Structural Engineering & Statics
where the load goes
Stability

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.

Assumes Strong enough and still falls over, The roof that jumps and The load that makes itself worse.

16 min read 8 figures Which failure arrives firstScale changes everything

Euler’s column is the best-behaved thing in structural mechanics. Its critical load is exact, a column built with an initial bow of a thousandth of its length reaches 95% of it, and a century of testing has confirmed the formula to within the scatter of the steel.

A cylindrical shell in axial compression has a critical stress that is just as exact, just as easy to derive, and it is wrong by a factor of three or more on every specimen ever tested. The theory has not failed. What differs is something neither critical load calculation contains.

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. 1 Three systems with identical critical loads, each drawn with the same 0.02 radian imperfection. The first climbs past the critical load and an imperfection is a nuisance. The second falls symmetrically and the imperfect version peaks at 89% — an imperfection is a demolition. The third falls one way and climbs the other, peaks at 77%, and which way it leans is decided by whatever crookedness was there first.

The critical load is the same number in all three. What decides the strength of the real structure is the slope of the equilibrium path immediately past that load, and that slope is invisible to every eigenvalue calculation ever performed.

Three shapes, and only one of them is forgiving

The three cases have names and they are worth attaching to objects.

Stable symmetric. The path rises after the critical load, so the structure can carry more than its critical load once it has bent — it just cannot do so without deflecting. A pin-ended column is this case, and so is a plate, which is why a plate’s post-buckling strength is a design resource rather than a hazard: a slender plate carries several times its critical load by shedding stress to its edges.

Unstable symmetric. The path falls after the critical load and does so symmetrically, so it does not matter which way the structure goes. A cylindrical shell in axial compression is the standard example, and a spherical shell under pressure is worse.

Asymmetric. The path falls one way and climbs the other. A portal frame buckling in sway is the structural example — the same sway that second-order analysis chases — and the consequence is peculiar: the strength depends on the sign of the imperfection, so two nominally identical frames built to opposite tolerances have different capacities.

A column that was never straightLoad against lateral deflection at mid-height, for a column starting with an initial bow of 0.002. There is no critical value to reach: the deflection grows from the first increment, slowly at first and then without bound as the ratio approaches 1.00 — which is the Euler load, and which the column therefore never attains. The perfect column, drawn for comparison, sits on the vertical axis until it arrives there and then has no answer at all.00.0050.010.0150.020.02500.20.40.60.81lateral deflection at mid-heightload ÷ P꜀ᵣP ÷ P꜀ᵣ = 1.00, approached and never reachedinitial bow: δ₀ = 0.002
Fig. 2 The stable case in detail: a column that was never straight, whose deflection grows from the first increment of load and heads for infinity at a critical load it never reaches. There is no maximum on this curve — the column does not fail by buckling at all, it fails when its stresses reach yield, and the critical load is an asymptote rather than an event.

The exponent is the whole subject

Take the unstable case and vary the imperfection. The maximum load falls, and the interesting question is how.

Why a tiny imperfection costs so muchThe load an imperfect structure reaches, as a fraction of the perfect critical load, against the size of the imperfection. Neither curve is a straight line through the origin: fitting the computed maxima gives an exponent of 0.662 for the unstable symmetric system and 0.488 for the asymmetric one — two thirds and a half, which is Koiter's result arrived at by measuring rather than by expanding. Both have infinite slope at zero, which is the whole of imperfection sensitivity: the first thousandth of crookedness costs more than the next hundredth.00.010.020.030.040.0500.20.40.60.81imperfection (radians of initial lean)load reached ÷ critical loadunstable symmetricdeficit ∝ ε^0.66asymmetricdeficit ∝ ε^0.49
Fig. 3 The load an imperfect structure reaches, against the size of the imperfection, for both falling cases. Neither curve is a straight line through the origin: fitting the computed maxima gives an exponent of 0.662 for the unstable symmetric system and 0.488 for the asymmetric one. Both have infinite slope at zero, which is what imperfection sensitivity means — the first thousandth of crookedness costs more than the next hundredth.

Those two numbers are two thirds and one half, and they are Koiter’s, from a doctoral thesis written in occupied Holland in 1945 and unread outside it for fifteen years. What is worth noticing here is how they were obtained: by computing the maximum of an exactly solved equilibrium path at twenty-four imperfection amplitudes and fitting a straight line through the logarithms. The exponents are measured on this site rather than quoted, which is the only arrangement in which agreement with the theory is evidence of anything.

