Structural Engineering & Statics
where the load goes
Concept

Torsional constant — where it appears

The section property that decides how much a member twists under a torque. It varies over three orders of magnitude between sections of similar bending stiffness, because it depends on whether the material forms a closed loop and on very little else — which is why a slit tube is hundreds of times softer than the same tube closed.

Named by 17 essays across 4 fields — each of them below, with the objects they name alongside it.

The length at which a beam stops being a beam. Elastic critical moment against the distance between lateral restraints, with the section's plastic capacity drawn across it. The two cross at 4086 — beyond that length the beam buckles sideways before it reaches the strength its cross-section has, and the capacity is set by the restraints rather than by the steel.

The load that moves with the twist

A beam about to buckle sideways is beginning to rotate, and everything attached to it rotates with it. A load hung from the top flange swings out over the side and drives the rotation on; the same load hung underneath swings back and stops it. Two identical beams, two different capacities, and the only difference is a height.

stability · Load height A soap film over a hole, and its volume is the torsion constant. The stress function for Saint-Venant torsion of a square of 100 mm, relaxed on a 97 by 97 grid until it stopped moving — 385 sweeps. Its contours are the lines the shear stress runs along, its slope is the magnitude of that stress, and twice its volume is the torsion constant: 1.4053e+7 mm⁴ against a closed form of 1.4058e+7, an error of -0.035 per cent. The steepest slope is 67.515 at the boundary, at the point on it nearest the centre — which is why the peak shear in a solid section is at the middle of the longest side and never at a corner, where the film comes down to zero from two directions and its slope vanishes.

Two volumes, and both of them are torques

Torsion of a solid section that is not a circle has no elementary answer, and for thirty years after Saint-Venant posed it the only way to get one was to blow a soap film over a hole cut in a plate and measure it. The film is not an illustration of the solution. It is the solution, and so is a heap of sand poured on the same hole.

sections · Membrane analogy The web down the middle carries no torsion at all. A 2-cell box 3.0 m wide and 1.5 m deep under 4000 kNm of torque. Statics gives one equation, T = 2 Σ q_i A_i, and there are 2 unknown flows — so the section is torsionally redundant and the missing statements are that every cell twists by the same amount. Solving that system gives 444.4 and 444.4 N/mm in the cells, so the internal web carries 0.00 N/mm — 0.00 per cent of the outer wall's. J is 2.3679e+12 mm⁴ against 2.3679e+12 for the same outline with no internal web at all, a ratio of 1.000000. The same walls slit open would give 7.413e+10, so closure is worth 32 times and the internal web is worth what the ratio says.

Two cells, one equation, and a web with nothing in it

Bredt's formula answers a single closed cell because a single closed cell has one unknown and one equation. Put a web down the middle and there are two unknowns and still one equation — and the answer, when the missing statement is supplied, is that the new web carries exactly nothing.

sections · Multicell torsion The deformation with no limit against it. A 8 m open section carrying 12 kN/m at an eccentricity of 75 mm from its shear centre. The torque is small — 900 Nmm per mm — and the twist is not: 2.55° at mid-span with the ends restrained against warping, against 9.09° if they are not, a factor of 3.57. What that angle does is move the flange tip sideways by 8.9 mm — 62 per cent of the member's own vertical deflection, and 28 per cent of the span/250 that vertical deflection is checked against. The same load on a closed section of the same depth moves it 0.15 mm, 60 times less. No code gives a limit for this quantity, so it is the one movement in the collection that is computed only after somebody has complained about it.

The movement with no limit against it

Every code in the world gives a deflection limit. None gives a twist limit — and a beam loaded off its shear centre twists. On an open section a modest eccentricity moves the flange tip sideways by three fifths of the sag that does get checked, and nothing anywhere says whether that is acceptable.

deflection · Twist serviceability Two diagrams for one load, and the second one has no straight-beam ancestor. Bending moment and torsion round a 90° arc of radius 6 m under a uniform load, built in at one end. The bending peaks at 576 and the torsion at 329, 57% of it. Both are zero at the free end and largest at the support, which is where a curved cantilever's bearing has to hold a torque it was probably not asked for.

