Shaft Keyway and Fatigue Failure Under Combined Loading
A 45 mm diameter solid steel shaft transmits 15 kW at 720 rpm through a spur gear and is supported by two bearings. A 14 mm wide by 9 mm deep keyway is cut into the shaft at the gear. The shaft material has an ultimate tensile strength of 600 MPa and an endurance limit (rotating bending, polished) of 300 MPa. The shaft experiences steady torque plus a reversing bending moment from the gear mesh. After 4000 hours of service, a crack initiated at the keyway corner and the shaft failed. The designer argues the shaft was safe because the nominal bending stress was only 40 MPa, well below the 300 MPa endurance limit.
1. Calculate the nominal stresses and show why the designer's comparison is invalid.
Read the worked answer
Torque and torsional shear stress
- Power P = 15 kW = 15 000 W, speed N = 720 rpm.
- Torque T = P / ω = 15 000 / (2π × 720/60) = 15 000 / 75.40 = 199.0 N·m.
- Polar section modulus for a solid shaft: Zp = πd³/16 = π(0.045)³/16 = 1.789 × 10⁻⁵ m³.
- Nominal torsional shear τ = T/Zp = 199.0 / 1.789 × 10⁻⁵ = 11.1 MPa.
Bending stress is given as 40 MPa nominal.
Why the comparison is invalid. The 300 MPa endurance limit applies to a polished, unnotched specimen under rotating bending. The real shaft differs in four ways:
- Stress concentration at the keyway. The keyway corner has a theoretical stress concentration factor Kt of roughly 2–3 for a square-ended keyway in torsion and bending. The local stress is therefore 80–120 MPa in bending, not 40 MPa.
- Notch sensitivity and fatigue notch factor. Kf = 1 + q(Kt − 1). For steel with a notch radius of ~0.5 mm, q is high (0.8–0.9), so Kf approaches 2–2.5.
- Size effect. A 45 mm shaft has a size factor below 1 (typically 0.85) relative to the small test specimen.
- Surface finish. A machined keyway corner is not polished; the surface factor is typically 0.8–0.9.
Corrected endurance limit: Se = 300 × 0.85 (size) × 0.85 (surface) = 216 MPa, before the notch factor.
Effective local alternating stress in bending = 40 × Kf ≈ 40 × 2.2 = 88 MPa. Add the torsional component through an equivalent-stress criterion (von Mises or Tresca), and the combined alternating stress rises further. The margin is far smaller than the naive 40 vs 300 comparison implies, and with a mean torsional stress superimposed, the Goodman line is approached. The failure is a classic fatigue initiation at the keyway corner.
2. Recommend design changes to prevent recurrence.
Read the worked answer
Reduce the stress concentration
- Replace the square-ended keyway with an end-milled (sled-runner) keyway with a generous run-out radius; this lowers Kt substantially compared with a broached square end.
- Specify a fillet radius at the keyway corner of at least 0.5 mm and a surface finish of 1.6 µm Ra or better in the keyway.
- Consider a keyless connection - a shrink-fit or a taper lock - which eliminates the keyway entirely and removes the notch.
Increase section strength
- Increase shaft diameter from 45 mm to at least 55 mm at the gear seat; stress falls with d³, so a 22% diameter increase cuts bending stress by about 47%.
- Use a higher-strength alloy steel with improved fatigue properties, or apply shot-peening to the keyway region to induce compressive residual stress.
Reduce the applied loading
- Improve gear alignment to reduce the reversing bending moment; check that the gear is centred between bearings so the bending moment diagram is balanced.
- Reduce torque fluctuations by using a flexible coupling.
Verification
- Recalculate with the corrected endurance limit, Kf, size and surface factors, and apply the Goodman or Soderberg criterion with a design factor of at least 2 on the combined mean and alternating stresses.
- Specify a fatigue test or a finite-element analysis of the keyway region to confirm the local stress.
- Add a scheduled inspection (dye penetrant) of the keyway corner during overhauls.