Shaft Diameter Combined⚠ unverified

Understanding the Parameters

• Bending moment $M$ — appears as $4M^2$ inside the root, i.e. weighted by $2^2$, because bending stress ($32M/\pi d^3$) has twice the coefficient of torsional stress ($16T/\pi d^3$). Bending therefore dominates the required diameter when $M$ and $T$ are comparable.

• Torque $T$ — appears as $3T^2$, the factor 3 inherited from the von Mises weighting of shear stress.

• Allowable stress $\sigma_{\text{allow}}$ — the equivalent-stress ceiling. For a rotating shaft the bending part is fully reversed, so a rigorous design uses a fatigue-based allowable (endurance limit / safety factor) and often the ASME formula that treats mean and alternating parts separately (see Shaft Fatigue Factor).

• Cube root — the same $d^3$ strength law as the pure-load cases; combining loads changes only the quantity under the root, not the cube-root form.

Derivation (Approaching a Proof)

From Combined Stress Shaft, the von Mises equivalent stress of a shaft under bending $M$ and torque $T$ is

$$\sigma_{\text{eq}} = \sqrt{\sigma^2 + 3\tau^2}, \quad \sigma = \frac{32 M}{\pi d^3}, \quad \tau = \frac{16 T}{\pi d^3}.$$

Substitute the stresses and factor the common $16/(\pi d^3)$:

$$\sigma_{\text{eq}} = \sqrt{\left(\frac{32 M}{\pi d^3}\right)^2 + 3\left(\frac{16 T}{\pi d^3}\right)^2} = \frac{16}{\pi d^3}\sqrt{4M^2 + 3T^2}.$$

Set $\sigma_{\text{eq}} = \sigma_{\text{allow}}$ and solve for $d$:

$$d^3 = \frac{16}{\pi\, \sigma_{\text{allow}}}\sqrt{4M^2 + 3T^2} \;\Longrightarrow\; d = \left(\frac{16}{\pi\, \sigma_{\text{allow}}}\sqrt{4M^2 + 3T^2}\right)^{1/3}.$$

The $4M^2 + 3T^2$ grouping is the fingerprint of the distortion-energy theory applied to a round shaft; using the maximum-shear-stress theory instead gives $\sqrt{M^2 + T^2}$ with a different coefficient. Both reduce to the pure-bending (Shaft Diameter Bending) or pure-torsion (Shaft Diameter torsion) cases when $T = 0$ or $M = 0$.

Dimensional check. $\sqrt{4M^2 + 3T^2}$ has units of N·m; dividing by $\sigma_{\text{allow}}$ gives m³; the cube root gives m. ✓

History and Development

The combined-stress shaft equation is central to the ASME shaft-design code and Shigley, built on von Mises' 1913 distortion-energy criterion. Modern shaft design extends it to fatigue via the Soderberg/Goodman-based ASME equation, which separates alternating (bending) and mean (torque) components and applies the Marin factors (Marin Modification Factors, Shaft Fatigue Factor) — but this static von Mises form is the essential first sizing pass.

Related Concepts: Combined Stress Shaft, Shaft Diameter Bending, Shaft Diameter torsion, Von Mises Stress, Shaft Fatigue Factor, Marin Modification Factors

Notes: Static von Mises sizing. For rotating shafts use a fatigue-based allowable and the ASME fatigue equation (mean/alternating split). Add stress-concentration (Keyway Stress Reduction) and safety factors.