A cloth held both ways settles higher

A tensioned cloth loses its load found a floor, and made the floor a design rule. Clamp a cloth at one length, and its crossings rearrange until the load has fallen to what friction can hold. That level is friction times a contact force times the sine of a weave angle, and none of the three depends on how hard the cloth was pulled. For a sheeting at a friction of 0.3 it is 0.0756 newtons an end. Re-tensioning gets nowhere, because the load it settles at is the floor it settled at before.

That account clamped one direction. Its reasoning took the crossing’s contact force from the cloth at rest, and treated it as something the clamped load does not change, which is correct for a strip with its other edges free. An awning, a stretched screen, a printing frame, a sail or a tent wall is held on all four sides. So the question left unanswered by that essay is simple and its answer is not a detail: what is the floor when both systems are under load? Each one presses the crossings the other is trying to slide through.

Each system presses for the other #

A pick under tension bends round the ends it crosses, and to bend it they have to push on it. That is every crossing being a force. With tension $T_2$ along it and its path turning through an angle $\theta_2$ either side of a crossing, each crossing carries an extra force $2T_2\sin\theta_2$.

The ends sliding through that crossing feel it, as a pressing they have to overcome friction against. So the resting contact force the earlier account used gets the other system’s own contribution added on. The floor for the ends is then

$$T_1 = \mu\,(N_0 + 2T_2\sin\theta_2),$$

and by exactly the same reasoning, with the indices swapped, the floor for the picks is

$$T_2 = \mu\,(N_0' + 2T_1\sin\theta_1).$$

Here $\mu N_0$ and $\mu N_0'$ are the one-way floors, unchanged. Neither floor is known until both are, and the pair is linear. So they solve in one line:

$$T_1 = \frac{\mu N_0 + 2\mu\sin\theta_2\,\mu N_0'}{1 - 4\mu^2\sin\theta_1\sin\theta_2},$$

with $T_2$ the same with the indices swapped. Two features of that expression matter, and the first is what is not in it. The load the cloth was pulled to does not appear. Pull a framed cloth as hard as it will bear and it settles to this pair, and pull it harder and it settles to this pair. So the one-way account’s rule survives whole: on a frame as off it, re-tensioning buys the time between the pull and the settling and nothing else. What changes is the level.

Half as high again on a sheeting #

Solved at a friction of 0.3, the sheeting’s two floors go from 0.076 and 0.070 newtons a thread to 0.115 and 0.113. That is a gain of 1.52 and 1.62, and it brings the two nearly level, where the warp had a small lead. Its crimp angles, 39 and 36 degrees, are large and nearly equal, so each system gets a substantial push from the other.

A duck gains 1.40 and 1.49, a filter cloth 1.43 and 1.51, a muslin 1.27 and 1.42, and a batiste 1.26 and 1.37. The size of the gain follows the crimp angle of the other system, which is what the equation says it should do. On a cheesecloth, whose weft is almost straight at six degrees, the warp’s floor rises by three per cent and no more, since a straight weft presses nothing, while the weft’s rises by half.

A poplin is the sharp case. Its warp is crimped at 41 degrees and its weft at 11, and those are construction facts, a warp-dense shirting with its weft nearly straight. Its warp’s floor rises by 8 per cent, and its weft’s by a factor of 2.47. The weft is being pressed by a steeply crimped warp, and itself has little to offer in return. So on a frame the poplin’s weft goes from a little over a quarter of its warp’s floor to about two thirds. A frame evens the two floors of a cloth that a strip test would show as wildly unbalanced.

That last point has a practical edge. A cloth’s weft floor from a strip test understates by more than half what it holds on a frame, and it understates the warp by less. Strip tests of a structural fabric, in each direction with the other free, rank its directions by a difference the frame largely removes, and a specification written from strips is designed for a loading the structure never sees.

Weight lifts a floor, and tension does not #

The one thing the first account could not offer was a way to raise a floor from outside the fabric. It said raise friction, or set the cloth closer, or leave it inside its band, and those are choices made before the cloth is cut. The equation for the ends offers a fourth, and it comes from having two different kinds of boundary.

