The seam that pressure protects

The weld that is weakest at fifty-five degrees found a rule for a soft heat-affected zone crossing a plate. Welding softens a band of the parent metal; pulled across the load, that band would like to neck on its own, but its neighbours either side are far below their own yield and hold its length along the weld fixed. With its strain along its own line held at zero, the band yields by von Mises at a load higher than its own uniaxial strength — 15 per cent higher across the load. Turn the band, and the help changes, until at 54.7° to the load it vanishes: the band carries exactly its own strength and its neighbours lend it nothing. Shallower than about 24° it cannot neck alone at all.

That essay ended on the tube. A spiral-welded tube winds its seam round itself at the angle the forming mill sets, typically 50 to 70 degrees to the tube’s axis — squarely in the trough. But a tube in service is rarely pulled along its axis alone. A pipe carries pressure, which pulls round its circumference twice as hard as along it, and the question is what that second stress does to the trough.

The same band, in a different stress #

The band condition never needed the load to be uniaxial. On the band’s own axes — normal to the seam and along it — the remote wall stress has a normal component $\sigma_n$ and a shear $\tau$, and those are what the band carries; its stress along its own line is whatever the zero-strain condition makes it, which for von Mises flow in a sheet is half its normal stress. So the band yields when

$$Y_z^2 = \tfrac34\,\sigma_n^2 + 3\tau^2,$$

whatever produced $\sigma_n$ and $\tau$. For a tube wall carrying an axial stress $\sigma_a$ and a hoop stress $\sigma_h$, a seam at angle $\phi$ to the axis has

$$\sigma_n = \sigma_a \sin^2\phi + \sigma_h \cos^2\phi, \qquad \tau = (\sigma_a - \sigma_h)\sin\phi\cos\phi.$$

The parent wall, meanwhile, yields by von Mises at $\sigma_a^2 - \sigma_a\sigma_h + \sigma_h^2 = Y^2$. The tube is as strong as the weaker of the two.

Take the alloy of the earlier essays, 6082-T6, whose parent is 310 N/mm² at ultimate and whose welded zone is softened to 185 — 60 per cent, part of which no amount of welding skill gives back. Wind the seam at 60° to the axis. Pull the tube along its axis and the seam lets it keep 60 per cent of the wall’s strength: 60° is close enough to the trough that the parent lends the band almost nothing. Close the tube’s ends and pressurise it, so that the hoop stress is twice the axial, and the same seam lets it keep 78 per cent. Same tube, same seam, same weld; the loading has moved the trough.

What the parent can lend, and when it cannot #

The cleanest way to see it is to ask, for each seam angle, how much the parent lends the band: the equivalent stress the whole wall carries at the moment the band yields, over the band’s own strength. One means no help at all; 1.15 is the help a band gets straight across a uniaxial pull; anything above the ratio of the two materials’ strengths, 1.68 here, means the band is so well held that the parent fails first.

Under axial tension the curve has its trough at 54.7° and the help rises to 1.15 at 90°, which is the earlier essay’s result, and the reason two welds of one zone can be decided differently by nothing but their direction. Under a closed tube’s pressure the curve is turned round: the trough is at 0°, along the axis, where a longitudinal seam is lent nothing at all, and the help rises steadily to 2 at the girth, where a seam round the tube carries only the axial stress, half the hoop, and is held in the plane-strain condition the earlier essay found so favourable — the same condition that makes a fillet weld stronger across the load than along it. A spiral seam at 60° under pressure is lent 1.32 and keeps 78 per cent; at 70°, 93 per cent.

Under hoop stress alone — an open-ended tube, a ring — the trough is at 35.3° to the axis, which is 54.7° to the hoop direction: the uniaxial result turned through a right angle. And under equal biaxial stress, a sphere’s, every seam is lent exactly the same 15 per cent. There is no trough at all.

The line nobody stretches #

The pattern has a single explanation, and it is the one the uniaxial case already contained. The band is lent nothing exactly when the condition its neighbours impose — no strain along the seam — is a condition the remote field satisfies anyway: when the seam lies along a line the wall’s stress field does not stretch. On such a line the band’s zero strain is not a restraint; the neighbours are doing nothing they would not do in a wall with no seam, so they lend nothing. Off it, holding the band’s length means holding the band against a stretch the field would give it, and that costs the field strength — which is the help.

A sheet under uniaxial tension has two such lines, at 54.7° either side of the load, because it stretches along the load and contracts across it by half as much, and somewhere between the two the extension is zero: $\tan^2\phi = 2$. A closed tube under pressure does not stretch along its axis at all — the axial stress is exactly half the hoop, and half the hoop is exactly the Poisson-like contraction that cancels it — so its unstretched line is the axis itself. A sphere stretches equally in every direction and has no unstretched line anywhere.

