An inflated beam wrinkles at a moment with no cloth in it

An inflated cylinder wants an unbalanced cloth, because a pressure inside a closed tube pulls its wall twice as hard around the circumference as along the axis. That essay was about the tube holding its pressure. It left out what inflated tubes are mostly for: an air beam in a tent frame, the side tube of an inflatable boat, an emergency shelter’s arch, a drop-stitch paddle board’s rails. They are beams, and a beam made of cloth ought to be impossible.

It ought to be impossible because a cloth cannot carry a push. Bend any beam and one side is compressed; bend a tube of cloth and that side has nothing to resist with. What makes the air beam work is the other half of the two to one: the pressure’s pull along the axis, which stretches the whole wall before any load arrives. Bending does not have to put the inside of the bend into compression. It only has to take some of that pre-tension away.

That gives the beam a strength with an exact value, and the value contains the pressure and the radius and nothing else. The cloth is in the answer twice, and neither time where the strength is.

The pressure pulls the wall along the tube #

A closed tube of radius r at a pressure p has a force on each end cap equal to the pressure times the area of the cap, $\pi r^2 p$. That force is carried by the wall, spread round the circumference, as a pull along the axis of p r over two newtons for every metre of circumference.

Take a tube 200 millimetres across at half a bar, 50 kilopascals above the air outside, which is an ordinary pressure for a low-pressure air beam or an inflatable’s main tube. Its end caps are pushed out with 1,571 newtons. Its wall carries 2.5 kilonewtons a metre along the axis and twice that, 5 kilonewtons a metre, around the circumference — the hoop load the unbalanced cloth was sized for.

Nothing has been bent yet, and the whole wall is already taut in both directions.

Bending takes the pull away from one side #

Bend the tube with a moment M. As in any thin tube, the bending adds a force to the wall that varies as the cosine of the angle round the section: tension on the outside of the bend, and on the inside a force of the same size in compression. For a thin tube its peak is the moment divided by $\pi r^2$.

On the outside of the bend the two pulls add. On the inside the bending subtracts from the pressure’s pull, and while there is any of it left the inside is still in tension and the tube is an ordinary beam, stiff in exactly the way a steel tube is.

The inside runs out of tension when the bending’s peak equals the pressure’s pull: the moment over $\pi r^2$ equal to $pr/2$. So the inside of the bend goes slack at

$$M = \frac{\pi p r^3}{2}$$

For the tube at half a bar that is 78.5 newton metres. It has a pressure in it and a radius cubed, and nothing that belongs to the cloth — not its modulus, not its weave, not its strength.

The same arithmetic as prestressed concrete #

A reader who has met prestressed concrete has met this result the other way round. Concrete carries almost no tension; a steel tendon stretched through a concrete beam squeezes it along its length, so that bending has to undo the squeeze before any part of the section is in tension, and the beam does not crack until it has.

An air beam is prestressed concrete turned inside out. The cloth carries no compression, the air stretches it along its length, and bending has to undo that stretch before any part of the wall is slack. The decompression moment of a prestressed section is its prestress force times its section modulus over its area; for a thin tube that ratio is half the radius, and the prestress force is the pressure’s end force, so the moment is $\pi r^2 p$ times $r/2$ — the same $\pi p r^3/2$. The engineer tensioning a tendon and the pump filling a tube are doing the same job to two materials that fail in opposite directions.

Past the first wrinkle the tube does not fail #

A concrete beam that cracks does not collapse, and an air beam whose inside goes slack does not fold. What happens is that the slack side wrinkles — at a wavelength set by the cloth’s bending stiffness, which is the one place bending stiffness enters — and the taut part of the wall has to carry both the pressure’s end force and the moment on its own.

Keep plane sections over the taut arc and put the edge of the slack at an angle $\varphi$ either side of the outside of the bend. The taut wall’s force then rises from nothing at that edge as the cosine does, and two conditions fix it: its total must still balance the pressure’s end force, and its moment about the axis must be the applied moment. Together they give the moment as a fraction of the end force times the radius, and that fraction is a half when the whole circumference is taut and one as the taut arc closes to a line.

So the tube carries more after it wrinkles, up to twice as much. With a fifth of its wall slack it carries 1.14 times its wrinkling moment; with half slack, 1.57 times; with two thirds slack, 1.8. It folds at $\pi p r^3$, when the whole of the pressure’s end force is carried by a line of wall at the outside of the bend a full radius from the axis — 157 newton metres for the tube at half a bar.

