Structural Engineering & Statics
where the load goes
Concept

Freeboard — where it appears

The height left between a liquid's surface and the roof or rim of its container. It has to exceed the sloshing wave, which is set by a spectral displacement rather than a force, and a wave that reaches the roof pushes on a part of the tank that was never designed to be loaded sideways.

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

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.

dynamics · Sloshing 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.

dynamics · Sloshing

Named alongside it

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

Base shearConvective massImpulsive massModal massMode shapeNatural periodResponse spectrumSloshingEigenvalueFree bodyHoop tensionLoad path

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