The tank that sloshes the tower

The liquid has a period of its own splits a tank’s contents into two masses. The impulsive part moves with the wall as though it were solid; the convective part sloshes at a period fixed by gravity and the radius, several seconds long, and sits so far out on the falling branch of a design spectrum that it supplies a few per cent of the base shear while deciding the freeboard. The two periods are orders apart — a quarter of a second against four and a half — and that separation is what lets the two answers be computed independently and added.

That essay ended by saying that a tank on legs is not a tank on the ground. The difference is the separation. A tower is a spring, and with a tank of water on top of it its period is not a quarter of a second but one or two or three — the same order as the sloshing. The question is what happens to the two masses, and to the habit of adding them, when the gap between them closes.

Two masses on two springs #

The tank here is 12 m across and holds up to 8 m of water, 905 tonnes full, in a container of 150 tonnes on a tower 25 m high. On the ground, full, its liquid splits into 589 tonnes that move with the wall and 304 tonnes that slosh at 3.65 s. The design spectrum is a generic one: a ground acceleration of 0.3 g, a plateau two and a half times that up to 0.5 s, falling as the inverse of the period to 2 s and as its inverse square beyond.

On the tower, the container and the impulsive water ride on the tower’s stiffness, and the convective mass rides on its own spring — not anchored to the ground but to the container, which is moving. That is the model Housner proposed for elevated tanks in 1963: two masses, two springs, in series. The tower’s stiffness is fixed by the period it would have with the tank full and the sloshing ignored, and two towers are compared, a stiff one of 1.2 s and a flexible one of 2.5 s.

The stiff tower behaves as expected: the fuller the tank, the larger the base shear, 2,141 kN full. The flexible tower does not. Its base shear rises with the fill to a peak of 688 kN with the tank 84% full, and then falls — to 665 kN with the tank full. A tank on a flexible tower is at its worst a little before it is full.

Why the full tank is not the worst one #

Two things happen as water is added to a tank on a flexible tower, and both lower the force it draws.

The first is the spectrum. Each tonne of impulsive water lengthens the tower’s period, and on the 2.5 s tower the shorter mode’s period crosses the spectrum’s last corner, 2 s, at about 84% full. Below the corner the spectral acceleration falls as the inverse of the period, so the force grows as the square root of the mass; beyond it the acceleration falls as the inverse square, which is exactly as fast as the mass grows, and a heavier mass on the same spring draws the same force. That is why the dashed curve of all-rigid water is flat from 37% full: past that fill a single mass on the tower’s spring is beyond the corner, and the shear it draws no longer depends on how much water there is.

The second is where the water goes. Adding depth to a broad tank adds more convective mass than impulsive at first, and the tower’s coupling then moves mass into the longer mode — which, on a falling spectrum, is the mode that draws the least acceleration. Beyond the corner the first effect stops the shear growing and the second makes it fall.

The fall is small: 688 kN to 665, a little over three per cent. But it means that the full tank, the load case every calculation is written for, is not the governing one on a flexible tower. The case that governs is the fill at which the tower’s mode reaches the corner, and that is a property of the tower and the spectrum together, not of the tank.

The fill that governs belongs to the site #

The corner at which the force stops growing is a property of the ground, not of the tank, and moving it moves the governing fill by a long way. Keep the tank and the towers and change only the period at which the spectrum turns from falling as $1/T$ to falling as $1/T^2$.

With the corner at 2.5 s or later, the full tank governs on all three towers from 2 to 3 s: the tower’s mode never gets past the corner, and every tonne of water adds force. With it at 2 s, the 2.5 s tower peaks at 84% full, as above, and a 3 s tower at 63% — 463 kN against 435 full. With it at 1.5 s, nearer where some codes put it for the smaller, nearer earthquakes that dominate low-seismicity regions, the 2 s tower peaks at 68% full, 908 kN against 869; the 2.5 s tower at 45% full, 574 kN against 499, fifteen per cent above the full tank; and the 3 s tower with the tank barely a sixth full, 423 kN against 326 — thirty per cent more force with almost no water in it.