The torque that has nowhere to go

A curved beam on two supports splits its torsion between them, and the two halves cancel at mid-span. A curved cantilever has one end, so every increment of torque accumulates toward it — and the largest action at the root of a curved balcony is one that a straight beam does not have at all.

internal-forces · Curved in plan The shear centre of a channel. A channel of 100 by 250, with the shear flow in its flanges drawn. Those flows form a couple, so the load has to be applied 38.0 outside the web to leave the section untwisted — a point in the air, outside the material entirely.

The eccentricity a purlin cannot avoid

A channel's shear centre is outside the material, so a load applied anywhere on the section misses it. The distance is fixed by the proportions rather than by the detailing, it is 38 mm on an ordinary purlin, and the torque it produces is not an error anybody made.

sections · Shear centre Warping stiffens a short member and nothing at all a long one. The stiffening 1/[1 − tanh(κ)/κ] against kL, both axes logarithmic, over kL from 0.05 to 200. At the low end the curve is a straight line of slope −2, because for small kL the bracket is κ²/3 and the stiffening is 3/kL²: it reaches 1201 at kL = 0.05, falls to 1.005 at the top, and every open section ever rolled sits somewhere on it. The same three plates arranged three ways are marked: the 533 by 190 mm I-section at kL 2.76 and ×1.562, the tee at kL 37 and ×1.027, the angle at kL 34 and ×1.030. A tee's warping constant is 288 times smaller than the I-section's and an angle's 236 times, because their plates meet at a point and there is no pair of flanges to bend against each other — so they have no warping resistance to offer at all, and that is the reason an angle is a poor thing to twist.

The restraint that beats the gradient

A moment-gradient factor is worth up to 2.7 on a beam's critical moment and is tabulated everywhere. Holding the ends against warping is worth more, is achieved by a detail rather than by a load case, and appears in no table at all.

stability · Moment gradient The shear centre waits for a thick plate. Where the shear centre of a channel 300 mm deep with 100 mm flanges, all 10 mm thick sits, measured from the web towards the toes, against the thickness of a plate closing the toes as a fraction of the channel's own, on a logarithmic scale. Open, it is 33.3 mm outside the web. Closed by a plate as thick as the rest it is at the middle, 50.0 mm inside the web, towards the toes. With a plate a thousandth as thick it has moved 0.3 per cent of the way; a hundredth, 3 per cent; a tenth, 23 per cent. It is half way at 0.28 of the thickness and crosses the web at 0.22.

The plate that closes the circuit

Close a channel with a plate across its toes and its shear centre is expected to come in from the air beside the web to the middle of the box. It does — once the plate is about a quarter as thick as the channel. A plate a hundredth as thick moves it three per cent of the way while multiplying the torsion constant by eight, so the load keeps its lever, the section stops twisting, and the whole torque goes round the circuit as a shear flow through the thinnest plate in it.

sections · Shear centre The film over a keyed shaft, and the corner it climbs. Contours of Prandtl's stress function — the height of a soap film blown up over a hole of the section's shape — for a 50 mm shaft with a keyway 14 mm wide and 5.5 mm deep, its bottom corners rounded to 0.40 mm; the shear stress is the slope of the film, so it is highest where the contours crowd. Left, the whole shaft: the contours follow the circle and bend round the keyway, and the keyway costs 11 per cent of the plain shaft's torsional stiffness. Right, a window 3.0 mm across around the keyway's corner, solved on its own fine grid with its edges taken from the whole film: the contours crowd into the fillet, where the stress is 2.81 times the plain shaft's surface stress under the same torque.

The corner a soap film cannot finish

Cut a keyway into a round shaft and its torsional stiffness falls by only 11 per cent. The stress does not fall; it moves. It leaves the surface beside the keyway's mouth almost unloaded and crowds into the keyway's bottom corners, where a soap film stretched over the section would have to stand vertical. Computed on finer and finer grids, a sharp corner's stress never settles — it grows by the cube root of two every time the grid is halved — and a fillet of a few tenths of a millimetre is what turns an infinite answer into three times the plain shaft's.

sections · Membrane analogy A deck holds the middle of the span. The twist along an 8 m open section on forks, free to warp, carrying 12 kN/m at 75 mm from its shear centre — 900 Nmm of torque per mm of span. With nothing fastened to it the beam twists 6.25° at mid-span; a deck resisting the top flange's rotation at 5.0 kNm per metre per radian holds it to 4.20°, and one four times as stiff to 2.10°. The deck works where the beam is weakest, in the middle; near the supports the beam's own torsional stiffness does most of the work whatever is fastened to it.

