Fatigue, Endurance Limit and Goodman Line

Most machine parts fail not under one big load but under millions of smaller repeated loads. This is fatigue: cracks start at stress raisers and grow until the part breaks, often far below the yield strength.

Fluctuating stress

A stress that varies between σmax and σmin has:

• Mean stress σm = (σmax + σmin)/2
• Alternating (amplitude) stress σa = (σmax − σmin)/2
• Fully reversed loading (a rotating shaft in bending): σm = 0.

S–N curve and endurance limit

Plotting stress amplitude against cycles to failure (log scale) gives the S–N curve. For steels it flattens at about 10⁶ cycles: below the endurance limit S_e the part lasts indefinitely. For steels, the test-specimen value is roughly S_e' ≈ 0.5 S_ut.

The real part's limit is lower: S_e = k_a k_b k_c k_d … S_e', with factors for surface finish, size, load type, temperature and reliability.

Notches in fatigue

• Fatigue stress concentration factor: K_f = 1 + q(K_t − 1), where q (0 to 1) is the notch sensitivity.
• Multiply the alternating stress by K_f (or divide S_e by it).

Mean-stress criteria

With S_e (endurance), S_y (yield) and S_ut (ultimate), and factor of safety N:

Cumulative damage

Miner's rule: if a part runs nᵢ cycles at stress levels whose lives are Nᵢ, it fails when Σ nᵢ/Nᵢ = 1.

Worked example

A steel bar has S_ut = 600 MPa and S_e = 250 MPa (corrected). It sees σm = 100 MPa and σa = 100 MPa. Find N by Goodman.

• 1/N = 100/250 + 100/600 = 0.4 + 0.1667 = 0.5667.
• N = 1.76.

Practice