The practical content of an exponent below one is the infinite slope at the origin. For the unstable case,

1−λmax⁡≈1.45 ε2/31 - \lambda_{\max} \approx 1.45\,\varepsilon^{2/3}

so an imperfection of a ten-thousandth costs 0.3%, a thousandth costs 1.5%, and a hundredth costs 7%. Each factor of ten in the imperfection costs only a factor of 4.6 in the deficit, which is why the last increment of straightness is the one that buys nothing and why tolerance specifications on shells reach a point of diminishing returns quickly.

Why a cylinder is the worst case there is

The shell is worse than the model above, and the reason is a genuinely different phenomenon: a cylinder has an enormous number of buckling modes at almost exactly the same critical load.

For an axially compressed cylinder, every combination of axial half-waves and circumferential waves satisfying one relation gives the same critical stress. Dozens of modes are coincident, they interact, and the interaction of several unstable-symmetric modes is far more sensitive than any of them alone. That is why measured strengths cluster at a fifth to a third of theory rather than at the 70% the single-mode calculation would give.

The fraction of the theory anybody dares useThe lower-bound knockdown factor for a cylinder in axial compression, against its radius-to-thickness ratio. A shell at R/t = 500 is designed to 32% of its classical buckling stress — 82 MPa against a theoretical 254 MPa. The curve is empirical, drawn under decades of test results rather than derived, and its shape is the argument: the thinner the shell, the further below theory it falls, because the imperfection that matters is measured against the thickness and a thinner shell is a rougher one in the only units that count.5001000150020002500300000.20.40.60.81radius ÷ thicknessusable fraction of the classical stressR/t = 500 → 32%
Fig. 4 The lower-bound knockdown factor for a cylinder in axial compression against its radius-to-thickness ratio. At R/t = 500 the design stress is 32% of the classical one — 81.8 MPa against 254.2. The curve is empirical, drawn under the test data of the 1930s to 1960s rather than derived, and its shape is the argument: the thinner the shell, the further below theory it falls.

The shape of that curve is the part that carries meaning, and it is the same statement as the exponent. The imperfection that matters is measured against the thickness — a dent of half a millimetre is nothing on a 20 mm plate and is a sixth of a 3 mm one — so a thinner shell is a rougher shell in the only units that count, and the knockdown falls accordingly.

The fraction of the theory anybody dares useThe lower-bound knockdown factor for a cylinder in axial compression, against its radius-to-thickness ratio. A shell at R/t = 100 is designed to 58% of its classical buckling stress — 739 MPa against a theoretical 1271 MPa. The curve is empirical, drawn under decades of test results rather than derived, and its shape is the argument: the thinner the shell, the further below theory it falls, because the imperfection that matters is measured against the thickness and a thinner shell is a rougher one in the only units that count.5001000150020002500300000.20.40.60.81radius ÷ thicknessusable fraction of the classical stressR/t = 100 → 58%
Fig. 5 The same curve with a much stockier shell marked: R/t = 100 gives a knockdown of 0.58 rather than 0.32. The classical stress is 1,271 MPa, which is far past yield for any steel, so this shell fails by squashing and the buckling calculation never governs. The knockdown factor matters exactly where the shell is thin enough for buckling to be the failure — which is where it is worst.

A design rule that is an empirical envelope under sixty years of test scatter is not a theory that failed. It is an admission that the quantity governing the strength is the geometry of the real shell, which is not on the drawing, and that the cheapest way to bound it is to measure a great many shells.

Where the sensitivity comes from

It is worth being precise about the mechanism, because “imperfections reduce strength” is true of everything and explains nothing.

In a stable-symmetric system the imperfect structure deflects early and then finds more stiffness as it goes, so the deflection stabilises and the load keeps climbing. In an unstable one the imperfect structure deflects early and finds less stiffness, so the deflection accelerates and the load reaches a maximum while the structure is still moving in the direction it was pushed.

The second is the same event as a snap-through: a maximum on an equilibrium path, at which the structure has nowhere nearby to go.

