The reaction that is made of short waves

The flange that is not all there found that a wide flange works over less than its width, because stress reaches it only through shear along its junction with the web. The flange works least where the shear is largest then found that a wide flange’s effective width is not a property of a beam but of a position along it, and that on a simply supported span it falls from its mid-span value toward the supports. It named the region over an interior support of a continuous girder as the place where all of this concentrates — the shear largest, the moment reversing — and guessed that the flange there might be half what a naive calculation gives. It did not compute it, because its model was a single simply supported span and an interior support is not in one.

This essay computes it, with the same model, by a change of view. A continuous girder is a simply supported girder with a load pushing up at each interior support. Two continuous 20 m spans are one 40 m beam resting on its two ends, with the middle support replaced by the reaction it supplies, $1.25wL$, acting upward on the web. The moment of that beam is the uniform load’s moment less the reaction’s, both can be expanded as sine waves along the 40 m, and every wave has its own effective width exactly as before. Nothing about the flange needs to know that one of the loads is a support.

What comes out is that the flange over the support works over a third of its width, against four fifths at the sagging peak, and that the reason is the shape of the reaction itself.

The girder #

Two equal spans of 20 m, carrying a uniform load, with a flange overhanging the web by 3 m each side — a box girder’s top flange, or the slab of a composite deck, treated as one elastic plate of steel. The overhang is 0.15 of the span, wide enough for shear lag to matter and within the proportions bridge decks have. The middle support’s reaction is spread uniformly over a bearing 1 m long, which is about what a diaphragm and its bearing occupy; the figures come back to that length, because it turns out to matter.

In each span the flange behaves much as a simply supported one does. At the sagging peak, 7.6 m from the end, it works over 0.81 of its overhang, a little less than the 0.88 a single span has at mid-span. Toward the support the curve does something no simply supported span shows: it falls to 0.33 — a third of the flange working, at the section carrying the largest moment in the girder.

The dashed lines are the European rule for the same girder. EN 1993-1-5 replaces the span by an effective length, $0.85L = 17$ m for an end span’s sagging region and $0.25(L_1 + L_2) = 10$ m over the interior support, and reads a reduction factor off each: 0.83 in the span and 0.34 over the support. The harmonic calculation and the rule agree to within a few per cent at both places. That is reassuring about the rule, and it is the check on the model, which was built for simply supported spans and never told what a continuous girder is.

A reaction is made of short waves #

Why the support should be so much worse than the span is the part worth understanding, because it is not that the shear is larger there — the earlier essay’s mechanism — although it is.

Every load on a beam can be written as a sum of sine waves along its length, and each wave makes its own moment. A load spread evenly over the whole girder is smooth, and its waves die away quickly — as the cube of their number. The sagging moment, built mostly from the uniform load, is within 2 per cent of its value after nine harmonics, and the shortest of those is a wave 4.4 m from crest to node.

A reaction is not smooth. It is a large force concentrated over a metre, and a concentrated force is made of waves of every length in nearly equal measure: the moment it makes has harmonics that fall only as the square of their number, against the cube for an even load, and spreading it over a bearing hastens their fall only for waves shorter than the bearing. The moment over the support is made largely by the reaction, because the reaction is what turns the girder’s sagging into hogging there. To get within 2 per cent of it takes 32 harmonics, down to waves 1.25 m long.

Now recall what the earlier essays found about a single wave. Its effective width depends on the ratio of the flange’s width to the wave’s length, not the span’s: a long wave spreads its stress across the whole flange, a short one cannot get it further than a fraction of the way out before the shear feeding it reverses. On this flange the wave as long as the girder works over 0.96 of the overhang, the one a third as long over 0.71, and a wave 1 m long over about 0.05. The support’s moment is built from exactly the waves that use the least of the flange. The sagging moment is built from the waves that use most of it.

That is a more precise version of the earlier essay’s “the moment reverses, so its harmonic content is short-wavelength by construction”. The moment does reverse, but the reversal is not the cause; the cause is that the reversal is produced by a concentrated reaction, and concentration is short-wavelength content. The same point can be read straight off the moment diagram, which has a sharp peak over the support and a rounded one in the span — the moment over the support is the one that appeared from nothing when the girder was made continuous, and it arrives as a cusp.

Continuity moves the worst stress and multiplies it #

Continuity is usually sold on its moments, and the moments are as advertised. The sagging peak falls to 0.56 of a single span’s, and the support moment — with the reaction spread over its 1 m bearing — is 0.94 of the single span’s mid-span moment. Drawn as if the whole flange worked, the stress follows the moment: a gentle hump in each span and a dip of about the same depth over the support.

