Fatigue Stress Assessment

Introduction

Resistance to fatigue was long thought to be a property of the metals because some metals react worse than others. Steel is good against fatigue, titanium is not. Titanium though is preferred by engineers because it is lighter and more exotic. So much for designing against fatigue failure! Lucky for us punters titanium is too scarce.

Total Life Fatigue Design

Fatigue we are constantly reminded is responsible for about 80% of failures. Straight from the department of guesswork or should I say from the sales dept. All we really know is that all structures are dynamic and undergo cycling so unexplained failures are probably caused by a fatigue mechanism. Just as well we have well-defined and long-standing techniques for calculation the onset of fatigue. Ho ho ho! Let us recap. Actually the reason that fatigue is blamed so often is that it is the most likely miscalculation in the design path. Yes I know the software salesmen tell you that their fatigue software can predict failure to an incredible accuracy. As you will see though, when you know the end result it's very easy to fiddle the in-between calculations to get there. In fact fatigue calculations are an endless series of fiddle factors.

Try this one Kf=Kt*Ks*Ke*Km*K.......

Kf is the fatigue strength reduction factor, introduced because the mathematics of the Kt calculation doesn't actually match with real life. So you need a material factor, an environment factor, a shape factor, a temperature factor etc. This equation alone is enough to invalidate your results.

Try this one too; Miner's cumulative damage rule:

n1/N1 + n2/N2 + n3/N3 ....... <=1.0

Where n is the number of cycles in each cyclic load condition and N is the corresponding number of cycles which it needs to fail. Simple and universally utilised but, unfortunately it doesn't compare well to real life situations. Instead of 1 you can substitute C. Wikipedia says "C is experimentally found to be between 0.7 and 2.2. Usually for design purposes, C is assumed to be 1", which Miner suggested on the basis of logic. I have seen data though which suggests C can be as low as 0.1. Ergo this calculation is thoroughly useless. Furthermore, N has to be adjusted for temperature by yet another fiddle factor. Why do we use this formula you may ask? Because no one has come up with anything better. There is really no mystery that it doesn't work because it ignores the effect of a previous cycle on the next cycle and it assumes uniaxial loading which only ever happens in a laboratory tensile test.

As if the level of guesswork wasn't enough we come across the "what you see is what you fiddled" miracle of rainflow counting in which you can manipulate the number of peaks and troughs of the load cycle by changing the sampling amount or "bucket size". Rainflow counting is not even logical because once you have opened a crack in one cycle, a further cycle which opens it half as much has no actual effect. So adjusting the bucket size is just a technique to magically reproduce already known results which is why those fatigue computer programs seem so accurate. It is a lot easier to predict failure when you already know what happened but we really want to predict failure before it happens.

Useless fatigue calculation summary; Find your SCF based on a fillet radius and thickness from Peterson's handbook of hopelessly limited 2D shapes, then multiply it by a variety of guess factors for environment, loading type, etc. Next reduce the factor by the fracture toughness factor which corrects for the fact that SCF's hopelessly overpredict the onset of fatigue in real life. Of course this value is only of use if the test was done on your actual structure, which it wasn't. Apply the final factor to your field stress, which is the general stress away from the discontinuity. The field stress concept is also based on simple 2D shapes and simple loading. It is not possible to obtain a field stress in any real situation unless you linearise the highly nonlinear stresses. A technique which is controversion and idiosyncratic, even impossible in a 3D situation. Finally find the expected life from a suitable endurance curve. This curve was produced for simple 1D specimens under simplistic loading and it had originally a phenomenal scatter which someone plotted on a logarithmic scale and drew a couple of straight lines through it: They could easily too have drawn a dancing bear through it. Of course you must further adjust the lines according to the extent of compression in the load cycle. For every cycle find a life fraction and add these fractions to get a total life value for the part using Miner's cumulative damage rule which has long been proven to be total nonsense. If you have a load history which is not conveniently sinusoidal then use rainflow counting to capture each mini-cycle within the larger cycles, then disregard the majority of these cycle "buckets" so as to not to be too conservative (because rainflow counting doesn't represent real life). Finally you will arrive at the conclusion you need. In this case, if it is someone elses design you must fail it by including for all eventualities, but if it is your design you must consider all the unnecessary assumptions until you manage to pass it.

