The panel that carries more after it has failed

A plate girder’s web is a large piece of very thin steel, and under shear it develops a diagonal compression and a diagonal tension at right angles to each other. The compression diagonal is a strut with almost no stiffness across it, and it does what a strut does: it goes. The web ripples, visibly, in a set of parallel folds running one way across the panel, and the girder does not fall down. It carries on, and it carries on carrying nearly twice as much again — which no other member in this collection does.

Failure and buckling have come apart, and that is unusual enough on this site to be worth stating carefully. Every other stability argument here treats the critical load as the answer. A plate is the exception, and the exception has a mechanism.

Why a plate is different from a strut #

A column that buckles has nowhere to go: its whole cross-section is on the same buckle, and once it starts to bend the load’s own eccentricity makes things worse. A plate is not one strut. It is a great many parallel strips, and they are joined side by side.

The strips near the middle of a buckled panel are the ones that have gone; the strips near a stiffened edge are held straight and can pick up more. So a plate has an internal load path available after buckling, and the effective-width idea is that path counted for a plate in direct compression.

Under shear the redistribution is not to the edges but across the diagonal, and it is more dramatic, because the two diagonals of a shear field carry opposite signs. Losing one leaves the other entirely intact.

Which free body produced the number #

Take the panel, cut it on a vertical line, and take the piece to one side. What crosses the cut before buckling is a uniform shear stress on the full depth: $V = \tau\, d\, t$, and the state of stress at every point is pure shear — equal tension and compression at 45°, which is what Mohr’s circle says about a point in pure shear.

The critical value of that shear comes from the plate’s own eigenvalue problem:

$$\tau_{cr} = k_\tau\,\frac{\pi^2 E}{12(1-\nu^2)}\left(\frac{t}{d}\right)^2, \qquad k_\tau = 5.34 + \frac{4}{(a/d)^2}$$

For the panel above, $k_\tau = 9.34$ and $\tau_{cr} = 63.8$ N/mm². A web that never buckled would have gone at $f_y/\sqrt3 = 158.8$, so this one buckles at 40% of the shear its material could have carried.

Now cut the same panel after buckling, and the free body is different. The compression diagonal contributes nothing across the cut, so the shear it used to carry has gone somewhere. What remains is a band of tension running corner to corner, anchored on the flanges above and below, and its vertical component is the extra shear.

The angle, which is not assumed #

The band is anchored between the flanges, so its width is limited by geometry: a band at angle $\theta$ to the horizontal, running between the panel’s corners, has a clear width of

$$s = d\cos\theta - a\sin\theta$$

and its vertical component per unit stress is $s \sin\theta$. Maximising that product is one line:

$$\frac{d}{d\theta}\left[\tfrac{d}{2}\sin 2\theta - \tfrac{a}{2}(1-\cos 2\theta)\right] = d\cos2\theta - a\sin2\theta = 0 \quad\Longrightarrow\quad \tan 2\theta = \frac{d}{a}$$

A square panel bands at 22.5°, not 45°. The result is Basler’s $\theta_d/2$ recovered from the geometry rather than quoted, and it is worth pausing on because the 45° is such a natural thing to expect — the yield lines in a slab are not at 45° either, for a related reason: the principal tension in the pre-buckling field really is at 45°, and the band is not the principal direction of anything. It is the direction that gets the most vertical force out of a strip of finite width anchored on two flanges, and the flanges are what make the answer smaller.

The trend is the reverse of the intuition too. A longer panel bands at a flatter angle, because a flat band in a long panel is wide, and width is what the vertical component is being traded against.

What the material has left #

The band cannot be stressed to yield, because the critical shear is still there underneath it: the panel does not unload when it buckles, it stops taking more shear in the buckled diagonal. Superposing a uniaxial tension on a state of pure shear at $\tau_{cr}$ and going to yield gives

$$\sigma_t = \sqrt{f_y^2 - 3\tau_{cr}^2}$$

which is 252 N/mm² here against a yield of 275 — so the buckling that has already happened has cost only 8% of the membrane stress available. Multiply out: $\sigma_t\, t\, s \sin\theta = 313$ kN of extra shear, on top of the 383 kN the panel buckled at, for a total of 696.

Where the reserve is largest #

That divergence is the practical content of the whole subject.

At $d/t = 106$ the two limits coincide: $\tau_{cr}$ has climbed to $f_y/\sqrt3$ and the web yields before it buckles, so there is no post-buckling reserve to have because there is no buckling. Below that ratio a web is a stocky web and this page has nothing to say about it.

Above it, the thinner the web the larger the fraction of its strength that arrives after it has visibly failed. A 4 mm web reaches only 53% of what its material could carry, but it reaches three times what a linear buckling calculation would have allowed it — and it does so on 40% of the steel a stocky web would have needed for the same shear.

The shape of the load path afterwards #

A plate is the stable-symmetric case: its path rises after the bifurcation, so the structure stiffens rather than sheds and the critical load is a lower bound on its strength instead of an upper one. A shell is the opposite and loses two thirds of its theoretical load to a defect nobody can see.

How much of a bargain that is depends entirely on how slender the panel is, and the dependence runs the opposite way to intuition.

So the thinner the web the larger the reserve and the smaller the answer. The multiplier and the capacity move in opposite directions, because the multiplier is measured against a buckling load that is itself collapsing — which makes “carries more after it fails” a true sentence and a bad design criterion at the same time.

The whole of this page depends on which of those curves a shear panel is on, and the answer is the top one. Stability is not a single quality: two structures with identical critical loads can be a bargain and a trap, and the difference is a fourth-order term in the energy that no eigenvalue calculation contains. A plate girder web is the most useful example of the bargain in the whole of structural engineering, and a cylindrical shell in compression is the standing example of the trap.

The truss it turns into #

The analogy is not decorative. After buckling, a plate girder has: two chords carrying the bending moment as a couple; a set of verticals carrying compression, because the tension band pulls the flanges together and the stiffeners hold them apart; and a set of tension diagonals at a shallow angle. A stiffened plate girder is a Pratt truss with the diagonals made of sheet.

Which explains the one detail that otherwise looks arbitrary: the intermediate stiffeners of a plate girder do almost nothing before buckling, and after buckling they are struts carrying real axial load. They exist to divide the web into panels — which raises $k_\tau$ — and to anchor the band, in the way a brace need not be strong to be worth having, and the second job only begins on the day the first job stops working.

The analogy has a limit, and it is worth stating. A truss panel that loses its diagonal is a mechanism; a web panel that loses its compression diagonal still has the other one. Triangulation in a single direction is enough here because shear has a sign, and a girder is only ever asked to carry it in the direction the band was drawn for.

That last observation is the honest boundary of the truss picture. A tension field is one-directional. Reverse the shear and the band has to reform along the other diagonal, which it will — but the panel has to go through its unbuckled state to get there, and under repeated reversal the folds work back and forth and fatigue becomes the governing question rather than strength.

What it does to the flange #

The band pulls, and it pulls on something. Its transverse component is delivered to the flange as a distributed pull of $\sigma_t\, t \sin^2\theta$ — 221 N per millimetre here, which over a 1 m panel is 221 kN trying to bend the flange inward between the stiffeners.

So a plate girder relying on tension field action has a flange doing a second job it was not sized for, and the amount of band that can be anchored is limited by the flange’s own bending capacity. That is the difference between the models: Basler assumed the flanges were infinitely flexible and anchored nothing, the Cardiff models compute an anchored length from the flange’s plastic moment, and the truth is in between and depends on a member nobody thinks of as a beam.

Which is the argument for stiffeners closer together, and it works on the flange for a reason that has nothing to do with the flange.

A factor of five off the flange’s transverse load for a change of panel proportion is the reason stiffener spacing is a flange decision as much as a web one. It also runs against the web: the long panel buckles earlier and finishes lower. The stiffener spacing that is best for the web is not the spacing that is best for the flange, and a plate girder’s transverse stiffeners are placed where those two arguments settle rather than where either would put them.

The stiffener becomes a compression member #

A transverse stiffener on a plate girder is usually introduced as a device for making the panels shorter, and its requirement is written as a stiffness. Once tension field action is claimed, it acquires a second and quite different job.

The band of tension pulls diagonally across the panel and has to be anchored at its ends. The horizontal component is taken by the flanges; the vertical component is taken by the stiffener, which is therefore carrying an axial compression down its own length — it is the vertical post of the truss the web has become.

So a stiffener that was sized to divide a panel has to be re-checked as a strut, with a length equal to the web depth, an effective section made of the stiffener plus a participating width of web either side, and an axial force that is the excess of the applied shear over what the web could carry before it buckled. On a heavily loaded girder that force is a substantial fraction of the panel shear, and it is a force the stiffness rule says nothing about.

