Timoshenko & Lekhnitskii's puzzle ‒ Rule of swapping for the torsional rigidity of a rectangular bar

Timoshenko presented the torsional rigidity of an isotropic rectangular bar, and Lekhnitskii presented that of an orthotropic rectangular bar. The solutions of Timoshenko and Lekhnitskii (T&L) are functions of the bar's length, width, thickness and shear modulus or moduli. However, the functions of T&L solutions become different from their original ones when the width and thickness are swapped. Swapping the width and thickness definitions does not alter the bar's physical properties, named the “rule of swapping” by the authors. In the last century, no research has shown the T&L solutions to satisfy the rule of swapping, an observation hereinafter referred to as the “Timoshenko & Lekhnitskii Puzzle”. Roughly 90 years later, Tsai et al. re-solved T&L cases using the TSAI technique. The derived solutions are nearly if not completely identical to T&L's numerically and satisfy the rule of swapping automatically. The rule of swapping is a novel issue and has never been mentioned before. Based on the Weierstrass factorization theorem, this study mathematically proves that they are identical for isotropic and orthotropic bars and satisfy the rule of swapping. The result of a torsional pendulum test is analyzed to support the rule.