A Derivation of Bending Free Toroidal Shell Shapes for Tokamak Fusion Reactors

Abstract

This paper describes an extension of mathematical solutions to bending free shapes of toroidal pressure vessels. Previous work on this subject [1] derived a pure tension shape of a toroidal membrane which was subjected to a uniform internal pressure. This research was directed toward the finding of a pure tension shape of a toroidal membrane which is subjected to a pressure inversely proportional to the square of the toroidal radius. Such a relationship is possible when electromechanical forces are considered. This family of shapes, which are nearly circular for aspect ratios greater than 2.5, is derived by solving the equilibrium, constitutive, and kinematic relationships for a uniform toroidal shell using the linear membrane theory. Serious consideration should be given to these toroidal shapes, as they provide alternative methods of design for Tokamak fusion reactors.