Variable Amplitude Reality
Real structures rarely experience constant-amplitude loading. An aircraft wing encounters gusts, maneuvers, and ground-air-ground cycles of varying severity. A bridge endures trucks of different weights in random order. Miner's rule, proposed by Palmgren in 1924 and formalized by Miner in 1945, provides a simple framework for predicting fatigue life under these variable-amplitude conditions by summing the fractional damage from each stress level.
The Linear Damage Hypothesis
Miner's rule assumes that each loading cycle at a given stress level consumes a fraction 1/Nᵢ of the total fatigue life at that level. The total damage D = Σ(nᵢ/Nᵢ) accumulates linearly, and failure occurs when D reaches 1.0. This elegant simplicity — requiring only the S-N curve and cycle counts — has made it the most widely used cumulative damage rule in engineering practice for over 75 years.
Limitations and Load Sequence Effects
Miner's rule ignores the order in which loads are applied, which can lead to significant errors. High-to-low loading sequences (overloads followed by lower stresses) typically cause failure at D < 1 because the overloads nucleate cracks that propagate faster under subsequent loading. Low-to-high sequences often give D > 1 because the low stresses create crack closure effects that retard growth under subsequent higher loads. This sequence sensitivity is Miner's rule's primary weakness.
Engineering Practice
Despite its limitations, Miner's rule remains the standard in many design codes (Eurocode, DNV, AWS) because it is simple, requires no additional material constants, and is often conservative when combined with appropriate safety factors. Many codes specify a reduced critical damage sum (D = 0.5 or D = 0.7) to account for sequence effects and scatter. Combined with rainflow cycle counting for extracting cycles from irregular load histories, Miner's rule is the workhorse of fatigue design.