Strength and Stiffness Design of Shafts

Shaft design in torsion requires separate checks of strength from shear stress and stiffness from angle of twist.

Strength condition

For a solid or hollow circular shaft:

\[\tau_{max}=\frac{|T|}{W_p}\le\tau_{allow}.\]

The critical section has the largest ratio \(|T|/W_p\), not necessarily the largest torque alone.

Stiffness condition

For a uniform segment:

\[\varphi=\frac{TL}{GJ}.\]

For a stepped shaft, segment twists are added algebraically:

\[\varphi=\sum_i\frac{T_iL_i}{G_iJ_i}.\]

The specified limit may apply to total relative rotation or twist per unit length.

Design procedure

1. Construct the torque diagram.
2. Calculate \(J\) and \(W_p\) for each segment.
3. Find maximum shear stress and check strength.
4. Calculate segment twists with their signs.
5. Check stiffness.
6. Select a practical standard size and repeat both checks.

Practical considerations

Keyways, shoulders, grooves, and holes introduce stress concentrations. Noncircular sections require the appropriate torsion constant instead of the circular polar second moment.