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 10 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 unfoldsOne 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.as builtthe same fold under axial pull, exaggeratedE along the girder = 222 N/mm², against 210 GPa for the plate itselfso the web takes 0.106% of the bending stress and all of the shear
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.

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 anyA 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.the fold, seen from aboved = 1230 mm between flange centroids203 N/mm²a flat web would take a share9% of the second momentM ÷ d = 2439 kN in each flange, with no second moment in it anywhere
Fig. 2 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.

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 happensThe 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.00.20.40.60.810%20%40%60%80%100%shear, as a fraction of the web's capacitymoment capacity leftthe sectionthe web alone√(1 − v²)98.3%92.8%web 12.8% of the plastic modulus · M_pl 6007 kNm · V_pl 1476 kN
Fig. 3 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 shearShear 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.24681012050100150200250300web thickness (mm)shear buckling stress (N/mm²)localglobalinteractionyield82 N/mm² at 6 mmapplied 69 N/mm² · utilisation 0.85
Fig. 4 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.

A 6 mm plate, and the width it can beThe elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 321 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 420 mm only 64 per cent of it is still working.1002003004000200400600plate width (mm)slender beyond 321 mmyieldcritical stress — inverse square in the widthwhat the plate actually delivers, over its full width
Fig. 5 The plate-buckling problem the local mode is, against the panel width rather than the web depth. This is 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.

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.

Shear stress across a sectionThe 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 0.39 against a mean of 0.20 — a ratio of 1.97 — and it falls at the neutral axis, where the bending stress is zero.neutral axispeak 0.4stressflow, q = VQ ÷ Imean stress 0.20 — the value a shear divided by an area would givepeak 1.97× that, and in the place bending ignoresthe flow is continuous; the stress jumps wherever the width does
Fig. 6 The flow that has to leave the web and enter the flange. On a flat girder that transfer happens along a straight line and generates nothing; 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 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 length at which a beam stops being a beamElastic critical moment against the distance between lateral restraints, with the section's plastic capacity drawn across it. The two cross at 3990 — beyond that length the beam buckles sideways before it reaches the strength its cross-section has, and the capacity is set by the restraints rather than by the steel.20004000600080001000012000140001600002004006008001000120014001600distance between lateral restraintsthey cross at 3990the plastic capacity of the sectionelastic critical momentSt Venant torsion alone — what is left at long lengthswarping dominates here
Fig. 7 The curve the flange is checked against. For a corrugated-web girder the web’s contribution to the term that resists the flange’s sideways movement is a fraction of a flat web’s, so the same girder sits further to the right on this axis than its dimensions suggest.

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 notStress 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.prestress alone-8.0523.29tensionat transfer-2.8018.04tensionin service10.811.38with no prestress17.25-17.25tension
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.

One slit, and the torsional stiffness falls by a factor of hundredsA 250 by 250 box of 10 mm wall, drawn closed and then slit along its length. Closed, the torque runs round the wall as a shear flow and the torsion constant is 1.38×10⁸ mm⁴; slit, the loop is broken and only each wall's own thickness resists, giving 3.20×10⁵ mm⁴. The ratio is 432 to one, so the same torque twists the slit section 432 times as far and raises a peak shear stress 36 times as high. Nothing about the material changed.closedJ = 1.38×10⁸ mm⁴twist 0.154° over 5.0 mpeak shear stress 5.2 N/mm²slit along its lengthJ = 3.20×10⁵ mm⁴twist 66.315° over 5.0 mpeak shear stress 187.5 N/mm²J closed ÷ J open = 432
Fig. 9 The comparison the fold sits between. 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 round a loop.

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.

The same material, three waysThree cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.09 × 10⁶1.0× the firsttall rectangleI = 23.66 × 10⁶259.1× the firstI-sectionI = 47.86 × 10⁶524.1× the firstevery section here has an area of 4200 — only the shape differsthe bar is the second moment of area, to scale
Fig. 10 Three sections of the same area, for the comparison the corrugated girder is really making. It is an I-section whose web has been removed from the bending calculation and kept for the shear — which is a section with slightly less second moment than the I, and considerably less fabrication.

Where this goes next

flat: I = 6750 mm⁴folded: I = 67.50×10⁶ mm⁴300gain 10000× = (100)²limited by buckling of the leg: the leg goes at 27 N/mm² 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. neutral axispeak 0.4stressflow, q = VQ ÷ Imean stress 0.19 — the value a shear divided by an area would givepeak 2.15× that, and in the place bending ignoresthe flow is continuous; the stress jumps wherever the width does 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. 1002003004005006007000200400600plate width (mm)slender beyond 370 mmyieldcritical stress — inverse square in the widthwhat the plate actually delivers, over its full width 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. stiffeners at 1000 mmthe band at 22.5°the truss it has becomestiffener in compression,web in tensionbuckles at 383 kN · carries 696 kN · a stocky web would reach 953 kNσ in the band 252 N/mm² over 541 mm 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.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

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