Structural Engineering & Statics
where the load goes
Concept

Euler buckling — where it appears

Elastic instability of a straight compression member, at a load proportional to its flexural rigidity and inversely to its length squared. It is a bifurcation of a perfect member, so a real one never reaches it — the deflection grows from the first newton and the amplification is one over one minus the load ratio.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The column curve. Failure 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.

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.

stability · Buckling A column that was never straight. Load 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.

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.

stability · Buckling Four guesses at one buckling mode. A pin-ended column, with four assumed shapes and the load each of them gives. The reference is a ten-term Ritz expansion solved as an eigenvalue problem, at 9.8696 EI/L² — which is π², as it must be. a half sine gives 9.870, its own sag shape gives 9.882, a mid-span sag gives 10.000, a parabola gives 12.000. Every one of them is high and none of them is low, because an assumed shape is a constraint on the column and a constraint can only stiffen it.

Guessing the shape, and getting the load anyway

A column's buckling load can be had from a shape that is wrong everywhere, because the energy criterion is stationary at the true mode. The error in the load is the square of the error in the shape, and it is always high.

stability · Stability energy

Named alongside it

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

Critical loadEffective lengthImperfectionResidual stressAmplification factorBifurcationBuckled mode shapeConservation of energyEigenvalueFlexural rigidityGeometric stiffnessRadius of gyration

All concepts