Structural Engineering & Statics
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Sloshing — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A broad tank sloshes and a tall one does not. The liquid's division into the part that moves with the wall and the part that sloshes, against the tank's proportion. The convective masses come from the potential-flow solution and the impulsive mass is whatever is left, so the two sum to the liquid's mass exactly at every proportion rather than approximately over part of the range. A tall tank at H/R = 3 is 84% impulsive and behaves almost like a solid; a shallow one at H/R = 0.5 is 72% convective and most of its contents never notice the earthquake. This tank sits at H/R = 1.33, which is 65% impulsive.

    The liquid has a period of its own

    Shake a tank and its contents do not all go with it. Part of the liquid moves as though it were bolted to the wall and part sloshes at a period fixed by gravity and the radius, which the tank's stiffness has no influence over whatever. The split is decided by one proportion, and the two parts then take entirely different amounts of the earthquake.

    part 1 · dynamics
  2. On a flexible tower the full tank is not the worst one. The tower's base shear against how full the tank is, for a 12 m tank holding up to 8 m of water on a 25 m tower: a stiff tower (period 1.20 s with the tank full) and a flexible one (2.50 s), the liquid and tower coupled (solid) and with all the water bolted to the container (dashed). The stiff tower's shear rises all the way to full, 2,141 kN, against 2,728 kN rigid. The flexible tower's peaks at 84% full, 688 kN, and falls to 665 kN full; rigid water says 884 kN, flat from 37% full, because past the spectrum's last corner a heavier mass on the same spring draws the same force.

    The tank that sloshes the tower

    On the ground a tank's liquid splits into a part that moves with the wall and a part that sloshes at its own long period, and the two answers can be added afterwards because they are orders apart. Lift the tank onto a tower and the tower's period moves up towards the sloshing period, the two masses stop being separate, and the system has two modes that belong to neither. Adding the ground answers then overstates the base shear by a third, treating the water as solid by as much again, the wave grows by half, and a flexible tower is worst with the tank about five-sixths full.

    part 2 · dynamics
  3. The water splits the tower's resonance, and a baffle flattens it. For a 12 m tank 8 m deep on a tower of period 2.5 s with 2% damping, 150 t of container, full: the base shear per unit of harmonic ground acceleration against the shaking's angular frequency, as a share of the frozen-water tower's peak. Frozen to its container, the water makes one resonance; free, it is a second mass on a spring tuned 0.69 times the tower's frequency, against Den Hartog's optimum of 0.71, and it splits the resonance in two — with half a per cent of damping both peaks are higher than the frozen one, 1.83 times it. A baffle giving 10% brings the peak to 0.21; the best damping for the rms, 27%, to 0.11.

    The water tower that could damp itself

    A full water tower's sloshing water is a second mass on a spring, hung from the container at very nearly the tuning a designed tuned-mass damper would be given, and it weighs two-fifths as much as what it hangs from. Plain water has half a per cent of damping, so it splits the tower's resonance into two sharp peaks and helps nothing. A screen across the tank that gives it a tenth makes the water a damper: the tower's rms base shear falls to four-tenths of what it would be with the water frozen, and the same screen keeps working as the town empties the tank.

    part 3 · dynamics
  4. A screen damps a tank on a tower and barely touches a standpipe. The rms drift under broadband ground shaking, as a share of the same structure with its water frozen to the wall, against the damping a screen gives the sloshing: a steel standpipe 10 m across and 30 m tall with a 12 mm shell, full (solid), and the water tower of the earlier essay, a 12 m tank 8 m deep on a tower of 2.5 s (dashed). Plain water, at half a per cent: 0.82 and 1.01. A screen giving 10%: 0.76 and 0.40. The standpipe's gain comes almost entirely from leaving the lens out of the shaft's motion, which plain water already does.

    The tank that is its own tower

    A standpipe holds its water in its own shaft, so the water fills a height several times its width. Only a lens of it near the surface sloshes — 0.455ρπR³, 178 t of the 2,199 t in a steel standpipe 10 m across and 30 m tall — and that lens stops growing once the water is deeper than a radius, so its period is set by the width alone: 3.3 s, against the shaft's 0.38. The lens is 8% of the water but a quarter of the shaft's moving mass, because it sits where the shaft moves most. It is far too slow to be tuned as a damper, and a screen can do little with it; what it does instead is stay still while the shaft sways, which takes nearly a fifth off the drift a frozen-water calculation predicts.

    part 4 · dynamics
  5. In a core, the depth has to tune the tank, and only shallow water is slow. The sloshing period of rectangular tanks 6, 10, 15 m long against the depth of their water (logarithmic), with building periods of 4, 5, 6 and 8 s (dotted). Deep, each tank is as fast as it can be: 2.77 s, 3.58 s, 4.38 s. To be tuned to a 5 s building the 10 m tank needs 1.80 m of water; to 4 s, 3.50 m; to 8 s, 0.65 m. A building faster than 3.58 s cannot be reached by the 10 m tank at any depth.

    The tank that must be shallow to be slow

    A deep tank's sloshing period is set by its width alone, so a tank in a tall building's core ought to be tuned by choosing how wide to make it. It cannot be. A deep tank fast enough to fit is too fast for the building: 3.6 s for a core 10 m across, where a 5 s building would need a tank 19.5 m long and an 8 s one 50 m. The depth has to tune it, and only shallow water is slow — 1.73 m for 5 s, 0.63 m for 8. Shallow water is the efficient kind, with four fifths of it sloshing. But at the accelerations the tank exists to reduce, the wave it makes is a quarter of its depth on a 5 s building and seven tenths on an 8 s one, and the linear tuning it was designed by has stopped describing it.

    part 5 · dynamics
  6. Pumping water in or out keeps the damper, and the lost water barely matters. The peak amplification of a building with 1 per cent damping and a 10 m tank sized for a 5.0 s building, 1.73 m of water weighing 2.0 per cent of the modal mass, its sloshing damped at 7 per cent, against the period the building actually has: the tank left at its design depth (dashed), retuned by its water level with the water's mass following the depth (solid), and retuned with the mass held at the design value (light); dotted, the design period; bare, the building would reach 50. At 4.5 s, 21.7 left alone and 9.6 retuned; at 5.5 s, 22.6 and 10.8; at 6.0 s, 34.0 and 11.7, of which the water let out costs 1.7.

    The tank that follows the building

    A sloshing damper is tuned to a period computed before the building exists, and the building that is built has another one — stiffer at handover, with its partitions and cladding working, softer once its concrete has cracked. A tank whose plan is fixed and whose period is set by its depth can be brought back by opening a valve. Ten per cent off tune, the tank left alone lets the building's peak response more than double; pumped back to the right depth, it holds within a tenth of the design value, and the water it gains or loses barely matters. What does matter is that a slower building needs shallower water, and in shallow water the wave the damper is made of grows towards the depth it is sloshing in.

    part 6 · dynamics

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