Calculating Shaft Diameter
Shafts transmit torque and support loads. Designing a shaft means finding the minimum diameter that can handle the combined bending and torsional stresses with an adequate safety factor. Here’s the practical approach.
Pure Torsion (Torque Only)
For a solid circular shaft transmitting torque T:
Shear stress: τ = 16T / (πd³)
Minimum diameter: d = ∛(16T / πτ_allow)
Example: 500 in-lbs torque, 4140 steel (τ_allow = 15,000 psi)
d = ∛(16 × 500 / (π × 15,000)) = ∛(0.1698) = 0.554 inches → use 5/8″ shaft
Pure Bending (No Torque)
Bending stress: σ = 32M / (πd³)
Minimum diameter: d = ∛(32M / πσ_allow)
Combined Bending + Torsion (Most Common)
Real shafts experience both bending and torsion simultaneously. Use the Maximum Shear Stress Theory (Tresca) or Distortion Energy Theory (von Mises):
ASME Shaft Design Equation
d = ∛[ (32/π) × √((K_b × M)² + (3/4)(K_t × T)²) / S_e ]
Where:
- M = bending moment (in-lbs)
- T = torque (in-lbs)
- K_b = bending fatigue factor (1.5 for keyways, 1.0 for smooth shafts)
- K_t = torsional fatigue factor (1.0 steady, 1.5 fluctuating)
- S_e = endurance limit (psi) — typically 0.5 × tensile strength for steels ≤ 200 ksi
Common Shaft Materials & Allowable Stresses
| Material | Tensile (ksi) | Yield (ksi) | Allowable Shear (ksi) | Endurance Limit (ksi) |
|---|---|---|---|---|
| 1018 Cold Drawn | 64 | 54 | 8 | 32 |
| 1045 Normalized | 91 | 56 | 12 | 45 |
| 1045 Q&T | 115 | 95 | 15 | 50 |
| 4140 Q&T | 148 | 130 | 20 | 68 |
| 4340 Q&T | 185 | 170 | 25 | 80 |
Practical Shaft Design Checklist
- Calculate combined loading — identify the cross-section with the highest combined bending moment and torque
- Apply stress concentration factors — keyways (K=1.5–2.0), shoulders (K=1.2–2.5 depending on fillet), cross-holes (K=2.0–3.0)
- Check deflection — even if stresses are acceptable, excessive deflection can cause gear misalignment, bearing wear, and vibration. Typical limit: 0.001″ per inch of bearing span.
- Check critical speed — for high-RPM shafts, ensure operating speed is well below the first natural frequency (typically at least 20% below).
- Verify bearing fits — shaft diameters at bearing locations need to match standard bearing bore tolerance classes.
- Specify surface finish — bearing journals typically need 16–32 µin Ra; seal surfaces need 10–20 µin Ra (Surface Finish Guide).
Related: Keyway & Key Sizes | Press Fit Chart | Bolt Torque Chart