The box that is a trough until its deck is cast

A trapezoidal box has its shear centre a little below its centroid, because the webs that lean carry part of a horizontal shear and their lean lowers the point it acts through. That essay closed every cell from the start. A real composite trapezoidal box is not built that way. It is fabricated as an open steel trough — two leaning webs, a bottom flange, and a narrow flange on top of each web to carry the shear studs — and it is closed only when the concrete deck is cast across those top flanges and has hardened. Between delivery and that day it is lifted, set on its bearings and loaded with wet concrete, and through all of that it is a trough.

This essay is about the trough, and about the light truss that is bolted across its top to hold it together until the deck takes over.

A channel on its back #

The section drawn is the box of the earlier essay, opened: 1,500 mm deep, 3,000 mm between the tops of its webs, 2,000 mm across its 12 mm bottom flange, 12 mm webs, and a 400 × 20 mm flange on top of each web.

The open trough’s shear centre is not inside it. It is 706 mm below the bottom flange, in the air beneath the girder, where nothing can be attached and no load can be applied. That is the point that is not in the section — the shear centre of a channel, which lies outside its web on the side away from its flanges — turned through ninety degrees. An open trough is a channel lying on its back: its bottom flange is the channel’s web, its two webs are the channel’s flanges, and its opening faces up, so its shear centre lies down.

The analogy is exact rather than suggestive. Make the webs vertical and remove the top flanges, and the trough is a channel 3,000 mm deep with 1,500 mm flanges, whose shear centre for a load parallel to its web is the textbook $e = H^2 B^2 t / 4I$ from the web. With $I$ = 1.08 × 10¹¹ mm⁴ about its vertical axis, that is 562.5 mm, and the general calculation gives the same 562.5 to the millimetre. Lean the webs inward and fit the top flanges and it moves to 706.

The figure sweeps the shape. A trough whose webs lean steeply, with a narrow bottom flange, has its shear centre closest to the steel, 398 mm below; vertical webs put it furthest, 793 mm below. The top flanges push it further down too, because like a channel’s flanges they carry shear flow round the outside of the section and add to the moment the flow makes about the bottom. No flange at all gives 554 mm at the 2,000 mm bottom flange; 800 mm flanges give 745. The shear studs need the top flanges, and the top flanges move the shear centre further away. There is no proportion in the figure for which an open trough’s shear centre is inside it.

Why does that matter while the trough is waiting for its deck? A load through the shear centre bends the girder without twisting it, and a load anywhere else does both. The wet concrete is mostly symmetric and its resultant passes through the vertical line of the shear centre, so it does not twist the trough on its own. But the deck overhangs are carried on brackets cantilevered from one web and are often poured before the other side; wind blows on the side of the girder; a girder curved in plan carries its bending partly as twist, because its weight is offset from the line joining its bearings; and the girder can buckle sideways. Each of these is a horizontal force or an offset vertical one, and each acts about a point 2.2 m below the top flanges, with the twisting resisted by whatever torsional stiffness the trough has.

An open trough has almost none. Its torsion constant is that of its plates laid flat — the sum of each plate’s length times its thickness cubed, over three — and for this section it is 5.1 × 10⁶ mm⁴. The finished box, with its deck, has twenty thousand times that.

A truss that is a plate for shear only #

The top lateral bracing is the standard remedy: a horizontal truss bolted between the two top flanges along the length of the girder, its chords the top flanges themselves, its diagonals steel angles. It makes the section a closed circuit for shear flow, and the question is how good a circuit.

The answer comes from asking what plate would store the same energy under the same shear. One panel of the truss, $a$ long and $b$ wide with a single diagonal of length $d$ and area $A_d$, carries a shear flow $q$ round its edges by putting a force $qd$ into the diagonal, which stores $q^2 d^3 / 2EA_d$ of strain energy. A plate of thickness $t^\ast$ under the same flow stores $q^2 ab / 2Gt^\ast$. Setting them equal,

$$t^\ast = \frac{E}{G}\,\frac{A_d\,ab}{d^3}$$

with the chords taken as rigid, which makes it an upper bound, and twice that for crossed diagonals, which share the shear.

