Concept

Bifurcation — where it appears

The load at which a second equilibrium shape becomes available, so that a structure which was straight discovers it can be bent instead.

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

00.0050.010.0150.020.02500.20.40.60.81lateral deflection at mid-heightload ÷ P꜀ᵣP ÷ P꜀ᵣ = 1.00, approached and never reachedinitial bow: δ₀ = 0.002

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
050100150200250300350-150-100-50050100150movement of the apex (mm)load (kN)limit point: 133.4 kNit jumps 260 mmas builtat the limitafter it goes

The roof that jumps

Every stability failure in this collection so far has been a bifurcation — a straight thing discovering it can be bent. A shallow frame does something else entirely. It stays perfectly symmetric, deforms steadily, and at some point the load it can carry starts to fall while it is still moving in the direction it was pushed.

stability · snap through
stable symmetric — a columnan imperfection is a nuisancecritical89%unstable symmetric — a shellan imperfection is a demolitioncritical77%asymmetric — a frameand it matters which waycritical

A third of what the theory promised

A column with a small crookedness reaches almost its full Euler load. A cylinder with the same relative crookedness reaches a third of its classical one, and the theory is not wrong — what separates them is the slope of the path just past the critical load, which no calculation of the critical load itself can see.

stability · imperfection sensitivity

Named alongside it

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

BucklingEquilibrium pathStabilityAmplification factorCritical loadEuler bucklingGeometric nonlinearityImperfectionImperfection sensitivityKnockdown factorLimit pointPost buckling

All concepts