Structural Engineering & Statics
where the load goes
Stability

The tension has to pull on something

A buckled web carries its shear on a diagonal band of membrane tension, and the band pulls sideways on the flanges and stiffeners that bound it. That pull is the design output nobody plots — it runs from 72 to 603 newtons per millimetre across ordinary panel proportions, it is largest exactly where the panel is most efficient, and at the end of the girder there is nothing beyond to take it.

Assumes The panel that carries more after it has failed, The web that carries no bending and The rib that is a boundary condition.

15 min read 7 figures The load must go somewhereWhich failure arrives firstGeometry beats material

A buckled panel is a truss nobody drew: the compression diagonal has gone, the tension diagonal takes over, and the web carries more than twice what it visibly failed at.

The hero is that panel — 1,500 mm deep, 2,000 mm between stiffeners, 8 mm thick. It buckles at 492 kN and finishes at 1,187, a reserve of 2.41. The band runs at 18.5° with a membrane stress of 348 N/mm² over a width of 788 mm.

And it pulls on the flange at 280.2 newtons per millimetre. That is the number this rung is about, because a diagonal tension is a force, and a force has to be resisted by something that was drawn.

The band’s boundary is four members

The tension band is a strip of membrane. It has no bending stiffness worth the name and it cannot end in mid-air, so at each of its ends it is anchored by whatever the panel is bounded by.

The two flanges take the vertical component of the pull along their length, as a distributed transverse load. A flange sized for the bending in the girder is now also a beam spanning between stiffeners under 280 N/mm.

The two stiffeners take the horizontal component, as a distributed load along their length that puts them into compression and bending. A stiffener drawn as a plate to stop the web buckling is now a strut with a transverse load on it.

And every one of those four is shared with the panel next door, which is the reason the interior of a girder works at all: the horizontal pull from the panel on the left is balanced by the horizontal pull from the panel on the right, and the stiffener between them carries only the difference.

That balance is the whole of the anchorage problem, and it fails at exactly one place — the end.

A buckled panel is a truss that nobody drew. A 1500 × 800 panel of 8 mm web, at d/t = 188. It buckles in shear at 122.9 N/mm², which is 1475 kN — and it then carries 2498 kN, 1.69 times as much, because the tension diagonal takes over from the compression one that has gone. The band runs at 31.0° with a membrane stress of 284 N/mm² over a width of 874 mm, and it pulls on the flange at 602.8 N per millimetre of its length. A web that never buckled at all would have reached 2460 kN, so the panel ends at 102% of a stocky web's capacity on a fraction of its steel.
Fig. 1 The same web with the stiffeners brought in to 800 mm. The panel buckles at 1,475 kN and carries 2,498 — and the pull on the flange is 602.8 N per millimetre, more than twice the hero’s. The band is steeper at 31.0°, its membrane stress is lower at 284 N/mm², and the panel now finishes at 102 per cent of what a web that never buckled would have reached.
A buckled panel is a truss that nobody drew. A 1500 × 4500 panel of 8 mm web, at d/t = 188. It buckles in shear at 31.2 N/mm², which is 375 kN — and it then carries 716 kN, 1.91 times as much, because the tension diagonal takes over from the compression one that has gone. The band runs at 9.3° with a membrane stress of 351 N/mm² over a width of 757 mm, and it pulls on the flange at 72.5 N per millimetre of its length. A web that never buckled at all would have reached 2460 kN, so the panel ends at 29% of a stocky web's capacity on a fraction of its steel.
Fig. 2 And with the stiffeners taken out to 4,500 mm. The panel buckles at 375 kN and carries 716 — 29 per cent of the stocky web’s capacity — and the flange pull is 72.5 N per millimetre. The band has flattened to 9.3°, which is why so little of its force is vertical.

Across those three panels the capacity rises by 3.5 times and the flange pull by 8.3. The anchorage demand grows faster than the thing it is buying, which is the single most useful sentence about tension field design and appears in no code as a sentence.

The reason is in the angle. A short panel bands steeply, so more of the membrane force is vertical — good for the shear — and the vertical component is precisely what the flange has to hold. A long panel bands flat, contributes little shear, and asks little of its flanges.

Which free body produced the pull

The 280.2 N/mm is a distributed force and it comes out of one cut, so it is worth doing.

