The spectrum a floor hands on

Every essay in this field treats the structure as the thing being shaken. That is the wrong emphasis for most of what an earthquake actually damages.

Structural collapse is rare and catastrophic. Far more common, and far more expensive in aggregate, is the failure of everything a building contains: transformers that walk off their pads, chillers that shear their holding-down bolts, server racks that topple, suspended ceilings that come down, sprinkler pipes that whip and break. Every one of those is mounted on a floor, and the motion of a floor is not the motion of the ground.

The building is a filter between the two, and it is a filter with a very sharp tooth in it.

What a floor spectrum is #

A response spectrum answers one question: for an oscillator of period $T$ and damping $\zeta$, subjected to this ground motion, what is the largest acceleration it reaches?

A floor spectrum asks the same question with the input changed. Take the ground motion, run it through the building, extract the acceleration history of one floor, and compute the spectrum of that. The answer is what a component mounted on that floor would experience.

Computing it means solving two problems in sequence. The building’s response is a modal superposition — each mode is a single-degree oscillator driven by the ground, and the floor’s absolute acceleration is the ground’s plus the sum of the modal contributions at that level. Then the resulting history drives another oscillator, one period at a time, and the peaks are the spectrum.

Nothing about that is subtle. What is worth attention is the shape that comes out.

The tooth, and its height #

The floor spectrum has a peak at the building’s own period, and the peak is very large.

For the eight-storey frame drawn — first mode at 0.93 s, five per cent structural damping, a component at two per cent — the peak floor spectral acceleration is 37.5 m/s² at 0.91 s. The ground spectrum at the same period is a fraction of that.

Expressed as a ratio, a component tuned to the building sees 6.09 times what the same component would have seen bolted to the ground.

The reason it is so large is that two amplifications multiply rather than add. The building amplifies the ground motion at its own period, producing a floor motion that is nearly harmonic at that frequency. The component then resonates with a nearly harmonic input, which is the most favourable possible condition for a resonance — far better than resonating with a broadband one.

A resonance inside a resonance is not twice as bad; it is a product, and the product is decided by the two dampings rather than by the ground motion.

The two-spectrum picture is the one a designer is handed, and it hides the mechanism because it never draws the component. Drawn as what it actually is — two masses on two springs, a large one shaken from below and a small one hanging off it — the same situation reports different numbers.

That is what “a resonance inside a resonance” means when it is measured rather than described. The building’s own magnification is ten; the small mass hanging off it moves sixty-four times the building’s static deflection, and it does so because the thing shaking it is not the ground but a nearly harmonic motion at exactly its own frequency.

Notice also which of the two masses the picture flatters. The building barely notices the component — its peak moved from 10.01 to 8.19 — while the component is doing something extreme. An interaction can be negligible at one end and dominant at the other, and reading it off the large mass is how equipment gets signed off.

The floor is worse everywhere, not only at the peak #

The tooth is what everybody remembers and it is not the whole of the effect.

At the very short-period end — a component so stiff it simply follows its support — the amplification is 1.79. That is not a resonance; the component has no dynamics at all. It is the ratio of the floor’s peak acceleration to the ground’s, which for this building is 8.01 against 3.50, a factor of 2.29 measured on the histories themselves.

Rigidity removes the resonance and not the amplification. A component bolted rigidly to a roof is being carried by a mass that is itself accelerating harder than the ground, and no amount of stiffness changes that.

At the long-period end the ratio falls back toward one — 1.12 at four seconds — because a component much more flexible than its support does not care what the support does; it stays where it is while the building moves around it.

So the shape is a broad elevation with a sharp peak on it, and the two features have different causes and different remedies.

Which free body produced the number #

The free body is the component, cut from its mounting, and the question is what crosses the cut.

