The Torsional Rigidity of a Rectangular Prism

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.

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

“…Tsai et al mentioned “rule of swapping” for an isotropic prism under torsion. 5 Figure 1 shows that the rigidity is derived under the arrangement that the x -coordinate is defined in the length-direction, the y -coordinate is defined in the width-direction, and the z -coordinate is defined in the thickness-direction. The rigidity is a mechanical parameter of the bar.…”

“…The derived torsional rigidity apparently satisfies the rule of swapping, in which the definitions for width and thickness are just for mathematic process, and cannot change the prism’s mechanical properties such as the torsional rigidity. 5 Surprisingly, the rule of swapping was not discussed in Timoshenko’s or Lekhnitskii’s process or any other research, even though it makes perfect sense physically.…”

“…This work tries to verify Lekhnitskii’s solution for the torsional rigidity of an orthotropic bar of unidirectional composite laminate under torsion (Figures 1 and 2). 5 In 2008, Chang, Wang and Tsai developed the unique TSAI technique to derive the advancing boundary of moisture diffusion in a composite laminate. 6 The technique was applied in tracking the hygric strain of composite under varying humidity 7,8 and quantifying chloride concentration of concrete plate.…”

Characterizing Shear Modulus by Torsional Pendulum Test Based on Different Theoretical Torsional Rigidities

“…2.1 The coefficient of torsional rigidity from Timoshenko [12] The coefficient of torsional rigidity from Timoshenko is…”

“…In 1934, Timoshenko presented an exact series form solution of the torsional rigidity of a rectangular isotropic bar, which is a function of the shear modulus of the material, along with the width, thickness and length of the bar [11]. In 2022, Tsai et al presented the exact closed-form solution of the torsional rigidity of a rectangular isotropic bar, which is a function of the shear modulus of the material, along with the width and thickness of the bar [12]. The torsional rigidities from Timoshenko and Tsai et al are numerically identical.…”