Structural Engineering & Statics
where the load goes
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. A perfect structure bifurcates and a real one never does: the imperfection turns a branch point into a smooth curve that approaches the same load from below.

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

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 A load with a maximum in it, and nothing bifurcates. Load against apex movement for a two-bar frame of half-span 1000 mm and rise 150 mm. The load rises to 133.4 kN at a movement of 64 mm — well short of the 150 mm that would bring the apex level — and then falls. Past that point the frame can only be held by taking load away, so under a dead weight it goes: 260 mm of movement at constant load, arriving inverted and in tension. The minimum on the path is -133.4 kN, the exact negative of the maximum, because the geometry is symmetric about the flat position and the arithmetic knows it.

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 Three paths out of the same critical load. Load against sideways movement past the critical load, for three systems whose critical loads are identical. The stable one climbs, so a real structure with a small crookedness reaches nearly the full load and keeps going. The unstable one falls symmetrically, so the imperfect structure has a maximum below the critical load and it matters not at all which way it leans. The asymmetric one falls one way and climbs the other, so the direction of the imperfection decides everything. All three are drawn at an imperfection of 0.02 radians.

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