Bending stiffness and the drape coefficient

Everything on this site’s mechanics ladder so far has treated cloth as a pin-jointed net: two families of inextensible threads, free to rotate where they cross. That model is powerful and it has one enormous hole in it. It has no stiffness. It says what shapes a cloth can take and refuses to say which one it will take, because taking a shape is a matter of energy and there is no energy anywhere in it.

The two measurements in this essay are the trade’s attempts to fill that hole, and they are worth putting on a site about structure because both of them turn out to be more geometric than they look.

The cantilever test #

The idea is elegant. A strip of fabric pushed out over an edge bends under its own weight. A stiff fabric goes a long way before it droops; a limp one droops at once. So push it out until the tip has drooped to a fixed angle, and the overhang at that moment is a measure of stiffness with the units of a length.

That length is the bending length, and Peirce’s expression for it is

$$c = L\left(\frac{\cos(\theta/2)}{8\tan\theta}\right)^{1/3}$$

where $L$ is the overhang and $\theta$ the droop angle at the tip. The flexural rigidity then follows as $G = W c^3$, with $W$ the mass per unit area.

The cube is the whole reason the test is designed around a length rather than around a rigidity. A ten per cent error in the bending length is a thirty-three per cent error in the rigidity, so the measurement is arranged so that the quantity read off the apparatus is the one where the error is smallest.

The convention that is not quite exact #

The test is universally described in a shorthand: push the strip out until the tip reaches 41.5°, and the bending length is half the overhang.

Run the formula at 41.5° and the factor is 0.5093.

Not a half. Two per cent out in the bending length, and — because of the cube — about six per cent out in the flexural rigidity. The angle at which the factor really is a half is 42.94°.

Six per cent is not a scandal. It is smaller than the spread between specimens cut from the same piece of cloth, and it is entirely consistent within a laboratory because every laboratory uses the same angle. What it is not is exact, and describing the test as “the bending length is half the overhang” states an equality where there is an approximation.

The reason to say so is not pedantry. It is that a reader who believes the equality will use it to convert between droop angles — and the factor is not constant, so a measurement taken at a different angle and converted with the one-half rule is wrong by considerably more than six per cent.

The drape coefficient #

The other measurement is different in kind. A circular specimen of fabric is laid over a smaller circular pedestal; it falls into folds; and the number reported is the fraction of the annulus that its shadow still covers.

$$F = \frac{A_{\text{shadow}} - A_{\text{pedestal}}}{A_{\text{specimen}} - A_{\text{pedestal}}}$$

A perfectly stiff cloth does not drape at all, its shadow is the whole specimen, and $F = 1$. A perfectly limp one hangs vertically from the pedestal’s rim, its shadow is the pedestal, and $F = 0$. Real fabrics sit between, and the number is quoted as a percentage.

What can be done with that geometrically is more than it first appears, and it turns on one constraint.

The hem does not stretch #

The draped specimen’s edge is a closed curve of cloth. Cloth does not stretch. So whatever shape the hem takes, its length is the flat specimen’s circumference, and that single fact fixes almost everything.

Model the hem as a wave with $n$ folds,

$$r(\varphi) = \bar{R}\,(1 + a\cos n\varphi)$$

and the amplitude $a$ is not a free parameter: it is whatever makes the arc length come out at $2\pi R_{\text{specimen}}$. Solving for it is a one-dimensional bisection on a monotone function, and the result can be checked by re-integrating — which the figures do, to within a part in a billion.

The projected area then follows by integration, and so does the coefficient. It is the drape coefficient’s own construction, taken literally.

What the coefficient is actually sensitive to #

Now the finding, and it is a negative one.

The fold count barely matters. Going from six folds to fourteen — which is a larger range than real fabrics show — changes the coefficient by about two points. Going from two folds to six changes it by about seventeen, and two-fold drapes do not occur in the apparatus.

The hem radius matters enormously. Moving the mean hem radius from just outside the pedestal to nearly the flat specimen’s edge takes the coefficient from nine per cent to ninety-eight.

So the apparatus is measuring how far the hem has come in, and almost nothing else. That is a defensible thing to measure — it is a real and relevant quantity — but it is not what the number is usually described as capturing. The drape coefficient is routinely presented as summarising “how the fabric falls”, with the fold count and the fold depth as part of the story. On the geometry, the fold count is nearly invisible in the answer.

The consequence is practical. Two fabrics can have identical drape coefficients and drape visibly differently, because one falls into five deep folds and the other into eleven shallow ones. Anyone who has compared two fabrics with the same measured drape and disagreed with the instrument has met exactly that.

