Calculations: Torsion in Orthotropic Rectangular Beams

Last updated June 27, 2022
By Ian Story

The following equations come from Lekhnitskii, S.G. 1981. Theory of elasticity of an anisotropic body (page 285).

(1)   \begin{equation*}d = \frac{h/b}{\sqrt{G_2/G_1}}\end{equation*}

(2)   \begin{equation*}\beta = \frac{32d^2}{\pi^4}\sum_{m=1,3,5,\cdots}^{\infty}\frac{1}{m^4}\left(1-\frac{2d}{m\pi}tanh\frac{m\pi}{2d}\right)\end{equation*}

(3)   \begin{equation*}k_1 = \frac{8d}{\pi^2\beta}\sum_{m=1,3,5,\cdots}^{\infty}\frac{(-1)^{\frac{m-1}{2}}}{m^2}tanh\frac{m\pi}{2d}\end{equation*}

(4)   \begin{equation*}k_2 = \frac{d}{\beta}\left(1-\frac{8}{\pi^2}\sum_{m=1,3,5,\cdots}^{\infty}\frac{1}{m^2cosh\frac{m\pi}{2d}}\right)\end{equation*}

(5)   \begin{equation*}\tau_{max,1} = \frac{T}{db^2}k_1\end{equation*}

(6)   \begin{equation*}\tau_{max,2} = \frac{T}{db^2\sqrt{G_2/G_1}}k_2\end{equation*}

An excel calculator with Lambda functions to solve for each of these parameters as a function of d is available here: Torsion Factors for Orthotropic Rectangular Sections.xlsx

Input Values

Shear modulus ratios (g) can be found in research by Hindman: Hindman et al. 2000. Orthotropic Behavior of Lumber Composite Materials. Trus Joist publishes shear modulus (G) values in their product data guides: LSL Product Data. Generally, it looks like G (in the principal plane) for LSL members is approximately equal to E divided by 16.