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}}\, $$