Numerical Analysis of the Axi-Symmetric Bending of a Toroidal Shell

Abstract

Finite difference approximations are used to analyse a symmetrically loaded toroidal shell. It is shown that the equations derived by E. Reissner in terms of H_ϕr₀ (the product of the radial force per unit circumference normal to the axis of revolution and the radius measured normal to the axis) and V (rotation of the meridional tangent) avoid the difficulties which occur, owing to a singularity at the pole circle of a toroidal shell, when Meissner's equations in terms of U = Q_ϕr₂ (the product of the lateral shear force per unit circumference and the second principal radius of curvature) and V are used (Fig. 1). A numerical example is solved with both forms of the equations. The values obtained for H_ϕ, Q_ϕ and V differ by less than 1 per cent near the edge of the shell but in the region of the pole circle there is no correlation between the two sets of results. Comparison with Turner's analytic solution shows that the values and positions of the maxima and minima differ by up to 6 per cent and 5° respectively.