The toughness that belongs to the plate

The crack between the two checks found that the usual check for a cracked plate — the lower of a fracture calculation that ignores yielding and a yield calculation that ignores the crack — overstates the plate by up to 23 per cent, at the crack length where the two lines cross. Dugdale’s strip-yield model fills the corner, and in units of that crossing length its curve is the same for every steel.

That essay used one toughness for each steel, and closed by asking whether that is right. Toughness is measured on test pieces of a standard thickness, and a real structure is made of plates of many thicknesses. The essay on the transition had already found that a thicker plate of the same steel is more brittle, and turned the thickness into a shift of the transition temperature. This essay puts that thickness back where the crack is: into the strip-yield assessment, beside the 23 per cent, to find whether the two effects are separate corrections or one.

A crack front that samples #

A crack through the thickness of a plate has a front — the line along which the crack will advance — as long as the plate is thick. In the ductile-to-brittle transition of a ferritic steel, cleavage starts at a single spot on that front: whichever carbide or inclusion is worst placed in the small volume ahead of it where the stress is highest. The toughness a test measures is the stress intensity at which the weakest spot in the test piece’s front gives way.

A longer front passes through more candidates, so it is more likely to contain a weak one. That is a weakest-link size effect, the same statistics that make a long chain weaker than a short one of the same links, applied along a line. Its form is set by how the toughness of one link scatters: in the master curve that the transition is now described by, the toughness of a 25 mm front has a Weibull distribution with a shape of 4 above a floor of 20 $\mathrm{MPa}\sqrt{\mathrm{m}}$, and the rule for any other front length $B$ follows at once:

$$K(B) = 20 + \big(K(25) - 20\big)\left(\frac{25}{B}\right)^{1/4}.$$

The steel used throughout is a grade 355 at −20 °C whose reference temperature — the temperature at which a 25 mm front has a median toughness of 100 $\mathrm{MPa}\sqrt{\mathrm{m}}$ — is −38 °C. Its assessments use the 5 per cent toughness, the value one test in twenty would fall below.

The quarter power is gentle, and it is still a large effect in the range structures use. A 12 mm web and a 60 mm flange of one girder, made from the same heat of steel and certified with the same test, differ in toughness by a third. The crossing length, which goes as the square of the toughness, differs by 80 per cent — as large a change as moving between the grades of steel that the crossing length was first used to compare, made here within one grade by a dimension of the plate.

Why a quarter #

The exponent is not fitted separately; it is the Weibull shape of the toughness scatter, turned into a length. If each 25 mm of front survives a stress intensity $K$ with probability $\exp\{-[(K - K_{min})/(K_0 - K_{min})]^4\}$, a front made of $B/25$ such lengths survives only if every one of them does, with probability $\exp\{-(B/25)[(K - K_{min})/(K_0 - K_{min})]^4\}$. That is the same distribution with its scale divided by $(B/25)^{1/4}$. A shape of 4 means the toughness scatter is moderate — the 5 per cent value about half the median above the floor — and a moderate scatter means a gentle length effect: sixteen times the front length for half the toughness above the floor.

The floor is the other half of the rule. Below about 20 $\mathrm{MPa}\sqrt{\mathrm{m}}$ no cleavage is observed however long the front, because a crack cannot extend by cleavage without first loading a region of steel larger than any single weak spot. The weakest-link term acts only on the toughness above the floor, which is why the thickest plates approach 20 slowly and never reach it.

Three plates of one steel #

Each curve is the strip-yield curve of the previous essay, drawn with the toughness of its own plate. All three begin at the yield stress for very short cracks and follow the fracture line for long ones, and each rounds the corner between the two lines at its own crossing length. Read at one crack, 20 mm, they spread from 267 N/mm² for the web to 214 for the flange — a difference of 25 per cent, the same size as the whole strip-yield correction. So the answer to the question the previous essay closed on is that the two are the same size, and whether they add or cancel depends on which side of the specimen’s thickness a plate lies.

The simple check sees none of this. It uses the certificate’s toughness, measured at 25 mm, and returns 274 N/mm² for a 20 mm crack in all three plates.

One error, or two that meet #

At the specimen’s own thickness the ratio is the previous essay’s curve exactly, peaking at 1.23. For every other thickness a second factor multiplies it: the simple check has the specimen’s toughness in it, and the plate has its own. In a thick plate the plate’s toughness is lower, the two factors are both above one, and the errors compound: 36 per cent at 60 mm, 45 at 100. In a thin plate its toughness is higher, the second factor is below one, and it pays back part of the first: the web’s worst error is 14 per cent, and for cracks longer than about twice the crossing length the simple check becomes conservative, because the extra toughness of a short front is by then worth more than the strip-yield penalty costs.

