Miner's Cumulative Damage Rule (Three Blocks)

Real service loads are not one amplitude, they are a histogram. Miner's rule, proposed by Arvid Palmgren for ball bearings in 1924 and generalised by Milton Miner at Douglas Aircraft in 1945, is the simplest possible bookkeeping: each block of $n$ cycles at a given stress consumes the fraction $n/N$ of the life that stress alone would allow, and failure is predicted when the fractions sum to 1. Three blocks — 10,000 of 100,000, then 20,000 of 200,000, then 50,000 of 1,000,000 — give $0.10 + 0.10 + 0.05 = 0.25$, so a quarter of the life is gone and the same duty cycle can be repeated three more times.

What the rule ignores is sequence, and that is not a small omission. A few large cycles applied first leave compressive residual stress at the notch root that retards the small cycles that follow, so high-then-low ordering commonly gives $D$ well above 1 at failure, while low-then-high gives $D$ well below. Test scatter routinely spans 0.3 to 3.0. Welded-steel design codes handle this by simply mandating $D \le 1$ with the conservatism buried in the S-N curves; aerospace practice often designs to $D \le 0.3$ or lower and inspects anyway.

The most consequential trap is what you leave out. Cycles below the endurance limit contribute nothing under a strict reading of the rule, and if you truncate a load history there you can discard most of the damage in a spectrum where those small cycles are the vast majority — real service data shows they do accumulate damage once larger cycles have started a crack. Modern practice extends the S-N line past the knee with a shallower slope rather than cutting it off. And the cycles must be counted properly out of a variable history, which means rainflow counting, not simply counting peaks.