Structural Engineering & Statics
where the load goes
Stability

Two ways of buckling at once

A thin-walled column can bow as a whole or ripple in its plates, and each has its own critical load. The received advice is that the worst arrangement is the one where the two are equal. The arithmetic says the opposite — at coincidence the interaction costs two per cent, and the expensive region is where the plates go first.

Assumes The plate that ripples, and the width that is left, Strong enough and still falls over and What is left after it ripples.

19 min read 8 figures Which failure arrives firstGeometry beats materialScale changes everything

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 π2EAr2/L2\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.

Not where the two loads meetHow much a column loses below the weaker of its two single-mode capacities, against the ratio of its local critical load to its global one. The received claim is that the worst place is where the two coincide; the arithmetic says otherwise. The erosion is largest at a ratio of 0.47 — 23% — sits within a per cent of that for every ratio below about a half, and at exact coincidence is only 2%. What the curve does say is the useful half of the folk claim: once the plates are stocky enough that the local critical load is twice the global one, the interaction is nothing at all, and the section is worth thickening only up to there.0.100.323.1610.000%5%10%15%20%25%30%local critical load ÷ global critical load (logarithmic)capacity lost below the weaker single modethe two coincide hereworst: 23% at 0.47at coincidence the loss is 2%
Fig. 1 The interaction, measured as the capacity lost below the weaker of the two single-mode answers. The received claim puts the worst point at a ratio of one, and the curve is nearly flat there. The expensive region is everywhere the plates go first, and it is expensive by about the same amount throughout.

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 Py=AfyP_y = Af_y and the Euler load PcreP_{cre} of the whole member. The column curve reduces the first for the slenderness λc=Py/Pcre\lambda_c = \sqrt{P_y/P_{cre}} and returns PneP_{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 PcrℓP_{cr\ell} against PneP_{ne}, not against PyP_y. The slenderness that enters the local reduction is λℓ=Pne/Pcrℓ\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.

The load did not move; the section didA lipped channel 200 by 65 mm at 2 mm thick, drawn twice on top of itself: the outline as fabricated, and the part of it still working once the plates have buckled. The web is held on both edges, so it loses its middle; the flanges are held at the web, so an unlipped one would lose its free edge. What survives is not symmetric with what was drawn, so the centroid moves 8.0 mm — and a load applied along the axis it was designed to arrives 8.0 mm off the section that has to carry it. At the 177 kN this section will take, that is 1.42 kNm of bending nobody applied.8.0 mmas drawnas it workswebλ 2.11 · keeps 43%flangeλ 0.68 · keeps 100%lipλ 0.61 · keeps 100%area left69%eccentricity8.0 mm256 kN of section, 177 kN of it effective, and 1.42 kNm that was not in the load case
Fig. 2 The section after the plates have gone. Local buckling removes the middle of each plate and leaves the edges working, which changes the area, the second moment and — the part that matters for interaction — the position of the centroid. A section that has buckled locally is not the section the global check was run on.

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 PneP_{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 PneP_{ne}, or the stub column’s local capacity Pnℓ0P_{n\ell 0} computed against the squash load. Measure against min⁡\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.

The local check depends on the column's lengthThree capacities against slenderness, all of one section. The upper curve is the global check alone — a squash load reduced for the member's own bowing. The middle one is the stub column, which is the local check alone and has no length in it at all, so it is flat. The lower curve is what the section actually carries, and it is below both: the plates ripple under whatever stress the global check has already allowed, so the local reduction is applied to a smaller number for a short column than for a long one. That is the interaction, and it is why a local buckling check that does not know the length is not a check on this member.5010015020025000.20.40.60.81slenderness L ÷ rcapacity ÷ squash loadglobal alonethe stub columnwhat it carriesat L/r = 75 the section loses 21% below the weaker single-mode answer
Fig. 3 Three capacities of one section against slenderness. The stub column’s line is flat, because local buckling knows nothing about length. The global line falls as a column curve. What the section carries is below both, and the gap between it and the lower of them is the interaction — largest where the two lines are of comparable height and different, rather than where they cross.

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.

Three paths out of the same critical loadLoad against sideways movement past the critical load, for three systems whose critical loads are identical. The stable one climbs, so a real structure with a small crookedness reaches nearly the full load and keeps going. The unstable one falls symmetrically, so the imperfect structure has a maximum below the critical load and it matters not at all which way it leans. The asymmetric one falls one way and climbs the other, so the direction of the imperfection decides everything. All three are drawn at an imperfection of 0.02 radians.stable symmetric — a columnan imperfection is a nuisancecritical89%unstable symmetric — a shellan imperfection is a demolitioncritical77%asymmetric — a frameand it matters which waycritical
Fig. 4 The post-buckling paths that the sensitivity argument is about. A stable symmetric path shrugs off an imperfection; an unstable one collapses at a fraction of the theoretical load. Mode coupling turns the first into something closer to the second, and it does so without changing either mode’s critical load.

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.

