The column that was never straight

Euler’s analysis asks a strange question and gives a beautiful answer to it. Given a perfectly straight, perfectly elastic column loaded exactly along its axis, at what load does a deflected shape become possible? The answer is $\pi^2 EI/L^2$, and below that load the only equilibrium position is the straight one.

The strangeness is in the setup. Nothing about a real column matches it. Rolled sections come out of the mill with a bow of perhaps a thousandth of their length; the load is applied through a connection whose centroid is not exactly the section’s; the section carries residual stresses from cooling that make one part of it yield before another. The perfectly straight column is not an idealisation of a real column in the way a frictionless pin is an idealisation of a bolt. It is a different object, with a qualitatively different response.

Bifurcation and amplification are different phenomena #

The perfect column exhibits a bifurcation: below $P_{cr}$ there is one equilibrium position, above it there are three, and the straight one has become unstable. Nothing deflects at all until the critical load, and then the behaviour changes character discontinuously.

The imperfect column exhibits an amplification. It is bent from the start, so the load acting through that eccentricity produces a moment, which bends it further, which increases the eccentricity. The process converges for small loads and the deflection settles at

$$\delta = \frac{\delta_0}{1 - P/P_{cr}}$$

which is the whole of it. At a tenth of the critical load the bow has grown by 11%. At half, it has doubled. At nine-tenths, it is ten times its original size, and the moment $P\delta$ has grown by a factor of nine as well.

The important structural consequence is that the imperfect column does not fail by buckling at all. It fails when the combination of axial stress and the bending stress from the amplified bow reaches yield, and that happens at a load below $P_{cr}$ — often far below, for a column of intermediate slenderness. Euler’s load has become an asymptote rather than a capacity.

That dip is the whole reason design codes do not use Euler’s formula. They use a curve fitted to tests, of which several exist for different section types and manufacturing routes, and the differences between those curves are differences in the imperfections the sections carry rather than in any mechanics.

The plot that measures what cannot be reached #

Here is the part that is genuinely surprising: the critical load of a column can be measured accurately from tests that never load it above about half of that value, and never damage the specimen.

Rearrange the amplification equation. The quantity that has to be plotted is the growth in deflection, $y = \delta - \delta_0$, rather than the total. With that substitution the relation rearranges to

$$\frac{y}{P} = \frac{y}{P_{cr}} + \frac{\delta_0}{P_{cr}}$$

so plotting $y/P$ against $y$ gives a straight line whose slope is $1/P_{cr}$ and whose intercept is $\delta_0/P_{cr}$. Using the total deflection instead leaves a $\delta_0/P$ term on the right, which is not a constant and not a line — a distinction easy to lose and impossible to hide, since the resulting fit returns a negative critical load. So a handful of measurements of deflection against load, at loads well inside the safe range, give both the critical load and the initial bow — the second of which is a quantity nobody measured directly and which the specimen was not asked to report.

The plot is due to Southwell in 1932, and its practical virtue is that it is non-destructive. A structure in service can be instrumented, loaded within its working range, and its buckling capacity inferred. It has been used on aircraft fuselage panels, on cooling towers, and on historic cast-iron columns where breaking one to find out was not an option.

Its second virtue is subtler. The method does not require the imperfection to be known, or even for it to be a simple bow — any imperfection whose shape resembles the buckling mode produces the same hyperbola, because the mode is what gets amplified. The plot therefore tolerates ignorance about the very thing that makes the column imperfect.

Which free body produced the number #

The free body is the column’s upper half, cut at mid-height, with the axial load $P$ at the top and the internal actions on the cut.

Moments about the cut: the load $P$ acts at a lateral offset $\delta$ from the cut, so it applies $P\delta$. The internal bending moment at the cut must equal it. That moment produces curvature $M/EI$, whose double integral over the half length gives the deflection — and the deflection appearing on both sides of that statement is the feedback loop the amplification describes.

For the sinusoidal case it closes exactly. An initial shape $\delta_0 \sin(\pi x/L)$ under load $P$ takes the deflected shape $\delta \sin(\pi x/L)$ with $\delta = \delta_0/(1 - P/P_{cr})$, and substituting back satisfies the governing equation identically. The Southwell figure fits a least-squares line through six points computed from that relation and recovers the two constants it was built from, which is the check the generator is held to: the fit has to return the $P_{cr}$ and the $\delta_0$ it was given, and it does.

The assumption the figures rest on is that the material stays elastic throughout. The amplification equation has no yield in it, so the curve rises indefinitely toward the asymptote; a real column leaves the curve when its extreme fibre yields, which for a stocky one happens at a small fraction of the critical load. Every one of these curves is therefore truncated in reality at a height the drawing does not mark.

