What the shed costs, in newtons

The loom ladder ends in the same sentence three times. The shed is an extension: lifting a warp end off the straight line makes its path longer, and the strain is exact trigonometry with nothing fitted in it. The back shaft works hardest: a clean opening at the reed forces each shaft’s lift to be proportional to its distance from the fell, so the strain rises down the harness. A figure is harder on its warp: a weave that flexes some ends more than others spends that strain unevenly.

Every one of those is a ratio, and every one of them ends by saying it cannot give a force, because the site had no model of how hard a thread resists being stretched. It has one now, and the arithmetic is a single multiplication.

The claim #

On an ordinary broad loom, the shed alone puts a back-shaft warp end at about half its breaking load, and does it on every pick for the whole weaving of the cloth.

That is not a large number in absolute terms — two newtons is the weight of a small apple — and it is a very large number as a fraction. A thread held at half its strength has half of it left for everything else the loom does to it: the beat-up, the let-off, the abrasion at the heddle eye, the knot from the last break. And the fraction is not a property of the yarn or of the weave. It is a property of how far back in the harness the end happens to be threaded.

The multiplication, and what is in it #

A tension is a modulus times a strain times a linear density. The site has the strain; the other two come from the yarn’s own mechanics.

The specific modulus in newtons per tex is the bulk modulus in GPa divided by the density in g/cm³, exactly — the units work out with no factor left over, which is worth knowing because a factor of a thousand in either direction would look entirely plausible in a result. Cotton at 8 GPa and 1.52 g/cm³ gives 5.26 N/tex.

The obliquity factor is cos²α, the standard first-order correction for fibres lying on a helix rather than along the axis, and this site already computes α from the count and the turns per metre. At 700 turns per metre a 25 tex cotton yarn’s surface helix is at 22.3°, so cos²α is 0.856 and the specific modulus falls to 4.50 N/tex.

Multiply by the count and a 25 tex yarn has a stiffness of 113 N — meaning that a strain of one per cent is a tension of 1.13 N.

The breaking load comes the same way: the fibre’s tenacity, times the fraction of it a spun yarn realises, times the same obliquity. Cotton fibre at 0.35 N/tex, a staple yarn realising about half of it, gives 0.15 N/tex and a breaking load of 3.74 N for a 25 tex yarn — which is what such a yarn breaks at.

None of that is fitted to anything here. Both constants are stated, both are ranges in the literature, and every result below is either a ratio in which they cancel or is quoted with the range attached.

The numbers, and the one that is worth arguing about #

Three harness depths on the same loom, with the same yarn:

The front shaft’s figure does not move, because the front shaft is at the same distance from the fell whatever is behind it. Everything that changes, changes at the back.

Two of these numbers are worth separating. The ratios are firm. They are strains divided by strains, the modulus cancels out of them entirely, and the site’s own machinery asserts that the tension ratio and the strain ratio are identical to twelve decimal places when there is no background tension. A ratio of 3.75 across a twenty-four-shaft harness is as solid as the trigonometry it comes from.

The percentages are upper bounds, and the reason is worth stating rather than burying. This model is a straight line through the yarn’s initial modulus, and a real cotton yarn’s load–extension curve is concave: it reaches its breaking load at five to seven per cent extension, where a straight line through the initial slope would reach it at 3.3. So a strain that this model prices at half the breaking load is really somewhat less. The direction is known, the size is not, and the ratios are untouched.

Why warp breaks happen at the back #

Every weaver knows that ends break at the back of the harness more than at the front, and the explanations offered are usually about abrasion — the back shafts travel further, so their heddles rub more, so the ends wear faster.

That is true and it is the second effect. The first is that the ends at the back are being held nearer their breaking load on every pick, and by a factor that is arithmetic rather than lore.

It also explains a rule that is otherwise pure custom. A weaver puts the ground weave at the front and the figure at the back — or, put the other way, puts the ends that are lifted most often where the strain is least. That is the right instinct for the wrong stated reason: it is usually explained by shed cleanliness, and it is at least as much about how much of each end’s strength is being spent.

What a background tension does to the fractions #

The shed’s contribution is an increment on a warp already held under tension, and the two combine in a way that makes the harness depth matter more rather than less.

A let-off holds a warp at a working tension of its own, chosen so that the shed opens cleanly and the beat-up drives home — commonly a fifth to a third of the yarn’s breaking load. Add the shed’s increment to that and the front shaft’s ends sit at something like a third of breaking load and the back shaft’s at something over four fifths.

The ratio the essay is careful about does not survive that addition, and the reason is worth stating rather than leaving as a caution. A ratio of increments is 3.75; a ratio of totals is (background + 1.95) over (background + 0.52), which at a background of one newton is 1.94 rather than 3.75. The background tension compresses the harness’s spread, because it is common to every shaft.

But it compresses the spread in tension while making the margin worse everywhere. What decides whether an end breaks is how much of its strength is left, and the background eats the same amount from every shaft — so the back shaft, which had least to spare, is the one that runs out. At a background of a fifth of breaking load the back shaft of a twenty-four-shaft harness is at 72 per cent and the front at 34; at a third it is 85 and 47.

So the two readings say opposite-looking things and both are right. The spread narrows and the danger concentrates, because a spread is a ratio and a margin is a difference.

It also says which of the two a loom’s settings can help with. Lowering the background tension raises every margin by the same amount and widens the spread, so it helps the back shaft most in absolute terms and makes the harness look more uneven on an instrument. Raising it does the reverse. The setting that makes the tension trace look uniform is the setting that puts the back shaft nearest its limit, which is an unhelpful arrangement for anybody tuning a loom by watching one. A weaver reading a tension meter at the back rest sees the first and a weaver counting broken ends sees the second, which is a good part of why the two have never obviously agreed.

Where the shaft budget really comes from #

The loom ladder’s second finding was that a stated warp-strain budget is a whole number of shafts: invert the leading-order strain expression and one per cent buys thirteen shafts on an ordinary broad loom.

That result was arithmetic without a justification for its own input. Why one per cent? The number was stated as a tolerance and no reason was given for it, because none was available.

With a modulus the tolerance stops being arbitrary. One per cent strain on this yarn is 1.13 N, which is 30 per cent of its breaking load, and 30 per cent of breaking load is very close to where a textile engineer would set a working stress on a component that is going to be cycled several hundred thousand times. The shaft budget is a fatigue allowance in disguise, and the disguise is why it looked like a convention.

Two things follow that the strain form could not say. A stronger warp buys shafts: the same loom carrying a yarn of twice the tenacity can accept twice the strain and therefore a deeper harness, which is why a fine worsted and a filament warp are not held to the same harness depth. And the budget is fibre-dependent in a way the geometry is not — the strain limit is a fraction of breaking strain, breaking strain is tenacity over modulus, and those two do not vary together across fibres.

What a jacquard escapes, priced #

The loom ladder found that a jacquard is exempt from the harness-depth bound, because its cords all hang from one distance rather than from a ladder of them. That was a statement about geometry and it can now be stated as a saving.

Every hook on a jacquard sits at the same distance from the fell, so every end sees the front shaft’s strain: 0.52 N, 14 per cent of breaking load, whatever the design’s complexity. A twenty-four-shaft dobby doing the crudest possible figure has ends at 52 per cent. The machine that can lift every end independently is also the machine that treats every end equally, and the two are the same fact about where the lifting mechanism is.

That is a much stronger argument for a jacquard on a difficult warp than the usual one. The usual one is about pattern freedom. This one says that on a warp near the edge of what it can survive — a fine silk, a weak regenerated yarn, a warp that is already broken twice this morning — a jacquard is gentler by a factor of nearly four, and the design has nothing to do with it.

