The flange a slender web wants heavier

What is left after it ripples took a thin section in compression and showed that its plates, once buckled, keep their edges and lose their middles, so the section that carries the load is not the one that was drawn and its centroid is somewhere else. In compression every plate is equally stressed and the question of where the material is lost is the whole of it.

In bending there is a second question, because the stress is not the same everywhere. Only part of a girder’s web is compressed, and only that part buckles; how much of it is compressed is set by where the neutral axis sits; and where the neutral axis sits is set by how the girder’s steel is shared between its flanges. So the share of steel between top and bottom, a choice made before anything is checked, decides how much of the web the buckling takes away. This essay asks which share a slender web prefers, and how much the answer is worth.

A bottom-heavy girder in sagging #

The girder has a web 1,500 mm deep and 8 mm thick — 187 thicknesses, well past the point at which a web in bending stays whole — between flanges 400 mm wide, 15 mm thick on top and 25 mm below, in S355. That is the shape of the steel beam in a composite floor or bridge, where the concrete slab will sit on the top flange and provide the compression, so the steel puts its weight in the bottom flange, where the tension will be.

Until the slab hardens the steel carries the wet concrete alone, in sagging, with the thin top flange in compression.

Two things go. The top flange’s outstands are 196 mm long and 15 mm thick, a little more slender than a flange in uniform compression can be and stay whole, and they keep 91% of their width. And the web, compressed over 959 mm of its depth, keeps 44% of that as two strips — one against the flange and one against the neutral axis — and loses a strip 538 mm deep from the middle.

The hole is in the compression half, so the neutral axis drops: from 665 mm above the bottom, where the gross section has it, to 566. The elastic modulus that governs falls from 12.76 to 9.91 × 10⁶ mm³, and the girder’s elastic moment resistance in S355 falls with it, from 4,530 kN·m to 3,520. A fifth of the section has gone, and nearly all of it from the web, which was the thinnest steel in the girder before any of it was taken away.

Where the gross section and the effective one disagree #

Hold the steel fixed — 40 mm of flange in total, 400 mm wide, top and bottom — and move it between the two flanges.

The gross section is best with the flanges equal. Its smaller modulus belongs to whichever face is further from the axis, and both faces are equally far only when the section is symmetric; move steel either way and one face gets further out, and the modulus that governs falls. That is the ordinary answer and it is drawn dashed.

The effective section is best at 22 mm on top and 18 below, and it is a different curve on the compression side of that point. Below 20 mm on top the effective modulus falls far faster than the gross one, because every millimetre taken out of the top flange both lowers the axis and thins the flange, and both deepen the hole. Above 22 mm the two curves meet: the web is losing so little that the modulus that governs is the tension face’s, and the tension face does not care what happened in the web.

The bottom-heavy girder sits on the steep part of the curve. Its gross section is 91% of the best split’s — a modest penalty for putting the steel where the finished composite beam will want it. Its effective section is 70% of the best split’s.

The hole is set by where the axis sits #

The mechanism is the depth of the compressed web, which the flange split controls.

With 10 mm on top and 30 below, the neutral axis is so low that the web is compressed over 1,152 mm — more than three-quarters of its depth — and the buckling takes 717 mm of it. With equal flanges the web is compressed over half its depth and loses 377. With 30 mm on top it loses 114, and it never quite closes, because the web stays slender over whatever part of it is compressed.

This is the same feedback that makes the web’s own calculation a fixed point — the hole lowers the axis, which deepens the compression, which enlarges the hole — read from the outside. The flange split sets the starting point of the loop, and the loop amplifies it: a girder whose axis starts low loses web, which moves the axis lower still.

The same steel turned the right way #

The top-heavy split is not a curiosity of the modulus. It is a different section with a different web.

The 22 mm flange is stocky enough to keep all of its width. The axis starts at 812 mm, above mid-depth, so the web is compressed over 744 mm, and it loses 322 — a hole 40% smaller than the bottom-heavy girder’s, in the same web, from the same steel.

And something odd happens to the modulus. The hole still lowers the axis, from 812 mm to 774, but on this section that moves it towards the face that governs, the tension face, which is now the further of the two. The effective section’s smaller modulus is a fraction larger than the gross section’s. A section that is not symmetric has two strengths, and which one governs decides whether losing material on the other side costs anything at all.

