# Elasticity/Prandtl stress function

## Prandtl Stress Function φ

[edit | edit source]The traction free BC is obviously difficult to satisfy if the cross-section is not a circle or an ellipse.

To simplify matters, we define the Prandtl stress function using

You can easily check that this definition satisfies equilibrium.

It can easily be shown that the traction-free BCs are satisfied if

where is a coordinate system that is tangent to the boundary.

If the cross section is simply connected, then the BCs are even simpler:

From the compatibility condition, we get a restriction on

where is a constant.

Using relations for stress in terms of the warping function , we get

Therefore, the twist per unit length is

The applied torque is given by

For a simply connected cylinder,

The projected shear traction is given by

**The projected shear traction at any point on the cross-section is tangent to the contour of constant** **at that point.**

The relation between the warping function and the Prandtl stress function is

### Membrane Analogy

[edit | edit source]The equations

are similar to the equations that govern the displacement of a membrane that is stretched between the boundaries of the cross-sectional curve and loaded by an uniform normal pressure.

This analogy can be useful in estimating the location of the maximum
shear stress and the torsional rigidity of a bar.

- The stress function is proportional to the displacement of the membrane from the plane of the cross-section.
- The stiffest cross-sections are those that allow the maximum volume to be developed between the deformed membrane and the plane of the cross-section for a given pressure.
- The shear stress is proportional to the slope of the membrane.