Textile Structure
how cloth is put together
Concept

Warp tension — where it appears

How hard the warp is held on a loom, which the let-off sets and the shed adds to at every shaft. It is what the beat-up works against, and its size decides the pick density a loom can reach before the fell stops moving.

Named by 18 essays across 6 fields — each of them below, with the objects they name alongside it.

How much a cloth can give back. Relaxation shrinkage against how much of the warp crimp the loom took out, with the area shrinkage above it and the ceiling drawn across. The ceiling is the crimp itself, read as c/(1+c): it is all the length there is to give, and no tension reaches past it.

Why the warp shrinks more

A woven cloth almost always loses more length than width, and the reason is not in the cloth. It is in the machine — the warp is held under tension for the whole of weaving and the weft is held for a fraction of a second — so the two systems arrive at the finishing works with different amounts of crimp missing.

finishing · Shrinkage The force at a crossing. One warp end of a sheeting riding over three picks, with the weave angle Peirce's geometry solves for at that construction: 36.8°. An end held at 0.50 N presses each pick it crosses with 0.599 N, which is twice the tension times the sine of the angle and has no material constant in it. What the drawing cannot show is the relaxed case: a cloth with no tension in it still holds its threads together, and what does the holding then is the yarn's own resistance to being bent, which needs an elastica not available here.

Every crossing is a force

A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.

cloth · Contact What a shed costs, in newtons. The tension the shed puts into one end at each shaft of a 24-shaft harness, for a 25 tex cotton yarn. The strain is set by the loom's geometry alone; the tension is that strain times the yarn's modulus. The front shaft holds 0.52 N and the back 1.95 N, which is 14 and 52 per cent of the yarn's breaking load. What the chart cannot show is the rest of the warp tension, which the let-off adds on top of all of these and which no geometry decides.

What the shed costs, in newtons

The shed's strain has been computed here and could not be priced: a strain is a length over a length and says nothing about how hard a thread is being pulled. A modulus turns it into a tension — and the back shaft of a twenty-four-shaft harness turns out to hold its ends at half their breaking load, all day, from the geometry alone.

mechanics · Yarn stiffness The beat-up, at the fell. The last picks of a sheeting at 26 picks per centimetre, with the beat-up zone shaded. Driving the fell forward makes the warp take more crimp and more crimp takes more thread, which the warp can only supply by stretching — so the force is the warp tension times the crimp's elasticity with respect to the pick spacing, 0.182 here. That is 0.205 N per end and 573 N per metre of reed. What the drawing cannot show is that the shaded band's width cancels out of the derivation exactly; it is drawn because a reader needs to see what is being compressed, not because the answer depends on it.

The blow that sets the pick

The take-up gear decides how far the cloth moves between picks and says nothing about the blow that puts each pick where it goes. That blow is a force, and a virtual-work argument gives it in one line — with the length of the beat-up zone cancelling out of the answer exactly, which is the part worth having.

setting · Beat-up How much too thick a round section is, and what reconciles it. For each cloth in the standard cloth table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

mechanics · Compression The two halves of a beat-up force. The force the reed must apply per metre, for a sheeting, against the number of picks that are still sliding against the warp. The elastic half — the warp tension times the crimp's elasticity with respect to the pick spacing — is 573 N/m and does not depend on the zone at all; that cancellation is exact and is the result the rung below established. The frictional half is 1133 N/m per sliding pick and is nothing but zone. Against a reported 400–1500 N/m, that leaves room for at most 0.82 picks sliding — so the fell region a weaver can see, ten to fifty picks deep, is not the same quantity as the picks that are still moving. What the rows cannot show is that this is a static friction throughout, and a beat-up is a blow.

The half of the beat-up that is all zone

The elastic half of the beat-up force is exact and the length of the beat-up zone cancels out of it, which is this site's own result and disagrees with every practical account of weaving. The frictional half is nothing but zone — and requiring the total to match the force a loom is actually built to apply puts the number of picks still sliding at about one.

setting · Beat-up plain over 5-end satin: what each layer's warp does. A warp end of each layer of a double cloth, in section over the same span of cloth, with the length of warp each eats drawn beneath. The crimps come from Peirce's geometry asked about a thread that bends at its own average rate rather than at every crossing, and they are 14.35% for the plain and 2.03% for the 5-end satin. They are not the same, so the two layers consume warp at different rates: a hundred metres of cloth takes 114.4 m of warp from one and 102.0 m from the other. A beam delivers one rate, so the difference has nowhere to go and accumulates with the length woven — one pick spacing after 4 mm. What the sections cannot show is the yarn's thickness, which is exaggerated so the path is legible; the crimps beside them are computed at the real diameter and are not read off these drawings.