There is no floor under that, which is worth showing rather than asserting.

Three tests, at 75, 35 and 15 per cent, returning the same critical load to four figures. That is the practical content of the method: a specimen can be tested to a load that damages nothing, and the number that comes out is the one the specimen would have failed at. It is why Southwell’s plot is the standard way of getting a buckling load out of a structure nobody is willing to break, and why it is used on full-scale members and on aircraft panels rather than only on laboratory columns.

What the imperfection is, physically #

“Initial bow” is a convenient single number for a collection of quite different things, and they are worth separating because they are controlled by different people.

Geometric bow. The member is not straight as delivered. Mill tolerances put this at about $L/1000$ for rolled sections, and it is the number codes use when they need one. A member five times worse than that is not five times worse off.

Which is the reassuring half of the picture and the misleading half at once. The deflection at any given load is proportional to the bow, so a member five times out of tolerance deflects five times as much — but the load at which it becomes unserviceable is not five times lower, because the amplification is what it is regardless. What the bow costs is spent in deflection rather than in capacity, and the capacity is taken away by the other three items in this list.

Load eccentricity. The load does not arrive on the section’s centroid. A beam framing into a column flange delivers its reaction at the face, not the centre, and the eccentricity is a design decision recorded — or not — on a connection drawing.

Residual stress. The section carries locked-in stresses from uneven cooling, and they cause yield to begin somewhere in the section at an applied stress well below the nominal yield. This behaves like an imperfection because it reduces the effective stiffness before anything visible happens.

Erection tolerance. The column is not plumb, and neither is the frame it stands in. This is an imperfection of the frame rather than of the member, and it is the notional load that sway calculations apply.

All four are handled in codes by a single equivalent imperfection calibrated against tests, which is a reasonable engineering compression of four unlike quantities and is worth knowing about, because it means the “imperfection” in a code is a fitted parameter rather than a measured length.

The same mathematics, three appearances #

The factor $1/(1 - P/P_{cr})$ is one of the most reused expressions in structural engineering, and its reappearances are worth collecting because they are the same phenomenon rather than an analogy.

Second-order effects in a frame. The frame version of the same loop. A sway frame carrying vertical load has its horizontal deflections amplified by exactly this factor, with $P_{cr}$ the frame’s elastic critical load. The notional-load method and the amplified-sway method are two ways of applying it.

Beam-columns. A member carrying both moment and axial force has its moment amplified by the same expression. Every interaction equation in every steel code carries a version of it, and lateral-torsional buckling has its own, with the critical moment in place of the critical load.

Ponding. A flat roof collecting rainwater deflects, the deflection makes room for more water, and the extra water increases the deflection. The governing ratio is different — it is the weight of water admitted per unit sag divided by the stiffness resisting it — but the equation is identical, and so is the conclusion: below a critical value the water depth converges, above it the roof fills until it fails. That case is the water that will not run off, and it is this essay’s figure with different axis labels.

The unifying description is that all three are positive feedback with a finite gain, and the critical load is the gain at which the loop closes. It is the structural version of a phenomenon that turns up in every discipline that has feedback, and the hyperbola is the signature.

Where the model stops #

Yield. Already noted, and it is the biggest omission. Everything the elastic curve says about loads near $P_{cr}$ is a description of a column that has long since passed its first yield. The elastic amplification is only a description until the first fibre yields, after which the effective stiffness falls, the effective $P_{cr}$ falls with it, and the amplification accelerates faster than the formula says. The tangent-modulus and reduced-modulus theories are attempts to carry the calculation past that point.

Residual stresses. A hot-rolled section cools unevenly, and the parts that cool last end up in tension with the rest in compression. Those stresses are self-equilibrating and invisible to any load calculation, and they cause yield to begin at a lower applied load than the nominal stress suggests — which is why two identical-looking sections made by different processes have measurably different column curves.

Slenderness range. The elastic amplification describes a slender column well and a stocky one hardly at all. For a column whose squash load is below the Euler load, yield arrives before the amplification has done anything interesting, and the failure is a squash with a small bending correction. The crossover slenderness is where the description changes hands.

Mode shape. The Southwell method assumes the imperfection resembles the mode being amplified. An imperfection orthogonal to the mode is not amplified at all, and a specimen with a symmetric bow tested for an antisymmetric mode produces a plot that is not a line.

