In a related blog "How to Calculate Fatigue Life When the Load History is Complex", the ground work for calculating fatigue life is discussed. In this post, I'll build upon the discussion of fatigue failure and apply it to a Random Vibration load history.

In a random vibration analysis, it is assumed that the loading and response is statistical in nature and it can be represented by a zero-mean normal (Gaussian) distribution. It is sometimes convenient to view this distribution from the perspective of the likelihood that a certain level of load or response will fall within a certain standard deviation from the mean. Typically, we consider the 1, 2, and 3 (standard deviation) levels. As an example, given a random Gaussian loading, x(t), the probability that x(t) lies between ± 1 is 68.3% (e.g the majority of the loading over time is assumed to occur near the nominal levels with the peaks occurring less frequently), the probability that it lies between ± 2 is 95.4%, and that for ± 3 is 99.7%, as shown in Figure 1 above.

Time-domain methods, using rainflow counting, as described in the aforementioned blog post, can also be applied to random processes. However, analysis in the frequency domain is usually preferred due to the significant advantage from the perspective of numerical computation. There are many frequency-based methods that have been developed over the years which calculate damage based on a random vibration loading. These different methods employ various techniques that calculate the fatigue life based on the 1 values that are typically calculated by the Finite Element Analysis. For example, an FEA calculated maximum stress of S_{max} = 5ksi indicates a 68.3% probability (1) that S_{max} will be 5ksi or less. A method to calculate fatigue life, therefore needs to be selected and applied to these results to account for the statistical variability (unless a 68.3% of survivability is acceptable). One of the simplest approaches that can be used for design is to just assume that ± 3 gives infinite life based on a Goodman diagram (i.e. a 99.7% probability (3) that S_{max} will be 3x5ksi = 15ksi or less). This approach can often be too conservative, in which case, more sophisticated methods are available.

All the common methods used today are based on the Palmgren-Miner hypothesis, which asserts that fatigue damage is cumulative, proportional to the applied levels of applied stress, and that the damage is independent of the order in which the stresses are applied.

Using this technique, structural performance is evaluated by comparing the calculated cumulative damage ratio to a specified cumulative damage index.

The Steinberg 3-band method for damage calculation is frequently used due to its simplicity (Ref 1). It uses a Miner's Rule approach to calculate cumulative fatigue damage by assuming that the stress amplitude response at a given location has a Gaussian distribution that's divided into the following three intervals:

- 68.3% of the time at 1
- 27.1% of the time at 2
- 4.3% of the time at 3

In each of these intervals, the number of cycles to failure (N_{1}, N_{2}, and N_{3}) can be determined from the material S-n curve. Then, if the total number of applied cycles "n" is known, we can use the Steinberg 3-band method as shown in Figure 2 to determine the cumulative fatigue damage index, Rn.

When all the life is used up, the value of Rn will be equal to 1. However, due to the uncertainties of this method, safety factors are built in via the failure index, C. The value of C to be used is dependent on the industry and application. For example, a value of 1 can be too high based on test data for electronics, and might be lowered to 0.7 or 0.3.

Stay tuned for Part 2 of this post, where I'll walk through a sample calculation using this method. I'll also discuss other methods, which can produce better correlation to results for a random response. If you have any experience with calculating fatigue damage, please leave a comment! I'd like to hear about your approach.

References

1. Steinberg, Dave S., “*Vibration Analysis for Electronic Equipment*,” 2nd Ed., John Wiley and Sons, 1988 .