Shaft Diameter (torsion)⚠ unverified

Understanding the Parameters

• Torque $T$ — the load the shaft must carry. Diameter grows only as the cube root of torque, so a shaft carrying eight times the torque needs just twice the diameter — a consequence of the $d^3$ strength dependence working in reverse.

• Allowable shear stress $\tau_{\text{allow}}$ — the design stress ceiling, set by dividing the material's shear yield (or torsional endurance) strength by a factor of safety. A higher allowable permits a smaller shaft, but leaves less margin against overload, fatigue, and stress raisers.

• The cube-root behaviour — because torsional strength scales with $d^3$ (the section modulus in torsion is $J/r = \pi d^3/16$), diameter responds weakly to both torque and allowable stress. Shafts are therefore relatively insensitive to modest load changes but must be sized carefully because a keyway or fillet can locally undo the margin.

Derivation (Approaching a Proof)

Start from the torsion formula at the surface (Torsional Shear Stress) with $r = d/2$ and the solid polar moment $J = \pi d^4/32$ (Polar Moment solid shaft):

$$\tau_{\max} = \frac{T (d/2)}{\pi d^4/32} = \frac{16 T}{\pi d^3}.$$

The grouping $\pi d^3/16 = J/r$ is the polar section modulus — the torsional analogue of the bending section modulus. Setting the maximum stress equal to the allowable and solving for $d$:

$$\tau_{\text{allow}} = \frac{16 T}{\pi d^3} \;\Longrightarrow\; d^3 = \frac{16 T}{\pi\, \tau_{\text{allow}}} \;\Longrightarrow\; d = \left(\frac{16 T}{\pi\, \tau_{\text{allow}}}\right)^{1/3}.$$

The cube root is the algebraic fingerprint of the $d^3$ torsional strength law. For fatigue or combined loading the same inversion is used but with an equivalent stress and endurance-based allowable (see Shaft Diameter Combined).

Dimensional check. $\left[\dfrac{T}{\tau}\right] = \dfrac{\text{N}\cdot\text{m}}{\text{N/m}^2} = \text{m}^3$, so $d = (\text{m}^3)^{1/3} = \text{m}$. ✓

History and Development

Shaft sizing by allowable shear stress is textbook 19th-century machine design (Coulomb/Saint-Venant torsion), codified in Shigley and Machinery's Handbook. Modern practice layers on the ASME shaft code, fatigue (endurance-based allowables), and stress-concentration factors, but this cube-root relation remains the first sizing pass every designer makes.

Related Concepts: Torsional Shear Stress, Polar Moment solid shaft, Shaft Diameter Bending, Shaft Diameter Combined, Shaft Fatigue Factor

Notes: Pure torsion, solid round shaft. Add a factor of safety in $\tau_{\text{allow}}$, a keyway factor (Keyway Stress Reduction), and use Shaft Diameter Combined when bending is also present. For fatigue, base $\tau_{\text{allow}}$ on the endurance limit.