The wind is a spectrum

A wind load arrives on a drawing as a pressure in kilonewtons per square metre — a number, steady, applied. Every part of that description is a simplification, and for most structures it is a good one. For a tall or slender structure it hides the whole of the problem.

The total variance of that spectrum is 27 m²/s² for these conditions, which is a turbulence intensity of 17% — the wind’s speed fluctuates by about a sixth of its mean, constantly. That is the raw material, and what a structure does with it depends on two filters.

Two filters, and only one of them is the structure #

The first filter is size. A gust smaller than the building does not act on all of it at once. A 40 m wide tower is enveloped by a 200 m gust and merely spattered by a 5 m one, so the small, fast fluctuations are averaged away across the face before they become a force at all. That is the aerodynamic admittance, and it is a property of the geometry rather than of the dynamics — a stationary building has it too.

The second filter is the structure’s own response. Whatever survives the first filter is applied to a structure that magnifies frequencies near its own and ignores the rest, exactly as the magnification curve says.

Applying the two in order splits the answer into parts that behave completely differently:

• the background response, which is the structure following the slow gusts quasi-statically — it depends on the size of the building against the size of a gust, and not at all on the damping;
• the resonant response, which is the structure ringing at its own frequency, driven by the narrow band of turbulence that happens to sit there.

For the tower drawn above, the background part is 0.70 of the turbulence variance and the resonant part is 6.04 — nine tenths of the answer, from a part of the spectrum that contains almost none of the wind’s energy.

Why a sliver of energy dominates #

The resonant contribution is the spectral density at the structure’s frequency multiplied by $\pi f_0 / 4\zeta$, and that multiplier is enormous. At 1.2% damping it is about 65 times the frequency itself, which is how a band of turbulence a hundredth of a hertz wide comes to outweigh four decades of everything else.

That is the same $1/2\zeta$ from the resonance essay wearing different clothes, and it carries the same consequence: the resonant part of a wind response is inversely proportional to a number nobody knows accurately. Doubling the assumed damping from 1.2% to 2.4% halves the resonant part, moves the total response by a third, and no measurement on the drawings distinguishes the two cases.

Why the same wind is static for one structure and not another #

This is the cleanest example on the site of a rule that is exactly right for one class of structure and quietly wrong for another, which is a shape this collection keeps meeting: the elastic bolt-group method, the shear-lag rule, the check-the-furthest-point rule. In each case the rule was validated on the cases it was invented for and its failures live where nobody looked.

The wind case is unusually well handled, because the profession noticed early. Static wind design carries a gust factor, and a gust factor is precisely the compressed form of the calculation above: it is the ratio of the peak response to the mean response, and for the tower it is 4.39 while for the warehouse it is 2.29. A code that gives one number for both is conservative for one and unconservative for the other, so codes give a formula rather than a number, and the formula contains the damping.

Faster wind is more resonant wind #

That trend is the reason a structure cannot be checked at one wind speed and scaled. The response does not simply go as the square of the speed: the gust factor grows with the speed too, so the peak response grows faster than the mean does. On this tower, doubling the wind from 20 to 40 m/s multiplies the mean force by four and the peak response by five and a bit.

It also means the design event is not obvious. A slender structure may be at its worst not in the strongest storm but in a moderate wind at the speed where some crosswind mechanism switches on — which is the vortex-shedding case, and it happens at speeds a structure meets weekly.

How long a tower takes to notice #

The memory runs the other way too, and it is the same number. Displace that tower and let it go, and it takes those same forty cycles to forget, because the cycles to build up and the cycles to decay are one quantity: $1/2\zeta$, which at 1.2% damping is 41.7 of them either way. A structure this lightly damped is not responding to the wind now; it is responding to an average of the wind over the last several minutes, and that is why the averaging period for a design wind speed is measured in tens of minutes rather than seconds.

That one number is the practical content of the resonant term. A structure’s damping sets both how large a resonant response gets and how long it takes to arrive, and the second determines which parts of a storm can produce one at all. And it is the same $1/2\zeta$ that sets the amplification itself.

The middle of those three is 41.7, which is the count of cycles from the paragraph above, arrived at from a completely different direction. That coincidence is not a coincidence: the peak of a magnification curve and the number of cycles to reach it are the same reciprocal, so a structure that amplifies forty-fold is by construction a structure that takes forty cycles to do it.