Take the band as a strip of web of width s=dcos⁡θ−asin⁡θs = d\cos\theta - a\sin\theta, carrying a uniform membrane stress σt\sigma_t along its own direction. Its total force is σtts\sigma_t t s — for the hero, 348 N/mm² times 8 mm times 788 mm, which is 2,194 kN acting at 18.5° to the horizontal.

Now cut along the flange. The band meets it over a length of flange equal to aa minus whatever the band’s end geometry takes off, and the vertical component of the band’s force, 2194sin⁡18.5°=6962194 \sin 18.5° = 696 kN, has to cross that cut. Spread over the panel’s 2,000 mm that is 348 N per millimetre, and the model’s own bookkeeping about where the band actually lands brings it to 280.

The horizontal component is much larger: 2194cos⁡18.5°=20812194 \cos 18.5° = 2081 kN. It goes into the stiffeners, and it is the number the end post has to deal with.

Two features of that free body decide the rest of the essay.

The pull is distributed, not a point load. So the flange is a beam under a uniformly distributed transverse load spanning between stiffeners, which is why the stiffener spacing appears squared in the flange’s own moment while it appears only linearly in the panel’s capacity.

And the horizontal component is between three and seven times the vertical one across the panel proportions on this page, because the band is flat. Every discussion of tension field anchorage that concentrates on the flange is discussing the smaller of the two forces; the larger one is being quietly passed along the girder, panel to panel, to the ends.

The end panel, where there is nothing beyond

At the last stiffener before the bearing, the balance breaks. The panel inside pulls horizontally on that stiffener and there is no panel outside to pull back.

So the end stiffener has to take the whole horizontal component of the end panel’s band, unbalanced, and deliver it to the bearing. That is a substantial force — for the hero panel, the horizontal component over 1,500 mm of depth is of the order of several hundred kilonewtons — applied to a plate that was drawn to be a web stiffener.

Practice deals with it two ways and both are worth naming.

Design the end panel without tension field action. Take its capacity as VcrV_{cr} alone — 492 kN rather than 1,187 for the hero — and make the panel short enough that this is enough. The girder’s end panel is therefore the one with the closest stiffeners and the lowest utilisation, which looks like over-design and is a load path.

Or provide an end post designed as a beam. Two stiffeners at the end with the web between them, spanning vertically between the flanges, resisting the unbalanced pull in bending. That is a real member with a real check, and it is the physical anchor for every panel in the girder — because the horizontal balance passes from panel to panel along the whole length and terminates there.

The second reading is the one worth carrying. The interior stiffeners are not anchoring anything; they are passing a horizontal force along. The girder’s whole tension field is anchored at its two ends, and the members that do it are the ones a drawing calls end posts.

Why the stiffener is not the anchor it looks like

A stiffener between two panels looks like the obvious anchor for both of them, and it is not anchoring anything.

Consider an interior stiffener with a panel each side. The left panel’s band pulls it left with some horizontal force; the right panel’s band pulls it right. If the two panels carry the same shear the two pulls are equal and the stiffener carries nothing horizontally at all — it takes only the vertical components that reach it and its own job of holding a nodal line in the web.

That is why interior stiffeners are sized by a stiffness rule rather than by a force. The rule asks that the stiffener be rigid enough to force a nodal line, which is a boundary condition rather than a member, and the force it carries is a second-order quantity arising from the difference between two adjacent panels.

The difference is not zero in a real girder, and it is worth knowing where it is largest. The shear in a simply supported girder falls from the support towards mid-span, so adjacent panels near the support differ most, and the stiffener there carries the largest unbalanced pull of any interior one. The stiffeners nearest the ends are working hardest on both counts — the largest shear and the largest difference — which is another reason end regions get closer spacing.

At the very end there is no difference to take: there is one panel and nothing beyond it, and the whole horizontal component arrives at one plate. The transition from “carries the difference” to “carries the whole thing” happens in one stiffener spacing, which is as sharp a discontinuity as this subject contains and is why the end panel is treated as a separate design.

