Textile Structure
how cloth is put together
Concept

Pressure vessel — where it appears

A container holding an internal pressure, whose wall carries twice the stress around the circumference as it does along the axis. The two-to-one stress ratio it imposes is what sets the angle a reinforcing helix wants, and that angle is a number rather than a choice.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

What a balanced cloth wastes under pressure. The fraction of a balanced fabric's fibre that is along for the ride, in a stress field of each ratio. A closed cylinder is exactly two to one — the ratio of the two areas the pressure acts on — so a balanced cloth reaches its limit around the circumference with the axial system at half its capacity, and a quarter of the fibre is doing nothing.

An inflated cylinder wants an unbalanced cloth

Balance is a virtue in almost every other cloth. Under pressure it is a defect with a size — a closed cylinder carries exactly twice the stress around its circumference as along its axis, so a balanced fabric reaches its limit in one direction with a quarter of its fibre doing nothing at all.

applied · Membrane The angle a pressurised hose wants. A fixed length of yarn wound on a cylinder at a stated angle, with the radius and length the geometry gives, beside the volume it encloses as the angle varies. The maximum is at arctan √2 — 54.74° — with no material constant anywhere in it, and the same angle comes out of balancing the hoop and axial stresses, which is a different calculation with the same answer.

The angle a hose wants

A braided hose has one angle at which pressure neither lengthens it nor shortens it, and the angle is arctan √2 — 54.74° — with no friction coefficient, no modulus and no fitted constant in it. Two arguments that share no algebra arrive at the same number, and which side of it a hose was braided on decides which way it moves.

applied · Bias Everywhere a muslin can go. Every state a muslin of 24 × 22 threads per centimetre in 20 and 20 tex can reach without a yarn changing length, solved from Peirce's plain-weave geometry. The set is a curve and not a region: 6.59 per cent of extension is available along the warp, and reaching it costs 21.82 per cent of the width.

Pulled both ways, only one can give

A cloth at constant thread length has one degree of freedom, so its reachable states are a curve rather than a region. Equal extension in both directions meets that curve at exactly one point — the state the cloth is already in — so the amount available is nought.

mechanics · Tensile What a knitted band presses a limb with. Pressure against extension for a 20 tex cotton band at a 3.5 mm loop, wrapped round a 30 mm radius — a wrist. The pressure is the fabric's own tension per unit width divided by that radius, and the tension is the loop's bending with the relaxed shape as the yarn's natural one, so nothing here is fitted. Over the range a cuff is actually used across it runs from a twentieth of a millimetre of mercury to 0.85. The shaded bands are what a compression garment is specified at, and the curve does not reach the lowest of them until 277 per cent — which is not a cuff, it is a fabric stretched almost to the point where its yarn runs straight.

What a cuff presses with

A rib cuff holds a sleeve on a wrist, so it must be pressing. Divide its own recovery force by the radius it is wrapped round and the pressure comes out at eight tenths of a millimetre of mercury — a fiftieth of the lightest medical compression, and two orders below what the same fabric resists being squashed with.

applied · Pressure A tube of an elastic band knitted 157 mm round, up one leg. A tube knitted 157 mm round from an elastic band, pulled up an illustrative leg and read at four stations. At each the tube is stretched by the leg's circumference over its own, pulls back with the band's tension at that stretch, and presses with that tension over the leg's radius: ankle 22 cm, stretched 40%, 20.0 mmHg; lower calf 29 cm, stretched 85%, 32.1 mmHg; calf 36 cm, stretched 129%, 39.4 mmHg; below the knee 34 cm, stretched 116%, 37.6 mmHg. The calf is pressed 1.97 times as hard as the ankle. What the bars cannot show is the leg's own give, which a firm tube flattens and which changes the radius the law divides by.

A tube of one size presses the calf harder than the ankle

A band presses a limb with its tension over the limb's radius, so it is easy to conclude that a band grips hardest where the limb is thinnest. That is true of a band held at one tension, and no knitted tube is. A tube knitted to one size is stretched further wherever the leg is thicker, and its tension rises faster than the radius does: an elastic tube pressing an ankle at twenty millimetres of mercury presses the calf at thirty-nine, and a cotton jersey tube presses its calf three and a half times as hard as its ankle. A stocking graduated the other way has to pull hardest where it presses less.

applied · Pressure An inflated tube in section at 0.5, 1, 1.6 times its wrinkling moment. The cross-section of an inflated tube of 100 mm radius at 50 kPa, bent by 0.5 times, 1 times, 1.6 times its wrinkling moment of 78.5 N·m, with the outside of the bend at the top. Each stroke is the wall's axial tension at that point: the pressure's even 2.5 kN/m with the bending's cosine added. At half the wrinkling moment the inside is still in tension; at the wrinkling moment it falls to nothing; past it the inside has gone slack over a dashed arc and the rest carries both the pressure's end force and the moment. What the drawing cannot show is the wrinkles themselves, whose wavelength a cloth's bending stiffness decides and this section leaves out.

