The bracing nobody took out

The box that is a trough until its deck is cast followed a steel tub girder through the weeks before its concrete deck hardens. Open, it is a trough of two leaning webs and a bottom flange whose shear centre hangs 706 mm below it, in the air, and whose resistance to twisting is its plates’ thickness cubed. A light truss of 100 × 100 × 10 angles across its top, in 4 m panels, closes it for twisting while the deck is poured: for shear the truss is a steel plate 0.48 mm thick, and that half-millimetre multiplies the trough’s torsion constant two thousandfold.

Then the deck hardens, and the deck is a far better lid. The essay ended by noticing that the truss is usually left where it is — taking it out from under a finished slab is more trouble than it is worth — and that it remains connected to the top flanges for the life of the bridge. When the composite girder bends, its top flanges strain, and so do the diagonals that join them; when it twists, the plane of its top shears, and the diagonals strain with that too. Whether the left-in bracing picks up enough from each passing lorry to fatigue its connections is a question about a member nobody designed for anything after the day the concrete went off.

A diagonal strains as the plane it lies in #

A diagonal bolted between the two top flanges, a panel apart, has no choice about its length. If the flanges stretch along the girder by a strain $\varepsilon_x$, the diagonal stretches by the projection of that stretch on its own line, $\varepsilon_x\cos^2\theta$, where θ is its angle to the girder’s axis. If the plane of the top flanges shears by $\gamma$, as it does when the girder twists, the diagonal stretches or shortens by $\gamma\sin\theta\cos\theta$, depending on which way it leans. In 4 m panels across 3 m between the web tops the diagonals lie at 36.9° to the axis: $\cos^2\theta = 0.64$ and $\sin\theta\cos\theta = 0.48$.

The diagonal’s own stiffness hardly matters. It is a small member in parallel with a large structure, so its strain is set by the structure and its stress is that strain times the steel’s modulus. The left-in bracing is a strain gauge on the top of the girder that carries the stress its readings imply.

So the question divides into two others: how much the top flanges strain under a passing lorry, and how much the plane of the top shears.

Two girders in one: the deck whole, and the deck cracked #

The girder is the earlier essay’s trough — 1.5 m deep, 3 m between its web tops and 2 m across its bottom — under a deck 250 mm thick and 6 m wide, continuous over two 40 m spans. Over most of each span the deck is in compression under the girder’s own weight and acts whole. Over the middle support the girder hogs and its deck is in tension and cracked; there only its reinforcement acts, 1 per cent of its section, over the 15 per cent of each span next to the support that design rules for continuous composite girders assume to be cracked, and that the slip that peaks where the moment turns found the hogging region to be.

The two are different sections. With the deck whole the neutral axis of the composite girder is 1,425 mm above its bottom flange, and the top flanges are only 75 mm above it: the deck has pulled the axis up almost to them. With the deck cracked the neutral axis falls to 835 mm and the top flanges are 665 mm above it, nine times as far. A top flange’s bending strain is its distance from the axis times the curvature, so for the same moment the cracked section’s top flanges strain several times as much — and the cracked section is also less stiff, so the same moment makes more curvature.

The plane of the top shears under torsion in proportion to the shear flow round the closed cell, $q = T/2A$, divided by its shear stiffness. The deck and the bracing share that plane. With the deck whole its shear stiffness is 3.4 MN per millimetre of width against the bracing’s 39 kN, and the bracing takes 1.1 per cent of the shear flow. A cracked deck’s shear stiffness is less, by an amount nobody knows well: its cracks transfer shear by aggregate interlock and by the bars crossing them, and the shear across a crack that is already there depends on how wide they are. Taken here as 0.3 of the whole deck’s, it leaves the bracing 3.6 per cent of the flow.

Harmless along the span, worked hard over the support #

Along the 34 m where the deck is whole, a passing lorry puts at most 5.5 N/mm² in a diagonal, most of it from the torque: the top flanges are so close to the neutral axis that the girder’s bending barely strains them. Over the last 6 m before the middle support the range jumps to between 33 and 42 N/mm², and both parts grow. The bending part, from the moment that swings from sagging to hogging as the lorry crosses one span and then the other, reaches 30 N/mm² because the cracked section’s top flanges are far from its axis. The torsion part triples to 15, because the cracked deck has handed the bracing three times its share of the twist.

