One plate and three structures

An orthotropic steel deck is a plate twelve or fourteen millimetres thick, welded to closed trapezoidal troughs running along the bridge at six hundred millimetre centres, sitting on transverse crossbeams every four metres, sitting on two main girders sixty metres apart.

Described that way it sounds like an ordinary hierarchy — a deck spanning to joists spanning to beams spanning to girders, which is the arrangement most floors have. It is not, and the difference is one sentence: the plate is the top flange of all three of them.

A wheel standing on the deck bends the plate between the trough webs, bends the trough between the crossbeams, and bends the crossbeam between the girders. Meanwhile the whole assembly is the top flange of a girder carrying the bridge. Four systems, one plate, and the stresses in it are the sum.

The systems, numbered #

The convention is to number them, and the numbering is the argument rather than a filing scheme.

System I is the main girder: span sixty metres, stress nearly uniform across the deck’s width except for shear lag at the edges. Its contribution at any point of the deck is a slowly varying membrane stress along the bridge.

System II is the crossbeam: span the twelve metres between girders, with the deck plate as its own top flange over an effective width. Its contribution is a bending stress that reverses over the crossbeam’s own span.

System III is the trough: span the four metres between crossbeams, with the deck plate as its flange again. Its contribution is a local bending stress, largest directly under the wheel.

There is a fourth, sometimes called System IV: the plate spanning between the trough webs, three hundred millimetres, under the tyre contact patch. It is a plate-bending problem rather than a beam one and it is what decides the plate thickness against permanent deformation.

Each system uses the whole plate and none of them knows about the others. The stresses superpose because they are stresses in one elastic body, and superposition is exact for a linear structure — which is the only easy thing on this page.

It is worth being clear about what is being superposed, because the word invites a mistake. The three systems are not three load cases; there is one load case, a wheel on a deck, and the three are three decompositions of the response to it. Splitting a structure’s response into parts that each look like a familiar member, solving each and adding, is a modelling device rather than a physical fact — and it is legitimate here only because the plate stays elastic and the geometry does not change. Both of those hold, and both are worth stating rather than assuming.

What they add up to #

For the deck drawn — 14 mm plate, 600 mm trough spacing, 4 m crossbeam spacing, 12 m between girders, a 100 kN wheel at midspan of a trough — the three contributions at the rib-to-deck weld are:

• System I: 120 N/mm²
• System II: 34 N/mm²
• System III: 13 N/mm²

and the total is 167 N/mm², of which the global system supplies 72 per cent and the two local ones 28.

Those proportions are the design. The global stress is large and slowly varying; the local ones are small and cycle with every axle. So the strength check is dominated by System I and is never close, and the fatigue check is dominated by Systems II and III and is the whole design of the deck.

Which free body produced the number #

Cut a small element out of the deck plate immediately beside a rib-to-deck weld, with the cut perpendicular to the bridge.

Crossing that cut is a single membrane force per unit width and a single bending moment per unit width. There is one of each, because there is one plate. Ask where they came from and the answer has three terms — the girder’s curvature, the crossbeam’s curvature and the trough’s curvature, each multiplied by the distance from the plate to that system’s own neutral axis — but the force crossing the cut is one number and the material responds to that number.

That is why the three cannot be checked separately. Each system taken alone gives a stress the plate is comfortably able to carry; the plate does not experience them one at a time. The free body is the element, the equilibrium is trivial, and all of the difficulty is in the bookkeeping of what contributes to it.

It also says exactly where the interesting point is. The three contributions have different signs at different places: the girder’s stress is tension in the deck over a pier and compression at midspan, the crossbeam’s reverses over its own span, and the trough’s reverses over its own. The worst point is not under the wheel and not over the pier; it is wherever the three happen to agree, and finding it is a search rather than an inspection.

Effective width, three times over #

Each of the three systems uses the deck plate as a flange, and each of them uses only part of it.

Shear lag means a wide flange is not fully effective: the stress falls away from the web, and the effective width depends on the ratio of the flange width to the span of that system. Since the three systems have spans of 60 m, 12 m and 4 m, they have three different effective widths of the same plate.

