The deck that cracks over the pier

A wide flange does not work across its whole width, because its stress arrives from the web through shear and shear takes distance to spread. The flange that is not all there measured how much is lost, and the flange works least where the shear is largest found that the loss is worst at the supports. The reaction that is made of short waves explained why: the moment over an interior support of a continuous girder has a kink, a kink is made of short waves, and a short wave of moment spreads only a short way into a flange before it dies out. Over the pier, a girder works over a fraction of its flange, and the reactions move because of it.

All four essays treated the flange as a steel plate, which does not crack. The commonest wide flange on a continuous bridge girder is a concrete deck on steel, joined by shear studs, and over every interior pier that deck is in longitudinal tension. It cracks across its width, at intervals of a few hundred millimetres, and what carries the tension is the longitudinal reinforcement. The rest of the deck is still there, between the cracks, but it no longer carries tension along the girder.

The question this essay asks is whether the cracked deck’s effective width over the pier is narrower or wider than the uncracked deck’s — and whether the two softenings, the one from cracking and the one from shear lag, add, multiply or replace one another.

The ratio that decides shear lag #

The physics of shear lag reduces to a contest between two stiffnesses of the flange. Its axial stiffness, along the girder, is what makes it carry stress at all; its in-plane shear stiffness is what moves stress outward from the web to the far parts of the flange. The governing equation for the flange’s longitudinal displacement $u$ is

$$E_x\,\frac{\partial^2 u}{\partial x^2} + G\,\frac{\partial^2 u}{\partial y^2} = 0,$$

and everything about how far a stress spreads sideways depends on the ratio $G/E_x$. A flange whose shear modulus is large compared with its axial modulus spreads stress easily and works over a wide width. One whose shear modulus is small lags more.

For a steel plate, or for uncracked concrete, the two moduli are tied by Poisson’s ratio, $G/E = 1/(2(1+\nu))$. For concrete with $\nu = 0.2$ that is 0.417. The uncracked deck’s effective width is the steel plate’s with that ratio, and the steel-plate arithmetic carries over directly.

A cracked deck is a different object. Along the girder, the cracks open and the concrete carries nothing in tension, so the axial stiffness is the bars’: $E_x = \rho_x E_s$, a fraction of the concrete’s own. In shear, the deck still works, but differently. Shear in a cracked membrane is carried the way a cracked web carries it: the concrete between the cracks forms diagonal struts, and the bars in both directions tie them. The shear stiffness is the stiffness of that truss — the bars stretching, the struts shortening.

Writing $a = 1/(\rho_x E_s)$ for the longitudinal bars’ flexibility, $b = 1/(\rho_y E_s)$ for the transverse bars’ and $c = 1/E_c$ for the concrete struts’, the strain energy of a unit shear stress, with the struts at an angle $\theta$ to the girder, gives

$$\frac{1}{G} = \cot^2\theta\,(a + c) + \tan^2\theta\,(b + c) + 2c.$$

The struts take up the angle that makes the deck stiffest, $\tan^4\theta = (a + c)/(b + c)$ — Kupfer’s angle — and there $1/G = 2\sqrt{(a + c)(b + c)} + 2c$. With equal bars both ways the angle is 45° and $1/G = a + b + 4c$.

The result is the striking thing. With 1.5% of steel each way, at a modular ratio of 6.06, the cracked deck’s shear modulus over its axial modulus is 0.423 — the uncracked plate’s 0.417 to within a per cent and a half. Cracking has softened the deck’s axial stiffness about elevenfold, and its shear stiffness by almost exactly the same factor, because both now come from the same bars. The two meet where $1 + 1 + 4n\rho = 2(1 + \nu)$, which for equal bars is $\rho = \nu/(2n)$: 1.7% at this modular ratio, within the range a deck over a pier actually carries.

That coincidence is real, but it depends on the bars being equal both ways. With half as much transverse steel as longitudinal the cracked ratio is 0.312; with twice as much it is 0.559. In the cracked deck, unlike the uncracked one, the ratio is something the designer chooses — by choosing the reinforcement.

Short waves see the bars #

The width over the pier is not decided by the deck’s properties uniformly, but by how each wave of the moment responds to them. A continuous girder’s moment can be written as a sum of sine waves, and the short ones carry the kink over the support. Each wave has its own effective width: a wave much longer than the flange is wide works over nearly all of it, and a wave short compared with the flange works over a strip near the web.

