The plate that ripples, and the width that is left

A column buckles because it is long. A plate buckles because it is wide, and its length hardly comes into it.

That difference is the whole subject. A flange too wide for its thickness ripples into waves whose length is comparable to the plate’s width, at a stress set by geometry rather than by strength — and the material never reaches yield anywhere.

Why width, and not length #

A column bows in one direction and nothing resists it but its own bending stiffness about the weak axis. A plate supported along both long edges cannot do that: bowing out of plane means stretching in the transverse direction as well, because the edges are held.

So the plate resists in two directions at once, and the relevant length is the distance between the supported edges — the width. The critical stress comes out as

$$\sigma_{cr} = \frac{k\pi^2 E}{12(1-\nu^2)}\left(\frac{t}{b}\right)^2,$$

which is the Euler expression with the slenderness $L/r$ replaced by the width-to-thickness ratio $b/t$, and with two extra pieces: the factor $12(1-\nu^2)$, which accounts for the plate resisting in two directions rather than one, and the buckling coefficient $k$, which encodes what is holding the edges.

The plate does buckle into waves along its length, and the number of them adjusts so that each is about as long as the plate is wide. That is why length drops out: a plate twice as long simply forms twice as many waves at the same stress.

What the coefficient encodes #

$k$ is where the edge conditions live, and its values are worth carrying because the spread is large.

Both long edges supported — a web between two flanges, or the flange of a hollow section: $k \approx 4.0$.

One long edge supported and one free — the outstand flange of an I-section, a channel or an angle: $k \approx 0.43$.

A factor of nine, from whether the far edge is held. That single number explains the geometry of the whole section catalogue: an outstand flange has to be roughly three times stockier than an internal plate to have the same critical stress, since the ratio enters squared. Compare a rolled I-section’s flange outstand — perhaps eight or nine times its thickness — with a hollow section’s face, which is routinely thirty times its thickness, and the difference is not a manufacturing accident.

The lipped cold-formed section exists for exactly this reason. Turning a small return along the free edge of a flange converts it from an unsupported edge into a supported one, and $k$ jumps from $0.43$ toward $4$. A few millimetres of steel bent through ninety degrees multiplies the flange’s critical stress several-fold, which is a remarkably good return and is why almost every cold-formed section has lips on it.

Buckling is not failure here #

This is the property that separates plates from columns entirely, and it is the reason the subject has its own vocabulary.

When a column reaches its critical load, the deflection runs away and there is no further capacity. When a plate reaches its critical stress, it ripples — and then goes on taking load. The middle of the plate, which has bowed out of plane, sheds its share; the strips near the supported edges, which cannot bow because they are held, keep working and take up the difference.

So the stress distribution across a buckled plate is no longer uniform. It sags in the middle and peaks at the edges, and the plate goes on carrying load until the edge stress reaches yield. That reserve beyond the critical stress is called post-buckling strength, and for a slender plate it can be several times the buckling load.

The device used to account for it is the effective width. Rather than model the varying stress, the plate is replaced by two narrow strips at the edges carrying the full edge stress, with the middle discarded entirely. The effective width follows Winter’s expression, which is an empirical fit that has survived seventy years:

$$\frac{b_{\text{eff}}}{b} = \frac{1}{\lambda_p}\left(1 - \frac{0.22}{\lambda_p}\right), \qquad \lambda_p = \sqrt{\frac{f_y}{\sigma_{cr}}}.$$

The gap between the two curves in that figure is the whole of the post-buckling reserve. The critical stress runs away downward as the plate widens; the delivered capacity falls much more slowly, because the edge strips go on working long after the middle has given up. A plate three times wider than its slender limit still carries something like a third of what a stocky one would, which for a column at three times its limiting slenderness would be closer to a ninth.

The section that is checked is therefore not the section that was drawn. It has holes in the middle of every slender plate, and its second moment of area, its section modulus and its centroid all differ from the gross values — which for an unsymmetric section means the effective neutral axis moves, and the calculation has to be iterated.

Where the section classes come from #

The whole apparatus of section classification is this argument turned into four bands, and knowing where they come from makes them stop looking arbitrary.

Class 1 — plates stocky enough that the section can reach its plastic moment and rotate at it long enough for a hinge to form and redistribute. Plastic design permitted.

Class 2 — the section can reach its plastic moment but cannot sustain the rotation. Plastic capacity available, plastic analysis not.

Class 3 — the section can reach first yield at its extreme fibre but not the plastic moment, because a plate buckles first.

Class 4 — a plate buckles before even first yield, and the effective-width calculation is required.

The limits are all expressed as width-to-thickness ratios, differ for internal and outstand elements by roughly the ratio the coefficient $k$ implies, and scale with $\sqrt{235/f_y}$ — because a stronger steel reaches a higher stress before yielding and is therefore more likely to buckle locally first. Which is the same observation as a stronger steel crossing over to buckling at a lower slenderness, at the scale of a single plate.

The theoretical limit, and the one that gets used #

Two numbers are worth computing, because comparing them shows what a design limit is actually made of.

Set the critical stress equal to yield and solve for the width-to-thickness ratio. For steel at $355$ with an internal element — both edges supported, $k = 4.0$ — the answer is

$$\frac{b}{t} = 46.2.$$

For an outstand, with $k = 0.43$, it is $15.2$. The ratio between them is $\sqrt{4.0/0.43} = 3.05$, which is exactly the factor of three quoted earlier and arrives here as arithmetic rather than as a rule of thumb.

Now the limits actually used. The corresponding code figures for a class 3 element in uniform compression are $42\varepsilon$ and $14\varepsilon$, with $\varepsilon = \sqrt{235/f_y} = 0.814$ for this steel — so $34.2$ and $11.4$.

The design limits are about a quarter stricter than the theory. That gap is the whole content of the difference between an elastic critical stress and a usable limit, and it is made of three things: initial out-of-plane distortion from rolling and welding, residual stresses that put parts of the plate into compression before any load arrives, and the requirement that the section reach yield with something in reserve rather than exactly.

The $\varepsilon$ factor is worth reading too. A stronger steel has a lower permitted ratio, because $\varepsilon$ falls as $f_y$ rises — the plate has to be stockier to reach a higher stress before buckling. Strength does not help a stability problem, and here it actively costs: the same plate in a stronger steel is a worse class.

What it costs to keep them flat #

The whole argument of getting material away from the neutral axis drives plates toward being thin and wide, and this essay is the bill.

The remedy is stiffeners, and they are a strange item on a drawing: pieces of steel carrying no bending moment, no shear and no axial load, present only so that something else keeps the shape it was drawn with. A deep plate girder carries transverse stiffeners at intervals along its web, bearing stiffeners over its supports, and for very deep webs longitudinal stiffeners running along it — each one subdividing a panel and raising its critical stress by the square of the reduction in width.

That last point is the economics. Halving a panel’s width quadruples its critical stress, so one longitudinal stiffener down the middle of a web is worth more than doubling the web’s thickness. Stiffening is the cheap answer and it is not free: every stiffener is a piece to cut, fit and weld, and fabrication cost is labour rather than material.

The consequence when it is got wrong is the most severe episode in the modern history of the subject. Between 1970 and 1971 four large steel box-girder bridges failed during construction — Milford Haven in Wales, West Gate in Melbourne, Koblenz in Germany, and Zeulenroda — with the West Gate collapse killing thirty-five people. The failures were plate buckling failures: webs and support diaphragms of exactly the slenderness this essay is about, in structures whose overall stresses were nowhere near yield.

What the investigations found was not a mistake in a formula. It was that box girders had grown steadily thinner and wider through the 1960s, that the design rules had been written for stockier plates and were being extrapolated, and that erection conditions — a partly assembled girder, temporarily supported, with distortions from welding — were being checked against rules meant for the finished structure. The British response, the Merrison rules of 1973, was the most detailed treatment of plate stability ever put into a design document, and it changed box-girder design permanently.

Both halves of that economics are computable, and neither behaves the way the sentence above suggests. A stiffener is not a quantity of steel; it is a boundary condition, and it is bought at a threshold.

That is the same shape as a brace on a column and it has the same consequence: a stiffener is adequate or it is nearly useless, and there is no partial credit. The threshold here is a required rigidity of 31.5, which asks for an outstand of 144 mm; the 150 mm one drawn gives 35.5, a margin of 1.13 and nothing more to be gained by making it larger.

The second half is what happens when the answer to a slender panel is another stiffener, and then another.

At four stiffeners there is more steel in the ribs than in the plate they are stiffening. So “stiffening is the cheap answer” is true once and false by the third repetition: the gain is a square and the price is a fourth power, and they cross. A panel that needs four longitudinal stiffeners is a panel that should have been thicker, and the arithmetic that says so is on that figure rather than in anybody’s judgement.

The web, which has its own version #

Everything above concerns compression. A web carries mostly shear, and shear buckling is the same phenomenon with a different stress field.

A web in shear has a principal compression at forty-five degrees, and a thin web buckles along it into diagonal ripples running corner to corner. The critical stress follows the same $\left(t/b\right)^2$ law with a coefficient appropriate to shear.

And it has the same post-buckling reserve, in a form that is even more striking. After the diagonal compression has buckled, the web goes on carrying shear by tension field action: a band of diagonal tension forms across the panel, anchored by the flanges and the transverse stiffeners, and the panel behaves as a truss — the flanges as chords, the stiffeners as vertical compression posts, and the diagonal tension band as a tie.

