Stresses and Deformations of the Thin-Walled Pressurized Torus

Abstract

The thin-walled pressurized torus is an example of a configuration that cannot be treated properly by means of linear membrane theory. Though the stress distributions that result from this theory do not exhibit any irregularities themselves, they lead to deformations which are incompatible with the assumption of a continuous shell. The deformation incompatibilities are such that superficially they would seem to require the introduction of bending and shear stresses. If this suspicion were correct, as has been assumed previously, this would not only invalidate the physical concept of an ideal membrane but it would also lead to a more complicated torus analysis than appears to be warranted by the nature of the problem. Indeed, such an analysis seems never to have been performed. In the present paper the concept of an ideal membrane is revalidated by showing that an adequate nonlinear membrane solution exists. This membrane solution can be derived with relatively little numerical effort. Numerical results are given which demonstrate the required corrections to the results of linear membrane theory. As a side result, a lemma is derived concerning the invariance with respect to Poisson's ratio of certain deformations which arise as thin shells of revolution are pressurized.