# Do I use alternating, mean or max/min stresses for Miner's rule?

I am trying to work out fatigue life of a part using Miner's rule, however I am having some trouble working out which stress values to use. I have seen some examples but there seems no rhyme or reason to the values they have used in the accumulative Miner's rule, where fatigue failure occurs when :

$$\sum_{j=1}^{j=k}\frac{n_j}{N_j}=1$$

Where $n_1, n_2...n_k$ represents the number of cycles at specific overstress levels and $N_1, N_2...N_k$ represent the life (in cycles) at these overstress levels.

The example below just used the maximum positive values, with no regard for negative values, even if they were nearly the same as that maximum positive value. Surely this negative stress would have some impact on fatigue, more so than a zero based loading?

So if I have a stress vs time plot (shown below) that varies from positive to negative quite randomly, where do I get these overstress levels from? Are they the alternating stresses, mean stresses, or max/min stresses that are over the endurance limit of the material?

I used Miners rule to summarized the fatigue life of different stress cycles for various times . As : stress A for B cycles + stress C for D cycles + stress E for F cycles. I don't think it applies to single stress applications.

• Yes it is accumulative damage, but my textbook doesn't say what stress to use. Jun 16, 2018 at 12:28
• I used maximum stress for each cycle, Jun 17, 2018 at 14:56
• I think I have worked out the answer. My textbook seems to plot the mean stress and alternating stress on an AM diagram and draw a line from the ultimate tensile strength on the mean axis through that point. Then use some trig to find the y-intercept on the alternating axis. They use this value as their equivalent stress value for the Miner's rule Jun 18, 2018 at 9:41

For a special case of variable stress history, where stress tensor at any point $$T_\sigma(t)$$ can be calculated as linear combination of two fixed tensors with just one variable $$k$$ (proportional loading): $$T_\sigma(t) = k\cdot T_{\sigma1}+(1-k)\cdot T_{\sigma2}$$ you can express the stress history using just one stress component as in your question. In such case, for cumulative fatigue damage evaluation, you have to first identify equivalent cycles using cycle counting methods such as range-pair, rainflow or reservoir (I highly recommend very short ASTM E1049-85 standard just for this). My favourite is the range-pair, as it uses just two relatively simple steps:

• identify peaks and valleys in the stress history, mathematically every $$\sigma_m$$ for which the neighbouring stresses are either both lower or both higher: $$\sigma_{m-1}<\sigma_m>\sigma_{m+1} \lor \sigma_{m-1}>\sigma_m<\sigma_{m+1}$$
• identify full cycles in the peak and valley plot, for which there is mathematical four point criterion: $$\left|\sigma_{n-1}-\sigma_n\right|\ge \left|\sigma_{n}-\sigma_{n+1}\right|\le\left|\sigma_{n+1}-\sigma_{n+2}\right|$$ The four point criterion can be visualised like this (you can see the full cycles between points $$x_2$$ and $$x_3$$):

After identifying a full cycle, you should delete the two full cycle points ($$\sigma_n$$ and $$\sigma_{n+1}$$) from the list and try to apply the four point criterion again. Here is a stress history after all full cycles have been identified:

The result is list of stress ranges with associated mean stresses, for each of which you can calculate number of allowable cycles $$N_j$$. There is also remaining sequence of stresses, where the four point criterion cannot identify any full cycles. Consecutive couples of stresses in this remainder can be treated as "half" cycles, contributing half the damage ($$0.5/N_j$$) compared to full cycles.

In general case, where the stress history cannot be described by varying a single parameter in time, the above mentioned methods cannot be used. For more information, I would suggest WRC 550 bulletin.

Miner's Rule is simply a ratio of specified cycles to total cycles at a given stress range. The "Stress Range" is determined by the analyst and should be selected to best describe or otherwise predict failure. The result of this ratio is called the cumulative damage. When the damage equals unity, in theory, 100% of the parts life has been consumed.

For example, say you determine that a part stressed to 10,000psi for 10,000 cycles in a zero-based loading scenario will fail on the 10,000th cycle. According to Miner's rule you can prove this because when you input 10,000 cycles into the ratio, the resulting value will be unity.

It is important to understand a few fatigue concepts:

1. That fatigue, or rather, fatigue failure is a complex group of subjects with a multitude of embedded theories.

2. Many fatigue assessment strategies are based on empirical data and valid only under specified conditions.

3. Many s-N curves are based on a single load scenario and therefore the data presented on the curve may not represent your scenario accurately.

4. Theoretical fatigue of perfect specimen can vary vastly from real life fatigue of an imperfect specimen (reference Shigley or Marin)

Best of luck.