Two welds, and the one that decides

The soft zone that takes all the stretch found that a heat-treated aluminium member welded across stretches only in the heat-affected zone, because the zone reaches its ultimate strength before the parent reaches its proof stress: a two-metre 6082-T6 tie that would extend 140 mm unwelded extends 12 mm welded. It closed on a member welded twice, and on a guess about it: “if the zones are identical, the neck forms in one of them and the other unloads with it, so the second weld adds nothing to the stretch and nothing to the strength.” It then asked whether the scatter between welds, rather than their number, decides how the member breaks.

The guess turns out to be half right, and the half that is wrong is the interesting part. The scatter does decide — but by taking away stretch that identical welds would have given, and by taking away strength that one weld would have kept.

Two identical zones stretch twice as far #

Two soft zones in series carry the same force. Each follows its own stress–strain curve, and if the two curves are the same, the two zones are at the same strain at every load. They reach their ultimate strength together, at the same load, and at that moment each has used its whole uniform elongation. The member’s extension at maximum load is therefore the parent’s small elastic stretch plus two zones’ uniform stretch rather than one.

For the 2 m 6082-T6 member with 60 mm zones that is 19.4 mm against one weld’s 12.3: the stretch beyond the elastic, 7.1 mm per zone, is exactly doubled. What the guess got right is what happens after. Once the load peaks, one zone begins to neck, since a neck is an instability and only one place can take it; the load falls, and the other zone unloads elastically along its own curve. So two identical welds double the stretch the member gives before its load starts to drop, and do nothing for the stretch in the neck. That is the first refutation.

The distinction is the one that matters for a structure. The ductility that depends on the ruler separated the uniform elongation, which is spread along a length and scales with it, from the necking elongation, which is local and does not. It is the uniform part that lets a member redistribute load to its neighbours, since a member whose load is falling has stopped helping; the necking part is what happens after it has stopped.

What the stretch is for #

Millimetres of stretch before the load drops are not a curiosity of a tensile test. They are what lets a member share. Two ties in parallel carrying one load divide it by stiffness until one of them reaches its capacity; after that, the other can pick up the difference only if the first can keep its load while it stretches. The bolts that do not share is the same question in a joint, where the end fasteners of a long row are loaded first and the row’s capacity depends on whether they can deform enough for the middle ones to catch up. And a frame designed to yield in an earthquake is asked for a displacement rather than a force, which a pushover turns into a demand on each member’s deformation before its load falls away.

So a welded aluminium member’s usable ductility is its uniform stretch, and for a heat-treated alloy that stretch lives in its zones. Anything that changes how many zones reach their ultimate together changes what the member can offer the structure around it.

A few per cent between them is most of it #

Welds are not identical. The heat input, the travel speed, the joint’s thickness and the time the metal spends above its dissolution temperature all vary from one weld to the next, and so does the strength of the zone they leave.

Suppose the second zone is 5 per cent stronger than the first — both its proof stress and its ultimate raised by the same factor. The member’s load is capped by the weaker zone’s ultimate, 185 N/mm². At that load the stronger zone is at 185 on a curve that peaks at 194, and it has used only part of its uniform strain.

How much is set by how flat the top of the curve is. The zones here harden by a power law, with plastic strain growing as the stress to the power $n$; for the 6082-T6 zone, with a proof stress of 125 N/mm², an ultimate of 185 and a uniform elongation taken as 12 per cent, $n = 10.4$. A zone whose ultimate is $1 + g$ times the weaker’s therefore reaches

$$\left(\frac{1}{1+g}\right)^{n} = \left(\frac{1}{1.05}\right)^{10.4} = 0.60$$

of its own uniform plastic strain when the weaker zone peaks.

The second weld’s contribution falls fast with the gap. At 3 per cent it gives 74 per cent of its stretch; at 7 per cent, half; at 15 per cent, under a quarter. The flatter the top of the curve, the less a second zone gives for a given difference in strength. A high exponent is the signature of a material that hardens little near its ultimate, and aluminium heat-affected zones are such materials: most of their strain happens in the last few per cent of their strength.

