Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

# Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

Version 1 : Received: 15 December 2020 / Approved: 17 December 2020 / Online: 17 December 2020 (09:07:55 CET)
Version 2 : Received: 27 February 2021 / Approved: 2 March 2021 / Online: 2 March 2021 (09:22:01 CET)
Version 3 : Received: 18 April 2021 / Approved: 19 April 2021 / Online: 19 April 2021 (12:03:23 CEST)

A peer-reviewed article of this Preprint also exists.

Bohua Sun, Small symmetrical deformation of thin torus with circular cross-section, Thin-Walled Structures 163 (2021) 107680 Bohua Sun, Small symmetrical deformation of thin torus with circular cross-section, Thin-Walled Structures 163 (2021) 107680

## Abstract

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, 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.

## Keywords

toroidal shell; deformation; Gauss curvature; Heun function; hypergeometric function; Maple

## Subject

Computer Science and Mathematics, Mathematics

Comment 1
Commenter: Bohua Sun
Commenter's Conflict of Interests: Author
Comment: correction of typos in Equation 22 and Table II
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