The rule that points sideways

A detail category is a number attached to a shape. A fillet-welded attachment is category 80, a transverse butt weld ground flush is 112, a bolt in tension is 50 — a table of drawings, each with a stress range beside it, and nothing in the table is a property of the steel.

There is one correction to those numbers and it does not look like it belongs. If the plate is thicker than 25 mm, multiply the category by

$$\left(\frac{25}{t}\right)^{0.2}.$$

That is strange in a way worth pausing on. The crack grows through the thickness, so the thickness is the dimension along the crack’s path and the rule says the longer the path, the weaker the detail. Every other size effect here runs the other way round: a bigger specimen is weaker because it is more likely to contain a large flaw, which is a statement about volume and not about a direction.

There is a mechanism, and it is not statistical.

The magnification is a ratio #

A weld toe is a notch, and a notch raises the stress near it and leaves it alone further away. The distance over which it does so is set by the notch’s own geometry — the toe radius, the weld leg, the attachment size — and all of those scale with the plate the weld is on. A 12 mm plate gets a small fillet and a 100 mm plate gets a large one, because that is how welds are specified.

So the elevated field is geometrically similar: a plot of the magnification against $a/t$ is one curve for all thicknesses. BS 7910 fits it as

$$M_k = \max\left(1,\; 0.83\,(a/t)^{-0.15}\right) \quad\text{for } a/t \ge 0.05,$$

with a steeper branch below, and a floor at one where the crack has escaped the notch’s influence entirely.

Now put the same flaw in two plates. A 0.2 mm weld-toe flaw is about the same size whatever the plate — it comes from the welding process, not from the thickness. In a 12 mm plate that is $a/t = 0.017$ and a magnification of 1.81. In a 100 mm plate it is $a/t = 0.002$ and a magnification of 3.50.

The flaw has not moved and the stress at it is nearly twice as large, because in the thick plate it is deeper inside a field that reaches further.

What the integration gives #

Integrate

$$N = \int_{a_0}^{t} \frac{da}{C\left[M_k(a/t)\,Y\,\Delta\sigma\sqrt{\pi a}\right]^m}$$

from a 0.2 mm flaw to the plate’s own thickness, at a fixed stress range, and convert the resulting lives into the stress ranges that would give equal lives. The answer is a power law with an exponent of 0.13.

Three things about that number.

It has the right form. Nothing in the integration assumes a power law; it is what comes out of a two-branch fit integrated over a range of thicknesses, and it is straight on log axes over a factor of eight in $t$. The design code’s form is not a convenient approximation to something else.

It has the right sign, and the sign is the whole question. The pale curve is the same integration with $M_k$ set to one — a crack growing through a uniform stress field from a fixed flaw to the plate’s thickness. That gives a life of 2.05 million cycles at 100 mm against 1.95 million at 25: the thick plate is very slightly better, because the crack has further to go and the extra distance is worth more than the higher $\Delta K$ at the end. The entire thickness effect is the toe’s stress gradient, and the intuition that a thicker plate is worse because the crack has further to travel is exactly backwards.

And it is two thirds of the design code’s. That gap is real and it is worth being precise about rather than fitting around. The exponent 0.2 is an empirical fit to test data on real details, and the crack-growth calculation here contains one mechanism out of at least three that contribute to it.

What the other third is #

The three candidates are all real and none of them is the statistical one.

The crack’s shape. This integration treats a through-thickness crack of depth $a$. A real weld-toe crack starts as a semi-elliptical thumbnail and grows in two directions with two different stress-intensity factors, and its aspect ratio evolves. Thicker plates let the thumbnail grow deeper before it breaks through, which changes the shape history and adds to the effect.

The residual stress field. A thicker plate takes more weld metal and more passes, and its residual stresses reach further and sit closer to yield over a larger depth. A crack in a thick plate spends more of its life in a tensile residual field, which raises the effective stress ratio and lowers the threshold — and the stress that was there before the load is already the reason a welded detail’s fatigue strength has no mean-stress term.

The specification of the weld itself. The rule assumes geometric similarity and welds are not perfectly similar: attachment sizes, toe angles and undercut do not scale exactly with the plate, and thicker plates in practice carry proportionally larger attachments.

And the statistical argument is the one that does least. A thicker plate has more weld length and therefore more chances of a bad spot, which is a real effect and a weak one — the life depends on the largest flaw, and the distribution of largest flaws grows very slowly with the sample. The figures below say how slowly.

