Structural Engineering & Statics
where the load goes
Concept

Critical load factor — where it appears

The factor by which a structure's loads would have to be multiplied for it to buckle elastically, written αcr. Its reciprocal says how much second-order effects amplify sway, roughly 1/(1 − 1/αcr), and a value below ten is the usual signal that they must be included.

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

Also named here as stability coefficient — the same set of essays touches all of them, so they are one junction rather than several.

Wrong in both directions until the building racks. Across buildings from a pure bending tower to a pure racking frame (αH on a logarithmic scale), each 20 storeys with gravity putting its own critical factor at 5.0: solid, the least storey sway factor over the building's own, which is what taking the least storey as the building's costs; dashed, the ground storey's second-order increase estimated from its own sway factor, over the true increase. At αH = 0.1 the least storey factor is 0.77 of the true one and the ground storey's estimate 0.09 of the true increase; at αH = 10, 0.86 and 0.37; at αH = 100, 0.94 and 0.98. The first error is on the safe side and the second is not, and both vanish only for a building that racks.

The storey that cannot see the building lean

A sway check made storey by storey asks each storey how much it drifts under a push and how much gravity sits on it, and reads a critical load factor for the storey from the two. For a frame whose storeys rack like a stack of shelves that is exact. For a building that bends — a braced core, a wall — it is wrong twice. Taking the least storey as the building's reads a critical factor of 3.9 for a building whose own is 5. And amplifying each storey by its own factor finds a 2 per cent second-order increase at the ground storey, where the truth is 20, because the ground storey hardly drifts and carries the lean of everything above it.

stability · Sway stability The buckling mode moves up to where the spine is not. The building's first buckling mode, each floor's sideways movement as a fraction of the roof's, up the height of a 20-storey building of a bending spine and a racking frame (αH 1.0), its gravity scaled to an elastic critical factor of 5.0, under an equal lateral load at every floor: the spine to the roof, and stopping at storeys 15 and 10. With the spine to the roof the mode is the bending tower's, growing all the way up. With the spine stopping, at storey 15 its top moves less than a thousandth as far as the roof, and at storey 10 its top moves less than a thousandth as far as the roof: the building buckles in the first storey of frame above the spine, every floor above that storey moving with it as one, and the spine takes no part.

The storey above the core decides

A building whose bending core runs to the roof shares one sway mode between its core and its frame, and the storey-by-storey stability check reads it wrongly twice. Stop the core five storeys short and the check becomes exact: the first storey of bare frame above the core reads the building's own critical factor to three figures, because the whole building buckles there and nowhere else. The check stays wrong below the core, and it goes wrong again when the frame above is more than about twice as stiff.

stability · Sway stability

Named alongside it

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

Amplification factorBuckling modeLateral systemP-deltaStability coefficientStorey driftSway stability

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