Shaft sizing is one of the most common mechanical design calculations. For a shaft carrying torque, torsional shear stress provides a useful first sizing check before more detailed fatigue, bending, keyway, bearing, and deflection calculations.
Torsion Formula for a Solid Circular Shaft
For a solid round shaft, the maximum torsional shear stress is given by τ = 16T/(πd³). Rearranging gives d = [16T/(πτ)]1/3, where T is torque, d is shaft diameter, and τ is allowable shear stress.
Worked Example
Assume a solid steel shaft carries 500 N·m of torque and an illustrative allowable shear stress of 50 MPa.
Convert torque: 500 N·m = 500,000 N·mm.
Then d = [16 × 500,000 /(π × 50)]1/3, giving approximately 37.1 mm. A real design would then select a practical standard diameter and perform additional checks.
Why the First Calculation Is Not the Final Diameter
A shaft rarely experiences pure torsion. Gears, pulleys, couplings, and overhung loads introduce bending. Keyways and grooves create stress concentrations. Variable loading introduces fatigue. Bearings and seals impose additional geometry requirements.
Checks to Add
| Check | Why it matters |
|---|---|
| Bending stress | Combined loading can raise shaft stress substantially |
| Fatigue | Repeated torque or bending can control life |
| Keyway effects | Material removed from the shaft creates a local stress concentration |
| Torsional deflection | Excessive twist can affect alignment and mechanism performance |
| Critical speed | Rotating shafts can experience resonance |
Hollow Shafts
For a hollow shaft, the polar second moment changes and the stress equation must use the inner and outer diameters. Hollow shafts can reduce mass while retaining useful torsional stiffness, but the geometry must be selected with manufacturing and local stress requirements in mind.
Design Workflow
- Determine peak and cyclic torque.
- Calculate a preliminary diameter.
- Add bending loads from gears, pulleys, or couplings.
- Check combined stress and fatigue.
- Check twist, bearings, keys, shoulders, and critical speed.
- Select a manufacturable diameter and verify with detailed analysis.
Conclusion
The torsion equation is an excellent starting point, but a production shaft should never be released from a torsion-only calculation. Use the preliminary diameter as the beginning of the design process, then validate the real load case.