For the main girder, whose span is enormous compared with the deck’s half-width, the plate is nearly fully effective. For the crossbeam it is partly effective. For the trough the flange is only 600 mm wide against a 4 m span and is fully effective again.

So the same plate is counted at three different efficiencies in three different calculations, and the sum of the three fractions has no meaning. That is not double counting — each system’s effective width is the correct answer to its own question — but it is the point at which anybody hand-checking the design loses confidence, and it is why orthotropic decks are analysed with finite elements in practice.

The weld that decides everything #

The rib-to-deck weld is a fillet or partial-penetration weld running along both edges of every trough, for the whole length of the bridge. On a 60 m span with troughs at 600 mm centres over a 12 m width that is 2.4 kilometres of weld, and every millimetre of it is a fatigue detail.

It is a poor one. The weld is loaded transversely by the local bending of the deck plate, the root is unfused by design in a partial-penetration detail, and a crack can start either at the toe — growing into the deck plate, where it will eventually appear on the road surface — or at the root, growing through the throat, where it is invisible from either side until it has crossed.

The stress range at that detail on the deck drawn is 47 N/mm²: the local contribution, since a single axle does not change the global bending appreciably. Against a detail category in the seventies that is not comfortable, and the entire geometry of an orthotropic deck — plate thickness, trough spacing, crossbeam spacing, weld penetration — is chosen to make that number acceptable.

The trough is closed, and that is the point #

The trapezoidal trough looks like a stiffener and behaves like a member, and the difference is that it is closed.

An open stiffener — a flat bar or a bulb welded to the plate — has a torsional constant that is negligible, so it stiffens the deck in one direction and does nothing in the other. A closed trough has a torsional constant larger by a factor of hundreds, because it encloses an area, and that torsional stiffness is what makes the deck genuinely two-way. A wheel between two troughs is carried partly by the plate spanning across and partly by the two troughs twisting, and the twisting share is not small.

The consequence is that a deck with closed troughs distributes a wheel over four or five of them, while one with open stiffeners loads one or two. The peak local stress falls accordingly, and so does the deflection under the axle — which is what the surfacing cares about.

The price is fabrication. A closed trough can only be welded from the outside, which is why the rib-to-deck weld is a partial-penetration detail with an unfused root in the first place. The closed section that makes the deck work is the reason its worst detail cannot be made properly, and that trade has been the subject’s central difficulty for sixty years.

Why the plate got thicker #

The history of the type is a history of that weld, and it is unusually well documented because the failures were expensive.

Early orthotropic decks used 10 or 12 mm plates, because the strength calculation permits them and the material saving is the reason for the type. A generation later a substantial fraction of them had cracked at the rib-to-deck weld, and the repairs — grinding out, rewelding, in some cases bonding a second layer of steel or a concrete overlay on top — cost far more than the plate saved.

The response was to thicken the plate. Modern practice uses 16 to 20 mm, and the reason is arithmetic: System III’s local bending stress goes as the inverse square of the plate thickness for a given wheel and spacing, so going from 12 mm to 18 mm cuts it by more than half. Two thirds of the local stress range was removed by changing a number that the strength calculation had never objected to.

That is the shape of the whole subject. The strength check has never governed an orthotropic deck; every dimension in it is set by a fatigue detail, and the calculation that appears to size the plate is not the calculation that did.

The grillage, and why stiffening can hurt #

The deck panel between the girders is a grillage: troughs one way, crossbeams the other, sharing load in proportion to their stiffnesses. That is not a hierarchy, and the difference matters when something is changed.

In a hierarchy — deck to joist to beam — making the beam stiffer reduces its deflection and leaves the joists alone. In a grillage, making the crossbeam stiffer attracts load to it, because load goes to the stiffest path. Its own stress may fall or rise depending on whether its section modulus grew faster than its share did, and the troughs’ stress falls.

So the intuitive move — deepen the crossbeam because System II’s contribution looks large — has two effects with opposite signs, and which one wins depends on the spacings rather than on the crossbeam. The reliable levers are the ones that change the demand rather than the distribution: a thicker plate, closer troughs, closer crossbeams.

