The cut-off belongs to the water

Three essays about this detail have rested on the same feature and none of them has questioned it. The spectrum essay found that the cut-off deletes most of the traffic. The crack-growth essay found the same feature in a different form — a threshold stress intensity below which a cycle grows nothing — and used it to show that the deleted cycles are deferred rather than harmless.

Both are facts about steel in air.

What the curve owes to the atmosphere #

An S-N curve for steel in air has two knees in it and both are real experimental features.

The constant-amplitude limit at five million cycles is where the slope changes from three to five. Below it, a crack that has started can still grow but does so more slowly than the extrapolated line predicts, because a growing fatigue crack partly closes on itself and the effective stress range at the tip is less than the applied one.

The cut-off at a hundred million cycles is where the line stops. Below it, a constant-amplitude test does not fail however long it is run, because the stress intensity at the tip of whatever flaw the detail has never reaches the threshold $\Delta K_{\mathrm{th}}$ needed to make a crack advance at all.

Both mechanisms need the crack to be a crack in steel and nothing else, and both are what makes the S-N curve a property of the detail rather than of the material. Closure needs two faces that can touch. A threshold needs a crack tip at which nothing is happening between cycles.

A crack tip that water can reach fails both conditions. Corrosion product wedges the faces apart, which removes the closure; and the chemistry at the tip — dissolution, hydrogen entering the metal ahead of it — does not require a stress intensity to proceed. So the two knees go, and what is left is a single slope with no lower end: for steel in seawater the offshore codes give a curve of slope three with no cut-off and no endurance limit whatever.

What that costs, on the same traffic #

The lorries and vans that were deleted come back, and they come back as more than a third of the damage.

Worth reading carefully is how they come back, because it is not proportional to their count. Under free corrosion the ordinary lorry at 27 N/mm² is the largest single contributor after the train, and the car at 7 is still nearly nothing — a slope of three means a cycle a quarter as large does a sixty-fourth of the damage, and no removal of a knee changes that. The cut-off was not deleting cars; it was deleting the band just below the line, and it is that band which returns.

That has a design consequence with a sharp edge on it. Two spectra that look similar — one whose ordinary lorries produce 27 N/mm² and one whose produce 30 — are identical in air, because the first is deleted by a hair and the second is not, and the second’s band does a fifth of the damage. In water they differ by the ratio of the cubes and nothing more. Corrosion makes a fatigue calculation less sensitive to where exactly a band sits, which is the one thing about it that is easier.

Six lives, one bridge #

Not one of those six is a mistake, and the spread between them is the honest width of the subject.

The three Miner lives — 300, 256 and 148 years — differ only in what the curve says about cycles below the constant-amplitude limit. Nothing about the traffic, the detail or the steel changes between them.

The three crack-growth lives — 196, 135 and 45 years — differ in whether there is a threshold and how fast the crack grows. Removing the threshold costs 31 per cent; tripling the rate, which is a modest figure for seawater, costs another two thirds. Neither is affected by the order the cycles arrived in, which is a per-cent effect beside these.

Nor does the plate’s thickness rescue any of it: the thickness correction is a twenty-per-cent adjustment to a category, against a factor of two here. And the pairs do not line up, which is the useful part. The air Miner life is 300 and the air crack life is 196; the free-corrosion Miner life is 148 and the no-threshold, faster-growth crack life is 45. Corrosion widens the disagreement between the two methods as well as shortening both, because the S-N route absorbs the environment into a curve shape and the crack-growth route absorbs it into a rate, and the two are not the same statement.

The two ways an environment gets into a calculation #

It is worth separating the two mechanisms cleanly, because a knock-down factor handles one of them and not the other.

A rate change scales the answer. Corrosion raises the Paris coefficient — by about three for seawater, more for some conditions — and every life scales inversely with it. On an S-N curve that is a downward shift of the whole line, which is exactly what a knock-down factor does, and a factor is the right tool for it.

A shape change does something a factor cannot. Removing the cut-off does not move the curve down; it extends it sideways into a region where it previously did not exist. No multiplication of the category reproduces that, because the effect is concentrated entirely in the bands that were below the line and is zero for the bands above it.

The distinction has a practical edge. A designer who applies a factor of two to the category of a wet detail has reduced the strength of the train and the heavy lorry by that factor and has still assigned zero damage to the ordinary lorry, the van and the car — because the cut-off scales with the category and moves down with it. The correction that was applied and the correction that was needed act on disjoint parts of the spectrum.

