A Modified Linear Membrane Theory for the Pressurized Toroid

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Abstract
A modified linear membrane theory for the pressurized toroid is presented for which the displacement field is nonsingular. The derivation rests on a nonlinear formulation, but the equations are linearized by exploiting the insensitivity of the meridional stress resultant to the deformation. The resulting equilibrium equations are not statically determinate, containing two geometric variables. Two compatability equations complete the formulation. It is possible to construct a quadratic functional such that its vanishing first variation generates the derived boundary-value problem. Approximate solutions are then obtained by direct methods of the variational calculus. The results are compared with previously published nonlinear solutions and show very good agreement for a wide range of the load parameter. The present formulation readily permits generalization to orthotropic toroids.

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