The locking angle

Pull a bias-cut strip and it gives easily, then more easily, and then stops. Not gradually — abruptly, as if it had hit something.

It has. The threads are touching each other, and the mechanism that was providing the extension has run out of travel.

The geometry #

Shearing a woven cloth closes the angle between the two thread systems. As it closes, the perpendicular distance between parallel threads shrinks: from the pitch $p$ at ninety degrees to $p\sin\theta$ at an included angle $\theta$.

The threads have a diameter $d$. They touch when $p\sin\theta = d$, so the cloth locks at an included angle of $\arcsin(d/p)$, which is a shear of

$$\gamma_{\text{lock}} = 90° - \arcsin(d/p).$$

The ratio $d/p$ is the cover in one direction — the fraction of the width the threads occupy. So the locking angle depends on cover and on nothing else: not on fibre, not on strength, not on weave except insofar as the weave decided the cover.

A cloth at forty per cent cover locks at sixty-six degrees. At half cover, sixty. At seventy per cent, forty-six. At ninety, twenty-six. The relationship is steep at the dense end, which is why a small change in setting makes a large change in drapeability.

Why this is the number that matters #

Locking sounds like a curiosity and it is the constraint that decides a great deal of what cloth can be made to do.

Draping over a curve. A flat sheet cannot cover a doubly curved surface without shearing, and the amount of shear required grows with how curved the surface is and how far the cloth travels over it. If the requirement exceeds the locking angle anywhere, the cloth cannot lie flat there and must wrinkle instead. That is why garments need darts.

Composite manufacture. A woven reinforcement being laid into a mould has exactly this problem, and there it is a production issue with a cost. The locking angle of a given fabric is measured — by a picture-frame test or a bias-extension test — and used to decide whether a preform can be draped in one piece or must be cut and joined.

Garment fit. A fabric with a high locking angle conforms; one with a low one resists. That is much of what people mean when they say a cloth is fluid or stiff, and it is available in any fibre because it is a function of setting.

Denser is not better #

The trade-off is direct and it catches people out.

A densely set cloth is firmer, more windproof, more abrasion-resistant, heavier and more opaque. It is also, for exactly the same reason, less drapeable — because the space between the threads that a shear needs to close is the space a dense setting has already used up.

So a cloth cannot be simultaneously dense and drapeable, and a designer choosing a sett is choosing a position on that axis whether they know it or not. It is one of the clearer cases in this subject where a single geometric quantity sets a trade nobody can escape.

The whole curve #

Extension against shear is worth seeing in full, because the shape of it is not what a material’s stress-strain curve looks like.

The curve rises steeply at first — the first fifteen degrees of shear buy twelve per cent of extension — and then flattens, because the geometry gives diminishing returns as the cells close. So most of the bias’s usable stretch is available early, and the last few degrees before the lock give very little.

That is why a fabric can feel as though it has plenty of give left when it is in fact close to jamming. The extension is nearly exhausted before the resistance rises, and the two do not warn each other.

What the lock feels like #

The subjective experience is worth describing, because it is the only evidence most people have and it is diagnostic once interpreted.

Take a bias strip between the hands and pull. At first the cloth resists a little and moves a lot — that resistance is friction at the crossings, not the mechanism, and it is roughly constant. Then a long easy stretch where the shear runs freely. Then, quite suddenly, the strip stops moving and the force in the hands rises steeply.

Nothing broke and nothing is near breaking. The threads are touching, and further extension would have to stretch fibre rather than change shape.

The steepness of that final rise is the clearest evidence that the limit is geometric. A material approaching its breaking strain stiffens gradually; a mechanism reaching the end of its travel stops. Anyone who has pulled bias binding has felt the difference, and it is the same sensation as a trellis fence reaching full extension.

Measuring it #

Two standard tests, and the difference between them is instructive.

The picture-frame test clamps a square of fabric into a hinged frame and pulls the frame’s diagonal, shearing the cloth uniformly. It gives a clean shear force against shear angle curve, and the locking angle shows as a sharp rise in force.

The bias-extension test simply pulls a rectangle cut at forty-five degrees. It is far easier to do and the deformation is not uniform — the middle shears fully, the ends barely at all, and the analysis has to account for three regions. It is the common test in practice because the apparatus is a tensile tester.

Both give a locking angle somewhat different from the geometric prediction, and always larger — real cloth shears further than the circular-thread model says it should, because threads flatten and slide sideways rather than simply touching and stopping. Typical measured values run from about forty degrees for a densely set fabric to sixty-five or more for an open one.

That discrepancy is the same one that appears in setting calculations, and it has the same cause: yarn is compressible and the model treats it as rigid.

