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

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

which has the solution

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

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

The stress field is

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

Thus, the maximum shear stress is

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

Elasticity