Elasticity/Torsion of thin walled closed sections

Torsion of thin-walled closed sections

The Prandtl stress function $\varphi$ can be approximated as a linear function between ${\varphi }_1$ and $0$ on the two adjacent boundaries.

The local shear stress is, therefore,

${\sigma }_{3s}=\frac{{\varphi }_1}{t}$

where $s$ is the parameterizing coordinate of the boundary curve of the cross-section and $t$ is the local wall thickness.

The value of ${\varphi }_1$ can determined using

${\varphi }_1=\frac{2\mu \alpha A}{{\oint }_S\frac{dS}{t}}$

where $A$ is the area enclosed by the mean line between the inner and outer boundary.

The torque is approximately

$T=2A{\varphi }_1$

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