Shell buckling fundamentals

Squeeze a thin ruler end to end and, at a certain load, it suddenly bows sideways. The metal has not run out of strength; the stress in it can still be well below yield. What has run out is stability, the ability of the straight shape to spring back when it is nudged.

Every compressed member has a load at which its straight, or round, shape stops being stable. For a slender column it is the Euler load, P = π²EI/L² for pinned ends, which depends on the stiffness E, the section's second moment of area I and the length L, and not on the yield strength at all. Thin shells behave the same way, with more complicated shapes.

A cylinder under outside pressure, such as a vessel under vacuum or a jacketed shell, buckles into lobes around its circumference: two for a very long tube, more for a short one held round by heads or rings. Under axial compression, from weight, wind or earthquake bending on a tall vessel or skirt, it buckles instead into a band of small inward dimples.

In the elastic range both buckling loads depend on the metal's stiffness and the shape, not on its yield strength. A stronger steel therefore does not help a thin shell that buckles elastically; a thicker wall, a shorter unsupported length or added rings do.

The ratio of radius to wall thickness, R/t, and the length between stiffeners set how easily a shell buckles. For a very long cylinder under outside pressure, buckling into two lobes, classical theory gives the elastic buckling pressure as:

The pressure falls with the cube of t/R: double the slenderness and it drops to about one-eighth. Shorter lengths between heads or rings raise it, because the shell has less room to form a lobe. For axial compression the classical elastic buckling stress of a perfect cylinder is:

The classical loads assume a perfect shape. Real shells have small out-of-roundness, dents, flat spots and sinkage at welds. Under axial compression a thin cylinder is very sensitive to these: an imperfection about as deep as the wall is thick can cut the buckling load to well under half the classical value, and test results scatter widely below it. Outside pressure is less sensitive, but still needs allowing for.

Design rules therefore apply reduction factors based on large sets of tests, and set fabrication tolerances on how far out of round a shell may be. The two go together: the factors are valid only if the tolerances are met. Where the elastic buckling stress approaches yield, yielding and buckling interact and the capacity falls further below either value alone. The factors and tolerances are code-specific.

Stiffening rings split a long shell into short bays that each buckle on their own. Because a short bay must form more, smaller lobes, its buckling pressure is much higher, and the rings add little weight. The rings must themselves be stiff enough not to buckle with the shell, which is why design rules ask for a minimum second moment of area of the ring, usually counting a strip of shell with it.

A thicker wall works too, because the wall's bending stiffness, E·t³ / 12(1 − ν²) per unit width, grows with the cube of its thickness. Heads count as stiffeners for the shell next to them, so the unsupported length used in design is usually measured between heads, rings or other stiff features, with part of each head's depth included as the governing code directs.