Length costs more than it looksThe same column section at four lengths, with the buckling capacity of each drawn as a bar. Capacity falls as the inverse square of the length, so a column three times as long carries a ninth as much.1× the length100% of the capacity1.5× the length44% of the capacity2× the length25% of the capacity3× the length11% of the capacityidentical section, identical material, identical end conditions
Fig. 5 What length costs a compression member, which is the pressure that makes a designer thin the plates in the first place. A section pushed outward for global stiffness has thinner plates for the same weight, and the two modes converge as a direct consequence of the optimisation.

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 PP 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.

Twice the metal is more than twice the sectionThe squash load of the effective section against its thickness, with the gross section's above it. The gross line is straight, because area is linear in thickness. The effective one is not: a thicker plate is both larger and less slender, so it keeps a greater fraction of itself as well as being bigger, and the capacity goes as t^1.43 fitted over the whole sweep. The gap between the two lines is what local buckling has taken — 31% of the section at 2 mm — and it closes only at a thickness at which nobody would be cold-forming anything.1.02.03.04.05.06.07.08.002004006008001000thickness (mm)squash load of what is left (kN)the section drawnthe section that works177 kNfitted power 1.43 · a section that never buckled gives exactly 1
Fig. 6 The section’s capacity against its plate thickness, which is the parameter the whole interaction is a function of. It rises as t1.43t^{1.43} rather than as tt, because thickening the plates raises both the area and the stress those plates can reach — two effects of one change, which is what makes thin-walled sections behave unlike solid ones.

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 2 mm plate, and the width it can beThe elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 93 mm the plate reaches yield before it buckles; above it the plate ripples first, and the fraction of the width still carrying load falls away — at 400 mm only 22 per cent of it is still working.501001502002503003504000200400600plate width (mm)slender beyond 93 mmyieldcritical stress — inverse square in the widthwhat the plate actually delivers, over its full width
Fig. 7 The mode this all starts from. A plate’s critical stress depends on its width-to-thickness ratio and its edge conditions, and not on its length — which is what makes local buckling a section property and what makes the interaction with a length-dependent mode interesting at all.

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 fyf_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.

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.

Where the class limits come fromThe width-to-thickness ratio at which two kinds of plate reaches its own elastic critical stress at the yield stress, for three steel grades. A flange outstand (buckling coefficient 0.43) derives to 18.6, 15.2, 13.3 at 235, 355, 460 N/mm², against quoted limits of 14.0, 11.4, 10.0; A web, in bending (buckling coefficient 4) derives to 56.8, 46.2, 40.6 at 235, 355, 460 N/mm², against quoted limits of 42.0, 34.2, 30.0. The derived number is the larger every time, and by the same factor at every grade — flange outstand 1.33, web, in bending 1.35 — because both the derivation and the quoted limit go as one over the root of the yield stress. A constant ratio is what a fixed knockdown looks like: the derivation is for a perfect plate and the quoted limit is for a rolled one, carrying residual stress and not quite flat.flange outstandk = 0.4318.6 at 23515.2 at 35513.3 at 460quoted: 14ε1.33× the quoted limit, at every gradeweb, in bendingk = 456.8 at 23546.2 at 35540.6 at 460quoted: 42ε1.35× the quoted limit, at every grade0102030405060width ÷ thickness
Fig. 8 The hot-rolled version of the same question, reduced to four classes and a table. 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.

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, π2EI/L2\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.

Where this goes next

1002003004005006007000200400600plate width (mm)slender beyond 370 mmyieldcritical stress — inverse square in the widthwhat the plate actually delivers, over its full width The plate that ripples, and the width that is left A wide thin plate in compression buckles at a stress that has nothing to do with the strength of the material. It then goes on carrying load — the middle drops out, and the edges work harder. 8.0 mmas drawnas it workswebλ 2.11 · keeps 43%flangeλ 0.68 · keeps 100%lipλ 0.61 · keeps 100%area left69%eccentricity8.0 mm256 kN of section, 177 kN of it effective, and 1.42 kNm that was not in the load case What is left after it ripples A thin plate that buckles locally has not failed. It has stopped taking load in its middle and gone on taking it near its edges, so the member is now made of a different section from the one that was drawn — and the new one has its centroid somewhere else, which turns a concentric load into an eccentric one. stable symmetric — a columnan imperfection is a nuisancecritical89%unstable symmetric — a shellan imperfection is a demolitioncritical77%asymmetric — a frameand it matters which waycritical A third of what the theory promised A column with a small crookedness reaches almost its full Euler load. A cylinder with the same relative crookedness reaches a third of its classical one, and the theory is not wrong — what separates them is the slope of the path just past the critical load, which no calculation of the critical load itself can see. 200030004000500060007000800090000200400600800length of the column (mm)critical load (kN)torsionalflexural about yflexural about zthe mode changes at 3442 mmthe lowest root — what the column actually does The column that twists instead of bending Euler's column has one mode. A real column has three, and which of them governs is settled by where the shear centre sits. A cruciform strut buckles by rotating about its own length at a load that does not change no matter how short it is made.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

Cold formedCritical loadDirect strength methodEffective widthEigenvalueGlobal bucklingImperfection sensitivityLocal bucklingMode interactionOptimisationPost bucklingSection classificationSlendernessSquash loadStub column