The angle that forgets the liner

The wire that cannot push built a thick tube by winding wire on a liner under tension, each turn square to the axis, each carrying hoop stress and nothing else. Its bore swung from $-\sigma_y$ at rest to $+\sigma_y$ at work, and that swing of twice the yield stress was the most any prestressed wall could give. It ended by pointing out what those hoops leave unattended. A tube with closed ends has an axial force, a gun barrel its breech load, and a wire wound square does nothing about either. Filament-wound pressure vessels are wound in helices, so that each fibre carries some of both.

There is a famous number for how. In a thin closed tube the hoop stress is exactly twice the axial stress, and a fibre at angle $\alpha$ to the axis, with a balanced partner at $-\alpha$, carries hoop and axial in the ratio $\tan^2\alpha$. Setting that to two gives $\alpha = \arctan\sqrt{2} = 54.7°$, which is why that angle is found on every glass-fibre pipe in the world. So the natural question for a jacket over a thick liner is whether one helix can both put the bore at its floor and carry the axial load, or whether a thick wall’s hoop stress, falling from the inside out, wants a different angle at every radius. The second half has a simple answer, and the first turns out to depend on something the number $54.7°$ was not designed to consider.

A wall that is not all fibres #

The tube has a 100 mm bore, a 15 mm liner of steel yielding at 900 N/mm², and a jacket built up in layers of balanced carbon fibres, here 30 mm thick. The fibres carry a stress that depends on the strain along them, $\sigma_f = F(\varepsilon_\theta \sin^2\alpha + \varepsilon_z\cos^2\alpha)$, with $F = 140{,}000$ N/mm² for the fibres’ modulus times their volume fraction, and by netting theory they contribute $\sigma_f \sin^2\alpha$ to the hoop stress and $\sigma_f\cos^2\alpha$ to the axial. A small transverse stiffness keeps the layer solid through its thickness. The wall is in generalised plane strain with a single axial strain, and a pressure $p$ brings an axial force $p\pi a^2$ for the closed ends to deliver. A radial finite element solves this with one radial displacement function and that one axial strain.

A note on what the wall contains. Netting theory is the statement that there is nothing else there: all load on the fibres. The 54.7° is a theorem about that case. With a liner, the first question is how much of each load the fibres receive at all.

Who carries what #

At the netting angle the liner carries 60 per cent of the axial force and 68 per cent of the hoop force. Steel is stiff, it starts from the bore, and at this wall thickness the jacket’s stiffness is the smaller. The fibres are not presented with a two-to-one loading. They are presented with what the steel has left, and that is in a ratio of its own.

The figure also shows the far end of the range. As the fibres swing towards a hoop winding, the liner’s share of the axial force goes up, to 95 per cent at 90°, since hoop fibres have no axial stiffness and the liner must take the end load alone. Its share of the hoop force falls only to 59 per cent. So a steel-lined tube already has something perfectly good at carrying its axial force, and what the jacket has to offer is help round the hoop.

Within the jacket the fibres are nowhere near sharing equally, and that is the thick-wall effect. A 30 mm jacket on a 115 mm radius has its outer fibres at 63 per cent of its inner. A 120 mm jacket has its outermost layer at about a tenth of its innermost: those fibres have been laid on a wall that hardly moves. Radial strain in the jacket is much softer than the fibres’ stiffness along their length, and so the hoop strain falls off quickly with radius. This is the falling hoop stress of a thick wall, as the force that is only a radius set it out for a ring, in its most extreme form, and it means thick jackets wound slack waste most of their outer fibres. That is the same as the discovery that drove autofrettage and wire winding: a thick wall cannot be brought into use by thickening it. The ceiling the bore sets makes the same point about a solid one.

Slack, the best helix is nearly a hoop #

The lower curve is the plain answer. With no winding tension at all, the pressure at which the liner first yields rises as the fibres swing towards the hoop: 139 N/mm² at 30°, 177 at the netting angle, 191 by 70° and flat after that. The famous angle is not the best angle here, and it is not close. The liner carries the axial load perfectly well, and it is the hoop that needs help, so the fibres do most good laid nearly round.

