The web that leans and lifts the centre

The web that moves the centre and not the twist put an interior web into a rectangular box and found a division of labour. The web, being the middle path between two cells, carries more of a vertical shear than either outer web and pulls the shear centre toward itself — about a quarter of its offset from the middle — while the torsion constant hardly changes. Every wall in that box was vertical or horizontal, and the essay ended on the shape most steel box girders actually have: a trapezoid, its webs leaning in toward a narrower bottom flange, its interior webs sometimes vertical and sometimes a leaning pair.

A leaning wall does something no vertical or horizontal wall can. It carries shear flow with both a vertical and a horizontal component. That is enough to move the shear centre in a direction the rectangular box never did.

A general box, and the book-keeping it needs #

The rectangular box was solved by cutting each cell once, integrating the open section’s shear flow from the cuts, and then adding a constant flow round each cell chosen so that no cell twists — the circulation the closing plate makes possible. Nothing in that procedure needs the walls to be square. For any network of straight walls the same steps hold: cut one wall per independent loop, integrate the open flow along what is left (a tree of walls), add one constant circulation per loop, and fix the circulations by requiring every loop’s twist, $\oint q/t \, ds$, to vanish. The shear centre is then the point the shear must pass through for the flows to be in moment equilibrium with it — its horizontal position from a vertical shear, its height from a horizontal one — and the torsion constant follows from the same loops by Bredt’s equations for several cells at once.

Applied to the two-cell rectangle of the earlier essay, this general calculation returns its shear centre, its torsion constant and its interior web’s share to six significant figures. Applied to a trapezoid, it answers questions the rectangle could not ask.

The box drawn throughout is 1.5 m deep, 3 m across the top and 2 m across the bottom, so that its webs lean in at 18.4 degrees, with a 15 mm top plate and 12 mm plates everywhere else. That is the proportion of a modest steel bridge box with its deck cantilevers left off.

The shear centre goes below the centroid #

The single-cell trapezoid’s shear centre is on its axis of symmetry, as it must be, and each web carries half of a vertical shear. Its height is the first surprise: 791 mm above the bottom flange, 106 mm below the centroid.

Start from the rectangle of the same depth and top width. With its heavier top plate its centroid sits a little above mid-depth, 808 mm up, and its shear centre higher still, 847 mm up, 39 mm above the centroid. Now narrow the bottom flange and let the webs lean. The centroid rises, because steel is taken out of the bottom; the shear centre falls. They cross at a lean of under six degrees, and by the bridge box’s 18.4 degrees they have swapped by 145 mm.

The fall of the shear centre is the inclined webs’ doing. A rectangle’s horizontal shear is carried only by its flanges, top and bottom in proportion to how much bending each does, and its resultant lies between them. A trapezoid’s leaning webs carry horizontal shear too: each web’s flow runs along its own line, and a line that leans has a horizontal component. The webs lean in toward the bottom, so their flow adds horizontal shear capacity lower down than the top flange, and the resultant of a horizontal shear — which is what fixes the shear centre’s height — moves down with it.

The height matters wherever a load acts sideways. Wind on a box girder’s side, the centrifugal and braking forces on a curved bridge, the horizontal component of a cable’s pull — each twists the box by its height above or below the shear centre, and a box whose shear centre is 145 mm lower than a rectangle-based estimate says has that much more lever on any load that acts above it.

A vertical web moves the centre sideways only #

Put a vertical interior web into the trapezoid. At the middle it changes the vertical shear’s distribution — each outer web now carries 31 per cent and the interior web 38, more than either, as in the rectangle — but the shear centre does not move at all: not sideways, because the box is still symmetric, and not in height, because a vertical web cannot carry horizontal shear, and the horizontal shear alone decides the height. Even the torsion constant is unchanged, because under a twist the two equal cells circulate equally and the web between them carries their difference, which is nothing.

Move the vertical web 500 mm off the middle and it behaves as in the rectangle: the shear centre moves 159 mm toward it, a third of its offset, and stays within 2 mm of the same height. The rectangular box’s rule survives the inclined outer webs intact. It does not survive an interior web that leans.

An inclined web lifts as it pulls #

Run an interior web from the middle of the top flange down to a point 500 mm left of the middle of the bottom flange. The shear centre moves 69 mm to the left — and 10 mm up. Neither number is what a vertical web would give. A vertical web standing where the inclined one is at mid-depth, 250 mm left of the middle, would pull the shear centre 81 mm sideways and leave its height alone.

