The liquid has a period of its own

Every dynamics figure on this site so far has put the mass inside the structure. A column has mass, a floor plate has mass, and when the ground moves they move with it because they are attached.

A liquid is not attached to anything. It is contained, which is a different relationship, and the difference is the whole of this essay. Shake a tank and the liquid near the bottom goes along as though it were solid, while the liquid near the free surface does something else entirely — it sloshes, at a period that has nothing to do with the tank’s stiffness and everything to do with gravity and the shape of the surface.

Which free body produced the number #

The split comes from the potential-flow solution for a rigid circular tank, and it is worth saying which half of it is computed and which half is a remainder.

The convective modes are the sloshing modes of the free surface, and each has a mass, a height and a frequency that fall out of the flow solution:

$$\frac{m_{c,n}}{m} = \frac{2\tanh(\lambda_n H/R)}{(\lambda_n H/R)(\lambda_n^2 - 1)}, \qquad \omega_n^2 = \frac{\lambda_n g}{R}\tanh\!\left(\frac{\lambda_n H}{R}\right)$$

with $\lambda_n$ the roots of a Bessel derivative. The impulsive mass is then whatever is left, which is what makes the accounting exact: the parts sum to the whole by construction rather than by an approximation that stops being true for a tall tank.

The height the impulsive mass acts at is not tabulated either; it comes from a statics argument. Accelerate a rigid tank of rigid liquid sideways at $a$ and the pressure on the wall is $\rho a x$ — independent of depth — so the resultant is $ma$ acting at $H/2$. The modal masses and heights therefore have to reproduce that:

$$m_i h_i + \sum m_{c,n} h_{c,n} = m\,\frac{H}{2}$$

and $h_i$ falls out. It is the one place in this field where a free body of the whole body settles a modal quantity, and it is worth having because it makes the decomposition checkable: a set of masses and heights that does not reproduce a resultant at mid-height is wrong, whatever it was derived from.

Tall tanks and broad ones are different objects #

The split depends on one number, $H/R$, and it depends on it strongly.

A tall tank at $H/R = 3$ is 84% impulsive: most of its contents are too deep to know there is a surface, and the tank behaves almost as though it were carrying a solid. A broad shallow one at $H/R = 0.5$ is 72% convective: most of its contents slosh, and most of them therefore never notice the earthquake at all.

That is a genuinely useful design fact and it runs the opposite way to intuition. A broad reservoir attracts far less seismic force than its weight suggests, and a tall slender tank attracts nearly all of it — the same total mass, twice the base shear, decided by a proportion.

The two markers are the whole of the design consequence. Nothing has been added to the reservoir to make it lighter seismically and nothing has been taken away; it has been made broad, and the part of it that participates in the impulsive response has fallen by more than half. A designer who sizes a tank by capacity alone has chosen that number without noticing.

The proportion enters through a hyperbolic tangent, which is the signature of a decay with depth. Below about one radius from the surface the sloshing motion has effectively died away, so the convective mass is roughly whatever liquid lies within a radius of the top and the impulsive mass is the rest. That reading makes the whole figure predictable without any of the algebra: a tank deeper than about two radii has a fixed convective mass and an impulsive one that grows with every further metre of depth, which is exactly the shape the curves take.

Two periods, orders apart #

The reason the split matters is what the two masses meet when they arrive at a spectrum.

The impulsive period is a structural quantity — the tank wall’s flexibility with the impulsive mass on it — and it is a quarter of a second, which is on the plateau of any ordinary design spectrum. The convective period is $2\pi/\omega_1$, gravity and radius only, and it is 4.47 seconds, which is well past the corner where a spectrum has begun to fall as $1/T^2$.

The two accelerations therefore differ by a factor of twenty, and 35% of the liquid supplies 3% of the base shear.

That is the spectrum not being a load doing real work: the same ground motion is enormous for one part of the same tank and nearly absent for another, because the two parts ask it different questions. A spectrum is a curve of answers, one for each oscillator that might be asked, and a tank asks it twice — once at a period its own wall decides and once at a period only gravity and the radius decide.

The second of those is the unusual one. Every other period in this collection is a stiffness divided by a mass, so anything done to the structure moves it. The convective period contains no stiffness at all: thicken the wall, stiffen the base, change the steel, and 4.47 seconds stays 4.47 seconds. It is the one natural period in structural engineering that a structural engineer cannot alter.

The overturning, which needs two heights #

The base shear is one of two answers a tank needs, and the other one is where it acts.

