Every crossing is a force

This site’s central check asks whether a draft describes one cloth or several, and it answers exactly, by connectivity: draw an edge from every lower thread to the upper thread at each intersection and ask whether the digraph is strongly connected. It has been the spine of the collection since its first essay.

It also has a standing limitation, recorded in an essay of its own. The criterion cannot see friction: it says whether the threads are topologically interlocked and knows nothing about how hard they are pressed together, so a fabric it calls two cloths may behave as one because its layers are held by surface fibre, and a fabric it calls one may come apart in the hand because nothing is gripping.

That essay ends by saying the missing quantity is a force. Here it is.

The claim #

A thread under tension T pressing over another at weave angle θ presses it with N = 2·T·sin θ.

Nothing else is needed and nothing is fitted. The tension is whatever is pulling the thread; the angle is Peirce’s, which this site has solved for every cloth in its table since the setting field was built.

The consequence worth stating first is a ratio rather than a value. For an ordinary sheeting θ is 36.8°, so 2 sin θ is 1.20 — and the crossing carries more than the thread running through it. A cloth is not a structure in which the joints are gently held while the members do the work; in a closely set cloth the joints are the most heavily loaded part of it.

Why the factor is what it is #

The derivation is a triangle. Consider the piece of warp in contact with one pick. It comes in along a straight run inclined at +θ to the plane of the cloth, wraps the pick through an arc, and leaves along a straight run at −θ. The tension is the same at both ends — nothing is being lost to friction in this idealisation — so the vector sum of the two pulls is a single force directed into the pick, of magnitude 2T sin θ.

The radius does not appear, and neither does the diameter. That is the same reason the capstan equation is usable here and it is worth noticing twice: the geometry that matters at a crossing is the angle turned through, not the size of the thing turned around.

So the whole variation across a table of cloths is variation in θ, and θ is decided by the sett and the yarn diameters through Peirce’s solution.

The factor passes one somewhere around thirty degrees, which for cotton yarn is somewhere around a cover factor of two-thirds. Below that a crossing is more lightly loaded than the thread; above it, more heavily.

What a closely set cloth is doing to itself #

Reading the table the other way round gives something a weaver would recognise and would not have expressed this way.

Setting a cloth more closely loads its crossings harder for the same thread tension. A cheesecloth’s crossings carry two-fifths of what its threads do; a sheeting’s carry six-fifths. That is a factor of three between two cotton cloths, from geometry alone, with no change in the yarn.

Three consequences follow and the first two are already elsewhere on this site under different descriptions.

The first is that a closely set cloth grips its own threads harder, which is why a thread is harder to pull out of one, why it frays less, and why its seams slip less. That is the next rung and the quantity it needs is exactly this one.

The second is about wear. Floats and abrasion argues that a long float wears faster because it stands proud and takes the rubbing. There is a second mechanism visible here: the interlacing points of a closely set cloth are being pressed together hard, and a pressed contact that moves is a wearing contact. The most tightly interlaced cloth is not simply the most durable one — it has fewer exposed floats and more heavily loaded crossings, and which effect wins is not decidable from either alone.

There is a third, and it is about how a cloth is held together at all. What holds a pick in argues the topological half of that question: a pick is held because it is interlocked, and interlocking is a matter of what goes over what. The force half says the interlocking is worth a specific number of newtons per crossing, and the number is not the same for two cloths with identical drafts.

The case this cannot do, and why the refusal is honest #

The force above is the contact force in a cloth that is under tension — on the loom, in a seam being pulled, in a fabric under load. A relaxed cloth lying on a table is not under tension, and its crossings are still pressed together, or it would fall apart in the hand.

What does the pressing then is the threads’ own resistance to being bent. A crimped thread is a bent rod trying to straighten, and the things stopping it are the threads it is wrapped around; the force between them is whatever it takes to hold the shape.

