An inflated cylinder wants an unbalanced cloth

A fire hose, an inflatable boat, an airbag, an air-supported roof, a lay-flat duct: in every one of them a fabric is holding a pressure, and the pressure does not load the fabric equally in its two directions.

It loads it in the ratio two to one, and the two is exact. Not a coefficient, not an approximation, not a property of any material: a closed cylinder of radius R under pressure p carries pR per unit thickness around its circumference and pR/2 along its axis, because those are the ratios of the two areas the pressure pushes on. Cut the cylinder across and the pressure acts on a circle of area πR²; cut it lengthwise and it acts on a rectangle whose length is what the seam has to hold. The circumference wins by two.

Balance, which is a virtue everywhere else #

This site has treated balance as a good thing since the essay that named it, and for good reasons. A balanced cloth wears evenly, hangs the same way in both directions, shrinks predictably, and does not curl or bow when it is finished. It is the default for a reason.

Under pressure the reason evaporates, and it evaporates by a computable amount. Strength per unit width goes as the fibre per unit width, which is the sett times the count. A field that loads one direction twice as hard wants twice the fibre in it — twenty ends against ten picks, or the same sett in twice the count — and a fabric that supplies equal fibre both ways is using three quarters of what it is made of.

The waste is exactly (1 + 1/R)/2, so it is 25 per cent at two to one and it rises: a third of the fibre wasted at three to one, three eighths at four. A spherical vessel, where the ratio is one, wastes nothing — which is why a sphere is the shape a pressure vessel wants and why an inflatable that has to be a cylinder pays for the shape twice, once in stress and once in fibre.

Where the fabric puts its two systems is not where the load is #

A cylinder’s two principal directions are the hoop and the axis, and they are at right angles. A woven fabric’s two systems are at right angles. It is tempting to conclude that the fabric can simply be laid with its warp around the circumference and be done.

Two things complicate that, and both are on this site already.

The first is that a woven fabric under biaxial load does not divide the load the way its setts suggest. The two systems share a thickness, so tension in one straightens it and crowds the other: the crimp interchange this site has computed since the foundation. A cloth pulled hard warpwise gives up warp crimp, gains weft crimp, and narrows. Under a 2:1 field the warp is straighter, so it takes a larger share of the load than its fibre content alone would predict — which helps, and which is not a design method, because the amount depends on the crimps and therefore on the state the fabric was finished in.

The second is the bias. A cylinder under pressure alone has no shear in its principal directions, but a real hose is bent, twisted, dragged and pressurised at once, and any of those puts a shear into the fabric. A woven cloth has almost no shear stiffness until its threads jam, so an off-axis load finds a mechanism rather than a material.

Which is why a hose is braided rather than woven #

The construction that answers all of this at once is not a weave.

A braid lays its yarns at an angle to the axis rather than along it, so a single system can carry both components of the stress: the yarn’s tension resolves into sin²α around the circumference and cos²α along the axis, and choosing α chooses the ratio. There is exactly one angle at which a single set of yarns carries a 2:1 field with nothing left over, and it is the subject of the next rung.

That is why fire hose is woven as a circular jacket with a heavy warp and a light weft rather than as a flat cloth seamed, why reinforced rubber hose is braided or spiral-wound rather than woven, and why an airbag — which is a sphere-ish shape at 1:1 — is the case where a balanced fabric is right.

The airbag is the case where balance is right, and it is worth saying why #

A quarter of the fibre wasted sounds like a reason never to use a balanced cloth under pressure, and the exception is instructive.

An airbag is not a cylinder. It is closer to a flattened sphere, its stress ratio is near one over most of its surface, and it is inflated once for thirty milliseconds. So the fabric wants equal strength in both directions — a balanced cloth, plain woven, tightly set — and the design problem moves entirely to two other quantities: how little air gets through, which is the cover factor again — though a cloth stops having holes before it stops passing air, and an airbag is close enough to its own floor for that to matter — and how the fabric behaves when it is folded into a housing for ten years and then loaded in a hundredth of a second.