The folding moment also has no cloth in it. A tube’s strength, from first wrinkle to fold, is a pressure and a radius cubed, and two numbers — a half and one — that come from the geometry of a circle.

One curve for every tube of every cloth #

The cloth does appear, once the question is how far the tube bends rather than how much it carries.

Below wrinkling the tube’s bending stiffness is its cloth’s tensile stiffness per unit width times $\pi r^3$, exactly as for a thin metal tube, so the curvature is the moment divided by that. Past wrinkling only the taut arc resists, and its stiffness is the cloth’s tensile stiffness per width times $r^3$ times a function of the arc that falls to nothing as the arc closes. Doubling the cloth’s stiffness halves the curvature at every moment and changes no moment at all.

Divide the moment by the wrinkling moment and the curvature by its value there, and every inflated tube there is lies on one curve. Its height is fixed by the pressure and the radius; its horizontal stretch by the cloth. It is straight to one, then bends over — 1.37 times the moment at 1.8 times the curvature, 1.57 times at 3.1, 1.8 times at about 9 — and rises towards two without reaching it, because the last of the taut arc is the stiffness that runs out.

A cantilever wrinkles and folds at the same loads whatever its cloth #

Stand the tube up as a cantilever two metres long and hang a load on its tip. The moment is largest at the root, so the root wrinkles first, at the wrinkling moment over the length: 39.3 newtons, about four kilograms. It folds at the root at twice that, 78.5 newtons.

What the cloth decides is the deflection on the way. For a cloth stiffness of a meganewton a metre, an illustrative figure for a coated technical fabric, the tip has gone 33 millimetres when the root wrinkles. At 1.6 times that load the root is wrinkled over 0.75 metres and the tip has gone 70; at 1.9 times, 0.95 metres wrinkled and 185 millimetres. A cloth half as stiff doubles every one of those deflections; one twice as stiff halves them. None of them moves either load.

That is the practical shape of the result. A designer who chooses a stiffer cloth for an air beam gets a beam that sags less under its working load, and a beam that wrinkles and folds at exactly the same loads as before.

The cloth sets the strength through the pressure #

The cloth is kept out of the moments at a given pressure. It comes back through the pressure the tube will hold.

A tube bursts, or its seams fail, when its hoop load reaches what the cloth and its joints will carry, and the hoop load is the pressure times the radius. So a cloth rated at a hoop strength T allows a pressure of T over r at most, and at that pressure the wrinkling moment is $\pi T r^2/2$ — the cloth’s strength per width times an area. For a cloth good for 20 kilonewtons a metre round a 100-millimetre radius, that is a pressure of two bar and a wrinkling moment of 314 newton metres, four times the half-bar tube’s.

The two slopes are the argument for large, low-pressure beams and the argument against them at once. At a fixed pressure a tube twice as wide is eight times as strong, which is why emergency shelters use fat arches at a few kilopascals. At the most pressure a cloth will hold, a tube twice as wide is only four times as strong, because the wider tube has to run at half the pressure — and a safety factor, which divides the burst pressure, divides this line and leaves every thin line alone.

A kinked hose straightens when the tap is opened #

Everybody has done the experiment. A garden hose lying empty kinks at the first sharp bend, and the kink springs out when the tap is turned on.

An empty hose has a pressure of nothing, a wrinkling moment of nothing and a folding moment of nothing, so it folds wherever it is bent, with only its own rubber resisting. Opening the tap raises the folding moment from nothing to $\pi p r^3$, and a kink held by less than that is pushed straight. A hose of 8 millimetres inside radius at 3 bar has a folding moment of about half a newton metre, added to whatever its rubber wall resists with on its own. The hose essay found the braid angle a pressurised hose wants with no material constant in it; the moment that holds the hose straight has none either.

A drop-stitch board is two membranes held apart #

A paddle board or an inflatable floor is not a tube but two flat sheets joined by thousands of threads, and it is stiff for a related reason. Its pressure pulls both sheets taut in their own plane, and bending the board has to take that pull away from the sheet on the inside of the bend before that sheet can go slack. The board’s wrinkling moment per unit width is therefore the sheet’s pre-tension times the distance between the sheets, and it too has no cloth modulus in it. Its pressure is much higher than an air beam’s, typically around a bar, because the distance between the sheets is only ten or fifteen centimetres and a person standing on a board puts a large moment into it.

