The wire that cannot push

The ceiling the bore sets compared two ways of putting a thick tube’s bore into compression before it is pressurised. Overstraining it past yield and letting it spring back carries at most twice the pressure at which a plain tube first yields, or $1 + \beta$ times if the steel’s reverse yield has been cut by the overstrain. Building it from layers shrunk one on another never yields the steel, and with many layers approaches the same twice — and never passes it, however many layers. The reason was the same for both. Once assembled, the tube answers its working pressure as one elastic wall, so the bore’s stress swings through the whole wall’s range; a prestress can choose where the range sits, never shorten it. The range belongs to the wall; the floor it may hang down to belongs to the steel.

The essay ended on how the many-layer limit is reached in practice: by winding. A liner is wrapped in wire or tape under tension, and every turn is a layer shrunk on by the tension it was wound at. A drawn wire’s yield strength can be twice a forged liner’s. Whether that lets the wound tube past the ceiling, or puts the bore’s floor lower than the liner could hold, and how the winding tension should change from the first turn to the last, were the questions left.

A hundred layers, each its own fit #

The tube is the earlier essay’s: a 100 mm bore, a wall 2.5 times the bore, a liner of steel yielding at 900 N/mm². A plain tube of that steel first yields at 378 N/mm², and the ceiling is twice that, 756. The liner runs from the bore to some radius $b$, and the wire, yielding at 1,800 N/mm², from there to the outside at 125 mm.

The design is Gadolin’s principle made continuous. Gadolin chose each shrink fit so that at the working pressure every layer’s bore reached yield together; a wound jacket has a layer at every radius, so every turn of wire is put at its yield at work. A wire is a steel like any other, and the radial pressure of the turns outside it adds to its hoop tension in Tresca’s criterion, so “at yield” means $\sigma_\theta - \sigma_r = \sigma_{yw}$. Equilibrium of the jacket, $d(r\sigma_r)/dr = \sigma_\theta$, then gives

$$\sigma_r = -\sigma_{yw}\ln\frac{c}{r}, \qquad \sigma_\theta = \sigma_{yw}\left(1 - \ln\frac{c}{r}\right)$$

through the jacket at work. The liner is a Lamé tube between the working pressure inside and $\sigma_{yw}\ln(c/b)$ outside, and its bore must reach the liner’s yield at work. That fixes $b$: 98.9 mm, so the jacket is 26 mm of a 75 mm wall. At rest, every stress is the working one less the whole wall’s Lamé field.

The bore does exactly what the ceiling requires and no more: −900 N/mm² at rest, +900 at work. The wire works in a narrower band near its own yield, between 1,341 and 1,512 at rest and at 1,800 everywhere at work. The wound tube carries 756 N/mm², twice the first-yield pressure, and that is the ceiling. The wire is twice as strong as the liner and has not lifted it.

It could not have. The bore is the liner’s, its range is the whole wall’s Lamé range for 756 N/mm², and that range is already the liner’s yield in both directions. A stronger jacket can push the bore no further than the liner’s own reverse yield, and a liner that had been overstrained first, its reverse yield cut to $\beta\sigma_y$, would cap the wound tube at $1 + \beta$ exactly as it capped the combination of an overstrain and a single shrunk-on jacket. The wire’s strength goes somewhere else.

Each turn wound harder than it will rest #

The rest state is fixed by the design; the winding is how it is reached, one turn at a time from the inside out. A turn wound at radius $s$ with a stress $w$ over a thickness $ds$ squeezes everything inside it, liner and earlier turns together, as an external pressure $w\,ds/s$ on a Lamé tube from the bore to $s$. That takes

$$w\,ds\,\frac{s}{s^2 - a^2}\left(1 + \frac{a^2}{r^2}\right)$$

off the hoop stress of every turn at $r < s$. So each turn ends at rest with its winding stress less what every later turn took from it, and the winding needed for a given rest state is found from the outside in: the last turn is wound at its rest stress, and each turn inside it at its rest stress plus everything the turns over it will take.

The answer to how the tension should change is that it should hardly change at all. The first turn is wound at 1,568 N/mm² and the last at 1,511, a fall of under 4 per cent, while what the turns keep at rest rises from 1,006 to 1,511. The innermost turn loses 562 N/mm² to the twenty-six millimetres of wire wound over it; the outermost loses nothing.

So a winding machine set to a nearly constant tension produces a rest state that rises steeply outward, and the steep rise is not something the winding does but something the later turns do to the earlier ones. The same arithmetic appears wherever a structure is prestressed one element at a time. A post-tensioned member whose tendons are stressed one after another loses force in every tendon already anchored as each new one shortens the concrete, and the tendons stressed first are left with least. The order the loads arrived in decides a force in each case, and the wound tube is the case with a hundred stages.

The winding also says why the rest stress cannot simply be chosen freely. The first turn has to be wound at 87 per cent of the wire’s yield to be left at 56; a design that asked for more at rest from the inner turns would have to wind them past yield.

