Structural Engineering & Statics
where the load goes
Sections and stress

The web that carries no bending

A corrugated web needs no stiffeners, because the folds give it in one direction a depth it does not have in its thickness. In the other direction the same folds make it an accordion — and a web that cannot be stretched cannot carry a bending stress at all.

Assumes The material far from the middle does nearly all the work, The shear nobody draws and The plate that ripples, and the width that is left.

19 min read 8 figures Geometry beats materialThe load must go somewhereWhich failure arrives first

A plate girder’s web is the part of it that costs the most to make. It is thin, it buckles in shear, and preventing that takes transverse stiffeners every metre or so — each of them a piece of plate, four fillet welds and a fitting operation, on a member whose flanges are two flat bars.

Fold the web instead. Run it up and down the girder in a shallow trapezoidal wave and the stiffeners are not needed at all: across the fold the web now has a second moment it never had in its thickness, and it holds its own shape.

The price is on the other axis, and it is not small.

Pull it along the girder and it just unfolds. One period of a 30° trapezoidal corrugation, 300 mm of flat and 260 mm of incline, and the same period pulled along the girder's axis. The fold opens by bending the inclined panels out of the web's own plane, so the axial flexibility contains the plate's t³ where a flat web's would contain t — and the effective modulus that comes back from solving the cell as a frame is 222 N/mm², which is 10.6 parts in ten thousand of the steel's 210 GPa. A web with a thousandth of the stiffness carries a thousandth of the stress, which is why the flanges of a corrugated girder carry the whole moment and why the section has 9 per cent less second moment than the flat-webbed girder it replaces. The fold buys freedom from stiffeners and pays for it here.
Fig. 1 One period of a 30° trapezoidal corrugation, and the same period pulled along the girder’s axis. The fold opens by bending the inclined panels out of the web’s own plane, so the axial flexibility contains the plate’s t3t^3 where a flat web’s would contain tt. The effective modulus that comes back is 222 N/mm², against 210,000 for the steel it is made of.

Which free body produced the number

The number above is solved rather than asserted, and the free body is one period of the corrugation.

Take a strip of the web one millimetre high — a folded strip, running along the girder. Its shape repeats, and the middle of each flat panel is a mirror plane of that shape, so a half-period between two such planes is a complete free body with the rotation fixed at each end and the offset free. Model it as three plane-frame members — half a flat panel, an inclined panel, half a flat panel — pull it along the girder with a force NN, and read the extension.

What resists that pull is not the plates’ axial stiffness. It is their bending: the inclined panel has to swing, and swinging it means bending it out of the web’s plane at a rigidity of Et3/12(1−ν2)Et^3/12(1-\nu^2) per unit width. So the flexibility contains t3t^3, the stress contains tt, and

Eeff∝t2E_{\text{eff}} \propto t^2

Doubling a 4 mm web to 8 mm takes the effective modulus from 99 to 395 N/mm², which is a factor of exactly four. That is a claim about the family rather than about any one drawing, and it is the cleanest evidence that the mechanism is bending rather than anything else.

Pull it along the girder and it just unfolds. One period of a 30° trapezoidal corrugation, 300 mm of flat and 260 mm of incline, and the same period pulled along the girder's axis. The fold opens by bending the inclined panels out of the web's own plane, so the axial flexibility contains the plate's t³ where a flat web's would contain t — and the effective modulus that comes back from solving the cell as a frame is 395 N/mm², which is 18.8 parts in ten thousand of the steel's 210 GPa. A web with a thousandth of the stiffness carries a thousandth of the stress, which is why the flanges of a corrugated girder carry the whole moment and why the section has 11 per cent less second moment than the flat-webbed girder it replaces. The fold buys freedom from stiffeners and pays for it here.
Fig. 2 The same cell at 8 mm rather than 6. The effective modulus comes back as 395 N/mm² — 18.8 parts in ten thousand of the steel’s 210 GPa, against 10.6 at 6 mm and 4.7 at 4. Thickening the plate makes the accordion stiffer as the square, and it is still three orders of magnitude away from being a web. What it does change is the price: the flat web this one replaces would have contributed more at 8 mm than at 6, so the section gives up 11 per cent of its second moment rather than 8.7.