All the peaks sit at the same crack, 11.9 mm, because that is where the simple check’s own corner is — the 25 mm toughness’s fracture line meeting the yield stress. The corner is a property of the check rather than of any plate, and every plate’s real curve passes below it by an amount that depends on how tough that plate is.

Collected over thickness, the worst error rises steadily. It is the strip-yield constant, 1.23, only at 25 mm, and the reason is not that 25 mm is special to the steel. It is special to the test. A correction that looked like a universal constant of the model is a constant of the model at the thickness the toughness was measured at.

The scatter moves too #

The 5 per cent toughness is a choice about how cautious to be, and the weakest-link rule moves the whole distribution, not a single fractile of it.

The two curves are nearly the same shape, shifted: the flange reaches each probability at a stress 40 to 55 N/mm² lower than the web. Near the yield stress the web’s curve steepens, because the strip-yield stress intensity grows without limit as the applied stress approaches yield, and there the difference between the plates narrows. At a working stress well below yield it is the full shift.

This is also a reminder of what the 5 per cent value is. The flange’s median — the stress at which half of such flanges would cleave — is 302 N/mm², above the web’s 5 per cent value of 268. A web and a flange are not a strong plate and a weak one; they are two overlapping populations, and the thickness moves one of them bodily toward lower stresses.

The crack each plate will tolerate #

Assessment is usually run the other way: at a known stress, how long a crack can the plate carry?

At 250 N/mm² the simple check with the certificate’s toughness allows a 24 mm crack in any plate of this steel. The strip-yield model with each plate’s own toughness agrees for a 12 mm web — the two errors cancel almost exactly there at this stress — and allows only 13 mm in a 60 mm flange, little more than half. A crack found in an inspection at 18 mm is acceptable in the web and unacceptable in the flange of the same girder.

That difference matters more than the stress figures suggest, because tolerable crack length is the quantity an inspection regime is set from. A damage-tolerant design inspects often enough to find a crack between the size an inspector can detect and the size the section can survive. Halve the second and the interval halves with it, or the detectable size has to fall.

A crack that walks from the web into the flange #

The web and the flange are not separate structures. A fatigue crack that starts at the toe of a stiffener weld on a girder’s web grows through the web, and if it is not found it reaches the flange — and the moment it turns into the flange its front stops being 12 mm long and becomes as long as the flange is thick. In the terms of the last figure, a crack tolerable at 24 mm in the web is, a few millimetres later, a crack in a plate that tolerates 13.

That is the arrangement in which the thickness effect is least forgiving, because nothing about the crack has changed except the plate it is in. An inspection interval set from the web’s numbers, or from the certificate’s, is set for a crack that becomes more dangerous the moment it reaches the part of the girder that carries most of the bending.

Penalised twice, by two rules that point sideways #

Fatigue design already charges a thick plate for its thickness. Every fatigue code puts the same detail in a thicker plate into a lower category, by a factor of $(25/t)^{0.2}$ — a rule about a dimension at right angles to the crack, which turns out to come mostly from the way a weld toe’s raised stress field scales with the plate, and partly from statistics of the same weakest-link kind as here. Both rules are normalised to 25 mm, and both penalise thickness with a small power.

They act at opposite ends of a crack’s life. The fatigue rule says a crack in a thick plate arrives sooner. The fracture rule says it can be allowed to grow less far before it is fatal. A damage-tolerant design lives in the window between the crack an inspector can detect and the crack the section can survive, and the window closes from both ends as the plate gets thicker: the growth is faster, and the end of the window is nearer. A 60 mm flange has a shorter window than a 12 mm web by more than either rule alone suggests.

What a steel’s reference temperature would have to be #

The weakest-link term can be bought off with a tougher steel. The flange’s 5 per cent toughness at −20 °C equals the 25 mm specimen’s at a temperature 12.8 °C colder, and the web’s at a temperature 10.5 °C warmer. So to give the 60 mm flange the crack tolerance the 12 mm web has, its steel’s reference temperature has to be about 23 °C lower than the web’s — roughly one step between the sub-grades a steel is ordered by, whose Charpy temperatures are set 20 °C apart.

That is the sense in which the tables that limit thickness by sub-grade already contain this essay’s result. What they do not do is carry it into an assessment: a girder whose web and flange were both ordered as the same sub-grade, as nearly all are, has a flange with a quarter-power less toughness, and the check that is run when a crack is found in it has to put that back in by hand.

