Which reversal closes the loop

The reduction of a traffic spectrum to one equivalent range began with a histogram of stress ranges and a count against each. A structure does not supply that histogram. A strain gauge on a weld toe supplies a record: stress against time, rising and falling, with no marks in it saying where one cycle ends and the next begins. Something has to decide which rises go with which falls, and fatigue damage goes as the cube of the ranges that decision produces.

A record is reversals, not cycles #

The record is built the way a strain gauge sees a short bridge. Each vehicle adds the mid-span influence line, a triangle that rises while the vehicle approaches mid-span and falls while it leaves, scaled to its own peak. Each also sets the deck ringing at its natural frequency, with an amplitude of 15 per cent of that peak, decaying with two per cent damping. Two ordinary lorries meet at 12 seconds, and the heaviest vehicle crosses at 20.

Everything a fatigue calculation can use from that record is in its reversals, the points where the stress stops rising and starts falling or the reverse. Between two reversals the stress is monotone, and nothing between them changes the damage. With a threshold of 0.5 N/mm², below which a wiggle is not recorded as a reversal, this record has 185 of them.

A cycle is something else. Physically, what drives a fatigue crack forward is a closed hysteresis loop at the crack tip: the local material is strained one way and then back to where it started, and the crack grows a little on each loop. A stress range is only half of that loop. So counting cycles is pairing reversals into loops, and the question is which pairing reproduces the loops the material actually traces.

The rule that closes loops #

The rule the profession settled on is called rainflow, after a picture of rain running off a pagoda roof drawn by Matsuishi and Endo in 1968. It is set out as an algorithm in ASTM E1049, and the standard gives a nine-reversal example with the answer tabulated, which makes it the natural place to watch the rule work.

The reversals go onto a stack one at a time, and after each one the rule compares two ranges: X, the newest, between the top two points, and Y, the one below it. If X is at least as large as Y, then Y is a closed loop. The excursion Y has been completely enclosed: the stress went out, came back, and has now gone past where Y started. Y is counted, and its two reversals come off the stack. If Y contains the earliest reversal still waiting, it cannot be a closed loop, because nothing before it was seen; it is counted as half a cycle and only that earliest point comes off.

On the example, the first two ranges, A–B and B–C, are each exceeded by the range after them while holding the earliest point, and go as halves. Then D is pushed, E, F, and G. When G arrives, the range F–G of 7 exceeds E–F of 4, and E–F is a whole cycle: the stress went from −1 up to 3 and back past −1, enclosing that excursion completely. With E and F gone the stack reads C, D, G, and D–G of 9 now exceeds C–D of 8, so C–D goes as a half. What is left at the end is the residue: D–G, G–H and H–I, each half a cycle.

The range of 9, from D at 5 to G at −4, is the largest in the record. No two successive reversals span it. E and F sit between them, and the excursion from −1 to 3 interrupts the fall from 5 to −4 without ending it. Counted between neighbours, the same history gives ranges of 6, 4 and 7 in that stretch and never a 9.

Why an interrupted cycle is still one cycle #

The rule is not a convention chosen for convenience. It reproduces a property of metals under cyclic plastic strain that was measured long before the algorithm was written: material memory. When a small loop interrupts a large excursion and closes, the material returns to exactly the stress–strain path it was on before the interruption and carries on along it, as though the small loop had never happened. A yielding brace cycled both ways traces loops of this kind at full scale. At a weld toe they are tiny, local and invisible, and they are what the crack responds to.

So the fall from D to G is one branch of one large loop, with the small loop E–F hung on it. A counting method that ends the large branch at E has cut a loop the material never cut. The three-point rule’s test — has the newest range gone past where the previous one began? — is exactly the test for whether a loop has closed on that path.

The lorry range counting cannot see #

On the bridge record the difference is not a detail of a textbook example. It is most of the damage.

The heaviest vehicle’s loop runs from its peak at 20.6 seconds, 69.0 N/mm², to the dip the ringing drives below zero at 21.2 seconds, a range of 74.3. The heavy lorry’s runs from 1.5 to 2.2 seconds and is 45.1. The two lorries meeting at 12 seconds make one loop of 38.5, from their combined peak to the trough after, where each lorry’s own influence line peaks at 27 — which is why meeting vehicles are what fatigue load models are built around. The van’s is 16.5.

