The period nobody chose

The last essay’s answer depended on one number about the structure and nothing else: how long it takes to swing once. A quarter of a second was nearly instantaneous for that platform, and it would have been leisurely for a bracket and geological for a suspension bridge.

That number is not on any drawing. Nobody specified it, no client asked for it, and it does not appear in any calculation done for strength. It is a consequence — of a mass fixed by what the building is made of and what goes in it, and a stiffness fixed by a deflection limit that was checked for an entirely different reason.

The frequency was in the deflection all along. It takes one line of arithmetic to get it out, and the line is the same for a beam, a floor, a frame or a mast.

Why mass over stiffness, and nothing else #

Draw the free body once more: a mass, a spring, and no applied load at all. The spring pulls back with $ku$ and the mass resists acceleration with $m\ddot u$, so

$$m\ddot u + ku = 0$$

and the only motion that satisfies it is a sine wave at $\omega = \sqrt{k/m}$. Two properties of the structure go in and one number comes out; nothing else can enter, because nothing else is in the equation.

That is a stronger statement than it looks. The amplitude is not in it. A structure swings at the same rate whether it is nudged or shoved, which is why a period can be measured at all — hit the thing however hard is convenient and time the swings. It is also why the whole of this field can be organised around a single number per structure, in a way that no strength calculation ever can.

The square root is the part that misleads. Frequency responds to stiffness and mass weakly, and both directions have been measured on the same beam:

Doubling the stiffness of a beam raises its frequency by a factor of 1.4142, which is √2 and is the entire return on doubling the material working in bending. Adding half again to the mass drops it by 0.8165, which is 1/√1.5. Doubling the span, by contrast, quarters the frequency, because span enters both terms at once — the stiffness falls as the cube of it and the mass rises in proportion to it.

So the honest summary of how to change a structure’s period is: change the span. The rest is square roots.

Rayleigh’s method, which is free #

The exact frequency of a beam comes from a characteristic equation with hyperbolic functions in it. The approximate one comes from a quantity every serviceability check already produces.

Rayleigh’s argument is an energy statement and it takes three lines. Assume the structure vibrates in some shape; at the extremes of the motion all the energy is strain energy, and at the middle all of it is kinetic. Equate the two, and the frequency drops out:

$$\omega^2 = \frac{g\sum w\delta}{\sum w\delta^2}$$

with $w$ the weight at each point and $\delta$ its deflection under those weights. Every quantity in it is in the static analysis.

Two things about this are worth knowing, and both are provable rather than empirical.

It is always an over-estimate. Assuming a shape is imposing a constraint, a constraint stiffens, and a stiffer structure has a higher frequency. So Rayleigh’s answer bounds the true one from above, always — 4.915 against 4.909 here, and never the other way round. A method that is guaranteed to err in one direction is worth far more than one that is more accurate on average.

It is astonishingly close. The static deflected shape under self-weight is not the first mode, but it is very nearly the first mode, because the loading that produces it is distributed the same way the inertia forces are. A tenth of a per cent is typical, and the error is second-order in the difference between the assumed shape and the real one — which is the general property of a Rayleigh quotient and the reason it was worth a name.

The rule of thumb every engineer carries falls straight out of it. Take a simply supported beam under self-weight, put the closed-form deflection into the quotient, and the result is

$$f \approx \frac{18}{\sqrt{\delta}}$$

with $\delta$ the mid-span deflection in millimetres. On the beam above that gives 4.977 Hz against 4.909 — 1.4% high, and high for the same reason. A floor that sags 13 mm is a 5 Hz floor, and that sentence is all most problems need, since what a floor does about footfall turns on which side of 5 Hz it lands.

The ends matter more than anything in the middle #

The characteristic equation is where the end conditions enter, and they enter hard.

That factor of 6.36 is the same phenomenon as the factor of sixteen between a column’s end conditions, and it arises the same way: the boundary conditions decide the shape, the shape decides the curvature, and the curvature is what the stiffness acts on. In both cases the number that changes is a root of a transcendental equation, and in both cases the practical consequence is that the least well-known part of a structure is the part that decides the answer.

Because the frequency goes as the square of that root, small differences in end fixity are magnified. A beam whose ends are neither pinned nor fixed — which is every beam, as the connections field spends its time establishing — has a frequency somewhere between 4.91 and 11.13 Hz, and a measurement on the finished floor is worth more than any calculation.

Higher modes exist, and are not evenly spaced #

The characteristic equation has infinitely many roots, and each is a different way the beam can vibrate.

The ratio 1 : 4 : 9 is worth pausing on because it is the reason a simply supported beam is a poor musical analogy. A string’s overtones are 1 : 2 : 3, which is why a string sounds like a note; a beam’s are 1 : 4 : 9, which is why a struck beam sounds like a clank. The difference is that a string’s restoring force comes from tension and a beam’s from bending, so one is governed by a second derivative and the other by a fourth.

A frame is not a beam, and the period is still a division #

For a building the mass sits at the floors and the stiffness is in the columns between them, and the same square root applies at a different scale.

