Bending Stress Shaft Diameter

Calculate solid or hollow shaft diameter under bending moment loading.

Inputs

M

Bending Moment

N·mm

\sigma

Allowable Bending Stress

MPa

Formula Interpretation

Solid Shaft — Formula ①

d = \sqrt[3]{\dfrac{10M}{\sigma}} \quad \text{(mm)}

$M$ is the maximum bending moment (N·mm); $\sigma$ is the allowable bending stress (MPa); $d$ is the solid shaft diameter (mm).

Hollow Shaft — Formula ②

d_2 = \sqrt[3]{\dfrac{10M}{(1-k^4)\sigma}} \quad \text{(mm)}

$k$ is the inner-to-outer diameter ratio ( $d_1$ / $d_2$ ); $d_2$ is the outer diameter (mm); $d_1$ = $k$ · $d_2$ gives the inner diameter.

Knowledge: Shaft Design Under Bending

When calculating shaft diameters under both torsion and bending, the larger of the two computed values should be used. For a shaft under bending, the maximum bending stress occurs at the outer surface. In shaft design, torsion, bending, and shear must all be considered. For longer shafts, resonance due to rotation must also be evaluated.

Worked Example

A hollow shaft has an inner-to-outer diameter ratio of $0.5$ , and an allowable bending stress of $50\,\text{MPa}$ . Find the inner and outer diameters when the maximum bending moment is $8{,}000{,}000\,\text{N}{\cdot}\text{mm}$ .

Step 1 — Apply Formula ② to find outer diameter d₂

d_2 = \sqrt[3]{\dfrac{10M}{(1-k^4)\sigma}} = \sqrt[3]{\dfrac{10 \times 8{,}000{,}000}{(1-0.5^4) \times 50}} \approx 120 \text{ (mm)}

Step 2 — Find inner diameter d₁ from k = d₁/d₂

d_1 = k \cdot d_2 = 0.5 \times 120 = 60 \text{ (mm)}

Therefore, outer diameter d₂ = $120\,\text{mm}$ and inner diameter d₁ = $60\,\text{mm}$ .

Extended Knowledge

•In determining the shaft diameter, take the larger of the values obtained from Formulas ① and ②.
•Shafts are typically subject to both torsion and bending. When keyways or diameter steps are present, stress concentration factors must also be considered.
•High-speed rotating shafts can excite vibration; when the critical speed is exceeded, abnormal vibration and failure may occur. Both static and dynamic balancing are essential.