The direction a plate was never tested in

A steel plate arrives with a certificate reporting a yield strength, a tensile strength and an elongation, all measured on a bar cut along the direction it was rolled in. Nothing on the certificate is untrue and nothing on it describes the direction that matters here.

Rolling does two things to a slab. It reduces the thickness, which is the point, and it stretches whatever the ladle left behind — sulphide and silicate inclusions, mostly — into flat stringers lying in the plane of the plate. The steel between the stringers is unchanged. What has changed is that a crack running through the thickness meets a plane of them every few tenths of a millimetre, and a crack running along the plate has to cut across them.

The strengths barely notice. The ductilities are three different numbers.

The measure is a ductility, and it converts exactly #

The quantity that decides this is not a strength but a reduction of area at fracture: how much narrower a tensile specimen’s neck is when it comes apart, as a fraction of where it started. It is quoted as a percentage — Z15, Z25, Z35 in the specification of through-thickness quality — and it looks like an arbitrary index until it is converted.

True strain is $\ln(A_0/A)$, so a reduction of area $Z$ is a true fracture strain of

$$\varepsilon_f = \ln\!\left(\frac{1}{1-Z}\right),$$

exactly and with nothing fitted. Fifteen per cent is 0.163. Sixty per cent is 0.916. A factor of four in the reported percentage is a factor of 5.6 in the strain the material can take before it comes apart, and the logarithm is what makes the gap so much larger than the numbers suggest.

That conversion is worth doing every time, because the percentages are close enough to look comparable and the strains are not. It is also the reason the property is quoted as a reduction of area rather than as an elongation: elongation depends on the gauge length it was measured over and reduction of area does not, so it is the one ductility measure that means the same thing on a specimen cut through a plate as on one cut along it. It also explains why a plate with a bad inclusion cluster — Z down at five per cent, which is a true strain of 0.051 — is in a completely different situation from one at fifteen, rather than being three times worse.

The demand is a shrinkage, and it is not elastic #

A weld is a volume of metal placed molten and cooled by six or seven hundred degrees while the plate around it holds it. It contracts and it is not allowed to, so the strain goes somewhere; in a T-joint pulling on the face of a plate, some of it goes through the plate’s thickness.

The natural way to compute that is the way this collection computes every other restrained contraction: a free strain $\alpha\Delta T$, multiplied by the fraction of it the surroundings refuse to allow. Doing so under-predicts the shrinkage a real weld produces by nearly an order of magnitude — a factor of 7.9 on the joint drawn here — and the discrepancy is a finding rather than an error.

The reason is that welding is not one elastic contraction. It is many thermal cycles, each of which yields the hot metal in compression on heating and leaves it shorter on cooling, so the movement accumulates plastically over the passes. The measured relation, established over sixty years and every process, is

$$\Delta \approx C \frac{A_w}{t}, \qquad C \approx 0.2 \text{ per mm},$$

with $A_w$ the deposited area. That is empirical and this page says so.

Why the ratio is what matters, and why it is squared #

Force the transverse shrinkage into the thickness of the plate being pulled on and the strain demand is

$$\varepsilon_z = \frac{\Delta}{t} = \frac{C A_w}{t^2}.$$

Two consequences, and the second is not obvious.

The deposited area goes as the square of the leg, so a fillet welded both sides deposits $2a^2$ per unit length and doubling the weld size quadruples the strain demand. There is no other variable in the joint with that leverage. A designer who specifies a larger fillet than the calculation needs, for comfort, has multiplied the through-thickness demand by four for a strength increase of two.

And the thickness appears squared. The 30 mm plate drawn, with a 12 mm throat both sides, takes a shrinkage of 1.92 mm and a strain demand of 5.4 per cent — a third of what an ordinary Z15 plate can supply. The same weld on a 60 mm plate demands a quarter of that.

So the arithmetic says thick plates are safer, and experience says thick plates are where lamellar tearing happens. Both are right, and the reconciliation is the next section.

Thickness helps the arithmetic and creates the mechanism #

The strain demand falls as $1/t^2$. The restraint rises with thickness, and restraint is what turns a shrinkage into a strain in the first place.

