pressure stiffness is neglected in linear shell theory. This deficiency of linear theory has two aspects. Both are illustrated by the model, one half of which is shown in Fig. 1, a pressurized, complete circular torus shell $ subject to an axisymmetric system P of external loads. Consider the equilibrium of the vertical forces on either one of the two toroidal shell segments that connect the two crowns of S. If we assume that S is a membrane shell, then the only vertical forces are P and the vertical components of p. These are