The best design is the most sensitive one

Fix the amount of steel. Fix the length and the end conditions. The only decision left about a circular hollow section is its proportion — how much of the area goes into radius and how much into wall — and it is the cleanest optimisation problem in the subject, because everything else has been held.

It also has an answer that is a warning rather than a recommendation.

Two curves, going opposite ways #

With the area $A$ fixed, the wall thickness follows from the radius: $t = A/2\pi r$. Everything else follows from that.

The second moment of a thin tube is $I = \pi r^3 t$, and substituting gives $I = A r^2/2$. So the Euler load rises as the square of the radius:

$$N_E = \frac{\pi^2 E A r^2}{2L^2}.$$

That is the whole reason for hollow sections, and it is why the instinct is always to make the tube bigger.

The local buckling stress of a cylinder in compression is $\sigma_{cl} = \alpha E t/(r\beta\sqrt3)$, and substituting the same thing gives $\sigma_{cl} = \alpha E A / (2\pi r^2 \beta\sqrt3)$: the local buckling load falls as the inverse square of the radius. A larger tube of the same area has a thinner wall, and a thinner wall ripples sooner.

The capacity is the lower of the two, capped by squashing. A minimum of a rising and a falling curve has a maximum, and it is at their intersection.

Where the maximum is #

Setting the two equal and solving for the radius gives

$$r^* = \left[\frac{\alpha A L^2}{\pi^3 \beta \sqrt3}\right]^{1/4},$$

a closed form with a fourth root in it, and the sampled capacity curve peaks within half a per cent of it. For the member drawn — 8,000 mm² of S355 over 24 m — that is $r^* = 405$ mm, a wall of 3.14 mm, a proportion of $r/t = 129$, and a capacity of 2,364 kN against a squash load of 2,840.

Two things about that expression are worth reading rather than evaluating.

The length appears, to the power a half. A longer member wants a proportionally fatter, thinner-walled tube — which is the opposite of the intuition that a long member is at risk and should be made robust.

And the knockdown factor $\alpha$ appears, to the power a quarter. That factor is the number nobody can predict — a statement about how round the tube is and how straight its seam, decided in a rolling mill. So the optimum proportion of a column depends on the quality of its manufacture, which is not a variable any design office has.

The load is not in the answer #

There is a quantity conspicuously absent from $r^*$, and its absence is the most useful thing on the page for anyone actually sizing a member.

The applied load does not appear. The optimum proportion depends on the area, the length, the modulus and the knockdown factor, and on nothing about what the column is carrying. That is not an accident of the algebra: the load enters only through the requirement that the capacity be sufficient, and the capacity’s shape as a function of proportion is fixed once the area is.

The design procedure that follows is therefore two steps rather than one. Choose the proportion from the length — it is the same proportion whatever the load — and then choose the area from the load. Most sizing is done the other way round, by picking a section from a catalogue that satisfies the load and checking its proportions afterwards, which arrives at a member that is adequate and is not on the curve at all.

It also means the optimum proportion is a property of a family of members rather than of one. Every column of a given length in a given structure wants the same $r/t$, at whatever area its own load requires — which is why a well-proportioned mast or tower looks self-similar down its height, and a building’s columns, sized individually from a catalogue, do not.

The plateau, and when it exists #

Everything above assumes the two buckling modes cross below the squash load. Often they do not, and then the answer changes shape completely.

If the material can be squashed before either mode arrives, there is a whole range of radii that all reach $f_y A$ — bounded below by column buckling and above by local buckling — and inside that range the capacity is flat. The design is free: any proportion in the band gives the same capacity, and the choice can be made on other grounds entirely.

For the same 8,000 mm² of S355 at ten metres rather than twenty-four, the plateau runs from $r/t = 27$ to $r/t = 106$. That is nearly every hollow section in a catalogue, which is why this whole problem is invisible in ordinary building design.

A stronger steel closes it sooner #

The plateau’s upper edge is set by local buckling reaching yield, at $r/t = \alpha E/(f_y\beta\sqrt3)$ — inversely proportional to the yield strength. The lower edge is set by column buckling reaching yield, which also tightens as the strength rises. So the whole band narrows with strength, from both sides at once.

