The cycles that do not count

A fatigue check runs on a stress range and a detail category, and neither of those is what a structure actually experiences. What it experiences is a spectrum — a histogram of ranges, each with its own count — and something has to reduce it to one number. That reduction is not an averaging, it deletes a third of the data outright, and the number it produces is closer to the worst vehicle than to the ordinary one.

The cube is the whole of it #

The S–N line has a slope of three on log axes, so the life at a range $\Delta\sigma$ is proportional to $\Delta\sigma^{-3}$ and the damage per cycle is proportional to $\Delta\sigma^{3}$.

A cycle twice as large does eight times the damage. A cycle a third as large does a twenty-seventh. That single exponent is what makes the two bars in the figure disagree.

Miner’s rule then sums the damage: $\sum n_i/N_i$, with failure notionally at one. Substituting the cube gives

$$D \propto \sum n_i \Delta\sigma_i^3$$

and the equivalent constant range is the value that would produce the same sum in the same total number of cycles:

$$\Delta\sigma_e = \left(\frac{\sum n_i \Delta\sigma_i^3}{\sum n_i}\right)^{1/3}$$

which is a cube-weighted mean rather than an arithmetic one. For the spectrum drawn it comes out at 25.5 N/mm², against an arithmetic mean of the five bands well below that.

Two features of that expression are worth pausing on before going further.

It has no count in it. The equivalent range is a property of the spectrum’s shape, so a busier road and a quieter one with the same mix of vehicles have the same equivalent range and different lives. The count enters separately, as the number of cycles the check is made at.

And it is not bounded by the largest band. A cube-weighted mean lies between the smallest and largest values, so the equivalent range is always inside the spectrum — but it can sit very close to the top of it. A spectrum of one heavy cycle and a thousand tiny ones has an equivalent range near the heavy one’s, which is the correct answer and is not what an average suggests.

Which free body produced the number #

There is no free body, and the substitute is worth naming because it is where the whole method’s confidence comes from.

Fatigue damage is a crack growing, and a crack does not know about cycles it has not yet seen. Miner’s rule asserts that damage is a scalar that accumulates linearly and independently of order — that a thousand large cycles followed by a million small ones does the same damage as the reverse.

That is not true. A large cycle leaves a plastic zone at the crack tip that retards subsequent small ones; a small cycle after a large one may do less damage than the same cycle would have done first. Sequence effects are real and Miner’s rule deletes them, which is why the rule’s sum at failure is observed between about 0.3 and 3 rather than at 1.

The rule survives because the alternative is intractable and because the spread is inside the safety factors — and because, for the cube-weighted case, most of the damage comes from a small number of large cycles whose order matters least.

What the equivalent range is worth knowing about #

The reduced number carries three properties that are worth having before it is used.

It is not a stress the structure sees. 25.5 N/mm² is not the range of any vehicle in the spectrum; it is the constant range that would do the same damage over the same count. Looking for it in a measured record is a category error.

It scales with the load and not with the traffic. Doubling every vehicle’s weight doubles the equivalent range; doubling the number of vehicles leaves it exactly where it was and doubles the count instead. The two enter the Miner sum in different places — the range cubed and the count linearly — so a heavier traffic is far worse than a busier one, by a factor of eight against two for the same total tonnage.

And it is dominated by a part of the spectrum nobody measures well. The heaviest few per cent of vehicles are the hardest to count, the most likely to be overloaded, and the most likely to change over a structure’s life. A spectrum’s mean is well known and its tail is not, and the tail is what the cube is weighting.

Together those say where the uncertainty in a fatigue life actually lives. Not in the S–N curve, not in Miner’s rule, and not in the category — in a histogram’s upper tail, forecast a century ahead.

The cut-off, which deletes a third of the traffic #

The line is not straight all the way down, and the kink is what makes a spectrum manageable.

At the constant-amplitude limit — five million cycles — the slope changes from three to five, and at the cut-off limit — a hundred million — the line stops and cycles below it are taken as doing no damage at all.

That is why 35 per cent of the crossings in the figure contribute nothing. They are cars, they produce a range of 7 N/mm², and the curve says a range that small never causes failure whatever its count.