The reason is simple enough to state directly. A bore of steel under a closed tube’s pressure has a hoop stress about twice its axial, and by von Mises the axial stress is then helping, since a two-to-one state yields later than the hoop stress alone would. Fibres that take hoop load off the steel without touching the axial lower the larger stress and leave the smaller, and the gain stops once the two have come level and the axial has begun to govern. That is why the curve flattens near 70° rather than climbing to 90. In any case slack jackets are a poor deal: all of these fibres are far below their strength at the point where the liner yields. The inner layer is at about 335 N/mm² when the liner first yields, under a quarter of its 1,500.

Wind it, and the order reverses #

The upper curve in that figure winds each layer under tension, enough to squeeze the bore at rest to the point of yield, its floor, and asks how much pressure the tube then reaches. Everything changes. At the netting angle it reaches 352 N/mm²; as plain hoops, 265, a third less. The best is 363 at 60°. The pressure has doubled across the range, from the winding alone, and now it is the helix that wins.

Two things are going on, and the figure separates them. At a given winding stress, plain hoops do slightly better for a while, since each newton of fibre tension goes entirely round the hoop. So on the first part of the figure the hoop line is above the helix line, and the argument that a helix wastes part of its tension has something in it. But the lines stop at different places. A hoop winding reaches its floor at a winding of 710 N/mm², and there it has to stop. A helix can be wound to 978. And while it goes on, its line goes on rising, past the hoop’s stopping place.

The long axis of the ellipse #

This is the whole of the reason, and it is a matter of where on a yield surface one chooses to stand. A steel element has a yield ellipse in its two principal stresses, longer along the diagonal where both have the same sign. A hoop winding drives the bore along the left-hand axis, straight into the ellipse at $-\sigma_y$ on the hoop with nothing on the axial: the point $(-908, -7)$. It has gone as far as it can. A helix presses along the length of the wall too, since every turn it lays down is pulling the tube’s ends together as well as squeezing it round. So its rest state goes off along the ellipse’s long axis, reaching $(-1{,}045, -609)$ before it yields. The hoop compression at rest is greater than the steel’s uniaxial yield, and the bore is still elastic.

So a helix is not giving up prestress for axial capacity. It is the only winding that can put more compression on the bore than the bore can take from a hoop winding, and it can do so because it also loads the axis the fibres were laid to carry. The essay on the wire’s ceiling set the swing at $2\sigma_y$. For a uniaxial swing that is right. A biaxial swing along the diagonal can exceed it, and that is one more reason why “the ceiling” was a statement about a single stress.

At the working end the same geometry applies. The hoop winding works at $(228, 778)$ with most of its stress on the axis. The fibres’ stress is nowhere near limit, the hoop stress is modest, and it is the end load, which no hoop fibre was ever going to touch, that brings the liner to yield. A hoop-wound tube is limited by the one thing it has no fibres to help with. The helix works at $(688, 59)$, its axial stress having been pre-compressed to the point of barely appearing, and the only stress left is the hoop.

That is why the best angle in the upper curve is not exactly 54.7°. The winding has reasons for its own preference, and it is not the netting ratio. The optimum is around 60° at this wall and 60° to 65° at thicker ones, where the axial component’s job is to give enough biaxial pre-stress for the working end, and any more fibres at the axis take away from the hoop. The 30° on the far left is a poor winding by comparison, but it shows the other wall of the valley: too little hoop to raise the working state at all, and at 30° wound to its floor the tube reaches only 34 N/mm². It would be easy to read the upper curve’s peak near 55° as a vindication of the netting angle, but it has arrived there by a different route.

Winding also evens the fibres #

The other half of the question was whether a thick wall wants a different angle at every radius, because its hoop stress falls from the bore outward. A slack jacket’s fibres do fall off steeply. When the winding is applied the answer changes, for a reason the wire that cannot push already found for hoops. Each new layer squeezes everything already wound, so the inner layers lose stress at rest. At the floor, the first layer rests at a much lower fibre stress than the last, and it picks up more under pressure, and so at the working pressure the fibres come out nearly even: 1,275 N/mm² at the inside and 1,396 at the outside, within a tenth of each other. The winding has done the job a graded angle was going to be asked to do. These fibres are all near the limit, and this is what a jacket looks like when it is working.

