The reaction moves one per cent and the moment five

The reaction that is made of short waves found why the flange of a wide continuous girder works over so little of its width at an interior support. The support moment is made by a reaction, a reaction is a kink in the moment diagram, its sine harmonics fall only as the square of their number, and a short harmonic puts its stress next to the web. Over the support of two 20 m spans with a 3 m flange overhang, the flange works over a third of its width, against four-fifths at the sagging peak.

Every reaction in that essay was found from the girder as an ordinary beam, with the same stiffness everywhere. Its last section pointed out that this is not true: over the support the flange is working over a third of its width, so the girder is less stiff there than its section suggests, and a continuous girder softer over its supports should shed moment into its spans. It asked whether that redistribution is enough to lower the support moment noticeably, which would ease the spike that caused it, and whether the girder’s stiffness is then “an iteration between its moment and its shear lag rather than a property of its section.”

The answer to the second question is no, and the reason makes the first one easy to answer.

A girder stiffer to long waves than to short #

The effective width of a flange is a ratio, the force the flange carries over what it would carry if the whole width worked at the stress at the web. The earlier essays on effective width found that ratio is a property of the wave of moment acting on the flange. A sine wave of bending moment along the girder whose wavelength is long compared with the flange’s width spreads its stress almost evenly across the flange; one whose wavelength is short concentrates it beside the web, because the flange’s shear stiffness cannot carry the strain out to the edges before the wave reverses.

If a wave of moment uses only part of the flange in stress, it uses only that part in stiffness too. The flange’s share of the girder’s bending stiffness, for that wave, is its share of the width. So a wide-flanged girder has no single stiffness. It has one for every wavelength.

The girder drawn is the earlier essay’s: two continuous 20 m spans, flanges 6 m wide in all and 30 mm thick, 1.2 m apart on 24 mm of web, so that the flanges are 97 per cent of the girder’s stiffness. A wave as long as the whole girder, 80 m, uses 95 per cent of its flange and bends it with nearly its full stiffness. A wave the length of two spans, the girder’s own sagging shape, uses 84 per cent. A 2 m wave uses 5 per cent, and the girder is a twentieth as stiff to it as its section says.

That is a linear girder. Its stiffness to a given wave does not depend on how large the wave is, so doubling the load doubles everything. The stiffness depends on the shape of the moment, not its size, and a linear structure whose stiffness depends on shape is solved in one step, wave by wave. The iteration the question imagined — moment gives widths, widths give stiffness, stiffness gives a new moment — is what happens if the widths are first collapsed into a single number per section. Kept as a number per wave, the widths never have to be revised.

One compatibility solve #

The girder is treated as the earlier essay treated it: one beam simply supported over its whole 40 m length, carrying its uniform load downward and its interior reaction upward, spread over a 1 m bearing. Each of these is a sum of sine waves. Each wave of moment bends the girder with its own stiffness, so each wave of deflection is that wave’s moment over its own $EI$ and over the square of its wavenumber. The reaction is whatever makes the deflection over the bearing zero. That is one equation for one unknown, written in waves rather than in sections, and it is the same compatibility statement the force method makes for any indeterminate beam — with the flexibility of each part of the moment depending on how fast it varies along the girder.

Two checks hold the calculation. A prismatic girder, one with the same width for every wave, gives the three-moment reaction, $1.25wL$, whatever that width is, because a uniform stiffness cancels from a compatibility equation. And the same prismatic girder’s deflection in the span comes out at $0.00542wL^4/EI$, the textbook figure for two equal spans.

What softness over the support does #

With each wave bending the girder by its own stiffness, the interior reaction falls by 0.88 per cent. The girder over the support is softer to the short waves the reaction is made of, so the support takes a little less of the load and the spans a little more.

The support moment falls by 4.6 per cent, from 47.0 to 44.8 kN·m for each kN/m of load, and the sagging peak in each span rises by 2.9 per cent. A change of under one per cent in the reaction has become a change of nearly five in the moment over it. The reason is arithmetic, not mechanics. The moment over the support is the uniform load’s moment on the 40 m simple span, $wT^2/8 = 200$, minus the reaction’s, $RT/4$ less a little for the bearing: about 200 minus 250 times the reaction per unit load. The support moment is the difference of two numbers four or five times its own size, and a small change in one of them is a large change in the difference.