A deck is a spring, not a wall

An open-section beam loaded off its shear centre twists, and the usual reassurance is that the deck fastened to its top flange will stop it. The deck resists the flange's rotation with a stiffness per metre of span, and that stiffness has to be compared with the beam's own. On an 8 m beam a screwed deck of ordinary stiffness removes a third of the twist; on a 16 m beam the same deck removes three quarters, because the beam's torsional stiffness falls with the square of its span and the deck's does not.

deflection · Twist serviceability What the fillets are worth across the section tables. For seven standard rolled sections, the torsion constant the film gives over the rolled outline, beside the published one (UB 203×133×25 5.96 against 5.96 cm⁴; UB 305×165×40 14.74 against 14.70 cm⁴; UB 457×191×67 37.14 against 37.10 cm⁴; UB 610×229×101 76.97 against 77.00 cm⁴; UC 152×152×23 4.63 against 4.63 cm⁴; UC 254×254×73 57.68 against 57.60 cm⁴; UC 305×305×97 91.22 against 91.20 cm⁴). Solid bars: what the fillets add to J over the same plates with sharp corners, 9 to 13 per cent. Second bars: how far the fillet's peak stress exceeds the flange face's, 45 to 65 per cent. Σbt³/3 is 7 to 12 per cent below the published constant.

The corner the rolling mill rounds

The torsion constant of an I-section is taught as Σbt³/3, the sum over three thin plates, and a rolled section is not three plates. Where its web meets each flange the mill leaves a fillet, a curve of steel about one per cent of the section's area, and the soap film whose volume is the torsion constant rises into it. Solved over the real outline, the film reproduces the published constant of seven standard sections to a fifth of a per cent and says the fillets are worth nine to thirteen per cent of it. It also puts the section's largest torsional stress on the fillet, half as much again as the flange formula every design check uses.

sections · Torsional constant The interior web pulls the shear centre toward itself. The shear centre of a two-cell box 600 mm wide and 300 mm deep on its centrelines, 10 mm walls, its interior web 200 mm from the left, its distance from the left against the interior web's thickness as a multiple of the other walls', on a logarithmic scale. The single box's middle is 300 mm (dashed) and the web is at 200 (dotted). A web a hundredth as thick as the walls moves it 0.6 per cent of the way; a tenth, 5; as thick as the walls, 27; half way at 5.4 times; a hundred times, 92.

The web that moves the centre and not the twist

A thin plate closing a channel into a box changes its torsion at once and its shear centre hardly at all. An interior web dividing a box into two cells does the reverse. Off the middle of a 600 mm box, a web as thick as the walls takes 38 per cent of the vertical shear — more than either outer web — and pulls the shear centre 27 mm toward itself, while the torsion constant changes by less than one per cent. The two walls are the same kind of wall; what differs is which flow they are part of.

sections · Shear centre The film over a bored shaft has an island that floats. Contours of Prandtl's stress function — a soap film blown over the section — for a 50 mm shaft with a 10 mm hole 15.0 mm off its axis, at the same pressure; the shear stress is the film's slope. Left, the bored shaft: the film is pinned to the outline and to a flat island over the hole, shaded, which floats at 55 per cent of the solid shaft's peak height, where the film's pull round its edge balances the pressure on it. The shaft keeps 93 per cent of the solid shaft's stiffness, and its largest stress, at the dot, is 1.70 times the solid shaft's surface stress under the same torque. Right, the same shaft slit along its length from the hole to the surface: the island is pinned at the outline's height, the film sags into it, and the shaft keeps 67 per cent, its largest stress 2.81 times.

The island the film floats

Bore a hole along a shaft, off centre for a lubrication passage or a cable, and the soap film that solves its torsion has a second edge: a flat island over the hole whose height nobody knows in advance. Let it float where the film's pull balances the pressure on it, and the shaft keeps nearly all its stiffness and doubles the stress beside the hole, as a hole in any shear field does. Pin it at the outline's height instead and the same shaft has been slit to its hole, and loses a third of its stiffness. The island's height is the whole difference between a bore and a cut.

sections · Membrane analogy A building that crosses the trough at an angle stands on a warped surface. Plan of a 30 × 12 m building 10 m high, its walls 300 mm thick, over a trough 30 mm deep with its inflection 15 m from the tunnel's axis, its long axis 45° from square across the tunnel. The lines are the greenfield trough's settlement contours, 5, 10, 15, 20, 25 mm, parallel to the tunnel's axis (dashed). The corners settle A 27.4, B 18.4, C 27.4, D 18.4 mm, so they do not lie in a plane: A and C together settle 18.1 mm more than B and D. A building spanning square across the trough has no such warp, and one lying along it has none either; between the two, the trough twists the building as well as bending it.

