MecMovies 4.0 : M6.17: Shaft Diameter Based on Power

The elastic torsion formula $(τ = Tc / J)$ can be used to solve for the minimum required diameter. The torque and the shear stress are known, but the polar moment of inertia $J$ and the shaft radius $c$ are unknown. Manipulate the torsion equation so that the torque and the shear stress are on the right-hand side and $J$ and $c$ are on the left-hand side.

$τ = \frac{Tc}{J} ∴ \frac{J}{c} = \frac{T}{τ}$

With just one available equation and two unknowns $(J and c)$ , we must simplify $J / c$ so that it can be expressed in terms of only one variable — the outside diameter $d$ .

For a solid circular shaft, $J$ is defined as:

$J = \frac{π}{32} d^{4}$

where $d$ is the outside diameter. The shaft radius $c$ is simply $d / 2$ . Thus, the term $J / c$ can be expressed as:

$\frac{J}{c} = \frac{π}{16} d^{3}$