Well that procedure was for high cycle, elastic stresses. There is a low-cycle, strain-based calculation for materials in the plastic regime. However it is currently carried out by using elastic stresses and assuming a Neuber plasticity curve. NAFEMS adroitly points out the inadequacy of this approach in a book on it's website, wherein it is pointed out that plastic computations would not only be possible but far more desirable. For me this approximation alone is enough of a fudge to render the calculation useless so I avoid discussing the remaining fiddle factors. However this technique is universally used in current Fatigue software. In fact the prostitution of several prominent academics in presenting this approximation technique as the last word in fatigue design is really quite disagreeable. I suggest you avoid said software and get instead the AFGROW program from the net. It's probably no better but it is well documented, well used and free. We may do a user interface for it one day.

Meantime, there is a module in FEMdesigner to plot the fatigue strength of the material but we use stress ratios instead of life predictions. Here there is only a maximum and minimum stress, an endurance strength (obtained with consideration of the design life) and a fatigue strength reduction factor. The software reads all the stresses in the current output file, calculates the maximum & minimum equivalent stresses, makes them tensile or compressive (+/-) according to the sign of the largest principal stress and applies the Goodman mean stress correction, then finds the endurance limit and adjusts for temperature of the material and presents the result as a red/green contour plot of Actual to Allowable stresses at each point in the model. It repeats this for every load step in this file and in selected other files presenting the worst values in all cycles. This test was developed and tested with performance forged pistons and it works well. Assuming only one maximum and one minimum stress for all cycles is a perfectly valid, well-used technique and avoids both the discredited rainflow-counting technique, Miner's rule and log-log plots. Goodmans stress correction has a lot of actual tests to back it up. The addition of the temperature correction is crucial as it seriously degrades fatigue life in many materials. In FEMdesigner that is made easy. In other fatigue codes it is not. In the nuclear engineering field we also used to prepare low-cycle, strain-based fatigue curves, for which this technique becomes usable. Fatigue tests are best done with the actual component under the actual loading of course. That may seem nonsensical since you may think you don't then need the computational test, but the key idea is to computationally replicate the actual results for the old design, identify the failure areas, modify the design to improve it, then compare the old design to the new one.

The alternative to total life calculation is fatigue from Fracture Mechanics considerations.

Fracture Mechanics and Fatigue

Fatigue it is now accepted is really just fracture in disguise. Hence fatigue is really the initiation of a crack from a material or geometric imperfection and propagation of that crack. This should have seemed obvious but for many years fatigue has been regarded as something that happens by repeated cycling of a body in an elastic state. Although it was material related, it was considered as load-controlled. Fracture mechanics grew up separately by looking at material behaviours at low temperature and then at what happens to notched specimens. We had realised that fatigue and fracture both happen at sharp corners so we invented Stress Concentration Factors for fatigue and Stress Intensity factors for Fracture. Still the penny never dropped because fatigue and fracture calculations were done separately. For both types of calculations we had so many assumptions so we still uultimately fall back on material testing of the actual component whenever possible.

There are two official stages Crack Initiation and Crack Growth. A crack growth calculation seems silly. You know there is a crack but instead of repairing it you calculate how long it will take before catastrophic failure occurs under a variable multiaxial load. When it does come, the crack apparently proceeds at 1/3 of the speed of sound, hence the term "catastrophic". Now I ask you, in all seriousness, would you get on an aircraft if you knew it had a crack in the wing? The answer is obvious, so a crack propagation calculation is largely an academic exercise. In practice if you see a crack you should stop using the component and repair it. Unfortunately, that was the easier calculation of the two. Crack growth calculations are summarised below:

Useless fracture calculation summary; You receive your NDT report which either shows you a crack in an X-ray from one angle, from which it is impossible to tell the real shape, or from an ultrasonic report which states that there is an "indication of size below 3mm". As you then don't know a crack shape you must do a "sensitivity analysis". That is, try every pertinent shape from the list of impossibly clean and mathematically perfect crack shapes to get your SIF. Ignore the extreme unlikelihood of not having an elliptically shaped crack. Then guess the positive residual stresses adjacent to the crack (because cracks in compression won't grow) assuming some fraction of yield. The final report will state that the crack will undoubtedly grow, and hence the structure will fail, under at least some of the fake scenarios you have been forced to use. Again you can happily pass your own design but fail someone elses depending on your assumptions and your degree of malevolence.

You can use your FE code to calculate the SIF and a certain Dr. Pook wrote a paper on it using FEMdesigner. For the future though we hope to consistently compute the crack path computationally.

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