Two layers need two beams

A layer weaving a metre of cloth eats one plus its crimp metres of warp, and a beam delivers one rate. A plain face over a five-end satin back differs by twelve percentage points of crimp, which is twelve metres of warp over a hundred-metre piece and one pick spacing of slack after four millimetres of weaving. The difference has nowhere to go and does not settle — so the only double cloth that can share a beam is two layers of the same weave at the same sett.

fancy · Double cloth How many layers the repeat, the harness and the beams each allow. Three ceilings on the number of layers a double cloth can have, for five layer weaves. The repeat's bound is half its ends and is a property of the notation. The harness's is the strain budget — 13 shafts on an ordinary broad loom at a 1.0% warp strain limit — divided by the shafts one layer of that weave costs. The beams' is how many warps the loom carries, which is 2. The shortest of each three is marked, and it is the beams at the coarse end of the table and the harness at the fine end; the repeat is never the binding one except at two-end layers, where it happens to coincide with the harness. A double cloth of eight-end satin layers needs 16 shafts and the budget is 13, so it is a jacquard construction by arithmetic rather than by choice. What the bars cannot show is the pick rate: a k-layer cloth needs k times the picks per centimetre of finished cloth and takes k times as long to weave, which is a cost rather than a ceiling and is the reason four-layer cloths are rare even where they are possible.

The repeat allows four layers and the loom allows two

A repeat of eight ends can hold four separable cloths, and there is a witness that reaches the bound exactly. No loom weaves four. The harness's strain budget buys thirteen shafts and a layer costs its own weave's shaft count, so five-end satin layers stop at two and eight-end satin layers cannot be doubled on a dobby at all — and two differing layers already want a beam each. The notation's ceiling is the only one of the three that is never binding.

fancy · Double cloth What the crossing end asks for, against what a cam gives it. The length a leno's crossing end needs at each fraction of the crossed shed's opening, against an easer driven in proportion to the shed. The demand is nothing for the first eighth of the opening and then rises steeply, because the kink's extra length is a hypotenuse less its run and grows as the square of the climb. The proportional easer runs ahead of it everywhere, worst at 0.375 of the way open, where it has given 16.5 millimetres more than the end can use.

A cam easer gives its slack too early

The essay before it left the crossing end with fifteen millimetres it could not use at half the shed, and three things it might do with them. All three are decidable. It cannot reach the shuttle, because the same eyes that trapped the kink keep the slack behind the harness — and the margin is negative, so that friction is not a nuisance here but the reason a leno weaves at all. It cannot snarl, because the span is under half this account's own threshold. It sags seventy-two millimetres where the easer gives it, and the repair is a cam cut to the demand rather than to the shed.

fancy · Leno What a figured warp pays in tension to share one beam. For six pairings of figure and ground, the difference in warp tension that lets their ends share one beam: the force per end that takes the take-up difference out of the ground's crimp, along that region's own constant-length locus, with the yarn's rigidity at both ends of the band two tests place it in. a damask: satin on its own complement, 0.00% apart: none needed; an eight-end satin figure on a five-end satin ground, 0.31% apart: 0.006 to 0.020 N; a five-end satin figure on a 3/1 twill ground, 0.90% apart: 0.021 to 0.069 N; an eight-end satin figure on a 2/2 twill ground, 3.02% apart: 0.304 to 0.995 N; an eight-end satin figure on a plain ground, 14.15% apart: past what the ground's crimp can give; a 2/2 twill figure on a plain ground, 11.14% apart: past what the ground's crimp can give. The loom's front shaft holds 0.52 N an end. What the chart cannot show is how uneven a warp's tension may be before it shows in the cloth.

A figured warp pays in tension before it needs a beam

Figure and ground take up warp at different rates, and counted against a fixed slack, any difference at all — a third of a per cent — made a round figure need four beams. A real let-off holds tension, not length. Ends that consume more pull harder, and an end pulled harder gives up crimp, until every end consumes alike. For an eight-end satin figure on a five-end satin ground that costs a hundredth of a newton an end, and one beam serves. For a satin on plain it costs more crimp than the ground has, and no tension will do.

fancy · Jacquard The crimp split of a muslin against the warp's tension. The warp's crimp and the weft's at the least-energy state of a muslin at its own sett, with the warp held at tensions from 0.0001 to 1.0 newtons, and the yarn's rigidity 4.0 times the free bound. With no pull the warp holds 28.5 per cent and the weft 0.0; they are equal at 0.019 N, and by the front shaft's 0.52 N the warp holds 0.00 per cent. What the plot cannot show is friction: a real cloth is held wherever its crossings stop it, not at the least state.

The loom hands the crimp to the weft

Bending alone gave the fixed-sett energy nothing to say: six of eight cloths fell to the end of their interval with the weft dead straight. Put the warp's tension in and the answer is not a well but a switch. The crimp changes hands across a factor of two or three in tension, at a few hundredths of a newton — and the loom holds its warp at half a newton, so on the loom every cloth's warp is as straight as its geometry allows.

setting · Crimp The springs a leno easer can be. The stiffest spring, per end, that an easer can carry without any span of the crossing end rising above an ordinary end's working tension at any fraction of the crossed shed, against how far its preload sits above the warp's resting tension of 0.5 N. Behind the harness, where an easing bar acts, it is 1.1 N/m at a preload equal to the resting tension and falls to nothing by 15 per cent above it; in front of the harness it is 4.8 N/m. Everything under a curve works and nothing above it does. The window is a corner: a spring barely stiffer than a dead weight, with a preload matched to the warp's own tension. What the chart cannot show is the bar's mass, which the loom's speed makes matter and which the next views take up.

A leno easer should be a light weight

A spring easer gives length when the crossing end pulls, so it cannot give it too early, which was the fault in a cam driven off the shed. The question left was its rate. The answer is that it hardly has one. Behind the harness the easer feels the back span, and two gripping eyes keep that span within fifteen per cent of its resting tension while the kink carries its load. So the spring must hold its span almost constant over a sixty-eight-millimetre stroke: at most 1.1 newtons a metre per end, a dead weight in all but name. At speed the bar's own mass is what limits it, and it falls with the square of the loom's speed: 2.3 grams an end at a hundred picks a minute, 0.6 at two hundred.

fancy · Leno A disc woven on one beam, as it catches a lamp. A disc of 5-float satin on a ground of float 3, twelve blocks square, woven on one beam so that every column is pulled to the tension that makes it consume what the others do, and drawn as its blocks catch a lamp. A block is lit when its warp's crossings still reflect into the eye; outlined blocks are figure. At the mirror every block is lit. Tilted by 10.8° along the warp, the ground in the columns that carry most ground — the tightest — has gone dark, while the ground in line with the figure's widest part, pulled least, still shines: a band as wide as the figure runs down the ground above and below it. What the drawing cannot show is the diffuse light every block also returns, which dilutes the contrast and does not reverse it.

A figure woven on one beam shows its columns off the mirror

A figured warp shares one beam by letting the columns that want more yarn pull tighter until every column consumes alike. With the figure allowed to give crimp as well as the ground, the tightest column of a satin disc on a twill needs 0.013 to 0.042 newtons, under a tenth of the working tension. Its warp lies tens of micrometres lower, a relief too small to see. At the mirror it shines exactly as a slack column does. A few degrees off the mirror, along the warp, its crossings stop reflecting before a slack column's do, and the figure's own columns appear as bands in the ground. Only a damask has none.

fancy · Jacquard How likely one staple length of yarn is to be weaker than a tension, under three laws. The probability that one independent try — a staple length of a 20 tex cotton yarn — is weaker than a tension, on a logarithmic scale, with the tension as a fraction of the yarn's 500 mm breaking load. Three laws are drawn, a normal, a lognormal and a Weibull, each fitted to reproduce the same two tensile tests at 100 and 500 mm exactly. Near the breaking load, where a tensile test samples at about one chance in 18, they lie together. At one chance in 1.4×10⁸, where a warp of 4,000 ends by 1,000 metres samples, they are far apart: at the back shaft's 52 per cent the three give normal 1.3×10⁻⁷, lognormal 1.1×10⁻¹¹, Weibull 4.0×10⁻⁶ per try. What the chart cannot show is which of the three a real yarn follows, which no tensile test can decide.

A warp breaks in a tail no tensile test reaches

A 500 mm tensile test is the weakest of about eighteen staple lengths of yarn. A warp of four thousand ends by a thousand metres is the weakest of a hundred and forty million, and an end breaks wherever one of them is weaker than the shed's tension. Three strength laws fitted to the same two tensile tests agree within five per cent at every gauge a tester can clamp, and disagree by a factor of 375,000 on how many ends a warp will break at the back shaft.

setting · Assembly Where the weight on a shared leno easer comes to rest. The tension in the back span of each end of a leno pair — the length between its last heddle and the back rest, where an easing bar acts — at each fraction of the crossed shed, with one bar under both ends weighted at 1.0 N a pair, twice the warp's resting tension, so that it rests on its stop with the shed closed. The two back spans always sum to the weight. Closed, they share it equally. The partner, an ordinary end lying in the bottom of the shed, can take only 11 mm of the bar's travel before it goes slack, which it does at 37.5% of the opening; from there the crossing end's back span carries the whole 1.0 N. On a bar of its own, dashed, the same span stays at 0.50 N. What the chart cannot show is what the crossing end's other spans do with the extra tension, which the next views take up.

