The theory of collapse and minimum weight design of cylindrical shells by Onat and Prager is applied in this paper to the development of a method for designing the variable wall thickness of a cylindrical shell under axial loading and arbitrary pressures to give maximum economy of material. The design is such that the shell does not fail plastically in the sense used in limit analysis. It will be assumed that the shell is supported at the ends by inextensible rings so that the circumferential strain rates vanish there.