The weld that is weakest at fifty-five degrees

A weld in a heat-treated aluminium member leaves a soft zone beside it, and every member drawn so far in this series has pulled straight across that zone. The soft zone then takes all of the stretch: it necks while the parent beside it is still elastic, and the member’s ductility is the zone’s width times its strain. With two welds the weaker decides. In each calculation the zone carried its own uniaxial strength, 185 N/mm² for a 6082-T6 zone against the parent’s 310.

Welds are not all square to the load. A plate girder’s web-to-flange welds run along it; a gusset’s welds meet a brace at whatever angle the geometry gives; a spiral-welded tube’s seam crosses its axis at fifty to seventy degrees. The question the series left was what a zone does when the load crosses it at an angle, and what it does when the load runs along it — partly in series and partly in parallel.

The answer turns on one fact about how a soft band can fail, and the fact puts the weakest weld somewhere nobody would have drawn it.

A soft band necks along its own line #

A soft zone between two strong parents can fail on its own in only one way: it necks along its own length, while the parent on either side is still well below yield. The parent on either side is then nearly rigid. Whatever the zone does, the two faces it is welded to cannot change their length along the weld line, so the zone cannot stretch along its own line. It may thin, it may open across, it may shear along — but its length along the weld is held.

That single condition is the whole mechanism. It was written down by Hill for the localised neck in a sheet, and it applies to a soft zone with no change, because a localised neck is exactly a narrow band that deforms while the material beside it does not.

Take the zone at an angle $\varphi$ to the load. A stress $\sigma$ along the member puts a tension $\sigma \sin^2\varphi$ across the zone and a shear $\sigma \sin\varphi \cos\varphi$ along it. Those two are fixed by equilibrium, because they cross the zone’s faces from the parent. The third stress — tension along the zone’s own line — is not: it is whatever the zone needs to keep its length, and a von Mises material with no strain along a direction carries, along that direction, half the stress across it. With that, the yield condition in the zone’s own axes is

$$Y^2 = \tfrac{3}{4}\,\sigma_n^2 + 3\,\tau^2,$$

and putting the two stresses in gives the member’s stress at which the zone necks:

$$\sigma = \frac{Y}{\sqrt{\sin^2\varphi \, \left(3 - \tfrac{9}{4}\sin^2\varphi\right)}}.$$

Straight across, the neighbours help #

Put $\varphi = 90°$ and the zone is straight across the load. All of the stress is tension across it, none is shear, and the expression gives $\sigma = 2Y/\sqrt{3}$: 1.155 times the zone’s own strength, 214 N/mm² for a zone that alone carries 185.

The extra fifteen per cent is the zone’s neighbours at work. A strip of soft metal pulled uniaxially contracts sideways as it stretches, and that sideways contraction is part of how it yields. Welded between two strong parents, a narrow transverse zone cannot contract along the weld, because the parent on either side does not; it is in plane strain, and a metal in plane strain needs $2/\sqrt{3}$ times its uniaxial stress to yield. Every number in the earlier essays took the zone at its uniaxial 185, which is the conservative reading and the one the design rules make.

The gain belongs to a narrow zone. Near a free edge the neighbours cannot hold the zone’s length, and a zone that is wide compared with the plate is not held at all, so a real member gains somewhat less than the full fifteen per cent.

At fifty-five degrees, nothing #

Turn the zone away from square, and the stress across it falls while a shear along it appears. The expression’s denominator, $\sin^2\varphi\,(3 - \tfrac{9}{4}\sin^2\varphi)$, rises from 0.75 at 90° to a maximum of exactly one at $\sin^2\varphi = 2/3$ — $\varphi = 54.7°$ — and falls again.

At that angle the member necks at exactly the zone’s uniaxial strength, 185 N/mm². The reason is that 54.7° is the angle along which a free sheet in uniaxial tension already has no strain: a strip pulled uniaxially contracts sideways by half its stretch, and along one particular direction the two exactly cancel. That direction is at $\arctan\sqrt{2}$ to the load, 54.7°. A soft zone lying along it is asking the parent for nothing — the parent would not have to stretch along that line even if it were yielding — so the zone necks exactly as if it were a free strip, at its own strength.

The weakest place to put a soft zone is not straight across the load. It is at 54.7 degrees to it, where a transverse weld’s fifteen per cent of constraint has been spent and nothing has replaced it.

The whole curve is a shallow trough with its floor at 54.7°. From straight across to the floor the member loses fifteen per cent; on the far side of the floor the strength climbs steeply, because the zone is lying closer and closer to the load and less and less of the stress crosses it. At 40° it carries 200 N/mm²; at 30°, 237.

