The angle a hose wants

Pressurise a braided hose and it moves. Which way it moves is decided by the angle its yarns were laid at, and there is exactly one angle at which it does not move at all.

The trade calls that the neutral angle and quotes it as fifty-four and three quarter degrees. It is arctan √2, it is 54.7356°, and there is no material constant in it anywhere — no friction coefficient, no modulus, no fitted parameter, nothing that has to be measured. It is the third result on this site with that shape, after the leno’s grip and the shrinkage ceiling, and it arrives twice by arguments that share no algebra.

The first argument: a mechanism seeking a maximum #

Take a braid of N turns using a total yarn length L, laid at an angle α to the axis, and treat the yarns as inextensible — the assumption this site has enforced with an assertion since its foundation.

The yarn’s path resolves into two components. Its circumferential component wraps the tube, so 2πrN = L sin α fixes the radius. Its axial component runs along the tube, so the length is L cos α. The enclosed volume is therefore

V = π r² h = L³ sin²α cos α ÷ (4π N²)

and with the yarn length fixed, the volume depends on the angle through sin²α · cos α and nothing else.

Differentiate: 2 sin α cos²α − sin³α = 0, so 2 cos²α = sin²α, so tan²α = 2. The volume is largest at arctan √2 and falls away on both sides.

Now the mechanism. A pressurised hose is a bag trying to enclose more volume — that is what pressure does — and the yarns cannot stretch, so the only thing available is the angle. So the braid rotates towards the angle that maximises the volume, from whichever side it starts on, and stops there.

Which way a hose moves, and why the answer is a sign #

The length is L cos α, which falls as the angle rises. Put that together with the rotation and the observable follows.

Below 54.74° the hose shortens and fattens. The angle is driven up towards the neutral value, and a larger angle is a shorter tube. Anyone who has pressurised a low-angle braid has watched it contract — and a braid laid at a very low angle contracts a great deal, which is the whole principle of the pneumatic artificial muscle: a bladder inside a braid at perhaps 20°, which shortens hard when it is inflated and pulls whatever it is attached to.

Above 54.74° the hose lengthens and narrows. The angle is driven down, and a smaller angle is a longer tube. A garden hose that grows when the tap is turned on has been braided above the neutral angle, usually because braiding at a high angle is faster and the elongation is a nuisance rather than a fault.

At 54.74° it does neither, to first order, because the volume is stationary there.

The second argument: two stresses in a ratio #

The other route arrives at the same angle with no volume in it at all.

A closed cylinder under pressure carries twice the hoop stress it carries axially, and the two is exact — it is the ratio of two areas. A yarn at angle α to the axis carries its tension in components: sin²α of it around the circumference, cos²α along the axis, because the tension resolves once for the direction and once for the area it acts across.

For the yarn to carry both components in exactly the proportion the vessel demands,

tan²α = hoop ÷ axial = 2

which is the same equation. Same angle, different physics, no shared algebra — one is a kinematic maximum of a volume, the other a static balance of two stresses — and the site asserts that the two agree to twelve decimal places rather than remarking that they do.

That agreement is not a coincidence and it is worth naming why. The volume argument says: with the yarn length fixed, this angle extracts the most volume from it. The stress argument says: at this angle the yarn’s tension matches the load exactly, so no other structural member is needed. Those are the same statement read through the principle of virtual work — a system does no work moving in a direction where the load is already balanced — and the neutral angle is where both readings become the same sentence.

A braid is an oblique interlacement, which the site has already drawn #

The construction under all of this is one an earlier essay here looked at from a different angle entirely.

A braid is an oblique interlacement: its strands are not two systems at right angles but several systems crossing each other at a bias, and it passes the site’s integrity criterion or fails it exactly as a weave does. That essay’s question was whether a braid word describes one piece; this one’s is what angle the strands should be at. The same object answers both, and neither answer contains the other.

The contrast with a woven cloth is the point of the pairing. A weave’s two systems are at right angles and its ratio is set by how much fibre goes each way; a braid’s single system is at an angle and its ratio is set by the angle. One is a decision about quantity and the other about direction, and only the second has a stationary point.