The units, and why they are awkward #

The two measurements are quoted in units that make them hard to compare, and the awkwardness is instructive rather than accidental.

Bending length is a length, in centimetres, and it is the length of cloth that bends under its own weight to the stated angle. That makes it directly comparable between fabrics of different weights, which is its great virtue: a heavy stiff cloth and a light limp one can have the same bending length and will drape alike.

Flexural rigidity is a length cubed times a mass per area, so in the traditional units it comes out in milligram-centimetres, and it is not comparable across weights. It is the quantity that enters a mechanical calculation and the wrong quantity for comparing fabrics.

The drape coefficient is a pure number with no units at all, which sounds like the cleanest of the three and is the most treacherous. It depends on the apparatus: the specimen diameter, the pedestal diameter, and the ratio between them. A coefficient measured on a 30 cm specimen over an 18 cm pedestal is not comparable with one measured on a 24 cm specimen, and the ratio matters more than either dimension. Quoting a drape coefficient without the apparatus is quoting a number that cannot be reproduced.

That is worth setting beside the thread count problem, because it is the same shape of error. A quantity that looks self-describing — a count, a percentage — turns out to depend on a convention nobody quotes, and comparisons made across the convention are meaningless.

Where the two tests came from #

Both are British, both are twentieth-century, and both were invented to replace a hand.

The cantilever test is Peirce’s, from a 1930 paper on the “handle” of cloth — the same Peirce whose thread geometry the setting ladder is built on. The problem he was addressing is the one every textile buyer had: judgements of quality were made by feeling the cloth, which is reliable within one person and not transferable between two. A measurement with a number attached could be written into a contract.

That is why the apparatus is so simple. It was designed to be usable in a mill by someone with no instruments: a flat plate, a ruler, and an inclined line at 41.5°. The specimen is pushed forward by hand until its tip touches the line. There is nothing to calibrate and nothing to go wrong, and the 41.5° is chosen so that the arithmetic afterwards is a halving.

The drapemeter is Cusick’s, from the 1960s, and it addresses the part Peirce’s test cannot reach: a strip bends in one direction and a garment does not. The original apparatus is a light source above and a ring of paper below, on which the shadow is traced and then cut out and weighed — a beautifully direct way of measuring an irregular area with no instrument at all. Modern versions photograph it.

Both tests, in other words, are shaped by what could be done in a mill in their decade, and both have survived long past that constraint. That is an ordinary fate for a standard, and it is why the 41.5° convention persists: changing it would break comparability with sixty years of measurements, and the two per cent it costs is smaller than the noise.

What the geometry declines to say #

Here is where the model stops, and it stops in a way worth naming precisely.

The geometry describes a family of draped shapes: any number of folds, any hem radius, all of them satisfying the inextensibility constraint. It says which are available. It says nothing at all about which one a given cloth will choose.

That choice is decided by energy — by the bending stiffness measured in the first half of this essay, against the fabric’s own weight. A stiff cloth resists curvature, so it makes few, large folds and keeps its hem far out; a limp one makes many small folds and lets the hem come in. There is no stiffness anywhere in the geometry, so there is no way for the geometry to pick.

That gap is the honest content of the essay. The cantilever test measures the thing that does the choosing; the drapemeter measures the outcome of the choice; and the geometry connects neither to the other. A complete account would need a bending energy and a minimisation, which is a mechanics problem this site does not attempt.

What decides the bending length itself #

One more layer down, and it connects this essay to the rest of the site.

A fabric’s bending stiffness is not a fibre property. It is dominated by whether the threads can slide past one another as the cloth bends — because a fabric bending as a solid plate is far stiffer than the same fabric bending as a stack of independent threads, and real cloth is somewhere between.

That puts the interlacing count squarely in the middle of it. A plain weave has the most interlacings of any weave, so its threads are most constrained and it is stiffest; a satin has the fewest, so its threads are freest and it is limpest. The same yarn at the same sett in the two weaves gives bending lengths that differ substantially, and the ordering is predictable from the matrix even though the magnitude is not.

The same argument explains why a double cloth drapes better than a single of the same weight: two thin layers free to slide over each other bend far more readily than one thick layer, and stitching them sparsely preserves most of that freedom.

So bending stiffness is one of the properties this site can predict qualitatively from the structure and not quantitatively. That is an unsatisfying position and it is the accurate one: the matrix decides the ordering, and the friction decides the numbers.

Why the two measurements are both needed #

A last observation, because it explains why the trade uses both despite the overlap.