Thickening the web, and turning the girder over #

The obvious repair for the bottom-heavy girder is a thicker web. It works, slowly.

The 15/25 girder keeps 77% of its gross modulus with a 6 mm web, 80% with 10 mm, 93% with 16 mm, and levels off a little below 97% once the web is stocky — the flange’s own reduction is still there. Doubling the web from 8 to 16 mm adds 2.8 tonnes of steel to a 30 m girder and raises its effective modulus from 9.91 to 14.92 × 10⁶ mm³. Moving 7 mm of flange from the bottom to the top adds no steel at all and raises it to 14.24.

Turned over, with the 25 mm flange in compression, the same girder loses nothing that governs at any web thickness from 5 mm up. Its web still buckles — at 8 mm it loses 244 mm from its compressed part — but the modulus that governs is the tension face’s, and the hole moves the axis away from it.

The dashed lines are the calculation done once, as EN 1993-1-5’s two steps do it: the flange reduced on the gross section, then the web on the section with that flange and a whole web. It is never more than 2% above the converged answer. The loop matters for understanding where the hole comes from; it is not where the 30% goes.

A stockier web leans less #

How far the best split leans towards the compression flange depends on how much of the web there is to lose.

With a 12 mm web the effective curve is closer to the gross one everywhere, and its peak has moved back towards the middle: 21.5 mm on top and 18.5 below, worth 15.87 × 10⁶ mm³ against 15.35 with equal flanges. The bottom-heavy girder keeps 76% of the best split, where with an 8 mm web it kept 70% and with 6 mm 68%. At 14 mm the best split is 20.5/19.5, and by 16 mm it is the symmetric one — the web is stocky enough over the depth that is compressed that it loses nothing that governs, and the gross section’s answer is the answer.

So the lean is a measure of slenderness. A girder whose web is stocky enough to be checked on its gross section should have equal flanges, as every textbook draws it; a girder whose web is slender enough to lose a strip of itself should not, and the more it loses the more the flange steel should move up. The rule that a girder’s flanges should be equal is a rule about the material far from the middle, and it stops being true at exactly the slenderness at which the middle starts to go missing.

The classification sees it first #

The same lean shows up a step earlier, in the table that decides whether a section is slender at all. A web’s class limit in bending depends on the stress gradient across it, and the gradient depends on the flange split, so the three girders of this essay reach the slender class at three different web thicknesses.

The 15/25 girder’s gross section has $\psi = -0.74$, for which the limit on a web’s depth over its thickness is $42\varepsilon/(0.67 + 0.33\psi) = 80.5$; its 1,500 mm web is class 3 only if it is at least 18.6 mm thick. With equal flanges $\psi = -1$ and the limit is $124\varepsilon = 101$, so 14.9 mm. With 22 on top and 18 below $\psi = -1.12$, the limit is $62\varepsilon(1 - \psi)\sqrt{-\psi} = 114$, and 13.2 mm will do.

The class limits are the plate buckling formula rearranged, with the same coefficient that depends on the gradient, so it is no surprise that they agree with the effective section about which way the steel should go. What the classification cannot say is how much a slender section loses once it is slender, and the effective section’s answer — 22% for the bottom-heavy girder with an 8 mm web, and nothing that governs for the same steel turned over — is the part that is worth knowing when choosing the flanges.

The section, by hand #

The flange first. Its outstand is $c = (400 - 8)/2 = 196$ mm, and with $\varepsilon = \sqrt{235/355} = 0.814$ and the outstand coefficient $k = 0.43$,

$$\bar\lambda_p = \frac{c/t}{28.4\,\varepsilon\sqrt{k}} = \frac{196/15}{28.4 \times 0.814 \times 0.656} = 0.862, \qquad \rho = \frac{0.862 - 0.188}{0.862^2} = 0.907,$$

so each outstand keeps 178 mm of its 196, and the top flange is 364 mm wide where it was 400.