A weld on a thin, free plate simply pulls the plate; the plate moves and almost nothing is strained. The same weld on a thick plate in a stiff assembly cannot move anything, so the whole of the contraction is taken as strain in the material immediately around the weld. The restraint depends on the axial stiffness $EA/L$ of whatever is holding the joint, which grows directly with plate thickness and with the size of the assembly it is part of.

That is why the incidents are concentrated where they are: heavy column-to-baseplate joints, thick tension flanges welded to webs, beam-to-column connections in the corners of moment frames, and offshore nodes — all of them thick, all of them stiff, and all of them welded late in the sequence when nothing can move.

The two effects have opposite signs and no calculation combines them, which is why through-thickness quality is specified by rule rather than computed. The rules are written in terms of a combined index that adds up contributions for weld size, plate thickness, restraint and joint type, and it is an index rather than a mechanics because the mechanics has two terms that cancel to an unknown degree.

Which free body produced the number #

Cut a block out of the plate directly beneath a fillet weld, with the cut faces parallel to the plate’s surfaces.

Across the top face, the weld is pulling upward: not with an applied load but with its own contraction, which the restraint converts into a tension. Across the bottom face, the rest of the plate holds it. Vertical equilibrium of the block says the tension crossing the top and the bottom are equal, and the block is being stretched through its thickness — the direction with a fracture strain of 0.163.

There is no external load anywhere in that free body. The tension crossing it exists because a volume of metal wanted to be smaller than the space it was welded into, and the whole of the mechanism is contained in that sentence. It is the same body a restrained temperature change acts on and the same body a residual stress lives in; the only novelty is the direction, and the direction is the whole problem.

When the plate is bad, and the number that says so #

The ordinary plate drawn — Z15, so a fracture strain of 0.163 — supplies three times the 5.4 per cent demanded. That is not much of a margin for a quantity nobody measured, and it disappears completely on the plates where it matters.

A plate with a heavy inclusion cluster can return five per cent reduction of area through its thickness, which is a fracture strain of 0.051. The same joint then demands 1.06 times what the material can supply, and the arithmetic says it tears. Working the same calculation backwards gives the weld size at which each grade runs out: 20.8 mm of throat for the Z15 plate, and 12.0 mm for the bad one — exactly the weld drawn.

The one plate that is always at risk #

There is a joint geometry that puts every one of the aggravating factors together, and it is one of the most common details in structural steelwork.

A beam framing into a column flange, with a full-strength moment connection, pulls its tension flange horizontally. That pull goes into the column flange through a weld across the flange’s face — so the column flange is loaded through its own thickness by the beam’s tension flange force, which at full strength is the beam flange’s yield load and is therefore as large as it can be.

Every term is at its worst. The column flange is thick, so the restraint is high. The weld is large, because it is developing a flange. The pull is perpendicular to the plate surface by geometry rather than by accident. And the joint is welded into an assembly that is already erected, so nothing can move.

That is why through-thickness quality appears in the specification for moment-frame column flanges and rarely anywhere else, and why the alternative details — an extended end plate bolted rather than welded, or a stiffened connection that spreads the pull — are chosen as often for this reason as for the arithmetic of the connection itself.

The tell is that the crack is not in the weld #

A lamellar tear has a signature that distinguishes it from every other welding defect, and it is worth knowing because it identifies the mechanism from a photograph.

It is in the parent plate, not the weld, and it lies parallel to the plate’s surfaces — a flat separation a few millimetres below the fusion line, often stepped, joining one inclusion plane to the next by short vertical shears. The stepped appearance is the mechanism made visible: the tear runs along one weak plane until it runs out, tears vertically through good steel to the next one, and continues.

It also has a characteristic timing. It appears on cooling, before any load is applied, and it is frequently found days later rather than immediately, because the hydrogen that helps it along takes time to diffuse to the tip. A joint that passed its inspection can fail it a week afterwards, which is why the specifications for critical joints require the delay.

The remedy is geometry, and it costs nothing to think of #

The strain demand is through the thickness because the joint pulls that way. Change what the joint pulls on and the demand moves to a direction where the material has 0.916 rather than 0.163 — a factor of 5.6, bought by drawing the detail differently.

