Structural Engineering & Statics
where the load goes
Concept

Hoop tension — where it appears

The circumferential force in a curved wall carrying internal pressure, equal to the pressure times the radius. It arrives with no lever arm, contains no wall thickness, and is settled by a free body of half a ring cut along a diameter.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

The free body that makes a hoop force a pressure times a radius. Half a ring cut along a diameter, with the pressure drawn normal to the wall wherever the wall is. Vertical equilibrium of the half ring is the whole derivation: the pressure acts over the projected width 2R whatever the shape of the arc, the two cut faces carry N each, so N = pR — 300 kN per metre here at 1 MPa on a 0.3 m radius. The result contains no wall thickness, no second moment, and no length along the pipe, which is why a hoop force is the one internal force in this collection that arrives with no lever arm attached to it. The stress does contain the thickness — 25 MPa at 12 mm — but the force does not, and a thicker wall carries exactly the same force at a lower stress.

The force that is only a radius

Every internal force in this collection arrives with a lever arm attached. A hoop force does not. Cut a cylinder along a diameter and the free body settles it in one line — pressure times radius, with no thickness, no second moment and no length in it — which is why a tank wall is thin and why its worst hoop force is not at the bottom.

internal-forces · Hoop tension A broad tank sloshes and a tall one does not. The liquid's division into the part that moves with the wall and the part that sloshes, against the tank's proportion. The convective masses come from the potential-flow solution and the impulsive mass is whatever is left, so the two sum to the liquid's mass exactly at every proportion rather than approximately over part of the range. A tall tank at H/R = 3 is 84% impulsive and behaves almost like a solid; a shallow one at H/R = 0.5 is 72% convective and most of its contents never notice the earthquake. This tank sits at H/R = 1.33, which is 65% impulsive.

The liquid has a period of its own

Shake a tank and its contents do not all go with it. Part of the liquid moves as though it were bolted to the wall and part sloshes at a period fixed by gravity and the radius, which the tank's stiffness has no influence over whatever. The split is decided by one proportion, and the two parts then take entirely different amounts of the earthquake.

dynamics · Sloshing The tendon is a load, pointing the other way. A 14 m beam with a parabolic tendon dropping 260 mm to midspan, stressed to 1440 kN after losses. Its curvature pushes the beam up along its whole length with an intensity of 8Pe/L² = 15.28 kN/m, against an applied 17.63 kN/m — so 2.34 kN/m is left to bend anything, and the beam carries 57.4 kNm where an unstressed one carries 432 kNm. What the section then feels is 6.40 MPa of uniform compression and very little else.

The load that comes from changing direction

A force that travels in a straight line asks nothing of anything. Bend its path and it asks for a transverse load of F over R along every millimetre of the curve, and that load is real, is nowhere on the load schedule, and is the same statement behind a prestressing tendon, a hoop force, an arch thrust and a web that buckles with nothing applied to it.

internal-forces · Deviation force The edge, and the length over which it is forgotten. A cylinder of radius 4.00 m and wall 12 mm under 0.6 N/mm² of internal pressure, held at its base. Away from the base the wall carries the pressure as pure hoop tension and bends nowhere, which is why a pressure vessel is a cylinder. At the base the hoop force is zero, because the wall cannot grow there, and the difference is made up by a boundary layer of bending that dies out inward. The length it dies out over is 1/β = 170 mm — 0.778√(Rt), a geometric mean of the radius and the thickness — and the moment is under a twentieth of its edge value by 3.07 of them. Nothing in that length is the load. The base moment is p/2β², and the bending stress it produces is 1.82 times the membrane hoop stress the whole design is about, at every pressure, every radius and every thickness: the ratio is √3/√(1 − ν²) and contains none of them. The hoop force overshoots by 4.3% at 3.2 lengths in, which is the wall springing back past where it was going.