Where the rules already know this #

None of this would surprise the rules for choosing a steel. The tables that set the thickest element a steel sub-grade may be used in, at a given lowest service temperature, allow a thinner plate at a colder temperature than a thick one, and the transition essay found the reason: thickness shifts the transition curve as a temperature would. What those tables settle is which steel to buy. They say nothing about how to assess a crack once it has been found, and there the practice is to use a toughness — often the certificate’s, or a value derived from its Charpy energy — in a check that knows nothing about thickness unless the assessor puts it in.

The fracture assessment procedures do provide for it. The master-curve route in them carries exactly the quarter-power adjustment used here, and requires the toughness to be adjusted to the length of the crack front being assessed rather than taken from the test piece. The adjustment runs both ways: it lowers the toughness of a thick section and raises that of a thin one. The figures above are what the adjustment is worth once it is put into the same calculation as the strip-yield correction.

The flange by hand #

For the 60 mm flange at −20 °C with $T_0 = -38$ °C: the 25 mm Weibull scale is $31 + 77\,e^{0.019 \times 18} = 139.4$ $\mathrm{MPa}\sqrt{\mathrm{m}}$, and at 60 mm it is $20 + 119.4 \times (25/60)^{1/4} = 115.9$. The 5 per cent toughness is $20 + 95.9 \times (-\ln 0.95)^{1/4} = 20 + 95.9 \times 0.476 = 65.7$ $\mathrm{MPa}\sqrt{\mathrm{m}}$, or 2,077 $\mathrm{N/mm^{1.5}}$.

For a 20 mm edge crack with $Y = 1.12$, the strip-yield failure stress is $\sigma_f = (2\sigma_Y/\pi)\arccos(e^{-R})$ with $R = \pi K^2/(8 Y^2 \sigma_Y^2 a) = \pi \times 2{,}077^2/(8 \times 1.254 \times 355^2 \times 20) = 0.536$, so $\sigma_f = 226 \times \arccos(0.585) = 226 \times 0.946 = 214$ N/mm². The simple check with the 25 mm toughness of 76.8 $\mathrm{MPa}\sqrt{\mathrm{m}}$, 2,428 $\mathrm{N/mm^{1.5}}$, gives $2{,}428/(1.12\sqrt{20\pi}) = 274$ N/mm², below yield, so 274 is its answer. The ratio is 1.28 at this crack.

A statistical front and a plane-stress strip #

The weakest-link rule is the whole of the thickness effect here. It is statistical: a longer front is less tough because it samples more, not because it is more constrained. The other reason thick plates are less tough — constraint, the triaxial stress at the tip of a crack in a thick section, which suppresses the yielding that would blunt it — is built into the master curve only at the standard specimen’s constraint. A thin plate, yielding through its thickness at the crack tip, is less constrained than the specimen and tougher again than the weakest-link rule says; the web’s numbers are therefore on the cautious side, and the flange’s are about right.

The strip-yield model is a plane-stress model. It assumes the yielded zone is a thin strip in the plane of the plate, which is what a thin plate does and a thick one does not. Its use for a 60 mm flange, as in every assessment procedure built on it, is an approximation that the procedures accept because the failure assessment line it produces is close to the ones measured on thick sections.

The steel is in its transition range at the temperature chosen. On the upper shelf, where the steel tears rather than cleaves, the weakest-link rule does not apply, and a thicker plate is not less tough by this mechanism.

What the pictures cannot show #

That the reference temperature is itself a measurement with an uncertainty of several degrees, and a few degrees of reference temperature are worth as much toughness as a doubling of thickness. Every curve here would move sideways by a few millimetres of crack if the steel’s $T_0$ were known a little differently, and the figures’ separation between plates is only meaningful because all of them share that uncertainty.

Nor can they show where along a real crack front the weak spots are. A crack running from a weld toe in a thick flange has a front that crosses the weld metal, the heat-affected zone and the parent plate, each with its own reference temperature, and the weakest-link rule then applies to a front whose links are not alike.

Still open: the front that is not straight #

Every crack here is a straight through-thickness crack whose front is the plate’s thickness. Most cracks found in service are not: a fatigue crack from a weld toe is a semi-elliptical surface crack whose front is a curve along which the stress intensity varies, largest at the deepest point for some shapes and at the surface for others. A weakest-link front of that kind is sampled unevenly, the most highly stressed part counting for more than its length. How much of a surface crack’s front the thickness adjustment should count, and whether the adjustment then moves the answer by as much as it does for a through crack, is the question this one leaves for the cracks inspections actually find.