Counted between successive reversals instead, the largest range anywhere in the record is 38.1 N/mm². The deck’s ringing is to blame. At 4 Hz and 15 per cent of the peak, the ringing’s rate of change is larger than the influence line’s, so every rise and every fall has small reversals on it, and each lorry’s excursion is recorded as a staircase of short ranges. Rainflow hangs the ringing’s loops on the lorry’s branch and counts them separately. Range counting takes the steps of the staircase to be the cycles.

Watching the stack while a lorry arrives shows how the rule keeps the branch whole. The stress starts near zero, which is the bottom of the stack. It rises a little, dips as the deck rings back, and rises again past the top of the first rise. The newest range now exceeds the dip before it, so the dip closes as a small loop and its two reversals come off, leaving the base of the branch at the bottom of the stack and the new high point on top. Every step of the staircase does the same: a ringing loop closes, comes off, and the base and the latest high point remain. When the lorry reaches mid-span the stack holds the base of the branch and the peak, with nothing between them. The falling branch then does the same thing in reverse, and when it goes below the base, the whole excursion from base to peak closes as one loop. The staircase has been counted as a dozen small loops and one large one, which is what the material at the weld toe traced.

The same number of cycles, and not the same cycles #

The two counts do not disagree about how many cycles the record contains.

Both counts hold 92 cycles, and they have to. Every range between successive reversals is used exactly once by both: range counting takes each as half a cycle, and rainflow assembles them into whole loops. Counting neither creates cycles nor destroys them. It decides which ranges they have.

That is why a histogram of counts cannot tell a good method from a bad one: the top panel of the figure looks similar either way, dominated by ringing. The difference is in the tail, and the tail is what the cube weights. One rainflow cycle carries 72 per cent of the record’s cube-weighted damage. Range counting has broken that cycle into pieces in the 20s and 30s, and a piece of half the range carries an eighth of the damage.

Three counts, three lives #

The consequence for a design check is larger again, because the damage is not the cube-weighted sum alone. A detail has a cut-off below which cycles do nothing, and a count that breaks large cycles into small ones pushes the pieces under it.

On category 71 the cut-off is at 29 N/mm². Rainflow’s three largest loops, at 74.3, 45.1 and 38.5, are all above it, and they carry the Miner sum to 2.97: the detail fails in 34 years. Range counting’s pieces are almost all below it, and its Miner sum is 0.24: the detail passes with four centuries to spare. The cube made the two counts differ by a factor of four; the cut-off made it twelve, because it deletes pieces and not loops. The same record, the same detail and the same traffic produce a failure and a comfortable pass, and nothing distinguishes them but the rule for pairing reversals.

Peak counting errs the other way, and in a way worth understanding. It sorts the peaks from highest to lowest and the valleys from lowest to highest, and pairs them in order, whether or not a peak and its valley were ever in the same loop. The heaviest vehicle’s peak is paired with the record’s lowest valley, the heavy lorry’s with the next lowest, and so on. For a record whose loops each use one peak and one valley, that is the most damaging pairing there is, because a sum of cubes of differences is largest when the largest values are set against the smallest. Peak counting is therefore an upper bound on rainflow, not an approximation to it. On this record it is 47 per cent above in Miner sum, and on a record with more independent events the gap has no reason to stay there.

A logger file is a cut through a loop #

Rainflow’s residue — the half cycles left on the stack at the end — looks like a bookkeeping remainder. It is the count’s memory, and a monitoring system that writes its data in files meets it directly.

The loss is small while the files are long and falls away once a file is shorter than the loops it is cutting. The heaviest vehicle’s loop takes 0.6 seconds from peak to trough, and its branches start earlier. A file boundary anywhere inside that interval puts the loop’s two reversals in different files, and each file counts its own half as half a cycle of a smaller range, because the reversal that would have closed the loop is in a file that has not been opened yet. At files of 0.41 seconds almost every loop larger than the ringing has been cut, and what is left is the ringing.