Bending length is a one-dimensional measurement: a strip, bent in one direction. Drape is inescapably two-dimensional, because a circular specimen falling into folds is bending in two directions at once and shearing while it does so.

That difference is not a technicality. A fabric bends easily in two directions only if it can also shear, because a doubly curved surface cannot be reached by an inextensible sheet without shear — which is the darts argument in another form. So a cloth’s drape depends on its shear stiffness as well as its bending stiffness, and the cantilever test measures neither of those together.

Which is why a fabric can be limp in a strip and stiff in a drape: a loosely woven cloth with high inter-thread friction shears badly, so it will not form folds, and it reports a high drape coefficient despite a short bending length. Sailcloth and coated fabrics behave exactly that way.

The factor is not a constant, and here it is #

The warning above — that converting between droop angles with the one-half rule is wrong by considerably more than six per cent — deserves the table that makes it usable, because the factor moves fast and in one direction.

Across the range a droop could plausibly be read at, the factor falls by nearly a factor of two, and since the rigidity goes as its cube, the same overhang read at 30° and at 70° differs by more than fivefold in reported rigidity. A conversion made with the half rule at a non-standard angle is not slightly wrong; it is wrong by more than the difference between fabrics.

That is the practical reason the standard fixes the angle by drawing a line on the apparatus rather than by asking the operator to judge it. The angle is the reading with teeth in it and the overhang is the tame one.

Which reading is the sensitive one #

The two sensitivities are worth comparing directly, since it decides what the apparatus should make easy.

The overhang enters linearly and is cubed, so a one per cent error in the length is three per cent in the rigidity. On a hundred-and-fifty-millimetre overhang, one per cent is a millimetre and a half — comfortably readable off a rule.

The angle enters through the factor. Differentiating across the table, the factor changes by about 1.26 per cent per degree near 41.5°, so one degree of angle error is 3.8 per cent in the rigidity — more than a whole millimetre of overhang, and far easier to commit by eye.

So the apparatus is arranged exactly the right way round. It removes the angle judgement entirely, by giving the operator a fixed inclined line that the tip either touches or does not, and leaves them a length to read against a rule. Nothing has to be estimated; the only two operations are pushing and reading.

That is a piece of instrument design worth admiring, and it is the reason the test survived. A version asking an operator to push the strip a fixed distance and then measure the droop angle would be reading the same physics through the sensitive variable, and it would scatter.

What limits the test in practice #

Putting the two together says where the measurement’s error actually comes from, and it is neither of the above.

A carefully read overhang gives the rigidity to about three per cent, and the fixed line removes the angle error altogether. Specimens cut from one piece of cloth scatter by considerably more than that — a fabric’s bending length varies across a width with the sett, with the finish’s evenness and with which warp ends the strip happened to include.

So the cantilever test is limited by the cloth rather than by the apparatus, which is the correct place for a measurement to be limited and is unusual. The six per cent the 41.5° convention costs sits below that floor, which is exactly why it has never been worth fixing — and why fixing it would break comparability with every measurement since 1930 for a correction nobody could detect on a single piece of cloth.

The place it does matter is a conversion between conventions, which is the one operation the floor does not protect: a systematic factor applied to every specimen does not average out over a piece the way a specimen’s own scatter does.

What neither test measures #

Four omissions, and they are the reason a buyer still feels the cloth.

Direction. Both tests are usually run on strips cut warpwise and weftwise, and a fabric’s bending stiffness on the bias is quite different from either — usually much lower, because bias bending recruits the shear mechanism as well. A garment panel cut on the bias behaves like a fabric neither strip measured.

Curvature sign. A cloth bent one way and the other way is not the same object: an unbalanced construction with more warp on one face is stiffer bent one way than the other. The cantilever test is run face-up by convention and does not report the difference.

Time. Both tests are static. A fabric’s response to being bent quickly and slowly differs, and a garment in motion is doing the fast one.

And surface. The whole reason bending stiffness was ever measured was to replace a judgement about handle, and handle is bending stiffness together with shear stiffness, compressibility, surface friction and thermal feel. Reducing it to one length was an enormous simplification that made contracts possible and did not make the judgement transferable. The later Kawabata system measures all five, at considerably more expense, and is used where the money justifies it.

Where the ladder goes next #

The rung below is why clothes need darts, where the impossibility a drape is negotiating is stated exactly. The rung above, in the neighbouring ladder, is shear locking in a preform, which is what happens when a cloth is asked to take a doubly curved shape and runs out of shear.

The companion measurement is the locking angle, which is the geometric limit on the shear all of this depends on — and which, unlike everything in this essay, is exact.