A bundle is weaker than its threads

A strip of cloth an inch wide holds a certain number of ends, and each of those ends breaks at a certain load. Multiply the two and there is a strip strength — the arithmetic every specification does, and the arithmetic that is wrong by about a quarter before any question of weaving, crimp, finish or grip has been raised.

The reason has nothing to do with cloth. It is what happens whenever things are loaded in parallel and are not identical.

The claim #

A bundle of parallel threads breaks at a load per thread strictly below the mean thread’s, by an amount decided by the shape of the distribution and not by its scale. At a coefficient of variation of fifteen per cent the bundle carries 73.8 per cent of what the mean thread would suggest; at eight per cent it carries 83.2; at twenty it carries 68.5.

Two consequences follow immediately and both are worth stating before the argument.

The loss cannot be measured away. Testing more threads, more carefully, gives a better estimate of the mean and does not move the bundle at all.

Evenness buys strength, exactly. Two yarns with identical mean strength and different spreads make fabrics of different strength, and the difference is a sixth between an even yarn and an ordinary one. Nothing about the fibre has changed.

The argument #

Load a bundle of n threads and raise the load slowly. Every thread carries the same share — a simplification, and one this essay returns to — so at a load of x per thread, every thread whose own strength is below x has already broken.

That is a fraction F(x) of them. The survivors are 1 − F(x), and they are carrying the whole load, so what the bundle holds at that moment is

$$n \cdot x \cdot (1 - F(x)),$$

and the bundle’s strength is the largest value that product ever reaches.

The shape of the product is the whole argument. At low x nothing has broken and the load rises in proportion; at high x almost everything has broken and the product collapses. In between there is a maximum, and the maximum is where the gain from raising the load per thread is exactly balanced by the loss from the threads that raising it destroys.

For a lognormal at fifteen per cent, that balance falls at 0.796 of the mean strength with 92.7 per cent of the threads still whole. The bundle breaks with more than nine threads in ten unbroken, at a load per thread below the average thread’s — which is why nothing about the failure looks like a population effect from the outside. It looks like a clean break.

What was counted, and how #

The maximum is found twice, by two routes that share nothing.

The first is the closed form: the product is evaluated over a grid, and the grid maximum is refined by golden section until the answer is a stationary point rather than the best of a list. That is Daniels’ result, applied rather than derived, and it belongs to probability rather than to this collection.

The second is a simulation of the thing itself: n strengths are drawn from the same distribution, sorted, and the largest of the loads a surviving group would carry is taken. It knows no theorem. It is run at four sizes of bundle, two hundred bundles apiece.

The two do not agree, and the disagreement is the interesting part. A finite bundle sits consistently above the limit — 0.803 at ten threads, 0.753 at a hundred, 0.744 at four hundred, 0.741 at sixteen hundred, against a limit of 0.738 — and the gap closes as the bundle grows. That is not a defect in either calculation. A finite bundle’s strength is a maximum taken over the sample it happens to contain rather than over the distribution the sample came from, so it is biased upwards exactly the way any sample maximum is, and the bias falls away as the sample grows.

The first version of this check asserted that the simulation would equal the limit at four hundred threads. It failed, correctly, and the assertion is now about the convergence: the gap is positive at every size, shrinks at every step, and is under a per cent of a mean thread by sixteen hundred. An assertion written against the number in front of it rather than against the claim is the commonest way an assertion goes bad on this site, and this is the fourth time it has been caught.

What it does to the strength of cloth #

The bundle result is about parallel threads and a fabric is more than that, so the transfer needs care in two directions at once — one that makes the fabric better than the bundle and one that makes it worse.

Better: a woven thread that breaks is not lost. It is gripped at every crossing by the threads of the other system, so a broken end can pick up its load again a few millimetres away. That is the mechanism this collection priced when it asked what holds a thread in a seam and again for a cloth with a hole in it, where the same friction is what stops a cut from being a catastrophe.

Worse: the load does not share equally. When an end breaks in a fabric, its load does not spread evenly over the remaining thousands; it goes to its immediate neighbours, because the crossings that transfer it are local. That concentrates stress exactly where the material has already failed, which is the mechanism behind every tear, and it makes the real bundle efficiency lower than the equal-sharing figure rather than higher.

So the honest statement is a bracket rather than a number: a fabric’s strip strength is below the sum of its threads’ mean strengths, by at least the bundle’s own shortfall. The equal-sharing figure is a ceiling on what the population argument costs, not an estimate of it.

Where this sits relative to everything else about strength #

It is worth being exact about what has and has not been added, because this collection already has a good deal of machinery about a cloth under load and none of it is this.

The locus answers how far will it go, and it answers it from geometry: where the crimp can move to, what the two systems must trade, what a given load does to the shape. Not one step of it depends on the threads being alike, and giving the locus a force did not change that — the force enters through stiffness, which is a mean quantity.

The bundle result answers when does it stop, and it is entirely a question about the population. The two are orthogonal, and a fabric’s load-extension curve is the first of them right up until it is the second: geometry all the way to the point where the ends of the distribution take over, and then a collapse the geometry has no term for.

The scatter runs the other way #

The mean strength of a bundle rises towards its limit as the bundle grows. The scatter falls, and much faster.

The consequence is a familiar laboratory oddity explained. A single-thread test and a strip test on the same yarn disagree about both numbers at once: the single thread reads stronger and scatters wildly, the strip reads weaker and repeats beautifully. Neither is wrong, and the temptation to trust the repeatable one as “more accurate” is a mistake — it is more precise, and it is measuring a different quantity, one that has the population’s shape baked into it.

What it is worth in newtons #

An example makes the size of it concrete, and this collection already has the machinery for every step.

Take a twenty tex cotton yarn at an ordinary twist. Its breaking load, computed from the fibre’s specific strength with the obliquity the helix angle costs and the translation efficiency a staple yarn suffers, comes out at 3.08 newtons. A strip test on a cloth set at twenty-four ends per centimetre, taken over the standard fifty millimetre width, engages a hundred and twenty ends.

The arithmetic that everybody does gives 370. The population takes a quarter of it away before the cloth is even woven — before crimp, before the grip of the jaws, before any of the reasons a strip test is known to read differently from the sum of its threads. And the difference between the even yarn and the coarse one, at identical mean strength, is 55 newtons: a fifth of the fabric’s strength, decided by a number that is measured, printed, and then discarded.

Nothing here is a correction to be applied afterwards. The efficiency is a function of the shape of the distribution alone, so it is knowable in advance from the yarn’s own test certificate, and a specification that carried the coefficient of variation alongside the count would let a strength be predicted rather than measured.

Why the failure looks nothing like a population #

The most misleading feature of the bundle result is that the failure it describes looks, from outside, exactly like the failure it is not.

At the peak, 92.7 per cent of the threads are still whole. Nine threads in ten have never been near their own limit. What happens next is not gradual: past the maximum, every break raises the load on the survivors, which breaks more of them, which raises it further — so the bundle unloads itself in a run, and what an observer sees is a sudden, clean, apparently simultaneous break.

So the visible evidence points the wrong way. A break that looks simultaneous is taken as evidence that the threads were alike, which is the exact opposite of what produced it: the more the threads vary, the earlier the peak, and the more violent the collapse afterwards. A perfectly uniform bundle would be the one that broke gently, thread by thread, at a load per thread of exactly one.

The same reasoning applies to a load-extension curve. A bundle of varying threads has a curve that rounds over before its peak — the early breaks are already happening — and a specification reading a “yield” off that rounding is reading the population rather than any property of the fibre.

Where the model stops #

Equal load sharing is an assumption and it is the generous one. Every real arrangement — a fabric, a rope, a bundle of fibres inside a yarn — shares load locally to some degree, and local sharing lowers the strength further. What is computed here is therefore an upper bound on the arrangement’s efficiency rather than a prediction of it.

The threads are treated as breaking at a load rather than at an extension. In a real test the threads are all stretched together, so what they share is a strain, and a thread that is stiffer reaches its own limit sooner regardless of its strength. Including that needs the joint distribution of strength and stiffness, which this collection does not have — and the direction is not obvious, because in most fibres a stronger specimen is also a stiffer one.

A yarn’s own strength is itself a bundle result. A staple yarn is a bundle of fibres held by twist, so the same argument has already been applied once before any of this begins, which is part of why a yarn’s tenacity is well below its fibres’. Applying the argument twice is not double-counting, but the two stages are not independent and nothing here separates them.

And the length of the specimen is in the answer. A thread tested over half a metre breaks at its own worst place over half a metre, so its measured strength falls as the specimen lengthens — the weakest-link argument applied along a thread rather than across a bundle. The bundle arithmetic above takes the thread strengths as given, and which gauge length gave them is part of what they mean.

The loss is sub-linear in the spread, and that inverts the usual advice #

The three efficiencies quoted above — 0.832 at eight per cent, 0.738 at fifteen, 0.685 at twenty — determine a relation, and the relation is worth extracting because its shape is the opposite of what a designer expects.

Write the loss as one minus the efficiency: 0.168, 0.262, 0.315. Against a spread that has risen by a factor of two and a half, the loss has risen by 1.88 — so

the bundle loss goes as the spread to the power 0.69, and a fit through all three points gives

efficiency ≈ 1 − 0.97 × CV^0.69,

which reproduces every one of them to within two per cent and is a usable formula from a yarn’s own delivery note.

The exponent below one is the finding. Each further point of evenness is worth more than the last, because the marginal return goes as CV to the power minus 0.31 — a third higher at eight per cent than at twenty.

That is an unusual shape for this collection, where nearly every lever saturates: tightness saturates, a seersucker’s beam ratio saturates under a square root, the felting ratio saturates, the setting ceiling has an asymptote at two. Evenness does not. There is no point at which improving it stops paying in this term, and the payment accelerates.

Which prices the spinning route #

The formula turns the argument into a specification rather than an exhortation.

A ninety per cent efficient bundle needs a coefficient of variation under four per cent. Setting the loss to a tenth and inverting gives 3.7 per cent, which is finer than any staple yarn reaches — an excellent combed cotton is eight to ten and a carded one fifteen to twenty.

So no spun yarn converts more than about eighty-five per cent of its threads’ strength into a fabric’s, ever, and the shortfall is a property of spinning rather than of weaving. A filament yarn does, because its variation is a fraction of a per cent, and that is a much sharper account of why filament fabrics test so much closer to their nominal strength than the usual appeal to fibre continuity.

It also prices the two big spinning decisions against each other. Combing takes a cotton from about eighteen per cent to ten, which by the formula is an efficiency of 0.70 to 0.79 — thirteen per cent more fabric strength from the same fibre, for a process whose usual justification is appearance and hairiness. Compact spinning takes ten to eight, a gain of four per cent, and is usually sold on strength; on this arithmetic the strength gain it claims is real and is the smaller of the two.

And it says where a strength specification should look #

The last consequence is about which number a buyer should be reading, and it follows from the exponent rather than from any value.

Two yarns of equal mean strength differing by ten points of CV differ by a sixth in the fabric they make. Ten points of mean strength — a fibre a tenth stronger — buys a tenth. So over any range a mill can actually move, the spread is worth more than the mean, and the crossover is not close.

Which makes the omission stark. A yarn is bought on its count and its tenacity; its evenness is measured, printed on the same certificate, and used to judge appearance. The number that decides how much of that tenacity survives into cloth is sitting on the page being read for something else.

The generalisation #

Anything carrying a load in parallel is worth less than the sum of its parts, and the discount is set by their variation rather than by their average. That is the transferable statement, and its practical form is sharper: in a parallel system, reducing the spread is worth more than raising the mean.

The arithmetic says how much more. Moving a yarn from a coefficient of variation of twenty per cent to eight raises the bundle efficiency from 0.685 to 0.832 — a gain of twenty-one per cent, for no change at all in the fibre, the count or the twist. Getting the same gain by improving the mean means finding a fibre a fifth stronger.

And the second lesson is about what a test measures. A single-element test and a whole-system test on the same population return different numbers, and the difference is not error. It is the system, and the more parallel elements a test engages, the more of the population’s shape it has folded into its answer. A specification that quotes one and predicts the other is silently assuming the spread is zero.

Who found it, and when #

The bundle result is Daniels’, from 1945, and it was published as a problem in probability rather than as one about materials — though the application to fibres was immediate and it has been the standard account of why a bundle underperforms its fibres ever since. Peirce had made the weakest-link argument for the length of a specimen a decade earlier, in the same collection of papers that gave this site its thread geometry.

What this collection adds is a small thing: the two arguments are the same population read at two different order statistics, and putting them beside the extremes in the jam and the thickness makes it clear that a cloth is being decided by the ends of its yarn’s distribution far more often than by its middle.

Where the ladder goes next #

Straight to the case where the sample is smallest and the scatter therefore largest: a tear asks only the few threads at the tip of the cut, which is the same distribution read at its minimum rather than at its bundle maximum.

Sideways, the argument that evenness is worth more than strength has a companion in the geometry: an even yarn also sets closer and passes less air than its spread suggests, so the same one number on a delivery note is deciding three unrelated properties of the finished cloth.