The figure is what a knock-down factor does to the spectrum rather than to the curve, and it is the reason a factor is not a substitute for the right curve. Halving the category moved two bands from zero damage to almost zero damage. It did not move the two bands that were already carrying all of it; those it weakened by the factor, which is what it was for. The correction went where the damage already was and the environment goes where the damage was not.

And the two interact. A lower category lowers the cut-off, which un-deletes a band; so a knock-down factor applied to a spectrum can change which bands count as well as how much they do, and the result can be discontinuous in the factor. That is a good reason to change the curve rather than to scale it, and it is what the offshore codes do.

What a design actually does about it #

Codes do not offer six answers. They offer one, and it is worth knowing which of these it is.

For structures in air with normal exposure — most bridges, most buildings — the air curve is used unmodified, and the argument for it is that paint and a drainage detail keep the water off. That argument is about maintenance rather than about materials, which is why a fatigue category and an inspection regime are the same subject.

For offshore structures the practice is explicit: a free-corrosion curve with no cut-off for unprotected steel, and for cathodically protected steel a curve between the two — the protection restores part of the threshold by suppressing the dissolution at the tip, though not the wedging.

For bridges in de-icing salt there is no separate curve at all, and the environment is handled by detailing: drainage, sealing, and keeping the fatigue-critical details out of the splash zone. That is not conservatism; it is an admission that no curve is available for a detail whose environment is unknown and varies along its own length.

The fourth case is the one to be careful about. A structure whose protection has failed locally has changed curve locally, and nothing in an inspection records that it has. A drain that blocks in year thirty puts a category 71 detail on a free-corrosion curve for the rest of its life, and the calculation that certified it was done with the other one.

Where the crack picture says the same thing #

The threshold and the cut-off are the same feature seen from two sides, and that is worth stating because it is the reason the two calculations move together.

A cut-off in an S-N curve is the statement that there is a stress range below which a detail does not fail. A threshold in crack growth is the statement that there is a stress intensity below which a crack does not advance. Given a detail with a starting flaw $a_0$, the second implies the first:

$$\Delta\sigma_{\mathrm{cut}} = \frac{\Delta K_{\mathrm{th}}}{Y\sqrt{\pi a_0}} .$$

For $\Delta K_{\mathrm{th}} = 63\ \mathrm{N/mm}^{1.5}$, $Y = 1.12$ and a 0.2 mm flaw, that is 71 N/mm² — which is above every band of this traffic and above the category’s own cut-off of 28.7. The design curve’s cut-off is set conservatively low relative to what a threshold argument would allow, and for good reason: the flaw is unknown, the residual stress is at yield, and the threshold used is the one for a fully tensile cycle.

Take the threshold away and the whole of that reasoning goes with it. There is no stress range below which nothing happens, so there is no cut-off to be conservative about, and the equation above returns zero for every flaw.

The disappearance of the cliff is the sharpest statement of what corrosion costs. In air a detail with a flaw under 0.26 mm had an unbounded life — not a long one, an infinite one, because no cycle in the spectrum could move the crack. That is a genuine safe harbour and it is where an enormous number of details actually sit: a sound weld in a lightly loaded member, in air, is not slowly using something up. It is doing nothing at all.

A wet detail has no safe harbour. Every flaw grows from the first cycle, so every detail is on a finite clock, and the only question is whether the clock is longer than the structure’s life. That is a different kind of design problem — one with no margin of kind, only of degree — and it is why offshore practice is built around inspection in a way bridge practice is not.

What the detail’s own geometry does about it #

One consequence of the shape change is worth following, because it reverses a piece of standard advice.

In air, improving a detail’s category is worth a great deal: the damage goes as the inverse cube, so moving from category 71 to 112 divides the damage of the counting bands by nearly four. It also raises the cut-off, from 28.7 to 45.3, which deletes the heavy lorry as well — so the better detail is better twice over, once by strength and once by deletion, and its life goes up by far more than the cube.

The two air curves are the same shape at different heights and the two free-corrosion curves are the same line at different heights, but only the first pair changed which traffic it counts. That is the whole of the asymmetry.

Under free corrosion the second half of that is gone. There is no cut-off to raise, so improving the detail buys the cube and nothing else. A wet structure gets less from a better detail than a dry one does, which is the opposite of the intuition that a hostile environment is a reason to specify more carefully.