So the case to check is not a fill but a range. A water tower’s level cycles every day between whatever the pumps allow, and the governing fill can be anywhere in that range; on a slender tower on a site whose spectrum turns early, it can be near the bottom of it. A calculation run once, for the full tank, is the right calculation on a stiff tower and can be nearly a quarter short on a flexible one.

The periods repel #

The two masses are not independent on a tower, and the clearest picture of it is what happens to their periods.

On a stiff tower the coupled periods sit on the uncoupled ones: 1.17 s against the tower’s 1.2, and 3.74 against the ground tank’s 3.65. The convective mass hardly notices that its anchorage is moving, because its anchorage moves fast and little. As the tower softens, its period climbs towards the sloshing period and the two do not cross. They repel. At a tower period of 2.5 s the system’s periods are 2.19 and 4.16 — the tower is stiffer than its own period says and the wave slower than its own. At 4 s they are 2.75 and 5.31, and neither of them is the tower’s and neither is the wave’s.

This is the veering of any two coupled oscillators, and it is the same arithmetic that a tuned mass damper is designed around — a small mass on a spring, attached to a structure, tuned to the structure’s period so that the two modes split around it. The tank’s convective mass is a damper nobody tuned, and it is not small. Here it weighs 304 tonnes against the 739 that ride on the tower: a mass ratio of 0.41, where a damper’s is a few per cent.

The heavy mode changes hands #

How much each mode moves decides how much force it draws, and the share of the mass in each mode changes completely across the range.

On a stiff tower the shorter mode is the tower swaying with its container and the impulsive water, and it carries nearly two-thirds of the mass; the longer is mostly the wave, carrying the rest. The two are equal at a tower period of 2.07 s, and beyond it the longer mode is the heavy one: 63% of the mass at 2.5 s, 91% at 4 s. In that longer mode the tower and the wave move together, slowly, and on a falling spectrum a slow mode draws little acceleration.

That is what the coupled answer captures and the separate one cannot. Treated separately, the tower is an oscillator carrying 739 tonnes at its own period, and draws the spectrum at that period on all of them. Coupled, most of that mass has gone into a mode with a period two-thirds longer, 4.16 s against 2.5, where the spectrum is barely a third as large. Most of the mass moves together in a building’s first mode; here the first mode, by period, is the one that most of the mass joins only once the tower is soft enough.

Both shortcuts overstate the force #

There are two ways to avoid the coupled calculation, and both give a larger base shear.

The first is to lift the ground calculation onto the tower unchanged: the tower carrying the container and impulsive water at its own period, the convective mass at the ground tank’s sloshing period, the two responses combined by the square root of the sum of squares. On the 1.2 s tower that overstates the coupled base shear by 8%, which is within what anyone would accept. On the 2.5 s tower it overstates it by 35%; near 3.25 s, where the tower’s period and the wave’s are closest, by 48%. The error is largest exactly where the two masses are least separate, which is where the method’s premise is most wrong.

The second is to bolt all the water to the container — one mass, one spring, no sloshing at all. It overstates the shear by 27% on the stiff tower and 33% on the flexible one, and by between a fifth and two-fifths everywhere on the range. Its curve is jagged where the single period crosses the spectrum’s corners. Neither shortcut is unsafe for the base shear, and that is the danger in them: an answer that is always conservative is never checked, and the margin it carries is spent on a foundation and a tower that are a third heavier than the tank needs.

The spectrum is not a load, and here that matters twice. The spectrum gives the peak response of a single oscillator; a coupled system has to be broken into its modes before the spectrum can be read at all, and a shortcut that reads it at the wrong periods reads the wrong values. The separate-masses method reads it at the tower’s period and the wave’s; the structure does not have either.

The wave on the tower #

The sloshing wave is the convective mass’s motion relative to the wall, and on the ground it is set by the spectral displacement at the sloshing period alone. On a tower the wall is moving too.

On the ground, this tank’s wave is 0.28 m. On the 1.2 s tower it is 0.35 m, and on the 2.5 s tower 0.43 m — the tower’s sway drives the liquid as well as the ground does, and when the two periods are close it drives it in step. The largest wave, 0.44 m, is at a tower period of 2.21 s, 55% above the ground tank’s. Only once the tower is softer than the sloshing period does it begin to isolate the tank, and the wave falls below the ground value.