A load with a maximum in it, and nothing bifurcatesLoad against apex movement for a two-bar frame of half-span 1000 mm and rise 60 mm. The load rises to 8.7 kN at a movement of 26 mm — well short of the 60 mm that would bring the apex level — and then falls. Past that point the frame can only be held by taking load away, so under a dead weight it goes: 104 mm of movement at constant load, arriving inverted and in tension. The minimum on the path is -8.7 kN, the exact negative of the maximum, because the geometry is symmetric about the flat position and the arithmetic knows it.020406080100120140-10-50510movement of the apex (mm)load (kN)limit point: 8.7 kNit jumps 104 mmas builtat the limitafter it goes
Fig. 6 The limit point of a shallow frame, which is the extreme case of the same behaviour: no bifurcation at all, just a path with a maximum in it. An unstable-symmetric bifurcation is this event reached by a different route — two paths meet, the structure takes the falling one, and the maximum it can carry is a point on that fall.

So the family is one family. A limit point, an unstable bifurcation and an imperfection-sensitive shell are three descriptions of a structure whose stiffness is falling faster than its load can be taken away, and the only question in each case is what put it on the falling branch.

The theory that does work, and what it is for

None of this makes the classical calculation useless, and it is worth stating what it is good for.

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. 7 The column curve: the envelope of squashing and Euler buckling, with real columns falling below both near the crossover. The rounding of that corner is imperfection sensitivity in the stable case — a few tens of per cent at worst, in the region where the two failures compete, and negligible at either extreme.

The critical load fixes the scale of the answer. A knockdown factor is a fraction of it, so an error in the classical stress — in the effective length, say, which the end conditions decide — is an error in the design stress in exact proportion, and the empirical curve is only usable because the theory it multiplies is exact. An empirical factor applied to an exact theory is a different thing from an empirical formula, and the difference is that the first generalises to a shell nobody has tested.

The critical load also identifies the mode, which decides where a stiffener has to go. And it tells the designer which regime the structure is in: below the crossover the shell squashes, above it the shell buckles and the knockdown applies. A calculation that is wrong by a factor of three about the strength can still be exactly right about which failure is coming.

Measuring the imperfection instead of assuming it

The modern alternative to the empirical envelope is to stop treating the imperfection as unknown. Measure the actual geometry of the shell, put it into a nonlinear analysis, and compute the maximum of its own equilibrium path.

That is now practical and it changes the character of the problem. The shell’s strength becomes a computed quantity with a measured input rather than a fraction of an ideal, and the sensitivity curve becomes a way of deciding how accurately the geometry needs to be surveyed. It also produces an uncomfortable finding — one this collection meets whenever it looks at what a section can absorb before it fails — which is that the shape of the imperfection matters as much as its size: an imperfection resembling the buckling mode is far more damaging than a larger one that does not, so a shell with a 5 mm dent in the wrong pattern can be weaker than one with a 15 mm dent in the right place.

The Southwell plot is the older version of the same instinct — extracting a critical load from measurements taken well below it.

Southwell: the critical load, from loads nowhere near itThe growth in deflection divided by the load, plotted against that growth, for six readings taken at up to 75% of the critical load. The relation is a straight line whose slope is the reciprocal of the critical load: the fit returns 1.000 against a true value of 1.000, and its intercept returns the initial bow as 0.0020 against 0.0020.00.0020.0040.0060.00800.0020.0040.0060.0080.01growth in deflection, y = δ − δ₀y ÷ Pslope = 1/P꜀ᵣ → P꜀ᵣ = 1.000intercept = δ₀/P꜀ᵣ → δ₀ = 0.0020
Fig. 8 Southwell’s construction: deflection growth divided by load, against that growth, which is a straight line whose slope is the reciprocal of the critical load. Six readings taken at up to 75% of critical return the critical load to three figures and the initial bow as a by-product. It works beautifully for the stable case and not at all for the unstable one, because there the structure never gets near its critical load before it fails.

The same shape of argument, in another field

This site has met a structure whose strength is decided by the largest flaw in it once already, and the parallel is close enough to be worth drawing precisely.

A brittle plate’s strength is not a material property in the way a yield stress is. It is Kc/(Yπa)K_c/(Y\sqrt{\pi a}), with aa the size of the worst crack, so the strength of the object is set by a defect nobody put on the drawing and the theory that predicts it is exact about a plate whose flaw size is known. Test a hundred nominally identical specimens and the results scatter over a factor of two, because the flaw sizes do.