That last number is the one to carry forward. Thickening a corrugated web makes the bending loss worse, not better, because the loss is measured against the flat web that was not built — and the flat web gets better with thickness at a rate the fold cannot follow.

What it costs, in the one quantity nobody was watching

A web with a thousandth of the steel’s stiffness along the girder carries a thousandth of the bending stress. In practice that is zero, and the usual phrasing — that the web’s contribution to the second moment “may be neglected” — makes it sound like a modelling convenience.

It is not a convenience. It is a loss.

The flanges carry all of it, because the web cannot carry any. A corrugated web girder's stress block beside the one a flat web of the same thickness would produce. Along the girder the fold behaves as an accordion — its effective modulus is 10.6 parts in ten thousand of the steel's — so the web takes no bending stress and the flange force is exactly M ÷ d, 2439 kN over a lever arm of 1230 mm. The flange stress is 203 N/mm² against 190 for the flat-webbed girder, because the flat web contributes 9 per cent of that section's second moment and this one contributes 9 per cent less overall. The accordion is usually sold as a benefit; here is its price.
Fig. 3 The stress block of the corrugated girder against the one a flat web of the same thickness would produce. The corrugated section’s whole moment goes into a couple between the flange centroids — 2,439 kN over a lever arm of 1,230 mm — and the dashed line is the linear distribution a flat web would have shared in. That web would have contributed 8.7 per cent of the second moment, and this one contributes nothing.

The flange force is then M/dM/d exactly, with no second moment in it anywhere. That is the same arithmetic as a bolt group’s lever arm and a truss chord’s: a moment divided by a distance is a force. What is unusual is that it is exact here rather than approximate, because the thing that would have made it approximate has been folded out of existence.

The size of what has been given up depends on the girder rather than on the fold, and it grows with exactly the dimension a designer reaches for first.

The flanges carry all of it, because the web cannot carry any. A corrugated web girder's stress block beside the one a flat web of the same thickness would produce. Along the girder the fold behaves as an accordion — its effective modulus is 10.6 parts in ten thousand of the steel's — so the web takes no bending stress and the flange force is exactly M ÷ d, 1478 kN over a lever arm of 2030 mm. The flange stress is 123 N/mm² against 108 for the flat-webbed girder, because the flat web contributes 14 per cent of that section's second moment and this one contributes 14 per cent less overall. The accordion is usually sold as a benefit; here is its price.
Fig. 4 The same corrugation and the same flanges on a 2.0 m web instead of a 1.2 m one. The lever arm rises to 2,030 mm and the flange force falls to 1,478 kN, so the girder is stronger in the ordinary way. But the flat web it is being compared against would now have carried 14 per cent of the second moment rather than 8.7, so the section gives up 14 per cent overall — the deeper the girder, the more the fold costs.

That is worth stating as a rule because it runs against the instinct. A deep web is where a flat plate earns most of its keep, since its contribution to the second moment goes as the cube of its depth while the flanges’ goes as the square; so the deeper the girder, the more of it a corrugation is declining to use. The form is at its most efficient on shallow girders and its most extravagant on deep ones, which is the reverse of where a slender web needs stiffeners most.

There is a second consequence, and it is the reason the arithmetic is worth doing rather than quoting. The two checks stop interacting. A flat web carries part of the moment and all of the shear from the same fibres, so a section under high moment and high shear together has to be checked for both at once — which is a real interaction and a real reduction. A corrugated web carries all of the shear and none of the moment, so there is nothing to interact: the flanges are checked for MM, the web for VV, and the two never meet.

Nothing happens, and then everything happens. The moment capacity left to a section already carrying shear, against the shear as a fraction of what the web can take. The web holds 12.8% of this section's plastic modulus and the flanges hold the rest, and only the web's share is reduced — by the factor √(1 − v²) that von Mises leaves it. So the curve is flat for most of its length: the first per cent of moment is not lost until v = 0.39, half the shear capacity costs 1.7%, and 7% is not reached until v = 0.9. The tangent at v = 1 is vertical, which is why the last tenth of the shear range costs more than the first eight.
Fig. 5 The interaction the fold removes. For a flat web the moment and the shear are carried by the same material and the curve is a real reduction near the support. The corrugated girder sits at the corner of this diagram permanently — full moment capacity in the flanges, full shear capacity in the web, and no term connecting them.

