Shaft Fatigue Failure at a Shoulder Fillet Machined Under Size
A 45 mm diameter drive shaft at the input of a bucket-elevator gearbox transmits 900 N·m at 250 rpm. It carries steady torque and a fully reversed bending moment of 900 N·m at a shoulder where the diameter steps from 60 mm at the bearing seat to 45 mm. The shaft is turned and ground from C45 carbon steel with Sut 630 MPa and Sy 370 MPa. The drawing specifies a fillet radius of 2.5 mm at the shoulder. The shaft failed after 14 months of two-shift operation; the fracture face shows beach marks and the origin is at the shoulder, on the side away from the chain pull. No overload was recorded - the motor draws 44 A against a rated 46 A - and the chain tension was checked and found within specification. A replacement shaft from the same workshop is on the shelf, but the fitter has measured its fillet radius as 0.5 mm and says it is 'close enough'.
1. Carry out a fatigue check at the shoulder for both the measured 0.5 mm radius and the specified 2.5 mm radius. State whether the failure is explained.
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Nominal stresses at the 45 mm section.
- Bending: σ_b = 32M/(πd³) = 32 × 900/(π × 0.045³) = 100.6 MPa (fully reversed)
- Torsion: τ = 16T/(πd³) = 16 × 900/(π × 0.045³) = 50.3 MPa (steady)
- Mean von Mises: σ_m = √3 × 50.3 = 87.1 MPa
Endurance limit. Se′ = 0.5 × 630 = 315 MPa. Surface factor 0.88 for ground finish, size factor 0.85 at 45 mm, load factor 1.0 for bending, temperature 1.0: Se = 315 × 0.88 × 0.85 = 235.6 MPa.
Case A - r = 0.5 mm. D/d = 1.33, r/d = 0.011, giving Kt ≈ 2.6 from a stepped-shaft chart; notch sensitivity q ≈ 0.80, so Kf = 1 + 0.80(2.6 − 1) = 2.28. Alternating stress = 2.28 × 100.6 = 229 MPa. Goodman: 229/235.6 + 87.1/630 = 0.974 + 0.138 = 1.112, giving n = 0.90 - infinite life is not achieved.
Case B - r = 2.5 mm. r/d = 0.056, Kt ≈ 1.65, q ≈ 0.85, Kf = 1.55. Alternating stress = 156 MPa. Goodman: 156/235.6 + 0.138 = 0.662 + 0.138 = 0.800, giving n = 1.25.
Static check. Peak stress with the sharp fillet is about 229 + 87 = 316 MPa, below Sy = 370 MPa. The shaft does not yield on first loading, which is exactly why the failure took 14 months rather than one shift.
Judgement. The sharp fillet predicts a fatigue life below infinite and the beach marks confirm high-cycle fatigue initiating at the shoulder. Chart read-off carries roughly ±10% uncertainty, but the conclusion is robust: the radius changes Kf from about 1.55 to 2.28, which is the difference between surviving and failing. The failure is explained.
2. What do you do with the replacement shaft on the shelf? Justify your decision in terms the workshop will accept, and list the checks you would make on the rest of the population.
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Do not fit it. 'Close enough' is not a defensible position here because the fillet radius is the governing variable in the design, not a cosmetic feature. The evidence chain is: fracture origin at the shoulder, beach marks indicating progressive crack growth, no overload in the records, and a notch factor that more than doubles the effective alternating stress when the radius is 0.5 mm instead of 2.5 mm. Fitting the shelf shaft would reproduce the failure with a high degree of confidence, and the cost of the second failure (unplanned outage, possible gearbox damage, injury risk from a rotating fracture) dwarfs the cost of re-machining.
How to present it. Show the two Goodman calculations side by side. Workshop personnel respond to the fact that the checked shaft has a calculated safety factor of 0.90 against 1.25 - a 40% deficit in notch severity, not a marginal 1 mm tolerance.
Action on the replacement. Re-machine to R2.5–3.0 mm, blended tangentially into both diameters with no visible tool marks or undercut at the blend. Verify with a radius gauge or a profile trace and record the reading on the inspection sheet. Then shot-peen the fillet region (0.006–0.010 A intensity) or induction-harden it to introduce compressive residual stress; either raises the fatigue limit by 20–40% at that feature and gives margin against a repeat.
Population checks.
- Measure the fillet radius and surface finish on every shaft supplied by that workshop, in service or in stores.
- Check for grinding marks, machining chatter and undercut at the shoulder - a sharp tool leaves a notch even at nominal radius.
- Verify material certification and hardness against C45 specification; a soft batch would compound the problem.
- Confirm chain tension and sprocket alignment against the manufacturer's figures, since even a compliant tension adds to the bending moment and reduces the margin.
- Consider a design change: increase the shoulder to 50 mm, or use a relief groove with a specified radius, if the gearbox envelope allows.
3. The plant manager asks whether the shaft material was defective and whether the supplier should be pursued. How do you answer, and what further evidence would you obtain?
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Answer. On the present evidence the material is unlikely to be the cause. The fracture shows beach marks, which are the signature of progressive fatigue crack growth under cyclic stress, not of a material defect or an overload. An overload would show a rough, fibrous final fracture with shear lips and measurable plastic deformation, and the motor current record shows no overload. A material defect such as a large inclusion or a forging lap would normally show as a distinct facet at the origin; it can only be confirmed or excluded by examination.
Evidence to obtain.
- Section the fracture face and examine the origin region under a scanning electron microscope or, at minimum, a metallographic microscope. Look for inclusion clusters, decarburisation, grinding burns or a pre-existing crack at the shoulder.
- Chemical analysis and hardness survey across the section, plus a microstructural check for banding or abnormal grain size.
- Check the certificate against the actual heat number, and check that the delivered bar was C45 and not a free-machining grade with higher sulphur.
- Review the workshop's inspection records for the failed shaft: was the radius ever measured, or was the drawing simply taken as done?
Attribution. The most probable root cause is a manufacturing non-conformance - fillet radius well below drawing - combined, secondarily, with a design that relied on a 1.25 safety factor with no peening or hardening at the notch. Even a correctly machined shaft at n = 1.25 is thin for a shock-loaded elevator drive, so the design review and the workshop quality issue should both be pursued rather than choosing one culprit. Pursue the supplier commercially only if the metallurgy confirms a material non-conformance; otherwise the corrective action is internal process control, with a first-article inspection requirement on all future shafts.