A plate girder in that state has spontaneously turned itself into a truss, which is a good demonstration that the distinction between a girder and a truss is one of degree. The design method that accounts for it is called exactly that, and the stiffener spacing follows from treating the panels as a Pratt truss.

Which way to run the stiffener #

The coefficient’s dependence on the panel’s proportions settles a practical question with an exact answer, and the answer is different for compression and for shear.

For a long plate in compression, $k$ dips to exactly 4.0 whenever the panel’s length is a whole number of times its width — the buckle forms in that many square half-waves — and rises slightly between. So making a compression panel longer changes nothing: it just accommodates another half-wave.

Making it shorter than its width is different, because then only one half-wave will fit and

$$k = \left(\frac{b}{a} + \frac{a}{b}\right)^2$$

which at $a/b = 1$ gives 4.0, at $a/b = 0.5$ gives 6.25, and at $a/b = 0.25$ gives 18.1. A transverse stiffener at half the panel width raises the critical stress by 56 per cent; at a quarter, by a factor of four and a half.

But a longitudinal stiffener does better still, because it halves $b$ — and $b$ is squared:

$$\sigma_{cr} \propto \left(\frac{t}{b}\right)^2$$

so halving the width quadruples the critical stress, against 1.56 for halving the length. For compression, run the stiffener along the stress.

For shear the coefficient is $k = 5.34 + 4(b/a)^2$, which rises as the panel is shortened and has no equivalent term for narrowing it. For shear, run the stiffener across.

Which is why a plate girder’s web carries transverse stiffeners at intervals — it is carrying shear — and a compression flange or a box girder’s compression panel carries longitudinal ones. Two families of stiffener, on two plates a metre apart, running at right angles to each other, for a reason that is entirely in which term of which coefficient the stiffener is allowed to touch.

There is a cost to the longitudinal one that the transverse one does not have, and it is why the transverse stiffener is the commoner of the two despite being the weaker lever. A transverse stiffener is a plate welded across a web, in a place where nothing else is happening, and it carries no direct stress. A longitudinal stiffener runs along the member, parallel to the flange, in the same direction as the stress — so it is itself a compression member, it attracts a share of the load in proportion to its area, and it has to be checked as a strut in its own right. A stiffener that buckles has stopped being a support, and the panel it was supporting reverts to its unstiffened width at the same instant.

Where the model stops #

Ideal edge conditions. $k = 4.0$ assumes simple supports along both edges. Real edges are partly restrained by whatever they connect to — a flange restrains a web, and a stocky flange restrains it more — so the true coefficient is somewhere between the simply supported and fixed values, and design uses the lower.

Perfect flatness. Plates are not flat. Rolling and welding leave out-of-plane distortion of the order of the thickness, and a plate with initial waviness starts deflecting from the first increment of load rather than at the critical stress. The transition is gradual, which is why design curves round the corner rather than meeting it.

Imperfection sensitivity. Plates are mild in this respect — nothing like a cylindrical shell, whose clustered modes cost it seventy per cent. A plate’s modes are well separated and its post-buckling behaviour is stable, which is why an empirical effective width works at all.

One plate at a time. Each element is classified on its own, with an assumed edge condition standing in for whatever it is attached to. A section is a set of plates that restrain each other, and treating the worst of them as governing is a convenience — the same convenience as judging a frame one column at a time, with the same caveat that the real mode belongs to the whole assembly.

Uniform compression. The coefficient assumes a uniform stress across the plate. A web in bending has a stress varying from compression to tension across its depth, and its coefficient is very much higher — around $24$ — because only part of it is in compression at all.

Elastic material. For a stocky plate the critical stress exceeds yield, and the buckling is inelastic. The elastic formula then returns a number the plate can never reach, which is precisely the class 1-to-3 region and is why those classes are handled by limits rather than by the formula.

The figures on this page carry a distortion worth naming. The critical stress is plotted as a smooth curve running well above yield at small widths, and that part of the curve is fictitious — a plate cannot reach a stress its material cannot sustain. The curve is drawn there because the crossing with yield is the quantity of interest, and everything to the left of it describes a buckling mode that never occurs.

The ladder from here #

Later rungs on this anchor: the plate buckling equation derived. Coefficients for other edge and stress conditions. Effective width and Winter’s formula. Section classification and its limits. Post-buckling behaviour and its stability. Shear buckling and tension field action. Web stiffener design. Cold-formed sections, which live almost entirely in class 4. Stiffened plates and orthotropic decks. And shell buckling, where the mild imperfection sensitivity of plates becomes savage.

Bryan solved the simply supported plate in 1891, and the result sat as a curiosity until thin-walled construction made it urgent — first in aircraft in the 1920s, then in welded plate girders and cold-formed steel. Winter’s effective-width expression dates from 1947 and remains, in essence, an empirical curve through test data that nobody has improved on.