Across a range of heights that model reproduces the rule of thumb every seismic engineer uses. Three storeys gives 0.387 s, five gives 0.605, eight gives 0.932, twelve gives 1.370, twenty gives 2.246 — which is 0.11 to 0.13 seconds per storey, against the standard estimate of about a tenth of a second per storey. The rule is not folklore; it is what $\sqrt{m/k}$ produces when mass and stiffness both scale with height in the ways that buildings make them.

It also shows where the rule stops. Soften that frame’s ground storey to 35% of the others — an open frontage, a car park, a hotel lobby — and the fundamental period goes from 0.605 s to 0.801 s, a third longer for a change to one storey out of five. The first mode’s shape changes with it: the ground-floor drift, 0.285 of the roof’s before, becomes 0.570. Half the building’s deformation has moved into one storey.

That is the soft-storey failure written as an eigenvalue rather than as a photograph. Nothing in the frame got weaker — every column is the one that was there before — and the deformation the earthquake will demand has been concentrated into the storey with the fewest columns to share it.

The end condition nobody drew is under the building #

Every frequency in this essay so far has assumed the structure stops somewhere and that the somewhere does not move. The beam’s supports were pins or encastrés; the frame’s columns were built in at the ground. But the ground is not a support condition. It is a material, and it deflects — and the four end cases drawn earlier are the ones a drawing can name rather than the ones a structure has.

The square in that term is the whole of why soil-structure interaction is described as a tall-building problem. A squat structure barely notices the foundation, because the only flexibility available to it is the sliding one, and sliding is nearly free against the stiffness of the structure above. Tip the same structure and the same foundation rotation moves the top by the rotation times the height, so the flexibility it contributes rises as the square of the height while the sliding term does not rise at all.

Two consequences follow, and both are the opposite of what the picture suggests. The first is that the mechanism nobody draws is the one doing the work: every sketch of a building on soft ground shows it sliding, and 89% of the effect at this height is a rotation of a footing through a fraction of a degree. The second is that the lengthened period is not free. A rocking footing pushes the ground down on one side and up on the other, the two nearly cancel in the far field, and almost nothing radiates away — so the extra flexibility arrives without the extra damping that a sliding foundation would have supplied with it.

Which makes the base a third guess of the same kind as the mass and the stiffness, and the least well-recorded of the three. A drawing states a section size to the millimetre and a shear wave velocity not at all.

Which mass is moving #

There is a quantity hidden in Rayleigh’s quotient that is worth taking out and looking at, because it answers a question that otherwise sounds unanswerable: when a beam vibrates, how much of it is moving?

Not all of it, and not at the same speed. The mid-span of a simply supported beam travels the full amplitude and the ends do not move at all, so the kinetic energy is less than it would be if the whole mass moved together. Do the integral for a half-sine shape and the answer comes out at half the total mass — that is the mass which, placed at mid-span on a spring of the beam’s stiffness, would have the same period as the beam.

For a cantilever in its first mode the number is about a quarter of the total, because most of the beam is near the root and hardly moves.

This is not bookkeeping. It is the quantity that decides whether a person walking on a floor can shift it at all, and it is why a long-span floor is worse than a short one for reasons beyond its lower frequency: the walker’s own weight is a larger fraction of the mass that is actually participating. A structure’s dynamic size is not its size, and the ratio between the two is a property of the mode rather than of the object.

Rayleigh, Dunkerley, and the shafts that came first #

The method is Lord Rayleigh’s, from The Theory of Sound in 1877, and it belongs to a period when the subject was acoustics rather than structures — the questions being asked were about bells, plates and organ pipes, and the structural applications came afterwards.

The engineering pressure came from rotating machinery. A turbine shaft has a speed at which it whirls violently, and it is the shaft’s own bending frequency; running through it on the way to operating speed is a design problem with an expensive failure mode. Dunkerley’s rule, published in the 1890s, gave a way to estimate the critical speed of a shaft carrying several rotors by adding the reciprocals of the squares of the frequencies each rotor would give on its own — an approximation that errs low, where Rayleigh’s errs high, so the pair of them bracket the answer.

The two rules bracketing from opposite sides is the useful part, and it is a pattern this site meets in other fields: the plastic collapse load is bracketed by an upper-bound mechanism and a lower-bound stress field, and both are worth more together than either is alone.

What is striking, from here, is that this apparatus was fully developed before it was needed for buildings at all. Structures were not analysed dynamically until the twentieth century — not because the mathematics was missing but because the loads that need it were rare when buildings were heavy and stiff, and because nobody had a way to measure the one number the theory cannot supply.

What the picture cannot show, and what the number assumes #

The figures above give a frequency to three decimal places for a beam whose real frequency nobody knows to two.

The mass is a guess about occupancy. A floor’s period depends on what is standing on it, and the load that will be there when the vibration matters is not the load the strength calculation used. Design codes ask for a fraction of the imposed load in the vibrating mass, and the fraction is a judgement.

The stiffness is a guess about cracking. A concrete floor’s second moment of area depends on how cracked it is, which depends on how it has been loaded since it was cast. A factor of two between the uncracked and fully cracked values is ordinary, and it moves the frequency by 1.4.

