A membrane is cut smaller than it is

A tensioned membrane roof is made of flat panels of coated fabric, cut to a pattern, welded together and then pulled into a doubly curved shape and held there. The shape it has to take is computed; the pattern is computed from the shape; and then the pattern is made smaller than the shape, by one or two per cent, because the fabric will grow when it is stressed.

The trade calls that compensation. It is measured biaxially, on a cruciform specimen, by everybody who does this for a living — and it is different warpwise and fillwise, always in the same direction. What this site can say about it is where the growth comes from, and there turn out to be two mechanisms with only one of them in the account usually given.

What a fixed thickness allows, and what it does not #

Compensation is the crimp coming out. That much is not controversial: prestress a woven fabric and its threads straighten, so the cloth gets longer with no fibre stretching at all. The relaxation essays ran that arithmetic in the other direction and it is the same arithmetic.

The question is what a prestress can do at a fixed cloth thickness, and the answer is: interchange, and nothing else.

Peirce’s closure condition h₁ + h₂ = D holds the two crimp heights to a constant sum. Fix the thread lengths — nothing stretches — and fix D, and the state has exactly one degree of freedom left: how the crimp divides. So a load that straightens the warp must crowd the weft, and the cloth grows in one direction and shrinks in the other.

At a 2:1 stress ratio in the cloth below that is a warp gaining 2.6 per cent and a fill losing 5.7. It is crimp interchange, it is real, it is what makes a fabric narrow when it is pulled — and it is not compensation, because a membrane pulled in both directions grows in both directions.

So the mechanism that is usually named cannot do the job on its own, and the finishing field hit the same wall from the other side: solving a loaded state with an equilibrium condition gives plausible numbers with the wrong structure. That is the second time, which is what makes it a rule.

The solver had to be taught which way to move, and the story is worth keeping #

Establishing that took a defect out of stateFromLengths — the routine that solves a cloth from its two thread lengths and a crimp ratio — and the defect is worth recording twice over.

First, the routine had no callers. It was written during the finishing field, the relaxation solver took an arithmetic route instead, and it shipped through every gate the fleet has, twice, without being run. Its first caller found two bugs in an afternoon.

The first was a missing refusal: the bisection kept the last state it managed to solve, so a request it could not satisfy came back as a state at some other crimp ratio with nothing saying so. It refuses now, and names the ratio it could reach.

The second was subtler and is the interesting one. A trial angle can fail in two opposite ways. Too large, and the first thread has no straight portion left — that is the jam. Too small, and closure hands the second thread a crimp height it cannot reach at any angle. The bisection treated both as “too large” and moved its upper bound down, so on a thinner cloth it walked away from the solution, ran to the bottom of its range, and reported that no state existed while a perfectly good one sat in the middle of the interval.

What caught it was a symmetry. A square-set cloth in a square-set state must compensate equally both ways, so the figure asking for a crimp ratio of one is the one case where the answer is known in advance — and the solver refused it. Two other results this essay first reported went with the bug: a “determinacy” claim that no strain was available at fixed thickness, and a minimum flattening below which no state existed. Both were artefacts of a bisection walking the wrong way, and both are deleted rather than quietly adjusted, because a result that came out of a defect is not a result.

What actually supplies the strain #

If a fixed thickness only permits an interchange, then growth in both directions needs the thickness to move — and under biaxial prestress it does. The threads are squeezed where they cross, the crossings flatten, D falls, and with less thickness to bend around both systems can straighten at once. Both dimensions grow. That is compensation.

At a flattening of 1.12 — the crossing 12 per cent wider than it is thick — a cloth whose warp carries 9.3 per cent crimp and whose fill carries 16.9 compensates 1.65 per cent warpwise and 2.80 per cent fillwise. Those are the numbers a fabricator’s table carries, which is why the flattening is set where it is; and the fact that a fitted parameter has been chosen to land the answer in the observed range is stated in the code and in this sentence rather than left for a reader to infer.

Why the warp always gives back less #

The directional split is not fitted. It falls out of the crimps, and it is the part of this arithmetic that predicts rather than reproduces.