What was counted, and how #

The strain half is the loom ladder’s and is unchanged: a loom stated once as five distances in millimetres, a half-shed tangent from the clear opening at the reed and the fell-to-reed distance, and each shaft’s exact extension as the sum of two hypotenuses over the straight line. The whole of it is scale-free — doubling every dimension of the loom changes nothing — and the machinery asserts that by rebuilding the loom at twice the size and requiring agreement to twelve decimal places.

The tension half is this rung’s and is one multiplication, with two assertions on it. The back shaft must hold more than the front, which would fail immediately if the strain rows were being read in the wrong order. And with no background tension the tension ratio must equal the strain ratio exactly — twelve decimal places — because any departure would mean the modulus had somehow entered per shaft, which would mean it had entered through the geometry.

The yarn is 25 tex cotton at 700 turns per metre throughout, which is an ordinary sheeting warp. Nothing in the ratios depends on it. Everything in the percentages does, and each of them moves as the reciprocal of the yarn’s tenacity.

Where the model stops #

This is the shed’s contribution and not the warp tension. A loom holds its warp at a background tension the let-off sets, and the shed’s strain is added to it. The figures above are therefore increments, and a real back-shaft end is at the sum of the two — which makes the picture worse rather than better, and by an amount only the loom’s own settings decide.

There is no fatigue in it. A warp end is cycled once per pick, tens of thousands of times across a warp, and what breaks it is not a static load. A fraction of breaking load is the input to a fatigue calculation rather than the answer to one, and the site has no S–N curve for a yarn and no way to get one from anything it does have.

The straight-line modulus is wrong in a known direction. Stated above, and worth repeating because it affects every percentage on this page and none of the ratios.

And nothing here knows about the heddle eye. An end at the back of the harness is being rubbed as well as stretched, the two interact, and the interaction is where a real warp break comes from.

The generalisation #

The shape of this rung is the plainest one in mechanics and it is worth naming because this collection has been living without it. A kinematic model gives ratios; the same model with one constitutive constant gives quantities. Nothing about the geometry changes. What changes is that the answers can be compared with something outside the model — a breaking load, a specification, a competitor — and comparison is where a number stops being internal.

The second half is less obvious. A ratio that survives is worth more than a quantity that does not, and this rung produces both. The 3.75 across the harness is exact and will still be right when somebody measures cotton’s modulus better; the 52 per cent is an upper bound that will move. A result written so that a reader can tell which is which is a different kind of result from one where they are mixed.

Who found it, and when #

The shed’s geometry is old and unattributed; it is in every loom-maker’s handbook as a construction rather than as a formula, and the strain it implies is the kind of thing that gets computed on the shop floor and not written down.

Warp tension as a measured quantity belongs to the loom-instrumentation work of the 1950s and 1960s, and Greenwood and Cowhig’s papers on the weaving cycle are where the numbers a modern account quotes come from. That literature measures the total tension at the back rest and its variation through a pick; it is not much interested in dividing it into a geometric part and a let-off part, because on a real loom the two are not separable.

The division is easy here for the reason it is hard there: this site has a loom with stated dimensions and no let-off at all, so the geometric part is the only part there is. What is this site’s is putting the answer beside the yarn’s breaking load — which is the comparison that turns a tension into a statement about whether the machine is being kind to the warp.

Where the ladder goes next #

The next rung of this ladder is the one the loom ladder could not reach either: the blow that drives the pick to the fell, which is a force in the other direction and turns out to be the warp tension multiplied by something the crimp decides.

Sideways, the same modulus prices the crossing force and, through it, what holds a thread into a cloth at all. And the same bending rigidity that produced the modulus is what closes the hole in Peirce’s geometry.

Further out, the missing piece is fatigue. Every result on this page is a static load on a component that is cycled, and the honest statement of what the harness costs the warp is a statement about cycles rather than about newtons.