The torsion of an irregular polygon

Abstract

This is the first paper in the development of a new exact method of solving linear boundary-value problems, of elliptic type, for regions bounded externally and, possibly, internally by polygons—problems previously attacked, in some simplified cases, by finite-difference and relaxation methods. The torsion problem is chosen as the most suitable medium to develop and exhibit the method when the boundary conditions do not involve derivatives. A subsequent paper on the corresponding elastic plate problems will remove this limitation. The method is based on representing the boundary effects by a series of exponentially decaying harmonics associated with each edge of the polygon. At the outset it is developed for the basic polygon—a solid polygon without re-entrant angles—the boundary conditions being satisfied by equating to zero the corresponding Fourier sine harmonics. Adjustment is necessary at the ends of each side of the polygon to ensure the required continuity of the solution. This also sharpens the convergence of the series involved. The resulting simultaneous equations are in a form especially suited to their generation and solution by an electronic computer. The stresses and torsional rigidity of the cross-section are obtained using complex variable methods. The method is then extended to solid polygons with re-entrant angles by subdividing these into basic polygons and formulating the necessary coupling conditions across the common boundaries. A slight extension brings the hollow polygon, containing any number of polygonal cavities, within range. The power and simplicity of the method is illustrated numerically by solving the torsion problem for the hollow square. This involves but 17 equations compared with the 432 required by Synge & Cahill (1957) for comparable accuracy. Five harmonics give the torsional rigidity correct to 1/10 5 .