Structural Engineering & Statics
where the load goes
Concept

Ring buckling — where it appears

The collapse of a ring or a tube under external pressure into a lobed shape, at a critical pressure proportional to the cube of the wall thickness over the radius. Restraint from surrounding ground raises both the critical pressure and the number of lobes, so an embedded ring is very much harder to buckle than a free one.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The cheapest way out of being round. A ring under uniform external pressure, drawn in its first four buckling modes with the pressure each one needs underneath it, in N/mm². The pressure has no direction: it stays normal to the wall wherever the wall goes, so it does work on any change of shape that reduces the enclosed area, and the ring buckles into whichever shape is cheapest. Bare, that is the oval — n = 2 at 3EI/R³ — and the modes rise as n² − 1, so three lobes cost 2.67 times as much. Nothing in the drawing prefers any orientation, which is the point — a column has an axis to buckle about and a ring has none.

The pressure that needs no direction

Every buckling problem in this collection has had a load with a direction — a column pushed along its axis, a plate along its edge, an arch by what is on it. A buried pipe has none. The pressure is the same everywhere, it stays normal to the wall as the wall moves, and it does work on any change of shape that reduces the area inside.

stability · Ring buckling The lining that carries less for being weaker. Bending moment and hoop thrust in a circular lining, against the lining's own bending stiffness, both as fractions of the free-ring values. The ground arrives already stressed — 500 kPa vertically and 300 horizontally at K₀ = 0.6 — and the difference between them tries to squash the hole into an ellipse. A lining stiff enough to refuse absolutely collects the whole distortion pressure, p₂R²/3 = 300 kNm/m; one flexible enough to go with the ground collects nothing, because there is no curvature change left to resist. The thrust is the flat line: it comes from the mean stress rather than the difference, so it does not move at all. Putting 8 joints in this ring drops the moment to 34% of the solid one and leaves the thrust exactly where it was, which is why a segmental lining is jointed and why the intuition carried over from a beam is inverted here.

The lining that is stronger for being weaker

A tunnel lining is not loaded. The ground arrives already stressed and the hole wants to squash into an ellipse; the lining's only job is to refuse, and how much moment it collects depends entirely on how hard it refuses. Make it stiffer and it takes more. Make it flexible — put joints in it, make it thin — and it takes almost none, while the hoop thrust it carries does not move at all.

structures · Tunnel ring The limit moment goes as the square root of the ring's stiffness, and pressure is part of it. Brazier's limit moment for a steel tube of radius 600 mm and wall 2.5 mm (r/t = 240), as a multiple of its value with no pressure, against the pressure as a multiple of the ring's own buckling pressure, pc = 3D/r³ = 4.07 kPa; positive inside. The dots are the peaks of the moment–curvature path found by minimising the energy at each curvature; the line is √(1 + p/pc). Half pc outside leaves 0.71 of the bending capacity, and the capacity is gone at pc; pc inside gives 1.41, three times pc 2.00. The pressure enters only by adding to the ring's stiffness against the oval, (3D/r³ + p) where it had 3D/r³.

Three kilopascals that halve a tube in bending

A long thin tube in bending flattens until its section can carry no more, at an oval of exactly two ninths of its radius. Put a pressure on it and nothing about that shape changes. What changes is how much bending it takes to get there — as the square root of one plus the pressure over the ring's own buckling pressure — and for a steel tube 1.2 m across with a 2.5 mm wall that unit is 4 kPa. Three kilopascals of suction halve its bending capacity; a fifth of an atmosphere inside doubles it and hands the tube to another failure entirely.

stability · Brazier buckling Which way the oval lies decides whether the pressure helps or harms. The limit moment of a steel tube of radius 600 mm and wall 2.5 mm (r/t = 240), whose ring buckles under 4.07 kPa, over its limit with no pressure (796 kN·m), against the angle between the bending plane and the diameter a non-uniform pressure pushes in on, for second harmonics of 0.10, 0.25 and 0.50 of that pressure; lines the closed form, dots the minimised energy. At 0.10, 0.93 aligned and 1.08 across; at 0.25, 0.82 aligned and 1.19 across; at 0.50, 0.65 aligned and 1.40 across; at 45° the oval does not touch the limit at all.

The oval the pressure brings

A uniform pressure changes how stiff a tube's ring is against the oval that bending squashes it into, and nothing else: the tube still gives way at a flattening of two ninths of its radius. A pressure that is larger on one diameter than the other has an oval of its own, and that oval does work on the bending's directly. Lying in the bending plane it has done some of the flattening before the bending starts, and a pressure difference a quarter of the ring's own buckling pressure takes a fifth off the tube's bending capacity; lying across it, it adds a fifth. At forty-five degrees it does nothing at all. And the two ninths turn out to have been a property of a round tube.

stability · Brazier buckling

Named alongside it

The objects these essays reach for when they reach for this one.

Geometric stiffnessHydrostatic pressureImperfectionLimit pointLocal bucklingSecond momentShellStiffnessBucklingBuckling modeCollapse loadConstruction sequence

All concepts