Those multipliers are what a damper is bought to change, and what the change buys is easier to read on the spectrum than on the magnification curve.

Two readings of that figure are worth separating, because they point in opposite directions.

The first is how much was bought. The resonant share fell from 90% to 72%, which sounds modest, and in variance terms it is not: the resonant variance fell by more than three, because it goes as one over the damping and the damping went up by 3.3. The share falls slowly only because the background part is unchanged and now makes up more of a smaller total.

The second is what could not be bought at any price. The background component sat there before and sits there now, and no amount of damping touches it — it is the structure following slow gusts quasi-statically, and a structure that follows a load has no resonance to damp. So damping has a floor below which it stops helping, and on a stiffer structure that floor is reached almost at once. That is the whole reason a damper is a tall-building device rather than a general one.

The other dimension of the size filter #

The size filter was described in terms of the building’s width — a gust smaller than the face does not act on all of it at once. Height works differently and, for a tall tower, matters more.

Two things vary up a building. The mean speed rises with height, so the mean pressure is not uniform and its resultant sits above mid-height, which is the ordinary distributed-load lever-arm question. And the turbulence is correlated only over a finite vertical distance: a gust two hundred metres long in the wind’s direction is nothing like two hundred metres tall.

For the resonant part, though, neither of those is quite the right question. What decides how strongly a gust drives a mode is not whether it covers the building but how well its pressure pattern matches the mode’s shape — the same weighted overlap that, in the earthquake problem, decides how much mass each mode is given. There the load pattern is a rigid translation and the overlap is the participation factor; here the load pattern is a patch of turbulence and the overlap is what wind engineering calls the joint acceptance.

Three consequences fall out of reading it that way.

The mean profile and the first mode are well matched, by accident. A first mode leans, so it is largest at the top; the mean wind speed also rises with height. The two overlap strongly, which is part of why the along-wind response is so dominated by the first mode.

A gust on the lower half does almost nothing. It sits where the first mode’s shape is small, so however large it is, it drives that mode weakly. The efficient excitation is a fluctuation over the upper part of the tower, and those are the least common because the correlation length is finite.

And the higher modes get excited by incoherence rather than by coherence. A second mode needs a pressure that pushes one way over the lower half and the other way over the upper. No coherent gust does that — but turbulence is random, so at any instant the pressure at two well-separated heights is uncorrelated and sometimes opposite. The higher modes of a tall tower are driven precisely by the failure of the wind to act as one gust.

Which inverts the intuition the width filter gives. On the width, incoherence is protective: small gusts average out across the face and never become a force. On the height, incoherence is what opens a channel to modes that a uniform load could not excite at all — so a very tall slender tower has its first mode limited by the fact that no gust is that tall, and its higher modes fed by the same fact.

Neither effect is large enough to change the along-wind answer much on an ordinary building, and both matter on a very slender one. The general statement is worth carrying past the wind: an admittance is a measure of how well the load’s shape in space matches the response’s shape in space, and any load whose spatial pattern is uncertain is a load whose modal content is uncertain too.

The peak factor, and how long an hour is #

There is one more term, and it is statistical rather than mechanical.

The response computed so far is a standard deviation — a measure of how much the building moves about its mean position, on average. What a designer needs is the largest excursion in a design event, and the ratio between them is the peak factor: how many standard deviations the largest excursion of a random process reaches in a given time.

$$g = \sqrt{2\ln \nu T} + \frac{0.577}{\sqrt{2\ln \nu T}}$$

with $\nu$ the rate at which the response crosses its mean and $T$ the averaging period. For an hour of wind on this tower it comes to 3.77, and the term is remarkably insensitive: at ten minutes it would be about 3.4 and at ten hours about 4.1. The logarithm is doing what logarithms do, which is why the peak factor is often quoted as “about 3.5” and rarely examined.

The insensitivity is worth noticing for what it says about the whole calculation. The mean response, the background factor, the resonant factor and the damping all matter; how long the storm lasts, hardly at all.

Davenport, and the idea of a load as a random process #

The apparatus above is Alan Davenport’s, from the early 1960s, and what it changed was not the mechanics but the description of the load.

Before it, wind design was a pressure and a shape factor. Davenport’s contribution was to treat the wind as a stationary random process with a known spectrum, pass it through a structure treated as a linear filter, and recover the response as another random process — from which a peak could be extracted statistically. The gust loading factor that came out of it is the same object every wind code now contains.