More welds, more stretch, by less than the count #

A member with many welds — a truss chord spliced every few metres, a frame of extrusions welded at every node — has many soft zones in series, and what they give together depends on how their strengths are spread. Draw each zone’s strength from a normal distribution with a given coefficient of variation, many times over, and average what the member does.

With no scatter every weld adds its full stretch, and ten welds give ten times one weld’s. With zone strengths scattered by 5 per cent, two welds give 1.6 times and ten give 5.1. At 8 per cent, 1.5 and 3.7. So the number of welds does add ductility, and more than the guess allowed, but the scatter takes a larger share of it with every weld added, because the strongest zones in a larger group are further above the weakest.

The rise is worth having. A member that must stretch — a tie that has to share load with a parallel one, a brace that must yield before a connection breaks — is more ductile with several welds than with one, on average. The average hides the individual member, though: the stretch depends on how close the second-weakest zone happens to be to the weakest, and that varies from member to member far more than the mean does.

The mean and the member #

The mean hides how much one member differs from the next. For two welds at 5 per cent scatter, the member’s stretch depends only on how far apart its two zones happen to be, and the difference between two independent draws is itself spread: in the seeded draws, one member in ten gets less than 1.31 times one weld’s stretch, the median member 1.61, and one in ten more than 1.91. The same design, built twice, can give one member half as much again of usable stretch as the other.

With ten welds the spread widens, from 3.65 to 6.50 times one weld’s stretch between the tenth and ninetieth members: over half the median, where two welds’ range was under two fifths of it, because each member’s stretch now depends on nine gaps above its weakest zone rather than one.

That spread is what a designer relying on a member’s ductility has to live with. The load that is never all there at once showed a sum of scattered quantities becoming more predictable as it gains terms; the stretch of a member with many welds does the same, since ten zones’ gaps average out more than two do. What does not average out is the weakest zone, because a minimum does not converge the way a sum does. Many welds make the member’s stretch more predictable and its strength less.

Every weld is another chance to be the weakest #

The other half of the question is strength, and here the guess was simply wrong. A member with several soft zones in series is as strong as its weakest zone. If the zones scatter, the weakest of two is on average weaker than one zone; the weakest of ten weaker still.

For two zones the expected shortfall has a closed form: the expected minimum of two standard normal values is $-1/\sqrt{\pi}$, so the weaker of two zones scattered by 5 per cent averages $1 - 0.05/\sqrt{\pi} = 0.972$ of the nominal strength. The seeded draws give the same. With ten welds the mean falls to 0.92, and the value 95 per cent of members exceed — the one a characteristic strength is built on — falls from 0.92 for a single weld to 0.87.

That is the bigger one is the weaker one in a form that has nothing to do with size: Weibull’s weakest-link argument, where a longer chain is weaker because it has more links to choose its weakest from, applies to welds in series exactly as it applies to flaws in a volume. The heat-affected zone’s strength in design rules is a property of one zone, measured on specimens with one weld; a member with ten welds has drawn ten times from that distribution, and its strength is its unluckiest draw.

Where the neck forms #

There is a practical consequence that the averages do not show. In a member with several welds, the neck forms at whichever zone happened to be made worst, and nothing in the drawing says which that is. Two nominally identical welds at the ends of a tie will neck at one end or the other depending on a difference of a few per cent that no inspection measures. A connection detailed to be the member’s fuse — the place where it is meant to yield first — is only reliably the fuse if its zone is weaker than every other zone in the member by more than the scatter.

That changes how a designer controls where an aluminium structure fails. It cannot be done by making one weld slightly weaker than the others, since a few per cent is inside the scatter. It has to be done by making every other zone clearly stronger: a thicker land at those welds, as the earlier essay found restores strength, or no weld at all where the member must not neck. The strength the welder gives back is a strength per weld; its scatter is a property of the fabrication, and the design has to allow for both.