The figure plots each wave’s effective width against its length for the uncracked deck, the cracked deck with equal bars, and the cracked deck with half as much transverse steel. For long waves the three are close together — a wave the length of the whole girder works over 0.96 of the flange in every case. For short waves they separate. A 2 m wave works over 0.053 of the overhang uncracked, the same cracked with equal bars, and 0.046 with half the transverse steel.

The reason is that a short wave’s stress spreads into a wide flange by a diffusion length, not by the flange’s width. From the governing equation, a wave of wavenumber $\alpha$ decays sideways as $e^{-\alpha y \sqrt{E_x/G}}$, so it reaches a distance proportional to $\sqrt{G/E_x}/\alpha$ before dying out. The short waves over the pier are the ones in that regime, and their effective width scales with the root of the modulus ratio. A deck whose ratio has fallen from 0.417 to 0.312 has a short-wave width $\sqrt{0.312/0.417} = 0.86$ times as large: 14 per cent narrower, which is what the next figure finds.

Over the pier #

Put the waves together and the local effective width along the girder emerges.

Take a deck overhanging 3 m each side of its girder, continuous over two 20 m spans, bearing on a 1 m diaphragm at the pier. In the spans, under sagging moment, the deck is in compression and uncracked, and it works over 0.81 of its overhang. Over the pier, uncracked, it would work over 0.325. Cracked, with 1.5% of steel both ways, it works over 0.328 — the uncracked value, within the drawing’s line width. With half as much transverse steel it works over 0.281.

EN 1994 gives one effective width for the region over the pier, an eighth of the effective length $0.25(L_1 + L_2)$ each side of the web — here 0.417 of the overhang — and uses it whether the deck is cracked or not. For the deck with equal bars that is exactly what the mechanics justifies: the bars’ arrangement has reproduced the uncracked plate’s modulus ratio, and the cracked width is the uncracked width. For any other arrangement it is not, and the rule does not say so.

The transverse bars decide it #

The transverse bars do two jobs in a deck over a pier. They span the slab between girders, carrying wheel loads across, which is what they are usually sized for. And in the cracked deck they tie the diagonal struts that carry shear from the web outward. The second job has no rule of its own.

Holding the longitudinal steel at 1.5% and varying the transverse, the effective width over the pier rises steadily: a quarter as much transverse steel gives 0.238 of the overhang, half gives 0.281, equal 0.328, twice 0.373 and four times 0.413 — nearly the rule’s value. A deck lightly reinforced across, which is common where the girders are close together and the slab span is short, works over a width a quarter narrower than the uncracked deck’s. One heavily reinforced across, which is common where the slab spans far between widely spaced girders, works over a wider one.

So the cracked deck’s effective width is a property of the reinforcement layout, not of the concrete. That makes it a design variable in a way the uncracked width is not: the uncracked width depends on the span and the overhang, which are fixed by the bridge, while the cracked width also depends on a ratio of bar areas that the designer sets for other reasons and could set for this one.

Two softenings that multiply #

There are two separate reasons the deck contributes less over the pier than its full section would. Cracking takes away the concrete’s axial stiffness and leaves the bars’. Shear lag takes away the far parts of the flange. It is natural to ask whether one of these makes the other redundant — whether, once the concrete has cracked, the shear-lag question still matters.

It does. The deck’s axial stiffness over the pier, as a fraction of the whole uncracked overhang’s, is 0.325 with shear lag alone and 0.091 with cracking alone — the bars, 1.5% of the section at a modular ratio of 6.06. With both, it is 0.030: the product of the two, within the small shift the cracked deck’s own modulus ratio makes to the lag. Neither softening replaces the other. Cracking reduces what each unit of width can carry; shear lag reduces how much of the width is reached; the two multiply, and over the pier the deck delivers 3 per cent of what a naive section would credit it with.

The multiplication has a consequence for the girder’s design that reaches beyond the deck. The stiffness the deck contributes over the pier sets how much of the hogging moment the girder carries as a composite section and how much the steel alone resists; it sets the support’s stiffness in the continuous analysis, and so the moment’s distribution between pier and span. An analysis that cracks the deck over 15 per cent of each span — the usual allowance — but counts all its bars over the full width overstates the deck’s contribution by a factor of three.

The concrete between the cracks #

Cracked concrete is not the bare bars. Between the cracks the concrete still grips the steel and carries tension, and a cracked tie is stiffer than its bars alone — stiffer than its cracked section says. The effect is tension stiffening, and the obvious way to include it is to raise the bars’ axial stiffness by some fraction.

