A tensioned cloth loses its load

Tension a fabric across a frame, note the load, and come back in the morning. The load is down. The cloth has not moved — a ruler across it reads what it read the night before, the clamps have not slipped, and the frame has not bent. Something has gone and it is not length.

The usual name for this is creep, or stress relaxation, and the usual account borrows a fibre’s own time-dependence: polymers flow slowly under load, so the fibres in the cloth have relieved some of their stress. That account is not available here, and this collection has no business borrowing it — a fibre’s viscoelasticity is a matter for a physics of polymers that is not this subject. It also turns out not to be needed. A woven cloth loses load at constant length for a reason that is entirely mechanical, and the amount it loses can be computed from two numbers this site already carries — a contact force and a friction coefficient.

The claim #

A cloth held at constant length settles at a load that is a property of the cloth and not of the tension that was applied to it.

The consequence follows in one line. If the settled load is fixed and the applied load is whatever was applied, then the fraction retained is the first divided by the second, and pulling harder buys a smaller share. A cloth tensioned to just past its resting band keeps almost all of its load; the same cloth tensioned a per cent further keeps a third.

And the second consequence is the one that decides what to do about it. Re-tensioning does not help, because the load it settles at afterwards is the same floor it settled at before. What re-tensioning buys is the time between the pull and the settling, and nothing else.

The mechanism, which needs no rate #

Every crossing in a woven cloth is held by friction. This collection established that when it gave the locus a force: a cloth’s resting length is not a point but a band, because a state can only slide along the locus when the energy it would release exceeds what friction takes to move the crossings, and inside the band it cannot.

Now clamp the cloth at a length past the band’s edge. The cloth as a whole cannot change length — the clamps see to that. But the cloth is not one crossing. It is a great many, and they need not all be at the same extension: one patch can take a little more and its neighbour a little less, with the total unchanged.

So the cloth rearranges internally. Each patch slides along its own locus in whichever direction relieves it, until the local driving force everywhere is down to what friction can hold. The mean length is untouched. The load, which is the average of the local loads, is not: it has fallen to the frictional level.

That level is friction times a contact force times the sine of a weave angle, and none of those three depends on how far the cloth was pulled. Hence the floor.

What the numbers are #

For a sheeting at a yarn-on-yarn friction of 0.3, the frictional floor is 0.0756 newtons per end and the resting band’s upper edge is at 3.68 per cent of strain.

Below 3.68 per cent nothing happens. The cloth was never outside what friction could hold, so there is nothing for it to give up and the load stays where it was put. That is a real and useful regime and it is easy to miss: a lightly tensioned fabric structure is dimensionally stable in a way a hard-tensioned one is not, for the same reason a lightly loaded bolt does not settle.

Above it the fall is fast. At 3.74 per cent the applied load is 0.0787 N per end and 96 per cent survives. At 4.22 per cent it is 0.113 and 67 per cent survives. At 4.94 per cent it is 0.262 and 29 per cent survives. The applied load is climbing steeply — the cloth is past its interchange and into its threads, so it is now stretching yarn rather than moving crimp, and that is expensive — while the floor stays exactly where it is.

The steepness of the load–extension curve past the jam is what makes the retained fraction collapse. It is the same steepness that makes a woven cloth feel firm when it is pulled hard, and it works against the person pulling it.

What re-tensioning is for, and what it is not for #

The design question underneath is usually put as a schedule: how often must an awning, a screen or a webbing sling be re-tensioned?

Put that way it has no answer here and this collection declines to invent one, because nothing in this ladder has a clock in it. What it can answer is the question underneath the schedule, which is whether re-tensioning gets anywhere.

It does not, in the following precise sense. Ask what tension would have to be applied so that the cloth settles at a stated working load. If the working load is below the frictional floor, no tension is needed at all — the cloth will hold it from inside its own band. If the working load is above the floor, there is no answer, because the settled load is the floor whatever was applied. Re-tensioning restores the load for as long as the rearrangement takes and then gives it back.

The consequences for design are three, and they are not the obvious ones.

Design to the floor, not to the pull. The load a structure can be relied on to carry is the frictional floor, which is a property of the cloth’s construction and its yarn friction. Everything above it is transient.

Raise the floor rather than the pull. The floor rises with friction and with the contact force at the crossings, which is to say with how firmly the cloth is woven. A closely set cloth of a high-friction yarn settles at a higher load than an open cloth of a slippery one, and no amount of tensioning changes the ordering.

Or leave the cloth inside its band. A structure tensioned to less than the band’s upper edge does not relax at all. That is a low tension by the standards of most fabric structures, and it is the only regime in which the answer to “how often” is “never”.

What the floor is made of, and which term to attack #

The floor is friction times a contact force times the sine of a weave angle, and the three terms are altered by three different people — so it is worth taking them apart before advising anybody to raise it.

The friction is the finisher’s, and it is the term with the widest range and the least control. Cotton on cotton runs from 0.2 to 0.4 depending on what has been put on it, and a softening finish moves it towards the bottom of that range. So a structure fabric that has been finished for handle has had its floor lowered by up to a third, by an operation performed for a reason that has nothing to do with load.

The contact force is the weaver’s, through the sett and the yarn. It rises with the cover, because a closer cloth crowds its threads harder, and it is the term a specification can genuinely be written against — a construction is a durable statement in a way a finish is not.

And the weave angle is the weaver’s too, through the same geometry, but it enters as a sine and is bounded above by one. A cloth cannot get more than a factor of two out of it however it is constructed, and ordinary cloths already sit between a fifth and four fifths of the way up that range.