Weld to the edge rather than the face. A tension member landing on the end of a plate pulls along the rolling direction. Where the geometry allows it, this removes the mechanism completely rather than reducing it.

Butt rather than fillet. A full-penetration butt weld with the preparation on the through-thickness side puts the fusion boundary where the stringers are and consumes them, rather than loading them.

Buttering. Weld a layer of low-strength, high-ductility metal onto the plate face first, then make the joint onto that layer. The soft layer takes the strain the plate cannot.

Reduce the weld. Since the demand goes as the square of the leg, the cheapest single move is to specify no more weld than the force requires, which is the opposite of the instinct.

What it is bought with, when it is bought #

Steel can be specified with a guaranteed through-thickness reduction of area — Z15, Z25 or Z35 — and it is tested by cutting specimens through the plate and pulling them, which is the only way to know.

It is not free. The quality is achieved by controlling sulphur to very low levels and by calcium treatment to change the shape of what is left, so the inclusions come out as small spheres rather than as stringers. That is a steelmaking decision made months before the joint is detailed, and it means through-thickness quality has to be ordered, not discovered.

The practical consequence is a sequencing one. The decision is taken at procurement and the need is identified at detailing, and those happen in the wrong order on most projects. A connection that turns out to need Z-quality after the steel has arrived has two options, both expensive: change the detail, or accept a much larger inspection burden and hope.

The check that could fail #

An assertion that has never rejected anything proves nothing, so it is worth writing down what would falsify the model above.

The claim is that the demand goes as $A_w/t^2$ and the capacity as $\ln(1/(1-Z))$. That makes a prediction with no free parameter in it beyond the shrinkage coefficient: for a fixed plate, the throat at which a given grade runs out should scale as the square root of the fracture strain. Going from Z15 to Z25 raises the strain from 0.163 to 0.288, a factor of 1.77, so the permissible weld size should rise by 1.33.

That is a testable statement about a specification whose grades were arrived at empirically, and the fact that Z-grades come in steps of roughly that ratio is either a confirmation or a coincidence. It is worth being explicit that this page cannot tell which, because both the ratio and the grades have been rounded.

The same reasoning also refuses something. If the mechanism were a strength being exceeded rather than a ductility, the demand would scale with the restraint stress and would saturate at yield — so the weld size would stop mattering above a threshold. It does not: the incidence goes on rising with weld size well past the point where the restrained stress has reached yield, which is evidence for the ductility model and against the strength one.

Where the model stops #

The shrinkage coefficient is empirical. Everything about the shape of the result — the square of the weld leg, the inverse square of the thickness — follows from the geometry and is solid. The constant in front of it is a fit to measurements across processes and preparations, and it varies by a factor of two between them.

Restraint is not in the calculation at all. It is the term that decides whether a shrinkage becomes a strain, and it is represented here by a single fraction. Real assessments use an index summing several contributions precisely because the mechanics of restraint in a three-dimensional weldment is not tractable.

Hydrogen is left out. It lowers the effective ductility, it arrives from the process rather than from the design, and it is the reason preheat and low-hydrogen consumables appear in the same clause as through-thickness quality.

The three directions are treated as three numbers. They are the principal values of a property that varies continuously with angle, and a weld that pulls at forty degrees to the surface is asking for something between two of them by a rule this page does not have. The polar form is how the same property is handled one field over, where the anisotropy is expected.

And nothing here is a fracture mechanics. The tear is treated as a ductility being exceeded, which is what the measurements support. Where the plane of inclusions is sharp enough to behave as a crack, a fracture mechanics is the right instrument and this arithmetic is not.

Where the ladder goes #

Later rungs on this anchor: the restraint index and what its terms are fitted to. Through-thickness testing, and why a Z-value measured on one specimen says so little about a plate. Calcium treatment and inclusion shape control, which is where the property comes from. Hydrogen-assisted cracking, which shares the geometry and has a different mechanism. Ultrasonic detection of tears, which is difficult precisely because they are parallel to the surface the probe sits on. Joint details that eliminate the through-thickness pull, drawn as a family. The same anisotropy in castings and forgings, where the flow lines are curved. And the wider question this belongs to: which of a material’s properties are on its certificate, and which of the ones that are not decide the design.