Structural Engineering & Statics
where the load goes
Concept

Divergence — where it appears

Loss of stability by the appearance of an adjacent equilibrium configuration, which is the buckling every conservative load produces. It is the load at which the stiffness matrix becomes singular. A load with no potential can have no divergence load at all, and fails instead by a growing oscillation.

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

The deflection goes on growing, and sometimes it does not stop. Second-order deflection of a sustained-loaded concrete column against age, on a log time axis. Creep takes the effective modulus down, which takes the buckling load down with it — from 11580 kN on the day to 3309 in the long term, 29 per cent of it — so the amplifier 1/(1 − N/N_cr) grows even though nothing was added to the load. At 2200 kN the column settles: 25 mm of eccentricity on the day and 60 mm at the end, a factor of 2.4 for a load that never changed. At 5294 kN — still only 46 per cent of the day-one critical load — it does not settle, and the divergence arrives at 55 days for no new reason at all.

The column that fails years later

A concrete column under sustained load goes on straining at constant stress, so its deflection grows — and because the second-order moment is the load times that deflection, the demand grows with it. There is a load below which the two settle and one above which they never do.

stability · creep buckling Two frequencies that meet, and a determinant that never moves. The two natural frequencies of Ziegler's two-bar column against the follower load Pℓ/k, with the determinant of its stiffness matrix drawn along the top. The determinant is k² at every load — it varies over this whole axis by 1.1e-16 of itself, which is round-off — so a static buckling analysis of this structure finds no critical load whatever and reports it as stable everywhere. The frequencies say otherwise: they approach, meet at Pℓ/k = 2.0858 — the closed form is (7 − 2√2)/2 = 2.0858 — and become a complex pair, which is oscillation that grows. Adding any internal damping at all drops the load at which that happens to 1.4643, which is 41/28 and 30% below the undamped value; the limit of the damped system is not the undamped system, which is the paradox Ziegler found in 1952 and which was taken for an arithmetic error for a decade.

The load it cannot buckle under

Every stability calculation on this site rests on an assumption nobody states: that the load has a potential, so a critical load is where a total potential energy stops being a minimum. A load that turns with the structure it is pushing has no potential, and the static analysis of such a column returns no critical load at all — a determinant that never vanishes, for a column that fails at a perfectly finite one.

stability · follower force

Named alongside it

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

BucklingAeroelastic instabilityAmplificationCreepCreep bucklingCritical loadDampingDuration of loadEccentricityEffective modulusEigenvalueEquilibrium path

All concepts