With the shear lag the dip is a spike. The stress at the web over the support is 2.52 times the single span’s peak stress, and 4.2 times the sagging peak in the same girder, for a moment only 1.7 times as large. The extra factor of two and a half is the flange that is not working: the same moment carried by a third of the flange instead of four fifths.

So the trade continuity offers a wide-flanged girder is not the one its moment diagram describes. It lowers the sagging stress by about two fifths, and it raises the girder’s worst flange stress to two and a half times what the simple span had, at the one section where a box girder’s top flange is in tension and its bottom flange — also wide, also lagging — in compression, where plate buckling is waiting. The support that is not a point found that a bearing’s width trims the support moment by $Rb/8$; here the same width decides much more than the moment.

The width over a support depends on what it bears on #

The 1 m bearing was a choice, and the model is honest about what the choice does.

Shorten the bearing and the reaction becomes more concentrated, its short waves stronger, and the width over the support smaller: 0.25 of the overhang for a reaction spread over 0.3 m, 0.33 over 1 m, 0.48 over 4 m. A true point reaction makes the stress at the web grow without limit — slowly, as a logarithm, but without limit — so there is no effective width over a point support at all. The width is a property of the girder and of the bearing together.

That is a genuine feature of elasticity rather than a defect in the model. A force delivered at a point on a plate produces an infinite stress at the point, and a wide flange fed by a web over a bearing is a plate fed along a line by a force concentrated over a short length of it. What saves a real girder is that the reaction is never a point: it comes through a bearing, into a diaphragm, and up through a web stiffened to receive it, and each spreads it over a length. The rules carry no bearing length, and the figure shows what they assume instead. EN 1993-1-5’s 0.34 is the width this girder has over a bearing about 1.2 m long; EN 1994-1-1’s simpler $L_e/8$, 0.42, is the width over about 2.5 m.

So for a box girder sitting on a short pot bearing under a single diaphragm, the European rule is near the computed width, and the composite-deck rule is about a quarter generous. For a girder continuous over a wide pier with a pair of diaphragms, both are conservative. The rule cannot tell those girders apart, and neither can an effective length.

Why a quarter of two spans is the right length #

The rule’s effective length over the support, $0.25(L_1 + L_2)$, looks like a fitted constant, and it is more than that. On two equal spans under a uniform load the moment changes sign at three quarters of each span from its end — 15 m along a 20 m span, 5 m short of the support. So the hogging region runs from 5 m before the support to 5 m after it, exactly $0.25(L_1 + L_2) = 10$ m. The rule takes as its span the length over which the support’s moment rises and falls, which is the honest length scale for the waves that make that moment.

Read that way, the rule treats the hogging region as a short girder of its own, 10 m long, loaded upward at its middle by the reaction and held at its two ends by the points of contraflexure, where the moment is zero as it is at a simple support. A 3 m overhang on a 20 m span is a modest flange; on a 10 m span loaded at its middle it is a very wide one. That is the whole of the support’s problem stated as a proportion: the flange is measured against the length of the moment it carries, and over a support that length is half a span.

The same reading says where the rule will drift. The points of contraflexure are not fixed. A live load on one span only moves them — the loaded span’s contraflexure point toward its support, the unloaded span’s away — and a short span beside a long one has a hogging region that is not centred on the support at all. The effective length in the rule is a property of the spans, written down once; the length it stands for is a property of the load case, and changes with it. Where the two part company the computed width and the rule’s will part company too, and the harmonic calculation, which knows only the loads, will follow the length that is really there.

The support loses its flange first #

The proportions decide how much of this a girder sees, and the figure says the support always sees it first. At an overhang of a twentieth of the span — a narrow flange, where nobody computes shear lag — the sagging peak keeps 0.96 of its flange, which is the reason nobody computes it. The support keeps 0.68. A third of the flange over the support is not working on a girder whose mid-span says the effect is negligible.

At a tenth of the span the two are 0.91 and 0.45; at a fifth, 0.63 and 0.25. The support curve falls from the narrowest flange drawn; the sagging curve stays near one until the overhang is about a tenth of the span. Throughout, EN 1993-1-5’s two factors track both curves closely, which is the rule doing what the earlier essay said it does — fitting the answer the harmonics produce rather than modelling them — and doing it well over the proportions it was fitted to.