That force is not available from the model this site has, and the reason is specific rather than a shrug. Peirce’s path is not an elastica. It is arcs of constant curvature joined to straight runs, so the curvature jumps at every join, and a real bent rod’s curvature is continuous. The bending energy of such a path is perfectly well defined — it is what the crimp-ratio rung and the load–extension rung both use — but the contact forces cannot be recovered by differentiating along it, because the discontinuity would put point couples at the joins and a smooth contact cannot deliver a couple.

The honest options were to solve an elastica, which is a different model and a larger piece of work, or to state the limitation and take the tension route where it applies. This rung takes the second.

So every number on this page is a number about a loaded cloth. Where a later rung needs a relaxed one — a tuft in a carpet that is not being pulled, a seam allowance before anything happens to it — the tension is stated as an assumption and the answer is proportional to it.

The factor has a crossing point and a ceiling, and the way to see both is to run the same crossing through a cloth at the other end of the range.

The factor has a crossing point and a ceiling, and both are exact #

Two numbers can be read straight off 2 sin θ without solving any geometry, and both are worth having because they bound the whole table.

The crossing point is exactly thirty degrees. 2 sin θ equals one when sin θ is a half, so a crossing carries precisely as much as the thread running through it when the thread arrives at 30° to the plane of the cloth — that is, when it changes direction by sixty degrees in passing over. Below that the joint is the lightly loaded part of the structure and above it the joint is the heavily loaded part, and the table’s transition between the filter cloth at 31.8° and the duck at 30.5° is that number rather than an accident of those two constructions.

The ceiling is exactly two. The sine is bounded by one, so no crossing anywhere can carry more than twice the tension in its thread, and it approaches that only for a thread arriving perpendicular to the cloth and doubling straight back on itself. An ordinary sheeting at 1.20 is therefore three fifths of the way to a limit no fabric can reach.

That the whole range of woven cloth occupies 0.39 to 1.20 out of a possible 0 to 2 is a more informative statement than the table alone. There is no construction available that would make a crossing dramatically more loaded than a sheeting’s, because the geometry runs out: reaching 1.5 needs 48.6°, and a weave angle that steep needs a cover far past where the threads jam. The factor is nearly saturated at the close end of ordinary cloth, and everything below it is spread over the open end.

What extension does to the grip #

The crossing force is a tension times a sine, and pulling a cloth changes both — in opposite directions, and in different systems.

Pull a cloth warpwise. The warp’s tension rises, which raises its crossing force; and the warp straightens, which is crimp interchange, so its weave angle falls, which lowers it. The two effects fight, and the product is not proportional to the load — a cloth extended warpwise presses its crossings harder than at rest but by less than the tension rose.

The weft does the opposite and does it with nothing pulling on it. Its crimp deepens as the warp’s shallows, so its weave angle rises through the whole extension, and the transverse force it delivers rises with it even though its own tension has not changed at all. A cloth pulled one way grips harder in the other, and the mechanism is the interchange rather than any load path.

That is the sharpest form of the asymmetry named in the limitations above, and it has a consequence for a seam. A seam allowance is a piece of cloth being pulled in one direction, and what stops its threads sliding out is the grip in the other direction — which the pull is increasing. So the fabric’s resistance to having a thread drawn out of it is not a constant of the fabric; it is a function of how hard the fabric is being pulled at right angles to the thread in question, and it rises with the load that is trying to pull it apart.

Nothing here computes how much, because the interchange gives the geometry at each state and the tension in the unpulled system is still whatever it happens to be. What the argument fixes is the sign, which is the direction nobody would guess: stretching a cloth tightens its grip on the threads it is not stretching.

What was counted, and how #

The angle comes from Peirce’s geometry, solved by bisection on how the cloth’s thickness divides between the two systems, and every solution is fed back through the equations it was solved from before it is used; the worst residual across the table is at machine precision.

The tension is stated. Every figure on this page uses half a newton in a warp end unless it says otherwise, which is a plausible tension for a cloth being handled rather than tested; on the loom the shed alone puts several times that into a back-shaft end. Every force here scales exactly with it.

