The tank that is its own tower

The water tower that could damp itself found that the water in a tank on a tower is, by a happy accident of proportion, nearly a tuned mass damper: its sloshing runs at close to the tuning a designed damper would be given, it weighs two fifths of what it hangs from, and a screen across the tank that gives the sloshing a tenth of critical damping takes the tower’s broadband response to four tenths of what it would be with the water frozen.

That essay ended by asking about a different structure. A standpipe — a tall steel or concrete cylinder that is its own reservoir, standing on the ground with the water filling its shaft — holds a height of water several times its width, and the water is spread along the very part of the structure that sways. Whether that column of water is still one tuned mass, whether its period can be chosen, and whether a screen could make the shaft damp itself as the screened tank damped its tower, are the questions here. All three answers are no, and the reason why is a fact about where in a tall tank the sloshing water is.

Only a lens near the surface sloshes #

The standpipe here is steel, 10 m in diameter and 30 m tall, with a shell 12 mm thick, full to 28 m. The water is divided as the liquid has a period of its own divided it: the potential-flow solution for a rigid cylinder splits the liquid into a part that moves with the wall, the impulsive part, and convective parts that slosh at periods of their own, of which only the first matters.

The sloshing mass does not grow with the water. In a shallow tank most of the water sloshes; as the depth passes the radius the sloshing part saturates, and beyond about a diameter it is a fixed quantity, $0.455\rho\pi R^3$, whatever is added beneath it. Here that is 179 t. The other 2,021 t of a full standpipe moves with the wall as though it were part of it. The sloshing water is a lens about a radius thick at the top of the column, and the column below it is ballast.

The reason is in the shape of the sloshing motion. A wave on the free surface moves the water beneath it in circles that shrink with depth, as $\cosh(\lambda z/R)$ measured up from the bottom, so the motion has all but died out a radius below the surface. Water deeper than that is too far down to slosh; it can only move with the walls that contain it.

The width sets the period #

The same saturation fixes the sloshing period. The first mode’s frequency is

$$\omega_c^2 = \frac{1.841\,g}{R}\tanh\frac{1.841\,H}{R},$$

and once $H$ is more than a radius or so the hyperbolic tangent is one and the depth has dropped out.

In a tank on a tower the town that draws water down changes the sloshing period, and the earlier essay found that this detunes the water as the tank empties. In a standpipe the town has no such influence: the period is 3.31 s from 5 m of water to the brim, and the only thing that sets it is the diameter, chosen decades earlier for the volume and the site.

The shaft’s period is a different story. Empty, the steel cantilever is stiff and light and sways at 0.084 s. As water is added the period rises, slowly at first, because water near the base hardly moves when the shaft sways, and then quickly as the water climbs into the part of the shaft that does. Full, it is 0.381 s. The two periods are a factor of nine apart at every depth.

Where the lens sits #

The fraction of the water that sloshes is small, but it is in the place that counts.

A cantilever’s first mode moves its top and barely its base, so a mass counts towards the mode in proportion to the square of how far its height moves. The impulsive water is spread uniformly up the shaft, and most of it is low, where the mode shape is small; weighted by the square of the shape, its 2,021 t count for about as much as a few hundred tonnes at the top. The sloshing lens is all near the top, where the shape is three quarters of its maximum. So the lens, 8% of the water by weight, is a quarter of what the shaft’s first mode carries.

A structure has more than one period made the general point that a modal mass is a weighted mass, and the weighting here is unusually lopsided. It is what makes the lens matter at all, and it is why everything that follows is about the top 5 m of a 30 m structure.

Too slow to be tuned #

A tuned mass damper is a mass that helps by being late: a mass on a spring whose frequency is close to the structure’s — Den Hartog’s optimum is $1/(1+\mu)$ of it, for a mass ratio $\mu$ — and it works by moving against the structure at the frequency where the structure wants to move. The lens is a mass on a spring. Its frequency is not close.

The lens runs at 0.12 of the shaft’s frequency where Den Hartog asks for 0.75 at this mass ratio. Nothing about the standpipe that a designer might vary closes the gap. A wider standpipe has a slower slosh, since its frequency falls as one over the root of the radius, and a stiffer shaft, since its bending stiffness grows as the cube of the radius and its mass only as the square: the two move apart. A narrower one is closer, but even at a radius of 2 m — a pipe, not a reservoir — the slosh is at 0.29. The depth, which tuned the elevated tank, cannot be used, because the period has stopped depending on it.

