A Diagonally Dominant Solution for the Cylinder End Problem

[+] Author and Article Information
T. D. Gerhardt

USDA-Forest Service, Forest Products Laboratory, P.O. Box 5130, Madison, Wisc. 53705

Shun Cheng

Department of Engineering Mechanics, University of Wisconsin, Madison, Wisc. 53706

J. Appl. Mech 48(4), 876-880 (Dec 01, 1981) (5 pages) doi:10.1115/1.3157749 History: Received June 01, 1980; Revised January 01, 1981; Online July 21, 2009


An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Copyright © 1981 by ASME
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