A warp is straightened twice before anybody stresses it: once by the loom, which holds it under tension for the whole of weaving, and again by the coating line, which pulls the fabric through under tension and sets the coating on it while it is stretched. So a coated base cloth arrives with less crimp in its warp than in its fill — that is what the 9.3 against 16.9 above represents — and it therefore has less crimp to give back.

Compensation is the crimp coming out, so the direction with less crimp compensates less. Warp compensation is smaller than fill compensation, in every fabricator’s table, for a reason that is one line of geometry rather than an empirical regularity.

The ceiling, and the case where a compensation goes negative #

Two boundaries come with the arithmetic, and both mean something.

The ceiling is the crimp. A direction cannot take up more length than its crimp holds, so compensation is bounded by c/(1 + c): 8.5 per cent warpwise and 14.4 fillwise for the cloth above. It is the same bound the finishing field derived for shrinkage, read in the opposite direction, and it is asserted at every row rather than mentioned.

The other boundary is a sign. At a lopsided stress ratio the interchange takes length out of the lightly loaded direction faster than the flattening puts it back, and that direction’s compensation goes negative — the pattern has to be cut larger than the shape across the fill. At a 2:1 ratio and a flattening of 1.12 the warp compensates 3.8 per cent and the fill by −1.4.

Fabricators do report negative fill compensations at extreme stress ratios, so the model’s sign change is a feature rather than an embarrassment — and the assertion in the code is conditioned accordingly. Positivity is asserted only at an equal stress ratio, because asserting it generally would have been an assertion about the default arguments rather than about the claim, which is this site’s commonest way for a check to go quietly bad.

Which length the per cent is a fraction of #

There is a place to lose a fifth of the answer here, and this site has already written an essay about the same trap in a different field.

The strain is measured against the unstressed length. The compensation is what has to be taken out of the pattern, which is measured against the stressed length — the shape the fabric is meant to end up as. Those are different denominators: a strain of 2.88 per cent is a compensation of 2.80.

Shrinkage and compensation are one calculation with the arrow reversed #

It is worth putting the two fields side by side, because the machinery is identical and the trades have no contact with each other.

Relaxation shrinkage is a cloth taking up crimp it was prevented from having: the loom held it straight, the constraint goes, the thread cannot lengthen, so the cloth shortens. Compensation is a cloth giving up crimp it was allowed to have: the prestress straightens it, the thread cannot lengthen, so the cloth grows. One expression, (c₂ − c₁)/(1 + c₂), with the two crimps swapped.

And both hit the same wall in the same way. The finishing field’s first attempt solved the loom state with Peirce’s closure condition and got a cloth that shrank in one direction and grew in the other; this essay’s first attempt redistributed the crimp at constant thickness and got no strain at all. Both failures are the closure condition being imposed on a state that does not satisfy it. An equilibrium condition applied to a loaded fabric produces plausible numbers with the wrong structure, and the second time is the one that makes it a rule rather than an anecdote.

What a patterner actually does with the number #

The arithmetic above produces two percentages, and it is worth being clear about what happens to them, because it is not a scaling.

A membrane panel is not shrunk uniformly. Warp compensation is applied along the panel and fill compensation across it, so the pattern is scaled by different factors in two directions — which means the flat panel’s shape is not similar to the shape it will take. Its edges are recomputed, and on a doubly curved surface where the warp direction wanders relative to the panel’s own edges, the compensation has to follow the fibre rather than the panel.

That is why the number matters more than its size suggests. Two per cent on a thirty-metre roof is six hundred millimetres, distributed unevenly around a boundary that is being welded to a cable; getting it wrong does not produce a slightly loose roof but a boundary that will not meet its fittings. And getting the direction wrong — applying the fill figure along the warp — produces an error of one per cent in each of two directions with opposite signs, which is the kind of mistake that shows up only when the last panel is being pulled into place.

So the ordering result above is the practically important one. Warp compensates less than fill, always, for a geometric reason, and a patterner who knows that has a sanity check on a table of measured numbers that arrived from a laboratory.