So of the three, one is bounded and small, one is a construction decision, and one is a finish that can silently undo the other two. The order of leverage is friction, then sett, then weave, and it is exactly the reverse of the order in which a fabric structure is usually specified — the weave is named first, the sett second, and the finish left to the finisher.

That is the practical form of the whole rung. A structural fabric’s reliable load is not written into its construction alone; a fifth of it is written into the last operation performed on it, by somebody who was asked for a handle.

And it explains a complaint the trade has without a mechanism for: that two rolls of nominally identical structural fabric behave differently on the frame. They can differ in nothing a specification records and still settle at loads a third apart, because the specification names the weave and the sett and leaves the friction to be whatever the finishing route produced.

What was counted, and how #

The load at a given strain is read off the computed load–extension curve by interpolation, and the machinery refuses a strain the curve does not reach rather than returning its end — this site has twice shipped a solver that handed back the last state it could evaluate as though it were an answer, and both times the symptom was plausible.

The retained fraction is asserted to fall monotonically across the sweep, which is a claim about the finding rather than about the numbers: a cloth tensioned harder must keep a smaller share, because the numerator is fixed.

The band itself comes from the frictional ladder and is asserted to straddle the least-energy state rather than sitting to one side of it, which is the check that catches a sign error in the friction term.

The two regimes, and which one a real structure is in #

It is worth putting the band edge and the working tension side by side, because the answer decides everything else and is usually not checked.

A sheeting’s band runs to 3.68 per cent of strain at a friction of 0.3. Fabric structures are commonly tensioned to a strain of a few per cent — enough to take the slack out, keep the surface taut against wind and stop it flapping — which puts the ordinary case squarely at the edge of the band and often past it. That is not an accident of the numbers. A cloth tensioned below its band edge is a cloth whose crossings have not been asked to move, and a cloth whose crossings have not moved is a cloth that still has its slack in it, which is the thing the tensioning was for. That is the same trade an awning shares with a garment: what comes back for nothing is the geometry’s, and what is left is the fibre’s.

So the two regimes are not “a well-designed structure” and “a badly designed one”. They are “taut enough to work” and “stable enough not to settle”, and for an ordinary woven cloth those two requirements very nearly exclude one another. The gap between them is the band’s own width, which is friction over stiffness, and a limper yarn makes it wider.

That last observation inverts the usual expectation and is worth stating on its own. The softest cloths have the widest bands and are therefore the most forgiving of being left tensioned, because a limp yarn stores less energy per unit of crimp released and friction holds it over a wider interval. A firm, stiff, high-modulus fabric — the sort chosen for a structure precisely because it is dimensionally reliable — has a narrow band and settles from almost anywhere.

Where the model stops #

The rearrangement is described and not solved. Nothing here computes how the local extensions distribute themselves, only where the process must stop. A proper account would be a statistical mechanics of a great many crossings with a distribution of frictional barriers, which is the same object the laundering ladder needs and which this collection has only in the crudest form.

There is no time anywhere in it. How long a cloth takes to settle is the question everyone asks and the one nothing here touches. The mechanism has a rate — it is set by how fast the local slips happen, which depends on vibration, handling and temperature — and this collection computes an endpoint.

The fibre’s own relaxation is real and is excluded. A polymer under load does relieve stress with time, and in a cloth held above its jam, where the threads are carrying the strain directly, that contribution is not small. What is claimed here is that the frictional mechanism is sufficient on its own to produce the observed shape, not that it is the only one acting.

And the floor is quoted at a friction coefficient that is a range. Cotton on cotton is reported between 0.2 and 0.4, so the floor is a range of the same width, and every retained fraction above should be read as a band a fifth wide either way. The orderings survive; the values are indicative.

The generalisation #

When a system’s settled state is set by a threshold rather than by an equilibrium, the settled value does not depend on the input that took it there. A bolted joint that relaxes to the level its threads’ friction can hold, a stack of stones that settles to its own angle of repose, a cable clamp, a magnetic domain wall pinned at a defect: in each case the final state is a property of the pinning and not of the disturbance, and the fraction of the disturbance retained therefore falls as the disturbance grows.

The design lesson generalises with it and is the useful half. In a threshold-limited system, more input buys less retention, so the correct move is to raise the threshold rather than the input. That is unintuitive in exactly the situations where it matters — the natural response to a slack structure is to pull it tighter, and pulling it tighter is the one thing that provably does not work.

The narrower lesson is about vocabulary. Calling this creep imports a material’s time-dependence into a problem that has a mechanical answer, and the import is hard to undo once made: a fabric structure specified with a creep allowance has a number in it that belongs to a polymer, when the number that governs it belongs to a friction coefficient and a weave angle.

Who found it, and when #

Load loss in tensioned fabric structures is thoroughly documented in that trade, and the practice of pre-stressing and re-tensioning is as old as the structures. The standard account attributes it to a combination of yarn creep and “crimp interchange under sustained load”, the second of which is this mechanism named without being computed.

The frictional band a cloth rests in is this collection’s own, from the ladder that gave the locus a force, and was originally about why a washed cloth’s relaxed length is a range rather than a value. Reading the same band as a floor on a held load rather than as an interval on a resting length appears to be new here, and it is the same object seen from ninety degrees away: one asks where a free cloth stops, the other what a clamped one keeps.

Where the ladder goes next #

The rearrangement this rung describes is the same one a wash performs deliberately, and the laundering ladder finds that a cloth shrinks five times rather than once because its frictional barriers are spread far wider than anybody’s friction table suggests.

Sideways, the strain a tensioned cloth is holding is strain spent from a budget, and spending it in one direction refunds it in the other — so a hard-tensioned cloth is not merely losing load, it is standing at the end of its own locus while it does so.