Fastener Fatigue Check (Modified Goodman Line)
Fastener Fatigue Check (Modified Goodman Line)
Fasteners in machinery subjected to cyclic loading — engine bolts, connecting rod fasteners, vibrating equipment mounts — fail not by exceeding a single static strength limit but by fatigue: repeated stress cycles below the ultimate strength gradually initiate and grow a crack until sudden fracture. The modified Goodman relation is one of the standard ways to check a fastener's fatigue margin when it sees both a steady (mean) stress and a fluctuating (alternating) stress component simultaneously.Fatigue failures are notoriously deceptive because the fastener can look completely fine right up until the final fracture, with no warning deformation the way a static overload often gives — this is why fatigue-critical fasteners (in engines, rotating machinery, aircraft) get design margins and inspection intervals that static-only strength calculations would never suggest are necessary.
The modified Goodman line combines the alternating and mean stress fractions of endurance and ultimate strength: 1/n_fs = sigma_a/S_e + sigma_m/S_ut, giving a factor of safety against fatigue failure. where sigma_a is the alternating (cyclic) stress amplitude, sigma_m is the mean (steady) stress, S_e is the endurance limit of the fastener material, S_ut is the ultimate tensile strength, and n_fs is the resulting fatigue factor of safety.
Inputs
sigma_a := 40 MPa=40 MPa
sigma_m := 60 MPa=60 MPa
S_e := 200 MPa=200 MPa
S_ut := 600 MPa=600 MPa
Each stress component is normalized by the strength limit relevant to it — alternating stress against the endurance limit, mean stress against the ultimate strength — and the two fractions are summed to find where the combined loading falls on the Goodman line, whose reciprocal is the factor of safety.
n_fs := 1/(sigma_a/S_e + sigma_m/S_ut)=3.33
Results
A fatigue factor of safety around 3.3, as found here, indicates a comfortable margin against fatigue failure under this combined loading — well above the 1.0 threshold where the Goodman line itself would be crossed. Because fatigue life is so sensitive to the alternating stress term (divided by the much smaller S_e rather than S_ut), reducing vibration or preload fluctuation is usually far more effective at improving fatigue life than upsizing the fastener, which mainly reduces both stress components proportionally. This check assumes a known, constant-amplitude alternating stress; variable-amplitude service loading would need a cumulative damage approach (like Miner's rule) instead of a single Goodman check.
Assumptions & limitations
- Constant-amplitude alternating and mean stress assumed — no variable-amplitude loading (Miner's rule) included.
- Endurance limit S_e already reflects applicable surface finish, size, and reliability correction factors.
- No stress concentration factor is applied here; thread roots and fillets typically need one applied to sigma_a and sigma_m.
- Purely axial fastener loading assumed — no combined bending or shear on the fastener.