Three ways for a folded web to buckle in shear

Removing the stiffeners does not remove the shear buckling; it changes which shear buckling.

Local buckling is a single flat panel rippling between its own folds. The panel is 300 mm wide and the fold lines are effectively rigid edges, so it is an ordinary plate buckling problem against the panel width rather than against the web depth — which is the whole reason the stiffeners are not needed. It gives 424 N/mm² here.

Global buckling is the whole web going as one plate. It is not an isotropic plate: the fold gives it a large second moment across the corrugation and a small one along it, so it is orthotropic, and both rigidities come from the profile. DxD_x is E∫z2t dsE\int z^2 t\,ds over a period, divided by the period — 3.92×1093.92\times10^9 N·mm here — and DyD_y is the plate’s own t3t^3 term scaled by the extra developed length the fold carries, 3.90×1063.90\times10^6, a thousand times smaller. Global buckling gives 82 N/mm².

And they interact. The two modes are not independent, and combining them by a reciprocal power law gives 82 as well — the interaction is doing almost nothing here, because the two criticals are a factor of five apart.

Three ways for a folded web to buckle in shear. Shear buckling stress against web thickness for the 1.2 m web drawn. Local buckling is a single flat panel between its folds, at the ordinary plate coefficient against the 300 mm panel. Global buckling is the whole web as an orthotropic plate, stiff across the corrugation because the fold gives it a second moment and soft along it, and both rigidities are computed from the profile rather than quoted. The interaction curve lies under both. At the 6 mm drawn, local is 424 N/mm², global 82 and the interaction 82, so interactive governs — and yield at 205 N/mm² is the ceiling none of them may cross.
Fig. 6 The three stresses against web thickness. Local buckling rises as the square of the thickness; global rises as its three quarters, because the orthotropic combination is Dx1/4Dy3/4D_x^{1/4}D_y^{3/4} and only DyD_y contains t3t^3. So thickening the plate closes the gap between the modes far more slowly than it raises either — and on a web where global governs, most of the money spent on thickness is spent on the mode that was never deciding.

The design consequence follows directly from the exponents. On this web the applied shear stress is 69 N/mm² against a critical stress of 82, and the utilisation is 0.85. Deepening the corrugation from 130 mm raises DxD_x as the square of the depth and moves global buckling up; thickening the plate raises it as t3/4t^{3/4}. The fold is the cheap variable and the plate is the expensive one, which is the opposite of the flat-web girder the section replaces.

Three ways for a folded web to buckle in shear. Shear buckling stress against web thickness for the 1.2 m web drawn. Local buckling is a single flat panel between its folds, at the ordinary plate coefficient against the 300 mm panel. Global buckling is the whole web as an orthotropic plate, stiff across the corrugation because the fold gives it a second moment and soft along it, and both rigidities are computed from the profile rather than quoted. The interaction curve lies under both. At the 6 mm drawn, local is 424 N/mm², global 94 and the interaction 94, so interactive governs — and yield at 205 N/mm² is the ceiling none of them may cross.
Fig. 7 The same sweep with the corrugation angle at 45° rather than 30°, which deepens the fold from 130 mm to 184 without adding a millimetre of plate. Local buckling does not move — the flat panel between the folds is still 300 mm wide — and global buckling rises from 82 N/mm² to 94, with the interaction following it. The fold angle moves the mode that governs and leaves the mode that does not exactly where it was.

The local mode is the ordinary plate-buckling problem with the panel width in place of the web depth, which is the whole of what the fold buys: a slender web whose plates are all short, so the coefficient is computed against 300 mm and not against 1,200. It is also why local buckling is so rarely the answer here — 424 N/mm² is above the yield stress the web can reach, so the mode that the stiffeners were removed to control has been removed from the problem along with them.

The flange picks up a moment the analysis never had

The shear the web hands to the flange is delivered along the fold line, and the fold line wanders. Over half a corrugation period it moves from one side of the flange centreline to the other, by the corrugation depth.

A shear flow of V/hwV/h_w delivered along a line that is offset by z(x)z(x) generates a transverse moment in the flange — about the flange’s own vertical axis — equal to the integral of the flow times the offset. On the girder drawn that comes to 11 kNm, which puts 14 N/mm² into a flange already carrying 203 from the girder’s bending.

