Thin-walled pressure vessels of minimum mass or maximum volume

This paper deals with two optimisation problems related to axisymmetric membrane shells made of homogeneous, linear and isotropic materials and subjected to an internal pressure. The design variables are the meridian shape and the thickness distribution (possibly not constant). In the first part of the paper, a general background on the mechanics of thin-walled shells is presented and the two optimisation problems concerning the search for meridian profile and thickness distribution which minimise the mass once fixed the internal volume (which, in the present work, is called direct problem) and which maximise the volume once fixed the mass (the dual problem) are formulated. The direct problem is studied and solved in closed form in several works available in the literature. All these works refer to the case of shells made of brittle materials. Herein, an extension to ductile materials is proposed. Moreover, the dual problem is solved in closed form, highlighting that the domain of boundary conditions for which a solution exists is more restricted with respect to that of the direct problem.