Where a knot breaks

Anybody who has broken a rope at a knot knows where it went. Not in the middle of the knot, where the thread is most tortured and most compressed. At the entry — on the standing part, at the first curve, just as the load enters.

That is not folklore. It follows from the two quantities a knot is made of, one of which falls monotonically from the first millimetre and one of which does not rise at all.

The two quantities #

Along the thread’s path through a knot, the strain at the outside of the bend is the sum of two terms.

The tensile strain, which is the tension divided by the modulus times the area. The tension falls by the capstan relation: e to the minus mu theta, monotonically, from the moment the wrap begins.

The bending strain, which is the radius of whatever is bending divided by the radius the thread is bent to. That is fixed as soon as the thread has taken up the knot’s curvature, and it does not change with the wrap angle.

So one term falls and the other is constant, and their sum falls. The maximum is at theta equals nought.

Why the answer is not obvious #

Stated that way it looks trivial, and it is worth saying why it is not.

The intuition that a knot breaks in its middle comes from the middle looking worse: the thread is crushed there, it is bent hardest there, its fibres are most displaced. All of that is true and none of it is what breaks a thread in tension.

A thread breaks where its fibres reach their breaking strain, and the fibres’ strain is the tension plus the bend. The middle of a knot has plenty of bend and very little tension left, because the wrap has already taken it.

Being crushed does not break a thread. It damages it, over time, under repeated loading — which is a different failure and a slower one — the same abrasion that wears a cloth rather than breaking it — and it is the reason a knot that has been cycled is weaker than a knot that has not.

What the observation is worth #

It is not an interesting prediction on its own, because everybody already knows the answer. What it is worth is as a check on the arithmetic, and this collection makes a habit of using known answers that way.

The capstan relation has a sign in it. Written with the wrong sign it produces a tension that rises through the wrap, which is a perfectly plausible-looking exponential curve that nobody would query on a plot. What would give it away is that the model would then predict a knot breaking at its exit, which contradicts something everybody has seen.

So the check that runs on this arithmetic is not a tolerance. It is: the maximum of the tension is at theta equals nought, asserted for three fibres, and the exit tension is below the entry tension. Both fail if the exponential is inverted.

That is a check with a known answer, and this collection’s whole method is built on preferring those to checks with tolerances, as an assertion that has never rejected anything proves nothing.

What it says about which part of a knot matters #

The practical consequence is that only the first curve matters for strength, and everything else about a knot matters only for security.

That divides a knot’s design cleanly. The turns after the first exist to make the tension small enough that the tail cannot pull through; they cost nothing in strength, because they are carrying almost no load. The first curve costs everything.

So the way to make a strong knot is to make its first bend gentle, and the way to make a secure one is to add turns after it. Those are independent and can be optimised separately.

That is exactly what the knots with the best measured efficiency do. A figure-of-eight bend leads the standing part round a large radius before any wrapping happens; a double fisherman’s does the same with two barrels. Both are bulkier than the knots they replace and both are stronger, and the bulk is where the gentle first curve went.

Why the capstan relation has no radius in it #

A detail worth surfacing because it is the reason the two effects are independent.

The capstan relation is tension out equals tension in times e to the minus mu theta, and there is no radius anywhere in it. That is not an approximation: the normal force per unit length is the tension over the radius, and the length wrapped is the radius times the angle, so the radius cancels exactly.

A thread wrapped a half turn round a pencil and a half turn round a telegraph pole are held equally well.

So the security of a knot depends only on the total angle wrapped and on the friction, and the efficiency depends only on the tightest radius. Two quantities, two independent inputs, and the fact that the radius cancels out of one of them is what makes them independent.

Where the standing part is #

One clarification, because “the entry” is not always where a reader expects.

A knot has two ends and usually only one of them is loaded. The standing part is the loaded side; the tail is the other. The failure is at the entry of the standing part.

In a bend — a knot joining two ropes — both ends are loaded, so there are two entries and two candidate failure points. Which goes first is decided by which has the sharper first curve, and in an asymmetric bend that is a real question with a real answer.

