The rib that is a boundary condition

A plate buckles at a stress set by its width-to-thickness ratio and not by its length: $\sigma_{cr} = k\pi^2D/(b^2t)$, with $k = 4$ for a long simply supported panel. The width is squared, so it is the variable worth attacking, and the only way to attack it is to put something down the middle.

That something is a longitudinal stiffener, and the way it is usually described — a member added to help carry the load — is wrong in a way that matters. A stiffener helps by not being there.

Two modes, and the plate takes the cheaper #

The whole result comes from comparing two ways the plate can buckle, and finding where they cost the same.

Mode (a): one half-wave across the whole width, with the stiffener riding on it. The stiffener bends, so it adds strain energy; it also carries axial load, so it adds to the work the load does. For a panel of aspect ratio $\alpha = a/b$ the buckling coefficient is

$$k_a = \frac{(1+\alpha^2)^2 + 2\gamma}{\alpha^2(1+2\delta)},$$

with $\gamma = EI_s/bD$ the stiffener’s rigidity relative to the plate’s, and $\delta = A_s/bt$ its area relative to the plate’s. Minimising over $\alpha$ — the derivative gives $\alpha^2 = \sqrt{1+2\gamma}$ — collapses that to

$$k_a = \frac{2\left(1 + \sqrt{1+2\gamma}\right)}{1+2\delta}.$$

Mode (b): the stiffener stays straight and each sub-panel buckles on its own. Referenced to the full width, $k_b = 4(n+1)^2$ for $n$ equally spaced stiffeners. This mode contains no $\gamma$ at all, because it does not deflect the stiffener.

The plate takes whichever is smaller. Below some rigidity mode (a) is cheaper and the capacity depends on $\gamma$; above it mode (b) is cheaper and the capacity does not. Setting them equal gives the threshold:

$$\gamma^* = \frac{\left(\tfrac{1}{2}k_b(1+2\delta) - 1\right)^2 - 1}{2}.$$

For one stiffener of the size drawn — $\delta = 0.0625$, so $k_b = 16$ and the bracket is 8 — that is $(64-1)/2 = 31.5$.

Why it is a threshold and not a trade #

Almost every other stiffness in this collection is a trade. More second moment gives more bending stiffness, more prestress gives more shear capacity, more bracing stiffness gives a higher critical load — up to a point, and then the return diminishes.

Here there is no diminishing return. There is a corner. Past $\gamma^*$ the plate is buckling in a mode the stiffener plays no part in, so the plate literally cannot tell whether the stiffener has been doubled or halved. A stiffener at $2\gamma^*$ is a stiffener at $\gamma^*$ with metal wasted on it.

The one difference is worth naming. A brace on a column reaches its plateau at a stiffness that depends on the column’s load; a stiffener reaches its plateau at a rigidity that depends only on the geometry, because both modes scale with the same $\pi^2D/b^2t$. So $\gamma^*$ is a pure number for a given arrangement, and it can be tabulated once and used for every plate of that shape.

The area is a cost, not a contribution #

$\delta = A_s/bt$ is in the expression, and it is in the numerator: a fatter stiffener demands a stiffer one.

The reason is that mode (a) is an eigenvalue problem in which the stiffener contributes energy in two ways with opposite signs. Its bending stiffness resists the buckle, which is the $2\gamma$ term. Its axial load drives the buckle, exactly as a column’s load drives its own — and that is the $(1+2\delta)$ in the denominator, dividing the capacity down.

A stiffener that carries load is a stiffener that has to be held up itself, so the more area it has the more rigidity it needs to be a node rather than a passenger. That is the opposite of what “adding material” suggests.

What each extra stiffener costs #

The gain is easy: the sub-panel width falls as $1/(n+1)$, so the buckling stress rises as $(n+1)^2$. Four, nine, sixteen, twenty-five.

The requirement is not easy. $\gamma^*$ contains $k_b^2$, so it goes as roughly $(n+1)^4$ — and the outstand a given $\gamma$ requires goes as $I_s^{1/3}$, so the stiffener’s height goes as $(n+1)^{4/3}$ and its area with it.

The first stiffener is the good one and everything after it is worse. That is why plate girder webs and box girder flanges carry one or two longitudinal stiffeners and not five, and why past that point designers change the geometry instead — a thicker plate, a narrower panel, a different arrangement entirely.

There is also a limit at the other end. At four stiffeners the plate’s buckling stress has reached 463 N/mm², above the yield strength of the steel: the plate is no longer slender, and adding stiffeners to a stocky plate buys nothing at all, because the material has become the limit and buckling has stopped being the question.

Why the width is the variable worth attacking #

It is worth being explicit about why anybody puts a rib on a plate rather than making the plate thicker, because both are available and one is much cheaper.

The critical stress is $k\pi^2 D/(b^2 t)$ with $D = Et^3/12(1-\nu^2)$, so it goes as $(t/b)^2$. Thickening the plate raises it as $t^2$ and costs steel linearly in $t$; narrowing the panel raises it as $1/b^2$ and costs only the rib.

For the plate drawn, going from 12 mm to 24 mm quadruples the buckling stress and doubles the weight of a 2.4 m wide plate — 226 kg per metre run of extra steel. One stiffener quadruples it too, and costs 6 per cent: 14 kg. The same gain, for a sixteenth of the material, and that ratio is the entire reason stiffened plates exist.

The catch is that the stiffener has to be slender itself to be cheap, and a slender rib is the very thing that fails to reach $\gamma^*$. The whole design problem is a rib deep enough to have the second moment and stocky enough not to buckle on its own.

Which free body produced the number #

Neither mode is found by cutting anything. Both are found by an energy comparison: for an assumed buckled shape, the strain energy stored in bending the plate and the stiffener is set equal to the work the in-plane load does as the plate shortens, and the load at which they balance is the critical one. The free body, if there is one, is the entire panel.

The two-mode comparison is the approximation, and it is conservative. The true buckling problem admits shapes intermediate between the two — the stiffener deflecting a little while the sub-panels also ripple — and admitting them lowers the critical load slightly at rigidities near the threshold, which raises $\gamma^*$ compared with an exact treatment. The value here, 31.5 for one stiffener, sits above the tabulated 16 to 25 for the same case in the codes, and the difference is that they solve the full eigenvalue problem and this compares two modes.

What it is actually for, which is often not strength #

There is a use of longitudinal stiffeners that has nothing to do with the buckling stress, and on box girders it is the commoner one.

A wide flange plate has to be handled, transported, welded and fitted. A 2.4 m × 12 mm plate is floppy: it sags under its own weight over any reasonable support spacing, it distorts when a weld runs along it, and it arrives on site with a wave in it that has to be pulled out. A rib turns it into a member that can be lifted by one end.

That is a construction requirement, and it produces stiffeners at spacings and sizes that no buckling check would ask for. It also produces them in the places where the buckling requirement is least binding — near the neutral axis of a girder, where the compression is small — because that is where the plate is otherwise unsupported.

So the honest ranking of what a longitudinal stiffener buys is: a fourfold critical stress, a plate that can be handled, a web that does not breathe — and, at the bottom of the list, an increase in the load the girder actually carries, which the post-buckling reserve had largely provided anyway.

Where the model stops #

Everything is elastic and perfectly flat. A real stiffened panel has an initial bow, a weld distortion along every rib, and residual stresses from the welding — and the residual stress alone can put the plate past yield before the load arrives. A design curve for a stiffened panel is a column curve, not a critical stress, and the $\gamma^*$ derived here is the input to it rather than the answer.

A plate does not fail when it buckles. It sheds its middle and carries on in an effective width, which is a large post-buckling reserve the whole of this page ignores. Where the reserve is being relied on, the stiffener has a second job — holding the effective widths apart — and the requirement changes.

The stiffener is one-sided. An outstand on one face of a plate is eccentric, so it bends the plate as it loads and its own second moment depends on where the neutral axis is assumed to be. Taking it about the plate face, as here, is a convention; taking it about the combined centroid gives a different $\gamma$ for the same rib.

The panel is long. $k_b = 4$ per sub-panel is the long-panel value. A short panel between transverse stiffeners has a higher $k$, sometimes much higher, so the sub-panel mode is cheaper than assumed and $\gamma^*$ rises.

The stiffener is straight and stays straight. The mode (b) argument assumes the stiffener line is a perfect node, which is what γ* is supposed to guarantee — but a welded rib arrives on site with an initial bow of its own, from the welding, of the order of L/500. A stiffener with an initial bow is a spring rather than a node, and the real capacity sits below the plateau by an amount nothing in this calculation measures.

And no drawing here shows the plate after it goes. The figures are of critical stresses, which is to say of the moment a perfectly flat plate ceases to be flat. The thing an inspector sees is a panel with a permanent ripple in it that has been carrying load for thirty years, and no number on this page describes that.

Where the threshold comes from, restated #

The corner in the curve is the whole content of this essay, and it is worth one more pass in words that have no algebra in them.

A buckle is a shape that costs the structure less energy than staying straight does. There are many shapes available, and the plate takes the cheapest. A stiffener changes the price of one family of shapes — the ones that deflect it — and leaves every other family’s price exactly where it was.

So there are two prices, one of which the stiffener controls and one of which it does not. While the controllable one is lower, making the stiffener bigger helps. Once it has been pushed above the uncontrollable one, making the stiffener bigger changes a price nobody is paying.

Every threshold in structural design has this shape: a quantity the designer can improve, a quantity nobody can, and a crossing. The ideal brace stiffness is the same argument, and so is the minimum reinforcement that stops a single crack taking everything, and so is the shear connection that turns two beams into one. In each case the wrong reading is more is better, and the right one is enough, and then stop.

The ladder from here #

Later rungs on this anchor: the transverse stiffener, which is a different member with a different job — it holds the tension field’s anchorage rather than dividing a panel, and its requirement is a force rather than a rigidity. Torsional stiffeners and the closed trough, where the rib’s own torsional stiffness matters and an open flat is replaced by a V. Orthotropic plate theory, where the stiffeners are smeared into the plate’s properties and the whole thing becomes one anisotropic sheet — good for many stiffeners and wrong for few. The post-buckling interaction of a stiffened panel, where local and overall modes arrive together and the coincidence is dangerous rather than efficient. And the stiffener as a fatigue detail, where the weld terminating at the end of a rib is one of the worst categories in the tables and has ended more bridges than any buckling ever did.