How much movement is available, which is the muscle’s whole specification #

The neutral angle is a zero, and the useful engineering quantity is the distance from it.

Length is L cos α, so a braid taken from α₀ to the neutral angle changes length by cos 54.74° / cos α₀ − 1. At 20° that is a contraction of 38 per cent; at 30°, 33 per cent; at 45°, 18 per cent; at 50°, 10 per cent. Above the angle the same arithmetic runs the other way: a braid at 70° lengthens by 69 per cent of its length on the way down, which is why a high-angle braid makes an expanding hose and a very high-angle one makes an unusable hose.

Those figures are the contraction available, not the contraction delivered, and the difference is a bladder. A pneumatic muscle reaches the neutral angle only if the bladder inside it can fill the volume the braid is trying to enclose, and the bladder’s own stiffness and the pressure decide how far along the geometric path the assembly actually travels. So the geometry supplies the ceiling and the mechanics supplies the fraction of it that is realised — which is the same division of labour as the shrinkage ceiling in the finishing field, where the crimp says how much length there is to give back and the process says how much of it comes.

There is a pleasing consequence. A braid’s contraction is bounded by 1 − cos 54.74° = 42 per cent of its length, whatever the yarn, the pressure or the bladder — because the braid can do no better than reach the neutral angle from zero. Real muscles reach 25 to 35 per cent, and the gap is the bladder, the end fittings and the fact that a braid at 0° is not a braid.

The angle is a bias, and the bias is the site’s oldest mechanism #

It is worth connecting the number to the field it is filed under.

The bias is a woven cloth’s shear degree of freedom: the trellis closing, extension with nothing stretching, the mechanism this site has drawn since its foundation. A braid at an angle is the same freedom used deliberately rather than suffered — the strands are laid at the bias in the first place, so the structure’s whole response to load is the rotation a woven cloth does only when it has been cut on the diagonal.

Read that way, the neutral angle is the bias’s stationary point. A woven cloth on the bias extends up to a hard ceiling of √2 − 1 as its trellis closes and then jams; a braid on a cylinder has a volume rather than a length as its objective, and the objective has a maximum in the middle of the range rather than at the end of it. Same mechanism, different objective, and the difference between a limit and an optimum.

That also explains why the two constructions fail differently. A woven cloth loaded off-axis distorts until it locks, and its fibre has been moved away from the load. A braid loaded on-axis rotates until the volume stops increasing, and its fibre has been moved towards where the load wants it. The braid is the construction that gets the mechanism’s cooperation.

Fifty-four and three quarters is one member of a family #

The angle is usually quoted as though it were a constant of nature. It is the answer to one particular question — a closed cylinder, carrying its own end load — and changing the vessel changes the number in a way the second derivation makes obvious.

The stress route says tan²α equals the ratio of the hoop stress to the axial one. So write that ratio as k and the neutral angle is arctan √k, and every vessel in the family reads straight off its own two stresses.

The two ends of that table are worth spelling out because both are things people build.

A sphere is 45°, since its two stresses are equal, and a spherically wound pressure vessel is filament-wound at 45° for exactly this reason. It is also why a filament-wound cylinder’s end domes are wound at a continuously varying angle: the stress ratio changes along the dome, so the neutral angle does too, and the winding path has to track it.

A pipe whose axial load is carried by flanges is 90° — pure hoop winding — because the fibre is being asked to do nothing along the axis. That is the limit the table’s third row names, and it is why a rigid pipe is reinforced circumferentially while a free hose is not.

And a hose under tension is below 54.74°. A hose being dragged, or hanging under its own weight, or pulled by whatever it is connected to, carries an axial load its own pressure did not supply. That raises the axial term, lowers k, and lowers the neutral angle — so the angle a hose wants depends on what is being done to it, and a hose braided at exactly 54.74° while slack is above its neutral angle the moment somebody pulls on it, and will try to lengthen.

That is a real design consequence and it points the same way as the note further down about a maximum being a flat place to stand. Braiding a little below the nominal neutral angle covers both cases at once: the hose is then below neutral when slack and nearer to neutral when it is being pulled, which is the condition it spends its working life in.

