Introduction to 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 ${\displaystyle b\,}$ is much larger than the thickness ${\displaystyle t\,}$, and that ${\displaystyle t\,}$ does not vary rapidly with change along the length axis ${\displaystyle \xi \,}$.

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

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

which has the solution

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

where ${\displaystyle \eta }$ is the coordinate along the thickness direction.

The stress field is

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

Thus, the maximum shear stress is

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