The Torsional Rigidity of a Rectangular Prism

Cho-Liang Tsai, Chih-Hsing Wang, Sun-Fa Hwang, Wei-Tong Chen and Chin-Yi Cheng
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Cho-Liang Tsai: Department of Civil and Construction Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan
Chih-Hsing Wang: College of Future, Bachelor Program in Industrial Projects, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan
Sun-Fa Hwang: Department of Mechanical Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan
Wei-Tong Chen: Department of Civil and Construction Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan
Chin-Yi Cheng: Department of Mechanical Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan

Mathematics, 2022, vol. 10, issue 13, 1-11

Abstract: Using the membrane analogy, in 1934 Timoshenko derived the torsional rigidity of a rectangular prism of isotropic material as a function of its material shear modulus, width and thickness. However, he did not consider the energy conservation criterion, as it could be either unnecessary or replaced by other criteria in Timoshenko’s process. To confirm the correctness of Timoshenko’s solution, this work re-derives the torsional rigidity by considering all the equilibrium conditions, boundary conditions, symmetric and anti-symmetric conditions of displacement and stress, the energy conservation criterion, and even the energy minimization criterion. Using the TSAI technique, exact solutions for the displacements, strains, stresses and the torsional rigidity are derived perfectly. The derived torsional rigidity is in a completely different form from that derived by Timoshenko and is numerically identical. Interestingly, the solutions derived in this work verify that, when the values of the width and thickness of the rectangular prism are swapped, the value of the torsional rigidity remains the same, which makes perfect sense physically but is not discussed in Timoshenko’s process or any other research. This work presents a procedure considering all the mathematical details and the results remain correct when the width and thickness of the prism swap. This fact makes perfect sense physically, though has never been expounded before in Timoshenko’s or other researchers’ solutions, either for torsional rigidity or for the induced shear stresses and displacements.

Keywords: energy conservation; rectangular prism; torsional rigidity; TSAI technique (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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