Introduction to Elasticity/Torsion of thin walled open sections

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[edit] 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 η 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}}\,