The physical reason is that a stress range below the threshold cannot grow a crack, because the stress intensity range at the crack tip is below the material’s threshold value $\Delta K_{th}$. That is a genuine effect and a fragile one: it holds only if the crack is small and the mean stress is low, and it disappears in a corrosive environment.

So the cut-off is an assumption with a condition on it, and the condition is exactly the one a welded structure fails — a weld toe carries residual tension at yield, so the local mean stress is as high as it can be. Codes handle that by placing the cut-off conservatively low, which is why it deletes cars and not lorries.

A better detail changes which cycles matter #

The most surprising consequence of the cut-off is what happens when the detail is improved.

Improving the category raises the whole line, which raises the cut-off with it, which deletes another band of the spectrum. The damage falls by more than the category ratio would suggest, because bands leave the calculation entirely rather than merely doing less.

The Miner sums say it: 0.40 against 0.03, a factor of 13, from a category ratio of 112/71 = 1.58. Cubed that would be 3.9; the remaining factor of 3.4 is the second band disappearing.

That is a strong argument for detail improvement and a warning about extrapolating it. The benefit is discontinuous in the spectrum, so a small improvement can be worth a great deal or almost nothing depending on where the bands sit relative to the cut-off, and the same improvement on a different traffic gives a different answer.

Where the equivalent range sits on the line #

The reduction to one number is only useful if the number can be read against the curve, and the reading has a trap in it.

An equivalent range is legitimate only against the curve it was computed for. The cube weighting used a slope of three, and the part of the line below the constant-amplitude limit has a slope of five — so a spectrum whose bands straddle the kink has been reduced with the wrong exponent for some of them.

Codes handle this by defining the equivalent range with the damage rather than with the ranges, which is exact by construction, and by specifying a reference number of cycles — two million — at which to quote it. The result is a number that is not a stress the structure ever sees and is not meant to be.

Why the cycles that count are not the ones that break it #

There is a distinction hiding in all of this that is worth making explicit, because it separates two calculations that use the same spectrum.

Fatigue damage is cube-weighted, so it is dominated by the heaviest few per cent of the traffic. That is the calculation on this page.

Fracture is not weighted at all. A member with a crack in it fails at the first cycle whose peak stress takes the stress intensity past the toughness, and which cycle that is depends only on the largest load, not on how often it comes.

So the same spectrum has two readings. For growth, the cube-weighted equivalent range says how fast the crack advances; for failure, the single largest peak says whether the crack that has grown is now critical. Damage tolerance needs both and they are answered by different statistics of one histogram.

That also explains a rule that otherwise looks arbitrary. A fatigue check uses a frequent load model and a fracture check uses a characteristic one — not two levels of conservatism, but two different questions asked of the same traffic.

What the spectrum itself is #

Everything above assumes the histogram exists, and obtaining it is the harder half.

A measured stress history is a continuous wiggle, not a set of cycles, and turning it into bands requires an algorithm. Rainflow counting is the one that works: it pairs each reversal with the one that closes the hysteresis loop it started, which reproduces the material’s own memory of its loading and is far less obvious than it looks.

Simpler counting methods — peak counting, level crossing, range counting — all mis-pair reversals and get the damage wrong, usually by under-counting the large slow cycles that matter most. Since those are the cycles carrying most of the cube-weighted damage, the counting method matters more than the accuracy of the measurement.

For design rather than assessment the spectrum comes from a code’s traffic model instead, which is a fictitious set of vehicles constructed so that its damage matches the real traffic’s — an upper contour rather than a description, and one designed against exactly the cube-weighted sum this page is about.

Where the category comes from, and what it hides #

The whole reduction ends by being compared with a category, and it is worth knowing what a category is a summary of.

A detail category is a fitted line through a set of test results on one geometry, and what varies between geometries is a local stress concentration of exactly the kind that figure shows. A weld toe, a bolt hole, a cope, a change of section: each raises the local stress above the nominal one that the check uses, and the category is what stands in for that ratio.

That has two consequences the check hides.

The stress range in the calculation is nominal. It is computed on the gross or net section, ignoring the concentration entirely, and the category carries the concentration instead. Using a local stress and a category would count the same effect twice.

And the concentration is geometric. The figure’s factor of 3.0 is independent of the hole’s size and depends only on its shape — which is why a category depends on a detail’s form and not on its dimensions, and why the detail decides and the steel does not.