In a loop knot — a bowline, say — the loop itself is loaded and the standing part is loaded, and again there are two entries. A bowline breaks at the standing part where it enters the collar, which is the sharper of the two.

So the rule is not “at the entry” but “at the sharpest bend that is carrying full tension”, and in a simple knot those coincide.

What this does to a fabric #

The same argument transfers to two failures this collection has computed separately.

A seam. A sewing thread wraps the fabric’s threads at every stitch, so the tension in it falls along the seam by the same relation. That means a seam breaks at its end — at the last stitch before the load, where the tension is full — and not in its middle, which is what is observed and which this collection derived from the same arithmetic without noticing it was the same argument.

And a frayed edge. A weft pick being pulled from a cut edge has its tension dropped by every crossing it still has, so the force needed rises exponentially with how far in it goes — and the pick fails, if it fails, at the edge rather than inside the cloth.

Three failures, one relation, and the same conclusion each time: the failure is at the end where the load enters, because that is the only place with full tension.

The one place the rule fails #

There is a case where a knot does not break at its entry, and it is worth naming because it is common and because it is the exception that shows the rule is about tension rather than about position.

A knot in a stiff, slippery cord — a monofilament, a coated line, a wire rope — can slip rather than break. The capstan relation with a low coefficient of friction leaves a great deal of tension at the tail, and if the tail is short the whole knot pulls through.

That is not a failure at the entry and it is not a failure of the thread at all: the thread is intact and the knot is gone.

So the rule has a precondition. A knot breaks at its entry if it holds, and whether it holds is the other half of the arithmetic. In a fibre with a coefficient of friction of a third, one turn is enough; at a tenth, three turns are needed for the same security, and a knot with fewer will slip before it breaks.

Anglers know this better than anybody and their knots reflect it: fishing knots have far more turns than rope knots, because monofilament is slippery, and the extra turns cost nothing in strength because they carry no load.

What a low-friction fibre does to the whole picture #

Following that through gives a prediction about a class of materials rather than about one knot.

The high-performance fibres — aramid, high-modulus polyethylene — are both slippery and brittle in bending. Their friction coefficients are low, so they need many turns; and their breaking strains are low, so a bend costs them a large fraction of what they have.

The arithmetic therefore says such fibres should be the worst possible things to knot, and they are: knot efficiencies of thirty to fifty per cent are usual, against seventy or eighty for a nylon, and the splices that avoid the bend entirely are used wherever the strength matters.

That is two independent properties of a fibre both pushing the same way, and neither is about the knot.

What was counted, and how #

The capstan relation is used unchanged and the coefficient of friction is the mid-point of the collection’s own range for the fibre.

The tension is evaluated at a hundred and sixty points round the wrap, and the maximum is located rather than assumed — the check finds the largest value and asserts that its angle is nought, so a wrong sign would produce a failing assertion rather than a wrong plot.

The bending strain is computed at the free end of the stiffness bracket, because the coherent end forbids knots entirely and a strain of fifty per cent against a breaking strain of seven is not a number to reason with.

The exit tension is checked to be below the entry tension for every fibre tested, which is the second half of the same guard.

Two failures that look the same and are not #

A practical distinction the arithmetic makes and inspection usually does not.

A rope that has broken at a knot shows a clean break at the entry to the standing part, with the knot itself intact and still tied. That is the failure this rung is about, and it is a tension failure at full load.

A rope that has failed at a knot after cycling shows a break inside the knot, at whatever point was most crushed, and usually with visible damage before the break. That is a fatigue failure and it is caused by the compression the tension arithmetic ignores entirely.

The two look similar in a photograph and are caused by different things. The first is fixed by using a knot with a gentler first curve; the second is fixed by not leaving a knot loaded and unloaded for months, or by inspecting it.

That is worth knowing because the remedy for one does nothing for the other, and both are commonly described as “the rope broke at the knot”.

