Elasticity/Torsion of thin walled closed sections

Torsion of thin-walled closed sections
The Prandtl stress function $$\phi\,$$ can be approximated as a linear function between $$\phi_1\,$$ and $$0\,$$ on the two adjacent boundaries.

The local shear stress is, therefore,

\sigma_{3s} = \frac{\phi_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 $$\phi_1\,$$ can determined using

\phi_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\phi_1 \, $$