Fatigue analysis per the Palmgren-Miner rule
Assess load spectra per the linear damage accumulation of Palmgren-Miner: spectrum steps and S-N curve yield partial damages, the damage sum, the utilisation against the allowable damage sum and the calculated fatigue life. Three Miner variants (original, elementary, modified per Haibach) can be compared directly, live with every input.
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Formulas and fundamentals
S-N curve in the finite-life regime
In the finite-life regime the S-N curve is a straight line on log-log axes (Basquin relation). With the knee point defined by the endurance-limit amplitude σ_D and the knee-point cycle count N_D, for amplitudes σ_a ≥ σ_D:
The slope exponent k describes how strongly the endurable cycle count drops with amplitude: the larger k, the flatter the line.
Damage sum per spectrum step
Each spectrum step i with amplitude σ_a,_i and cycle count n_i consumes the life fraction d_i, the damage sum is the total of all partial damages:
At D = 1 the theoretical life is used up; in practice the check is made against an allowable damage sum D_allow, because the Miner rule overestimates fatigue life on average.
Miner variants
The three variants differ only in how amplitudes below the endurance limit are treated: original Miner assumes they cause no damage at all (N_i = ∞). Elementary Miner extends the finite-life line unchanged with slope k - the most conservative simple assumption. The Haibach modification extends with the flatter slope 2k−1, because pre-damage from large amplitudes gradually lowers the endurance limit while small amplitudes damage less than the unchanged line would imply. It always holds that:
Utilisation and fatigue life
From the damage sum follows the utilisation:
Check passed for a ≤ 1. This yields the endurable spectrum repetitions H and the total cycles until D_allow is reached:
Model prerequisites: damage grows linearly and independently of load sequence, and S-N curve and spectrum must belong to the same stress ratio or mean stress.
Worked example
Four-step spectrum on a component S-N curve with σ_D = 200 N/mm², N_D = 10⁶ and k = 5 (typical guide values for non-welded steel components): step 1 with 400 N/mm² and 1,000 cycles, step 2 with 300 N/mm² and 5,000 cycles, step 3 with 250 N/mm² and 20,000 cycles, step 4 with 150 N/mm² and 100,000 cycles.
Steps 1 to 3 lie above σ_D and are identical in all variants: N₁ = 10⁶ · (200/400)⁵ = 31,250 gives d₁ = 0.0320; N₂ = 10⁶ · (200/300)⁵ ≈ 131,687 gives d₂ = 0.0380; N₃ = 10⁶ · (200/250)⁵ = 327,680 gives d₃ = 0.0610.
Step 4 at 150 N/mm² lies below the endurance limit and separates the variants: original does not count it (D = 0.1310), elementary computes N₄ = 10⁶ · (4/3)⁵ ≈ 4.214 · 10⁶ giving D = 0.1547, Haibach with slope 2k−1 = 9 gives N₄ ≈ 1.332 · 10⁷ and D = 0.1385.
With D_allow = 0.3 (practice recommendation for non-welded parts) the component endures H = 0.3 / 0.1547 ≈ 1.94 spectrum repetitions per the elementary variant, i.e. about 2.44 · 10⁵ cycles. Per original Miner it would be 2.29 repetitions - choosing the variant is an engineering decision, not a formality.
Frequently asked questions
Which Miner variant should I choose?
Without detailed knowledge of the spectrum, elementary Miner is the safe choice because it fully counts amplitudes below the endurance limit. The Haibach modification (slope 2k−1) is the common practical compromise. Original Miner tends to be non-conservative for spectra with a large share below the endurance limit.
Why is an allowable damage sum below 1.0 recommended?
Comparative evaluations show that the actual damage sum at failure scatters widely and that the Miner rule overestimates fatigue life on average. Without test verification, D_allow = 0.3 for non-welded and 0.5 for welded components is the common practice recommendation per the FKM guideline. Two common sources sit side by side: the general practice rule for the relative Miner rule (nominal-stress concept) gives D_allow ≈ 0.5 … 1.0, whereas the FKM value D_m = 0.3 (non-welded) is stricter and more conservative. The FKM value is not a simple allowable damage sum but is derived via its own conversion.
Do I enter stress amplitudes or stress ranges?
All stresses in this calculator are amplitudes σ_a, not stress ranges Δσ = 2·σ_a. Mixing them up is the most common error: a factor of 2 in stress becomes a factor of 32 in cycle count at k = 5.
How does mean stress enter the calculation?
It does not - this is a deliberate model limitation. S-N curve and spectrum must belong to the same stress ratio R or the same mean stress. If the spectrum differs, transform the amplitudes beforehand (e.g. via the mean-stress sensitivity).
Can I use the S-N curve of a polished specimen?
No, the component S-N curve is required, including all effects of notches, size and surface finish. Using a specimen curve massively overestimates fatigue life. Without test data, k = 5 and N_D = 10⁶ are common guide values for non-welded, notched steel components.
How accurate is the Miner life estimate?
The method yields orders of magnitude, not precise predictions: real damage sums scatter roughly between 0.2 and 6. Sequence effects such as residual-stress redistribution from overloads are inherently not captured by the linear hypothesis. For validated statements, fatigue testing remains decisive.
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