Add test for torsion of a rectangular section

[carlos-adir]

Let $a$ and $b$ be the width and the height of a section.

The Warping function can be computed as

$$ \omega(x, \ y) = xy - \sum_{n=0}^{\infty} \alpha_{n} \cdot \sin \left(k_{n} x\right) \cdot \sinh\left(k_n y\right) $$

With

$$\alpha_n = \dfrac{8a^2\left(-1\right)^{n}}{\pi^3\left(2n+1\right)^3 \cdot \cosh \left(\frac{1}{2}k_{n} b\right)}$$

$$k_n = \dfrac{\left(2n+1\right)\pi}{a}$$

Then, the torsion constant $J$ can be computed as

$$J = I_{xx}+I_{yy} - \int_{\Omega} \left(y \dfrac{\partial \omega}{\partial x} - x \dfrac{\partial \omega}{\partial y}\right) \ dx \ dy$$

$$J = \dfrac{a^3 b}{3} - \dfrac{64a^4}{\pi^5} \sum_{n=0}^{\infty} \dfrac{\tanh\left(\frac{1}{2}k_n \cdot b\right)}{ \left(2n+1\right)^5}$$

The stress field can be computed as

$$\sigma_{xz}(x, \ y) = \dfrac{M_z}{J}\left(\dfrac{\partial \omega}{\partial x} - y\right);\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sigma_{yz}(x, \ y) = \dfrac{M_z}{J}\left(\dfrac{\partial \omega}{\partial y} + x\right)$$

$$\sigma_{xz}(x, \ y) = \dfrac{M_z}{J} \left( -\sum_{n=0}^{\infty} \alpha_n \cdot k_n \cdot \cos\left(k_n \cdot x \right) \sinh \left(k_n \cdot y\right)\right)$$

$$\sigma_{yz}(x, \ y) = \dfrac{M_z}{J} \left( 2x - \sum_{n=0}^{\infty} \alpha_n \cdot k_n \cdot \sin \left(k_n \cdot x \right) \cosh \left(k_n \cdot y\right)\right)$$

Reference:

• Warping functions
• Wikipedia torsion constant