Sweep the web’s foot along the bottom flange and the shear centre traces a curve, not a line. With the web vertical it sits at the cross, 791 mm up on the middle line. As the foot moves either way the shear centre moves toward the side the foot went and climbs, slowly at first and then faster, until with the web leaning as far as the bottom flange allows it is 117 mm across and 825 mm up — 34 mm higher than any vertical web puts it. A vertical web at the inclined one’s mid-depth position would have pulled it further sideways, 152 mm, and left it on the level.

The reason is a couple. Under a horizontal shear the inclined web carries flow along its own line, and because the line leans, that flow has a vertical component as well as a horizontal one. Nothing vertical is applied, so the outer webs must carry an equal and opposite vertical force — and a vertical force in the interior web balanced by vertical forces in the outer webs some distance away is a couple, which shifts the line along which the horizontal shear’s resultant acts. The leaning web moves the shear centre’s height by turning part of a horizontal shear into vertical forces in the other webs.

Which way it leans matters #

The same couple explains why the direction of the lean matters. Run the web the other way round, from 500 mm left of the middle at the top to the middle of the bottom, so that its mid-depth position is the same 250 mm left. Now the shear centre moves 88 mm sideways and 12 mm down — further sideways than the first web, and in the opposite vertical direction. Two webs with the same mid-point and the same length and the opposite lean are, for this question, two different structures.

So there is no equivalent vertical web. An inclined web is not a vertical web at some representative position: its horizontal flow component depends on which way it leans, the couple that component makes has a sign, and the shear centre’s height follows the sign. The rectangle’s rule — a web pulls the shear centre toward itself by a fraction of its offset — has become a rule with two numbers in it, sideways and up, and the second is set by the lean.

A vee lowers it a long way #

The commonest arrangement of leaning interior webs in a wide box is a symmetric pair meeting at the bottom flange — a vee, which turns the middle of the box into a triangular cell and supports the deck plate at two more lines. Being symmetric, it keeps the shear centre on the middle line. Its effect on the height is large: with its two tops 1 m apart, the shear centre falls from 791 mm to 749.

Spread the vee and the shear centre keeps falling — to 596 mm when the webs’ tops are nearly at the outer corners, so that the vee’s webs almost duplicate the outer ones. That is nearly 200 mm below the single-web box and more than 250 below the centroid. The torsion constant, meanwhile, barely notices: 1.7 per cent more with the tops 1 m apart, under 10 per cent at the widest. As in the rectangle, the interior webs move the shear centre and not the twist.

The vee lowers the shear centre for the same reason the inclined outer webs did, and more strongly. Under a horizontal shear its two webs act like the diagonals of a truss, taking horizontal shear from the top flange straight down to the middle of the bottom one; the horizontal shear acquires a stiff path that ends low in the section, and its resultant follows the path down.

What the height is worth #

A worked case makes the heights concrete. Put a parapet and a deck on the box so that the wind on its side, deck edge and traffic together has its resultant 1.2 m above the bottom flange. On the rectangle-based estimate of the shear centre, 847 mm up, that wind twists the box with an arm of 353 mm. On the trapezoid’s own shear centre, 791 mm, the arm is 409 mm — 16 per cent more torque from the same wind. With a vee of interior webs spread to 2.9 m, the shear centre is 596 mm up and the arm 604 mm: 71 per cent more than the rectangle estimate. None of these is a large torque on a bridge box, which is very stiff in torsion; but the torque from wind adds to the torque from eccentric traffic and from curvature, and a calculation that places the shear centre in the wrong third of the depth has its horizontal-load torsion wrong by those fractions.

The same arithmetic applies to anything whose line of action passes above or below the shear centre, which is why the height of a load decides how it twists a beam: a load applied above the shear centre of a member free to twist sideways adds to the twist it causes, and one applied below it resists. A shear centre that is lower than the designer thought puts more of the deck’s loads above it.

Where the vertical shear goes #

The vertical shear’s distribution keeps one feature of the rectangle through every arrangement: an interior web always carries more than an outer one. A single vertical interior web takes 38 per cent against 31 for each outer web; an inclined one 36 against 30 and 33, the outer web on the side its foot went taking the less; the two webs of a vee 26 each against 24. The interior webs sit between two cells, each cell’s circulation adds to them, and they carry the sum of what the cells’ compatibility asks for.