What to carry away #

The reduction is cube-weighted. The equivalent range is nearer the heaviest band than the average, and a small number of large cycles carries most of the damage.

A third of the cycles may count for nothing. The cut-off deletes them, and the deletion is an assumption that holds for low mean stress and small cracks.

Improving the detail changes which cycles matter, not only how many the structure survives — bands drop below the raised cut-off and leave the sum entirely.

The two ends of the histogram are the ones nobody measures well, and the cube weights one of them heavily.

And the counting method is part of the answer. A spectrum is produced by an algorithm, and the wrong algorithm mis-counts the cycles that carry the damage.

Where the model stops #

Miner’s rule is linear and order-independent. Sequence effects are real, retardation after an overload is real, and the observed sum at failure ranges from about 0.3 to 3.

The cut-off assumes a threshold that a corrosive environment removes. In sea water there is no endurance limit and the small cycles resume counting.

The slope of three is fitted to welded details. Unwelded and bolted details have different slopes and different mean-stress sensitivity, and the same reduction with a different exponent gives a different equivalent range.

Residual stress is not in the calculation. It is why the mean stress does not appear, and it is also why the cut-off has to be placed low — one omission justifying another.

One detail is checked at a time. A member has many details and the traffic passes them all, and the spectrum at each is different because the influence lines differ — a detail near a support sees a different history from one at mid-span under the same vehicles.

And the spectrum is a prediction. A hundred-year traffic model is a forecast of what will use a structure, and it is the input with by far the widest uncertainty in anything on this page.

A worked reduction, in five lines #

The arithmetic is short enough to do here, and doing it once makes the weighting obvious.

Take five bands with ranges 62, 41, 27, 16 and 7 N/mm² and shares of the crossings 2, 8, 25, 30 and 35 per cent.

Cube each range: 238,328; 68,921; 19,683; 4,096; 343.

Weight by the shares: 4,767; 5,514; 4,921; 1,229; 120 — a total of 16,551.

Take the cube root: 25.5 N/mm², which is the equivalent range.

Now read the weighted column rather than the total. The top two bands supply 62 per cent of the sum on 10 per cent of the crossings; the bottom band supplies 0.7 per cent on 35 per cent of them. And that is before the cut-off, which deletes the bottom two entirely and takes the top two to 100 per cent of what is left.

Five numbers, one cube root, and the whole of the subject’s asymmetry. Anybody handed an equivalent range without its spectrum has been handed the answer to this calculation and no way of knowing which vehicles produced it — which matters the moment the traffic changes.

Counting is only half of a fatigue assessment. What the count is compared against is a detail category rather than a material strength, and what is done with the answer is an inspection interval rather than a factor of safety. Both are why a load that never came near failing anything can still be the load that ends the structure.

The ladder from here #

Later rungs on this anchor: rainflow counting set out as the algorithm it is, with the loop-closure rule and the reason simpler methods fail. Paris’s law integrated to a life rather than a category. Mean stress effects and the Goodman diagram, for the unwelded cases where they matter. Low-cycle fatigue and the Coffin–Manson relation, where the controlling variable is a plastic strain range. Corrosion fatigue and the loss of the endurance limit. Improvement techniques — toe grinding, hammer peening, TIG dressing — and how much each is worth. And the thickness effect, where a thicker plate of identical detail has a lower category for the same reasons that made it a worse fracture prospect.

There is a last reading of the whole subject worth carrying. Fatigue is the only limit state in this collection whose demand is a history rather than a state, and every difficulty on this page comes from that one fact: a history has to be summarised before it can be compared with anything, and every summary throws something away. The cube-weighting throws away the small cycles, the cut-off throws away a third of them outright, Miner’s rule throws away their order, and the traffic model throws away the future. What is left is one number, and it is a good one — but knowing which four things went missing to produce it is most of what a fatigue calculation is worth.

Miner published the rule in 1945, Palmgren had it in 1924 for ball bearings, and neither claimed it was more than a working hypothesis. That a rule known from the start to be wrong about sequence has survived eighty years of better alternatives says something about what a design method is for: the error it makes is small compared with the uncertainty in the spectrum it is applied to, and a more accurate rule fed a forecast of next century’s traffic would not be a more accurate answer.