Elasticity/Torsion of thin walled open sections

Torsion of thin-walled open sections

Examples are I-beams, channel sections and turbine blades.

We assume that the length $b$ is much larger than the thickness $t$, and that $t$ does not vary rapidly with change along the length axis $\xi$.

Using the membrane analogy, we can neglect the curvature of the membrane in the $\xi$ direction, and the Poisson equation reduces to

$\frac{d^{\varphi }}{d{\eta }^2}=-2\mu \alpha$

which has the solution

$\varphi =\mu \alpha (\frac{t^2}{4}-{\eta }^2)$

where $\eta$ is the coordinate along the thickness direction.

The stress field is

${\sigma }_{3\xi }=\frac{\partial }{\partial }\varphi \eta =-e\mu \beta \eta ;{\sigma }_{3\eta }=0$

Thus, the maximum shear stress is

${\tau }_{\text{max}}=\mu \beta t_{\text{max}}$