The only way to raise a deep tank’s sloshing frequency is to divide the water into narrower cells. A cell of width $b$ in deep water sloshes at $\omega^2 = \pi g / b$; to bring this standpipe’s water up to three quarters of its shaft’s 16.5 rad/s, the cells would have to be $\pi \times 9.81 / 12.4^2 = 0.20$ m wide. A lattice of 200 mm cells through the top of a reservoir is not a reservoir. Tuned liquid dampers in buildings are shallow tanks for exactly this reason: in shallow water the depth sets the period, and a depth can be chosen.

What the screen can do #

With the lens so far out of tune, the question of what a screen can do has a definite answer.

On the tower, plain water does nothing — 1.01 of the frozen response — and a screen is transformative: 0.40 at 10 per cent of damping. On the standpipe the curve is nearly flat. Plain water already gives 0.82 of the frozen response, and a screen improves that only to 0.76. Most of the standpipe’s 18 per cent comes before any screen is fitted, and it comes from a different mechanism from the tower’s.

The mechanism is isolation, the one the machine that shakes the building traced for machines on soft mounts. A mass on a spring whose frequency is far below the frequency it is shaken at does not follow the shaking: it stays where it is, and the spring takes up the difference. The lens hangs from the shaft on the spring of gravity acting on its free surface, at 0.12 of the shaft’s frequency, so when the shaft sways at its own frequency the lens hardly moves with it. The shaft’s first mode then carries three quarters of the mass the frozen calculation gave it, its period shortens from 0.381 s to 0.330 s, and the participation of the ground shaking in it falls. A screen adds a dashpot between the lens and the wall, which couples the lens back to the shaft a little; at heavy damping it acts as a damper to a nearly stationary mass and takes a few more per cent, but it cannot tune what is out of tune.

Why it is isolation and not damping #

The distinction matters because the two mechanisms have opposite requirements. A tuned damper works by resonance: the damper’s mass swings through large relative motions exactly when the structure is moving, and the damping between them takes energy out of every cycle. That needs the damper’s frequency near the structure’s, and it needs damping, because an undamped tuned mass only moves the structure’s resonance rather than flattening it — which is why plain water on the tower gave 1.01 and not less.

Isolation works by its absence. A mass on a spring shaken at a frequency $r$ times its own moves, relative to the ground it would otherwise follow, with a transmissibility of about $1/(r^2 - 1)$ when it has little damping. The shaft sways at 8.7 times the lens’s frequency, so the lens follows the shaft’s motion to about $1/(8.7^2 - 1) = 1.3$ per cent. In absolute terms it barely moves; the shaft moves under it, and the free surface tilts slightly as the wall slides past. For the purposes of the shaft’s own vibration the lens is not there.

That is a benefit which needs no damping — damping a mass that is meant to stay still couples it back to what it is isolated from — and needs no tuning: any frequency far enough below the shaft’s will do. The standpipe has it whether its designer intended it or not. What the standpipe cannot have is the elevated tank’s benefit, which needs the opposite proportions.

What a frozen-water calculation costs #

The comparison throughout is with the water frozen to the wall, which is not how a code would treat a standpipe; the European rules for tanks split the water into impulsive and convective parts exactly as this essay does, and so already let the lens hang on its own spring. The frozen calculation is the shortcut — the water treated as dead mass added to the shell — and it is worth knowing what the shortcut costs, because it is the one a quick check uses.

On a design spectrum — which is not a load but a list of peak responses by period — with a ground acceleration of 0.3 g and a plateau 2.5 times that, the shaft’s period with or without the lens sits on the plateau, so the spectral acceleration is the same 7.36 m/s² either way. What changes is the mass the first mode carries: 1,334 t with the water frozen, 1,187 t with the lens left free. The base shear falls from 9,817 kN to 8,735 kN, 11 per cent. The lens, on its own spring at 3.31 s, is far out on the spectrum’s falling branch at 0.67 m/s², and contributes 120 kN — just over one per cent of the total. The frozen calculation is conservative by about a tenth in shear and a fifth in rms drift, and in both cases the error is the lens put in the wrong place.

