# Introduction to Elasticity/Torsion of thin walled closed sections

## Torsion of thin-walled closed sections

The Prandtl stress function ${\displaystyle \phi \,}$ can be approximated as a linear function between ${\displaystyle \phi _{1}\,}$ and ${\displaystyle 0\,}$ on the two adjacent boundaries.

The local shear stress is, therefore,

${\displaystyle \sigma _{3s}={\frac {\phi _{1}}{t}}}$

where ${\displaystyle s\,}$ is the parameterizing coordinate of the boundary curve of the cross-section and ${\displaystyle t\,}$ is the local wall thickness.

The value of ${\displaystyle \phi _{1}\,}$ can determined using

${\displaystyle \phi _{1}={\frac {2\mu \alpha A}{\oint _{S}{\frac {dS}{t}}}}}$

where ${\displaystyle A\,}$ is the area enclosed by the mean line between the inner and outer boundary.

The torque is approximately

${\displaystyle T=2A\phi _{1}\,}$