Abstract
By introducing a variable transformation ξ=12(sinθ+1), a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun’s ODE, whose exact solution is obtained in terms of Heun’s functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner’s formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell. A general Maple code is provided as essential part of this paper.
Highlights
The torus or toroidal shell, in full or partial geometric form, is widely used in structural engineering
Changes its sign as principal radius of curvature Rθ when the angle θ goes from 0 to 2π. This means that the Gauss curvature has a turning point, K = 0 at θ = ±π
Recall that the partial differential equations governing the elasticity of elliptic shells K > 0 are themselves elliptic, while those for hyperbolic shells K < 0 are hyperbolic
Summary
The torus or toroidal shell, in full or partial geometric form, is widely used in structural engineering. When the torus was first studied, high-order and complicated governing equations of a torus under symmetric loads were reduced to a single lower-order, complex-form ordinary differential equation (ODE) by Hans Reissner (1912)[2] when he was a professor at ETH in Switzerland. In 1959, Tao introduced a variable transformation and successfully transformed the complex-form equation of a torus to a Heun-type ODE, and was the first person to find an exact solution that can be expressed in terms of Heun functions [12]. No exact solution in terms of special functions has been obtained for Novozhilov’s complex-form ODE of symmetrical deformation of a torus. We will shoulder this burden and propose such a solution.
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