For a component of mass $m$ on a floor with absolute acceleration $a_f(t)$, the equation of motion in terms of the relative displacement $u$ is

$$m\ddot u + c\dot u + k u = -m\,a_f(t),$$

which is exactly the ground-motion equation with the floor’s acceleration substituted for the ground’s. The component does not know it is on a building. Everything the building does to it arrives as a change to the forcing function.

That is why the two-stage calculation is legitimate at all, and it is also where its main assumption hides. The equation above treats $a_f$ as prescribed, which means the component’s own inertia is assumed not to affect the floor. That is true when the component is light relative to the floor and false when it is not — and a chiller on a roof, or a water tank, is often not.

A heavy component is a tuned mass whether anybody intended it or not, and if it happens to be tuned to the building it will reduce the very floor motion that is exciting it. The two-stage calculation gets that case badly wrong in the conservative direction, which is the only reason it is tolerated.

Three things follow from that comparison and none of them is in the two-stage calculation.

The heavy component moves less, by a factor of three, for no reason connected to its mounting — it is drawing energy out of the floor that is shaking it, so the excitation it receives is smaller than the excitation the empty building would have produced.

It also widens the tooth it sits in. The two fixed points move from within four per cent of the building’s frequency to within thirteen, so the range of periods that are genuinely resonant with the building has grown with the mass hung on it.

And it changes the answer for everything else in the building. The floor spectrum every other component is checked against was computed for a building without the chiller on it, and the chiller has moved the peak. A two-stage calculation cannot represent any of that, and it errs safe only for the heavy item itself.

Detuning has a direction #

The practical response to a peak is to move away from it, and the move has a sign that is easy to get wrong.

A component stiffer than the building — most equipment, most of the time — is on the short-period side of the tooth. Making it softer moves it toward the peak. So the instinct to “isolate” a piece of equipment by putting it on flexible mounts is, for most equipment on most buildings, a move in the wrong direction until the mounting is soft enough to carry it past the peak and out the other side.

A component softer than the building is already past it, and softening it further is safe.

That is a genuinely awkward result for anyone specifying anti-vibration mounts, because those mounts are chosen for a completely different problem: keeping a machine’s operating vibration out of the building, which wants the mount as soft as possible. The seismic requirement and the vibration requirement are both about the mounting’s stiffness and they do not agree, and the mount that solves the second can create the first.

The machinery underneath both stages is the same one that produced twice the deflection for the same load: one oscillator, one input, one peak read off the response. Nothing about the mathematics changes between a ground spectrum and a floor spectrum. What changes is the input, and a narrow-band input produces a far sharper resonance than a broadband one — which is the entire reason the tooth exists and the entire reason it is so much taller than the ground spectrum’s own hump.

Why the peak’s height is a damping ratio #

The tooth’s height is the quantity that decides everything and it is not a property of the earthquake. It is a property of two damping values, and the reason is worth setting out because it explains why the calculation is so sensitive.

A structure with damping $\zeta_p$ responds to a broadband input with an amplification of roughly $1/(2\zeta_p)$ at its own period. The floor motion is therefore nearly harmonic at that period, with an amplitude set by the structure’s damping.

A component with damping $\zeta_s$ driven by a nearly harmonic input at its own period amplifies again, and its amplification is limited by both dampings together rather than by its own alone — because the primary system is not an infinite source and the two exchange energy. The classical result for a light secondary system is an amplification of order

$$\frac{1}{2(\zeta_p + \zeta_s)}$$

at exact tuning, rather than $1/(2\zeta_p) \times 1/(2\zeta_s)$, which is what multiplying the two amplifications naively would give.

That is the one piece of good news on this page. The product would be a factor of 125 for the dampings used here; the correct combination gives about 7, which is the order of the six that the computation returns. The primary system’s damping protects the component, and it does so by an amount that has nothing to do with how strong either of them is.

The sum in that denominator is checkable by moving one number and watching two.