It is a small number here and it is not always. It goes up with the corrugation depth, which is the variable the shear buckling wants increased, and it goes up with the shear, which is largest exactly where the fold is doing the most work. A designer who buys shear capacity by deepening the fold is also buying transverse flange bending, and the two curves are not usually drawn on the same page.

What makes this one hard to see coming is that nothing about it is a shear problem. The flow that has to leave the web and enter the flange is the same quantity on both girders and it is checked the same way; on a flat girder that transfer happens along a straight line and generates nothing further, and on a corrugated one the line moves from side to side of the flange and the flange bends about its weak axis to carry the difference. The check that catches it is a flange check, and the quantity that drives it is a web dimension.

The flange has lost its restraint as well

A plate girder’s compression flange is held against buckling sideways partly by the web, which is a plate standing on edge and offering the flange a rotational restraint along its whole length. That restraint is what makes the lateral-torsional buckling of a plate girder a problem about the span between braces rather than a problem about the flange alone.

A corrugated web offers the same restraint out of its own plane — where it is stiff, because the fold gives it depth — and almost none in the direction the flange wants to move, which is sideways. The compression flange of a corrugated-web girder is closer to a free plate on an elastic foundation than to the flange of an I-section, and the foundation is soft.

In practice this is handled by the same thing that handles it for a plate girder — bracing at intervals — and the interval is shorter. What is worth noticing is the pattern, because it is the third appearance of the same sentence: the fold has traded stiffness along the girder for stiffness across it, and every quantity that depended on the first has moved.

The curve the flange is checked against does not change shape for a corrugated girder; the girder simply sits further along it than its dimensions suggest, because the web’s contribution to the term resisting the flange’s sideways movement is a fraction of a flat web’s. Reading a corrugated-web girder off the same curve at the same unbraced length is therefore optimistic, and by an amount nothing in the section properties announces.

What the fold costs in steel, which is almost nothing

A corrugated web has more developed length than projected length — 1,120 mm of plate for every 1,050 mm of girder here — so it uses 6.6 per cent more steel than a flat web of the same thickness.

Against that: the stiffeners are gone, and a stiffener is not a plate, it is a fitting operation. The comparison that matters is not steel against steel but plate against labour, which is why corrugated-web girders appear in exactly two places — cold-formed building beams where the web is a millimetre or two thick and would need stiffeners everywhere, and post-tensioned bridge girders where the web is a steel plate between concrete flanges and its axial softness is a feature, because a web that carries no longitudinal force also transmits none of the prestress into the flanges it is supposed to go into.

That second case is the interesting one. Prestressing a concrete section puts a compressive force into it and expects the whole section to take it; a corrugated steel web takes none, so all of the prestress reaches the flanges where it is wanted. The accordion effect, which is a defect from one direction, is the entire reason the form exists from the other.

Two triangles that cross zero, and a block that does not. Stress across a 300 × 700 mm section at each stage, compression positive. The prestress alone gives -8.05 MPa at the top and 23.29 at the bottom; at transfer, with only self-weight on it, the top is at -2.80 MPa and in service the section runs from 10.81 to 1.38 MPa — compression everywhere. The same beam with no prestress reaches -17.25 MPa at the bottom fibre, which is 5.8 times what the concrete can hold.
Fig. 8 The section the corrugated web is used with. Prestress applied to a concrete box with steel webs goes where the designer wants it, because the webs will not take any: a stiffness of a thousandth is an efficient way of declining a force.

And the fold makes it stiff in torsion, which nothing asked for

The corrugation has one more consequence and it points the other way from everything above.

A plate girder’s torsional stiffness is its St Venant constant, which for three thin rectangles is almost nothing — the reason an open section is hundreds of times softer in torsion than a closed one. Corrugating the web does not close the section, and it does raise the resistance substantially: the fold gives the web a depth out of its own plane, so a twist of the girder has to bend the corrugations rather than merely shear a flat plate.

