Shaft Diameter Bending⚠ unverified

Understanding the Parameters

• Bending moment $M$ — the peak internal moment from the transverse loads and supports (from a bending-moment diagram; see Beam Bending Stress). As with torsion, diameter scales as the cube root of $M$.

• Allowable normal stress $\sigma_{\text{allow}}$ — the design ceiling for bending stress. For a rotating shaft the bending stress fully reverses every revolution (a point on the surface goes tension→compression→tension), so $\sigma_{\text{allow}}$ should usually be based on the fatigue endurance limit, not the static yield strength — a critical subtlety in shaft design.

• Coefficient 32 vs 16 — the bending section modulus of a round shaft is $\pi d^3/32$, exactly half the polar (torsion) section modulus $\pi d^3/16$. So for equal load magnitude a shaft is more highly stressed in bending than in torsion.

Derivation (Approaching a Proof)

Start from the flexure formula at the surface (Beam Bending Stress) with $c = d/2$ and the round section's second moment of area $I = \pi d^4/64$:

$$\sigma_{\max} = \frac{M c}{I} = \frac{M (d/2)}{\pi d^4/64} = \frac{32 M}{\pi d^3}.$$

The grouping $\pi d^3/32 = I/c$ is the section modulus $S$ of the round shaft. Setting the maximum stress to the allowable and solving:

$$\sigma_{\text{allow}} = \frac{32 M}{\pi d^3} \;\Longrightarrow\; d = \left(\frac{32 M}{\pi\, \sigma_{\text{allow}}}\right)^{1/3}.$$

The cube root again reflects the $d^3$ strength law. Comparing with the torsion result, the only differences are $M \to T$ and the coefficient $32 \to 16$ (because $I = J/2$ for a circle), quantifying that round shafts are twice as strong in torsion as in bending.

Dimensional check. $\left[\dfrac{M}{\sigma}\right] = \dfrac{\text{N}\cdot\text{m}}{\text{N/m}^2} = \text{m}^3$, so $d = \text{m}$. ✓

History and Development

Bending shaft sizing is standard strength-of-materials practice (Navier's flexure formula, 1820s) applied to machine elements in Shigley and the ASME shaft-design code. The recognition that rotating shafts experience fully-reversed bending — and must therefore be sized on the fatigue endurance limit — was a key 20th-century refinement, tied to Wöhler's fatigue work and embodied in the fatigue shaft equations (Shaft Fatigue Factor, Marin Modification Factors).

Related Concepts: Beam Bending Stress, Section Modulus, Shaft Diameter torsion, Shaft Diameter Combined, Shaft Fatigue Factor, Rectangular Moment of Inertia

Notes: Pure bending, solid round shaft. For a rotating shaft use a fatigue-based $\sigma_{\text{allow}}$ (reversed bending). Combine with torsion via Shaft Diameter Combined. Add stress-concentration and safety factors.