The end conditions are the largest uncertainty and are drawn as though they were exact. The figure above shows four discrete cases. The real beam is between two of them, and the answer moves by a factor of two across that gap.

What saves the situation is that these errors do not scatter. Almost every one of them runs the same way, because almost every one of them is a stiffness that was deliberately left out.

Not one of those four is a modelling error. Cladding and partitions are stiffness nobody is allowed to rely on for strength, because they can be removed on a Tuesday; a nominally pinned connection is modelled as pinned because relying on its rotational stiffness would be relying on something nobody detailed; a cracked section is assumed because a designer cannot know where the cracks will be. Each omission is the right decision on its own terms.

The asymmetry is the finding. Leaving stiffness out makes a deflection larger than it will be, which is conservative, and it makes a frequency lower than it will be, which is not — because a lower frequency sits closer to the range a walker excites. The same modelling convention is safe in one check and unsafe in the other, and the correction that would have got the vibration check right is exactly the stiffness nobody is willing to count on for strength. A designer who wants the real number has to model the building twice, in two different spirits.

So the honest reading is that a computed natural frequency is a number with a band around it of perhaps ±20%, and the design decisions that turn on it should survive that band. This is exactly why a floor’s response is presented as a curve against frequency rather than as a single number: what a designer needs to know is not the frequency but whether the neighbourhood of it is a bad place to be.

The decay measures what nothing else can #

One property of a structure genuinely cannot be calculated from the drawing, and the same test that measures the period gives it away.

Pull the structure twenty millimetres out of position and let go. What follows is a decaying sine, and it carries two numbers rather than one. The period is the spacing of the peaks, and it is the quantity this whole essay has been computing from mass and stiffness. The damping is the rate at which those peaks shrink, and at two per cent of critical each swing reaches 0.882 of the one before — which is $e^{-2\pi\zeta/\sqrt{1-\zeta^2}}$ and nothing else. Everything about the first number was predictable from a drawing. Nothing whatever about the second was.

Which is the subject of the next essay, and the reason it is a separate one: the period comes from the structure’s geometry and its material, and the damping comes from neither.

The period as a diagnostic, and why the square root spoils it #

A natural frequency is one of the very few structural properties that can be measured without loading anything. Record twenty minutes of ambient vibration, take a spectrum, and the peaks are there. No jack, no datum, no reference beam.

That makes it the obvious quantity to monitor. Stiffness is what a structure loses when it is damaged — a crack, a corroded tendon, a bearing seizing, a scoured pier — so tracking the frequency over years should show the loss.

It does, and the square root that has been flattering every uncertainty in this essay now works against the measurement. Since $f \propto \sqrt{k}$,

$$\frac{\delta f}{f} = \tfrac{1}{2}\,\frac{\delta k}{k}$$

so a ten per cent loss of stiffness shows up as a five per cent change in frequency, and a loss of a fifth as a tenth. The instrument compresses the signal by two before anybody looks at it.

Which would still be usable if nothing else moved the frequency, and a great deal does. A bridge’s frequency varies by several per cent over a year, because concrete’s modulus rises as it cools, asphalt surfacing stiffens enormously in winter, and bearings that slide freely in July can be effectively seized in January. A floor’s varies with how many people and how much furniture are on it, through the mass term. Both effects are of the same order as the damage they are supposed to reveal.

So a monitoring scheme that records frequency and nothing else detects the seasons. The schemes that work do one of two things: regress the frequency against temperature and occupancy and watch the residual, which needs a year of baseline before it can say anything; or watch the mode shapes rather than the frequencies, because a local loss of stiffness changes the shape near the damage a great deal while changing the global frequency very little.

That second option is the more interesting one and it is the general lesson. A frequency is a global quantity — an average of the whole structure’s stiffness weighted by where the mode moves — so it is a poor detector of a local defect by construction, and the square root halves what is left. A mode shape is a distribution rather than a number, and a distribution can say where.

There is a cheerful corollary hiding in the same arithmetic, and it is the reason the method survives at all. The compression that hides damage also hides error: a frequency computed from a model whose stiffness is wrong by twenty per cent is wrong by only ten, so a period prediction is a good deal more reliable than the stiffness estimate behind it. The square root is a poor microscope and a forgiving one, and which of those matters depends entirely on whether the quantity is being predicted or being watched.

Which is the same asymmetry the check that cannot see the error records for a static analysis: a single global number closes over a wide class of local mistakes, and finding them needs a quantity with a position in it.

Where the ladder goes #

Three directions lead out of here, and each is an essay of its own.

The first is that a real structure has many periods and not one, and the question of which of them matters is the question of where the mass is — the next rung.

The second is what happens when something drives a structure at the period it has, which is where the number computed here turns into a number of millimetres.

The third is the practical one, and it is the reason engineers compute this at all: the period decides which of the loads a structure meets are static and which are not. A 5 Hz floor is dynamic under footfall and static under wind. A 0.2 Hz tower is dynamic under wind and static under footfall. Same two loads, same physics, opposite conclusions — and the only thing that decides which is which is a number nobody wrote down.