The numbers are small. A 100 × 100 × 10 angle, 1,920 mm² of steel, in 4 m panels across a 3 m trough is worth 0.48 mm of plate. Halving the panel length raises it only to 0.64, because shorter panels have proportionally longer diagonals for their area; crossing the diagonals doubles it, to 1.27 mm in 2 m panels. Even angles twice the size in crossed 2 m panels reach only about 2.5 mm. The truss is light because it is meant to be — it is temporary, bolted on and often left in place as dead weight — and against a 15 mm steel top plate, or a 250 mm concrete deck worth 39 mm of steel in shear, it is a sheet of foil.

What the truss is not is a plate for bending. Its diagonals carry no longitudinal stress worth counting when the girder bends, so it adds nothing to the section’s area or second moment, and the centroid of the braced trough is exactly the centroid of the open one. The truss is a wall in the shear-flow circuit and invisible to the bending. The calculation that follows treats it that way: a wall of thickness $t^\ast$ for shear and zero for normal stress.

The shear centre barely moves #

With the bracing in place the section is closed, and closed sections have their shear centres inside or near them. The question is how much closing a 0.5 mm wall does.

Very little. At 0.50 mm the shear centre has risen from 706 mm below the bottom flange to 538 mm below — 168 mm of a journey of 1,833 to where the finished box will put it. It reaches the bottom flange at 2.9 mm, which no ordinary truss provides, and it does not reach the centroid until the top is about 11 mm of real plate.

The reason is the way the flow divides. A horizontal shear on the open trough has to be carried by the two webs and the bottom flange, round the open circuit, and the flow it produces makes a moment about the bottom that puts the shear centre below it. Closing the top adds a circulating flow, constant round the circuit, whose size is set by compatibility: no twist means that the integral of the flow over thickness round the loop is zero. A very thin top wall makes that integral enormous for any flow through it, so the circulation compatibility allows is small, and the open flow is barely corrected. A thin closing wall is a weak correction to the shear flow and therefore to the shear centre. It takes a wall a few millimetres thick before the circulating flow is comparable to the open one, which is the S-shaped curve in the figure.

The torsion constant moves a great deal #

The same thin wall has the opposite effect on torsion, and this is the half of the story that makes the bracing worth bolting on.

The open trough resists twist only through the thickness of its plates, which is why its torsion constant goes as the cube of 12 mm. A closed circuit resists twist through the area it encloses: Bredt’s formula, $J = 4A^2 / \oint ds/t$, with $A$ the enclosed area — here 3.75 m² — and the integral dominated by the thinnest wall. A 0.5 mm wall makes that integral large, but $4A^2$ is so much larger again that the result is still 9.96 × 10⁹ mm⁴, nearly two thousand times the open value. The plate that closes the circuit was shown to do exactly this to a channel: a closing plate a hundredth as thick as the channel’s walls multiplied its torsional stiffness by orders of magnitude. The tub girder is the same arithmetic at the scale of a bridge.

Under a torque of 1 kN·m the open trough twists at 2.42 milliradians per metre; braced at 0.5 mm, at 0.00124. A sideways load on the top flanges acts about a shear centre 2.2 m below them when the trough is open and 2.0 m below when it is braced — nearly the same lever arm, so nearly the same torque — and the whole difference in how far the girder turns is the circuit the bracing has closed.

So the two halves come apart, and the decoupling is the useful finding. The bracing closes the box for torsion long before it closes it for the shear centre. At 0.5 mm the trough has 9 per cent of the finished box’s torsional stiffness and its shear centre has made 9 per cent of the journey. The shear centre’s 9 per cent is 168 mm and changes nothing about where loads act. The torsion constant’s 9 per cent is a factor of two thousand on the open trough and changes everything about how far it turns.

Three stages, three sections #

A tub girder is three different sections in its life, and the calculation for each stage should use the right one.

While it is lifted and set, the trough is open. Its torsional stiffness is negligible and its shear centre far below. A trough lifted by slings on its top flanges hangs upright, because its weight is below the hooks, but any twist the slings put into it — one end picked a little before the other, a sling a little shorter — is resisted only by its plates’ thickness cubed, and a long open trough visibly winds in the air under its own handling.

While the deck is poured, the trough is braced. Its torsional stiffness is a tenth of the finished box’s, which is enough for most purposes, and its shear centre is still half a metre below its bottom flange. The wet concrete is not yet composite with the steel, so it is pure load: the steel trough carries all of it, on the section it has that day. Every horizontal load and every offset vertical one acts about that point, so the torques are those of the open trough and the stiffness to resist them is that of something close to a box.

Once the deck has hardened the section is a composite box. Its shear centre is inside it, near its own centroid — the concrete has moved both up — and its torsion constant is ten times the braced trough’s again.

