Open carlos-adir opened 10 months ago
Let $a$ and $b$ be the width and the height of a section.
width
height
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
$$\alphan = \dfrac{8a^2\left(-1\right)^{n}}{\pi^3\left(2n+1\right)^3 \cdot \cosh \left(\frac{1}{2}k{n} b\right)}$$
$$k_n = \dfrac{\left(2n+1\right)\pi}{a}$$
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
$$\sigma_{xz}(x, \ y) = \dfrac{Mz}{J}\left(\dfrac{\partial \omega}{\partial x} - y\right);\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sigma{yz}(x, \ y) = \dfrac{M_z}{J}\left(\dfrac{\partial \omega}{\partial y} + x\right)$$
$$\sigma_{xz}(x, \ y) = \dfrac{Mz}{J} \left( -\sum{n=0}^{\infty} \alpha_n \cdot k_n \cdot \cos\left(k_n \cdot x \right) \sinh \left(k_n \cdot y\right)\right)$$
$$\sigma_{yz}(x, \ y) = \dfrac{Mz}{J} \left( 2x - \sum{n=0}^{\infty} \alpha_n \cdot k_n \cdot \sin \left(k_n \cdot x \right) \cosh \left(k_n \cdot y\right)\right)$$
Reference:
Let $a$ and $b$ be the
width
and theheight
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
$$\alphan = \dfrac{8a^2\left(-1\right)^{n}}{\pi^3\left(2n+1\right)^3 \cdot \cosh \left(\frac{1}{2}k{n} b\right)}$$
$$k_n = \dfrac{\left(2n+1\right)\pi}{a}$$
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
$$\sigma_{xz}(x, \ y) = \dfrac{Mz}{J}\left(\dfrac{\partial \omega}{\partial x} - y\right);\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sigma{yz}(x, \ y) = \dfrac{M_z}{J}\left(\dfrac{\partial \omega}{\partial y} + x\right)$$
$$\sigma_{xz}(x, \ y) = \dfrac{Mz}{J} \left( -\sum{n=0}^{\infty} \alpha_n \cdot k_n \cdot \cos\left(k_n \cdot x \right) \sinh \left(k_n \cdot y\right)\right)$$
$$\sigma_{yz}(x, \ y) = \dfrac{Mz}{J} \left( 2x - \sum{n=0}^{\infty} \alpha_n \cdot k_n \cdot \sin \left(k_n \cdot x \right) \cosh \left(k_n \cdot y\right)\right)$$
Reference: