A thread is held one crossing at a time

Three quite different pieces of trade knowledge say the same thing in three vocabularies.

A seam allowance is a centimetre or so, and an openly woven cloth is given more. A frayed edge runs a few millimetres into a close cloth and centimetres into a loose one. A tuft in a carpet is bound by how many ground picks it wraps, and a specification states the force it takes to pull one out.

All three are questions about how far a thread has to be held before holding it beats breaking it, and all three have the same number underneath.

The claim #

The force needed to withdraw a thread from a cloth is the friction at one crossing multiplied by the number of crossings gripped, and the length at which it equals the thread’s breaking load is the quantity that decides whether a cloth frays, slips or holds.

Call it the crossover length. For the eight cloths in this site’s table, at an ordinary friction, it runs from 4.3 millimetres to 73.7 — and the ordinary apparel cloths sit between 7 and 10, which is a seam allowance.

The arithmetic, and the two curves in it #

Every crossing is a force: a thread at tension T crossing another at weave angle θ presses it with 2T sin θ. Friction at that contact resists sliding with μ times as much.

A pick gripped over a length L of cloth crosses L/p₁ warp ends, with p₁ the end spacing. So the resistance is

F_out = μ · 2T sin θ · L / p₁

which is proportional to L. The thread’s own breaking load is its tenacity times its count, and that is independent of L. Two straight lines, one through the origin and one horizontal, and they meet at

L = (breaking load) · p₁ / (μ · 2T sin θ)*

Below L* the thread comes out. Above it, the thread breaks and the cloth keeps it.

Three things are worth reading off that expression before any numbers.

It contains the sett twice. Once explicitly, as p₁ in the numerator — a closer sett means more crossings per millimetre. And once inside θ, because a closer sett makes a larger weave angle and a harder press. Both push the same way, which is why the effect across a table of cloths is large.

It is exactly inversely proportional to friction. μ appears once and nowhere else, so μ·L* is a constant for a given cloth — asserted here to twelve figures rather than remarked on, because a hyperbola that came out of a computation with several friction-dependent steps would be worth checking.

And it does not contain the float length. That is a genuine omission rather than a simplification, and it is taken up below.

The three rules of thumb, as one number #

Why a cut edge frays as far as it does. A thread at a raw edge is held only by the crossings between the edge and wherever it is being pulled from. Nothing holds it beyond the crossover length, so a cloth’s threads can be worked loose over about that distance and no further — a batiste frays four millimetres and a cheesecloth frays seven centimetres. Ravel, fray and run argues the topology of that: a woven thread comes out one at a time and a knitted one takes its neighbours with it. This is the distance.

Why a seam allowance is what it is. A sewn seam loads the threads beside the stitching, and the allowance is the length over which they are gripped. An allowance shorter than the crossover means the threads slide out and the seam opens with the fabric intact; longer, and the fabric or the thread gives first. Ten millimetres — the ordinary allowance — sits right in the middle of the table, which is why some cloths slip and some break at the same allowance.

Why a tuft is bound the way it is. A tuft is a thread gripped over a very short length indeed — two or three ground picks — so it is far below any crossover and comes out by sliding. That is why the pile ladder’s answer is a capstan ratio rather than a strength, and what it takes in newtons is this arithmetic on a shorter grip.

The three have been described separately for as long as anybody has written about cloth. They are one inequality read at three lengths.

The one knob, and its three consequences #

The friction coefficient is the only thing in the expression a finisher can move, and it moves all three consequences together and in the same direction.

Soften a cloth — a silicone finish, a fabric conditioner, a mechanical softening — and μ falls. Mercerising and calendering both move it too, in the course of doing something else entirely, which is one of the reasons a finished cloth’s seam behaviour is hard to predict from its greige state. The crossover length rises in exact proportion. So the cloth frays further, its seams slip sooner, and its pile is less firmly bound, all by the same factor, and none of them can be traded against the others.

That is a stronger statement than the trade makes. The usual account treats seam slippage and fraying as separate faults with separate remedies, and there are separate remedies — a French seam, a bound edge, a resin finish that glues rather than lubricates. What there is not is a way to soften a cloth and keep the grip, because the same coefficient is in both.

Run the numbers on a sheeting: at μ = 0.15 the crossover is 14.9 millimetres and at μ = 0.4 it is 5.6. A soft finish nearly triples the length of cloth a thread has to be held over, which is the difference between a seam that holds and a seam that opens.

Why the crossover does not order the cloths by sett #

The table runs from four millimetres to seventy-four and the ordering is not the sett’s, which is worth explaining because the reason is a competition between two quantities that both rise with the yarn.

The crossover is a breaking load over a grip per unit length. The breaking load rises with the yarn’s count — a coarse thread is strong — and the grip per unit length rises with the sett and with the weave angle, both of which fall as the yarn gets coarser at a fixed cover. So a coarse open cloth has a large numerator and a small denominator and lands at the top of the table, and a fine close cloth has the reverse.

That is why the duck sits third from the top at thirty millimetres despite being the heaviest cloth in the collection, and why the batiste sits at the bottom at four despite being the lightest. Neither cloth is where its weight would put it, and a designer reasoning from weight or from firmness would rank them the other way round.

The quantity that does order them is the ratio of the yarn’s strength to the cloth’s grip, which is not a number anybody quotes and does not have a name. Its two halves are decided by different people — the spinner sets the first and the weaver the second — and the ordering of a table of cloths by their fraying and slipping behaviour is therefore an ordering by a quantity that no single specification carries.

The practical form is a warning against a substitution. A cloth substituted for another of the same weight, the same cover and the same fibre can have a crossover several times different, if its yarn is coarser and its sett correspondingly opener. Every appearance property is unchanged and the seam behaviour is not, which is exactly the substitution a mill makes when a yarn count is unavailable and the construction is adjusted to keep the weight.

That substitution has a direction as well as a size. Holding the weight while coarsening the yarn means opening the sett, and opening the sett raises the crossover twice over — through the spacing and through the weave angle — while the coarser yarn raises it again through the breaking load. All three effects push the same way, so the substituted cloth frays further and slips sooner by a compounded factor rather than a marginal one, and there is no arrangement of the trade in which it goes the other way.

That makes a count substitution the one change to a construction that a seam specification should always be re-checked against, and the only one whose direction is known before the check is run.

What the model leaves out, and it is the float #

The expression counts crossings and ignores what happens between them, and a float is precisely what happens between them.

A thread lying on the face for four picks is not gripped over those four picks. It is gripped at the ends of the float and free in the middle, so a weave with long floats has fewer effective crossings per unit length than a plain weave at the same sett. The model above counts every intersection as a grip, which for a plain weave is right and for a satin is not.

The direction is known and the size is not. A satin at the same sett as a plain weave has a quarter of the interlacing points and should therefore have something like four times the crossover length — which is exactly why a satin frays worse than a plain weave of the same cloth, why satin seams slip, and why a satin-faced ribbon has to be cut with a hot knife.

That correction is a multiplication by the interlacing rate, which this site computes for every weave. It is not applied here because doing it properly needs an account of how a float’s two anchoring crossings share the load, and a capstan argument along a float is not the same as a sum of independent contacts. It is recorded as owed rather than approximated.

What was counted, and how #

Each cloth’s state is Peirce’s solution at its quoted construction, checked against the equations it was solved from. The weave angle comes out of that; the end spacing is ten over the sett; the breaking load is the yarn’s tenacity times its translation efficiency times the twist obliquity, all stated constants from the yarn’s mechanics.

The tension in the gripped thread is stated at half a newton throughout, and every crossover length is inversely proportional to it. That is the least defensible number on the page and it is the one a reader is most likely to want to change, so it is an argument everywhere rather than a constant.

Two assertions guard the result. The crossover length must fall as friction rises, which would fail on a sign error and is the kind of sign error that produces an entirely plausible table. And μ·L* must be constant across the friction range to twelve figures, which checks that friction has entered once and only once — the computation runs through a contact force, a per-crossing grip and a per-millimetre rate, and a second appearance of μ anywhere in that chain would be invisible in any single value.

The friction range itself is quoted, not measured: 0.2 to 0.4 for cotton on cotton, from the table the pile ladder has used since it was built, and it is a range because yarn-on-yarn friction depends on fibre, finish, moisture and crossing angle and is not a material constant.

Where the model stops #

The float correction is missing, as above, and it is the largest known error in the model.

Every crossing is treated as an independent contact. In reality a thread being withdrawn is a capstan problem: the tension in it falls as it passes each crossing, so the far crossings are pressed by a smaller tension than the near ones and contribute less. The independent-contact sum is an over-estimate of the resistance for a long grip, and the discrepancy grows with the length — which means the crossover length computed here is, if anything, short.

The tension is externally applied. In a relaxed cloth the crossings are pressed together by the threads’ own bending rather than by any tension, and that force is not available from this site’s model. A seam allowance at rest is exactly that case, so the numbers should be read as applying to a cloth already under some load — which a seam about to fail is.

And a real thread is not smooth. A spun yarn’s surface is hairy, and a hairy surface’s resistance to being pulled through a crossing is not simply μN; it has an entanglement component that a friction coefficient does not describe. That is the same mechanism felting runs on, and it is why a wool cloth’s threads are harder to withdraw than its friction coefficient says.

The generalisation #

The shape is one of the commonest in engineering and it is worth naming because the textile version hides it.

A length-proportional resistance meeting a length-independent strength gives a characteristic length, and that length organises the whole problem. It is the pull-out length of a fibre from a matrix, the development length of a reinforcing bar in concrete, the splice length of a rope, the overlap length of a lap joint. In every case the design question is the same: is the engagement longer or shorter than the characteristic length, and therefore does the system fail by sliding or by breaking?

Two things about the textile case are worth carrying back. First, the characteristic length is not designed; it emerges from the cloth’s own construction, and a maker choosing a sett is choosing it without being told. Second, the single parameter that shifts it — friction — cannot be moved for one consequence without moving it for the others, which is a much more common situation than the separate-remedies framing suggests.

And there is a warning in the float. A model that counts contacts is wrong wherever the contacts are not independent, and the case where they are not is exactly the case a designer reaches for when they want a smooth face. The most useful cloths for the model are the ones it handles worst.

Who found it, and when #

Yarn pull-out as a measured quantity belongs to the fabric-mechanics literature of the 1960s, and the standard reference is the seam-slippage and yarn-slippage testing that grew up around it; the observation that pull-out force rises with the length of engagement is in every account of it.

The capstan correction is Euler’s, from 1762, and this site has used it for the pile case since that ladder was built, where the wrap angles are large and the independent-contact sum would be badly wrong.

What is done here is to put the three consequences on one axis. Fraying distance, seam allowance and tuft anchorage are discussed in three separate literatures — finishing, garment construction and carpet specification — and the observation that they are one inequality read at three lengths does not seem to be written down anywhere, probably because no one of those three specialisms has any reason to look at the other two.

Where the ladder goes next #

The float correction is the next thing this ladder owes, and it needs a capstan argument along a float rather than a sum of independent contacts.

Sideways, the same arithmetic at a stated allowance answers which cloths slip at a seam and which break, and at a much shorter grip it gives what holds a tuft in, in newtons — which is the pile ladder’s own standing question, unanswered since it was asked because it needed a force.

Further out, the missing relaxed-cloth contact force would remove the stated tension from every expression on this page and turn a family of proportionalities into a set of values.