Where the life is spent #

Both cracks spend most of their lives very shallow, which is the ordinary consequence of $\Delta K \propto \sqrt a$ and a cube law. What the magnification changes is how shallow.

The thick plate’s crack is at 0.2 per cent of the thickness on its first day and the thin plate’s at 1.7 per cent. Since the magnification is a function of that ratio, the thick plate begins in a field nearly twice as strong — and because most of the life is spent near the start, that factor applies to most of the life rather than to a corner of it.

The thickness effect is therefore a beginning effect. It is decided in the first few per cent of the crack’s depth and the first ninety per cent of its life, in a region no inspection has ever seen.

The same mechanism, doing something useful #

This is the figure that changes what the essay is about.

The essay that integrated the crack found that a fatigue life computed by crack growth is violently sensitive to the starting flaw: below a threshold length nothing grows at all, and just above it the life falls by a factor of three for a tenth of a millimetre. That was computed with a uniform stress field. Put the same crack in a weld toe’s magnified field and the sensitivity almost disappears — a flaw forty times larger costs a factor of two.

The mechanism is a cancellation. A smaller flaw has a smaller $\sqrt a$ and therefore a smaller $\Delta K$, which is why it lives longer. It also has a smaller $a/t$ and therefore a larger $M_k$, which works the other way. Over the range of flaws a weld actually produces, the two very nearly cancel.

That cancellation is what makes a detail category a number at all. A category is a single stress range attached to a shape, and it can only be that if the life does not depend strongly on the one quantity the shape does not fix. If life moved with flaw size the way a uniform-field calculation says it does, no table of categories could exist, and every weld would need its own calculation with its own inspection report attached.

So the toe’s magnification is not simply a penalty. It is the reason the whole S-N apparatus works — and the thickness effect is the small residue left over when the cancellation is not quite exact.

The whole of it, once, by hand #

Take the 25 mm and 100 mm plates, both with a 0.2 mm toe flaw, both at 80 N/mm².

At $a_0 = 0.2$ mm the 25 mm plate has $a/t = 0.008$, below the branch point, so

$$M_k = 0.51 \times 0.008^{-0.31} = 2.28,$$

and the 100 mm plate has $a/t = 0.002$ and $M_k = 0.51 \times 0.002^{-0.31} = 3.50$.

The stress intensities at that flaw are

$$\Delta K_{25} = 2.28 \times 1.12 \times 80 \times \sqrt{\pi \times 0.2} = 162, \qquad \Delta K_{100} = 3.50 \times 1.12 \times 80 \times \sqrt{\pi \times 0.2} = 249\ \text{N/mm}^{1.5},$$

so the initial growth rates differ by $(249/162)^3 = 3.6$.

That factor does not survive to the end — the magnification falls away as the crack deepens, and the thick plate has further to go — but it applies where most of the life is. The integration gives 0.74 million cycles for the 25 mm plate and 0.42 for the 100 mm, a ratio of 1.76, which at a slope of three is a strength ratio of $1.76^{-1/3} = 0.83$.

The design code says $(25/100)^{0.2} = 0.76$. Seven points of strength apart, on a calculation containing one mechanism.

How much of the answer is the flaw the calculation assumed #

The derived exponent is a function of the flaw the calculation is given, and the dependence is orderly: 0.150 at a 0.05 mm flaw, 0.134 at 0.2, 0.111 at 0.5 and 0.081 at 1 mm. The smaller the assumed flaw, the closer the answer comes to the design code’s 0.2.

That is a consistency check rather than a fudge, and it is worth reading as one. If the mechanism were wrong, changing the flaw would move the exponent in no particular direction. Instead it moves it monotonically and in the direction the mechanism predicts — because a small flaw is deep inside the notch field, where $M_k$ is large and varies fast, and a large flaw is near the edge of it where $M_k$ is nearly one and varies slowly.

It also says which way the remaining gap is likely to close. The flaws used in a fitted crack-growth assessment of a weld toe are usually taken between 0.1 and 0.25 mm, which is where this calculation sits; assuming a smaller one raises the exponent toward the design code’s without changing anything else. Whether the true effective flaw is smaller than 0.1 mm, or whether the remaining gap belongs to the crack’s shape and the residual field, is not decidable from this calculation — which is why the essay says two thirds rather than claiming the rule.

Where else geometric similarity does this #

The mechanism generalises beyond fatigue and it is worth naming the family, because the family is what makes the argument credible.