What was counted, and how #

Three computations and three assertions that are not formalities.

The delivered state is Peirce’s geometry at a stated crimp ratio, and its solution is fed back through assertPeirceConsistent, which re-runs the model’s own equations forwards and requires the residual below 10⁻⁹. That check has been in yarn.js since the foundation and it is the reason a solver failure here would be loud.

The prestressed state is the same two thread lengths at a flattened thickness. The warp’s cloth dimension is the weft spacing and the weft’s is the warp spacing — the crossing-over that produced a cloth which grew when it was relaxed, during the finishing field — and the code names that in a comment beside the line, so the next reader does not have to rediscover it.

Then three assertions. The identity at an equal stress ratio — the delivered ratio at the delivered thickness must return the delivered cloth, to a part in a million, which is what the corrected solver now does and what the broken one could not. The symmetry at equal crimps, which is the check that found the defect. And the ordering when the crimps differ, which is the claim the essay turns on and the one a sign error would invert.

What the picture cannot show #

The compensation figure draws two curves against a flattening, and the flattening is the one quantity in this essay that is not computed.

So the picture cannot show its own most important caveat: that its horizontal axis is a parameter standing in for a mechanical model this site does not have. A reader who takes the curves as a prediction has taken more than the figure offers, and the caption says so on its own face — which is the only place it can be said.

Nor can it show the coating, which carries much of a membrane’s shear and holds its threads in place; or the creep that makes compensation a function of time as well as of stress; or the biaxial state itself, which is a surface rather than a curve. The figures that draw cloth in this field — the crossings flattening, the two rulers — carry geometry, and the trade-off figure carries a specification’s shape.

Where the model stops #

The flattening is a fitted number and everything scales with it. The strain a membrane takes under prestress needs the yarn’s transverse stiffness, the coating’s contribution, and the way a crossing deforms when it is squeezed between two coated systems. None of that is here. flatten stands in for all of it, and its default was chosen so that the answers land in the range fabricators report — which is a calibration, is labelled as one, and means no per cent in this essay is a prediction.

Real compensation is not a constant of the fabric. It depends on the prestress level, the stress ratio, the temperature, and how long the fabric has been held — because a coated fabric creeps. Fabricators measure it under the ratio and level the structure will actually see, and quote it per fabric type and per stress ratio. A single pair of numbers is a simplification of a surface.

The stress ratio has been left out of the state. In this arithmetic the crimp ratio is the fabric’s own, carried over from its delivered state, and the prestress only flattens. A full treatment would let the load redistribute the crimp as well — which the determinacy result says it cannot do at constant thickness, and which it can do once the thickness is free. That coupled problem is not solved here.

And nothing here is the coating. A membrane’s coating carries shear, seals the weave, holds the threads in place and does much of the mechanical work; the arithmetic treats the fabric as bare. The direction of every result survives that, and none of the magnitudes should be trusted past their stated calibration.

Who found it, and when #

Compensation as a practice is as old as tensioned-membrane engineering — the 1950s and 1960s, Frei Otto’s work and the cable-net roofs that followed — and it has always been a measurement rather than a calculation. The biaxial cruciform test exists precisely because the two directions cannot be predicted from two uniaxial tests, and the fabric literature is candid that the numbers are fabric-specific.

What is offered here is not a replacement for that measurement. It is an account of the mechanism’s structure: that the strain cannot be crimp redistribution at constant thickness, that it must therefore be the crossings flattening, that the direction with less crimp compensates less, and that the whole of it is bounded by c/(1 + c). Every one of those is a statement a measurement cannot make, and none of them is a number a measurement can be replaced by.

Where the ladder goes next #

Cutting a pattern smaller than its shape is one of two ways cloth and geometry disagree at the cutting table. The other is that a flat piece cannot cover a curved form at all, and the applied field’s next two rungs are about what a cutting room does with that: first the bias cut, whose famous waste turns out to be a statement about one panel and a selvedge rather than about the diagonal.