The web that moves the centre and not the twist

A channel loaded down its web twists, because the shear centre — the point a load must pass through to bend a section without twisting it — is out in the air beside the web. Close the channel with a plate across its toes, and the shear centre is expected to come in to the middle of the box. It does only when 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. The closing plate is decisive for the torsion and nearly irrelevant to the shear centre.

Put a web down the middle of a box and, for torsion, the same family of calculations found the opposite: a centred interior web carries no torsional flow at all and adds nothing to the torsion constant, and an off-centre one adds a few per cent at most. That essay left one question about the web it had shown to be useless in torsion. Does an interior web matter to anything?

It does, and to exactly the thing the closing plate did not.

A box with its web off the middle #

Take a box 600 mm wide and 300 mm deep on its wall centrelines, every wall 10 mm thick, and put an interior web 200 mm from the left: two cells, one 200 mm wide and one 400.

Without the interior web the section is a single box, symmetric about its middle, and its shear centre is the middle: 300 mm from the left. With the interior web as thick as the other walls the shear centre is at 273 mm — 27 mm toward the web, a quarter of the way there. A web a hundred times as thick, which stands for a web far stiffer than everything else, puts it at 208, nearly on the web.

The box is symmetric top to bottom, so the shear centre lies at mid-depth whatever the web does. It is its horizontal position that the web moves, and it moves it toward itself.

How the shear finds its way down #

The shear centre is where the vertical shear’s flows have their resultant, so to find it the flows have to be found, and in a closed section they are indeterminate. The textbook route is to cut each cell once, so the section becomes an open tree whose flows follow from statics alone, and then to add a constant flow round each cell — one unknown per cell — chosen so that neither cell twists. For each cell, the integral of the flow over the wall thickness, taken round the cell, must be zero.

Each cell’s equation contains every wall of that cell, and the interior web belongs to both. Its flexibility, its length over its thickness, appears in both equations, weighted by the flow it carries in each. That is the mechanism. The outer webs each answer to one cell; the interior web is the wall through which the two cells’ circulations meet, and it ends up carrying the difference between what each cell needs.

The answer does not depend on where the cells were cut. Cut both cells at their top outer corners, or at their bottom ones, and the shear centre comes out at 272.57 mm either way — which is the check that the bookkeeping of signs round two loops sharing a wall has been done right.

The shared web takes the most #

The share of the shear each web carries is the clearest way to see it. With every wall the same thickness, the interior web carries 38 per cent of the vertical shear, the right web 35 and the left web 28. The interior web is not a third wheel. It is the busiest of the three, because the shear flowing round both cells is delivered down it.

Make it thinner and it carries less, smoothly: a tenth as thick as the walls, 8 per cent; a hundredth, under one. Make it thicker and it takes over: ten times as thick, 73 per cent; a hundred times, 95. Every share it takes is taken from the outer webs, and from the left web faster, because the left web belongs to the smaller cell, whose circulation the interior web is most able to shortcut.

The shear centre follows the share #

The shear centre is where the three webs’ shares balance, so it moves as they do. Against the interior web’s thickness, on a logarithmic scale, it falls along a smooth S from the box’s middle to the web. A web a tenth as thick as the walls already moves it 5 per cent of the way; as thick as the walls, 27; it is half way at 5.4 times. A hundred times as thick, it is 92 per cent of the way, 8 mm from the web.

That is the closing plate’s curve read backwards. A closing plate a hundredth as thick as the channel moved the channel’s shear centre three per cent of its way and changed its torsion eightfold. An interior web a hundredth as thick moves the box’s shear centre 0.6 per cent of its way — and changes its torsion, as the next figure shows, by nothing anyone could measure.

And the torsion does not notice #

The torsion constant of the two-cell box is the single box’s to within 0.8 per cent with the web as thick as the walls, and 1.3 per cent with it a hundred times as thick. A torque on a closed section runs round the outside as a constant flow, and dividing the box into two cells only asks the two cells to share it, and where that flow meets a web already carrying vertical shear, the two add on one side of the box and subtract on the other. The interior web carries the difference between their two flows — 13 per cent of the larger, here — and since both cells are twisted by the same amount, that difference is small whatever the web’s thickness.

So a wall of the same material and the same proportions does two opposite things depending on where it is in the section’s circuit. A closing plate makes the circuit, so it is decisive for torsion, and carries little of the vertical shear, so it barely moves the shear centre. An interior web divides a circuit that already exists, so it barely changes the torsion, and it is a shared path for the vertical shear, so it moves the shear centre a long way.

Where the web stands hardly matters #

The share of the way the shear centre moves is almost a property of the web’s thickness alone. With the web as thick as the walls, it moves 23 per cent of the way when the web is near one side and 28 when it is close to the middle; with the web ten times as thick, 60 and 62. The distance it moves in millimetres is that share times the distance from the box’s middle to the web, so a web near the edge moves the shear centre further — but the proportion of the way is nearly fixed.

That gives a rule of thumb the calculation supports: an interior web as thick as the box’s other walls draws the shear centre about a quarter of the way from the box’s middle to itself, wherever it stands.

At the scale of a bridge #

Nothing in the calculation has a length in it that is not a ratio. Scale the box by ten — 6 m wide and 3 m deep, 20 mm plate, the interior web 2 m from one side — and every share is the same, and the shear centre moves by ten times as much: 274 mm from the box’s middle toward the interior web, a quarter of the 1 m between them.

That is a number a bridge designer meets. A box girder’s torsion is computed from the eccentricity of each lane load about the shear centre — an internal force with no diagram of its own on most drawings — and the lane loads on a 6 m box sit a metre or two either side of its middle. A torque computed about the middle rather than the shear centre is wrong by the vertical load times 274 mm — a seventh to a quarter of the eccentricities that matter, added on one side of the box and subtracted on the other. The side toward the interior web is the side whose torque the middle overstates; the side away from it is the side it understates, and that is the side of the larger cell, whose outer web is the one carrying the torque’s flow most heavily.

