Textile Structure
how cloth is put together
Mechanics and drape

The locking angle

The bias runs out at an angle that yarn diameter and thread spacing decide between them. It is the number behind whether a cloth will go round a curve, and it has nothing to do with how strong the fabric is.
14 min read 7 figures A mechanism, not a materialMeasured, not claimed

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.

How far the bias goes, and where it stopsExtension along the bias against shear angle, with the angle at which the threads jam marked for three settings. The curve is geometry and so is the wall — a more closely set cloth reaches it sooner.0%10%20%30%40%0°15°30°45°60°cover 40%locks at 66°, +38%cover 60%locks at 53°, +34%cover 80%locks at 37°, +26%extension along the biasshear anglethe yarns never stretch; the net changes shape
Fig. 1 Extension along the bias against shear angle, with the wall marked for three settings. The curve is the trellis geometry and the wall is where the threads jam, and a more closely set cloth reaches it sooner.

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 pp at ninety degrees to psin⁡θp\sin\theta at an included angle θ\theta.

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

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

The ratio d/pd/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.

A trellis sheared 25°The net at an angle, with every thread segment exactly the length it started at. The extension along the diagonal is the bias stretch, and it is a change of shape rather than a change of length.shear 25°bias +19.3%across -24.0%area 91%locks at 41°every segment checked against its own lengthno thread stretches
Fig. 2 A closely set cloth at twenty-five degrees of shear, already near its limit. The mechanism is still there; there is simply almost nowhere left for the threads to move.

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.

A trellis sheared 45°The net at an angle, with every thread segment exactly the length it started at. The extension along the diagonal is the bias stretch, and it is a change of shape rather than a change of length.shear 45°bias +30.7%across -45.9%area 71%locks at 60°every segment checked against its own lengthno thread stretches
Fig. 3 The net at forty-five degrees, well into the useful range. The cell sides are unchanged, the long diagonal is up by about a third, and this is roughly where a moderately set cloth would be approaching its limit.

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. There is a first phase where 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 phase 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.

A cloth laid over a sphereEvery thread segment is exactly one pitch and none has stretched. What has changed is the angle at each crossing, and the amount is decided by the surface — a developable one costs nothing and a curved one costs more the further the cloth goes.a sphereworst shear 22.2°cells closelocks at 57°nothing jamsthe fishnet construction, every segment checkeddoubly curved
Fig. 4 A cloth laid over a sphere, with the shear it costs. The requirement grows outward from where the cloth was first placed, and where it exceeds the locking angle the cloth cannot lie flat at all.

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.

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)90° - \arcsin(d/p), where dd is the yarn diameter and pp the thread spacing. That ratio d/pd/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.

The same thread count, twiceTwo cloths with identical thread counts and different yarn. The count is the same number in both; the fraction of the surface the threads actually occupy is not, and that fraction is what thread count is usually taken to mean.100 ends and 100 picks per inch in bothfine yarncover 78%coarser yarncover 97%cover from diameter and spacing, circular sections
Fig. 5 The controlling quantity. Two cloths of the same thread count and different yarn have different covers — and therefore different locking angles, different bias, and different drapeability, from a difference that thread count does not report.

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.

How far the bias goes, and where it stopsExtension along the bias against shear angle, with the angle at which the threads jam marked for three settings. The curve is geometry and so is the wall — a more closely set cloth reaches it sooner.0%10%20%30%40%0°15°30°45°60°cover 35%locks at 70°, +39%cover 55%locks at 57°, +35%cover 75%locks at 41°, +29%extension along the biasshear anglethe yarns never stretch; the net changes shape
Fig. 6 The budget for three settings. An open cloth has more than sixty degrees to spend and a dense one barely more than forty, and that difference decides which curves each can be persuaded round.

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.

How far the bias goes, and where it stopsExtension along the bias against shear angle, with the angle at which the threads jam marked for three settings. The curve is geometry and so is the wall — a more closely set cloth reaches it sooner.0%10%20%30%40%0°15°30°45°60°cover 35%locks at 70°, +39%cover 55%locks at 57°, +35%cover 75%locks at 41°, +29%extension along the biasshear anglethe yarns never stretch; the net changes shape
Fig. 7 What that test is measuring. Three settings, three budgets, and the point at which each stops — the number under a reader’s fingers when the cloth goes solid.

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.

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.

Where this goes next

shear 30°bias +22.5%across -29.3%area 87%locks at 60°every segment checked against its own lengthno thread stretches The bias is a mechanism Cloth cut at forty-five degrees stretches by a third and springs back, while the threads in it stretch by nothing at all. Almost every explanation given for this is wrong, and the right one is not about elasticity. a sphereworst shear 22.2°cells closelocks at 57°nothing jamsthe fishnet construction, every segment checkeddoubly curved Why clothes need darts A flat cloth goes round a cylinder for nothing and cannot go round a sphere at all without shearing. The amount of shear is decided by the curvature, and when it exceeds what the threads allow, something has to be cut out. weaveinterlacingsdensest settingplain1.0020 per inch2/2 basket0.5027 per inch2/1 twill0.6724 per inch2/2 twill0.5027 per inch3/1 twill0.5027 per inch5-end satin0.4029 per inch8-end satin0.2532 per incha geometric jamming model, circular sections, yarn diameter 1/40 inch How close can threads be set There is a maximum. Push more threads into a cloth than the geometry allows and they simply will not go, and the limit depends on the weave as much as on the yarn.