So the coupling cuts both ways. The base shear is smaller than either shortcut says, and the wave is larger than the ground tank says. A freeboard sized from the ground tank’s formula — a sensible-looking 0.3 m for this tank — would be overtopped on most of the towers on the figure, and a wave that reaches the roof turns sloshing mass into impulsive mass against a roof that was never designed to be pushed sideways. On a tower the roof is the top of the structure, where the lever arm to the foundation is longest.

The tower’s sway #

The top displacement follows the base shear’s pattern and makes the case for the coupled calculation more strongly, because a displacement is what a tower’s second-order checks and its pipework depend on.

The stiff tower sways furthest when full, 104 mm. The flexible one sways 145 mm at 84% full and 140 mm full; rigid water says 186 mm, flat from 37% full because beyond the corner the spectral displacement no longer depends on the period. A slender tower’s design is often governed by its sway, through the extra moment the tank’s weight makes when the top has moved, and the rigid-water answer overstates that by a third. The riser that runs up the tower to the tank is held at the bottom by the ground and at the top by a container that moves by the coupled displacement, not by either shortcut’s — the situation of a pipe held at every floor, driven by the difference between its supports, with one support a tank. And anything fixed to the container itself feels the spectrum the tower hands on rather than the ground’s.

The periods, by hand #

The coupling can be checked with two numbers from the ground tank. With the 2.5 s tower, the tower’s stiffness is $k = 4\pi^2 \times 739/2.5^2 = 4{,}670$ kN/m, and the convective spring is $k_c = 304 \times (2\pi/3.65)^2 = 901$ kN/m. The two masses’ frequencies squared, alone, are $k/739 = 6.32$ and $k_c/304 = 2.96\ \text{s}^{-2}$; coupled, the eigenproblem is

$$\begin{vmatrix} (k + k_c)/739 - \omega^2 & -k_c/739 \\ -k_c/304 & k_c/304 - \omega^2 \end{vmatrix} = 0,$$

with diagonal terms 7.54 and 2.96 and off-diagonal product $1.22 \times 2.96 = 3.61$. Its roots are $\omega^2 = 5.25 \pm 2.98$, so $8.23$ and $2.27\ \text{s}^{-2}$, and the periods 2.19 s and 4.17 s — the figure’s two, moved outward from the uncoupled 2.50 and 3.65 by the off-diagonal term. That term is the convective spring divided by the tower mass: it is large when the sloshing liquid is heavy against what the tower carries, which a broad, shallow tank on a slim tower is.

Where two masses are not enough #

The model is the simplest that captures the coupling, and its choices limit it.

One convective mode. The second sloshing mode holds about one per cent of this tank’s liquid, at a period of 2.1 s — right beside a flexible tower’s — and on such a tower it couples too; it is left out, and its mass is too small to move the base shear much, though it adds to the wave.

A rigid container on a shear spring. A real tower bends, rocks on its foundation and has mass of its own; its own modes add to the two here, and a tower of slender legs has a second period close enough to the first to matter.

Damping is the spectrum’s 5%. Sloshing is damped at well under one per cent, and a coupled mode that is mostly wave should be read off a spectrum for half a per cent, which is higher; the long mode’s share of the shear is underestimated by the figures, which matters most where that mode is heavy.

Linear sloshing. A wave of 0.44 m in a tank 12 m across is small enough for the linear theory; on a softer tower or a stronger earthquake the wave steepens and breaks, and the convective mass stops being a spring.

The ground does not move the foundation. A water tower on soft ground sits on a spring of its own, which lengthens its period and moves it further towards the wave.

Still open: whether the water can be tuned #

The convective mass on a tower behaves as a damper that nobody tuned and nobody damped. It already pulls the system’s mass into a slow mode; with the right depth and the right baffles it could be tuned to the tower’s period on purpose and given damping by a screen in the liquid, which is the tuned liquid damper some tall buildings carry in their roof. A water tower is a tuned liquid damper with a very large mass ratio and no damping. Whether a tower’s own water, with a baffle chosen for one fill level, can cut its base shear more than the coupling already does — and how much that reduction is worth when the fill level is whatever the town used yesterday — is the question of whether the tank can be designed to help the tower that holds it.