Multi-phased solutions of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion and the general rule of swapping

Abstract

Based on the conventional solution of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion, one can derive displacements, shear strains, shear stresses and torsional rigidity. The conventional solution of Prandtl's stress function has a hyperbolic function for the coordinate in the bar's thickness direction, and a trigonometric function for the coordinate in the width direction. This paper raises questions about the solution. Why is the solution not arranged in the opposite way? Why is the hyperbolic function not for the coordinate in the width direction and the trigonometric function for the coordinate in the thickness direction? How is it that these obvious questions have never been addressed? This study rearranges the solution of the conventional Prandtl's stress function using the TSAI technique and finds that the solution is multi-phased, indicating that the coordinates in the width and thickness directions and their corresponding parameters are swappable, a phenomenon proposed as the general rule of swapping.