A notch’s field scales with the notch. The hole that multiplies the stress raises it by a factor that depends on the hole’s shape and not its size, and the raised region extends about a hole radius. Everything about a notch is a length ratio, which is why a stress concentration factor is dimensionless and a stress gradient is not.

Which is why a small notch is less damaging than its factor says. The notch the crack does not feel in full is the same observation from the crack’s side: a crack of a given size in a small notch’s field escapes it quickly, and in a large notch’s field does not. The fatigue thickness effect is that statement with the notch’s size replaced by the plate’s.

And it is why concrete’s size effect has a different exponent. A larger concrete beam is weaker by a mechanism that also involves a fixed length — the fracture process zone — competing with a scaling geometry, and Bažant’s law falls out of that competition with an exponent of its own. The two are the same kind of argument: a dimensionless answer needs two lengths, and a size effect is what appears when only one of them scales.

That is the deepest reading of the thickness rule. It looks like a rule about a dimension at right angles to the crack, and it is really a rule about the one length in the problem that does not scale — the flaw — measured against the one that does.

Which free body produced the number #

The free body is a crack of depth $a$ growing from a weld toe through a plate of thickness $t$, with $\Delta K = M_k(a/t)\,Y\,\Delta\sigma\sqrt{\pi a}$, $Y = 1.12$, and $M_k$ BS 7910’s two-branch fit for a toe floored at one. The integration runs from a fixed 0.2 mm flaw to $a = t$, in equal steps of $\log a$ because that is what a $1/\sqrt a$ integrand wants, and reports the cycle count.

Lives are converted to stress ranges through the growth law’s own exponent — $N \propto \Delta\sigma^{-m}$ at fixed geometry, so equal-life stress ranges scale as $N^{1/m}$ — and the exponent quoted is a least-squares fit of $\log(\text{strength})$ against $\log t$ over the seven thicknesses drawn.

The control is the same integration with $M_k$ identically one, which isolates the gradient from everything else the geometry does.

What the picture cannot show #

Two dimensions. The crack is a line of depth $a$ and a real one is a thumbnail with a length as well, growing under two stress-intensity factors that are not equal. That simplification is why the exponent is a lower bound on the effect rather than an estimate of it.

The weld that does not scale. The whole argument rests on geometric similarity, and a 100 mm plate with the same 6 mm fillet a 12 mm plate would have is not similar to anything. Such a detail has a weaker thickness effect and a worse category for a different reason.

Anything below the threshold. The integration stops if $\Delta K$ falls under 63 N/mm^1.5, which for these ranges and flaws it does not. A thinner plate at a lower range would, and its life would be unbounded — a discontinuity in thickness that no power law can represent.

And the first crack. Everything here is growth from a flaw that already exists. In a detail with no flaw — a machined surface, a ground weld — the initiation life dominates, thickness has almost no effect on it, and the correction is not applied.

The assumption that carries the argument #

That $a_0$ does not scale with $t$. The entire effect comes from holding the flaw fixed in absolute size while the stress field scales with the plate, and if the flaw scaled too — if a thick plate’s welding process left proportionally larger defects — the magnification at the starting flaw would be the same in every plate and the effect would vanish.

Is it true? Approximately, and for a reason rather than by luck. A weld-toe flaw is an undercut or an intrusion of slag at the toe line, and its size is set by the arc, the travel speed and the electrode — process variables that do not change much with the plate. What does change is the number of passes, so a thick plate has more toe length at risk rather than deeper flaws at each point.

That is the statistical argument arriving at the end rather than at the beginning, and it is a modest one: more chances at the same distribution of flaws, and the life depends on the worst of them. Doubling the length at risk moves the expected worst flaw by a few per cent, which moves the life by a few per cent — real, and an order of magnitude smaller than the gradient effect that this essay is about.

Still open: the cut-off that the sea removes #

The threshold has appeared in every essay about this detail as a fixed property of steel: below about $63\ \mathrm{N/mm}^{1.5}$, a cycle does nothing, and most of the traffic falls under the S-N curve’s matching cut-off. Every life computed here and in the two essays before it depends on it.

It is not a property of steel. It is a property of steel in air. A crack tip in seawater, or in a de-icing salt solution, or in anything that can reach it, has chemistry available at the crack tip that does not wait for a threshold — and the endurance limit that is the S-N curve’s most useful feature goes with it. What a spectrum does to a detail with no cut-off at all, how much of the ninety per cent of the traffic that did nothing comes back, and why a structure in the splash zone is designed by a different curve, is the question after this one.