Each mass has its own height, and the two are very different. The impulsive mass sits at about 41% of the liquid depth; the first convective mass sits at 66%. So the overturning moment is not the base shear times any single height, and the shares that make it up are different from the shares that made the shear.

The convective mass supplies 3% of the shear and rather more of the moment, because it is acting nearly two thirds of the way up. That is a small effect here and it is the general shape of a result worth carrying: a mass that is unimportant for force can be less unimportant for moment, because a moment weights everything by its height.

There is a further subtlety in what the moment is being computed for. The moment just above the base plate decides the wall’s own vertical compression and its anchorage; the moment below the base, including the pressures on the tank’s floor, decides the foundation. The two differ, and the modal heights for them are different numbers derived from the same flow field — which is why published tables give two sets and why using the wrong one is a common and entirely invisible error.

None of that is a refinement. On a slender tank the overturning moment is what governs, because it decides whether the shell lifts off its foundation on the windward side — and a shell that lifts and sets back down is a shell doing something no elastic analysis of it contains.

The wave, which is a displacement #

The sloshing mass takes almost no force and it decides something else.

A wave height is proportional to $R \cdot S_a(T_c)/g$, and in the falling branch $S_a$ goes as $1/T_c^2$ while $T_c^2$ goes as $R$ — so the two dependencies cancel exactly and the wave stops growing. A tank twice the size gets the same wave.

The consequence for design is that freeboard is not scaled with the tank. It is a roughly constant height, set by the ground motion’s long-period content rather than by the vessel, and a large tank therefore needs proportionally less of it.

Where the freeboard is insufficient, the wave hits the roof — and then the convective mass, which had been contributing almost nothing, is stopped by a structure. The force it delivers over that short contact is impulsive by every argument about a load that is over before anything has moved, and a roof designed for a snow load meets a mass of liquid arriving with a velocity.

The pressure on the wall #

The two masses put pressures on the wall with opposite shapes, which is the clearest picture of what the split means physically.

The impulsive part is uniform with depth, because a rigid body accelerated sideways produces a pressure that does not know how deep it is, and it reaches 43.1 kN/m² — 37% of the hydrostatic pressure at the base. The convective part is the opposite: a disturbance of the free surface, dying away downward, and largest at the top.

So the two masses load different parts of the wall, and a wall designed for the total force in the wrong distribution is wrong in both halves. It is also why the two cannot simply be added: their peaks occur at different times, being responses at periods twenty times apart, and the accepted treatment is to combine them as the square root of the sum of their squares rather than arithmetically — which for this tank makes almost no difference to the shear and rather more to the wave.

What the wall does with it #

A tank wall carries a horizontal pressure as a hoop force, and that is where this essay meets the static half of the collection.

The force that is only a radius is $N = pR$, and it arrives with no lever arm — the one internal force in this collection with none. The seismic pressure adds to the hydrostatic one and the hoop force follows directly.

The complication is at the base. A wall cast into its slab cannot move outward there, so it has no hoop strain and therefore no hoop force, and the pressure is carried in vertical bending instead at a fixing moment of 83.8 kNm/m. The peak hoop force is a quarter of the way up. Reinforcement placed by the membrane triangle is in the wrong place and misses the moment.

The component that does not slosh at all #

Everything above is about horizontal shaking, which is where the two-mass decomposition lives. The vertical component has no decomposition, and on a tank it is not the minor case it usually is.

Shake the ground vertically and the liquid cannot get out of the way. There is no free surface to relieve it — a rising floor raises the whole column — so the entire contents respond, and the pressure on the wall is simply the hydrostatic one recomputed in a stronger gravity:

$$p(z) = \rho\,(g + a_v)\,z$$

For this tank the hydrostatic pressure at the base is 117.7 kN/m². A vertical acceleration of $0.4g$ adds 47.1 kN/m² to it — more than the 43.1 the horizontal impulsive mass delivered, and the horizontal number was the one the whole first half of this essay was computing.

Two things make it worse than that comparison already suggests. The horizontal pressure acts on one meridian and relieves the opposite one, so a hoop force it adds on the windward side it subtracts on the leeward; the vertical pressure acts on the entire circumference at once and has no favourable side anywhere. And it is in phase with itself all the way up the wall, so it scales the hoop force at every level by the same factor rather than peaking somewhere and vanishing elsewhere.

In hoop force, at the base: 1,059 kN/m hydrostatic, 424 from the vertical component, 388 from the horizontal impulsive one on the meridian where it is worst. The seismic addition is roughly three quarters of the static value, and the larger half of it comes from a load case with no sloshing, no modal split, no spectrum reading past a corner period, and nothing about it that requires this essay’s machinery at all.