The length a structure was never given

A disturbance applied at one place dies out over a distance, and the distance is not something anybody chose. A beam forgets a badly applied load over its own depth. A beam on the ground forgets a point load over the fourth root of its stiffness against the soil's. A shell forgets a held edge over the square root of the radius times the thickness — a geometric mean of two lengths three orders of magnitude apart, which is neither of them and is not near either.

internal-forces · Edge disturbance The lining that carries less for being weaker. Bending moment and hoop thrust in a circular lining, against the lining's own bending stiffness, both as fractions of the free-ring values. The ground arrives already stressed — 500 kPa vertically and 300 horizontally at K₀ = 0.6 — and the difference between them tries to squash the hole into an ellipse. A lining stiff enough to refuse absolutely collects the whole distortion pressure, p₂R²/3 = 300 kNm/m; one flexible enough to go with the ground collects nothing, because there is no curvature change left to resist. The thrust is the flat line: it comes from the mean stress rather than the difference, so it does not move at all. Putting 8 joints in this ring drops the moment to 34% of the solid one and leaves the thrust exactly where it was, which is why a segmental lining is jointed and why the intuition carried over from a beam is inverted here.

The lining that is stronger for being weaker

A tunnel lining is not loaded. The ground arrives already stressed and the hole wants to squash into an ellipse; the lining's only job is to refuse, and how much moment it collects depends entirely on how hard it refuses. Make it stiffer and it takes more. Make it flexible — put joints in it, make it thin — and it takes almost none, while the hoop thrust it carries does not move at all.

structures · Tunnel ring The force nobody applied, and the speed it wins at. Lateral force per unit weight for a vehicle on a 400 m curve, against speed. The rising curve is what the free body demands — v²/gR, which is the body's own acceleration written on the other side of the equation — and the flat line is what 6.0° of cant supplies from the weight. They cross at 73 km/h, which is the speed the curve was set out for; below it the deficiency has the other sign and the rail is pushed the other way. The upper line is overturning, at b/2h = 0.399 — and there is no mass in that number, so a loaded vehicle and an empty one go over at the same 160 km/h and only the height of the load decides. At the 108 km/h drawn the deficiency is 0.124 of the weight, which is 49 kN on this 40 tonne vehicle.

The force that is really an acceleration

Every other load in this collection is applied by something. This one is applied by nothing at all — it is the body's own acceleration, written on the other side of the equation so that statics can be used on a problem statics has no business with. The move is legitimate, it is a hundred and eighty years old, and it is exactly half done more often than it is done.

equilibrium · Centrifugal load The confinement that slenderness switches off. The capacity of a concrete-filled tube against slenderness, divided by the plain sum of its two materials. Below about λ̄ = 0.5 the concrete is confined and the section is worth more than its parts — up to 29 per cent for a stub. Above it the bonus is gone, because confinement needs the concrete to dilate, dilation needs strain, and a slender column buckles before it gets there. The column drawn is at λ̄ = 0.42 and has 0.3 per cent of a bonus, which is to say none. What does not switch off is the other half: at d/t = 80 an empty tube buckles locally at 286 N/mm², below its own yield of 355, and the filled one reaches 508 because the wall cannot go inward. That is worth more than the confinement ever was, and it applies at every slenderness.

Each one stops the other failing

A concrete cylinder crushes by splitting outward and a thin steel tube fails by rippling inward. Put one inside the other and each material's failure mode requires a movement the other one prevents, which is a much stronger statement than composite action.

materials · Filled tube What a 60-degree seam leaves of the wall. The combinations of axial stress σa and hoop stress σh that a tube of 6082-T6 (parent ultimate 310 N/mm², softened zone 185) carries, with its seam at 60° to the axis (solid), inside the parent wall's von Mises ellipse (dashed). Where the solid line falls inside the ellipse the seam's zone fails first. axial tension: 60 per cent of the parent; closed pressure: 78 per cent of the parent; hoop alone: 76 per cent of the parent. The seam is weakest, at 60 per cent, along the loadings whose unstretched line lies on it.