What it buys, and what it costs #

The reason to build one at all is weight. An orthotropic deck weighs around 250 kg/m² against 500 or more for a concrete deck of the same span, and on a long-span bridge the deck’s weight is carried by the deck, so the saving compounds: a lighter deck needs less cable, which needs less tower, which needs less foundation.

That is why every very long span in the world has one, and why almost no short span does. Below about 200 m the deck’s own weight is not the problem the structure is solving, and the fatigue detail, the fabrication cost and the surfacing difficulties are all real.

The surfacing is the least discussed and among the worst. A thin bituminous layer on a plate that flexes visibly under every axle is a difficult material problem — it debonds, it cracks, it has to be replaced far more often than a surfacing on concrete — and its replacement is a lane closure on a bridge that usually has no diversion.

Three spacings, one decision #

The geometry has three free lengths in it — the plate thickness, the trough spacing and the crossbeam spacing — and they are not independent, because each of them changes a different one of the local systems.

Thicken the plate and System IV falls as $1/t^2$ and System III falls too, because the trough gains flange area. Close the troughs up and System IV falls as the square of the spacing while System III is barely touched. Bring the crossbeams closer and System III falls as the square of that spacing while System II rises, because a shorter crossbeam span means more of them.

So the design is a three-variable optimisation against one fatigue detail, which is why deck geometries across the world have converged so tightly: 14 to 20 mm plates, troughs at 600, crossbeams at 3 to 4 metres. Those are not conventions inherited from a first design; they are where the optimisation lands, and it lands there because the objective function is the same everywhere.

What a crack in one looks like #

The failure mode is specific enough to be diagnosed from the road surface, which is unusual.

A crack starting at the weld toe grows upward into the deck plate, and when it has crossed the plate it appears as a longitudinal crack in the surfacing directly over a trough web. That is the visible version, it is found by inspection, and it is repairable by grinding and rewelding from above.

A crack starting at the root grows through the weld throat, and it is invisible from the road and from underneath until it emerges — often after it has run a long way. It is found by ultrasonic testing that has to be done from a position the trough does not allow, which is why it is usually found by the toe crack it eventually turns into.

A crack at the trough splice or at the crossbeam cut-out is a different detail with the same consequence, and the cut-out cracks are the ones that grow fastest because they sit in a web carrying shear as well as the local effects.

The pattern across a bridge is diagnostic. Cracks concentrated in the wheel tracks are System III and IV — a plate or trough problem. Cracks distributed along the whole trough length are System I — a global problem, usually near a pier. Where the cracks are says which system produced them, and the repair differs accordingly.

Where the model stops #

The superposition is exact and the three analyses are not. Each system’s contribution is computed with an idealisation of its own boundary conditions, and the crossbeam in particular is neither simply supported nor fixed at the girders.

The wheel is a point. A tyre contact patch spreads the load over a few hundred millimetres through the surfacing, and System III and System IV are both sensitive to that spread. Treating it as a point overestimates the local bending substantially.

The effective widths are elastic. They are computed for a linear plate and they change once anything yields, which for local stresses under a heavy axle is not hypothetical.

The crossbeam’s own web is left out. A crossbeam in an orthotropic deck has cut-outs where every trough passes through it, and the stresses around those cut-outs are a separate and notorious problem — a second bad detail, at the free edge of a hole in a web that is also carrying shear.

And nothing here is a fatigue analysis. A stress range at one detail under one axle is the beginning of a fatigue calculation, not the end of one; the real check is a spectrum of axles, a damage sum and a detail category with its own scatter.

Where the ladder goes #

Later rungs on this anchor: System IV, the plate between trough webs, and the surfacing that spreads the wheel onto it. The Pelikan–Esslinger method, which is how these were analysed before finite elements. Rib-to-deck weld details and the root-versus-toe cracking they choose between. Fatigue assessment with an axle spectrum. Trough splices, which are the other bad detail. Retrofits of cracked decks, from grinding to bonded overlays. Orthotropic decks in movable bridges, where the weight saving is the whole design. And the comparison with the other way to make a plate span two ways, where the corrugation does the stiffening and the flange is left alone.