The length at which it closes goes as $1/f_y$: 20.0 m for S355, 15.4 for S460, 10.3 for S690, 7.4 for S960.

That is a genuinely awkward result, and it is a specific case of something general. Strength buys less as it gets larger, because the constraint that eventually governs is a stability one and stability does not know about yield strength. Doubling the yield doubles the squash load and leaves the Euler load and the local buckling stress exactly where they were, so the member arrives at the point where two instabilities meet at half the length.

High-strength steel is therefore not a drop-in substitution in compression members. It is a substitution that moves a member toward a configuration where the theoretical capacity is highest and the sensitivity to workmanship is worst.

Which free body produced the number #

There is no single cut here, because the quantity being computed is a comparison between two different free bodies — and saying which two is the whole content.

Euler’s free body is the whole member: cut it at midspan in its buckled configuration, take moments about the cut, and balance the internal moment $EI\,y''$ against the external $N y$. The length is in it because the buckled shape is a half sine over the length.

Local buckling’s free body is a small element of the wall: a patch of shell a few $\sqrt{rt}$ across, buckling into dimples, balancing its own bending stiffness $Et^3$ against the membrane compression and the curvature that resists it. The member’s length is not in it at all.

The two are different structures asked the same question, and the optimum is where they happen to give the same answer. Nothing physical connects them; the coincidence is arranged by the designer, and that is precisely why it is dangerous.

Why coincidence is the dangerous case #

A structure with one critical mode has a post-buckling path — a definite thing it does after buckling — and that path decides whether an imperfection matters. A column’s path is nearly flat, so its imperfection sensitivity is mild. A cylinder’s path drops steeply, so it is severe.

When two modes have the same critical load the paths interact, and the result is worse than either alone. Koiter showed that the interaction produces an imperfection sensitivity of a higher order: the capacity falls as a lower power of the imperfection than for either mode separately, so a small deviation costs more than it would in either pure case.

This site has that argument already, made about members where the coincidence was an accident of proportion. The point here is that it is not always an accident. An optimisation against two failure modes produces coincidence by construction, because the maximum of a minimum of a rising and a falling function is always at the crossing. Anyone who optimises will land there.

How much the peak is worth giving up #

The redeeming feature is that the peak is flat, and the arithmetic of how flat decides how much robustness costs.

Near a maximum, a function is quadratic: moving off the optimum by a fraction $\delta$ of the way costs about $\delta^2$ of the capacity. Moving ten per cent off the optimum proportion costs a per cent or so of capacity, and it moves the member decisively onto one side of the crossing, where one mode governs by a clear margin and the interaction sensitivity is gone.

Robustness is nearly free and the optimum is nearly worthless, which is the practical conclusion and the reason no code writes the optimum down. What codes write down instead is a limit on $r/t$ — a bound on the upper side of the crossing — which keeps the member on the column-buckling side where its behaviour is understood, its imperfection sensitivity is mild and its post-buckling reserve exists.

That is worth restating because it makes a familiar rule legible. A slenderness limit is not a strength requirement. It is a requirement that one mode governs clearly, and its purpose is to prevent exactly the coincidence an optimisation would seek.

The same argument in every other pair #

Once the shape of the argument is visible it is everywhere, and the examples are worth listing because none of them is usually presented this way.

A plate girder web has a shear buckling limit and a bending limit, and a web optimised against both has them coincide; codes impose a depth-to-thickness limit that keeps one clear.

A stiffened panel can buckle between stiffeners or as a whole, and the stiffener rigidity that makes the two coincide is exactly the threshold value — past which the whole-panel mode is suppressed. Designing at the threshold is designing at a coincidence.

A cold-formed section has local, distortional and global modes, and the proportions that make any two of them meet are the ones the signature curve shows as a shallow minimum.

Shape is a separate lever from proportion #

It is worth separating this optimisation from the one it is easily confused with, because the two answer different questions and only one of them has a maximum.

Putting the same steel in a different shape is an argument about where material sits relative to the axis being bent about, and it has no interior optimum at all — further out is always better for the second moment, and the argument runs until something else stops it. This page is about what stops it.