Nobody buys a corrugated web for that, and it arrives anyway. Where it shows up is in the lateral stability of the girder during erection, when the section is unbraced and its torsional stiffness is most of what stops it from rolling over. A corrugated-web girder is measurably more stable on its own than the plate girder it replaces, at exactly the moment in its life when it is least looked after — which is the sort of accidental benefit that never appears in a design calculation and occasionally decides whether one is needed.

The fold sits between the two ends of that comparison rather than at either. A closed cell and a slit one differ by a factor in the hundreds; a corrugated web is neither, and its resistance to twist comes from bending its own folds rather than from a shear flow running round a loop — which is the same mechanism the accordion is, read about a different axis and arriving as a benefit instead of a cost.

The accordion is why the prestressed version exists

The largest single use of a corrugated web is not a steel plate girder at all. It is a bridge with concrete flanges and a corrugated steel web — a form that exists precisely because of the property this essay has been treating as a curiosity.

Prestress a concrete flange and it shortens: immediately by its elastic strain, and over years by creep and shrinkage, to a total of five or six hundred microstrain. If the web joining the two flanges is a flat steel plate, it refuses. Its axial stiffness is EsAwE_sA_w, it is welded to both flanges along their whole length, and it restrains the shortening in exactly the way a restrained deformation always generates a force.

Put numbers to it. A 1,200 by 6 web has Aw=7,200A_w = 7{,}200 mm², so EsAw=1.5×109E_sA_w = 1.5\times10^9 N, and 600 microstrain of restrained shortening is a force of

600×10−6×1.5×109=900 kN600\times10^{-6}\times1.5\times10^9 = 900\ \text{kN}

— nine hundred kilonewtons of restraint, taken out of the prestress the flanges were supposed to keep, delivered into the concrete as tension, and crossing the web-to-flange weld along its whole length.

A corrugated web supplies none of it. Its longitudinal stiffness is the accordion’s, which this page has measured at a few per cent of a flat plate’s, so the flange shortens and the fold simply closes a little. The prestress goes where it was put.

That is the whole design argument for the form, and it inverts the way the accordion is usually introduced. A corrugated web is not a plate girder whose web has been made useless in bending; it is a member deliberately arranged so that its two flanges can move longitudinally without arguing, with a web that carries shear and refuses everything else.

The fold cannot take a stiffener

There is a practical consequence at the supports that decides how these girders are detailed.

A bearing stiffener on a flat web is a plate welded across it, from flange to flange, on a straight line. On a corrugated web there is no straight line to weld it to: the fold means the web’s surface is not planar in any transverse section, so a flat stiffener meets it along a zigzag.

The standard answer is to insert a flat panel at each support and at any other point that needs a stiffener — a load introduction, a cross-beam, a splice — and to make the transition from corrugated to flat within a fold or two. So a corrugated-web girder is not corrugated everywhere; it is corrugated between the places where something has to be attached to it.

Those flat panels are then ordinary plate-girder web panels, with ordinary shear buckling, ordinary stiffener requirements and ordinary detailing — and they are also where the girder is most heavily loaded. The form’s difficulty is concentrated in the parts of it that are not the form.

Where the model stops

The corrugation cell is solved as a plane frame. That treats the web as a folded strip of unit height and ignores what happens at the top and bottom of it, where the fold meets the flange and is restrained. Near the flanges the accordion is stiffer than the model says, and the effective modulus is therefore a lower bound rather than an estimate. It is a very low lower bound, which is why the conclusion survives.

The orthotropic buckling coefficient is quoted rather than derived. The 32.4 in the global expression is the orthotropic plate result for simply supported edges; the rigidities inside it are computed here from the profile, but that coefficient is not, and a web restrained by heavy flanges is somewhere between simply supported and fixed.

The interaction is a power law with an exponent somebody chose. Local and global buckling of a corrugated web genuinely interact and the reciprocal-cube combination is the standard way of saying so; it is calibrated, not derived, and on this web it makes a difference of a quarter of a per cent because the two criticals are far apart. On a web where they are close it would make several per cent, and the exponent would be carrying the answer.

And nothing here is about fatigue. The fold line is a weld between a plate that is trying to bend and a flange that is trying not to let it, at a detail that repeats every few hundred millimetres along a girder carrying moving loads. That is a fatigue question rather than a strength one, and it is what actually limits the form on bridges.