The step at the edge of the shaded zone is the model’s: a real deck’s cracking fades in over a metre or two rather than starting at a line, and the range rises across that distance rather than at a point. What the figure says is not where the step is but how large the two plateaus are, and they differ by a factor of seven.

The worst diagonal is not over the support but 5.5 m from it, at the start of the cracked zone. The live moment’s swing is largest there — the lorry on the near span sags it by 1,360 kN·m and the lorry on the far span hogs it by 1,590 — while at the support the near span’s lorry can only hog it, and the swing is smaller. A section that a passing load bends both ways collects the larger range, which is why the load that never came near failing anything found fatigue concentrating where the influence line changes sign.

The shortest-lived members of the girder #

A stress range of 40 N/mm² is a fraction of any steel’s strength, and nobody checks a bracing member for it. What a range does over a bridge’s life depends on how many times it arrives and on the detail it arrives at. A diagonal angle bolted to a gusset through one leg is a poor fatigue detail: an eccentric single-sided connection with its stress concentrated at the first bolt, in the low categories of EN 1993-1-9 — around 50 for ordinary bolts in clearance holes, and nearer 90 only for a preloaded joint that carries its load by friction. The detail decides, and the steel does not: the angle’s grade is irrelevant to its fatigue life, and a stronger steel in the same connection lasts exactly as long. What a fatigue category measures is the geometry of the stress at the connection’s worst point, and an eccentric single-sided connection has about the worst geometry a bolted member can have.

Where the deck is whole every category lasts without limit: 5.5 N/mm² is below even category 50’s cut-off, the range under which a constant-amplitude stress does no damage. Over the support the picture inverts. Under two million lorries a year — the flow EN 1991-2 assigns to the slow lane of a motorway carrying many lorries — a category 50 connection at the worst diagonal lasts two years, a category 71 one eight, and a preloaded category 90 one twenty-five. The members designed for a few weeks of construction are, over the supports, the shortest-lived members of the finished girder.

Two million identical 480 kN lorries a year is a severe assumption. Real traffic is a spectrum of lighter and heavier vehicles, and many of the lighter ones put ranges below the cut-off. But the cut-off is a constant-amplitude idea: in a spectrum that also contains heavy vehicles, the ranges below it are counted after all, at a reduced rate, and the life comes down rather than up from what the heaviest vehicles alone would give.

What the lane and the bars do #

Life is inversely proportional to the traffic, so a bridge carrying a twentieth of the heaviest flow gives a category 50 connection 34 years, a category 71 one 150, and a category 90 one nearly five hundred. A left-in bracing is a problem on a busy motorway and not on a quiet road, and the girder’s own design, which was done for the motorway, did not include it either way.

The deck’s reinforcement decides how far the cracked section’s neutral axis falls, and so how hard the top flanges strain. Doubling the bars from 1 to 2 per cent lifts the axis 113 mm and takes the range from 42 to 33 N/mm²; halving them to 0.5 per cent drops it 77 mm and raises the range to 50. Fatigue life goes as the cube of the range, so the bars that a crack-width check set are worth a factor of two in the bracing’s life either way.

The lorries’ lane decides the torsion part. On the girder’s centre line they twist nothing and the range is the bending’s 30 N/mm²; 5 m off it, which is where the slow lane of a single box carrying a two-lane carriageway runs, the torque adds as much again and the range is 55. A box that carries a wide deck on one girder is twisted by every lorry, and its bracing reads the twist.

Why the deck takes the torque and leaves the moment #

The two parts of the range behave differently enough to need saying separately. The torque’s part depends on how the deck and the bracing share the top plane’s shear, which is a ratio of stiffnesses: 3.4 MN per millimetre against 39 kN where the deck is whole, so the bracing gets 1.1 per cent and its strain is the deck’s. Cracking reduces the deck’s share in proportion to how much shear stiffness the cracks cost, which is the softest number here.

The bending’s part is independent of the deck’s shear: it comes from the top flanges’ strain, which the bracing cannot share or avoid, since it is bolted to them. That is the part a stiffer deck cannot take away. It is set by the girder’s geometry, and a deck that is cracked to its bars moves the neutral axis away from the flanges, which is the one thing a composite girder normally does in the bracing’s favour, undone.

