FEXLA

Buckling: imperfections and why design factors are large

Why a shell that looks perfect buckles far below its theoretical load, and how design factors allow for it.

Principle 19 of 20 · 5 min read ·

Perfect and imperfect shells under compressionLoad as a fraction of the classical buckling load against added sideways deflection. The perfect shell rises straight up the load axis to 1.0, then follows a falling path as it deflects. Two shells with small initial dents, a and b, bend from the start and peak lower, at about 0.76 and 0.52, where their paths meet the falling path. Illustrative curves from a simple model.classical buckling load0.51.0abPerfect: up to 1, then fallsa: small dent, peaks lowerb: larger dent, lower stillLoad ÷ classical buckling loadAdded sideways deflection
Figure 1. Perfect and imperfect shells under compression.
Contents

Thin shells under axial compression buckle far below the load that theory gives for a perfect shell, because small departures from the perfect shape, often smaller than the wall thickness, cut the buckling load sharply. Design rules allow for this with reduction factors and design factors that are much larger than those used against yielding or collapse.

This article explains why a shell is so sensitive to its imperfections, which loads are the most sensitive, and how design rules and computer models deal with it.

FEXLA shell model of a horizontal vessel with ellipsoidal heads on support lugs at its sides
Figure 2. A horizontal vessel on four support lugs as one shell model: how the weight shares between the lugs depends on the shell between them.

Theory and tests disagree

For a perfect, long cylinder under uniform axial compression, classical theory gives a buckling stress that depends only on the material and on the ratio of thickness to radius.

σcl = E·t / (R·√(3(1 − ν²))) ≈ 0.605·E·t / Rν = 0.3; R mean radius, t wall thickness

When real cylinders were tested, most buckled at a fraction of this value, and the results scattered widely between nominally identical specimens. The thinner the shell, the larger the gap tended to be. The theory was not wrong for the shell it described. The real shells were simply not that shell.

What happens after buckling

The answer lies in what the shell can carry just after it buckles. A column that buckles can still carry close to its buckling load as it bows. A cylinder in axial compression cannot: once it starts to dimple, the load it can carry falls steeply. The drawing at the top of this page shows that falling path.

A shell with a small dent does not wait for the bifurcation point. The dent grows from the first load, and the shell leaves the perfect path early, reaching a peak where its growing dent meets the falling post-buckling path. The steeper that path falls, the lower the peak for the same dent. A structure whose post-buckling path falls is called imperfection-sensitive.

Small dents, large losses

Koiter's theory gives the trend in closed form for an axially compressed cylinder with an imperfection in the shape of its buckling mode. With ρ the buckling load as a fraction of the classical value and δ the imperfection amplitude as a fraction of the wall thickness:

(1 − ρ)² = (3√3 / 2)·c·δ·ρ, c = √(3(1 − ν²))solve the quadratic for ρ; ρ = 1 at δ = 0
Buckling load against dent depth, Koiter's formulaThe buckling load of an axially compressed cylinder as a fraction of the classical value, against the depth of an imperfection in the shape of the buckling mode as a fraction of the wall thickness, from Koiter's formula with Poisson's ratio 0.3. The curve drops from 1.0 at no dent to about 0.53 at a tenth of the wall, 0.37 at a quarter and about 0.16 at a full wall thickness.0.250.250.50.50.750.7511a tenth of the wall: about halfperfect shellBuckling load ÷ classical loadDent depth ÷ wall thickness
Figure 3. Buckling load over the classical load against imperfection amplitude over wall thickness, from the formula above. An imperfection of a tenth of the wall already takes the load to about half.

No shop can roll and weld a 4 mm shell of 4 m diameter to a tenth of a millimetre. Dents of the order of the wall thickness are normal, which is why the test results sat so far below the classical line.

