Finite-Life Fatigue Strength and Life (S-N Line)
For a part that only has to last a known number of cycles, or a stress above the endurance limit.
Example
You enter
- Ultimate strength Sut (psi) 90000
- Corrected endurance limit Se (psi) 45000
- Fatigue strength fraction f (blank for Shigley's fit) 0.86
- Cycles N, 1,000 to 1,000,000 (for the strength) 10000
- Reversed alternating stress (psi, for the life) 55000
You get
- Strength at cycles (psi) 64600
- Life cycles 77500
- S-N line 133100
Checked against published worked examples from: McGraw-Hill (Shigley, Mischke, Budynas).
Details, formula, and sources
ScopeThe straight S-N line on log-log axes from f Sut at 1,000 cycles to the corrected endurance limit at 1,000,000. It gives the fatigue strength at a cycle count and the life at a reversed stress. A 90 kpsi steel with a 45 kpsi endurance limit and f = 0.86 holds 64,600 psi for 10,000 cycles, and at 55,000 psi it lasts about 77,600 cycles. Life swings hard with stress: 10% more stress cuts that life by about 70%. Steel, completely reversed stress. A design aid; test data govern.
Formula or tablef = entered, or 1.06 - 2.8e-3 Sut + 6.9e-6 Sut^2 (Sut in kpsi, 70 to 200), 0.9 below 70 kpsi; a = (f Sut)^2/Se; b = -(1/3) log10(f Sut/Se); Sf = a N^b (10^3 <= N <= 10^6); N = (sigma_a/a)^(1/b) for Se < sigma_a <= f Sut; infinite at or below Se.
Source and editionThe idealized finite-life S-N line for steel (Shigley, Mischke and Budynas, Mechanical Engineering Design, Ch. 6, Eqs. 6-13 to 6-16 and the Fig. 6-18 fatigue strength fraction), by name; it reproduces Shigley Example 6-2 (1050 HR, Sut 90 kpsi: a 133.1 kpsi, b -0.0785, Sf 64.6 kpsi at 10^4 cycles, 77,500 cycles at 55 kpsi).
Free accessStandard published fatigue relations; the strengths, the fraction f, the cycles, and the stress are the user's inputs.
What governsEstimate. AHJ and licensed professional govern.
API field namesultimate_strength_psi, endurance_limit_psi, fatigue_fraction, cycles, alternating_stress_psi, strength_at_cycles_psi, life_cycles, coefficient_a_psi
- S-N line log-log straight line through (10^3, f Sut) and (10^6, Se)Shigley Eqs. 6-13 to 6-16
- Fatigue strength fraction f = 1.06 - 2.8e-3 Sut + 6.9e-6 Sut^2 (kpsi) for 70 to 200 kpsi, 0.9 below 70, unless enteredShigley Fig. 6-18 fit
- Loading completely reversed uniaxial stress on steelscope of this tile