Why nobody has to know which knot #

An oddity worth pointing out, because it is what makes a rule of thumb possible at all.

Knot efficiency is quoted as a property of a knot — a bowline at seventy per cent, an overhand at fifty — as though the whole structure mattered. The arithmetic says only one bend matters, and only its radius.

So the efficiency figures in the tables are, in effect, measurements of each knot’s tightest loaded bend radius, expressed in units of strength. Two knots with the same first-curve radius should have the same efficiency however different they look, and two versions of the same knot dressed differently should not.

Both of those are testable and neither is in the literature in that form. The published tables treat the knot’s name as the variable, and the arithmetic says the variable is a radius nobody has measured.

That is the kind of reframing this collection exists to make: not a new measurement, but a statement that the measurements already made are measurements of something else.

Where the model stops #

The bending strain is treated as constant through the knot. It is not: a knot’s curvature varies along its path, and the first curve is not always the sharpest. Where it is not, the failure moves to wherever the product of remaining tension and local curvature is largest, and computing that needs a knot geometry this collection does not have.

The thread is treated as elastic to failure. A real yarn’s fibres reseat, creep and break individually, so the failure is progressive rather than sudden and the breaking strain is a population statistic.

And the compression is ignored. A knot squeezes its thread hard, and lateral compression reduces a fibre’s tensile strength. That effect is real, it is why a knot that has been loaded and released is weaker, and none of it is here.

Nothing dynamic is here either. A shock load travels along a rope and arrives at the knot as a wave, and the arithmetic above is quasi-static throughout.

The generalisation #

The rung is short and its value is a method rather than a result, and it is one this collection uses constantly without naming.

Check machinery against a fact everybody knows.

Most checks in this collection are tolerances: does the closed form match the quadrature, does the identity hold to twelve figures, does the energy fall as the order rises. Those catch arithmetic and they cannot catch a sign, an inversion or a reversed convention — because a wrong-signed exponential is a perfectly good exponential and agrees with itself.

What catches those is a check whose answer is known outside the model. A knot breaks at the standing part. Two rings threaded through one another have a linking number of one. Glass’s two moduli satisfy the isotropic relation. An untwisted thread needs no tension to stay straight.

None of those is a measurement and all four are certainties, and each of them is placed in this collection’s machinery at exactly the point where an inversion would otherwise be invisible.

The arithmetic in one paragraph #

For a reader who wants the whole thing rather than its consequences.

The strain at the outside of the bend, at wrap angle theta, is the tensile part plus the bending part: the entry tension times e to the minus mu theta, divided by the modulus and the area, plus the bending radius over the bend radius. The first term is strictly decreasing in theta for any positive friction; the second does not contain theta at all. So the sum is strictly decreasing, its maximum is at theta equals nought, and the thread breaks there.

Nothing in that requires knowing the knot, the rope’s construction, the number of turns or how tightly it was pulled. It requires only that the friction be positive and that the curvature not increase along the wrap.

The second of those is the assumption that can fail, and it fails in knots whose sharpest bend is not their first — which is a real class and includes several knots with poor measured efficiency for exactly that reason.

Who found it, and when #

Euler wrote the capstan relation in 1775. That a rope breaks at the knot, and at the standing part of it, is in every practical rope text and in the rigging manuals long before that.

The measurement literature on knot efficiency is large, scattered and hard to compare, because efficiency depends on the rope, the knot, the dressing and the loading rate, and different investigators have controlled different ones.

What is this collection’s own is using the known failure location as a check rather than as a result — placing an assertion at the point where a sign error would otherwise produce a plausible curve.

Where the ladder goes next #

The location is settled and the magnitude is not. A knot costs about half a rope’s strength, which is a rule of thumb quoted for every rope and every knot, and the arithmetic here gives it a derivation with an unexpected consequence.

The half comes out only if the yarn’s fibres bend individually. If they bent as a solid section, a knot would be impossible. So a knot holding at all is evidence about a quantity this collection cannot otherwise pin down, and a knot halves a yarn and says why.