Two consequences follow. A stiffener adequate for panel division may be inadequate once the reserve is claimed, which makes the tension field a decision that reaches back into a component already chosen. And the stiffener’s own buckling matters: an outstand too slender to carry its share is a post that folds, and the truss it belonged to has lost a member.

The end panel has nothing to lean on #

The tension band pushes horizontally on the stiffener at each end of a panel. In the interior of a girder the panel on the other side pushes back with an equal and opposite force, the two cancel, and the stiffener carries only the vertical component.

At the end of the girder there is no panel on the other side. The horizontal component has nowhere to go, and the last stiffener has to carry it alone.

That makes the end of a plate girder a different structure from its middle, and there are two ways to build it.

A rigid end post: two stiffeners a short distance apart, welded to the flanges and to each other, forming a small vertical beam that spans between the flanges and anchors the horizontal pull. It is sized as a member in its own right — bending under the anchor force, with its own section and its own check — and it is the reason the end of a plate girder has a pair of stiffeners where the rest has singles.

Or a non-rigid end post, which is a single stiffener that cannot anchor the band. Then the end panel is simply not permitted to use tension field action: it is designed on its buckling capacity alone, which is a fraction of the interior panels’. That is a legitimate and common choice, and it means the end panel is often the shortest one on the girder, because the only way to raise a buckling capacity is to shorten the panel.

The two options are a trade between fabrication and web thickness, and the choice is made once per girder end. What is not available is the third option of ignoring the question, which is what applying the interior formula to the end panel amounts to — and it is a mistake with no warning in it, because the calculation returns a number and the number is simply for a panel that is not there.

Where the model stops #

Nothing here sizes anything. The band model above is a Cardiff-type one with the interaction simplified to $\sigma_t = \sqrt{f_y^2 - 3\tau_{cr}^2}$, and design codes use rotated-stress-field or Basler formulations that differ from it and from each other by 10–20%. The argument is about where the reserve comes from.

The flange is assumed to anchor the whole band. It cannot. A real anchored width is set by the flange’s plastic moment and the stiffener spacing, and a girder with light flanges reaches considerably less than the numbers above.

The end panel has no neighbour to lean on. A tension field pushes horizontally on the panel next to it, and interior panels balance out — but the panel at the end of a girder has nothing beyond it, so either it is designed without tension field action or the end stiffener has to be a substantial vertical beam. The most common error in this subject is applying the interior-panel formula to the end one.

The web still has to carry the bending moment. Everything above treats a shear panel in isolation. In a real girder the same web carries direct stress from the flexure of the whole member, that stress interacts with the shear, and the interaction is a curve rather than a pair of separate checks.

And the deformations are large. A web at its ultimate shear has out-of-plane folds of the order of its own thickness or more. Nothing in the elastic buckling calculation applies to that geometry; the critical stress is used only as a marker for where the redistribution begins.

What the pictures cannot show #

The hero draws the band as a clean parallelogram with a sharp edge, and the panel as flat. Neither is true. The membrane stress varies across the band and dies gradually rather than at a line, the folds are out of the plane of the page by several millimetres, and the drawing is a plan of a surface that is no longer plane.

The reserve figure plots two loads on one axis as though they were the same kind of quantity. They are not: the lower curve is a bifurcation load computed from linear elasticity, and the upper one is an ultimate load computed from a plastic mechanism. Nothing continuous connects them, and the vertical distance between the curves is a comparison of two theories rather than a path a panel travels along.

And no figure here shows the deflection. A girder working in tension field action has a shear stiffness far below the elastic value, so it sags more than a shear calculation predicts, and the extra movement arrives suddenly at the buckling load. The strength argument and the serviceability one point in opposite directions, and only the first is drawn.

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, and the plastic hinges in the flange that decide the anchored length. End panels, bearing stiffeners and the horizontal thrust a tension field applies to whatever is beyond it. The interaction of shear and bending in a slender web, which is a surface rather than a curve. Corrugated webs, which do not buckle in shear at all and therefore have none of this — and pay for it in fabrication. Tension field action in aircraft structure, where Wagner published the complete diagonal-tension theory in 1929 for a skin far thinner than any girder’s and where the reserve is not an extra but the whole design basis. And repeated shear reversal, which turns a strength argument into a fatigue one.

The historical shape of the subject is worth noticing. Wagner had the complete theory in 1929 for thin aircraft skins, where webs are so slender that they were never expected to do anything else. Civil engineering, whose webs are thicker and whose factors of safety are larger, went on treating buckling as failure for another thirty years — and only started using the reserve when the arithmetic of a welded plate girder made a thinner web worth the trouble.