That extension is stated rather than computed here: a drop-stitch board’s section is two flat skins rather than a circle, the threads between them carry the pressure’s pull across the thickness, and the sheet’s pre-tension is set by those threads rather than by an end cap.

What the cloth’s other properties do #

The cloth is not idle in a real air beam, and it is worth saying where its properties go if not into the strength.

Its tensile stiffness sets the deflection, as above, and it sets how much the tube grows when it is pumped up, which a membrane has to be cut smaller to allow for. Its balance sets whether the tube lengthens or shortens as it inflates, which is the neutral angle again when the cloth is laid at a bias, and whether a biaxial load can move one direction at all. Its bending stiffness sets the wrinkles’ wavelength and nothing about the load they start at. And its coating decides whether the air stays in, which is a separate failure at the holes.

Its shear stiffness is the one property this beam theory leaves out and a real air beam cannot. A woven cloth has almost none until its threads lock — the bias is a mechanism, not a material — so a short air beam deflects by shearing as well as by bending, and its deflection is larger than the lines above. Pressure stiffens that shear too in the fuller theories of inflated beams, but it does not enter the moments at which the tube wrinkles and folds.

Two conditions and a circle #

The pre-tension is the pressure’s end force spread round the wall. The bending adds the thin-tube cosine, and the wrinkling moment is where the two cancel on the inside of the bend. Past it, the wall force over a taut arc of half-angle $\varphi$ rises from nothing at the arc’s edge as the cosine does; requiring it to carry the end force and the moment gives the moment as $\dfrac{\varphi - \sin\varphi\cos\varphi}{2(\sin\varphi - \varphi\cos\varphi)}$ times the end force and the radius, and the bending stiffness over the arc as the cloth’s tensile stiffness per width times $r^3(\varphi - \sin\varphi\cos\varphi)$. The cantilever’s tip deflection is that curvature integrated along the beam against the distance to the tip, with the moment falling linearly from the root.

The closed forms were checked against the thing they summarise. At five taut arcs the wall force was integrated numerically round the section and carried the pressure’s end force and the closed-form moment to a part in a thousand. The moment was confirmed to be $\pi p r^3/2$ with the whole wall taut, with the curvature continuing the straight part, to rise monotonically as the arc closes, and to reach $\pi p r^3$ within a part in a thousand as it vanishes. Doubling the cloth’s stiffness was confirmed to halve every curvature and every tip deflection and to move no moment; doubling the pressure to double the moments and doubling the radius to multiply them by eight; and below wrinkling the cantilever’s tip to be the ordinary $FL^3/3EI$.

Where the beam is only a beam #

The section stays round. Past wrinkling a real tube’s slack side buckles into folds and its section flattens towards an oval, which moves the taut arc nearer the axis and lowers what the tube carries before it folds. The folding moment of $\pi p r^3$ is the membrane result for a section that keeps its shape, and a real tube, whose section does not, should fold somewhat before it; by how much is not computed here.

Plane sections are assumed over the taut arc. It is the ordinary beam assumption, carried into a wall part of which is slack; it is what makes the closed forms possible, and it ignores the way load spreads round a thin wall near the edge of a wrinkle.

The pressure does not change as the tube bends. A sealed tube bent hard loses a little volume, and its pressure rises; a tube connected to a pump or a large reservoir holds its pressure. The difference is small until the fold.

Shear, seams and ends are left out. The shear flexibility of a woven cloth, the seam that runs along most tubes, and the end caps and fittings where a cantilever is held all change a real beam’s deflection and several change where it fails, and the stiffnesses quoted for the cloth are illustrative bracketing values rather than measurements of a particular fabric.

Still open: where along a curved arch the first wrinkle forms #

Every result here is for a straight tube. The commonest air beam is an arch, curved before it is loaded, and a curved tube under pressure is not uniformly pre-tensioned: the outside of the curve has more wall than the inside, and a tube of one size presses and pulls differently where its circumference differs. An arch loaded by wind therefore starts with its inside wall carrying a different pre-tension from its outside, and the first wrinkle should form where the arch’s own curvature and the load’s moment combine worst — which need not be where the load’s moment alone is largest. Where that is, and how much it lowers the arch’s wrinkling load below a straight tube’s, needs the curved tube’s pre-tension followed round its bend, and has not been done here.