Why it decides drapeability #

The number’s importance comes from a fact about surfaces rather than about cloth.

A flat sheet of inextensible threads cannot cover a doubly curved surface without shearing, and the shear required grows with the curvature and with the distance travelled. So covering a curve is a budget problem: the surface demands a certain shear, and the cloth has a certain amount available.

If the requirement stays inside the budget, the cloth lies down. If it exceeds it, the cloth buckles instead, and that is what a dart is for — a wedge cut out so that the shear the cloth cannot supply is replaced by cloth that was never there.

An open, loosely set fabric has a large budget and drapes over almost anything. A dense one has a small budget and has to be cut into pieces. That single number is why some fabrics tailor and others do not, and it is a function of the setting rather than of the fibre.

What locking is not #

Three confusions worth heading off.

It is not a strength limit. Nothing has broken and nothing is close to breaking. The cloth has reached a configuration and stopped, and it will come back unchanged when released. Pushing past it damages the fabric — the threads are forced over one another and the cloth buckles out of plane — but the limit itself is geometric.

It is not the setting limit. Jamming is threads touching their neighbours in the undeformed cloth, which is what limits how many can be woven in. Locking is threads touching after a shear. A cloth well below its jamming sett still locks at some angle, and the two are related without being the same.

It is not the same in both directions. An unbalanced cloth has different covers in warp and weft, so the locking angle depends on which way the shear runs. Bias in one direction and bias in the other are not equivalent in such a fabric, which is a real nuisance in cutting.

Which cover, when a cloth has two #

The formula takes one cover and an ordinary cloth has two, so the question of which one binds has to be answered before the number means anything — and the answer is not the average and not the total the trade quotes.

Shearing closes both perpendicular distances by the same factor. The gap between the ends goes from p₁ to p₁ sin θ and the gap between the picks from p₂ to p₂ sin θ, so whichever system was closer to touching touches first. The condition is

sin θ_lock = max(K₁, K₂),

and the locking angle is 90° less its arcsine. The denser system decides, and the other one is irrelevant until it becomes the denser.

That has a consequence for how a cloth is opened out to make it drape. All of the improvement has to come from the denser direction. A denim at a warp cover of 0.9 and a weft cover of 0.5 locks at 25.8 degrees; opening the warp to 0.7 takes it to 45.6, and opening the weft from 0.5 to 0.3 changes nothing whatever.

Yarn spent opening the open direction buys no drape at all. That is a specific instruction and it is the opposite of the usual response to a cloth that will not tailor, which is to open the construction generally.

And it is not the cover a specification quotes #

The trade’s combined cover factor is the two covers less their product, which is a statement about how much of the area is thread. The locking angle depends on a maximum, and a maximum and a sum are not the same statistic — which is the same complaint this collection makes about a thickness and about a jam, arriving in a third place.

Two cloths at the same total cover of 0.84:

Balanced, at 0.6 and 0.6 — locks at 53.1 degrees.

Unbalanced, at 0.8 and 0.2 — locks at 36.9 degrees.

Sixteen degrees apart, at the same quoted cover, which is a large difference in drape from a number that reports them as identical. And the direction is unambiguous: a balanced cloth always locks later than an unbalanced one of the same total cover, because a maximum is minimised when the two are equal.

That is worth having as a design rule because it is free. Two constructions using the same amount of yarn, arranged evenly or unevenly, differ in how far they can be draped, and the even one is always the better draper.

What is actually asymmetric #

The locking angle itself is the same whichever way a cloth is sheared, because sin(90° − γ) and sin(90° + γ) are equal — so the two bias directions reach their wall at the same place.

What differs between directions is the extension the shear delivers on the way there. The long diagonal of a rectangular cell grows by a factor involving 2r/(1 + r²) with r the ratio of the two spacings, so an unbalanced cloth yields less extension per degree of shear than a balanced one — and its most compliant direction is not at forty-five degrees either.

So the two quantities the word bias runs together separate cleanly. The angle at which a cloth locks is a maximum over its two covers and is direction-free. The extension it gives before locking is a function of its cell’s aspect ratio and is not. A cutter meets both and has one word for them.

Cover, which is the controlling ratio #

The formula depends on one ratio and it is worth naming properly, because it connects this essay to the setting one.

The locking angle is $90° - \arcsin(d/p)$, where $d$ is the yarn diameter and $p$ the thread spacing. That ratio $d/p$ is the cover in one direction: the fraction of each inch occupied by thread rather than gap.

So locking is a function of cover and of nothing else. Not fibre, not weave except through the cover it permits, not strength, not finish. Two cloths with the same cover lock at the same angle whatever they are made of.

