# Miner’s Rule for Cumulative Fatigue Damage

When spectral loading is present, Miner's rule is often used as a way to determine the amount of cumulative damage that has occurred.  In this article we will discuss the use of Miner's rule and how it is used to determine either damage that has already occurred, or to provide an estimate of fatigue life.

Spectral loading is when applied loads are of different amplitudes.  This presents a challenge when trying to determine fatigue life, or how much fatigue damage has already occurred.  Miner's rule is a method to solve this problem.

With alternating stress where the amplitude is the same for each cycle, all we have to do is make a correction for mean stress, and look up the fatigue life in a strength vs. cycles (S-N) curve.  Miner's rule does this by repeating this process for each stress amplitude and summing the results.

Before determining fatigue damage, we need to count the number of stress cycles and place the amplitudes and mean stresses in bins.  There are several cycle counting algorithms, but the most common is rain flow cycle counting.  MEboost uses the rain flow counting algorithm and is implemented per ASTM 1049-85.

When a cycle is counted, its stress level (peak to valley or valley to peak) is placed in a bin for that level.  The bin value, which covers a range of values, is used in the Miner’s rule calculation.  Bin values are the midpoint of their range.

A stress cycle count matrix is shown below, where stress range bins are rows and mean stress bins are columns.  In this example, 32 bins are used.

## Miner's Rule

Now that stress cycles have been counted, we can employ Miner's rule to determine fatigue damage.   We need to determine how we handle stress below the fatigue limit, or if one even exists.  There are two options:

1. Assume no damage below the fatigue limit.
2. For all stress at or below the fatigue limit, assume cycles to failure is the same as the fatigue limit.

For materials with a fatigue limit, damage is typically not calculated if the stress amplitude below this threshold.  For materials without a fatigue limit, damage can be calculated using the highest N value supplied.  This is somewhat conservative since actual cycles to failure at stress levels below the last point is likely greater than the last point.  In the S-N diagram below, we see the two options.  For option 2, fatigue life is assumed to be 10 million cycles.

To learn more about how MEboost interpolates S-N data above the fatigue limit, refer to the article on Interpolating S-N Data.

### Mean Stress Correction

Assuming the S-N data was generated with stress that has a mean of zero, a correction can be applied to the stress amplitudes to account for non-zero mean stress.  MEboost has three corrections available: Goodman, Gerber, and Soderberg.

### Miner's Rule Defined

Miner’s rule adds cumulative damage associated with different stress magnitudes.  For each magnitude, the number of cycles is multiplied by the stress level.  These products are summed to determine the fraction of fatigue life incurred, C.

In the counts matrix above, MEboost calculates the C value and it is shown in cell B7.  C = .0003 or .03% of the fatigue life has been used.

## Limitations of Miner's Rule

Miner's rule has limitations which should be considered.

• S-N data will have a random element that results in scatter. The S-N curve will not be exact.
• Miner’s rule does not consider the order in which high stress vs. low stress cycles occur. The order can affect fatigue life.

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