The Torsion of an Irregular Polygon
Proceedings of the Royal Society A
T his is th e first p ap e r in th e dev elo p m en t o f a new e x a c t m e th o d o f solving lin e a r b o u n d a ryvalue problem s, of elliptic ty p e , for regions b o u n d ed e x te rn a lly a n d , possibly, in te rn a lly b y polygons-problem s p reviously a tta c k e d , in som e sim plified cases, b y finite-difference a n d relax atio n m ethod s. T he torsion problem is chosen as th e m o st su ita b le m ed iu m to develop a n d e x h ib it th e m eth o d w hen th e b o u n d a
... n th e b o u n d a ry conditions do n o t involve d eriv ativ es. A su b seq u en t p a p e r on th e corresponding elastic p la te problem s will rem ove th is lim itatio n . T he m eth o d is b ased on rep resen tin g th e b o u n d a ry effects b y a series o f e x p o n en tially d e caying h arm onics associated w ith each edge o f th e polygon. A t th e o u ts e t it is developed for th e basic polygon-a solid polygon w ith o u t re -e n tra n t angles-th e b o u n d a ry con d itio n s being satisfied b y eq u a tin g to zero th e corresponding F o u rie r sine harm o n ics. A d ju stm e n t is necessary a t th e ends of each side o f th e polygon to ensure th e req u ire d c o n tin u ity o f th e solution. T his also sharp en s th e convergence o f th e series involved. T he resu ltin g sim u ltan eo u s e quations are in a form especially su ited to th e ir g en eratio n a n d solu tio n b y a n electronic com puter. T he stresses a n d to rsio n al rig id ity of th e cross-section are o b ta in e d using com plex v ariab le m ethods. T he m eth o d is th e n ex ten d e d to solid polygons w ith re -e n tra n t angles b y su b d iv id in g these in to basic polygons an d fo rm u latin g th e necessary coupling conditions across th e com m on boundaries. A slight extension brings th e hollow polygon, co n tain in g a n y n u m b e r o f polygonal cavities, w ith in range. T he pow er a n d sim plicity of th e m eth o d is illu stra te d n u m erically b y solving th e to rsio n problem for th e hollow square. T his involves b u t 17 eq u atio n s com pared w ith th e 432 required b y Synge & C ahill (1957) for com parable accuracy. F iv e h arm onics give th e torsional rig id ity co rrect to 1/ 105. . I n t r o d u c t i o n Explicit solutions of the torsion problem for beams with multiply-connected crosssections are not numerous. A general method based on conformal mapping and using a series of orthogonal functions has been proposed by Bergman (1950), but it appears very difficult to apply to irregular polygonal cross-sections. Abramyan (1950) has given a solution of the torsion problem for a rectangular beam with a longitudinal cavity by reducing the problem to the solution of a system of linear ordinary differential equations of the second order with constant coef ficients. Using the Hyper circle method, Synge & Cahill (1957) obtained upper and lower bounds for the torsional rigidity of a hollow square, obtaining in the process close approximations to the values of (J ) and ijr at the grid-points of the calculation network. However, it is difficult to see how either of the above methods can be generalized to give a practical computational approach to the hollow polygon with one or more cavities.