The building the trough twists

A building that crosses a tunnel's settlement trough at an angle stands on ground that is warped as well as sagged. Its four corners do not settle into a plane, and beyond about twenty degrees of skew the warp is larger than the sag. Whether the building follows that warp or resists it depends on a property its bending never needed — whether its walls and floors close into a tube — and the same walls, closed or open, differ a thousandfold in their stiffness against it.

deflection · Differential settlement Two islands in one film: level with each other, or not. Contours of Prandtl's stress function for a 50 mm shaft bored with two 8 mm holes, their edges 2.0 mm apart; the shear stress is the film's slope, and each hole is an island, shaded, floating where the film's pull round it balances the pressure on it. Left, the holes side by side at 12.0 mm from the axis: by symmetry both islands float at the same height, 219, the film between them stays nearly flat, and the steel there carries almost nothing. The shaft keeps 95 per cent of the solid shaft's stiffness, and its largest stress, at the dot, is 1.13 times the solid shaft's surface stress. Right, one hole inside the other on one radius: the inner island floats at 237 and the outer at 156, so the film between them is steep and the steel between the holes carries up to 1.96 times; the shaft's largest stress, 2.03 times, is at the outer hole's outer edge, with the steel between the holes close behind.

Two islands in one film

Bore a hole along a shaft and its torsion film acquires an island that floats at a height of its own. Bore two and there are two islands, and the steel between them is a strip of film stretched from one to the other. Side by side at one radius, the islands float level, the strip is flat, and two holes cost twice one hole's stiffness and none of its stress. One inside the other on a radius, the islands float at different heights, the strip is steep, and the steel between the holes becomes the shaft's weak point.

sections · Membrane analogy The film over an angle's corner. Contours of Prandtl's stress function over the corner of an equal angle 100 × 100 × 10 mm with a 12.0 mm root radius and 6.0 mm toe radii: left as rolled, right with sharp corners. Along each leg the contours run parallel to its faces; at the corner they bulge into the metal the fillet adds, round the outside of the L. The rolled angle's J is 6.80 cm⁴ and the sharp one's 6.18 cm⁴. The dot marks the steepest slope: on the fillet in the rolled angle, in the re-entrant corner in the sharp one, where it grows without limit as the grid is refined.

The corner with one fillet

A rolled I-section's web meets its flanges in two T junctions, four fillets, and its torsion constant is 7 to 12 per cent above Σbt³/3 because of them. An angle meets itself in one L, and a channel's web meets its flanges in two. Solved over the real outline, an L junction's fillet adds about two-thirds of what a T junction's does with the same plates. The sum then errs on either side: low by 8% for an I-section with a channel's plates, high by 0.8% for the channel itself, and anywhere from 5% low to 2% high for angles, depending on which length of leg is counted. The fillet's peak stress is still the inscribed circle's, to a few per cent.

sections · Torsional constant A passage opposite a corner, and opposite a flat. The section of a 50 mm shaft with a hexagonal bore whose flats are 8 mm from the axis and whose corners are rounded to 1.5 mm, with a passage 5 mm across centred 13 mm from the axis, opposite a corner of the bore (left) and opposite a flat (right), and the largest shear stress each leaves round the passage and round the bore, as a multiple of the plain round shaft's surface stress under the same torque. Opposite a corner the passage peaks at 1.25 and the bore at 1.15; opposite a flat, 1.19 and 0.54. With no passage the bore's corners carry 0.54.

The bore with corners and the passage beside it

A hollow shaft's bore is not always round: a hexagonal socket or a square drive turns the island in Prandtl's soap film into a polygon, and every corner of the bore is a corner of the steel that the film cannot finish. Its corners carry twice the stress of its flats. A passage drilled through the wall beside it can sit opposite a corner, where it doubles that stress again, or opposite a flat, where the bore does not notice it at all.

sections · Membrane analogy

Named alongside it

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

TorsionWarpingShear centreMembrane analogyShear flowStress concentrationShear stressOpen sectionClosed sectionCompatibilityFree bodyRestraint

All concepts