Investigation of the elastic torsion problem of circular hollow shafts using the boundary element method

Elastic pure torsion behavior is critical in designing load-bearing components such as shafts, curved beams, and eccentrically loaded bridge girders, making it a key topic in mechanical and civil engineering. This paper investigates the torsional response of hollow circular shafts using the Prandtl stress function, focusing on the effects of eccentricity and diameter ratio. A general differential equation was solved using the boundary element method (BEM), in which the boundary domain was discretized into Linear elements. A custom FORTRAN code was developed and validated against the exact solution of Prandtl stress function for concentric shafts, showing strong agreement across the radius, except at the boundaries. The error decreased with finer discretization. Diameter ratios from 1.25 to 5 were analyzed under a constant rotation of 5°/m. For each case, induced torque and maximum shear stress were calculated with increasing eccentricity from 0 to 0.9. The results, presented in figures and contours, show that the maximum and minimum Prandtl stress function values occur at the midpoint of the maximum and minimum thicknesses along the eccentric centerline. These values increase with diameter ratio for any eccentricity. Shear stresses concentrate on the thicker side of the eccentric centerline, intensifying with higher eccentricity and diameter ratios. Torque also rises with increasing eccentricity or diameter ratio, highlighting the importance of geometric parameters in torsional performance. This method is first step for more significant topic of analyzing torsion problem for non-prismatic shafts or tubes through dealing with such sahfts as series of portions of small prismatic shaft in order to apply this method for each portions since every small portion can be assumed as small prismatic shaft.