Shallow enough, and the zone cannot neck alone #

As the zone lies down toward the load, the stress its own neck needs grows without limit, since a zone lying along the load would have to neck while carrying none of it across. At some angle the zone’s neck needs more than the member can carry through any other route, and the zone stops being the place the member fails.

The other route is a neck across the parent that takes the zone’s share of the section with it — the way the member fails when the zone runs straight along it. With the zone a quarter of the section, that member carries 278 N/mm². Below 24.2 degrees, the zone’s own neck needs more than 278, and the member reaches 278 instead: an inclined weld that shallow behaves, for strength, as if it ran along the member.

The critical angle depends on how soft the zone is. A zone at half the parent’s strength must lie steeper than 20° before it can neck alone; 6082-T6’s zone, at 0.60, steeper than 24°; a zone at nine tenths of the parent, steeper than 40°. The softer the zone, the shallower the weld that it governs. The weakest angle does not move at all: whatever the zone’s strength, its worst orientation is 54.7°, because that angle is a property of how metal yields and not of how much it has lost.

A steel weld, whose heat-affected zone is usually no softer than its parent, never governs at any angle by this mechanism, which is why steel design rules have no reason to ask the question. It is an aluminium problem — and a problem for every alloy that gets its strength from heat treatment and loses it to a welding arc.

Along the member, the zone stretches with the parent #

At $\varphi = 0$ the zone runs along the member, and the question changes. There is no band to neck along. The zone and the parent sit side by side across the section, both stretch together because they are one piece of metal, and neither can neck while the other still holds.

That last clause is the whole difference from series. In series, the weakest zone reached its ultimate first and necked, and the rest of the member stopped stretching. In parallel, a zone that would have necked alone at some strain cannot, because the parent beside it is still hardening and refuses to thin locally; the member necks only when the sum of the two materials’ forces stops rising.

The two materials peak at different strains. The parent alone would reach its greatest load at 7.0 per cent and start to neck; the softer zone, which work-hardens for longer, would reach its greatest load at 12.0 per cent. Joined, the member peaks between them, at 7.8 per cent. The soft zone running along the member has made the member more ductile, not less — by about an eighth, in the uniform elongation that sets how far it stretches before it necks.

The price is strength, and it is close to the area share. With a quarter of the section soft, the member carries 278 N/mm², 90 per cent of the parent’s. The simple mix of the two ultimates would be 279; the member falls slightly short because, at the strain where the member peaks, the zone has not yet reached its own ultimate and the parent has just passed its own.

The trade holds at every share. At half the section the member keeps 80 per cent of the parent’s strength and 126 per cent of its uniform elongation; at three quarters, 70 and 146. The strength line hugs the area mix and the elongation line climbs steadily toward the zone’s own. A longitudinal weld costs exactly what the earlier essay priced it at — the softened area times what it lost — and nothing else.

The scarf joint, and where its rule stops #

There is an old rule that says the opposite of this. A joint that is weak across its own plane — a glue line, a splice in timber, a brittle braze — is made stronger by cutting it at a shallow slope, a scarf, so that the load crosses it obliquely and the stress across the joint is only $\sin^2\varphi$ of the stress in the member. Every halving of the slope’s angle buys more, and the rule holds for as long as the joint fails by being pulled apart.

A ductile soft zone does not fail by being pulled apart. It fails by flowing, and its flow is driven by the shear along it as much as by the tension across it. Incline it and the tension falls, but the shear that was zero across the load rises to meet it — and until the angle is shallow, the shear wins.

The crossover can be found exactly. A zone at angle $\varphi$ matches the transverse zone’s strength when the term under the square root equals its transverse value of three quarters, which happens at $\sin^2\varphi = 1/3$: 35.3 degrees. Steeper than that, an inclined weld is weaker than a square one; between 35.3° and 90° every inclination costs strength, and the worst, at 54.7°, costs the whole fifteen per cent. A scarf at 1 in 2 — about 27° — gets back to 259 N/mm², 21 per cent above the square weld, and only because it is shallow enough to have crossed the line.

So the scarf rule is right for a joint that is brittle across its plane and wrong, over most of its range, for a joint that is soft. A designer who inclines an aluminium butt weld to 45° in the belief that a scarf is always kinder has made the weld 11 per cent weaker than leaving it square — 191 N/mm² against 214.

Why the same zone does such different things #

Three welds, one alloy, one zone, three behaviours.

Straight across, the zone is in series with the parent and necks alone, and its neighbours hold its length and lend it fifteen per cent. The member’s strength is a little above the zone’s; its ductility is the zone’s width times the zone’s strain, which is the small number the earlier essays found.

At 54.7°, the zone is still in series and still necks alone, but along the one line where a free strip has no strain the neighbours have nothing to resist. The member’s strength is the zone’s own, the lowest of all, and its ductility is no better.