Which is exactly why it gets left out. The interesting analysis is the horizontal one — it has masses that have to be found, periods orders apart, a wave height, a decomposition worth drawing. The vertical case is a multiplication, and a multiplication does not look like an analysis. The period it needs is the shell’s own axisymmetric breathing period, which is short enough to sit at the plateau or above it, so the amplification over the ground’s vertical acceleration is modest and the answer is close to $\rho\,a_v z$ with $a_v$ read straight off the vertical spectrum.

The generalisable form: a decomposition says which parts of a load are hard to compute, and says nothing about which are large.

What the anchorage is being asked #

The last part of a tank’s seismic design is at its perimeter, and it is a connection problem of exactly the kind this collection has already met.

A tank shell resists overturning by a couple: compression on the leeward side, tension on the windward. The tension has to be delivered into the foundation by holding-down bolts or by the weight of the shell and its contents over the uplifted zone — and where the bolts do the work, their capacity is a cone of concrete rather than anything about the steel.

Many tanks are not anchored at all. An unanchored tank resists overturning by lifting one side of its base plate, at which point the base plate acts as a membrane, the shell’s compression on the far side concentrates over a short arc, and the whole thing rocks. That is a perfectly acceptable design and it is not one any linear analysis describes: the mechanism is a large displacement, the restoring force comes from the liquid’s own weight over the uplifted area, and the period lengthens as the rocking grows.

The failure it protects against is worth naming. A rigidly anchored tank that cannot rock puts its whole overturning moment into the shell as vertical compression, and a thin cylinder under vertical compression buckles into the diamond pattern known as elephant’s foot at a stress far below anything the material would suggest. Letting the tank lift is a way of limiting the force it can attract — made weaker on purpose, applied to a vessel.

Where the model stops #

The tank was assumed rigid. The convective solution is exact for a rigid wall, and a steel tank’s wall is not — its flexibility raises the impulsive period, and for a thin shell it can also couple with the shell’s own breathing modes, which the two-mass model has no representation of at all.

Only the first convective mode matters, and only mostly. The higher sloshing modes carry a few per cent of the mass between them, and they matter for the same reason higher modes matter anywhere: most of the mass moves together is a statement about a first mode, and the remainder has to go somewhere.

Nothing here is about the tank’s own modes. A thin shell has breathing modes of its own, and a structure has more than one period applies to the container as much as to the contents.

A tank on the ground is not a tank on legs. Everything above assumes the tank’s base is on the ground and the impulsive period is short. An elevated tank has a long structural period, the impulsive mass sits somewhere quite different on the spectrum, and the two periods approach each other.

That figure is the same view as the earlier one and it is a different design problem. Lengthening the impulsive period halved the force on the part of the liquid that carries nearly all of it, which is the ordinary benefit of a flexible support; but it also brought the two periods within a factor of four of each other in time, and the two-mass model assumes they are far enough apart to be treated as independent. Where they are not, the tower and the liquid respond as one system, and the decomposition that this whole essay is built on stops being a decomposition of anything.

What the picture cannot show #

The two masses are a fiction. There is no part of the liquid that is impulsive and no part that is convective; there is one flow field, decomposed into modes for convenience, and the decomposition is exact in what it predicts and imaginary in what it depicts.

Nor does any figure show what a partially full tank does. Everything here is a function of $H/R$, so a tank draws a path across the first two figures as it empties, travelling from the second’s proportion towards the first’s and past it, and its convective period lengthens while its impulsive mass falls — so the worst case is at some fill level between empty and full, and which one it is depends on the spectrum rather than on the tank. The two markers drawn are two frames of that journey, and there is no figure here of the journey itself.

The generalisation #

The habit worth carrying is to ask, of any mass in a structure, whether it is attached.

A liquid is the extreme case and it is not the only one. Grain in a silo moves partly with the walls and partly not, and it also has friction, so its answer is different again. A suspended ceiling, a raised floor, a stack of stored goods, a crane’s load hanging on a rope: each is a mass that goes with the structure at some frequencies and not at others, and each is included at its full weight in almost every model ever built.

The general principle is that a mass contributes to a seismic force only in proportion to how much of it moves with the structure at the structure’s own period. Most of the mass moves together is the essay about how much of a building’s mass participates in its first mode; this is the same question asked of the contents rather than of the frame, and the answer is that a great deal of what a structure weighs is not, dynamically, part of it at all.