Suppose the picks are not clamped at a length but are held by a dead weight: a hung bar, a spring of low rate, an air pressure, or a catenary rope. A dead weight does not relax. The picks can rearrange as they like, and their tension stays where it was put. Then the first equation stands alone, with $T_2$ a given number.

A newton’s dead weight on each pick lifts the ends’ floor from 0.076 to 0.424 newtons, a factor of more than five, and the rise is a straight line with slope $2\mu\sin\theta_2$. That is 0.35 of the weight at a friction of 0.3.

So the two boundary conditions are very different things. A clamp is a load that a cloth will take away. A weight is a load it cannot take away, and it also supports the other direction’s clamp through every crossing. A structure with one direction under constant load and the other clamped can settle a clamped load far above its own friction floor, and the reason is nothing more than a pressed crossing grips better. An inflated structure is the extreme of that. Its hoop and its axial load are both set by a pressure, which relaxes nothing, and it never had this problem to begin with. A clamped tent wall with its ridge on a sprung pole is much closer to the arrangement here.

A frame is a loading with a memory of its own edges, and the distinction between the two kinds of boundary is one a drawing of a clamped cloth does not show, because the two look identical while nothing has yet relaxed. It takes a night to tell them apart.

It is also a test the equations would have to pass. If the slope against dead weight really is $2\mu\sin\theta_2$, then a cloth clamped one way with weights on the other reads its own friction coefficient off a straight line, since the crimp angle is a matter of geometry. That is a friction measurement at a working crossing rather than the sled-and-plate figure the two coefficients have to be taken from, and it is a new way of getting the one number all of this depends on.

A friction at which nothing relaxes #

The denominator is the other feature of the solution. It is $1 - 4\mu^2\sin\theta_1\sin\theta_2$, and it can reach nought.

Past a stated friction there is no finite pair of floors at all: each system’s grip on the other more than repays what it took, and no rearrangement relieves anything, so the frame holds whatever it was given. For a sheeting that friction is

$$\mu^* = \frac{1}{2\sqrt{\sin\theta_1\sin\theta_2}} = 0.83,$$

and it is 0.98 for a duck and 1.20 for a muslin. A cloth with shallower crimp locks at a higher friction and a steeper one at a lower. Cotton on cotton runs from 0.2 to 0.4, and no ordinary textile yarn gets near 0.83, so this is a limit rather than a state any cloth is found in.

The curve on the way to it is where the interest lies, though. At a friction of 0.2, the sheeting gains 1.29; at 0.3, 1.52; at 0.4, 1.87. The range of an ordinary cotton’s friction, from softened to harsh, changes the one-way floor by a factor of two, and it changes the frame floor by a factor of about 2.9: 0.065 to 0.188 newtons. The first essay ordered friction first in leverage. On a frame it has more leverage than that, since it enters the gain as well as the base. So a softener applied for handle costs a framed fabric more than it costs a strip of it, and a finish that raises friction buys a framed fabric more.

That is one more reason a cloth’s two rolls can behave differently on a frame, and more sharply than on a strip. Two rolls nominally identical, one with a third less friction than the other, settle a third apart on a strip and more than two fifths apart on a frame.

What a frame does to the share kept #

Most of what is written about a cloth’s retained tension is the fraction, because the fraction is what falls.

The frame does not change what kind of curve it is. A sheeting clamped at 4.46 per cent keeps 54 per cent of its load one way and 82 per cent on a frame; at 4.82 it keeps 35 per cent and 54. The ratio between those is the gain of 1.52 wherever the frame curve is below one, and above that the frame is holding all of what it was given. Below about 4.2 per cent it would not relax at all; the one-way floor had that band’s top edge at 3.68. Pull past a point and the fall is as steep as before, for the same reason: the cloth is past its interchange and stretching yarn, so the applied load is rising much faster than the floor can.

