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 #

The whole of this page depends on which of those three 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.

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.

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.