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The Quarterly Journal of Mechanics and Applied Mathematics 1958 11(2):172-184; doi:10.1093/qjmam/11.2.172
© 1958 by Oxford University Press
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THE APPLICATION OF RELAXATION METHODS TO THE SOLUTION OF DIFFERENTIAL EQUATIONS IN THREE DIMENSIONS

III. THREE-DIMENSIONAL STRESS ANALYSIS

D. N. DE G. ALLEN and S. C. R. DENNIS

( University of Sheffield
Queen's University of Belfast )

Previous parts of this series have been concerned with the solution of Laplace's equation in three dimensions. In this paper the methods already developed are extended in order to include in their scope the solution of stress problems in three dimensions; from the relaxational point of view this requires the simultaneous satisfaction of four Poisson-type equations in four dependent variables. The paper firstly derives these four governing equations, with corresponding boundary conditions which express given values of the boundary tractions, and then demonstrates their solution for a particular example of stress distribution in a cube.


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