Torsion in Bars

This notes sheet covers torsion in bars, be they thin-walled, hollow or solid.

• Thin-Walled Shafts
• Solid & Hollow Shafts
• Summary

When a shaft (stationary or rotating) experiences a torque, it may begin to twist about its axis.
The torque gives rise to a shear stress, , and the twist generates shear strains, .

These are related in terms of the shear modulus, , of the material:

The shear modulus is given by the Poisson’s ratio and Young’s modulus of the material:

Thin-Walled Shafts

Just like with thin-walled pressure vessels, the key assumption is that the thickness of the wall is
significantly smaller than the radius. Another important assumption, however, is that the angle of
twist, , is also small.

This only applies to circular cross-sections.

The shear strain is given as the arc length of twist over the length of shaft:

Using the shear stress-strain relationship above, shear stress is given as:

We know that stress can also be written as force over area, and torque, , as force times radius:

Approximating area as the circumference times the thickness, , these two equations can be
combined to eliminate :

Solid & Hollow Shafts

Generally, you cannot assume a shaft is thin-walled. This means the variation in torsion with
radius is not negligible:

To find these in terms of torque, the second polar moment of area, , is required:

This is given as:

• is the diameter of the shaft

Combining the equations above gives us:

\frac{\tau}{r}=\frac{T}{J}=\frac{G\theta}{L}\]

This is equivalent to the fundamental equation for beam theory.

Torsional Stiffness

The quantity given by torque over angle of twist is sometimes called torsional stiffness, :

Torsional stiffness values can be added:

• For thin-walled shafts:
  • 
  • 
• For solid/hollow shafts:
  • 
  • 
  • 
• Second polar moment of area:
• Torsional stiffness: