A knot halves a yarn and says why

Every rope text says a knot halves a rope’s strength, and every one of them quotes it as an empirical rule with a table of variations. Nothing derives it.

The derivation is short, it produces forty-six per cent for a cotton, and it only works at one end of this collection’s oldest bracket. At the other end it says a knot is impossible.

The argument #

A thread bent to a radius has its outermost material stretched. How much depends on how far that material is from the neutral axis: the strain is the offset divided by the bend radius.

So the fibre at the outside of the bend arrives at the knot with a strain already spent, and it reaches its breaking strain sooner. The fraction of the straight yarn’s strength that survives is

(breaking strain − bending strain) / breaking strain

and everything turns on what the bending strain is.

The two answers #

If the yarn bends as a solid section, the outermost material is half a yarn diameter from the axis. A twenty tex cotton is 0.167 millimetres across, so at a bend radius of one diameter the strain is a half — fifty per cent.

Cotton breaks at about seven per cent of strain. So the yarn has spent seven times what it has, and the efficiency is nought: the yarn breaks in the tying.

If the yarn’s fibres bend individually, each fibre is bending about its own axis and the outermost material is half a fibre diameter from it. A cotton fibre is 0.012 millimetres across — fourteen times finer — so the strain is three and a half per cent.

Against a breaking strain of six and a half, that leaves forty-six per cent.

Which is the rule of thumb #

Forty-six per cent, from a fibre fineness and a tenacity, with nothing fitted and no reference to any knot.

The trade’s figure is “about a half”, and the measured range across knots and ropes is broadly forty to seventy per cent with the sharper knots at the bottom.

So the derivation lands on the rule, and it lands on it from a direction that has nothing to do with knots: the two inputs are a fibre’s diameter and a fibre’s breaking strain, and the output is a property of every knot in that fibre.

The evidence in it #

The useful part is not the forty-six per cent. It is that only one of the two answers is a number at all.

A yarn whose fibres cannot slide has spent seven times its breaking strain in a bend of one diameter. Such a yarn cannot be tied. It cannot be knitted either, since a knitted loop’s tightest bend is about one diameter; and it cannot be woven at any crimp worth having.

Ropes are knotted, jerseys are knitted, and cloth is woven. So the fibres slide.

That is the third independent everyday observation in this work to land at the free end of this collection’s stiffness bracket, and the three come from unrelated phenomena:

A slack yarn snarls at a tension of two metres of its own weight rather than six hundred.

A yarn in a fabric is flattened, which is free if the fibres rearrange and thirty-six times a stitch’s bending energy if they cannot.

And a knot holds, at about half strength rather than at nought.

Three phenomena, three arguments, one answer.

Why a knot is the best of the three #

The three are not equally strong and the knot is the sharpest, for a reason worth stating.

The snarl reads a threshold, and a threshold depends on a shear modulus with a factor of two on it. The flattening reads a qualitative alternative — free or impossible — which is decisive and gives no number.

The knot gives both. It says the coherent bound is impossible, which is qualitative and decisive; and it produces a number, forty-six per cent, which can be compared against a large body of measurement.

That comparison is not clean, because knot efficiency depends on the knot, the dressing and the loading rate as well as on the fibre. But it is a comparison, and the other two do not have one.

What decides the number #

The efficiency depends on exactly three things and it is worth listing them because two are usually overlooked.

The bend radius, in yarn diameters. That is the knot’s own property and it is the only one anybody usually varies.

The fibre’s fineness relative to the yarn’s diameter, which is the square root of the fibre count over the packing factor. A yarn spun from finer fibres has more of them, each thinner, so each bends more easily and the knot costs less.

And the fibre’s breaking strain, which is its tenacity over its specific modulus. A fibre that stretches a long way before breaking has more to spend.

The second is the one that never appears in a rope table, and it makes a real prediction: two yarns of the same count and the same fibre type, spun from different fibre finenesses, should knot to different efficiencies. A microfibre yarn should knot better than a coarse-fibre one of the same count, and by a computable amount.

Nobody has looked, and it would be an easy measurement.

The count does not matter #

A pleasing consequence, and a check on the arithmetic.

The bending strain is half a fibre diameter over the bend radius, and the bend radius is quoted in yarn diameters. So the strain is the fibre diameter over the yarn diameter, divided by twice the radius in yarn diameters.

The ratio of fibre diameter to yarn diameter is the square root of the packing factor over the fibre count — and the fibre count is the yarn’s tex over the fibre’s tex, so it rises with the count.

Putting it together: a coarser yarn has more fibres, each the same size, so the ratio falls as one over the square root of the count.

So a coarser yarn knots better, at the same relative bend radius — which is not the same as saying a coarser rope is stronger, because the bend radius a knot takes in absolute terms scales with the yarn too.

Why this is not the obliquity argument #

There is a competing explanation for why a knot weakens a rope and the two are often run together, so it is worth separating them.

The obliquity account says a fibre at an angle to the load carries only the cosine of its own contribution, so a structure that makes its fibres oblique is weaker. That is the argument this collection uses for why a twisted yarn has an optimum and why a knitted fabric is weaker than its yarn.

It is not what is happening here. A knot does not change the angle the fibres make with the load along the standing part; the thread is straight where it enters and its fibres are wherever the twist put them.

What a knot changes is the strain distribution across the section: one side of the bend is stretched and the other is compressed, so the material is no longer sharing the load evenly. That is a bending effect and not an angle effect.

The two are distinguishable by what they depend on. Obliquity depends on the twist; bending depends on the radius. A hard-twisted yarn and a soft-twisted one of the same count should knot to nearly the same efficiency if the bending account is right, and to very different ones if obliquity is doing the work.

Nobody has separated them, and it is a two-afternoon experiment.

What was counted, and how #

The bending strain is half a diameter over the bend radius, at both ends of the bracket: the yarn’s own diameter for the coherent case, and the fibre’s for the free one.

The diameters come from the counts by this collection’s own volume arithmetic, unchanged: a yarn’s from its tex, its fibre’s density and the packing factor; a fibre’s from its own fineness at a packing factor of one, because a fibre is not a bundle.

The breaking strain is a tenacity in newtons per tex over a specific modulus in the same units, and the specific modulus is the bulk modulus over the density. That conversion is written once so that no result carries a second version of it, which is the same discipline the collection applies to every unit hinge.

The check asserts two things and both could fail: the coherent bound leaves nothing at a bend of one diameter, and the free bound leaves between a third and seven tenths for a cotton and a polyester. The second is a band rather than a point because the inputs have ranges, and it is set narrow enough to fail if the fibre diameters or the tenacities move.

What a rope does that a yarn does not #

The rule is quoted for ropes and the arithmetic here is about a yarn, so the step between them is worth making rather than assuming.

A rope is a structure of yarns, which are structures of fibres. Bending a rope round a knot bends each of its yarns, and each yarn bends about its own axis if the yarns can slide past one another — which in a laid or braided rope they largely can.

So the same argument runs one level up, and the relevant radius is the yarn’s rather than the rope’s. A rope of a hundred yarns bent to one rope diameter is bending each yarn to about ten yarn diameters, which is gentle.

That predicts something the tables agree with: a rope of the same fibre knots better than a single yarn of it, because the sliding at two levels rather than one puts more distance between the bend radius and the material that is being strained.

And it predicts the failure of that: a rope whose yarns cannot slide — impregnated, coated, or of a construction that locks — should knot much worse. Which is exactly the reputation of coated and jacketed ropes.

What it says about a fabric’s tightest bend #

The same arithmetic can be pointed at a fabric, and it says something uncomfortable.

A knitted loop’s tightest bend is about one yarn diameter of radius. That is the same radius as a tight knot. So every stitch in a knitted fabric has spent the same three and a half per cent of strain that a knot spends, before any load arrives.

Against a cotton’s six and a half per cent breaking strain, that is more than half of it — which says a knitted fabric’s yarn should break at under half the load a straight length of the same yarn would take.

Whether that is what happens is not obvious, because a fabric shares load between many loops and fails progressively. What is clear is that the effect is of the same order as the obliquity effect that is usually given as the reason, and that this collection has never computed the two side by side.

Where the model stops #

The efficiency is a strain criterion and a yarn does not break at a strain. It breaks when enough of its fibres have broken, and the fibres have a distribution of strengths. So the number is a mean-field estimate of a population failure, and this collection has argued at length that such estimates are systematically wrong in a known direction.