Read those three together and the practical instruction is unusually clean. Between 0.5% and 10% component damping the stroke changes by a factor of 2.7, and nothing about the component’s strength, its mass or its mounting stiffness has changed at all. Damping is the only property of the component that moves its own answer without moving its tuning, and it is the property nobody specifies.

It is also why the same picture is a tuned mass damper when it is deliberate. The mass that helps by being late is exactly this arrangement built on purpose: a small mass tuned to a large one, moving enormously, limited by the sum of two dampings, exchanging energy with the structure. The difference between a damper and a casualty is whether the large motion was intended and whether something was built to accommodate it.

Height, and why the roof is the worst place #

The floor spectrum is different on every floor, and the variation is close to the mode shape.

The first mode’s contribution at level $i$ is proportional to $\phi_i$, which for a regular frame grows nearly linearly with height. So the amplification at the peak grows with height and is largest at the roof — which is where the plant is.

That is a genuinely unfortunate coincidence of building layout. The heaviest and most fragile equipment in a building is usually on the roof, which is where the floor spectrum is worst, and it is there for reasons — noise, access, air intake — that have nothing to do with structural dynamics.

The second and third modes complicate the picture near the middle of the building. They have nodes, so a floor near a node of the second mode sees almost none of it, and a floor near its antinode sees a secondary peak at 0.31 s. On a tall building the higher-mode peaks in a floor spectrum can exceed the first-mode one at upper-middle levels, which is why floor spectra are computed per floor rather than scaled from the roof.

Where codes put the answer #

The full calculation — a ground motion, a building model, a modal history, a spectrum of that history — is expensive and is done for nuclear plants and for very little else.

What codes give instead is a formula for the peak floor acceleration as a function of height, of the form

$$a_f = a_g\left(1 + \frac{z}{H}\right)\!\cdot\!\text{something},$$

multiplied by an amplification factor for components near the building’s period. It is a two-parameter fit to the shape described above: a linear rise with height, and a bump at tuning.

That fit is adequate for the broad elevation and poor for the tooth, because the tooth’s height depends on both dampings and on how narrow-band the floor motion is. A code formula gives the floor’s motion well and the component’s resonance badly, and the resonance is the part that produces the failures.

What the fit cannot carry is that a floor spectrum is the mode shapes read at one level. The first mode grows toward the roof, so the first-mode peak grows with height and a linear rise is a fair fit to it; the higher modes have nodes, so their peaks appear and disappear as the level changes and no monotonic formula in z/H can produce them.

And the building is not the first filter in the chain. The soil filters the bedrock motion before it ever reaches the foundations, with a tooth of its own at four times the layer depth over its shear wave velocity. Each stage has a period and each has a sharpness, and a component tuned to a building that is tuned to its site is the worst arrangement the chain can produce — three resonances in series, none of which was chosen with the others in view.

The mass that is not counted #

There is a bookkeeping consequence that connects this page to the structural analysis it sits beside, and it runs in the opposite direction to everything above.

The equipment on a floor is part of the building’s mass. A roof plant deck carrying transformers and chillers can be several hundred tonnes, which on an eight-storey frame is a substantial fraction of a floor’s seismic mass — and it is at the top, where the mode shape gives it the most influence.

So the same objects that are being damaged by the floor spectrum are helping to determine it. Adding mass at the roof lengthens the building’s period, which moves the tooth, which changes which components are tuned to it.

That circularity is not usually acknowledged. Seismic mass is taken from the structural drawings plus an allowance; the equipment schedule arrives later and from a different consultant; and the two are reconciled, if at all, after the analysis. A building’s period is decided partly by objects that were chosen for reasons unrelated to it, and a late change to the plant layout is a change to the floor spectrum every other piece of equipment is being checked against.

What is actually done about it #

Three remedies, and they map onto the three features of the spectrum.

For the broad elevation, the answer is strength: anchorage designed for the floor’s acceleration rather than the ground’s, which is what the code formulae supply. Most equipment failures are anchorage failures, and most anchorage failures are the result of using the wrong acceleration rather than of using it wrongly.

