Structural Engineering & Statics
where the load goes
Stability

Strong enough and still falls over

A column can fail at a fraction of the load its material could carry, by going sideways. Buckling is a failure of stability rather than of strength, and it is decided by geometry.
11 min read 6 figures Which failure arrives firstGeometry beats material

Press down on a metre rule stood on its end and it will not crush. It will bow sideways, suddenly, at a load far below anything the material minds — and if the load is removed it springs back, undamaged.

That is buckling, and it is a different kind of failure from everything else on this site. Nothing broke. No stress reached a limit. The column simply stopped being able to stay straight, and the load at which it stops is set by geometry rather than by strength.

The column curveFailure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it.5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither
Fig. 1 Failure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet.

Two failure modes, and the lower one wins

A column has two ways to fail and takes whichever arrives first.

Squashing is the material running out. The load divided by the area reaches the yield stress, and it happens at P=σyAP = \sigma_y A regardless of length.

Buckling is the column running out of stability. Euler’s formula gives the load at which a perfectly straight, pin-ended column ceases to prefer being straight:

Pcr=π2EIL2.P_{\mathrm{cr}} = \frac{\pi^2 EI}{L^2}.

The two are plotted against each other in the figure. The horizontal line is squashing, which does not care about length; the falling curve is Euler, which cares about nothing else. The governing capacity is the lower of the two, so a column is stocky and crushes or slender and buckles, and there is a crossover between.

The formula’s contents are worth reading. EE and II — stiffness, and how the material is arranged — appear on top; the length appears squared underneath. The yield stress does not appear at all. A column of high-strength steel and one of ordinary steel, identical in shape, buckle at exactly the same load. Buying stronger steel for a slender column buys nothing.

Why the length is squared

The mechanism is a feedback loop that either converges or does not.

Give a straight column a small sideways nudge. The load, now applied to a bent member, has a lever arm equal to the deflection, so it produces a moment. That moment bends the column further, which increases the lever arm, which increases the moment.

Against it works the column’s own bending stiffness, which resists curvature. The competition has two possible outcomes: the stiffness wins and the column returns to straight, or the load wins and the deflection runs away.

The critical load is where they exactly balance. The restoring moment goes as EIEI divided by the square of the length — a longer member has more curvature for the same tip deflection — while the disturbing moment goes as the load times the deflection. Setting them equal gives the inverse square, and π2\pi^2 falls out of the shape the buckled column takes, which is a half sine wave.

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. 2 The 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.

That inverse square is harsher than anything in strength calculations, where the relationships are linear or at worst quadratic in the right direction. It is why the height of a storey matters so much more to a column than its load does, and why a column spliced mid-height without lateral restraint at the splice is a very different member from the one on the drawing.

Slenderness, which combines all of it

Comparing columns needs one number rather than three, and slenderness is it.

Writing I=Ar2I = Ar^2, with II the second moment of area, defines the radius of gyration rr, which is the distance at which the whole area could be concentrated to give the same second moment. Dividing the Euler load by the area gives a critical stress:

σcr=π2E(L/r)2,\sigma_{\mathrm{cr}} = \frac{\pi^2 E}{(L/r)^2},

and λ=L/r\lambda = L/r is the slenderness ratio. One dimensionless number that contains the length, the size and the shape.

The crossover with yielding happens at λ=πE/σy\lambda = \pi\sqrt{E/\sigma_y}, which for ordinary structural steel is about 75 and for high-strength steel is lower — the stronger material crosses over sooner, because its squash line sits higher and meets the same Euler curve earlier. Another way of saying that strength does not help a slender column.

A column is judged by its largest slenderness, which means its weakest axis. An I-section has two very different radii of gyration, so as a column it can only use the smaller — which is the argument for hollow sections, where every axis is equally good.

The same material, four waysFour cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 3 Four sections of identical area. As beams they are ranked by second moment about one axis; as columns they would be ranked by their worst axis, which reverses much of the order.

Why real columns are worse than both curves

The Euler load belongs to a perfectly straight column of perfectly uniform material loaded exactly on its axis. Nothing is.

A real column has an initial bow of perhaps a thousandth of its length, residual stresses from rolling or welding, and a load that arrives slightly off centre. Each of those means the column is bending from the first newton rather than from the critical load, and the bending grows as the load rises.

The consequence is visible in the third curve on the first figure: real columns fall below both the squash line and the Euler curve, and the shortfall is worst near the crossover. That is where the two mechanisms interact — the member is bending enough to matter and stressed enough to yield — and it is where most practical columns sit.

Every steel design code contains a set of column curves derived from testing thousands of real members, precisely because neither theoretical line is safe on its own in that region. The curves differ by section type and by axis, because residual stresses differ by how the section was made.