Why aluminium, and not steel #

None of this arises for a structural steel weld made to a normal procedure. There the weld metal is overmatched — deliberately stronger than the parent — and the heat-affected zone of an ordinary grade is, if anything, harder than the parent beside it. The zones are not the weak links, the parent carries the stretch along its whole length, and the number of welds does not enter. The earlier essay’s section on steel found the exceptions, the quenched and tempered grades whose zones are softened by welding, and those are exactly the grades for which this essay’s series argument returns.

Heat-treated aluminium is the other way round by nature. Every weld dissolves the precipitates the temper was made from, every weld leaves a zone at about half the parent’s proof stress, and the zones, not the parent, are the member’s weak links. So a welded aluminium member is a chain whose links are all its welds, and the chain’s length is the number of welds rather than the member’s length. That is why a spliced chord behaves differently from one welded only at its ends, even when every weld passes its own check.

What the rules take as given #

EN 1999-1-1 designs a heat-affected zone with reduction factors on the parent’s strength — $\rho_{o,haz}$ on the proof stress and $\rho_{u,haz}$ on the ultimate — which for 6082-T6 are the 0.48 and 0.60 behind the 125 and 185 N/mm² used here. They are characteristic values, set so that the great majority of zones exceed them, and they are the same for a member with one weld and a member with twenty. The partial factor applied to a heat-affected zone’s ultimate then covers, among other things, the scatter of the zone itself.

What it cannot see is that a member with several zones takes the minimum of several draws. The 5 per cent value falls from 0.92 of nominal for one weld to 0.87 for ten — a shortfall of about 5 per cent that is inside the partial factor’s reach and is spent from it without being named. For most members that is fine, since the partial factor has room. For a member whose design depends on reaching its heat-affected strength at a known place — a capped force is the extreme case, where the whole point is a resistance that is neither lower nor higher than intended — the fuse needs a margin over every other zone in the member larger than the scatter, and the rule that sets the zone’s strength says nothing about that margin.

By hand #

The member is 2 m of 6082-T6 with an elastic modulus of 70,000 N/mm², and each zone is 60 mm long. At the zone’s ultimate of 185 N/mm² the whole member’s elastic stretch is $185/70{,}000 \times 2{,}000 = 5.3$ mm. One zone’s plastic stretch is its uniform plastic strain, $0.12 - 185/70{,}000 = 0.117$, times 60 mm, which is 7.0 mm; with the parent’s few microstrain of plasticity the single-weld member reaches 12.3 mm. Two identical zones add another 7.0, for 19.4. A second zone 5 per cent stronger adds $0.60 \times 7.0 = 4.2$, for 16.6. And the weakest of two zones scattered by 5 per cent averages $185 \times 0.972 = 180$ N/mm².

Where the model stops #

Each zone is a uniform piece of one material. A real heat-affected zone is graded, softest near the fusion line and recovering over its width, and the strain concentrates in the softest millimetres. The model’s 60 mm of uniform zone is a stand-in for that gradient, and it overstates the stretch a real zone gives.

Strength scatter is all the scatter. The proof stress and ultimate are scaled together and the uniform elongation is held constant. Real zones scatter in ductility as well, and a zone with low uniform elongation adds little stretch even when it is weakest.

The zones’ strengths are independent. Welds made by the same welder, on the same day, on the same thickness are correlated, and correlated zones behave more like identical ones: the stretch is closer to the count and the strength closer to one zone’s. The independent case is the pessimistic one for strength and the optimistic one for the spread between members.

What the pictures cannot show #

That the scatter is a property of a fabricator, not of an alloy. The coefficient of variation used here, 5 per cent, is an assumption, and a controlled process with qualified procedures and consistent heat input will do better than one welding in the field. The alloy and its temper fix the mean strength of a zone; the shop fixes how much zones differ, and with it how much of the member’s ductility and strength survives having more than one.

Still open: the zone the load crosses at an angle #

Everything here pulls the member along its length, straight across every weld. A longitudinal weld runs along the member instead, and its soft zone is in parallel with the parent rather than in series with it: the two strain together, and the soft zone cannot take all the stretch because it cannot stretch more than the metal beside it. Whether a zone in parallel costs any ductility at all, and what happens to a weld that crosses the load at an angle and is partly in series and partly in parallel, is the next question.