The permeability requirement is the interesting one, because it is the filter’s arithmetic with the sign of the requirement reversed. A filter is specified for a minimum open area; an airbag fabric is specified for a maximum permeability, which is to say a maximum open area — and the arithmetic connecting sett, diameter and open area is identical. One trade wants (1 − nd)² large and the other wants it small. Both are reading the same expression.

So the applied field’s rule holds here as elsewhere: the requirement decides which side of the inequality matters, and the geometry underneath does not change. A fabric engineer moving from geotextiles to airbags has not learned a new quantity, only a new direction to push it.

The waste is not the only cost of balance #

A quarter of the fibre doing nothing is the cost that shows on a weighbridge, and there is a second cost that does not, which follows from the same inequality and is usually the one that decides.

A balanced cloth in a 2:1 field reaches its limit around the circumference while the axial system is at half. So the failure is a hoop failure, and a hoop failure in a cylinder is a longitudinal split — the fabric opens along the tube rather than across it, which is the most damaging way a pressurised vessel can fail because the crack runs in the direction the load keeps feeding it.

An unbalanced cloth built to the ratio fails in both directions at once, which sounds worse and is not. A structure whose two limits coincide has no preferred crack direction, so what happens at the limit is decided by whatever defect is largest rather than by the geometry — and a failure that has to find a defect is a failure with more warning in it than one the stress field is steering.

That is a second and independent argument for the 2:1 construction, and it is the one a safety case is written from rather than the fibre economy. It also explains why the trade’s tolerance for the waste is asymmetric: a hose overbuilt in the hoop direction is a hose whose axial system fails first, which is a circumferential parting at a coupling — an inconvenient failure rather than a dangerous one — so a designer in doubt overbuilds the hoop, which wastes fibre in the direction that was already carrying least.

And the waste is recoverable in one place the arithmetic does not see. A cylinder’s ends are not at 2:1; a hemispherical end cap is at 1:1, and a flat end is at neither. So a vessel built from one fabric throughout is at the right ratio over its barrel and the wrong one everywhere else, and a vessel built from two fabrics has a joint at the transition — which is the same choice, between a ratio and a seam, that runs through the whole of this field.

Two systems in a stated ratio, and how a loom supplies it #

The ratio the arithmetic asks for has to be woven, and there are two ways to supply it — with different consequences that the arithmetic does not distinguish.

By sett. Twenty ends against ten picks, in the same count. The warp then carries twice the fibre, and the cloth is markedly warp-faced with the picks widely spaced: an open fabric on one axis, which matters if the fabric also has to be reasonably impermeable or to resist abrasion across its face.

By count. Equal setts with the warp in twice the tex. The fibre ratio is the same, the surface is far more even, and the jamming is quite different because the heavier warp is thicker: a coarse warp at the same sett crowds the cloth and closes the gaps.

Two constructions with identical fibre ratios, different covers, different thicknesses, different crimps and different behaviour in every respect but the one being specified. The fibre ratio is one equation and a fabric is decided by several, which is why a pressure fabric’s specification names a sett, a count, a weave and a coating rather than a ratio — and why a designer who has computed only the ratio has done a quarter of the work.

What was counted, and how #

Very little of this needs a computation and one part of it needs an assertion.

The stress ratio is arithmetic on two areas and is asserted to be exactly two — within 10⁻¹², at every radius, pressure and wall thickness the routine is asked about. That is a check on the implementation rather than on the physics, and it is worth having for a reason this field keeps meeting: the whole argument rests on the two being exact rather than approximate, so a routine that returned 1.97 for some inputs would be quietly undermining the claim.

The utilisation of a balanced cloth is (1 + 1/R)/2, and the assertion on it is the boundary rather than the value: a balanced cloth is fully used only in a balanced field, and the waste rises monotonically with the ratio. Both halves would be trivial to check by hand and neither is trivial to keep true through a change of parameterisation, which is what an assertion is for.