Large deflections. The equation is derived on small-deflection theory. A very slender elastic strut can be pushed far past $P_{cr}$ and will follow the elastica, a large-displacement solution in which the load rises slowly with deflection rather than remaining at the critical value. This is real and mostly irrelevant to steel columns, which yield long before.

The figures cannot show the thing that makes this subject dangerous, which is how sudden the end is in practice. The hyperbola looks gentle: it rises smoothly and its steepness is a matter of scale. A column at 85% of its capacity has deflected visibly and a person could point at it; a column at 95% has deflected a great deal more and is still standing. What the curve does not convey is that the last increment of load is applied by something — a gust, a crane, another storey — that does not know it is the last increment.

Only the part along the mode is amplified, and that is why shells are different #

The observation that an imperfection orthogonal to the mode is not amplified is more useful than it looks, because it explains the single largest division in the whole subject of stability.

A real imperfection is not shaped like anything in particular. Decompose it into the structure’s buckling modes and each component is amplified by its own factor, $1/(1 - P/P_n)$, using that mode’s own critical load. The response is the sum.

For a column that sum has one term in it. The modes are at $P_1$, $4P_1$, $9P_1$ — well separated — so at 80 per cent of the first critical load the amplifiers are 5.0, 1.25 and 1.10. The first mode’s component is magnified five times and everything else is essentially untouched, which is why a column’s behaviour is described so well by a single hyperbola and why its imperfection sensitivity is mild.

A cylindrical shell is the opposite. Its modes are not spread out: dozens of different buckling patterns — different numbers of circumferential waves, different axial half-wavelengths — have critical loads within a few per cent of one another. So at 80 per cent of the lowest, dozens of components are each being amplified by something near 5, and they add.

That is the mechanism behind the notorious result that a real cylinder buckles at a fraction of its theoretical load. It is not that shells have larger imperfections; it is that they have clustered modes, so any imperfection whatever has a large projection onto something near-critical, and the amplifications accumulate rather than being dominated by one.

Which gives a way of predicting imperfection sensitivity before computing anything. Ask how far apart the modes are. A structure whose lowest critical load is well below its second is forgiving; one whose modes crowd together is not. Columns, beams and plates are in the first family; cylindrical shells, spherical caps, stiffened panels and closely braced frames are in the second, and every one of them is a form where the theoretical capacity has to be knocked down by a factor rather than corrected by a term.

It also says which imperfection to assume when one has to be. The code’s answer — a bow shaped like the buckling mode, of amplitude $L/1000$ or thereabouts — is not a guess at what a member looks like; it is the component that matters, applied at a magnitude chosen to cover the population. An imperfection specified in any other shape would have to be larger to do the same work, and one specified in the mode shape is the smallest assumption that is safe.

The generalisation #

The move from bifurcation to amplification is a general one and it changes what kind of question is being asked.

A bifurcation analysis asks for an eigenvalue: at what load does an alternative equilibrium appear? It is a linear problem, it is cheap, and its answer is a number with no deflection attached. An amplification analysis asks for a response: given the imperfections, what does the structure actually do? It is a nonlinear problem whose answer is a curve, and the critical load appears in it only as an asymptote nothing reaches.

Almost every stability problem in the subject can be posed either way, and the two questions have different uses. The eigenvalue is what a code needs to classify a frame as sway or non-sway, and it is what the Southwell plot measures — it is also what the effective length is a repackaging of. The response is what decides whether the structure is adequate. The habit worth carrying is that an eigenvalue is a property of a structure that does not exist, and its usefulness comes from being a good predictor for the structures that do.

Robert Hooke and Euler both worked on struts; Euler’s 1744 result was regarded for a century as a mathematical curiosity, because the columns being built were stocky enough to crush, and the formula gave capacities so far above observed strengths that engineers concluded it was wrong. It was not wrong; it was answering a question about a column nobody had. Ayrton and Perry supplied the correction in 1886 by putting an initial curvature into the analysis — a move that converts an eigenvalue problem into a stress problem and, in doing so, makes the answer depend on a quantity that has to be assumed rather than derived — which is the origin of every modern column curve, and Southwell’s plot arrived nearly fifty years after that as a way of measuring the number the correction was correcting.

The ladder from here #

Later rungs on this anchor: inelastic buckling and the tangent modulus. Residual stresses, measured rather than assumed. The interaction of axial force and moment, and the surfaces that describe it. Imperfection sensitivity as a property of the structural form, where shells are the extreme case and can fail at a fraction of their theoretical load. Snap-through buckling, which is a different instability entirely. And the effect of a restraint partway along, which changes the mode and therefore the whole question — the subject of the brace that need not be strong.