The generalisation is worth stating separately, because it is what makes the family a family. The angle carries no material constant because it is a ratio of two loads, and any structure whose members can only carry tension will orient them at the arctangent of the square root of whatever ratio it is asked to carry. A braid is one instance; a filament winding is another; and the reason the answers agree is not that they share a mechanism but that they share the arithmetic of resolving one tension into two directions.

What was counted, and how #

The maximum is found by search over ninety thousand angles and then compared with arctan √2, and the search agrees to four decimal places. That is the wrong way round from how a mathematician would do it and the right way round for a check: the closed form is what the argument uses, and the search is what would notice if the closed form had been differentiated wrongly.

The stress route is computed from the vessel’s own two stresses rather than from the number two typed in, so the agreement is between two computations and not between one computation and a constant. The stress ratio itself is asserted to be exactly two — at every radius, pressure and wall thickness — because the whole argument rests on its being exact.

The sign of the length change is computed by differencing the volume numerically either side of each angle, and two things are asserted about the resulting column: that it changes sign exactly once across the range, and that the change falls between the two angles that bracket 54.74°. A count of sign changes is a cheap assertion and it catches the expensive mistake, which is a derivative whose sign is inverted — a figure that would then confidently report every hose behaving backwards.

Where the model stops #

The yarns are inextensible and a real braid’s are not. A polyester braid stretches a per cent or two under load, and the angle it settles at is the one where the yarn tension, the elongation and the pressure balance — not quite the geometric optimum. The neutral angle is where the kinematics is stationary; a real hose has a small elastic correction to it.

Nothing here is the yarn’s tension. The volume argument says which way the angle moves and not how hard; the stress argument gives the ratio the yarn must carry and not the force. A braid designer needs both, and getting from here to a burst pressure requires a yarn strength and a bundle model that this site does not have.

The braid’s own mechanics are missing. Real braid yarns rub where they cross, and that friction resists the rotation the volume argument predicts: a braid does not slide freely to its optimum, it creeps there, and a stiff braid at high pressure may never arrive. The capstan is the tool for that and it needs a measured μ, so any number from it is a number at a stated friction — the same boundary the fancy weaves ran into.

And a hose is not a bare braid. It has a liner, a cover, sometimes several braid layers at opposite angles, and a rubber that carries shear between them. Two layers at ±54.74° are the standard construction precisely because a single layer’s tendency to rotate has to be balanced by another one, which is a fact about the assembly rather than about the angle.

Who found it, and when #

The angle is old and its attribution is spread thin. It appears in the filament-winding literature of the 1950s and 1960s as the netting-analysis result for a cylindrical pressure vessel, where it is derived by the stress route; it appears in the hose trade as the neutral angle, generally quoted rather than derived; and it appears in the pneumatic-muscle literature of the 1950s — McKibben’s actuator — where the volume route is the natural one because contraction is the point.

Its most famous appearance is in a third field altogether, and the coincidence is exact: arctan √2 is the magic angle of nuclear magnetic resonance, where a sample spun at 54.74° to the field has its second-order broadening averaged away. The same number arrives there because the same expression, 3cos²θ − 1, vanishes at it. Nothing physical connects a spinning sample to a braided hose; what connects them is that both are asking when a particular quadratic in cos θ is stationary or zero, and there is only one angle in the first quadrant that does it.

One more consequence is worth recording because it is a design rule rather than a curiosity. A hose whose angle sits at the neutral value has no first-order length change, but it also has no restoring tendency — the volume is stationary, so nothing drives the angle back if something disturbs it. Hoses are therefore often braided deliberately a little below the neutral angle, so that pressure holds the braid against its end fittings rather than leaving it indifferent. The optimum is a maximum, and a maximum is a flat place to stand.

Where the ladder goes next #

A pressure fabric is cut before it is loaded, and it is cut smaller than the shape it is meant to take, because stressing it makes it grow. How much smaller is a question about the crimp coming out — and it turns out that with the thread lengths and the cloth’s thickness both fixed, no prestress can strain the fabric at all. The next rung finds where the strain actually comes from.