The numbers close exactly. The reaction is 25.0 per unit load on the prismatic girder and 0.88 per cent less on the softer one, 22 hundredths of a unit; times the 10 m lever of the 40 m simple span’s midpoint, that is 2.2 kN·m per kN/m — the whole of the difference between 47.0 and 44.8. Nothing else has moved. The uniform load’s own moment is the same on both girders, because a load does not care how stiff the beam it sits on is. Only the redundant does.

The moment over the support found that continuity’s whole benefit lives in that difference; this is the same fact read the other way, as a sensitivity. Any small softening near a support — a cracked slab over a pier, a bearing that settles a millimetre, a flange that works over less of its width — is amplified by the same factor in the moment there.

The spike follows the reaction #

The reason the question was worth asking was the stress. Over the support, the flange’s stress beside the web is far higher than the support moment would suggest, because the reaction’s short waves put their stress there. If the girder sheds moment from the support, does the spike ease?

Three stress diagrams are worth putting side by side. A prismatic girder, stiffness and stress both from one width, gives a smooth diagram with no spike; over the support it is wrong by a factor of more than two, which was the earlier essay’s point. The design route — the moments from a prismatic global analysis, which EN 1993-1-5 allows, then each wave’s stress from its own effective width — puts the spike back, at 2.33 times the prismatic stress over the support. The consistent calculation, moments and stresses both from each wave’s own flange, is the solid line.

It is 2.3 per cent below the design route over the support. The support moment fell by 4.6 per cent; the spike over it falls by half as much. The spike is made of the reaction’s short waves, and the reaction changed by 0.88 per cent. Part of the support moment’s fall is in its long waves, the ones the uniform load and the reaction share, and those spread their stress across the flange where it does not concentrate. In the span the stress rises by 2.5 per cent, with the moment that was shed into it.

Magnified in the moment, diluted in the spike #

The chain — a small change in the reaction, a larger one in the moment, a smaller one in the spike — holds across the range of flanges.

For flanges a twentieth of the span wide, the effect is nothing at all: shear lag barely touches a narrow flange, and every number is below a tenth of a per cent. For flanges 0.3 of the span wide, the case above, the reaction falls by 0.9 per cent, the moment by 4.6 and the spike by 2.3. At the widest drawn, flanges 12 m wide on 20 m spans — a deck plate on a box girder — the reaction falls by 2.8 per cent, the moment by 14.9 and the spike by 5.3.

So the answer to the earlier essay’s question has two halves. The redistribution is a few per cent of the moment that a design can ignore for ordinary flanges, and approaching a sixth for very wide ones. The spike it might have eased eases by about a third as much as the moment. A designer who wanted the support region’s peak stress lower cannot get it from the analysis; it comes from the section — a wider web, a stiffened flange, a longer bearing.

Three and four spans #

The two-span girder is the one where an interior support carries the most. With more spans the same chain holds, and its size changes in a way that does not follow the count of spans in order. Three continuous 20 m spans with the same flanges shed 2.0 per cent of the moment at each interior support, from a reaction that falls by only 0.17 per cent, and the spike there eases by 0.8. Four spans shed 3.8 per cent of the first interior support’s moment, from a reaction 0.95 per cent smaller, and the spike eases by 2.0.

In every case the reaction moves least, the support moment most and the spike in between. The magnification from reaction to moment is larger for three spans than for two — twelve times rather than five — because the moment over a three-span girder’s support, close to 40 per unit load, is the difference of the load’s 400 on the 60 m simple span and the two reactions’ 440. The shed moment is smaller only because the reaction itself changes less, 0.17 per cent against 0.88; how two softened supports share the change between them is not something these figures separate. None of these numbers is large. All of them are bigger in the moment than in anything that drives it.

