A pump shaft that fractured at a diameter step. The shaft was 45 mm on the impeller side and 35 mm on the coupling side. The shoulder fillet was 1 mm radius. The shaft ran at 2900 RPM for 2 years, then a fatigue crack started at the fillet and propagated until the shaft broke. The customer thought the shaft material was wrong. It was C45 steel — correct for the application. The issue was the fillet radius. A 1 mm radius on a 10 mm step creates a severe stress concentration. This is about shaft shoulder design and fatigue life.
The stress concentration factor
At a diameter change, the stress isn’t uniform. The material at the fillet sees a local stress much higher than the nominal stress. The stress concentration factor (Kt) depends on the ratio of diameters and the fillet radius:
Kt ≈ 1 + 2 × √(D/d – 1) × (r/d)^(-1/2)
For D=45, d=35, r=1 mm: D/d = 1.29, r/d = 0.029. Kt ≈ 1 + 2 × √0.29 × (0.029)^(-0.5) = 1 + 2 × 0.54 × 5.86 = 1 + 6.33 = 7.3. That means the local stress at the fillet is 7.3x the nominal stress. If the nominal bending stress is 50 MPa, the fillet sees 365 MPa. C45 steel fatigue limit is about 200 MPa in bending. The fillet stress exceeds the fatigue limit by 80%. The crack starts.
What was changed
1. Increased the fillet radius to 3 mm. With r=3 mm: r/d = 0.086. Kt ≈ 1 + 2 × 0.54 × (0.086)^(-0.5) = 1 + 1.08 × 3.41 = 4.6. The local stress drops to 230 MPa. Still above the 200 MPa fatigue limit — but close. With shot peening at the fillet, the fatigue limit rises to 280 MPa. Now safe.
2. Used a undercut relief. When the shoulder must fit against a bearing inner ring (can’t increase the radius beyond the bearing corner), a relief undercut is machined. The undercut is a small groove at the root of the shoulder. It allows a larger fillet on the shaft while the bearing seats against a square shoulder. The Kt drops to 2.5. The shaft survives.
3. Changed to a stronger material. For a new design, C45 normalized is replaced by 42CrMo4 quenched and tempered. The fatigue limit rises to 400 MPa. Even with Kt=4.6, the local stress of 230 MPa is well below 400 MPa. The shaft doesn’t crack. The material costs 30% more but eliminates the failure mode.
The fillet radius rule
| Step ratio D/d | Minimum fillet r | Kt at min r |
|---|---|---|
| 1.1 | 0.02 × d | 2.5 |
| 1.25 | 0.05 × d | 3.0 |
| 1.5 | 0.10 × d | 3.5 |
| 2.0 | 0.15 × d | 4.0 |
For a rotating shaft subject to bending, the goal is Kt under 3.0. That requires a generous fillet. If the design forces a sharp shoulder (bearing seat), use the undercut relief or a stronger material. Never use a sharp-cornered shoulder on a rotating shaft — it’s a fatigue crack waiting to happen.
The fillet rule: r = 0.05 × d minimum on rotating shafts, 0.10 × d preferred. The broken pump shaft wasn’t wrong material — it was a 1 mm fillet on a 10 mm step with Kt=7.3. Increase the radius, add shot peening, or use alloy steel. For bearing seats, use an undercut relief. Fatigue cracks start at geometric discontinuities, not in smooth material.
