Elasticity/Torsion of thin walled closed sections

Torsion of thin-walled closed sections

The Prandtl stress function $ϕ$ can be approximated as a linear function between $ϕ_{1}$ and $0$ on the two adjacent boundaries.

The local shear stress is, therefore,

σ_{3 s} = \frac{ϕ_{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 $ϕ_{1}$ can determined using

ϕ_{1} = \frac{2 μ α A}{\oint_{S} \frac{d S}{t}}

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

The torque is approximately

T = 2 A ϕ_{1}