Nothing here computes a strength. The step from fibre per unit width to force per unit width needs a fibre strength, a translation efficiency and a statistical model of how a bundle fails, and none of the three is on this site. What is claimed is the ratio the fabric should be built in, which is where the geometry ends and the materials begin.

The seam runs the wrong way round, which follows from the same two areas #

One consequence of the 2:1 ratio is a rule of thumb about seams that is worth deriving rather than remembering.

The hoop stress is the larger, and it acts across any seam that runs along the cylinder — a longitudinal seam is being pulled apart by the bigger of the two loads. A circumferential seam, running around the tube, carries only the axial stress and therefore half as much. So a lay-flat duct with a longitudinal seam has its weakest line on its most loaded axis, and a duct made from a circular-woven jacket has no longitudinal seam at all.

That is why fire hose is woven circular — as a seamless tube, on a loom whose warp runs around the circumference — rather than woven flat and seamed. The construction removes the seam from the direction that would have loaded it most, and it does so at the cost of a much more awkward loom. Two areas, a factor of two, and a whole machine designed around it.

The same reasoning decides the layout of a large air-supported roof, where the fabric arrives in panels and every joint is a decision about which stress it will carry. Panels are laid so that their seams run along the direction of lower stress wherever the geometry allows, and where it does not, the seam is doubled. Nothing in that is a fabric property: it is the pressure’s two areas, read as a specification for where to put a joint.

Where the model stops #

A membrane is not a fabric with two independent directions. The interchange above is the visible part of a deeper problem: the biaxial response of a coated fabric is measured on a cruciform specimen precisely because it cannot be predicted from two uniaxial tests. The 2:1 stress ratio tells a designer what to aim for and not what a given cloth will do.

The coating is doing work this arithmetic ignores. A pressure fabric is almost always coated, and the coating carries shear, seals the surface, and holds the threads where they are. It is why a coated cloth’s bias behaviour is nothing like the bare net drawn above — and none of it is in a geometric model of a weave.

The seam is usually the weakest part, and it is not here at all. A hose’s or an inflatable’s failure is far more often at a joint than in the middle of a panel, which is why two rungs of this field are about seams and why a specification for a pressure fabric always names a seam strength.

Nothing here knows what a fabric does when it is folded for ten years. An airbag fabric is packed into a housing, kept at whatever temperature a car interior reaches, and then asked to work once. Every property that matters over that history is a material property — the coating’s ageing, the yarn’s set at a crease, the fabric’s permeability after compression — and a geometric model of a weave has nothing to say about any of it.

And the geometry is a cylinder. A real inflatable is a cylinder with ends, bends, taper and fittings, and every one of those redistributes the stress locally. The 2:1 ratio is the answer for the straight part far from anything, which is the same caveat that attends every thin-shell result.

Who found it, and when #

The hoop-and-axial arithmetic is old enough to have no useful attribution — it is the thin-cylinder result any engineering course derives in a page, from Barlow’s formula for pipe onward — and this essay quotes it rather than deriving it, because it belongs to statics and not to cloth.

What belongs to cloth is the step nobody bothers to write down: that a 2:1 stress field asks a woven fabric for a 2:1 fibre ratio, that a balanced cloth in it wastes exactly a quarter of its fibre, and that the shape of that waste is what makes braiding rather than weaving the right construction for a hose. Fire hose has been woven unbalanced since the nineteenth century, and rubber hose has been braided at an angle since not long after, so the practice preceded the arithmetic by a comfortable margin. It usually does.

Where the ladder goes next #

A single system of yarns can carry both components of a 2:1 field if it is laid at the right angle, and there is exactly one right angle. It is arctan √2, it has no material constant in it, and two entirely different arguments arrive at it — the next rung computes both and finds they agree.