The innermost turn would have to push #

The rest state is the working state less the whole wall’s Lamé field, and the Lamé field is largest at the inside of the jacket. With weak wire or a thin wall, the working hoop stress at the liner’s outside, $\sigma_{yw}(1 - \ln(c/b))$, can be smaller than the Lamé increment there, and then the innermost turn’s rest stress is negative: the design asks it to sit in hoop compression with no pressure inside the tube.

A shrunk-on layer can be in hoop compression at rest; a solid ring pushes back. A turn of wire cannot. Wound under tension, it either keeps some of that tension or goes slack, and a slack turn carries nothing until the pressure stretches it back into tension, which leaves the turns inside it to carry more. The design asked of them is not the one they get.

So the strength of the wire decides whether the design exists at all. Wire as strong as the liner’s steel keeps its innermost turn in tension only on walls at least 3.01 times the bore. Wire 1.5 times as strong, on walls of 1.69. Wire twice as strong, on any wall: at the limit of a thin tube the liner swings between −900 and +900 and the wire between 900 and 1,800, and the wire’s floor is still a tension.

The wire’s strength buys the wall. It does not raise the ceiling; it lowers the wall ratio at which the ceiling can be reached, from 3.01 for wire of the liner’s steel to 1.69 for wire half as strong again, and to nothing for wire twice as strong.

The dotted line is the comparison the earlier essay invites. Layers of one steel shrunk one on another reach the ceiling only where the whole wall, yielding right through, could carry it — where $\sigma_y\ln k$ is at least twice the first-yield pressure, which is a wall of 2.22. Below that even infinitely many layers fall short, because they would need more strength than a fully plastic wall has. Wire of the same steel needs more than the layers do, 3.01 against 2.22, and the whole of the difference is that a layer can be compressed and a wire cannot. Winding is the many-layer limit only for a material that can push; for a wire, it is the many-layer limit with a floor at zero.

How much of the wall the wire needs #

Once the design exists, the wire’s strength decides how much of the wall it takes. The jacket’s task is to supply the bore’s compression at rest and to carry its share of the pressure at work, and both go as the wire’s stress times its thickness. On the 2.5 wall, wire half as strong again as the liner takes nearly half the wall; twice as strong, a third; three times, under a quarter. The liner takes the rest, and since the liner’s bore is at yield both ways and its outside well inside it, a thicker liner is a liner more of whose steel is lightly worked.

The share is the economic result, because the wire and the liner are different products. Drawn wire is cheap per tonne and strong; a forged liner is expensive and has to be bored and finished. A design that puts more of the wall into the wire puts more of it into the cheaper, stronger material, and the ceiling, which belongs to the liner, is the same in every one of them.

Wound or shrunk, the same ceiling #

Set against the wall ratio, the four constructions show the earlier essay’s ceiling from a new side. Every one of them is capped at 2, and the overstrain at $1 + \beta$, 1.70 here. Four shrunk-on layers climb towards 2 and reach it only on walls of about 3.4. The wound tube with strong wire sits at 2 on every wall, which is what a hundred perfectly chosen layers would do if the layers could be as strong as the designer liked. The wound tube with wire of the liner’s steel sits at 2 only from 3.01; below that, the turns it would need cannot be wound.

The dotted curve, $\sigma_y\ln k$ over the first-yield pressure, is the most a wall made entirely of the liner’s steel could carry by yielding right through. It passes 2 at 2.22, and every construction of the liner’s steel alone is under it. The wound tube with strong wire is not: on a wall of 1.5 it carries twice the first-yield pressure where a fully plastic wall of liner steel would carry 1.46, because the jacket is not liner steel. That is the wire’s strength showing itself — not by lifting the ceiling, but by reaching it where one steel could not reach it at all.

The winding is a residual field built in stages #

Every stress the tube carries at rest is a residual stress in the sense of the stress that was there before the load: a field in equilibrium with nothing outside the body. The overstrain finds its residual field by yielding the bore and letting it spring back; the shrink fit builds it from a few interferences; the winding builds it from a hundred tensions, one stage at a time. All three are answers to the question Melan’s theorem asks of a structure under a load that comes and goes — is there a residual field that keeps every state of loading inside yield — asked at the bore of a tube, where the answer is yes up to twice the first-yield pressure and no beyond.

What the stages add is that the field cannot be chosen all at once. A turn’s stress is fixed when it is laid on and then changed by every turn after it, and only through the Lamé squeeze, which reaches every earlier turn and the liner with a weight that falls off as $1/r^2$. That is the shape of every construction sequence in which each new piece loads the ones already in place, as each new floor of a concrete frame loads the young floors below it through the props: the last stage is the only one whose stress is exactly what it was given, and every earlier one carries the sum of what came after.