Along the member, the zone is in parallel, cannot neck alone, and strains with the parent. The member’s strength is the area mix and its ductility rises.

The three are the same metal under the same load. What differs is which deformation the parent will allow, and the parent allows exactly the deformations that do not require it to stretch along the weld. This is the material that has a direction appearing from geometry instead of from rolling: a welded aluminium member’s strength has an orientation, and it is the weld’s.

The assumptions underneath #

The parent is rigid while the zone necks. That is true while the parent is elastic, which it is at 185 N/mm² against its 260 proof stress, but the parent’s elastic stretch is not zero, and a parent that gives a little lets the zone’s length change a little and the constraint relax. The gain straight across is therefore an upper bound.

The zone is narrow. Plane strain along the weld needs the weld to be long compared with the zone’s width, and a short weld, or the ends of a long one at the plate’s free edges, is not held. For a zone 30 mm wide in a 200 mm plate the middle of the weld is held and the edges are not.

The plate is thin. The analysis is in plane stress, the plate free to thin through its thickness. A zone narrow compared with the plate’s thickness is held through the thickness too and gains more; that is the constraint that makes an undermatched weld in thick plate stronger than its weld metal, and it is a separate calculation.

And the strengths are ultimates. The band neck is a failure at the greatest load, and the comparison is between ultimate strengths: 185 for the zone, 310 for the parent. The design rules work in proof stress, and a zone at an angle reaches its proof stress by the same factors, but the rules do not credit the constraint at any angle — and, the figure above says, should not credit a transverse weld’s gain to a weld that is not transverse.

What the strip drawings do not show #

They do not show the parallel route’s geometry. Between 0 and 24° the member’s strength is taken as the parallel member’s, a quarter of the section soft. A real inclined weld puts a different share of the zone into each cross-section, and the true strength in that range depends on the plate’s width and the zone’s length as well as its angle; the figure’s flat shoulder is the right height at 0 and an estimate elsewhere.

They do not show where the neck starts. A band neck forms along the whole zone at once in the analysis. In a real plate it starts where the constraint is weakest — at a free edge — and runs inward, and a member with a crack-like defect at the edge of a weld may never reach the band strength at all.

And they do not show the stretch. A zone that necks along its own line takes the member’s stretch with it, as the transverse zone took all of it; the inclined zone’s stretch is its width along the load times its own strain, and a zone at 55° is longer along the load than a transverse one by about a fifth. That does not change the strength and changes the ductility by little.

The numbers by hand #

The member’s strength at each angle is one square root. With $Y = 185$ N/mm²:

• 90°: $\sin^2\varphi = 1$, the term is $1 \times (3 - 2.25) = 0.75$, and $\sigma = 185/\sqrt{0.75} = 214$.
• 54.7°: $\sin^2\varphi = 2/3$, the term is $\tfrac{2}{3} \times 1.5 = 1$, and $\sigma = 185$.
• 45°: $\sin^2\varphi = 1/2$, the term is $0.5 \times 1.875 = 0.9375$, and $\sigma = 191$.
• 30°: $\sin^2\varphi = 1/4$, the term is $0.25 \times 2.4375 = 0.609$, and $\sigma = 237$.

And the parallel member’s strength is close to the area mix: $0.75 \times 310 + 0.25 \times 185 = 279$, against 278 when the two materials’ peaks are allowed to fall at their own strains.

What it comes to #

A soft zone that the load crosses necks along its own line, because the parent on either side holds its length.

Straight across, that hold lends it 15 per cent. A 6082-T6 zone of 185 N/mm² carries 214 in a transverse weld.

At 54.7° it lends nothing. The zone carries its own 185, and that is the weakest orientation for any soft zone in any alloy.

Below 24° it cannot neck alone, and the member reaches 278, the strength it has with the zone running along it.

Along the member, the zone costs strength and gives back stretch. A quarter of the section soft keeps 90 per cent of the strength and raises the uniform elongation from 7.0 to 7.8 per cent.

Still open: the seam that winds round a tube #

A spiral-welded tube carries its seam at the angle its forming mill set, often between 50 and 70 degrees to its axis — squarely in the trough this essay found. Under axial load alone the seam’s soft zone would sit near its weakest. But a pressurised tube is pulled round its circumference twice as hard as along it, and under that biaxial stress the line along which a sheet has no strain is no longer at 54.7 degrees to anything. Whether a seam angle exists at which a spiral-welded tube’s soft zone is held by its neighbours under the combined pull of pressure and axial load — and whether the angles mills actually use are near it or far from it — is a question about the same band in a different stress, and it is the one a tube designer meets first.