Structural Engineering & Statics
where the load goes
Series

Floor spectrum — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The building is a filter with one very sharp tooth. What a component mounted on the roof of a eight-storey building feels, divided by what the same component would have felt on the ground. A rigid item sees the floor's peak acceleration rather than the ground's, which is already 2.29 times as much. An item whose own period matches the building's sees 6.1 times as much, at 0.94 s — a resonance inside a resonance, and the two amplifications multiply rather than add. Far above the building's period the ratio falls back toward one, because a component much softer than its support simply follows the ground. The shape of this curve is a property of the building and not of the earthquake, which is why equipment is qualified against a floor spectrum rather than a ground one.

    The spectrum a floor hands on

    Most of what breaks in an earthquake is not the structure. It is a transformer, a chiller, a rack, a ceiling, a pipe — and every one of them is bolted to a floor rather than to the ground. The motion it feels has already been through a filter with one very sharp tooth in it.

    part 1 · dynamics
  2. The same riser, held three ways. The peak bending stress at each floor of a water-filled DN100 riser running the full height of the eight-storey building, under the same record, held three ways. Anchored against rotation at every floor it reaches 131 N/mm², at the lower floors where the drift is largest. Guided at every floor — held in line and free to turn — the same pipe never exceeds 3 N/mm². Guided everywhere but anchored at its base, it carries 77 N/mm² at the base and 21 N/mm² one floor up, and the guided values from there on.

    The pipe held at every floor

    A riser runs the height of a building and is fixed at every floor, so each of its supports moves with a different floor. The floor spectrum prices the pipe's inertia, and for a pipe anchored at every floor the inertia is nearly irrelevant: the drift puts 131 N/mm² into a 100 mm riser at two thirds of a per cent, thirty times its inertia, and passes yield at 200 mm. Guide the same pipe instead of anchoring it and the drift almost vanishes — the pipe then feels only how much the drift changes from one storey to the next.

    part 2 · dynamics
  3. A branch braced along its length holds the riser to a floor. Elevation along the branch: a water-filled DN100 riser guided at every floor of the eight-storey building, with a DN50 branch taken off 500 mm below the first floor's slab and run 2.0 m to a brace on the slab above. The riser leans with the storey's drift between its guides; the branch's brace holds the branch's far end to its slab. Under the building's two-way record the riser alone carries 2.8 N/mm² at the tee's height. With the branch braced so that it cannot slide along its length, it becomes a strut tying the riser to the slab: the riser carries 71 N/mm² at the tee and the branch 10. Let the branch slide at its brace and the riser is back to 4.3, while the branch, now turned by the riser's lean, carries 54.

    The branch that holds the riser to a floor

    A riser guided at every floor leans with each storey's drift and is barely bent by it. Take a branch off it and brace the branch to a slab, and the branch becomes a strut that holds the riser to that slab partway up the storey. The riser, which was carrying 3 N/mm², carries 71 at the tee. Let the branch slide at its brace and the riser is spared, but the branch is then bent by the riser's lean. The lean has to go somewhere, and the brace decides where.

    part 3 · dynamics
  4. A short leg past the elbow ends the strut and moves the stress. Peak stresses in the two-direction earthquake against the length of the leg past the elbow, for a DN50 branch off a water-filled DN100 riser, its tee 500 mm below the first floor's slab, run 2.0 m to an elbow of 76 mm radius and on to a brace on the slab above: the riser at the tee, the branch at the tee with the welding tee's stress intensification of 1.58, and the elbow with its own of 1.73. Straight and held, the riser carries 71 N/mm² (dashed). With a 100 mm leg it carries 19, the tee 82 and the elbow 146; with 300 mm, 4, 82 and 84; with 1 m, 3, 53 and 39. From a leg of about 400 mm the tee carries more than the elbow.

    The elbow that takes the strut away

    A branch braced to a slab is a strut that holds a guided riser to that slab, and the riser, carrying 3 N/mm² alone, carries 71 at the tee. Turn the branch through an elbow before its brace and the strut is gone with 300 mm of second leg: the riser falls to 4. The storey's lean does not vanish. It bends the first leg and the elbow instead. With the stress intensification the piping rules give each fitting, the elbow carries 146 N/mm² with a 100 mm leg and matches the tee's 82 at 300 mm. Past that the tee governs again. An elbow can rescue the riser, but it cannot rescue a brace on the farther slab, where it reaches 347 N/mm².

    part 4 · dynamics

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