This notes sheet covers torsion in bars, be they thin-walled, hollow or solid.
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:
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:
This is equivalent to the fundamental equation for beam theory.
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: