Two ways of buckling at once

A cold-formed steel column is a thin sheet folded into a shape. It has two entirely different ways of failing by instability, and they have almost nothing in common.

It can bow as a whole, over its full length, exactly as any column does — a load $\pi^2EA r^2/L^2$ that depends on the length and on where the material sits in the section.

Or its plates can ripple, in waves a few times the plate width long, at a load that depends on the width-to-thickness ratios and not on the member’s length at all. The section keeps carrying load afterwards, which is what makes cold-formed design possible.

Each has a well-established check. What they do to each other has a folk answer that is wrong, and the correct answer is more useful.

Which free body produced the number #

The arithmetic here is a pair of reduction curves rather than a free body, and it is worth being explicit about the order they are applied in, because the order is the interaction.

The global step. Take the squash load $P_y = Af_y$ and the Euler load $P_{cre}$ of the whole member. The column curve reduces the first for the slenderness $\lambda_c = \sqrt{P_y/P_{cre}}$ and returns $P_{ne}$ — the load a column of that length would carry if its plates could not buckle.

The local step. Now compare the local critical load $P_{cr\ell}$ against $P_{ne}$, not against $P_y$. The slenderness that enters the local reduction is $\lambda_\ell = \sqrt{P_{ne}/P_{cr\ell}}$, and the reduction applied is a Winter-type expression in that ratio.

That second step is where the modes meet. The local check is not asking whether the plates buckle under the squash load; it is asking whether they buckle under the stress the global check has already conceded. A longer column allows less stress, which makes its plates relatively safer, which is why the local reduction on a given section depends on the member’s length.

Measured against what #

Here is where the folk claim goes wrong, and it is a question about the denominator rather than about the physics.

If the erosion is measured against the global capacity $P_{ne}$, it grows without limit as the plates get thinner — 22 per cent, 40, 60 — and the worst case is a section with no plate stiffness at all. That is a true statement and it is not interaction. It is local buckling, which the local check was going to find anyway.

The interaction is what the two modes cost beyond what either would have cost alone. So the reference has to be the weaker of the two single-mode answers: the global capacity $P_{ne}$, or the stub column’s local capacity $P_{n\ell 0}$ computed against the squash load. Measure against $\min$ of those and the picture changes completely.

• At a local-to-global ratio of 10 — stocky plates — the erosion is zero. The plates never buckle and the column curve is the whole answer.
• At a ratio of 1 — the two criticals equal — it is 2.1 per cent.
• At 0.45 it is 23.2 per cent, the worst point in the range.
• At 0.01 — a sheet rather than a section — it is 21.8, and falling.

The curve is a broad plateau with a cliff on its right-hand side.

What the useful half of the folk claim is #

The received advice is not useless; it is the right advice for the wrong reason.

What the curve does say is that once the local critical load is about twice the global one, the interaction is nothing. That is a design rule with real content: thickening the plates past that point buys no interaction benefit at all, and the section should be optimised on other grounds.

What it does not say is that the two criticals should be kept apart to avoid a peak. There is no peak at coincidence to avoid. A section proportioned so that its plates buckle at exactly its member’s Euler load is a perfectly reasonable section, and it is losing two per cent to interaction.

The advice that is genuinely correct in the neighbourhood of coincidence is about something else entirely — imperfection sensitivity. A structure with two coincident buckling modes has a two-dimensional space of buckled shapes available at one load, and coupled-mode structures are far more sensitive to imperfections than either mode is alone. That is Koiter’s result, it is about the shape of the post-buckling path rather than about the capacity a design curve returns, and it is a real reason to be careful. It is not the reason usually given.

An optimised section is a coupled one #

There is a reason the coincidence case keeps being singled out, and it is a fact about how sections get designed rather than about how they fail.

A designer minimising material for a given capacity pushes every check toward governing at once. Material sitting in a plate that will never buckle is material that could have been moved outward to help the global mode; material in a section too slender to reach its global capacity is wasted the other way. The optimum is where the two limits meet, and the optimum is therefore a coupled-mode design by construction.

That is why the warning exists and why it is worth keeping despite the arithmetic above. Optimised structures are imperfection-sensitive not by coincidence but by definition: optimisation drives modes together, and coincident modes are sensitive. The correct statement is that an optimised section needs a larger imperfection allowance, not that it has a lower critical load.

The pressure that makes a designer thin the plates in the first place is what length costs a compression member: a section pushed outward for global stiffness has thinner plates at the same weight, and the two modes converge as a direct consequence. What that larger allowance has to cover is not linear in the imperfection either.

The centroid moves, which is the other coupling #

There is a second and completely different way the two modes interact, and it is geometric rather than energetic.

When a plate buckles locally, its middle stops carrying and its edges continue. The section that remains is not the section that was drawn: a lipped channel that loses its web’s middle has its centroid move 8 mm away from the web. Take the lips off and the flanges become outstands, lose their free edges, and the centroid moves 6.8 mm toward it.

A concentric axial load applied to the original centroid is therefore eccentric to the effective one, and the column is a beam-column rather than a column. The moment that produces is $P$ times the shift, and it is fed straight into the global check.

That is a genuine local-global interaction with nothing statistical or energetic about it: local buckling changes the section, and the global check is run on the section. The direct-strength arithmetic above does not model it explicitly — it is buried in the calibration — and an effective-width calculation does model it, which is one of the few things the older method does better.

Three modes, not two #

Cold-formed sections have a third mode that is neither of the two above, and it is often the governing one.

Distortional buckling moves a whole flange-and-lip assembly sideways, rotating about the flange-web junction, in waves several times longer than the local ones and shorter than the global. It is not local — the fold lines move — and it is not global — the member’s axis stays straight. It has its own critical load, its own reduction curve, and its own interactions with both of the others.

Its existence is the strongest argument for the direct-strength approach over the effective-width one. An effective-width calculation asks what is left of each plate; distortional buckling is not a plate phenomenon and there is nothing natural to remove. A method that takes each mode’s elastic critical load from a numerical analysis of the whole section and applies a strength curve to it handles all three the same way.

A short column and a long one, of the same section #

The clearest way to see what the interaction actually is is to take one section and change nothing but the length.

At half a metre the member is a stub. Its global critical load is enormous, the column curve takes almost nothing off the squash load, and the plates are being asked to carry stress right up to $f_y$. The local reduction is applied against nearly the full squash load, and it bites hard — but every bit of that is the local check doing its ordinary job, and the interaction term is nil, because the stub column’s capacity is the weaker single-mode answer.

At three metres the column curve has already taken 33 per cent off. The plates are being asked for less, so the local slenderness is lower and the local reduction is milder in absolute terms. The section carries 447 kN against a global-alone 569 and a stub-alone 583: 21 per cent below the weaker of the two, and all of it interaction.

At ten metres the column curve has taken nearly everything. The plates are barely stressed, the local reduction is close to one, and the interaction is small again — the member is simply a slender column.

So the interaction is an interior effect, largest in the middle of the practical slenderness range, which is exactly where cold-formed columns are used. It is not the behaviour of an extreme case; it is the behaviour of the ordinary one.

The same coupling, one scale up #

The argument is about a section with two modes, and nothing in it is about sections. Wherever two instabilities of a structure have critical loads close together, the same erosion applies and for the same reason.

The largest instance of it is not a frame at all.

A cylinder has very many buckling modes at very nearly the same load — that is what makes it the worst case in the subject rather than merely a bad one — and the two-thirds discount on that figure is what coupled-mode sensitivity looks like when the coupling is not between two modes but among dozens. Everything on this page is that phenomenon with the count turned down.

A braced frame whose bracing system’s own buckling load is close to the frame’s sway critical load is a coupled system. A truss whose chord’s local buckling between panel points coincides with its overall buckling is another. A lattice mast whose leg buckles between nodes at about the load the whole mast leans at is a third. In every case the two modes interact, the imperfection sensitivity is worse than either mode’s alone, and the actual capacity is below both criticals.

And in every case the coupling is what an efficient design produces. A structure with a mode well below the others has capacity going to waste in the modes that are not governing; removing that waste means raising the low mode until it meets the next one, which is exactly the arrangement with the worst sensitivity.

So there is a design constraint here that never appears as one: keep the critical loads apart. A ratio of 1.2 or 1.5 between the lowest two costs a little material and buys back the interaction, and it is a decision that has to be taken deliberately because no optimisation will take it.