So the two are halves of one statement. The shape argument says spread the material; the proportion argument says how far, and answers with a number that depends on the length and on the manufacturer. A section catalogue is the record of a compromise between them made by somebody else, for a range of lengths nobody specified.

The same split appears whenever a section is being chosen rather than checked. Depth is nearly always worth having and is limited by something that is not bending; width is nearly always worth having and is limited by something that is not compression. The interesting number is never the benefit; it is the constraint that ends it.

What a real catalogue does about it #

Manufactured hollow sections do not span the range this page plots. A structural CHS catalogue runs from about $d/t = 10$ to $d/t = 50$, which is $r/t$ from 5 to 25 — well below the theoretical local buckling limit of 107 for S355, and far below the optimum of 129.

That is not conservatism about buckling. It is the manufacturing process: a tube much thinner than that cannot be rolled, welded and handled without denting, and a dent is precisely the imperfection $\alpha$ is a measure of. The section range available is a statement about fabrication that happens to keep every ordinary member on the safe side of this problem, and the members that leave that range — masts, spun concrete poles, spacecraft booms, silo walls — are exactly the ones where the problem becomes real.

That is also why the effect is unfamiliar to most structural engineers and entirely familiar to anyone who designs launch vehicles, where the area is genuinely fixed by mass and the optimisation is genuinely performed.

The one place ordinary structural practice does meet it is in fabricated members, where the proportion is not chosen from a catalogue. A welded box column, a tubular mast made from rolled plate, a spun concrete pole: in each of those the wall thickness is a free variable and somebody has to pick it. The rules that constrain the pick — the class limits, the $d/t$ caps — are then doing the whole of the work described here, and they are usually read as though they were about local strength rather than about keeping two modes apart.

The evidence, and what would refuse it #

The claim being made is falsifiable in two ways, and both are worth stating because a stability argument that cannot be checked is a story.

The closed form against the sampled curve. The expression for $r^*$ was derived by setting two algebraic expressions equal; the peak was found by evaluating the capacity at 241 proportions and taking the largest. They agree to 0.5 per cent. If the algebra were wrong, or if the capacity were not the minimum of the two curves, they would not.

The plateau’s existence and disappearance. The model predicts a band of proportions all reaching the squash load for short members and no band at all for long ones, with the transition at a computable length. At ten metres the band is $r/t = 27$ to 106; at twenty-four there is none. That is a qualitative change predicted from the same two expressions, and a member designed at twenty-four metres that reached its squash load would refuse the model outright.

What the model cannot be checked against here is the sensitivity claim, because imperfection sensitivity is a statement about scatter rather than about a mean, and scatter needs a population. The evidence for it is elsewhere — in the shell test data where the classical stress is reached by nothing — and this page borrows it rather than establishing it.

Where the model stops #

The local buckling stress is a classical value with a knockdown. The classical stress is exact for a perfect cylinder and is never reached; $\alpha$ carries all of the discrepancy and varies between manufacturers by more than the optimum is sensitive to.

The two modes are treated as independent until they meet. Real interaction begins before coincidence, so the capacity near the peak is lower than the minimum of the two curves, and the peak is flatter than drawn.

The material is elastic. Near the squash load neither buckling stress is elastic, and both curves should be reduced by an inelastic factor that this arithmetic does not have.

Only one section shape is considered. A circular tube has one proportion to choose. A square hollow section, an I-section or a built-up member has two or three, and the optimisation becomes a surface with ridges rather than a curve with a peak — but the structure of the argument, that the summit sits where modes coincide, survives the extra dimensions.

And the area is not really fixed. It is a proxy for cost, and cost is not proportional to area: a very thin wall costs more per kilogram to roll, to handle and to weld, so the real objective function has a term this one does not.

Where the ladder goes #

Later rungs on this anchor: Koiter’s interaction theory and the order of the sensitivity at coincidence. Optimum proportions for I-sections, where there are three variables rather than one. Slenderness limits read as mode-separation requirements rather than as strength ones. Simultaneous-mode design in aerospace, where the optimisation is performed and the sensitivity is managed rather than avoided. Manufacturing tolerance as the input to $\alpha$. The same argument for stiffened panels and for cold-formed sections. And the general question this belongs to: what an optimum is worth when the objective is a lower envelope of several failure modes, and every one of them is uncertain.