Plotted against the direction of the loading, the unstretched line’s angle sweeps round as the loading turns. It sits at 54.7° under pure axial tension and moves through the mills’ band of 50 to 70 degrees only for loadings close to it: axial tension with a hoop stress between about a third of it in compression and four tenths of it in tension. Between a hoop-to-axial ratio of a half and two — which includes every pressurised closed tube with axial tension added — there is no unstretched line, and the soft seam is helped at every angle it could be wound at.

So the trough is not a property of the angle 54.7°. It is a property of the loading that makes 54.7° the unstretched line, and a different loading has its trough somewhere else, or nowhere.

The wall’s strength, in stress space #

The whole of it fits in one picture. In the plane of the two wall stresses the parent’s strength is von Mises’s ellipse, tilted along the line of equal stresses. The seam cuts a smaller curve inside it, and the gap between the two is the strength the seam costs at each loading. It is widest where the loading’s unstretched line falls on the seam — near pure axial tension and pure axial compression for a 60° seam — and narrowest along the ellipse’s long axis, where both stresses pull together and the band is held everywhere. A pressurised tube’s loadings lie in that narrow part of the gap.

The longitudinally welded tube is the comparison that matters, and it is the mirror image. Pulled along its axis, its seam is parallel to the load — the case the earlier essay found to be no weakness at all, the zone stretching with the parent — and the tube keeps its full strength. Pressurised, its seam lies on the unstretched line, and it keeps 60 per cent, the soft zone’s own share and nothing lent. Under pressure the longitudinal seam is the exposed one and the spiral seam the protected one — the reverse of the intuition that a seam at the “weakest angle” must be the weaker tube.

The intuition is not entirely wrong; it is wrong about the loading. The old rule of boilermaking, that a longitudinal seam carries twice the stress of a circumferential one, says the same thing in the elastic language of hoop tension: the girth seam is relieved because it carries only the axial stress. The band argument adds the other half — that the girth seam is also held, lent twice its strength, because the field it lies across stretches it along its line and its neighbours refuse.

A pull undoes the protection #

The protection belongs to the pressure, so anything that changes the ratio of axial to hoop stress changes it. A buried pipeline cooled after installation, a pipe in a bend, a riser hanging from a platform, a pile pulled out of the ground: each adds an axial tension to whatever the pressure provides. Add an axial stress of half the hoop stress, so that axial and hoop are equal, and the 60° seam still keeps 69 per cent. Add one and a half times the hoop stress, and it keeps 61 — almost exactly back to the softened zone’s own share. The pull has swung the unstretched line from the axis, through the region where there is none, and out the other side onto the seam.

Taking axial stress away does the opposite for the steep seams and the opposite again for the longitudinal one. A pressurised pipe whose ends are not closed — held by its thrust blocks rather than by its own end caps — carries hoop stress alone, and there the 60° seam keeps 76 per cent and a longitudinal seam 69. The seam’s safety is a property of the load path along the pipe, not only of the pressure in it, and a pressure test with closed ends tests the loading in which a spiral seam is best protected.

The mill decides the angle, and the diameter decides the mill #

A spiral mill does not choose its seam angle freely. It forms a strip of fixed width into a helix, and a helix of angle $\phi$ to the axis on a tube of diameter $D$ advances by $\pi D/\tan\phi$ per turn; the strip’s width, measured square to the seam, is that advance times $\sin\phi$, which is $\pi D\cos\phi$. So

$$\cos\phi = \frac{w}{\pi D},$$

and one strip makes steep seams on large tubes and shallow seams on small ones. From a strip 1.5 m wide, a tube of 0.6 m diameter has its seam at 37° to the axis, one of 1.0 m at 61.5°, one of 1.5 m at 71.4° and one of 2.0 m at 76.2°.

Under a closed tube’s pressure those four keep, in the same alloy, 63, 80, 95 and 100 per cent of the wall’s strength: the seam is lent enough to match the parent at every angle steeper than 74.3°. In axial tension the same four keep 67, 60, 64 and 66 — all close to the trough, because between about 35 and 75 degrees every seam is near a line a uniaxial pull leaves unstretched. The large spiral pipe is the well-protected one under pressure, for a reason that has nothing to do with the weld and everything to do with the strip it was made from; and the small spiral tube, whose seam winds shallow, is exposed under pressure much as a longitudinal one is.

