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 $b\,$ 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

$\tau_{\text{max}} = \mu\beta t_{\text{max}}\,$