The flange’s share of the girder #

Everything above is for a girder whose flanges are 97 per cent of its stiffness, the extreme of a thin-webbed plate girder or box. A thicker web carries more of the bending in plane sections, with no shear lag, and so dilutes the flange’s wave-dependence. At a flange share of 53 per cent the support moment falls by 1.8 per cent rather than 4.7. A concrete-filled or heavily webbed section is closer to that end; a steel box with a wide orthotropic deck — one plate and three structures — is at the far one.

The bearing changes the spike and not the shed #

The bearing’s length was, in the earlier essay, the thing that decided how high the spike was: a reaction spread over a longer bearing has its shortest waves damped out, and the flange over it works over more of its width. That is still true — on the design route the spike falls from 2.80 times the prismatic stress over a 0.3 m bearing to 1.56 over a 5 m one. But the redistribution barely notices. What is shed from the support moment stays at four and a half per cent whatever the bearing.

The two are controlled by different parts of the reaction. The redistribution depends on how much softer the support region is to the reaction as a whole, which is a matter of the reaction’s middle-length waves, those of a few metres. The spike depends on the very shortest waves, which only the bearing’s length can remove. A designer has a lever for each: the support detail for the spike, and the flange’s width for the shed moment.

What a global analysis can assume #

EN 1993-1-5 allows a global analysis of a girder with wide flanges to take one effective width per span, its sagging value, along the whole span, and then to check the sections with their own widths. That is the design route in the figures, and it is now possible to say how good it is. For flanges up to about a third of the span wide, it overstates the support moment by under five per cent and the spike over the support by about two; it understates the span’s moment by about three. Those are smaller than the uncertainty in the effective width itself, which the flange works least where the shear is largest found to depend on assumptions no rule records. For a very wide deck plate the errors are three times larger and worth computing.

What the simplification cannot be allowed to do is check the support section with the span’s width. That is the prismatic line in the stress figure, and it misses the spike by a factor of two or more. The global analysis can be prismatic; the section check cannot.

There is a familiar pattern in this. Solved by passing it around showed that moment distribution converges because each joint’s imbalance is shared out by stiffness, and that the answer depends on stiffness ratios rather than stiffnesses. Here the stiffness ratio between the support region and the span changes, and the support moment responds — but only through the reaction, which is the one quantity a continuous beam’s compatibility actually determines. Everything else is equilibrium on top of it.

What the calculation takes for granted #

The flange’s effective share for each wave is the one-term parabola, capped at the exact wide-flange ceiling, as in the essays before it. A more exact stress function would move each wave’s share by a few per cent, and since the result is a difference of differences, the shed moment would move by a similar fraction of itself. The shape of the chain would not change.

The girder is elastic and uncracked. A composite girder’s concrete deck over a support cracks in tension, which softens the whole support region rather than only its response to short waves, and composite design makes an explicit allowance for the support moment that cracking sheds into the spans for exactly that reason. Shear lag and cracking act in the same region in the same direction, and here only the first is modelled.

The load is uniform and static. A patch of traffic over one span has its own short waves, and a girder that is softer to short waves will spread a concentrated load less than a prismatic one does, which is the reaction’s essay applied to the load rather than the support.

The supports are rigid. A real pier has a stiffness, and the reaction’s sensitivity found here is the same sensitivity to a settlement or a soft bearing; one support too many made that point about every redundant support.

Still open: the box whose deck cracks over the pier #

The 6 m flange here is steel, and steel does not crack. The commonest wide flange on a continuous bridge girder is a concrete deck on steel, which works over a reduced width over the pier for two reasons at once: shear lag, which is a matter of the wave, and cracking, which is a matter of the sign of the moment. A cracked deck over a pier still carries its reinforcement, and the reinforcement’s own shear lag — bars spread across a width whose concrete no longer joins them in shear — is a different problem from the plate’s. Whether the cracked deck’s effective width over the pier is narrower or wider than the uncracked deck’s, and whether the two softenings then add or one simply replaces the other, is a question about the flange that every essay here so far has treated as uncracked.