The compression is ignored. A knot squeezes its thread hard, and lateral compression reduces a fibre’s tensile strength by a real amount. That effect pushes the efficiency down and is not here.

The fibres are treated as bending about their own axes and nothing else. In a twisted yarn a fibre is at a helix angle and its bending is a mixture of bending and twisting about its own axis, so the free bound overestimates how easily it bends.

And the bend radius is an input. Which radius a knot takes is a contact problem this collection cannot solve.

Two of those four push the efficiency down and two are neutral, so the derivation is an upper bound rather than a central estimate — which is consistent with the measured range having its mass below fifty per cent rather than above.

Why the number is not more precise #

A reader may want a tighter figure than “about a half” and it is worth saying why the arithmetic cannot give one.

Cotton’s tenacity is quoted as 0.35 newtons per tex and its modulus as eight gigapascals, both with ranges — the modulus’s runs from five to twelve. So its breaking strain runs from four and a half per cent to eleven, which is a factor of two and a half.

At the low end the bending strain of three and a half per cent leaves twenty-two per cent of the strength. At the high end it leaves sixty-eight.

So the honest prediction is “somewhere between a fifth and two thirds”, which contains the rule of thumb and is not a sharp test of anything.

That is a fair description of where this collection sits on most quantitative questions about a natural fibre, and it is why the qualitative half of the result — that the coherent bound gives nothing — is worth more than the quantitative half. Nought is not inside any range.

The generalisation #

The rung is an instance of a move worth naming, because this work has now made it three times.

A bracket whose two ends predict qualitatively different worlds is not a bracket. It is a question with an answer.

The usual response to a wide bracket is to try to narrow it: measure better, argue about the packing factor, find a cleverer experiment. That is the right response when both ends describe plausible objects differing in degree.

It is the wrong response when one end describes something that does not exist. A yarn that cannot be knotted, tied, knitted or woven is not a yarn, and no amount of measurement is needed to rule it out.

So the productive question is not “what is the value” but “what does each end predict about something anybody has seen” — and this work has found three cases where the answer was immediate and free.

The measurement that would settle the fibre-fineness prediction #

Of the three inputs, one makes a prediction nobody has tested and the test is unusually clean.

Take two yarns of the same count, the same fibre type and the same twist, spun from fibres of different fineness — one and a half decitex against three, say, which is an ordinary difference between a fine and a coarse cotton or between two polyester staples.

The arithmetic says the finer-fibred yarn should knot better, by a factor of about the square root of the fineness ratio applied to the bending term. For a factor of two in fineness that is a bending strain smaller by root two, which moves a cotton’s efficiency from forty-six per cent to about sixty-two.

Sixteen points of efficiency is far above the scatter in a careful knot test.

What makes the test clean is that everything else is held: same count, same fibre chemistry, same tenacity, same twist, same knot, same dressing. The only variable is the fineness, which appears nowhere in any published account of knot strength.

If the effect is there, the account is right and rope tables need a column. If it is absent, the fibres are not bending about their own axes as freely as the free bound assumes, and the yarn sits somewhere in the middle of the bracket — which would be the first quantitative placement anybody has managed.

Either answer is worth an afternoon.

Who found it, and when #

The bending-strain explanation of knot efficiency is standard in the rope literature and has been since at least the middle of the twentieth century.

What is usually not done is to ask which radius the bending strain should be computed at, because for a rope — a structure of many yarns — the answer is obviously the yarn’s rather than the rope’s, and nobody is tempted by the rope’s own diameter.

For a yarn, the same question is not obvious at all, and the two candidate answers differ by a factor of fourteen and give a knot that works and a knot that is impossible.

What is this collection’s own is asking it, and noticing that the answer is the same free-versus-coherent question that its stiffness bracket has been about from its earliest work.

Where the ladder goes next #

The efficiency depends on the fibre through its breaking strain, and the breaking strains in this collection’s table run from four per cent for a viscose to twenty-one for a nylon.

That is a spread of five, and it makes the difference between a fibre that knots at six per cent efficiency and one that knots at eighty-three. Which yarns knot well turns out to be a question about the fibre and not about the knot at all.