The practical reading is the reverse of the one a designer usually makes. Shear lag is checked by asking whether a flange is wide compared with its span, and the question is answered at mid-span. The question that matters is whether the flange is wide compared with the distance over which the support’s moment rises and falls, and over a support that distance is a few metres, not twenty.

Every interior support is the worst section #

The pattern holds for longer girders. Three and four continuous spans lower the sagging stress to about two thirds of the single span’s, and put the worst stress over an interior support at 2.2 to 2.5 times it — though their support moments are smaller than two spans’, 0.80 of the single span’s mid-span moment for four spans against 0.94 for two. A smaller support moment still makes a spike, because the spike’s shape is set by the reaction’s concentration and not by its size.

Two consequences follow for a continuous girder of any length. The governing flange section is over an interior support, every time, and its stress is set by a width the moment diagram cannot show. And the girder’s sagging regions, which most of the design attention goes to because most of the length is in them, are lightly stressed by comparison: the flange material there is working at a quarter of what it does over the supports. A girder whose flange thickness is constant along its length is a girder whose flange is mostly idle.

That is the structural argument for what long-span box girders actually do, which is to thicken their flanges over the supports and to stiffen them there more heavily — and it is why the four box-girder collapses of 1969 to 1971, which the earlier essay recalled, put shear lag into the same rules as plate buckling. The flange over a support is in compression on one side of the box, it is working over a third of its width, and the part that is working is the strip beside the web, where the stiffeners and the welds are.

The numbers over the support, by rule and by hand #

EN 1993-1-5 Table 3.1, with an effective length of $0.25(L_1 + L_2) = 10$ m over the support: $\kappa = b_0/L_e = 3/10 = 0.30$, and

$$\beta_2 = \frac{1}{1 + 6.0\left(\kappa - \dfrac{1}{2500\kappa}\right) + 1.6\kappa^2} = \frac{1}{1 + 1.792 + 0.144} = 0.341.$$

In the span, $L_e = 0.85 \times 20 = 17$ m, $\kappa = 0.176$, $\beta_1 = 1/(1 + 6.4\kappa^2) = 0.834$. EN 1994-1-1 takes $b_e = L_e/8$ per side: $10/8 = 1.25$ m over the support, $0.42$ of the overhang, and $17/8 = 2.1$ m in the span, 0.71.

The stress ratio follows from the widths and moments. The support moment over a 1 m bearing is $wL^2/8 - Rc/8 = 50 - 3.1 = 46.9$ (for $w = 1$, $L = 20$); the sagging peak is $9wL^2/128 = 28.1$. The stress at the web goes as moment over working width: $46.9/0.325 = 144$ against $28.1/0.810 = 34.7$, a ratio of 4.2.

One plate, one beam, and a reaction pushing up #

The flange is one elastic plate fed by the web along a line, with each harmonic’s stress across it given the one-term parabolic shape the earlier essays used, and capped at the exact wide-flange limit $1/2\alpha$ for the short waves, where the parabola would stop falling at a sixth of the flange and the true width goes on falling. Without the cap the support’s width would come out higher, and wrongly.

The girder is prismatic and its supports are rigid. The interior support enters only as its reaction, found by the three-moment equation for equal spans and spread uniformly over the bearing. A support that settles changes the reaction and therefore the spike, in proportion.

A uniform load over every span. A live load arranged to maximise the support moment is no different in kind: it is the reaction that makes the spike, and every arrangement has one.

No cracking. A composite deck’s slab over a support is in tension and cracks, and its effective width then becomes a width of reinforcement, which the earlier essay noted multiplies the two reductions. A steel box flange has nothing to crack.

A width is one number for a distribution #

Every figure reports a width, which is one number for the stress across the flange, and over a support the distribution is not the gentle parabola it is at mid-span. It is a sharp peak at the web, falling to a small fraction within a metre, with — in the regions either side where the moment’s sign changes — the free edge stressed more than the web, the negative shear lag the earlier essay warned looks like a faulty gauge. A plate with that much of its stress beside the web is a plate whose welds and stiffeners near the web are doing most of the work, which is the detail a width calculation is least able to describe.

Still open: the lag that moves the reactions #

Every reaction here was found from the girder as an ordinary beam, which assumes that its stiffness is the same along its length. It is not: 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 that is softer over its supports sheds moment from the supports into the spans. Whether that redistribution is enough to lower the support moment noticeably — which would ease the spike that caused it — or whether it is a few per cent that a design can ignore, and whether the stiffness a continuous girder seems to have is then an iteration between its moment and its shear lag rather than a property of its section, is the question this calculation leaves open.