The torsional rigidity of an orthotropic bar of unidirectional composite laminate part I: Theoretical solution

Abstract

In 1934, Timoshenko derived and published the torsional rigidity of a rectangular bar of isotropic material by using the membrane analogy. The rigidity depends on the bar’s shear modulus, width and thickness. In 1950, Lekhnitskii followed Timoshenko’s process to derive the torsional rigidity for an orthotropic bar of unidirectional composite laminate, where the rigidity depends on the laminate’s width, the thickness and the shear moduli. In 1990, Tsai, Daniel and Yaniv solved the same case, deriving a quasi-exact solution of torsional rigidity. In the same year, Tsai and Daniel verified their result through multiple experiments. All these rigidities become different when the definitions of thickness and width are swapped. However, they remain identical numerically, lacking mathematic proof for nearly a century. In 2022, Tsai et al. resolved Timoshenko’s case by considering all the conditions and energy minimization criterion. By a completely different approach, the torsional rigidity was derived in a completely different form from that of Timoshenko but numerically identical. Moreover, the solution satisfies the rule of swapping, which makes perfect sense physically. This work applies Tsai’s process to Lekhnitskii’s case. A general solution is derived that satisfies the rule of swapping involving the swapping of shear moduli.