Example 2 : Elliptical cylinder

Choose warping function

${\displaystyle \psi (x_{1},x_{2})=kx_{1}x_{2}\,}$

where ${\displaystyle k\,}$ is a constant.

Equilibrium (${\displaystyle \nabla ^{2}{\psi }=0}$) is satisfied.

The traction free BC is

${\displaystyle (kx_{2}-x_{2}){\frac {dx_{2}}{ds}}-(kx_{1}+x_{1}){\frac {dx_{1}}{ds}}=0~~~~\forall (x_{1},x_{2})\in \partial {\text{S}}}$

Integrating,

${\displaystyle x_{1}^{2}+{\frac {1-k}{1+k}}x_{2}^{2}=a^{2}~~~~\forall (x_{1},x_{2})\in \partial {\text{S}}}$

where ${\displaystyle a\,}$ is a constant.

This is the equation for an ellipse with major and minor axes ${\displaystyle a\,}$ and ${\displaystyle b\,}$, where

${\displaystyle b^{2}=\left({\frac {1+k}{1-k}}\right)a^{2}}$

The warping function is

${\displaystyle \psi =-\left({\frac {a^{2}-b^{2}}{a^{2}+b^{2}}}\right)x_{1}x_{2}}$

The torsion constant is

${\displaystyle {\tilde {J}}={\frac {2b^{2}}{a^{2}+b^{2}}}I_{2}+{\frac {2a^{2}}{a^{2}+b^{2}}}I_{1}={\frac {\pi a^{3}b^{3}}{a^{2}+b^{2}}}}$

where

${\displaystyle I_{1}=\int _{S}x_{1}^{2}dA={\frac {\pi ab^{3}}{4}}~~;~~I_{2}=\int _{S}x_{2}^{2}dA={\frac {\pi a^{3}b}{4}}}$

If you compare ${\displaystyle {\tilde {J}}}$ and ${\displaystyle J\,}$ for the ellipse, you will find that ${\displaystyle {\tilde {J}}<J}$. This implies that the torsional rigidity is less than that predicted with the assumption that plane sections remain plane.

The twist per unit length is

${\displaystyle \alpha ={\frac {(a^{2}+b^{2})T}{\mu \pi a^{3}b^{3}}}}$

The non-zero stresses are

${\displaystyle \sigma _{13}=-{\frac {2\mu \alpha a^{2}x_{2}}{a^{2}+b^{2}}}~~;~~\sigma _{23}=-{\frac {2\mu \alpha b^{2}x_{1}}{a^{2}+b^{2}}}}$

The projected shear traction is

${\displaystyle \tau ={\frac {2\mu \alpha }{a^{2}+b^{2}}}{\sqrt {b^{4}x_{1}^{2}+a^{4}x_{2}^{2}}}~~\Rightarrow ~~\tau _{\text{max}}={\frac {2\mu \alpha a^{2}b}{a^{2}+b^{2}}}~~(b<a)}$

Shear stresses in the cross section of an elliptical cylinder under torsion

For any torsion problem where ${\displaystyle \partial S\,}$ is convex, the maximum projected shear traction occurs at the point on ${\displaystyle \partial S\,}$ that is nearest the centroid of ${\displaystyle S\,}$.

The displacement ${\displaystyle u_{3}\,}$ is

${\displaystyle u_{3}=-{\frac {(a^{2}-b^{2})Tx_{1}x_{2}}{\mu \pi a^{3}b^{3}}}}$

Displacements (${\displaystyle u_{3}\,}$) in the cross section of an elliptical cylinder under torsion