An out-of-plane displacement solution in toroidal elasticity

doi:10.1016/0308-0161(94)90068-X

International Journal of Pressure Vessels and Piping

Volume 58, Issue 3, 1994, Pages 309-319

https://doi.org/10.1016/0308-0161(94)90068-X Get rights and content

Abstract

A displacement solution is given for the out-of-plane loading case in Toroidal Elasticity. The governing three-dimensional equations of elastostatics in toroidal coordinates are given in displacement form. A general series solution to these equations is then presented to cover out-of-plane loading cases. Finally an example, dealing with a 90° elbow subjected to out-of-plane loading, is considered in detail.

References (4)

D. Redekop
A displacement solution in toroidal elasticity
Int. J. Pres. Ves. & Piping
(1992)
A. Colle et al.
Pressure loading and bending of hollow tori
Int. J. Pres. Ves. & Piping
(1987)

There are more references available in the full text version of this article.

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This researcher employed specifically a stress approach, obtaining solutions and satisfying the equilibrium equations. Zhu and Redekop (1994) developed a displacement approach of TE for isotropic materials. Solutions satisfied Navier equations.
In this paper (Part 1), a new displacement-based method is proposed to investigate orthotropic curved tubes subjected to pure bending moment. A displacement approach of Toroidal Elasticity is chosen to analyze orthotropic curved tubes with a single layer. The governing equations are developed in three toroidal coordinates system. The method of successive approximation is used to find the general solution. Then, the governing equations are decomposed into different orders, based on a small parameter. The formulas for calculating different order displacement components are derived. The accuracy of the proposed method is subsequently verified by comparing the numerical results obtained using the proposed method with finite element method (FEM), stress-based Toroidal Elasticity and Lekhnitskii solution. The results show good correspondence. The proposed method provides advantages in terms of computational time compared to FEM.
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In Lang׳s works, the stress approach method was utilized, thus, he did not find deformation fields directly. The works of Lang had been advanced by Redekop [18,19]. He used the displacement components as the basic variables and developed the governing Navier equations in this coordinate system.
Simple beam theories may be applied to straight structures, while the behavior of curved structures subjected to mechanical loadings is complex. In the present study, a displacement approach of Toroidal Elasticity is chosen to analysis thick isotropic curved tubes under pure bending moments. The governing equations are developed in a toroidal coordinate system. The method of successive approximation is used to find the general solution. The accuracy of the presented method is subsequently verified by comparing the results with finite element method (FEM) and stress-based Toroidal Elasticity (SBTE). The results show good agreement. Also, high efficiency in terms of computational time is shown when the presented method is used as compared with FEM (ANSYS). Finally, several numerical examples of stress distributions in the thick isotropic curved tube subjected to pure bending are presented and discussed.
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An out-of-plane displacement solution in toroidal elasticity

Abstract

Int. J. Pres. Ves. & Piping

Int. J. Pres. Ves. & Piping