Halve the interior web — 10 mm in the 20 mm box, a web added for buckling of the deck rather than for shear — and the shift is 187 mm, still most of it. The shear centre does not forget a web for being thin; a web a fifth as thick as the walls still holds a tenth of the way.

The stress in the shared web #

The interior web is also the most heavily stressed of the three in shear, though not by much. At equal thickness its average shear stress — its share of the vertical shear over its area — is 1.13 times what an equal third of the shear would give, against 1.04 in the right web and 0.83 in the left. A designer who sizes all three webs for the same third of the shear has under-sized the interior one by 13 per cent and given the left one a fifth more than it needs.

That is the share figure read as a stress, and it explains one habit of box girder design that the torsion calculation could not. Interior webs are often made thinner than outer ones, on the reasoning that they carry no torsion — which is true — and the result is a web whose share of the shear falls with its thickness at a slower rate than its area does. A web half as thick carries 26 per cent of the shear rather than 38: two thirds as much force on half the area, nearly two fifths more stress than before. The force a web attracts falls more slowly than its thickness, because the cells keep routing their circulation through it, and a thinner interior web is a more highly stressed one.

The outer web of the larger cell is the other one to watch. It carries the second-largest share of the vertical shear and, under a torque, the larger of the two cells’ flows — 15 per cent more than the smaller cell’s here — so where bending shear and torsion add, it is the outer web on the side of the larger cell that adds them most.

A slice of the box, and one condition a cell #

The free body is a slice of the box one unit long, cut across, with the change in bending stress from one face to the other pushing on it — the same push that the shear nobody draws is made of in a solid beam. That push is what the open-section flows integrate, and it is pure statics: the flow at any point of an open tree is the push on everything beyond that point.

What closes the problem is not a free body at all. It is the requirement that the slice, in the act of bending, does not twist — one compatibility statement per cell. The shear centre is then a resultant: the line along which the vertical shear must act for those particular flows to be in moment equilibrium with it. A load anywhere else is that shear plus a torque, and the torque runs round the outside of the box, where the interior web has no say.

The single box by hand #

For the box with no interior web, everything is symmetric about the middle and the shear centre is there by inspection — 300 mm. The torsion constant is Bredt’s: four times the enclosed area squared over the wall’s length-to-thickness summed round it,

$$J = \frac{4 (600 \times 300)^2}{2 \times 600/10 + 2 \times 300/10} = \frac{4 \times 3.24 \times 10^{10}}{180} = 7.20 \times 10^{8} \ \text{mm}^4.$$

With the interior web as thick as the walls, the two-cell calculation gives $7.26 \times 10^8$, 0.8 per cent more, and moves the shear centre 27 mm. The torsion barely moves and the shear centre moves a great deal, and the arithmetic of the single box already says why: its shear centre comes from symmetry, which any interior web off the middle breaks, while its torsion constant comes from the outer circuit, which no interior web changes.

Thin walls, a section that keeps its shape #

The walls are thin and the flows uniform across them, and the section is free to warp — which a closed box hardly needs, though an open section cannot stay flat without it. For a steel box of 10 mm plate on a 300 mm depth that is a fine approximation; for a concrete box girder with webs a third of a metre thick it is not, and the interior web’s share of the shear would have to come from a solid analysis.

The section does not distort. A two-cell box loaded off its shear centre twists, and a box that twists without diaphragms also distorts, its cross-section racking out of shape. The interior web is one of the things that resists that distortion, and in a box girder that may be its most important job — a section that will not keep its shape is the other half of a box’s torsion, and nothing here measures it.

And the shear centre is a property of the section, not of the load’s path into it. A deck load reaches a box girder through its top flange and its webs, and how much of it arrives at the interior web depends on the deck as well as on the section.

What the drawings do not show #

They do not show shear lag. A wide flange does not carry the bending stress uniformly, and the flows from which the shear centre is built assume that it does. In a box wide enough for an interior web to be worth having, that assumption is the first to go.

They do not show fatigue. The interior web, carrying the largest share of the vertical shear, is the web whose welds to the flanges see the largest stress ranges under traffic. A web added as an afterthought, sized as the outer webs are, may be the one that cracks first.

And they do not show construction. A box made of two channels set back to back between a top and a bottom plate has an interior web of two plates, twice the thickness of the outer webs; by the figures above, its shear centre is about 37 per cent of the way to that web from the box’s middle.

What it comes to #

An interior web is part of both cells, so it carries the largest share of the vertical shear. As thick as the walls, 38 per cent of it, against 28 and 35 in the outer webs.

So it pulls the shear centre toward itself. A quarter of the way from the box’s middle at equal thickness, half way at 5.4 times, nearly onto itself at a hundred times.

And it leaves the torsion alone. The torsion constant changes by less than 1.3 per cent at any thickness.

That is the closing plate reversed. The plate that makes the circuit changes the torsion and not the shear centre; the web that divides it changes the shear centre and not the torsion.

Still open: the box with a web that is not vertical #

Every wall here is vertical or horizontal. A trapezoidal box girder — the commonest steel box in bridges — has inclined outer webs, and an interior web in a trapezoidal box is often a vertical stiffening web or a pair of inclined ones. An inclined web carries shear with a horizontal component, and two inclined webs meeting at the bottom flange make a cell that is a triangle rather than a rectangle. Whether an inclined interior web still pulls the shear centre toward itself by the same quarter, or whether its horizontal flow component moves the shear centre vertically as well — taking it off the line a symmetric box had kept it on — is a question about a geometry that the rectangular box has no way to ask.