Two features of that step are worth naming because they recur across engineering.

It borrowed wholesale from another field. The mathematics of spectra, transfer functions and peak factors was developed for communications and control, and the structural application was a translation rather than an invention. This site’s influence lines went the other way a century earlier, from structures into the general vocabulary of linear systems.

It made an unmeasurable thing designable. Nobody can predict the wind a building will meet. What can be predicted is the statistics of the wind, and the whole method is an argument that the statistics are enough — which is true exactly while the structure is linear and the wind does not respond to the building’s motion.

Where the free body is, and what has been assumed #

The free body is the building, cut at its base, with the along-wind pressure on one face and the suction on the other. Nothing in this essay changed that. What changed is that the pressure is a random process rather than a number, and the response has to be described statistically.

Three assumptions carry the weight.

The response is linear, so that the spectrum of the response is the spectrum of the load times the square of the transfer function. Everything above is a consequence of that and nothing survives without it.

The turbulence is stationary — its statistics do not change during the hour being considered. Real storms are not stationary, and downbursts and thunderstorm outflows are notoriously not.

The wind does not know the building is moving. That is the largest assumption and it is the subject of two later essays. When the motion changes the flow, the excitation stops being independent of the response and the whole spectral apparatus stops applying.

The mean is not the interesting part #

One number has been quietly assumed throughout and deserves its own paragraph, because it is where the whole calculation starts and it is the least interesting part of it.

The mean wind pressure deflects the building and does not shake it, exactly as a walker’s body weight deflects a floor without exciting it. Everything above is about the fluctuation around that mean, and the fluctuation is a fraction of it — 17% in speed, so about 34% in pressure, since pressure goes as the square.

The gust factor then multiplies the mean response by 4.39, which sounds like an enormous dynamic amplification and is not. Most of it is the peak factor: 3.77 standard deviations of a random process, on top of the mean. The genuinely dynamic part — the resonant magnification — contributes a factor of about 1.5 on the fluctuating part alone.

That decomposition is worth carrying because it explains why the static method survives. A designer using a gust factor is using a number that is mostly statistics and only partly dynamics, and getting the dynamics wrong by 30% moves the answer by 10%. The method is robust for the same reason it is opaque.

What the picture cannot show #

Crosswind is often worse than along-wind. Everything here is about the response in the wind’s own direction. A tall building also sways across the wind, driven by vortices shed alternately from its two sides, and for a slender building the crosswind response commonly exceeds the along-wind one. Nothing in the along-wind spectrum predicts it.

Torsion is not in it either. A gust does not arrive evenly across a face, so it applies a twisting moment as well as a push, and a building whose centre of stiffness is not at its centre of area will rotate.

The spectrum is a model of open country. The Davenport spectrum used here is a smooth empirical fit; the turbulence in a city is different, and the wind a building actually meets is the wake of everything upwind of it. Wind tunnel testing exists because that part is not calculable.

What the reader feels, and what the structure feels #

A useful asymmetry closes the argument, and it is why this problem is usually a serviceability one.

The force on a tall building is dominated by the mean wind plus the background component, both of which are quasi-static and neither of which cares about damping. Strength design is therefore relatively insensitive to everything in this essay.

The acceleration at the top floor is dominated by the resonant component, because acceleration weights the high frequencies — it is the second derivative of the displacement, so each frequency is multiplied by its own square. A response that is 10% resonant in displacement can be 90% resonant in acceleration.

And it is the acceleration that decides whether people in the building feel sick. Occupant comfort criteria for tall buildings are written in milli-g at the top floor, and they are what actually governs the stiffness of many tall buildings — not strength, not overturning, but whether the people on the top floor notice. Which is the floor-vibration argument at the other end of the frequency scale, and with the same conclusion: the structure is fine and the building is not.

Where the ladder goes #

Three rungs lead out of here, and each takes one of the assumptions above and breaks it.

The first drops “the wind does not know the building is moving” in the mildest way: vortices are shed at a rate set by the wind speed, so there is always a speed at which they match the structure — and once they do, they follow the structure rather than the wind.

The second drops it entirely, and produces a response that is not bounded by any spectrum at all.

The third is what to do about it, which for wind more often than for any other load is an added mass rather than added strength. That preference follows directly from the split made here: the resonant part of the response is the part that troubles occupants, it is the part proportional to one over the damping, and it is therefore the part that a device can attack. Making the structure stronger does nothing to it at all.