Two refusals are checked rather than assumed. A contact force with no tension in the thread is refused outright rather than returning zero, because zero would be a wrong answer — a relaxed cloth’s crossings are not unloaded — and a function that silently returns a plausible wrong number is worse than one that stops. And a friction coefficient of zero is refused for the same reason one step further on.

The one quantity that is a range rather than a value is friction, which does not appear on this rung at all and appears on every rung above it. It is quoted at 0.2 to 0.4 for cotton on cotton, from the same table the pile ladder has used since it was built.

Where the model stops #

The relaxed case is missing. Stated above at length, because it is the limitation a reader is most likely to walk into.

The two systems are not required to agree. The force the warp presses on the weft with, and the force the weft presses on the warp with, are computed from different tensions and different angles and are equal only if the cloth happens to be loaded in the ratio that makes them so. In a cloth pulled one way only, they are not equal — which sounds like a contradiction and is not, because the difference is carried by the cloth’s own curvature out of plane.

Nothing here is a pressure. A real contact is an area, the thread flattens over it, and the pressure distribution across that area is what actually decides friction and wear. The site models a flattened thread with the racetrack section and has no contact mechanics for it.

And the arcs are frictionless. The tension is taken as the same on both sides of the crossing, which is what makes the resultant 2T sin θ. With friction the tensions differ — that is exactly the capstan effect — and the next rung uses it, so the two rungs are using two idealisations of the same crossing for two different purposes. Both are standard and the inconsistency is worth naming.

The generalisation #

The statement that survives is about any structure made of tensioned members that deflect around each other.

A load-carrying member that turns through an angle delivers a transverse force proportional to the sine of half the turn, and the radius of the turn does not enter. That is the rope-over-a-pulley calculation, the cable-over-a-saddle calculation and the tendon-around-a-bone calculation, and it is why a small change of direction in a heavily loaded cable produces a very large force at the deviation.

The textile version has a feature the engineering versions usually do not: the deviation angle is set by the same geometry that sets everything else about the object. A cable’s saddle angle is a design choice. A cloth’s weave angle is decided by its sett and its yarn diameters, so a maker choosing a cover factor is choosing a crossing force without being told, and the two are not separable.

The second lesson is about the shape of the refusal in the middle of this rung. A model can be exactly right for one quantity and structurally unable to give another, and the boundary is not always where it looks. Peirce’s arcs give an energy and cannot give a force, which is counter-intuitive — energy is usually the harder thing — and the reason is a curvature discontinuity that costs nothing when integrated and is fatal when differentiated.

Who found it, and when #

The resolution of thread tension into a normal force at a crossing is elementary and is in every account of fabric mechanics that treats yarn friction at all; there is no discovery to attribute.

What is worth attributing is the problem it is being used on. The idea that a woven fabric’s mechanical behaviour is dominated by what happens at the crossings — rather than by the yarns as members — is the central claim of the fabric-mechanics tradition that runs from Peirce’s 1937 geometry through Olofsson and Grosberg in the 1960s, and yarn-on-yarn friction as a measured quantity belongs to the same period.

This site’s own contribution is a negative one that took some working out: that its existing geometry can give the loaded contact force exactly and cannot give the relaxed one at all, and that the reason is a property of the path rather than a gap in the arithmetic.

Where the ladder goes next #

The next rung counts crossings. A thread gripped over a length of cloth is held at every crossing it makes, so the resistance rises with the length held while the thread’s own breaking load does not — and where those two curves meet is a length that turns out to be a seam allowance.

Sideways, the same force is what holds a tuft into a carpet once the capstan is applied to it, and it is the missing frictional half of the beat-up, where the reed has to push a new pick past picks that are already gripped.

Further out is the elastica. A model that gives a relaxed cloth’s contact forces would close the standing limitation on this site’s central criterion, and it is the largest single thing missing from the mechanics field.