So grading the angle has only a modest contribution left. A 60 mm jacket at 60° throughout reaches 401 N/mm², and a jacket graded from 80° at the inside to 40° at the outside reaches 418. The sense of the grading is what one would expect: hoop fibres near the bore, where most of the hoop strain is, and axial ones outside where the axial strain is still the same as anywhere else, since an axial strain does not fall off with radius. Four per cent is real, and a winding machine can do it for nothing in material, but it is not what decides the tube. Choosing the winding stress first and the angle second has given most of what there was to give. The choice of 60° over the netting angle is worth more than the grading: 401 against 375.

The hand check #

The hoop wound at 710 N/mm² with 30 mm of fibre, in the thin-wall approximation: the jacket’s pressure on the liner is its hoop force over the mean radius. At the winding stress, $710 \times 30/130 = 164$ N/mm² of outer pressure, reduced by the compliance of the layers underneath, and the bore’s hoop stress from a Lamé tube $a = 100$, $b = 115$ under external pressure $q$ is $-2qb^2/(b^2 - a^2) = -8.2q$. So $q \approx 110$ N/mm² gets to the 908 computed, which puts the effective winding loss at about a third, reasonable for layers wound onto a wall that is shortening as they arrive.

The axial end of the helix check is simpler. The tube’s end force at 352 N/mm² is $352 \times \pi \times 100^2 = 11.1$ MN, carried by the liner’s 10,100 mm² of steel at a bore stress of 59 N/mm², and by the fibres’ axial component. The fibres, at about 1,330 N/mm² and $\cos^2 54.7° = 0.334$ over an area of 24,500 mm², carry 10.9 MN, and the steel’s 0.6 MN brings the sum within a few per cent. So at work the fibres are carrying the whole end load very nearly by themselves, as netting theory would say. They needed to be wound to their limits for the liner to get out of the way, and it is only there that the theorem recovers its terms.

What this wall leaves out #

A liner that never yields. All of this is set up so that the liner is elastic throughout. Real metal-lined composite vessels are usually autofrettaged, with the liner taken past yield in a proof test, and then the rest state comes from the plastic strain and not only the winding. The overstrain the release gives back and the fibres’ own preload could then share the same budget, and the argument about the ellipse applies to the rest state however it is produced. Taking the Bauschinger effect seriously would then lower the swing, and that is the argument a reversal in a hardening metal always raises, as after the first yield described.

Fibres that are perfect. The fibres here are linear to a limit, with no scatter, no waviness at crossings and no resin cracking. A helix’s crossings in particular produce real knockdown at the crossover bands, and a 1,500 N/mm² limit here takes no account of it. It may well bring the fibre limit down far enough that it governs the winding and not the floor. Fibre stress rupture under long-held tension is another limit: carbon holds well, and glass does not, and a 1,300 N/mm² fibre stress held for the life of a pipe might be asking too much.

Open ends and domes. Only the cylinder is modelled. The end dome’s geometry is what really forces a filament-winding angle: a geodesic path on a dome has an angle fixed by the opening’s size, and it often lands well below 54.7° at the cylinder. Many vessels add hoop layers on top for that reason, and the pair is a form of grading that this essay’s figures do not reach. Where a seam or weld occurs in the liner, the seam that pressure protects puts a limit of its own on the liner’s own biaxial state.

A static pressure. A vessel that is cycled has its fibres and its liner on different fatigue curves, and the liner’s range is what the swing of the bore sets. The seam that pressure protects and the cycles that do not count read that range in different ways, and neither has been asked about it here.

A radial stiffness that is small. The jacket’s transverse modulus sets how fast the hoop strain falls off with radius, and it depends on the resin. A stiffer resin would lift the outer fibres in the slack case and change little in the wound case. It is worth noting because the 48 N/mm² on the outer layer of the thick slack jacket depends on it heavily.

Still open: the winding path over a dome #

A cylinder lets a fibre be any angle anyone wants. A closed vessel’s ends do not. A fibre laid on a dome without slipping follows a geodesic, and its angle at the cylinder is set by the radius of the port it passes round, so the angle this essay has been treating as a free choice is a fact about the end fitting. To get a helix of 60° onto a cylinder the port would have to be 87 per cent of the cylinder’s radius. Whether a practical vessel can have the winding angle the ellipse wants, or whether the port fixes a smaller angle and hoop layers then have to be added, moving the design back towards the bore’s less favourable end of the figure, is the question a real vessel’s two ends put to its middle.