For design that is the useful half of the result. The interior web of a box is often the thinnest plate in it — a stiffening web, put in to carry the deck rather than the shear — and it is the one that carries the most shear. The vee’s two webs, each carrying more than an outer web, carry more than half the vertical shear between them.

The trapezoid by hand #

The single trapezoid’s torsion constant is Bredt’s single-cell formula, and it is worth doing because it is the number every arrangement here is compared with. The enclosed area is $\tfrac12(3{,}000 + 2{,}000) \times 1{,}500 = 3.75\ \text{m}^2$. The walls’ $\oint ds/t$ is $3{,}000/15 + 2{,}000/12 + 2 \times 1{,}581/12 = 200 + 167 + 264 = 631$, each web being $\sqrt{500^2 + 1{,}500^2} = 1{,}581$ mm long. So $J = 4A^2/\oint ds/t = 4 \times (3.75 \times 10^6)^2/631 = 8.9 \times 10^{10}\ \text{mm}^4$, which is the general calculation’s 89.3 × 10⁹.

The shear centre’s height has no comparably short formula, which is the reason for the general calculation; but its direction of travel can be checked by the argument above. Lean the outer webs and their horizontal flow adds low in the section; the resultant of a horizontal shear must come down.

Thin walls, rigid cross-sections, and no cantilevers #

The walls are thin and the cross-section keeps its shape. Shear flow and the shear centre are properties of a section that does not distort. A trapezoidal box without enough diaphragms does not keep its shape under an eccentric load, and one diaphragm is nearly none; the shear centre computed here is where a load produces no twist of a box that is held square.

No deck cantilevers. Real box girders carry the deck plate out beyond the webs as cantilevers, which add area to the top flange and raise the centroid; their effect on the shear centre is small, because a cantilever’s open flow carries little shear, but the gap between centroid and shear centre grows.

And the plates are unstiffened. A bridge box’s flanges carry longitudinal stiffeners, which change the section’s bending properties and therefore its open shear flow, though not its topology. The arrangement of webs, which is what this essay is about, is a matter of topology.

Warping, distortion and the curved box #

They cannot show warping. A box resisting torsion warps a little, and the warping stresses of a trapezoid with leaning interior webs are not those of the rectangle; the section that cannot stay flat is a separate calculation from the one that finds where the shear centre is.

They cannot show distortion. A load away from the shear centre of an open-topped or lightly braced trapezoid distorts the cross-section as well as twisting it, and a vee of interior webs is itself a very effective brace against distortion — one of the reasons it is used.

And they cannot show a curved box. In a box curved in plan, vertical load produces torsion along the whole span, bending that arrives as twist, and the torsion’s arm is measured from the shear centre. A curved trapezoidal box with a vee of interior webs has its shear centre 200 mm lower than the same box without, which moves the twist that horizontal loads add to the curvature’s twist.

Leaning walls carry horizontal shear #

Inclining the outer webs puts the shear centre below the centroid. In a box 1.5 m deep and 3 m across the top, from 39 mm above as a rectangle to 106 mm below with a 2 m bottom flange.

A vertical interior web moves the shear centre sideways only, by about a third of its offset, and leaves its height and the torsion constant alone.

An inclined interior web lifts the shear centre as it pulls it, by up to 34 mm here, and pulls it less far sideways than a vertical web at its mid-depth would; which way it leans decides whether the height goes up or down.

A vee of interior webs lowers it a long way — from 791 mm to as little as 596 — while the torsion constant rises by under a tenth, and interior webs always carry more of a vertical shear than outer ones.

Still open: the box whose cells are not closed until the deck is cast #

Every box here is a closed section from the start. A composite trapezoidal box is often built as an open steel trough — two leaning webs and a bottom flange, braced across the top by a light truss — and closed only when the concrete deck is cast on it. Until then it is an open section whose shear centre lies far below its bottom flange, and its torsion is resisted by the top bracing acting as a fictitious plate of some equivalent thickness. How far the shear centre of the open trough is from the closed box’s, how the equivalent plate thickness of the bracing decides where it lies during construction, and whether a vee of interior webs, closed only at the bottom flange, makes the open trough behave more like a closed one, is the question the box asks before its deck exists.