The seam that pressure protects

A soft weld zone pulled at 55 degrees to the load gets no help from the metal around it, because 55 degrees is the line a plate under uniaxial tension does not stretch. A spiral-welded tube winds its seam at about that angle. Pressurise the tube and the line that does not stretch moves to the tube's own axis, so the spiral seam is suddenly well protected and the longitudinal seam is the exposed one. The soft zone's weakness belongs to the loading as much as to the weld: a seam is in danger only where the field around it leaves its line unstretched.

materials · Heat-affected zone Past a wall ratio of 2.22, a thicker tube gains nothing more. The most autofrettage can raise a thick cylinder's elastic pressure, as a multiple of its first-yield pressure, against its wall ratio b/a. Yielding the whole wall would buy σy ln k ÷ py (faint), which keeps rising; the release caps it at 1 + β, where β is the reverse yield strength over the forward one. With β = 1.0 the two meet at b/a = 2.22, and the gain is 2.00 for every thicker wall; with β = 0.7 the two meet at b/a = 1.80, and the gain is 1.70 for every thicker wall; with β = 0.5 the two meet at b/a = 1.55, and the gain is 1.50 for every thicker wall. Thinner walls than that can be yielded right through and gain less.

The overstrain the release gives back

A thick tube is pressurised past yield once, at the factory, so that releasing the pressure leaves its bore in compression and the working pressure must overcome that before it does any harm. The obvious rule is to overstrain as far as possible — right through the wall. For a tube whose outside is more than 2.22 times its bore that is wrong: the release itself reverses the bore past yield, and every newton of overstrain beyond twice the first-yield pressure is given back on the way down. The best overstrain stops part-way through the wall, and a steel that yields early in reverse stops it sooner.

materials · Unloading Two ways to put a bore in compression. For a tube 2.5 times its bore in a 900 N/mm² steel: the stress difference σθ − σr through the wall, as a share of the yield strength, at rest (dashed) and at the working pressure (solid) — for the wall shrunk together from two layers so that every layer's bore yields together at work, and for the same wall autofrettaged to its best overstrain (thin lines). The layered wall carries 540 N/mm², 1.43 times the first-yield pressure, and at rest its bore sits at −0.43 of yield; the autofrettaged wall carries 756, 2.00 times, its bore at rest at −1.00.

The ceiling the bore sets

A thick tube can be put into compression at its bore two ways: overstrain it past yield once and let it spring back, or build it from layers each shrunk on to the one inside. The second never yields the steel at all. Two layers carry 1.43 times a plain tube's pressure where an overstrain carries 2; four layers 1.75; and no number of layers, no overstrain and no combination of the two carries more than twice. The ceiling is not in either method. It is the range the bore's stress swings through, which belongs to the assembled wall, and the floor that range may reach, which belongs to the steel.

materials · Unloading The pressure on the wall jumps where the bin meets the hopper. The normal pressure on the wall of a bin 8 m across holding 30 m of a solid weighing 9 kN/m³, wall friction 0.45 and K = 0.45, over a conical hopper of half-angle 20° with wall friction 0.40, the solid's internal friction 35°, down the bin and then down the hopper: in the bin, K times Janssen's vertical stress (solid) and that times the 1.35 discharge factor (dotted); in the hopper, filling (light) and emptying (dark). At the foot of the bin the wall carries 38.1 kPa, 51.4 with the discharge factor. On the hopper's top edge it carries 75.8 kPa filling and 95.7 emptying — 1.99 and 2.51 times the bin wall's. Half way down the hopper, 46.3 and 36.7 kPa.

The pressure that switches at the hopper

Below a silo's bin a conical hopper funnels the grain to its outlet, and the wall pressure there does not keep growing with depth as a liquid's would. It peaks at the hopper's top edge. Filling, the edge carries twice the bin wall's pressure just above it; emptying, the grain converges on the outlet, its largest stress turns from vertical towards horizontal, and the edge carries two and a half times. Half a metre above the apex the stress is 2.6 kPa whether the bin holds 5 m of grain or 60 — which is why a silo empties at the same rate full or nearly empty.

equilibrium · Silo pressure

Named alongside it

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

Free bodyMembrane actionEquilibriumAutofrettageBauschinger effectBending momentBiaxial stressBoundary layerDomeFrictionLoad pathOverturning

All concepts