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Pump Shaft Fatigue at a Shoulder Fillet

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Pump Shaft Fatigue at a Shoulder Fillet

A 45 kW centrifugal pump is driven through a 50 mm diameter 4140 shaft, quenched and tempered to 285 HB (Sut about 1000 MPa, Sy about 850 MPa). At the bearing shoulder the diameter steps to 60 mm with a 1.5 mm fillet; the shaft turns at 1800 rpm. Belt pull produces a steady bending moment of 2000 N·m at the shoulder, and torque is steady. The original design compared the nominal bending stress with 0.5 Sut and concluded infinite life. After 14 months in service the shaft cracked at the fillet root; the fracture surface shows beach marks and ratchet marks at the fillet, with the final overload zone about 30% of the section. The replacement shaft will be identical but ground to Ra 0.4 µm and shot-peened. A junior engineer proposes that shot peening alone will restore infinite life.

1. Calculate the nominal bending and torsional stresses at the shoulder and the equivalent completely reversed alternating and mean stresses. State your assumptions.

Read the worked answer

Assume steady torque, rotating bending that is fully reversed, and elementary beam theory for nominal stresses; stress concentration is treated separately.

  • Section modulus: Z = pi d^3/32 = pi(0.050)^3/32 = 1.23 x 10^-5 m^3.
  • Polar modulus: J/c = pi d^3/16 = 2.45 x 10^-5 m^3.
  • Bending stress: sigma_b = M/Z = 2000/1.23 x 10^-5 = 163 MPa.
  • Angular speed: omega = 2 pi (1800)/60 = 188.5 rad/s.
  • Torque: T = P/omega = 45,000/188.5 = 239 N·m.
  • Torsional shear: tau = T/(J/c) = 239/2.45 x 10^-5 = 9.8 MPa.

Because the shaft rotates, the bending stress is fully reversed: sigma_a = 163 MPa. The torque is steady, so tau_m = 9.8 MPa. Convert the mean shear to von Mises: sigma_m = sqrt(3) x 9.8 = 17 MPa. The nominal stresses are well below Sy, so first-cycle yielding is not the issue; the fillet stress is Kf times larger. The beach marks and ratchet marks confirm fatigue initiation at the stress concentration.

2. Estimate the modified endurance limit for the original machined shaft and for the proposed ground, shot-peened shaft. Use Marin factors, show each factor, and state where you are uncertain.

Read the worked answer

Baseline: Se' = 0.5 Sut = 500 MPa.

Original machined shaft:

  • Surface factor, machined: ka = 4.51 Sut^-0.265 = 4.51(1000)^-0.265 = 0.72.
  • Size factor for d = 50 mm: kb = 1.24 d^-0.107 = 1.24(50)^-0.107 = 0.82.
  • Notch: D/d = 60/50 = 1.2, r/d = 1.5/50 = 0.03. From a shoulder-fillet chart, Kt is about 2.3. For Sut = 1000 MPa and r = 1.5 mm, q is about 0.85, so Kf = 1 + q(Kt - 1) = 1 + 0.85(1.3) = 2.1.
  • Se = 500 x 0.72 x 0.82/2.1 = 140 MPa.

Proposed shaft:

  • Ground finish: ka = 1.58 Sut^-0.085 = 1.58(1000)^-0.085 = 0.88.
  • Same size and notch factors.
  • Shot peening adds a compressive residual stress; use a peening factor of about 1.25 if coverage and intensity are controlled.
  • Se = 500 x 0.88 x 0.82/2.1 x 1.25 = 213 MPa.

Goodman check, original: 163/140 + 17/1000 = 1.16 + 0.02 = 1.18, so n = 0.85 and finite life is predicted, consistent with failure. Proposed: 163/213 + 0.02 = 0.77 + 0.02 = 0.79, so n = 1.27. The largest uncertainties are Kt from the chart, q, and the peening factor; all should be confirmed by FEA or fatigue testing rather than treated as exact.

3. Decide whether shot peening alone is an adequate fix. Recommend at least two design or process changes and the verification testing you would require.

Read the worked answer

Shot peening alone is not adequate. It raises the calculated margin only to about 1.3, and it does not reduce Kt. The residual stress can relax if the pump runs warm or if the surface is damaged, and a single corrosion pit or handling nick would erase the benefit. It also does not protect against a future increase in belt pull.

Better changes:

  1. Increase the fillet radius from 1.5 mm to at least 3 mm. For r/d = 0.06 and D/d = 1.2, Kt falls to roughly 1.7, and Se rises to about 280 MPa, giving n near 1.7.
  2. Reduce the bending moment by moving the outboard bearing or changing belt tension; even a 20% reduction in M lowers sigma_a to about 130 MPa.
  3. Specify a ground finish and controlled shot peening as supplements, not as the primary fix, and add a relief groove or undercut if geometry permits.

Verification: FEA of the fillet, a fatigue test of the actual shoulder geometry, and first-article inspection of fillet radius and surface finish. Also monitor belt tension and vibration after restart.