On a solution of Laplace’s equation with an application to the torsion problem for a polygon with reentrant angles

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In this paper will be found a general method of solution of the two-dimensional Laplace equation with certain boundary conditions prescribed along the sides of any rectilinear polygon. The applicability of this method to the solution of technical problems will be illustrated by the treatment of the torsion problem for an infinite T-section. The mathematical solution of this problem, and of that for a finite T, is not to be found in the existing literature. I It will be seen that this scheme of solving Laplace's equation can readily be applied to any region which can be mapped conformally onto the upper half-plane in such a way that the boundary of the region goes into the entire real axis while the interior of the region transforms into the upper halfplane. Despite the considerable scientific interest in the behavior of structural members subjected to pure torsion, only a limited number of torsion problems have been brought within the range of mathematical analysis. A torsion problem is solved when one has determined a function 4) which satisfies the equation V2( =0, and which on the boundary of the section subjected to torsion reduces to? 4)* 4(X2 + y2). The determination of 4) for such simple regular sections as the circle, ellipse, equilateral triangle, and rectangle has been achieved with comparative ease.l1 In 1908, F. K6tter? succeeded in obtaining a solution of the torsion problem for an L-section by the use of the known solution for the rectangle and by application of the scheme of conformal transformation. K6tter's