The whole record has a residue of its own, and it matters even when nothing is cut. Counted as a single open record, the heaviest vehicle’s loop does not close before the record ends: the stress never falls back below the level it started from by more than the ringing, so the loop sits in the residue as two half cycles, one of 74.3 N/mm² and one of 72.4. Taking the record to repeat — starting the count at its largest excursion and ending it there — closes those two halves into one whole loop of 74.3, and raises the cube-weighted damage by 4.5 per cent. A record that genuinely repeats, a day of traffic that will be followed by another like it, should be counted closed. A record of a single event that will not recur should not.

Carrying the stack across the cut undoes all of it, exactly. The stack at the end of a file is everything the count knows about the past: the reversals whose loops are still open. Starting the next file’s count from those points rather than from nothing reproduces the whole record’s count to the last cycle, however many files the record was cut into. That is the correct treatment of a residue in general. It is not a set of half cycles; it is a set of loops waiting for their closing reversals.

What a count cannot know #

A rainflow count is the right summary of a record, and it is still a summary, with four things deliberately left out.

The order of the loops. Rainflow reports how many loops of each range a record contained, not when. Miner’s rule needs no more than that, and the rule’s indifference to sequence is its known weakness: a large loop leaves a plastic zone at the crack tip that retards the growth of the small loops after it. The count preserves exactly the information Miner can use and discards exactly the information that would correct it.

The mean of each loop. The algorithm records it, and a welded-detail check ignores it, because the residual stress at a weld toe is already at yield and the applied mean hardly moves it. For an unwelded component the mean matters, and the count supplies it for a later correction.

The threshold. Reversals smaller than 0.5 N/mm² were never recorded, and the choice of threshold shows something about the summaries a count feeds. Raising it from 0.1 to 10 N/mm² cuts the record’s reversals from 198 to 10 and its cycles from 98 to 5, and leaves the Miner sum on category 71 at 2.970 to four figures, because every cycle it removes is ringing far below the cut-off. The equivalent range, meanwhile, climbs from 18.0 N/mm² to 48.3: the same damage divided among fewer cycles is a larger equivalent range. An equivalent range quoted without the number of cycles it stands for says nothing, and a threshold is enough to move it by a factor of nearly three.

And direction. The count is of one stress component at one point. A weld whose principal stresses rotate as a vehicle passes over it sees loops in a plane that turns, and counting one component misses them. Multiaxial fatigue is counted on critical planes, and rainflow is run on each.

Where the model stops #

The record is stylised. A triangular influence line and exponentially decaying ringing are a legible stand-in for a gauge record, not a measurement. Real records have axle groups, road roughness and several modes of vibration, and the conclusions about pairing do not depend on any of that, but the particular numbers do.

The detail category is a constant-amplitude result. The S–N curve behind the Miner sum was fitted to specimens tested at one range. Applying it to loops of many ranges is the assumption the equivalent-range calculation set out, and rainflow does not remove it.

Seven vehicles stand for a day. The lives assume the record’s mix of vehicles repeats. What rainflow says about this record is exact; what it says about a century depends on whether the next century’s traffic looks like these 26 seconds, which is the forecasting problem every fatigue calculation inherits.

And damage tolerance asks a different question. A count feeds a crack-growth integration as readily as a Miner sum, and an inspection interval set from crack growth needs the same loops. The count is the same; what changes is the law it is fed to.

Still open: the loops a crack grows on #

The count now exists; what a crack does with it is the next question. Paris’s law integrated over a rainflow count, loop by loop, from a flaw an inspector can find to one the section cannot carry, turning this record into a crack length rather than a Miner sum. The retardation after an overload, measured against Miner’s order-blindness on the same record played forwards and backwards. The mean stress of each loop, and the Goodman correction for the unwelded details where it counts. The thickness effect, where a thicker plate of the same detail sits in a lower category. And the improvement techniques — toe grinding, hammer peening, TIG dressing — which move a detail’s line rather than its loops, and whose worth depends, as the cut-off does here, on how many loops sit just below the line they move.