The wave the lens makes #

The lens’s other job is to make a wave, and that is the one effect of the water a standpipe’s designer must provide for directly. The wave’s height is about $0.84 R\,S_a(T_c)/g$, which here is $0.84 \times 5 \times 0.67/9.81 = 0.29$ m. A standpipe whose roof sits less than that above the full water level will have its roof struck from below; many have, and a dented roof is the commonest earthquake damage to tanks of every shape. The wave depends on the lens and not on the column beneath it, so a standpipe needs no more freeboard than a ground tank of the same width; and since its 3.31 s period lies on the spectrum’s last branch, where the spectral acceleration falls as the square of the period while the period grows as the root of the radius, the wave there is the same height in a wider tank — which the first essay in this line found for tanks of every proportion.

As the town draws it down #

The standpipe’s water level changes daily, and it moves the lens up and down the shaft.

With a few metres of water the lens is near the base, where the shaft hardly moves, and leaving it free makes no difference: the ratio is 0.91 at 6 m. As the water rises the lens climbs into the moving part of the shaft and the ratio falls, to its least, 0.76, at about half full. Above that the lens keeps rising but the water beneath it, which moves with the wall, begins to dominate the shaft’s modal mass, and the lens’s share of it falls back; full, the ratio is 0.82. The screen’s dashed line follows the plain water’s a few per cent below it at every depth. Over the whole range of operation the free lens is worth between a tenth and a quarter off the drift, and the screen a few per cent more.

The lens and the shaft by hand #

Each number can be checked in a line. The lens is $0.455 \times 1{,}000 \times \pi \times 5^3 = 178{,}700$ kg, 179 t. Its period is $2\pi\sqrt{R/(1.841g)} = 2\pi\sqrt{5/18.06} = 3.31$ s.

The empty shaft is a cantilever with $EI = E \pi R^3 t = 2.1\times10^{11} \times \pi \times 125 \times 0.012 = 9.9\times10^{11}\ \text{N·m}^2$ and a shell of $7{,}850 \times 2\pi \times 5 \times 0.012 = 2{,}959$ kg/m. With the shape $1 - \cos(\pi z/2L)$, its generalised stiffness is $EI(\pi/2L)^4 L/2 = 1.12\times10^8$ N/m and its generalised mass $0.227 \times 2{,}959 \times 30 = 20{,}100$ kg, so $\omega = 74.5$ rad/s and $T = 0.084$ s. Full, the generalised mass of the water that moves with the wall and the shell together is about 15 times that, and the lens, at $0.76^2 \times 178.5 = 103$ t, adds a third to it again: $\mu = 0.33$, and Den Hartog’s tuning is $1/1.33 = 0.75$.

Where the cantilever model stops #

The shaft is modelled in its first mode only, as a cantilever on a rigid base, with its water split by the rigid-tank solution and its shell elastic and round. A tall steel standpipe’s shell is thin enough that it ovals under the water’s pressure when it sways, and the coupling of the shell’s own deformation to the water — the flexible-tank problem — raises the impulsive pressure near the base above what a rigid wall gives; that matters to the shell’s buckling at the base, which is how standpipes fail in earthquakes, and nothing here addresses it. Soil under the base lengthens the shaft’s period and so moves it a little towards the slosh, though nowhere near enough to tune it. Higher sloshing modes have periods a few times shorter than the first and masses much smaller, and they are left out. And the screen is a linear dashpot, where a real screen’s damping grows with the sloshing’s amplitude, which helps a damper that has to work hard and does nothing for one that has nothing to work on.

The tank that sloshes the tower treated the elevated tank’s sloshing mass as one mass on one spring, hung from a container on a tower, and that picture made the damper possible. The standpipe puts the same lens of water at the top of the same kind of cantilever, and the cantilever is too stiff for it.

Still open: the tower whose shaft is a hollow core #

A water tower with a concrete shaft often has a hollow core used for stairs, pipes and sometimes a second, smaller tank, and a modern tall building may hold a tuned liquid damper near its roof inside a core that is itself the building’s spine. In both, the structure is a tall, slender cantilever with a period of seconds, not tenths of a second, and the water is in a tank whose width is chosen rather than inherited. Whether a deep, narrow tank inside a slender core can be tuned by its width alone — since its depth cannot help — and whether the half a spring that a deep tank’s sloshing provides is stiff enough to reach a core’s frequency, is the question the stiff standpipe leaves for a structure soft enough to need the damper.