So the first essay’s advice stays exactly as it stood, only with a higher number in it. Design to the floor, and on a frame it is the frame’s own floor. Tensioning past it is transient. A cloth tensioned below it, and on a frame that now means a larger region, does not settle at all. A designer sizing a stretched panel from a one-way floor is being safe by about half, and it is possible to leave that margin on the table, or to spend it.

What it does and does not change about the locus #

The budget account says that a cloth has one budget for two directions: spend strain one way and the crimp interchange refunds it in the other, so a strip at its working length is at the end of its own locus. A frame takes that away. Both lengths are clamped, so the cloth cannot move along that locus, and the only rearrangement left is local: one patch stretched a little and its neighbour shortened, at unchanged means in both directions.

That is exactly the mechanism the floor needed, and the frame does not stop it, since patches moving locally do not need the whole cloth to move. What the frame does remove is the one-way account’s extra relief: the free direction’s shrinkage. So a frame’s relaxation should finish sooner and in less displacement, though the first essay’s account has no clock in it and neither has this one. That difference, and how much of a real cloth’s loss on a frame is local and how much is at the frame’s own slip, is not something this arithmetic can see.

How the numbers were reached #

The crimp angles and one-way floors are the first essay’s: the least-energy state on each cloth’s own locus, at balanced crimp, with the contact force at its resting working value and a friction of 0.3 unless a figure says otherwise. The frame is one linear pair of equations in the two systems’ floors, so it is solved in closed form, and both equations are checked against the solution. The retained shares use the one-way load at each strain, read off the same load–extension curve, so the frame figures hold the applied load as the one-way account gave it.

Required of it, not shown: that both equations hold at the solution; that a frame floor sit above the one-way floor in both systems; that a higher friction raise it; that no second system reproduce the one-way floor; and that the cloth lock on one side of its own stated friction and not on the other.

What this frame leaves out #

The crimp angles are held at their resting values. A cloth under real tension on a frame has its crimp altered, so the angle falls in the direction pulled, and a falling angle lowers what that system can press with. So the figures at high load are the upper end, and a self-consistent solve would pull them down somewhat. It would not change the structure of the result.

One contact force stands for every crossing. A real crossing population has a spread, and a fabric is a population of contacts rather than one contact repeated, so the friction limit would be approached first by the weakest crossings. A cloth nearing the locking friction would lock in patches.

It takes no time. Like the account it continues, this says where a cloth stops and not how long it takes. No rate has been computed for any relaxation here.

The fabric’s edges are not modelled. A frame’s clamps hold the edge threads differently from the middle, and a real panel near its fixings carries a concentration of tension. All of the above is about a cloth far from those. And the weave’s own locks are absent. A leno, a coated or a heat-set cloth has crossings that hold for reasons beyond friction, so a floor of this kind is a lower limit on them.

And the friction is held at one value for the two systems, though a warp and a weft are often differently sized and finished, and an unequal pair would simply change the equations’ two coefficients.

Who worked out which part #

Frame-tensioning, biaxial testing of structural fabrics and the idea that tension in one direction changes behaviour in the other are all long established, particularly for coated fabric structures. What is added here is the pair of equations for a frictional floor in both systems; the gains of the named cloths with the sharply different responses of a warp-dense one; the distinction between a clamp and a dead weight, with the straight line it gives; the locking friction; and the finding that a friction change has more leverage on a frame than on a strip.

Still open: what a shear does to the frame floor #

Everything here holds a frame square. Most frames are not, because a panel is not a rectangle, and a shear in a held cloth changes both crimp angles and how much of each system’s tension resolves onto the crossings. The bias is a mechanism with almost no tensile load in it, and it would be expected to change these figures rather than leave them.

The question it would answer is whether a floor can be lost by shear alone. A square-held sheeting at its floor, sheared by a few degrees at fixed mean strains, has altered neither length and yet may have moved its crossings below their friction limit. If so, a panel that settled at its floor would lose further load every time wind sheared it, with no slip and no stretching, and a frame’s load would creep down in a way no one-direction relaxation test could show. It needs the sheared geometry’s two crimp angles in these same two equations, and that much is already in hand.