The middle stage is the critical one, because it is where the girder carries most of its own load and least of its eventual stiffness. Lateral–torsional buckling of a single tub girder during its deck pour is the failure that the top bracing exists to prevent, and it has happened: a single-girder footbridge at Marcy, New York, twisted and fell during the pour of its deck in 2002, and the investigation pointed to top lateral bracing too light to give the trough the torsional stiffness the stage needed. A girder whose torsional stiffness comes from a truss is only as stiff as the truss, and the truss is a plate a millimetre thick. It is also the same problem as a row of torsional braces on an I-girder, with the difference that here the brace is spread along the whole length and is part of the section rather than a spring attached to it.

The trough, by hand #

The open trough’s shear centre with vertical webs and no top flanges is the channel formula. The bottom flange is the web of length $B$ = 3,000 mm, the webs are the flanges of height $H$ = 1,500 mm, all 12 mm thick, and the second moment about the vertical axis is

$$I = \frac{12 \times 3000^3}{12} + 2 \times 12 \times 1500 \times 1500^2 = 1.08 \times 10^{11}\ \text{mm}^4$$

so the shear centre lies $H^2 B^2 t / 4I = 1500^2 \times 3000^2 \times 12 / (4 \times 1.08 \times 10^{11}) = 562.5$ mm below the bottom flange.

The bracing’s equivalent thickness for a 100 × 100 × 10 angle in 4 m panels across 3 m: $d$ = 5 m, so $t^\ast = (210{,}000 / 81{,}000) \times 1920 \times 4000 \times 3000 / 5000^3 = 0.48$ mm.

Bredt’s constant for the braced box: the enclosed area is $(3000 + 2000)/2 \times 1500 = 3.75 \times 10^6$ mm²; the integral $\oint ds/t$ is dominated by the top, 2,600 mm of bracing between the flanges’ inner edges at 0.5 mm, giving 5,200, plus the webs and bottom at 12 mm, about 430, plus the 400 mm of top flange inside the circuit at 20 mm, 20. So $J \approx 4 \times (3.75 \times 10^6)^2 / 5{,}650 = 9.96 \times 10^{9}$ mm⁴, which is the full calculation’s figure; the plates’ own Saint-Venant constant, 5.1 × 10⁶, adds a twentieth of a per cent.

Rigid chords, no distortion #

The calculation rests on choices that limit it.

The bracing’s chords are rigid. The top flanges are the truss’s chords and they stretch and shorten as the panel shears, which makes the real truss softer than $t^\ast$ says; the formula with chord flexibility included gives a smaller number, sometimes much smaller for long panels.

The section does not distort. A trough’s walls are thin plates joined at corners, and under a torque that is not applied as a pure shear flow round the circuit the cross-section changes shape — the webs lean further, the corners open. Internal cross-frames or diaphragms at intervals are what keep the shape, and between them the girder distorts. The shear centre and torsion constant here are for a section that keeps its shape, which is to say a girder with enough internal bracing.

Warping is ignored. Like a web that moves the shear centre but not the twist, the shear centre here is a property of the cross-section alone, while the twist depends on the girder’s length and restraint too. The open trough resists twist mainly by warping — its flanges bending in their own planes — rather than by Saint-Venant torsion, and over a short length its warping stiffness can be far larger than its torsion constant suggests. On the spans of bridges the closed circuit’s Saint-Venant stiffness dominates once the bracing is in, but the open trough during lifting is a warping problem.

The bracing is continuous. The equivalent plate is spread uniformly along the girder. A truss with panels at 4 m is a plate on average; near a missing panel or at an opening for a temporary access it is not there at all, and the girder there is open over that length.

Still open: the bracing that is left in #

Most top lateral bracing is left in place after the deck has hardened, because removing it from under a finished slab is more trouble than it is worth. It then sits a few hundred millimetres below a deck that is forty times as stiff as it is in shear, and the deck carries nearly all the shear flow round the top. But the truss’s diagonals are still connected to the top flanges, and when the composite girder bends the top flanges strain — and so do the diagonals that join them, at an angle to the flanges and so at a fraction of their strain. Every passing lorry is one more cycle, and a load that never comes near failing anything is exactly what fatigues a bolted angle. Whether the left-in bracing then picks up longitudinal force from the girder’s bending, enough to fatigue its connections over the life of a bridge carrying traffic, is a question about a member nobody designed for anything after the day the concrete went off.