The worst diagonal by hand #

At 34.5 m the lorry on the near span sags the section by 1,363 kN·m and on the far span hogs it by 1,593, a swing of 2,956 kN·m. The cracked section’s second moment is $4.18 \times 10^{10}\ \mathrm{mm^4}$ and its top flanges are 665 mm above its axis, so their stress swings by $2.956 \times 10^9 \times 665/4.18 \times 10^{10} = 47.1$ N/mm², and the diagonal’s by $0.64 \times 47.1 = 30.1$.

The lorry 2.5 m off the centre line twists the span by a torque that swings by $480 \times 2.5 = 1{,}200$ kN·m as it passes. The cell encloses $3.75 \times 10^6\ \mathrm{mm^2}$, so the shear flow round it swings by $1.2 \times 10^9/(2 \times 3.75 \times 10^6) = 160$ N/mm. The top plane’s shear stiffness where the deck is cracked is $13{,}750 \times 250 \times 0.3 + 81{,}000 \times 0.478 = 1.07 \times 10^6$ N/mm, so it shears by $1.50 \times 10^{-4}$, and the diagonal’s stress by $210{,}000 \times 0.48 \times 1.50 \times 10^{-4} = 15.1$ N/mm². The two do not peak with the lorry in the same place, and leaning the worse way the diagonal’s total range is 42.

What a left-in bracing would have to be designed for #

If the bracing is to stay, its diagonals over the supports have to be treated as fatigue-loaded members, and the calculation says what that means. Two hundred million lorries in a hundred years is past the point at which a constant-amplitude curve flattens, so a connection lasts without limit only if its range is below its category’s cut-off: 20 N/mm² for category 50, 29 for 71, 36 for 90. The worst diagonal’s 42 is above all three. A preloaded joint of the best bolted category, 112, has its cut-off at 45 — the only ordinary connection the worst diagonal could have without a finite life.

The alternative is to release the diagonals where they are worked. A diagonal with one end in a long slotted hole cannot carry the girder’s strain because it is free to slide by it, and the deck no longer needs the truss to close the box. Slotting the diagonals in the cracked zones after the deck has hardened — a few minutes’ work per connection with the bolts loosened, compared with removing the members from under the slab — makes them the 5.5 N/mm² members they are along the rest of the span. It is the same move as the counter fitted after the load: a member that cannot be allowed to carry a misfit is fixed or freed at the moment its misfit stops mattering.

Neither option is in the bracing’s original design, because the bracing was designed for wind and the wet concrete’s eccentricity on the open trough, for a few weeks. The members that close the box for the plate that closes the circuit’s reason during construction become, unless someone decides otherwise, a set of strain gauges that fatigue.

What the count assumes #

The diagonals are tight in their holes. A diagonal bolted with ordinary bolts in clearance holes takes up load only once its bolts bear, as the joint that carries nothing until it slips found of a joint’s first millimetre, and 42 N/mm² over a 5 m diagonal is an elongation of 1 mm — less than the clearance. Such a diagonal may slip back and forth rather than carry the range, fretting its holes instead of cracking them, which is a different failure on a different clock. A preloaded joint carries the range in full and is the category 90 curve.

The dead load keeps the cracked zone hogging. The cracked section is used for both directions of the live moment because the girder’s own weight and deck hog it there by more than a lorry sags it; at the edge of the zone that is marginal, and the edge is where the worst diagonal is.

The girder’s cross-section does not distort. A tub girder with widely spaced cross-frames distorts under eccentric load, as the stiff frame among soft ones found, and distortion warps the top flanges differentially — a third way to strain a diagonal, left out here.

One lorry at a time. Two lorries side by side or in convoy over the two spans make larger swings less often, which counts in the damage as a cube.

Still open: the gusset welded to the flange #

Removing the diagonals over the supports — the 15 per cent of each span where they are worked — would take their connections out of the calculation, and the deck needs none of them by then. But a bracing is fixed to a girder by gussets, and a gusset welded to a top flange stays when the diagonal is unbolted. A gusset more than 100 mm long welded along a flange is a longitudinal attachment in one of the lowest fatigue categories, and it sits on the very flange whose stress swings by 47 N/mm² over the support, a range the flange’s own design never had to put through a detail that poor. Whether the gussets left on a girder are the bracing’s real legacy, and whether cutting them off — leaving a ground flange — costs less than the cracks they would start, is what the bracing leaves behind even when it has been taken out.