For the tooth, the answer is detuning, in the right direction, or damping. Adding damping to the component flattens the peak substantially, because the peak’s height depends on the sum of the two dampings — which is why equipment mounted on damped isolators does far better than the same equipment on undamped springs.

For the flexible tail, the answer is displacement rather than force: a component softer than the building does not see much acceleration and does see a great deal of relative movement, so what fails is the connection to it — the pipe, the duct, the cable tray, which has to accommodate the difference between two things that are moving differently.

The third of those is the one that produces the most spectacular losses relative to its cost. A sprinkler pipe crossing a movement joint, or running between two structures with different periods, is a connection between two supports that will not move together — and the water that follows its failure damages a great deal more than the pipe did. The cheapest component in the building fails first and does the most damage, which is a poor arrangement and a very common one.

The measurement that settles it #

Floor spectra are one of the few dynamic quantities that get measured routinely, because instrumenting a floor is much easier than instrumenting a structure.

An accelerometer on a floor records the history this page computes. Its spectrum is the floor spectrum, directly, with no model in between — and comparing it with a computed one checks the building model, the damping assumption and the modal superposition in a single test.

The comparison is usually done with ambient vibration or with a small event rather than with a design-level earthquake, and that limits what it settles. The period and the mode shapes come out well and are usually close to the model. The damping does not: measured damping at small amplitude is typically well below the five per cent a design assumes, because the mechanisms that produce structural damping — cracking, friction at connections, non-structural elements working — are amplitude-dependent.

So a floor spectrum measured from a small event has a taller, sharper tooth than the design one, and reading it as evidence that the design was unconservative is the standard mistake. The instrument is right and it is measuring a different building: one at an amplitude where nothing has begun to damp.

Why the peak is narrower than it looks #

There is a feature of the tooth that decides how much attention it deserves, and it is not its height.

The peak’s width in period is roughly $2\zeta$ of the period at which it sits — a few per cent for lightly damped systems. So the range of component periods that are genuinely in resonance with the building is narrow: for a 0.93 s building at five per cent damping, from about 0.88 to 0.98 s.

That has two consequences pointing opposite ways. It means the number of components actually caught by the tooth is small — most equipment is nowhere near — so a code approach that applies a single amplification factor to everything is conservative for nearly all of it. And it means the ones that are caught are caught badly and are found by chance rather than by design, because nobody knows a chiller’s mounted period to five per cent.

A narrow, tall peak is the worst kind of hazard to design against: it affects few things, affects them enormously, and its position depends on a quantity nobody measures. That is a fair description of why floor vibration problems are so often discovered after occupation rather than before.

Where the model stops #

The two-stage calculation assumes the component is light. A heavy one interacts with the floor and reduces its own excitation, and the calculation cannot represent that.

Everything is elastic. A building that has yielded has a longer effective period and a softer, broader response, so the tooth moves and flattens. Paradoxically, structural damage protects the equipment.

The damping values are the weakest input. The peak’s height depends on the component’s damping, which is rarely known to better than a factor of two and is usually assumed.

The floor is treated as rigid in its own plane and as a point. A large floor plate has its own flexibility and its own modes, and a component in the middle of a long span sees a motion different from one at a column.

And a single record has a single realisation of the tooth. The peak’s height varies substantially between ground motions with the same spectrum, because it depends on how much energy the record happens to contain near the building’s period.

Where the ladder goes #

Later rungs on this anchor: direct generation of floor spectra from a ground spectrum, without a time history. The interaction of heavy secondary systems with their supporting structure. Code formulae for component acceleration and what they are fitted to. Anchorage of non-structural components, which is where nearly all the losses are. Distributed systems — piping, ducts, cable trays — which are supported at several floors moving differently. Ceilings, partitions and facades as displacement-governed components. And the wider question this field mostly ignores: what an earthquake costs when nothing structural has failed.