It is also why the overstrain and the winding do not combine past the liner’s cap. A liner overstrained before it is wound has yielded one way and lost part of its reverse yield; its floor is $-\beta\sigma_y$, and the winding over it can push the bore no further than that floor allows. The jacket can supply some of the compression the overstrain would have had to supply, which lets the overstrain be gentler. It cannot lower the floor, which is a property of the bore’s steel and of what has already been done to it.

The liner by hand #

At work the jacket’s radial stress at the liner is $-1{,}800\ln(125/98.9) = -422$ N/mm². The liner is then a Lamé tube with 756 N/mm² inside and 422 outside, and its bore’s stress difference is

$$\frac{2(756 - 422)\,b^2}{b^2 - a^2} = \frac{2 \times 334 \times 9{,}781}{7{,}281} = 898 \text{ N/mm}^2,$$

the liner’s yield to rounding. That is the equation that fixes $b$.

The innermost turn at work carries $1{,}800(1 - 0.234) = 1{,}378$ N/mm² of hoop stress and 422 of radial compression, a Tresca difference of 1,800. The whole wall’s Lamé field adds $756 \times 2{,}500/13{,}125 \times (1 + 15{,}625/9{,}781) = 374$ N/mm² of hoop stress at 98.9 mm, so at rest the turn keeps $1{,}378 - 374 = 1{,}004$, which the figure gives as 1,006 after the jacket’s discretisation. With wire as strong as the liner the same arithmetic on the same wall gives a working hoop stress of 900(1 − ln(125/b)) against a Lamé increment that is larger at the smaller $b$ the weaker wire needs, and the innermost turn’s rest stress comes out at −281 N/mm²: a turn in compression.

What a winding is held by #

A shrink fit is a length: the jacket is bored 0.20 mm smaller on its radius than the liner and heated on. A winding is a force: each turn is laid on at a tension the machine measures. That is the practical reason winding displaced shrinking for high-pressure tubes. A tension can be held to a few per cent by a load cell on the payoff spool, and an error in one turn spreads over the hundred turns around it, where an error in the bore of a shrink-fitted jacket is the whole fit’s error — the problem a frame built to the wrong length has with every member.

A force held by a stretch is also a force that leaks. Every turn rests at between 56 and 84 per cent of its yield, which is exactly the range in which a high-strength wire relaxes under a strain it cannot escape, the same loss a prestressing strand suffers. A wire that loses 3 per cent of its stress over the tube’s life loses 3 per cent of the bore’s floor with it, and a tube designed to the ceiling then carries 1.97 times its first-yield pressure rather than 2. The design has to start a little deeper than the ceiling, by however much the wire will give back.

What the calculation takes for granted #

The jacket is a continuum of rings. Each turn is a ring carrying hoop tension and passing radial pressure inward, with nothing between turns but contact. Real wire is round or flat, its turns touch along lines, and the radial stiffness of a wound pack is lower than a solid’s; the jacket’s Lamé response at work is then softer than the solid wall assumed, and the liner takes more of the pressure.

Tresca everywhere, plane stress, no end load. The same simplifications as the essays before, kept so the comparison with them is like for like. A closed vessel’s axial stress changes the numbers a little and the structure of the argument not at all.

The liner never yields. Its bore reaches yield at rest and at work and goes no further, and the design asks exactly that. A liner that had been overstrained first would have a lower reverse yield and a lower ceiling, $1 + \beta$, whatever was wound over it.

The winding is perfect. Every turn is laid at exactly its computed tension, with no friction between turns and no change of the earlier turns’ stress except through the Lamé squeeze. Friction between turns resists the inner turns’ shortening and leaves them with more of their winding tension than the frictionless count, which is in the safe direction for the floor and the unsafe one for the wire’s yield.

What the wire was for #

The wire does not lift the ceiling; nothing prestressed can, because the range at the bore belongs to the wall and its floor to the liner. What it changes is everything around the ceiling. It decides whether the ceiling can be reached on a given wall at all — on any wall for wire twice as strong as the liner, from three times the bore for wire of the liner’s own steel — and how much of the wall the liner has to be. And it replaces a fit, set by a length to a hundredth of a millimetre, with a winding, set by a force to a few per cent, at the price of a force that relaxes.

Still open: the jacket that is not a ring #

Every turn here is a hoop, wound square to the tube’s axis, carrying hoop stress only. A tube carrying its own end load — a closed vessel, a barrel with a breech — has an axial stress the hoops cannot help with, and filament-wound pressure vessels are wound in helices, so that each fibre carries some of both. A helix at about 55 degrees to the axis carries the hoop and axial stresses of a closed thin tube in exactly the 2-to-1 ratio they come in, which is why that angle appears on every filament-wound pipe. Whether a helical winding over a thick liner can put the bore at its floor and carry the axial load in one jacket, or whether a thick wall’s hoop stress, falling from bore to outside, wants a different angle at every radius and so a different winding for every layer, is the question a jacket of fibres rather than rings puts to the ceiling.