The parallel holds in four places at once. Both are governed by a defect rather than by the nominal geometry. Both have a sublinear dependence on the defect size — a square root there, a two-thirds power here — so the first small defect costs disproportionately. Both produce large scatter in tests of identical specimens. And in both the engineering response has been the same: bound the defect by specification and inspection, then design to a lower envelope of the data.

Where they differ is instructive. A crack is local and can be found by looking; an imperfection in a shell is distributed, and the damaging ones are the components of it that resemble a buckling mode, which no inspection procedure is naturally shaped to find. And a crack grows under repeated load, so fatigue makes the defect worse with time, whereas a shell’s imperfections are fixed the day it is rolled — which is the one respect in which the stability problem is the easier of the two.

The general statement they share is worth having in plain words. Some structures are governed by their nominal properties and some by their departures from them, and no calculation of the first kind will ever indicate which sort a structure is. What indicates it is the shape of the path past the limit, and finding that costs an analysis nobody does unless they have been told to.

What the picture cannot show

The three models are rigid bars on springs. They reproduce the exponents and the shapes exactly, which is what they are for, and they contain no material, no stress and no real geometry. A real shell’s behaviour is a superposition of interacting modes and the single-degree-of-freedom picture is a caricature — an accurate one about the property being illustrated, and a caricature.

The knockdown curve is a lower bound and not a prediction. It is drawn under the data, so a real shell will usually be stronger than it says, by an amount nobody can compute in advance. Designing to a lower bound means accepting that the material is being used at an efficiency that is unknown and probably poor.

The imperfection is a single number. A real structure’s departure from its drawing is a field — a function over the whole surface — and reducing it to one amplitude is the step that makes the arithmetic possible and the prediction approximate. Two shells with the same measured maximum deviation can differ by 40% in strength, and everything above would report them identically.

Nothing here treats plasticity. The stockier shell in the second cylinder figure fails by yielding, and the transition between the two regimes is a region in which both matter — the same interaction the column curve rounds off and for the same reason.

Where the ladder goes

The first rung is the interaction of modes, which is where the shell’s factor of three actually comes from and which needs more than one degree of freedom to show. Two coincident unstable modes produce a sensitivity exponent lower than either alone, and a structure optimised so that two modes coincide — which is what optimisation naturally does — has been optimised into the most sensitive configuration available.

The second is the practical business of imperfection tolerances: how much out-of-roundness a silo may have, how it is measured, and what the specification is worth given that the shape matters more than the size.

The third is the connection this essay has been circling. A structure’s sensitivity to its own construction is a computable property, and it is not the factor of safety, not the stiffness and not the redundancy. It belongs beside them as a thing a designer should know about a structure, and it is the one that decides whether the drawing describes what was built. It also decides how much an inspection is worth, which is the question every combined-failure check eventually reduces to: which of the two things being combined is actually known.

Where this goes next

00.0050.010.0150.020.02500.20.40.60.81lateral deflection at mid-heightload ÷ P꜀ᵣP ÷ P꜀ᵣ = 1.00, approached and never reachedinitial bow: δ₀ = 0.002 The column that was never straight Euler's load is the load at which a perfectly straight column becomes indifferent to being bent. No column is perfectly straight, so no column ever reaches it — and the load it never reaches can still be measured. 5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither 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. 050100150200250300350-150-100-50050100150movement of the apex (mm)load (kN)limit point: 133.4 kNit jumps 260 mmas builtat the limitafter it goes The roof that jumps Every stability failure in this collection so far has been a bifurcation — a straight thing discovering it can be bent. A shallow frame does something else entirely. It stays perfectly symmetric, deforms steadily, and at some point the load it can carry starts to fall while it is still moving in the direction it was pushed. 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. 501001502000100200300400500600crack length, mmstress at failure, N/mm²275 N/mm² crosses at 33.5 mm460 N/mm² crosses at 12.0 mmfracture: the crack decides The flaw that sets the strength A member with a crack twenty millimetres long fails at its yield stress. Make the steel stronger and the crack that does it gets shorter, so the same flaw that was harmless in the weaker grade decides the stronger one.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

BifurcationBucklingEquilibrium pathImperfection sensitivityKnockdown factorPost bucklingShell bucklingStability