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, \tau, and the twist generates shear strains, \gamma.

These are related in terms of the shear modulus, G, 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, \theta, is also small.

torsion in thin-walled shafts

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, T, as force times radius:

    \[\tau=\frac{F}{A} \quad \quad T=FR_0\]

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

    \[A=2\pi R_0 t\]

    \[\tau = \frac{T}{2\pi R_0^2 t}\]

Solid & Hollow Shafts

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

    \[\gamma (r) = \frac{r\theta}{L}\]

    \[\tau = \frac{Gr\theta}{L}\]

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

    \[T = \frac{G\theta J}{L}\]

This is given as:

    \[J=\frac{\pi D^4}{32}\]

  • D is the diameter of the shaft

Combining the equations above gives us:


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, K_T:


Torsional stiffness values can be added:

torsional stiffness, torsion in bars


  • For thin-walled shafts:
    • \gamma=\frac{GR_0}{L}
    • \tau=\frac{GR_0\theta}{L}=\frac{T}{2\pi R_0^2 t}
  • For solid/hollow shafts:
    • \gamma (r) = \frac{r\theta}{L}
    • \tau = \frac{Gr\theta}{L}
    • T = \frac{G\theta J}{L}
  • Second polar moment of area: J=\frac{\pi D^4}{32}
  • Torsional stiffness: K_T = \frac{T}{\theta}=\frac{jG}{L}