That is also why the spiral seam was trusted for water and gas mains long before anyone wrote the band condition down. Those pipes are large, their seams steep, their loading pressure with ends held — the combination in which the seam is lent the most — and the tests that qualified them were pressure tests, which load the seam in exactly the way it is strongest.

A sphere has no weak seam #

The other end of the same argument is the sphere. A spherical tank under pressure stretches equally in every direction in its wall, so it has no unstretched line anywhere, and a soft seam in any orientation is lent the same 15 per cent a band gets straight across a uniaxial pull. In this alloy a spherical shell keeps 69 per cent of its wall’s strength across a seam of any orientation. Nothing about the seam’s direction can make it worse — and nothing can make it better either, which is why a sphere’s welds are made as good as the plate rather than placed where they are protected.

By hand, at sixty degrees #

The numbers come from two lines of arithmetic. For a seam at 60° under axial stress alone, $\sigma_n = \sin^2 60° = 0.75$ and $\tau = \sin 60°\cos 60° = 0.433$ per unit axial stress, so the band’s equivalent stress is $\sqrt{0.75 \times 0.5625 + 3 \times 0.1875} = \sqrt{0.984} = 0.992$. The band, at 185, therefore yields at an axial stress of $185/0.992 = 186.5$, while the parent would carry 310: the tube keeps $186.5/310 = 0.60$.

Under a closed tube’s pressure, per unit hoop stress the axial stress is a half. Then $\sigma_n = 0.5 \times 0.75 + 1 \times 0.25 = 0.625$ and $\tau = (0.5 - 1) \times 0.433 = -0.217$, so the band’s equivalent stress is $\sqrt{0.75 \times 0.391 + 3 \times 0.047} = \sqrt{0.434} = 0.659$, while the wall’s own is $\sqrt{0.25 - 0.5 + 1} = 0.866$. The help is $0.866/0.659 = 1.32$; the band yields at a hoop stress of $185/0.659 = 281$; the parent at $310/0.866 = 358$; and $281/358 = 0.78$.

A thin wall, a narrow band, and an ultimate that is a yield #

The wall is thin and the band narrow. The help depends on the parent holding the band’s length along the seam, which it does only if the band is narrow compared with the distance over which the parent can push back. In a thin-walled tube with a seam a few millimetres wide, it is; near the end of a tube, where the seam runs out, it is not.

The strengths are ultimates, treated as yield points. The band argument is a yield argument applied at the ultimate strengths, as the earlier essay applied it, so it compares two materials each at its most stretched before necking. It is a statement about which fails first and roughly at what load, not a prediction of the tube’s bursting pressure to the last per cent.

And the softened band is uniform. A real heat-affected zone grades from weld metal to parent across a few millimetres, with its softest strip somewhere inside it, and a spiral seam’s zone may vary along its length with the welding speed.

The weld metal, toughness and the bend #

They cannot show the weld metal itself. In steel line pipe the weld metal is usually stronger than the parent and the softening is a narrow band beside it, smaller than in this alloy. The geometry of the argument is the same; the 60 per cent is not.

They cannot show toughness. A spiral seam’s danger in a pipeline is as often a crack as a neck, and the seam’s resistance to a running fracture is a property of its toughness, which no strength argument decides.

And they cannot show bending. A pipe bent along a curve — which also flattens its own section as it goes — has axial stresses that vary round its circumference, tension on one side and compression on the other; a spiral seam crosses both, and at some point round the tube it meets the loading whose unstretched line lies on it, whatever the pressure.

The loading owns the trough #

The band condition holds in any plane stress, and the parent lends a soft seam strength in proportion to how far the seam’s line is from being one the wall does not stretch.

Under axial tension that line is at 54.7°; under a closed tube’s pressure it is the axis; between hoop-to-axial ratios of a half and two there is none.

So a spiral seam is protected by pressure and a longitudinal one exposed by it. At 60° in 6082-T6, 60 per cent of the wall in axial tension and 78 under pressure; along the axis, 100 and 60.

And an axial pull turns the protection off, taking the 60° seam from 78 per cent to 61 once the axial stress is twice the hoop.

Still open: the seam that crosses a bend #

A pipe bent into a curve carries an axial stress that changes sign round its circumference, from tension on the outside of the bend to compression on the inside, with the pressure’s hoop stress constant all the way round. A spiral seam winds through every one of those loadings in each turn, and at two points per turn it crosses a loading whose unstretched line lies along it. Whether those two points are where a bent spiral pipe’s seam fails, whether they move as the bend tightens, and whether the parent’s help is still available at a point that is only briefly on the line — the band is held by its neighbours along the seam, and the neighbours there are in a different loading — is the question a spiral seam asks of a pipeline that is not straight.