Which loads are most sensitive

Buckling fundamentals: Thin shells under compressionLeft, the cross-section of a cylinder with pressure pushing in from outside and its buckled shape with four lobes. Right, a cylinder squeezed along its axis with a band of small diamond-shaped dimples.Squeezed end to endIt snaps sideways into waveswhile the stress is still below yieldPressure outside: lobesAxial load: dimplesRadius / thicknessBuckling pressureR/t2 × R/tTwice as slender:about one-eighth the pressurelong cylinder, pressure outsideSideways deflectionLoadIdeal buckling loadPerfect shapeReal shape, small dentsThe reduction factorStiffening ringseach bay buckles on its ownLShorter L or thicker wall:higher buckling load
Figure 4. A thin cylinder under outside pressure buckles into a few lobes round the circumference; under axial load it forms many small dimples.

Sensitivity depends on the load and the shape. A cylinder in axial compression and a sphere under outside pressure are the most sensitive: both have post-buckling paths that fall steeply, and both have many buckling modes at almost the same load. A cylinder under outside pressure buckles into a few long lobes, its path falls less, and tests come much closer to theory. Shear and bending of a cylinder lie in between.

Spread-out and clustered buckling multipliersBuckling multipliers divided by the first one, for mode numbers 1 to 6. For a pin-ended column they climb as 1, 4, 9 and off the chart. For a thin shell, illustrative values bunch together between 1.0 and about 1.3.24681012345616, 25, 36Column: n² × the firstThin shell (illustrative): bunchedλ ÷ λ1Mode number
Figure 5. Buckling multipliers of a column climb quickly from mode to mode; those of a thin shell bunch together. Bunched modes can combine, and any small dent can trigger one of them.

Why design factors are large

A strength check can rely on the metal yielding and redistributing load before collapse, so a modest margin covers the uncertainty. A buckling check has no such cushion: the load can drop suddenly, and the imperfections that set it are not known in advance. Design rules therefore stack two allowances.

  • A reduction, often called a capacity reduction or knockdown factor, that brings the classical buckling load down to a value that test results stay above. It is smallest for axial compression of thin cylinders and falls as R/t rises.
  • A design factor on top, like the factor used against yielding, so the design load sits safely below the reduced buckling load.
  • Fabrication tolerances on out-of-roundness and local dents, checked on the finished shell. The reductions assume the shell meets them; a shell outside tolerance is outside the design basis.

Together these make the margin against a perfect-shell buckling load several times larger than the margin against collapse. The factors, how they depend on slenderness and load type, and the tolerances are code-specific, so check the governing code.

Imperfections in a computer model

A linear eigenvalue buckling analysis finds the buckling load of the perfect shell as modelled, so it shares the classical result's weakness: it is an upper bound, and it must be reduced by the code's factors before it is compared with the design load.

The more direct route is a geometrically nonlinear analysis of a shell that already has a dent. In a finite element model the dent is often seeded as the first buckling mode, scaled to an amplitude tied to the fabrication tolerance, or as a measured shape. The load is raised until the model reaches its peak. That peak still needs a design factor, smaller than in the linear route, and the shape, amplitude and factor to use are code-specific.

Key points

  • Real thin cylinders in axial compression buckle far below the classical stress 0.605·E·t/R.
  • The cause is a post-buckling path that falls steeply: a small dent lets the shell peak early.
  • Koiter's formula shows a dent of a tenth of the wall thickness can halve the buckling load.
  • Axial compression of cylinders and outside pressure on spheres are the most sensitive; outside pressure on cylinders much less so.
  • Design rules combine a reduction factor, a design factor and fabrication tolerances, all code-specific.

Where this matters in practice

  • Tall columns and skirts under wind or seismic bending, where the shell is in axial compression.
  • Vessels under vacuum or outside pressure, and spherical heads under outside pressure.
  • Thin shells near heavy local loads such as lugs, legs and saddles, where local compression adds to the overall load.
  • Out-of-roundness and dent checks on fabricated shells, which belong to the buckling design basis.

FEXLA's buckling check runs an eigenvalue buckling analysis of your shell model and applies the code's design factors. In early access, free to use.

See the buckling check

Related principles

Next principle: Pressure stiffening and softening

Educational article. It explains principles and is not design guidance; follow your governing code and edition.