Thinner is not cheaper

A buckled panel is a truss that nobody drew. A 1500 × 2000 panel of 4 mm web, at d/t = 375. It buckles in shear at 10.2 N/mm², which is 61 kN — and it then carries 416 kN, 6.77 times as much, because the tension diagonal takes over from the compression one that has gone. The band runs at 18.5° with a membrane stress of 355 N/mm² over a width of 788 mm, and it pulls on the flange at 142.8 N per millimetre of its length. A web that never buckled at all would have reached 1230 kN, so the panel ends at 34% of a stocky web's capacity on a fraction of its steel.
Fig. 3 The hero’s panel at 4 mm instead of 8. It buckles at 61 kN and carries 416 — a reserve of 6.77 times, the largest number on this page — and the flange pull is 142.8 N/mm, half the hero’s. The panel’s capacity has fallen by 65 per cent and its anchorage demand by 49.

The reserve ratio is the statistic that misleads here, and it misleads because it is a ratio.

A thin web buckles at a load proportional to t2t^2 and finishes at a load very nearly proportional to tt, so the ratio between them goes as 1/t1/t and is largest for the thinnest web. That says nothing about whether the thin web is a good design.

Compare the two panels by what they deliver per unit of anchorage. The 8 mm web carries 1,187 kN for 280.2 N/mm of pull, which is 4.24 kN of shear per N/mm. The 4 mm web carries 416 kN for 142.8 N/mm, which is 2.91. The thin web is 31 per cent worse on the quantity that has to be paid for elsewhere, while looking better on the quantity that gets quoted.

What a thin web is worth before and after it buckles. A 1500 mm panel at a/d = 1.33, with the web thickness varied. The lower curve is the load at which the panel buckles, which goes as the square of the thickness; the upper one is what it carries in the end, which is very nearly linear in it because the band's own force is a stress on an area. So the reserve is largest exactly where the buckling load is smallest: at d/t = 375 the panel carries 5.5 times the load it visibly failed at, and at d/t = 75 only 1.00 times. The flat line is the shear a web that never buckled would reach, which no panel here gets to.
Fig. 4 The two loads against thickness. The lower curve — the load at which the panel buckles — goes as the square of the thickness; the upper one is very nearly linear, because the band’s force is a stress on an area. So the reserve is largest exactly where the buckling load is smallest: 5.5 times at d/t=375d/t = 375 and 1.00 at d/t=75d/t = 75. The flat line is what a web that never buckled would reach, which nothing here gets to.

Where the angle comes from

The band finds its own angle, and it is not 45°. The shear the tension band contributes, swept over every angle it could take, for five panel proportions. Each curve is the band's width d·cos θ − a·sin θ times its own vertical component, so it is zero when the band is flat and zero again when it no longer fits between the flanges. The maxima are at 25.8°, 18.5°, 13.3°, 9.3°, 7.0°, which is ½·arctan(d/a) in every case — the optimum recovered from the geometry rather than quoted. A square panel bands at 22.5°, and a panel twice as long as it is deep at 13.3°.
Fig. 5 The shear the band contributes, swept over every angle it could take, for five panel proportions. The maxima are at 25.8°, 18.5°, 13.3°, 9.3° and 7.0°, which is 12arctan⁡(d/a)\tfrac12 \arctan(d/a) in every case — recovered from the geometry by sweeping rather than quoted from a formula.

The band’s angle is not a material property and it is not chosen. It is the angle at which the product of the band’s width and its vertical component is largest, and both of those are geometry: the width is dcos⁡θ−asin⁡θd\cos\theta - a\sin\theta and the vertical component is sin⁡θ\sin\theta.

That the sweep recovers 12arctan⁡(d/a)\tfrac12\arctan(d/a) exactly is worth a moment, because it is the one place in this subject where a closed form falls out cleanly. Everything else about tension fields is a model with a name attached to it — Basler’s, Cardiff’s, the rotated stress field — and they disagree with each other by twenty per cent and more. The angle is the part they agree on.

And it is the part that decides the anchorage. A steep band pulls hard on the flanges and little on the stiffeners; a flat one does the reverse. So the panel proportion chooses which boundary member carries the anchorage, and a designer who wants to keep the flange out of it makes the panels long — at the cost of most of the capacity.

What the flange is actually being asked

A buckled panel is a truss that nobody drew. A 1500 × 1200 panel of 8 mm web, at d/t = 188. It buckles in shear at 66.6 N/mm², which is 800 kN — and it then carries 1768 kN, 2.21 times as much, because the tension diagonal takes over from the compression one that has gone. The band runs at 25.8° with a membrane stress of 336 N/mm² over a width of 830 mm, and it pulls on the flange at 506.9 N per millimetre of its length. A web that never buckled at all would have reached 2460 kN, so the panel ends at 72% of a stocky web's capacity on a fraction of its steel.
Fig. 6 An intermediate case at 1,200 mm spacing: 1,768 kN of capacity, a band at 25.8°, and 506.9 N/mm on the flange. The panel reaches 72 per cent of the stocky web’s shear on half its steel.

Take 506.9 N/mm on a flange spanning 1,200 mm between stiffeners, with the ends restrained by the stiffeners themselves. As a fixed-ended beam that is a moment of wL2/12=60.8wL^2/12 = 60.8 kN·m, in the flange, about its own weak-ish axis, on top of the axial force it is carrying from the girder’s bending.

Two things follow that decide how much of the tension field is actually available.

The flange has to be stiff enough not to pull in. If it deflects towards the web, the band loses width, and the capacity computed from the full width was never there. This is the compatibility half of the problem and it is the reason the older models limit the tension field to a band narrower than the panel diagonal.

And it has to be strong enough not to hinge. The band anchors over the length of flange between two plastic hinges, and where those hinges form is what the flange anchorage calculation finds. A flange that hinges shortens the anchored length, which narrows the band, which lowers the capacity — a redistribution the panel calculation does not contain.

A buckled panel is a truss that nobody drew. A 1500 × 2000 panel of 8 mm web, at d/t = 188. It buckles in shear at 41.0 N/mm², which is 492 kN — and it then carries 1023 kN, 2.08 times as much, because the tension diagonal takes over from the compression one that has gone. The band runs at 18.5° with a membrane stress of 266 N/mm² over a width of 788 mm, and it pulls on the flange at 214.0 N per millimetre of its length. A web that never buckled at all would have reached 1905 kN, so the panel ends at 54% of a stocky web's capacity on a fraction of its steel.
Fig. 7 And the same panel in S275 rather than S355. The buckling load is unchanged at 492 kN, because buckling is a stiffness question and the modulus has not moved; the capacity falls from 1,187 kN to 1,023, and the flange pull from 280.2 N/mm to 214.0. A weaker steel is a smaller anchorage problem.

That last figure is the cleanest demonstration that the band is a yield phenomenon and the buckling is not. The grade moves everything after the buckle and nothing before it, which is why a high-grade slender web is a worse anchorage problem than a low-grade one of the same dimensions.

What this costs against the alternatives

A tension field is a way of buying shear capacity, and it is worth pricing against the other ways.

A thicker web. The hero’s 8 mm web reaches 1,187 kN; a stocky web of the same depth reaches 2,460 and needs no anchorage at all, no intermediate stiffeners and no end post. It also weighs more than twice as much over the whole length of the girder, which is why nobody does it on a long span.

Closer stiffeners. Cheap in material and expensive in fabrication — each one is two plates, four fillet welds and a cope at each corner — and it multiplies the anchorage as this essay has measured.

A corrugated web. It does not buckle in shear at all, because the corrugation gives it an out-of-plane stiffness a flat plate does not have, so there is no post-buckling reserve to argue about and no anchorage. It also carries no bending, which changes what the flanges are for, and it is a specialised fabrication.

Or accept the buckle and design the boundary. Which is the tension field, and its real economy is that the members it loads — flanges and end posts — exist anyway.

The comparison that decides it is not a weight comparison. It is a comparison between plate and labour, and the answer moves with where the girder is being made. That is why deep plate girders in high-labour economies have fewer, thicker webs and more in low-labour ones have thin webs and many stiffeners, and why the same span gets different answers in different decades with no change in the mechanics at all.

There is a fifth option that belongs in the list because it is what long-span bridges actually do: make the web deep enough that the shear stress is low and let the panel be stocky. Depth is free in a girder’s shear check in a way it is not in a Vierendeel — the shear area is dtd t, so doubling the depth halves the stress — and a girder deep enough is one where none of this arises. What stops it is the plate slenderness the depth produces, a transport limit, and a headroom limit, in that order.

What to carry away

The band pulls on four members and they are not the web. Two flanges take its vertical component and two stiffeners its horizontal one.

Closer stiffeners buy capacity and cost anchorage faster. 4,500 mm to 800 raises the capacity 3.5 times and the flange pull 8.3.

The end panel is a different structure. Its stiffener has nothing beyond it, so either the panel is designed without a tension field or an end post is designed as a beam.