That gives a satisfying closure with the thread count essay. Cover is the quantity thread count is mistaken for; it is also the quantity that decides how a cloth drapes; and it is computable from two numbers a mill knows. A great deal of what people mean by fabric quality reduces to it.

What the model does not know #

Two limits, and they run the same way.

Threads flatten. The prediction assumes circular threads that touch and stop. Real yarn is compressible: it flattens, and the cloth continues to shear past the geometric angle with rapidly rising force. So the model gives a lower bound rather than a value.

Friction is absent. The trellis has none, so it says the cloth shears freely up to the lock and then stops absolutely. A real fabric resists from the first degree and stiffens gradually, and the force curve is smooth where the model’s is a step.

Both mean the same thing about how these numbers should be read: the locking angle here is the geometry’s answer, and a measured one will be larger. The ordering — denser locks sooner — is robust across every model and every measurement.

Where the number is used in earnest #

Two industries treat this as a working number rather than a curiosity, and their reasons are instructive.

Composites. A woven glass or carbon reinforcement has to be laid into a mould, and the shear the mould demands has to stay within the fabric’s locking angle everywhere. Exceed it and the reinforcement wrinkles, and a wrinkle in a laminate is a defect that shows up as a strength reduction rather than as a cosmetic problem. So the locking angle of a given cloth is measured, drape is simulated before any tooling is cut, and where the simulation says a single piece will not go, the preform is designed in several.

The simulation used is the fishnet construction — the same one behind the drape figures on this site — and it has been standard since the 1950s precisely because it is deterministic and cheap.

Tailoring, which arrived at the same place by hand. A cutter choosing whether a curve can be eased in or needs a dart is making the same judgement, and the phrase “this cloth will not take it” is a statement about a locking angle. The difference is that a tailor measures with their hands and a composites engineer with a picture frame.

The number in the hand #

It is worth converting the arithmetic into something a reader can check without apparatus, because this is one of the few quantities on this site that can be.

Take a piece of woven cloth. Hold it between finger and thumb of each hand at forty-five degrees to the threads, an inch or two apart, and pull. Note where the easy movement stops.

For an open shirting, the strip will lengthen noticeably — a fifth or a quarter — before it firms up. For a densely woven cotton drill or a canvas, it will hardly move at all. For a fine open voile, it will move a great deal and feel almost slack.

That easy range is the locking angle expressed as extension, and the ordering across fabrics is exactly the one the geometry predicts. It is unusual for a structural quantity in this subject to be directly palpable, and it is worth taking the opportunity: most of what this site computes has no signature a hand or an eye can find.

The other sign of curvature #

One asymmetry in the argument is worth flagging, because a check written only against locking would miss half the problem.

A surface of positive curvature — a sphere, a dome — closes the cells of the trellis, crowding the threads together, and that is what runs into the locking angle. A surface of negative curvature — a saddle — opens them, moving the threads apart. Threads moving apart do not jam, so there is no locking angle in that direction at all.

What a saddle produces instead is a cloth pulled open: gaps between the threads, a fallen cover factor, and in a composite preform a resin-rich region that is weak and does not announce itself. A wrinkle is visible in the mould; an opened cell is visible only in a section afterwards.

So the locking angle bounds one sign of curvature and says nothing whatever about the other, which is a limit on the number rather than a criticism of it.

Where it stops being a curiosity #

The locking angle is a geometric limit and it becomes an engineering constraint the moment a cloth is asked to take a shape rather than to hang in one.

Laying a woven reinforcement into a mould is exactly that. The curvature of the mould demands a definite shear at every point, computed from the geometry and not negotiable; the cloth supplies a definite shear, computed from its own sett. Where the demand exceeds the supply the trellis has no travel left and the cloth buckles out of the surface — which is a wrinkle, and it is predicted rather than observed. Shear locking in a preform does the comparison cell by cell and finds a radius at which it happens.

The consequence for the cloth is the one worth carrying away. Reinforcement fabrics are woven openly on purpose, and the purpose is not resin flow: it is to buy shear. A cloth at seventy per cent cover locks at about forty-six degrees; the same yarn at ninety per cent locks at twenty-six, and the difference decides whether a part can be made in one piece.

Where the ladder goes next #

The consequence in three dimensions is why clothes need darts: curvature demands shear, and locking is the budget.

The mechanism itself is the bias, and the related setting limit is jamming.

What the pictures here cannot show. The figures draw threads as lines and a jam as a hard limit. A real cloth approaches its lock gradually, with force rising and threads flattening, and none of that gradual behaviour is in a kinematic model.