And the imperfection is relatively larger #

The second half of why cold-formed design is so test-based is the size of the departures relative to the thing departing.

Local buckling is sensitive to out-of-flatness measured against the plate thickness. A rolling line leaves a plate flat to some absolute tolerance — a fraction of a millimetre — and that fraction is a small share of a 10 mm hot-rolled web and a large share of a 1.5 mm cold-formed one.

So a cold-formed section arrives with a relatively larger imperfection and with modes close enough to interact, and the two compound. It is the reason the whole subject is built on curves fitted to tests rather than on a theory with a correction applied to it.

Where the model stops #

The reduction curves are calibrated, not derived. The 0.658 and the 0.877, the 0.15 and the 0.4 in the Winter expression are fitted to test data. They contain the interaction, the imperfections and the residual stresses of the sections they were fitted on, and a section outside that population is outside the calibration.

The two criticals are treated as independent inputs. In reality the local critical load of a member that is bowing is not quite the local critical load of a straight one, and a numerical eigenvalue analysis of the deformed member would give a different pair.

And the erosion measured here is a capacity ratio. It says nothing about the deformations, the ductility or the behaviour after the peak — and a coupled-mode section’s load-shortening curve falls away much more steeply after its maximum than either single mode’s does, which matters wherever a member is part of a redundant structure that needs it to keep carrying.

Why the older method hid this #

The effective-width method that preceded the direct-strength one handles the same physics and never states the interaction as such, and it is worth seeing how it disappears.

In that method, local buckling is dealt with by replacing each plate with a narrower “effective” one carrying uniform stress, and the resulting effective section — smaller area, different second moment, moved centroid — is then put through the column curve. Because the effective width itself depends on the stress, and the stress depends on the column curve, the calculation iterates.

That iteration is the interaction. Each cycle the column curve concedes a lower stress, the effective widths grow slightly, the section improves, and the process converges. Nothing in the method is labelled “interaction” and the whole of it is there, distributed across an iteration loop.

The two approaches agree well for the sections they were both calibrated on, and they disagree in their intelligibility. One of them produces a number and a converged effective section; the other produces two slendernesses whose ratio can be plotted, which is why the curve at the top of this page can be drawn at all.

The hot-rolled version of the same question is reduced to four classes and a table, and it can be, because where the plates are stocky the interaction vanishes and a classification is enough. Cold-formed sections live where a classification is not, which is why they need two slendernesses rather than one class — and why the ratio between those two slendernesses is a quantity the older method never had a name for.

What the pictures cannot show #

The curves are drawn against a ratio of critical loads, which is a number nobody has. A real section’s local and distortional critical loads come from a finite-strip analysis of the cross-section — a curve of critical stress against half-wavelength with two minima in it — and reading those minima off is the practical step that all of this depends on.

Nor can any of these figures show the buckled shape, which is the thing that makes the modes recognisable. A local buckle is a ripple a hand would cover; a distortional one opens the section up; a global one bends the whole member. They look nothing like each other and are drawn here as three numbers on an axis.

The interaction is also invisible in the one place a designer might expect to meet it, which is the deflection. A cold-formed column whose plates have buckled locally is softer than the analysis says — its effective second moment has fallen — and the extra sway that produces feeds back into the global mode as a second-order effect. Nothing in the strength calculation reports it, and a member checked only for capacity has no indication that its stiffness has changed at all.

The assumption the figure rests on #

The column’s global critical load is taken as flexural, $\pi^2EI/L^2$ about the weak axis. A cold-formed section is thin-walled and open, so its torsional stiffness is very small, and the governing global mode is usually flexural-torsional rather than flexural — at a load that can be well below the flexural one. Using the flexural load overstates the global capacity, which understates the erosion, in exactly the region where the erosion is largest.

The ladder from here #

Later rungs on this anchor: distortional buckling and its own interactions, which are the least settled part of this subject. The finite-strip signature curve, which is where the critical loads actually come from. Koiter’s coupled-mode sensitivity, and the amplification an imperfection gets when two modes share a critical load. The effective-width method’s treatment of the shifted centroid, which the direct-strength method hides. And the same interaction in hot-rolled sections, where the plates are stocky, the erosion is zero, and the whole question quietly disappears — which is why it took thin sheet steel to make anybody ask it.