What the pictures cannot show

The corrugation in the first figure is drawn at three cycles across the page. A real girder has a hundred, and the amplitude is 130 mm on a web 1,200 mm deep — so the fold that dominates every argument here is, at the scale a whole girder would be drawn at, a texture rather than a shape.

Nor can the stress block show the one thing a fabricator would ask about first, which is how the web is made. A trapezoidal corrugation is press-braked or roll-formed in a continuous line, and the geometry in the options above is not a free choice — it is whatever the machine that made the plate produces. The design variable is which supplier, and the sweeps are drawn over a range nobody can actually move along.

The assumption the figure rests on

Every number here assumes the fold lines stay straight and the corners stay sharp.

They do not. A press-braked corner has a radius of several thicknesses, which softens the accordion — making the effective modulus rather higher than 222 N/mm² and the web’s share of the moment rather more than nothing — and it rounds the fold line that the local buckling coefficient treats as a rigid edge, which lowers the local critical stress. Both effects are small on a web six millimetres thick with a 130 mm fold. Neither is small on a cold-formed web a millimetre thick with a 20 mm one, which is the other place this section is used, and where the corner radius is a substantial fraction of the panel.

Set against the ordinary comparison of sections of the same area, the corrugated girder is making a claim none of those four make. It is an I-section whose web has been removed from the bending calculation and kept for the shear — a section with slightly less second moment than the I-section it is drawn from, and with considerably less fabrication in it. Every argument on this page is about what that trade costs; the reason the trade is made at all is that the cost is paid in a quantity a mill sells by the tonne and the saving is made in one that a fabricator sells by the hour.

Where this goes next

The same sheet, twice, and a factor of ten thousand. A 3000 mm developed width of 3 mm sheet, covering 2400 mm in plan — so the legs sit at 36.9° and the fold is 300 mm deep. Flat, its second moment about its own mid-plane is 6750 mm⁴, which spans nothing. Folded, it is 67.50×10⁶ — 10000 times as much, which is exactly the depth in thicknesses squared. The material is identical, the plan cover has fallen by 20%, and the only thing that changed is where the material sits. What limits it is buckling of the leg: at this leg length the flat between the folds goes at 27 N/mm², well below the steel's 275. Folded until it spans A flat sheet has a second moment of area of B·t³/12 and will not span anything. Folded, the same material has B·t·h²/12, and the gain is exactly the fold depth over the thickness, squared — a ratio with no material in it and no width in it. Shear stress across a section. The distribution of shear stress over an I-section, computed as VQ/It by accumulating the first moment of the area above every height. The peak is 0.41 against a mean of 0.19 — a ratio of 2.15 — and it falls at the neutral axis, where the bending stress is zero. The shear nobody draws A stack of loose planks slides at its ends when it is loaded. Glue them and the sliding stops — and whatever the glue is now carrying is a stress that no bending calculation contains. A 8 mm plate, and the width it can be. The elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 370 mm the plate reaches yield before it buckles; above it the plate ripples first, and the fraction of the width still carrying load falls away — at 700 mm only 47 per cent of it is still working. The plate that ripples, and the width that is left A wide thin plate in compression buckles at a stress that has nothing to do with the strength of the material. It then goes on carrying load — the middle drops out, and the edges work harder. A buckled panel is a truss that nobody drew. A 1000 × 1000 panel of 6 mm web, at d/t = 167. It buckles in shear at 63.8 N/mm², which is 383 kN — and it then carries 696 kN, 1.82 times as much, because the tension diagonal takes over from the compression one that has gone. The band runs at 22.5° with a membrane stress of 252 N/mm² over a width of 541 mm, and it pulls on the flange at 221.3 N per millimetre of its length. A web that never buckled at all would have reached 953 kN, so the panel ends at 73% of a stocky web's capacity on a fraction of its steel. The panel that carries more after it has failed Everywhere else in this field a critical load is where the argument ends. A thin web is the exception — it buckles visibly, in waves anybody can see, and then goes on to carry nearly twice as much again by turning itself into a truss nobody drew.

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Accordion effectCorrugated webFabricationFlangeLever armLocal bucklingOrthotropic platePlate bucklingPlate girderSecond moment of areaShear bucklingShear flowStiffenerTransverse bendingWeb