Then the web. The section with that flange and a whole web has its axis 648 mm above the bottom, so the top of the web, at 1,525 mm, is 877 mm above the axis and the bottom, at 25 mm, is 623 mm below it: $\psi = -623/877 = -0.71$. The coefficient for that gradient is $7.81 - 6.29\psi + 9.78\psi^2 = 7.81 + 4.47 + 4.93 = 17.2$, and

$$\bar\lambda_p = \frac{1500/8}{28.4 \times 0.814 \times \sqrt{17.2}} = 1.96, \qquad \rho = \frac{1.96 - 0.055(3 - 0.71)}{1.96^2} = 0.478.$$

The compressed depth is $1500/(1 + 0.71) = 877$ mm and its effective part $0.478 \times 877 = 419$ mm, so the first pass cuts a hole of 458 mm. Taking that out lowers the axis to 574 mm, and the loop goes round three more times to 538. Each pass changes the hole by less than the one before, and the modulus by less than 2% in all.

The girder that is the wrong way up for a few weeks #

The bottom-heavy shape is not a mistake. In a composite girder — two beams made into one by the studs between them — the slab is the compression flange, a metre or more of concrete beside which the steel’s top flange is a detail, and the steel’s job in service is the tension. Its section is drawn for that, and for that it is right.

What is wrong is the order. Every structure is built in stages, and an unpropped composite girder carries its own weight and the wet concrete on the bare steel, which is the section this essay has been taking apart. The construction stage is shorter and its moment smaller than the finished girder’s, often a third or two-fifths of it — but it is carried on 70% of what the same steel would carry the other way up, by a compression flange that is also, until the concrete hardens, laterally free.

Put numbers on it. Let the girder span 30 m, simply supported, at 3 m centres under a slab 250 mm thick. The wet concrete is $0.25 \times 3 \times 25 = 18.8$ kN/m, the steel its own 28,000 mm² at 78.5 kN/m³, or 2.2 kN/m, and a nominal construction load of 1 kN/m² adds 3 kN/m. Factored at 1.35 and 1.5, that is 32.8 kN/m and a mid-span moment of $32.8 \times 30^2/8 = 3{,}690$ kN·m.

The gross section’s elastic resistance is 4,530 kN·m, and on its gross section the girder passes with a fifth to spare. Its effective section’s is 3,520, and it fails by 5% — before its compression flange’s lateral stability has been looked at, and on a load case that lasts a few weeks. The same 40 mm of flange split 22/18 would carry 5,060 kN·m, more than the gross bottom-heavy girder, and pass by a third. Nothing about the finished composite girder has changed except which flange is thicker, and the finished girder would want the steel at the bottom.

So the construction stage is the stage that decides whether a slender web is affordable on a bottom-heavy girder, and the answer depends on a calculation — the effective section — whose result the gross section gives no hint of.

In hogging, over an internal support of a continuous composite girder, the roles reverse. The slab is in tension and cracked, its reinforcement is the top flange, and the steel’s heavy bottom flange is in compression — the turned-over girder of the figure above, with nothing to lose.

What the effective section leaves out #

The calculation is the elastic one a class 4 girder is checked by, and its assumptions limit it.

No shear. A slender web near a support carries most of the girder’s shear, and a web in shear buckles in a different pattern with its own reserve; where both act together the interaction between them governs, and the hole computed here is not the whole of what the web gives up.

No lateral buckling of the compression flange. A 15 mm flange 400 wide, unrestrained between cross-frames during construction, buckles sideways long before its outstands matter, and that check usually governs the construction stage on its own.

No longitudinal stiffener. A stiffener in the compressed part of the web, at a fifth of its depth, divides it into two panels each far stockier than the whole, and closes most of the hole for a few hundred kilograms of steel. It is the usual answer, and it makes the flange split matter less.

Elastic throughout. The effective section gives the moment at which the extreme fibre yields on a section that does not exist. A girder that has buckled locally redistributes further before it fails, by an amount the elastic calculation deliberately does not count, and a stocky one yields progressively, with an axis that moves for a different reason altogether.

Still open: whether the flange split can be left to the stiffener #

A longitudinal stiffener closes the hole in the web whichever way the flange steel is shared, and once it is there the gross section’s preference for equal flanges returns. A girder designed for the finished composite state is bottom-heavy, a girder designed for the construction stage would be top-heavy, and a stiffener lets one section serve both. Whether the stiffener’s steel, its welding and its own buckling checks cost less than the few millimetres of flange that would have made it unnecessary — on a girder whose construction moment is a third of its service moment, and whose web is slender enough that the hole is real — is the question of whether a section should be drawn for the stage that governs its weight or the stage that governs its stability.