A finite element analysis in polar co-ordinates of the Saint Venant torsion problem

A characteristic feature of the variational functionals for several boundary value problems in polar co-ordinates is the fact that one independent variable occurs explicitly in the denominator. Therefore, the coefficients of the finite element equations for sectors of circular ring shaped elements are not constants but functions of the distance of the elements from the origin of co-ordinates.¹ We name them coefficient functions. In order to show the particular aspects of the calculations in terms of polar co-ordinates we deal here with the solution of the torsion problem by bilinear and bicubic Hermitian interpolation. The finite element equations are arranged according to Schaefer² in the form of block which can easily be transformed into ‘stars’³ or molecules^4,5 similar to those used in finite difference methods. The origin of co-ordinates requires a special consideration, firstly because of the coincidence of several nodes at that point and secondly because of the divergent behaviour of some coefficient functions. It turns out to be advantageous for the numerical calculations to expand the coefficient functions in power series. Besides, the expansions are required to deduce the equations for rectangular elements by limiting processes. The twisting moments and shearing stresses calculated for several cross sections illustrate the numerical suitability of the method. The finite element values are compared partly with exact solutions and partly with experimental results obtained by a moiré method using Prandtl's soap film analogy.⁶ Finally it is shown how the accuracy of the finite element values can be improved by the Richardson extrapolation⁷.