It runs the other way for the thing that actually helps. Keeping the water off is worth a factor of two on a good detail and a factor of two on a bad one, and it is the only intervention on this page whose value does not depend on the category at all.

The whole of it, once, by hand #

The spectrum is 800 crossings a day: 2 per cent at 62 N/mm², 8 at 41, 25 at 27, 30 at 16, 35 at 7.

In air, category 71: the constant-amplitude limit is $71(2/5)^{1/3} = 52.3$ and the cut-off is $52.3(5/100)^{1/5} = 28.7$. Only the train and the heavy lorry are above 28.7, so

$$D_{\mathrm{year}} = \frac{5{,}844}{2\times10^6 (71/62)^3} + \frac{23{,}376}{5\times10^6(52.3/41)^5} = 3.3\times10^{-3},$$

a life of 300 years.

Under free corrosion, the curve is $N = 2\times10^6 (71/\Delta\sigma)^3$ for every range. Now every band counts:

$$D_{\mathrm{year}} = \frac{2\times10^6}{71^3}\left[\frac{62^3}{2\times10^6} \cdot \frac{5{,}844}{2\times10^6} + \cdots\right] = 6.8\times10^{-3},$$

a life of 148 years. The two bands that did all the damage in air still do 62 per cent of it; the three that did none now do 38.

The arithmetic is worth doing because of what it shows about the shape of the answer. Removing the cut-off does not double the damage of the bands that were already counting — it does not touch them. It adds a new term, and the size of that term is set by how much traffic sits just under the line.

Which free body produced the number #

The free body is the same one every fatigue calculation here has used: one detail, one stress range per vehicle, and a Miner sum or a crack integration over the traffic. What varies between the six calculations is the constitutive statement — the S-N curve’s shape, or the growth law’s threshold and coefficient — and nothing else. The spectrum, the crossings a day, the category and the detail are identical in all six.

The three curves are the design curve as published, the same curve with its second branch extended instead of terminated, and a single slope of three anchored at the same two-million-cycle point. The three crack integrations are the same code with $\Delta K_{\mathrm{th}}$ at 63 and at zero, and with the Paris coefficient at its air value and at three times it.

What the picture cannot show #

Time. Corrosion is a rate and a fatigue calculation is a count. A structure cycled slowly in seawater is worse off than one cycled quickly, because the chemistry has longer between cycles — so the same million cycles is not the same damage at one hertz and at one cycle a day. Nothing in any of these six calculations contains a clock.

Pitting. The mechanism assumed here is that the environment changes how a crack grows. The other mechanism is that it makes the crack: corrosion that eats a pit into a smooth surface has created a starting flaw where there was none, and a pit is a far better crack starter than a sound surface. That path is not in these figures at all and it is often the one that matters.

Cathodic protection. Overprotection puts hydrogen into the steel and can be worse than no protection for a high-strength steel. The curve for a protected joint is not simply between the two drawn here.

Which environment. “Corrosive” is not one condition. Seawater, de-icing salt, sulphur dioxide and condensation under lagging produce different rates and different thresholds, and the free-corrosion curve is one point in that space chosen because it is the one that has been tested.

The assumption worth naming #

That the environment is known and constant. Every figure holds it fixed for a hundred years, and the thing being modelled is a coat of paint.

That is the sharpest form of the problem and it is not a modelling deficiency — it is the actual state of affairs. A structure’s fatigue environment is a maintenance outcome, decided by whether a drain was cleared and whether a coating was renewed, and it varies along the structure and over its life in a way no calculation can anticipate. The six lives on this page are not six estimates of one quantity. They are the lives of six different structures, and which one is standing there depends on what was done to it.

Which is why the useful output of the comparison is not a number. It is a sensitivity: a bridge whose fatigue life is comfortable in air and marginal under free corrosion is a bridge whose safety rests on its drainage, and that is a fact worth knowing at design time and not at the first inspection.

Still open: the crack that starts at a pit #

Every calculation so far has begun with a flaw that was already there — a weld-toe intrusion, an undercut, a defect from the process. The environment has been allowed to change how a crack grows and never allowed to make one.

Corrosion makes them. A pit in a smooth surface is a notch with a radius of a few tens of microns, and it grows by dissolution at a rate that has nothing to do with the cycling. At some depth it stops being a pit and starts being a crack, and where that transition sits — and whether the life of a plain member in a corrosive environment is a corrosion problem with a fatigue ending or the reverse — is the question left open.