An inflated beam wrinkles at a moment with no cloth in it

An air-filled tube can be used as a beam because the pressure pulls its cloth taut along its length, and a cloth that cannot carry a push can carry a bending moment for exactly as long as that pull outweighs it. The inside of the bend goes slack at πpr³/2 and the tube folds at πpr³ — seventy-nine and a hundred and fifty-seven newton metres for a tube a fifth of a metre across at half a bar — and there is no property of the cloth in either. The cloth decides how far the beam bends on the way, and how much pressure it can be pumped to.

applied · Membrane The pressure a band puts on a calf with a ridge down its front, point by point. The section of a calf with a ridge down its front modelled as the convex hull of 2 circles, 360 mm round, with a band on it tensioned so that a round limb of the same girth would be pressed at 20.0 mmHg. The band rides the hull: on each arc it presses with its tension over that circle's radius, and on the straight stretches between arcs it presses nothing. shin ridge 214 mmHg over 2.4% of the girth, calf muscle 21 mmHg over 68.9% of the girth; 29% of the girth carries no pressure at all. The spikes are drawn outward from the band with length proportional to the pressure.

A band presses where the limb turns

Every compression pressure is quoted as a band's tension over the limb's radius, and a limb has no radius. It has a curvature that changes all the way round, and a band presses each point with its tension times the curvature there. The round-limb number survives exactly — as an average over the band's length — and is the pressure almost nowhere. On a calf with a ridge down its front the ridge takes ten times it and the flat face beside it takes none; on an ankle, two bones and a tendon carry nearly three quarters of the force on under a fifth of the girth.

applied · Pressure A graduated stocking knitted in 8 equal steps. The pressure a stocking knitted as 8 steps of fixed girth puts on the illustrative leg from ankle to knee, against the smooth graduation from 20 mmHg at the ankle to 12 below the knee that it approximates. The steps are equal in height. The rings, the pressure change along each step, run 4.79, 3.96, 3.32, 2.82, 2.43, 2.12, 0.87, 0.92 mmHg from the ankle up; the largest is 4.79 mmHg.

A stepped stocking should step most at the ankle

A graduated stocking asks for a girth that rises smoothly up the leg, and a knitting machine gives it one girth, then the next. Over each step the tube is one size on a stretch of leg that is not, so it presses harder at the top of the step than at the bottom, and the pressure up the leg is a saw-tooth. Each tooth is the leg's change of girth across the step divided by the girth squared — so it is largest just above the ankle, where the leg is thinnest and widening fastest. Eight equal steps leave a ring of 4.8 mmHg there; eight steps spaced by the leg's own shape leave 2.7 everywhere.

applied · Pressure Where an air arch first wrinkles under snow over the whole span. A semicircular air arch 4 m across, pinned at both feet, of a tube 200 mm across at 50 kPa, under snow over the whole span; beside it, unrolled from the left foot to the right, the share of the tube's wrinkling budget πpr³/2 spent at each section when the first wrinkle forms, with the thrust's part, N·r/2, shaded darker. The first wrinkle forms 25° above the right foot, at 167 N per metre of span, where the moment of 60.2 N·m compresses the inside of the curve and the thrust is 336 N; there the thrust has taken 21 per cent of the budget. Judged by its moment alone the arch would carry 218.

An air arch pays for its thrust out of its pressure

An inflated tube bent into an arch was expected to start with its inside wall pulled differently from its outside, and to wrinkle where that difference and the load's moment combined worst. It does not: a curved tube is pulled along its length at exactly the straight tube's pr/2, all the way round. What the arch spends its pressure on instead is the thing every arch exists to make — its own thrust — which takes a fifth of the wrinkling budget at the haunch of a shelter arch under snow and more than half on a tight one.

applied · Membrane

Named alongside it

The objects these essays reach for when they reach for this one.

SpecificationContact pressurePrestressLoad-extensionAnisotropyBiasBucklingCrimp interchangeCurvatureElastic recoveryExtensibilityInextensible

All concepts