One weight under a leno pair ends on one end

A leno's crossing end wants about seventy millimetres eased to it at every crossed shed; its partner lies in the bottom of the shed as an ordinary end and wants none. One weighted bar under both is set where the closed shed shares its weight between them, but the partner can take only eleven millimetres before it hangs, and it is slack by three eighths of the shed. From there the crossing end carries the whole weight, twice the rest its own bar would hold it at, and its tightest span ends forty per cent over the working tension while the partner hangs thirteen centimetres behind the harness. No weight serves both, because a bar gives every end it carries one length.

fancy · Leno A patterning bar's thread through one repeat. One guide-bar thread followed through a 12-course repeat, needle places across and courses down, with each course's underlap drawn beside it as a bar against the beam's steady delivery, the repeat's mean of 1.984 mm. The shogs are 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3: 6 of 1 space (1.155 mm) and 6 of 3 spaces (2.814 mm), at 28 gauge and 14 courses a centimetre. The overlaps, the same on every course, are not drawn. What the drawing cannot show is the tension: a course whose bar runs past the mean is a course the beam has not yet paid for.

A patterned warp knit needs a stroke as long as its stripe

A guide bar that changes its shog from course to course takes yarn at a changing rate from a beam that delivers at one. The overlap is the same on every course, so it cancels again, and the running shortfall is the sum of each course's underlap less the mean — exact, with no loop in it. It returns to nought every repeat and strays inside it by a stroke that something must absorb. For a stripe the stroke is about the stripe's own height in the cloth, whatever the machine's gauge. The same twelve courses of tricot and satin need six times the stroke in two blocks as alternating, and of all 924 orderings exactly one pattern needs the least.

knits · Warp knit A leno partner lifted by its doup's loop. The tension in each span of a leno's partner end at the full crossed shed, against how far the doup's loop lifts it above its own straight line 6 mm in front of its heddle on the doup's shaft. The partner is then three spans: fell to loop, loop to heddle, heddle to back rest. With the loop clear it is an ordinary end, 1.07 N in its front span. At a 4 mm lift the front span carries 1.48 N, +38% on the untouched partner, the span between loop and heddle 1.26 and the back span 1.00; with nothing gripping, dotted, all three would carry 1.12. At 8 mm the front span reaches 1.90 N. What the chart cannot show is how far a real loop lifts its partner, which the doup's standard decides and which is swept here rather than computed.

A leno's slackener eases the partner, not the doup

In a bottom-doup leno the doup's loop passes beneath the partner, and if it lifts the partner on its way up it puts a second turn six millimetres in front of the partner's heddle. A four-millimetre lift bends the partner through 74 degrees in two places, and its front span goes from 1.07 newtons to 1.48 — a tenth of that from the bend and the rest from grip. The back span, which is all an easing bar can feel, does not move. A slackener that drops the doup's standard four millimetres clears the partner exactly, gives the crossing end six per cent of its easing, and costs the shuttle nothing.

fancy · Leno The partner's peak against how late a slackener starts. The largest tension the partner's front span reaches, against the share of the shed opened when the slackener starts to drop, at a 4 mm rise. It stays at 1.07 N for any start up to 0.748 of the shed, where the partner with no slackener passes that tension. Later than that, the peak is the touched tension at the moment it starts, up to 1.39 N at 0.95, against 1.48 N for no slackener at all.

A leno's slackener only has to arrive

A slackener that lets the doup down in step with the loop keeps a leno's partner clear at every moment of the shed, and it looked as though a real one would have to track that closely. It does not. A partner's tension only rises as the shed opens, so a slackener can leave its whole drop until three quarters of the way up and the peak does not move at all. Lateness is paid for elsewhere: starting later than 0.58 of the shed rubs the loop more than having no slackener, and above 234 picks a minute a falling weight cannot start as late as the partner would let it.

fancy · Leno

Named alongside it

The objects these essays reach for when they reach for this one.

CrimpCapstanShedEaserBeamDoupLenoSpecificationJacquardContact forceFrictionLoom-state

All concepts