The load that makes itself worseThe amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached.00.20.40.60.80246810applied load ÷ buckling load1.3×1.7×2.5×5.0×10.0×first-order analysis says the answer is always 1×one over one minus the ratio
Fig. 4 The amplification of a deflection against the ratio of applied load to buckling load. An imperfect column is already deflecting, and this factor is what happens to that deflection as the load approaches the critical value.

The amplification curve explains the shape of the failure. A column with an initial bow does not reach the Euler load and then buckle — it deflects progressively more as the load rises, and it fails when the combined bending and axial stress reaches yield somewhere. The load makes itself worse, and the buckling load is the asymptote of a process rather than an event.

What it does to a beam

Buckling is usually described as a column problem, and the same instability governs beams.

The deflected shape is the moment, integrated twiceA loaded beam with its deflected shape above and its bending-moment diagram below. The shape was obtained by integrating the moment twice and fitting the constants to the supports; the vertical scale is exaggerated enormously.the largest movement, at x = 4.00momentdrawn at roughly three hundred times the real deflection —a beam at its serviceability limit moves about a three-hundredth of its span
Fig. 5 A beam’s deflected shape in its own plane. A deep, narrow beam can also move out of that plane, twisting as it goes, at a moment well below its section capacity.

Lateral-torsional buckling is the compression flange behaving as a column while the tension flange holds it back. The remedy is restraint along the compression flange, and the requirement grows exactly as the section is made more efficient — a deeper, narrower beam is a better beam and a worse one.

The ends decide the length that mattersFour columns of identical height and section, buckling under four sets of end conditions. The effective length factor is the fraction of the column that behaves like a pin-ended one, and the buckling load goes as its inverse square.K = 0.5both ends fixedK = 0.7one fixed, one pinnedK = 1both ends pinnedK = 2fixed at the base, free at the topsame column, same section, four ways of holding the endsthe load at which each buckles goes as 1 ÷ K² — a factor of sixteen across this row
Fig. 6 Four end conditions for a column. The same idea applies to a beam’s compression flange, where the effective length is the distance between points of lateral restraint.

What the ends and the restraints are doing therefore decides a beam’s capacity as much as a column’s, and a beam adequate on a drawing that shows restraints is not adequate on a site that has not installed them.

Where the model stops

Perfectly straight, perfectly axial, perfectly elastic. Covered above, and the reason design uses empirical curves.

Pin ends. The formula as written is for a pin-ended column, which is an idealisation of a support like any other. Other end conditions change the effective length, and the change is large — a factor of sixteen in capacity between the best and worst of the standard four.

Elastic material throughout. Above about half the squash load, parts of a real steel section have yielded because of residual stress, and the effective stiffness is lower than EE suggests. The tangent-modulus theory addresses this and is why the real curve dips.

Overall buckling only. A thin-walled section can buckle locally — a flange rippling or a web folding — at a load unrelated to the member’s overall slenderness. Section classification exists to keep local buckling out of the picture, and a class 4 section cannot reach the capacity the Euler formula implies.

One buckling mode. A beam in bending can buckle sideways and twist at once, which is lateral-torsional buckling and has its own theory. An open section in compression can twist rather than bow.

The figures carry a distortion worth naming. The buckled shapes are drawn with a lateral deflection of tens of pixels against a length of a few hundred, an amplitude of perhaps a tenth of the length. A real column at its critical load has deflected by essentially nothing — the whole point of the critical load is that it is where deflection begins to run away, and the amplitude is indeterminate in the linear theory. Every drawing of a buckled column shows a post-critical state that the calculation cannot predict.

The ladder from here

Later rungs: the Euler derivation from the governing differential equation. Effective length and the four standard cases. The radius of gyration and slenderness. The tangent-modulus and reduced-modulus theories. Imperfections and the Perry–Robertson approach. Column curves and how they were obtained. Local buckling and section classification. Lateral-torsional buckling. Plate and shell buckling, where imperfection sensitivity becomes extreme. And the energy method, which gets the critical load without solving the differential equation at all.

Euler published the result in 1744, and it was ignored by engineers for a century because the columns of the time were stocky enough to crush. It became indispensable the moment iron allowed slender ones.

Where this goes next

K = 0.5both ends fixedK = 0.7one fixed, one pinnedK = 1both ends pinnedK = 2fixed at the base, free at the topsame column, same section, four ways of holding the endsthe load at which each buckles goes as 1 ÷ K² — a factor of sixteen across this row The ends decide the length that matters Four columns of identical height and section, buckling at loads sixteen times apart. Nothing differs but what is holding the two ends. 00.20.40.60.80246810applied load ÷ buckling load1.3×1.7×2.5×5.0×10.0×first-order analysis says the answer is always 1×one over one minus the ratio The load that makes itself worse A structure that has leaned carries its weight off the axis, which makes it lean further. The amplification is one over one minus the load ratio, and it runs away long before the buckling load. the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale The same steel in a different shape, and a factor of forty Four sections of identical area, identical weight and identical cost. The stiffest is dozens of times the stiffest of the flattest, and the only thing that changed was the arrangement.