If tension stiffening is credited to the axial stiffness alone, it narrows the deck. With equal bars both ways the width over the pier falls from 0.328 with none to 0.266 when the concrete adds half again to the bars’ stiffness, and to 0.230 when it doubles it. The mechanism is the same ratio: what stiffens the deck along the girder without stiffening it in shear raises $E_x$ and lowers $G/E_x$, and the short waves lag more.

Whether that crediting is right is not settled by the model. The concrete between the cracks that grips the bars along the girder also carries the diagonal struts, and if it stiffens one it plausibly stiffens the other; and shear across a crack that is already there is carried partly by the crack faces themselves, which this model ignores. So the curve is a bound in one direction: it shows how much narrower the deck could be if tension stiffening helps the deck along its length and not across it. It does not show that it is.

Across span lengths #

The narrowing from light transverse steel is not a feature of one span.

Over spans from 8 to 60 m the uncracked deck and the cracked deck with equal bars lie on one curve, and the deck with half the transverse steel lies below them by a nearly constant share: 14 per cent at 20 m, 11 per cent at 40 m. The share falls slowly with span because longer spans move more of the moment over the pier into longer waves, which see less difference between one deck and another.

The figure also carries EN 1994’s width for comparison, and it shows the rule standing well above the local width at the support, increasingly so as the span grows: at 40 m the rule gives 0.833 of the overhang and the uncracked deck’s local width over the bearing is 0.489. That gap is inherited from the uncracked analysis in the reaction that is made of short waves, which found the composite-deck rule about a quarter generous over a short bearing at 20 m: the rule treats the hogging region as a short girder of its own and gives one width for all of it, where the local width at the bearing is that region’s minimum. The gap grows with the span, because the rule’s width grows in proportion to the hogging region while the local width at the bearing grows more slowly. The cracked deck neither closes nor opens that gap when its bars are equal. When they are not, it widens it.

What the model holds fixed #

The whole girder here is given one set of deck properties, which is right over the pier and wrong in the spans, where the deck is in compression and uncracked. The width over the pier is made mainly of short waves, which see mainly the deck near the pier, so the error is modest; but the long waves that also pass over the pier see the cracked properties where the deck is really whole. The true narrowing for light transverse steel therefore lies between the uncracked value and the one drawn — closer to the one drawn, since the short waves dominate there.

The steel girder’s top flange, between the web and the deck, is not separately modelled, and the shear studs are taken as rigid. Studs that slip, as composite action always lets them a little, add a third softening in series with the other two, and the connection is busiest exactly near the pier, where the shear is largest.

And the deck is a single layer. Real decks carry bars near both faces, and the transverse steel in the top face over the girder is often heavier than in the bottom; what matters for shear transfer is the transverse steel that ties the diagonal struts across the region where they cross from the web, which is not always the steel the deck’s own design provided.

Narrower, wider, or the same #

The question the uncracked essays left open has a three-part answer.

The same, when the deck is reinforced equally both ways near $\rho = \nu/(2n)$, which for normal concrete and steel is about 1.7%. The cracked deck’s two stiffnesses then fall together, its modulus ratio is the uncracked plate’s, and EN 1994’s single effective width for cracked and uncracked sections is justified — though not for the reason a designer would guess. It is not that cracking leaves shear lag alone; it is that cracking happens to soften both stiffnesses by the same factor.

Narrower, when the transverse steel is lighter than the longitudinal, by roughly the square root of the ratio of the cracked to the uncracked modulus ratios: 14 per cent narrower at half, a quarter narrower at a quarter. And narrower still if tension stiffening stiffens the deck along its length and not across it.

Wider, when the transverse steel is heavier, by the same root.

And in every case the two softenings multiply. A designer who treats a cracked deck’s reinforcement as fully effective across its width, on the reasoning that steel has no shear lag, is treating the bars as if they were not embedded in the concrete that delivers their force — when that concrete, cracked as it is, is the only way the force gets there.

Still open: the deck whose bars are not where the struts cross #

Every deck here is a membrane with its reinforcement smeared through its thickness and across its width. A real deck over a steel girder has its transverse steel in two layers, its longitudinal steel bunched over the girder where the hogging is largest, and a strip near the web — over the steel flange and the studs — where the struts start and the bars may be thinnest. Whether the width over the pier is governed by the average transverse steel or by the steel in the first metre from the web, whether concentrating the longitudinal bars over the girder narrows the effective width as much as reducing them would, and whether a deck reinforced for its slab span alone is the narrowest deck a continuous girder can have, is the question a real reinforcement drawing asks of this one.