And the reserve ratio is not a design quantity. It is largest for the thinnest web, which carries least and costs most per kilonewton anchored.

Where the model stops

The band is a uniform strip at one angle. A real buckled web has a curved stress field whose angle varies across the panel and whose stress is not uniform, and every model on offer is a strip fitted to test data.

The flange’s flexibility is not in the calculation. The width dcos⁡θ−asin⁡θd\cos\theta - a\sin\theta assumes the boundary is straight and rigid; a flexible flange narrows it, and the models that account for this do so with a factor rather than a compatibility.

The stiffener is a boundary and not a member. What a stiffener actually has to be is a stiffness requirement first and a strength one second, and nothing here computes either.

And the interaction with bending is missing. Two actions on one section do not simply add, so a web carrying shear and moment together has less of both available, and the boundary between the two is a surface rather than a curve — including the flange that gets crushed from inside when the girder is curved in elevation.

The web’s boundary members carry the band’s pull, and a rib is a boundary condition before it is a member is the stiffness half of what they are being asked for.

The ladder from here

Later rungs on this anchor: Basler’s model and the rotated stress field set out side by side, with the assumptions that separate them named. The flange anchorage calculation proper, and the plastic hinges that decide the anchored length. Bearing stiffeners and end posts sized against the unbalanced horizontal pull. Shear and bending interaction in a slender web. Corrugated webs, which do not buckle in shear at all and therefore have none of this. And Wagner’s complete diagonal tension theory of 1929, written for an aircraft skin far thinner than any of these.

The mechanism was found by people who had no choice about web thickness. Aircraft structure in the 1920s used skins a fraction of a millimetre thick, they visibly buckled at a small fraction of the design load, and the aeroplanes flew — so the question was not whether to allow buckling but how to compute what happened afterwards. Civil engineering adopted the answer forty years later, cautiously, and the caution is still visible in the end panel: the one place where the profession declines to use the reserve it has spent decades justifying.

Where this goes next

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. A stiffener is a boundary condition, and it is bought at a threshold. The buckling stress of a 2400 × 12 mm plate with one longitudinal stiffener, against how rigid that stiffener is. Below γ the stiffener rides on the buckle and the plate takes the whole-width mode; at γ the stiffener stays straight and the plate buckles between stiffeners at 74 N/mm², 4.0 times the bare plate's 18.5. Above γ nothing further happens at all, because the sub-panel mode does not know the stiffener is there. The curve is a ramp and then a horizontal line, so a stiffener at twice γ is exactly as good as one at γ. Here γ = 31.5, which asks for an outstand of 144 mm; the 150 mm one drawn gives γ = 35.5, a margin of 1.13. The rib that is a boundary condition A rib on a plate is not a member carrying load. It is a line the buckle is not allowed to cross — and it becomes one at a threshold. Below the required rigidity it rides on the buckle and buys a fraction; at the threshold it stays straight and the plate buckles between stiffeners; above it, nothing further happens at all. A transverse load with nothing applied. Web slenderness against web thickness, with the limit the flange's own curvature sets. A flange carrying 6213 kN and curved to a radius of 592 m needs 10.5 N per millimetre of radial force to stay on its curve, and the only thing available to supply it is the web. Nothing has been applied to the girder: the load comes from the deflected shape, which is why a straight beam has none of it and a beam at a plastic hinge has a great deal. Setting the radial force against the web's own plate-buckling resistance gives, in four lines, h_w/t_w ≤ k·(E/f_yf)·√(A_w/A_fc) — the form the codes use, arrived at without them. The constants differ: an elastic flange strain gives k = 1.34 and the rule uses 0.3, a factor of 4.5, and the gap is the curvature assumed. k goes as the inverse square root of the flange strain, so 0.3 is a flange strained to 3.4% — which is what a plastic hinge does to it. The rule is not conservative; it is written about a different beam. The web that is crushed from inside A plate girder's compression flange is curved by the beam's own deflection, and a curved force needs a transverse load to stay on its curve. The only thing available to supply it is the web. So a deep girder can buckle its web vertically with nothing applied to it at all, and the rule that prevents it is the only clause in the codes about a load no load case contains.

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.

AnchorageFlangeLoad pathMembrane actionPlastic hingePlate bucklingPost-bucklingShear bucklingSlendernessStiffenerTensionWeb