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Mechanical
Miner’s rule cumulative damage (three load levels)
Linear damage sum for a load history made of blocks at different stress levels: the cycles applied at each level divided by the cycles to failure at that level.
- Ref
- MEC-91
- Rev
- v1
- Field
- Mechanical
- Result
- D
Miner’s rule, or linear damage accumulation, adds up the fatigue damage from several load levels: D = n₁/N₁ + n₂/N₂ + n₃/N₃, where ni is the number of cycles applied at level i and Ni is the cycles to failure at that level from an S–N curve. Failure is predicted at D = 1. Counts are in thousands of cycles; ratios are unaffected by the unit as long as n and N use the same one. For a single level that fails at the cycles N, D is n/N.
Linear damage sum for a load history made of blocks at different stress levels: the cycles applied at each level divided by the cycles to failure at that level. D near 1 is a nominal failure estimate, not a universal limit.
Calculator
- Damage from level 10.1
- Damage from level 20.1
- Damage from level 30.05
- Fraction of life remaining, 1 − D0.75
- Repeats of this load sequence to reach D = 14
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- These three load levels together use 25 % of the fatigue life (D = 0.25). The sequence could be repeated 4 times before the rule predicts failure.
- The level with the biggest share is the one to reduce: level 1 at 40 % of the damage,; that share, not the cycle count, decides which load level matters most.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| N3 Cycles to failure at level 3 | 1 to 100,000 thousand | 0.201 | 100.2 | 100 | +10% | −1.82 % |
| N2 Cycles to failure at level 2 | 1 to 100,000 thousand | 0.1505 | 50.15 | 50 | +10% | −3.64 % |
| n1 Cycles applied at level 1 | 0 to 5,000 thousand | 0.15 | 25.15 | 25 | +10% | +4 % |
| N1 Cycles to failure at level 1 | 1 to 100,000 thousand | 0.1502 | 20.15 | 20 | +10% | −3.64 % |
| n2 Cycles applied at level 2 | 0 to 5,000 thousand | 0.15 | 10.15 | 10 | +10% | +4 % |
| n3 Cycles applied at level 3 | 0 to 5,000 thousand | 0.2 | 2.7 | 2.5 | +10% | +2 % |
Sweep of Cycles applied at level 1
| Cycles applied at level 1 (thousand) | Accumulated damage |
|---|---|
| 0 | 0.15 |
| 625 | 3.275 |
| 1,250 | 6.4 |
| 1,875 | 9.525 |
| 2,500 | 12.65 |
| 3,125 | 15.78 |
| 3,750 | 18.9 |
| 4,375 | 22.03 |
| 5,000 | 25.15 |
Sweep one variable
Variables and how each one contributes
Accumulated damage
What the equation gives you. Everything else feeds it.
Cycles applied at level 1
Cycles applied at load level 1, in thousands. Raises the damage in direct proportion to its share n₁/N₁. Raising it by 5 thousand (to 25 thousand) moves the result by +10 %, from 0.25 to 0.275.
Cycles to failure at level 1
Cycles to failure at level 1 from the S–N curve, in thousands. Lowers the damage in inverse proportion: a higher life means each cycle does less harm. Raising it by 20 thousand (to 220 thousand) moves the result by −3.64 %, from 0.25 to 0.2409.
Cycles applied at level 2
Cycles applied at load level 2, in thousands. Raises the damage in direct proportion to its share n₂/N₂. Raising it by 5 thousand (to 55 thousand) moves the result by +4 %, from 0.25 to 0.26.
Cycles to failure at level 2
Cycles to failure at level 2, in thousands. Lowers the damage in inverse proportion. Raising it by 50 thousand (to 550 thousand) moves the result by −3.64 %, from 0.25 to 0.2409.
Cycles applied at level 3
Cycles applied at load level 3, in thousands. Raises the damage in direct proportion to its share n₃/N₃. Raising it by 10 thousand (to 110 thousand) moves the result by +2 %, from 0.25 to 0.255.
Cycles to failure at level 3
Cycles to failure at level 3, in thousands. Lowers the damage in inverse proportion. Raising it by 200 thousand (to 2,200 thousand) moves the result by −1.82 %, from 0.25 to 0.2455.
Worked example with the values above
- Level damages: n₁/N₁ = 20/200 = 0.1; n₂/N₂ = 50/500 = 0.1; n₃/N₃ = 100/2,000 = 0.05.
- Add them.
- D = 0.25, so 75 % of the life is left.
Assumptions and limits
- Linear damage accumulation: each cycle uses up the same fraction of life at a given level, whatever came before.
- The N values come from a consistent S–N curve for the actual part, and loads below the endurance limit cause no damage in this simple form.
Common mistakes
- Mixing units, with n in thousands and N in millions.
- Using the rule where load sequence effects are strong, such as an overload followed by small cycles.
- Treating D = 1 as a safe limit rather than the predicted mean failure.
Statistics and uncertainty
Miner’s rule is an approximation: test results scatter widely about D = 1 and the order of the loading matters, with high loads first usually more damaging than the rule suggests. Designs therefore keep D well below 1, and the N values taken from an S–N curve carry the scatter of the fatigue data.
Where it is used
- Estimating the fatigue life of a component under a mix of load levels, such as a duty cycle of a vehicle or a flight profile.
- Ranking load cases by how much of the life each takes.
Maths you need
- Stress cycle: the repeated rise and fall of stress
- Tension positive: compression is entered as a negative stress
- S–N curve: the plot of stress amplitude against cycles to failure