## 19 Oct 2021 Cumulative Fatigue Damage

A fatigue analysis (https://www.fea-solutions.co.uk/fatigue-analysis/) is straight forward if the structure is subjected to just one load case, which can consist of several loads, but with the loading and unloading acting simultaneous for all of them.

It gets more complicated if there is just one load case, however the load is cycling to different amplitudes, e.g. 1,000,000 cycles to 50% of the load, 100,000 cycles to 70% of the load and 10,000 cycles to 100% of the load.

It gets even more complex if there are different load cases, acting independently, e.g. a force in x direction, a moment along the y axis and an internal pressure. The peak stress location due to the force is different to those due to moment or pressure, but moment and pressure nevertheless generate certain stresses at the peak stress location of the force load case.

In these cases, the simplistic Endurance Limit Analysis (https://www.fea-solutions.co.uk/endurance-limit-analysis/) is no longer appropriate. Instead, the portion of the total life that has been used, the so-called damage, caused by each of the load cases and/or amplitudes, has to be calculated for any location of the component and subsequently summed up for all load cases and/or amplitudes. This is known as a cumulative fatigue analysis.

The most commonly used cumulative damage equation is Miner’s Rule. It assumes the damage done by each stress repetition (or load cycle) at a given stress level is equal. This means that the damage done by the very first load cycle is as damaging as the last.

This cumulative damage rule also assumes that the amount of useful fatigue life consumed by a given stress/load cycle combination is proportional to the total number of cycles in the fatigue life. For example, if a part undergoes 3,000 load cycles at a stress level that would cause failure in 100,000 cycles, 3% of the fatigue life has been expended. Repeated stress at another level would consume another similarly calculated portion of the fatigue life. Thus, the damage is cumulating in the component. When 100% of the fatigue life is expended in this way, the part would be expected to fail.

Using Miner’s Rule, the order in which individual stress/load cycles are applied is not taken into account and hence doesn’t affect the results. In reality, however, changing the sequence of the loading could have an effect, in particular if plasticity is present (https://www.fea-solutions.co.uk/elasticity-and-plasticity/). If a high load causing plasticity is applied for a number of cycles before a lower load is applied, the fatigue life of the component will be shorter than if the low load is applied before the high load. This is because the plastic strain from the high load would still be present in the component when the low load cycles are applied.

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