Structural Engineering & Statics
where the load goes
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Sway stability — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. One restraint, and several times the load. The same portal — the same columns, the same beam, the same steel — buckling with its head held against sway and with its head free to sway. The braced frame's critical load is 16.46 EI/L² and the swaying one's is 5.69 EI/L², a factor of 2.89, and the effective length factor that comes out of each eigenvalue is 0.774 against 1.317. Both are eigenvalues of the assembled frame at a beam-to-column stiffness ratio of G = 1.00; the buckled shapes are the mode vectors themselves, drawn at 18 per cent of the storey height so that the movement can be seen.

    Held, and not held

    One horizontal restraint at the head of a storey, carrying no vertical load whatever, moves the critical load of the columns beneath it by a factor of 2.89. Effective length is a property of the frame, not of the member.

    part 1 · stability
  2. A base plate, and when the bolts start working. A 550 × 450 mm plate carrying 900 kN and 140 kN·m, so the resultant sits 155.56 mm from the centre against a kern of 91.67 mm. The plate is in partial contact: bearing over 358.33 mm at a peak of 11.16 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 82.5 kN·m and crushes at 192.95 kN·m, and the bolts are not needed until 247.5 kN·m.

    The pinned base that is not pinned

    A column base drawn as a pin is a plate bearing on grout, and a plate in contact over its whole length resists rotation whether anybody wanted it to or not. The stiffness it delivers depends on the axial load, so the assumption is one a frame can leave and re-enter as its loads change.

    part 2 · stability
  3. 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.

    part 3 · stability

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