The bridge that was pushed by its own sway

On 10 June 2000 the Millennium Bridge over the Thames opened, several thousand people walked onto it, and it swayed sideways enough to make them hold the handrails. It was closed after two days and reopened twenty months later with dampers fitted — thirty-seven viscous units and fifty-two tuned masses, on a bridge whose structure needed no strengthening whatever.

The mechanism found was not one anybody had designed against, and it is not resonance.

Why a person pushes sideways at all #

Walking is not a vertical activity. Each step puts the body weight down slightly to one side of the centre line and the next step to the other, so a walker applies a lateral force that alternates at half the pacing rate — about 1 Hz for a 2 Hz walk, and a few per cent of body weight.

With a crowd walking on a rigid deck, those lateral forces are uncorrelated. Everyone has a slightly different pace and a random phase, so the sum over a few hundred people is a small random force that grows as the square root of the number rather than in proportion to it. That is the standard model of crowd loading and it is correct — on a deck that does not move.

What changes when the deck moves #

A person walking on a surface that is moving sideways under them adjusts. The adjustment is involuntary and immediate: to stay balanced on a floor that is sliding left, a walker puts the next foot further left and pushes right.

That push is in phase with the deck’s velocity. It is not in phase with its displacement, which would be a stiffness; it is not out of phase, which would be a damper. It is with the velocity, which is a damper with the sign reversed.

So the crowd contributes a term $-kNv$ to the equation of motion, where $N$ is the number of people and $k$ is a coefficient per person that Arup measured on the closed bridge by walking crowds across it while it was instrumented — about 300 newton-seconds per metre. The equation becomes

$$m\ddot u + (c - kN\,\overline{\phi^2})\,\dot u + ku = 0$$

and everything depends on the sign of the bracket.

Below the threshold there is no problem at all. The crowd is adding damping of the wrong sign but not enough of it, so a disturbance still dies away, and the bridge feels solid.

Above it, there is no equilibrium. Any disturbance — a gust, a jogger, somebody stepping on — grows exponentially. Nothing is forcing the bridge at its natural frequency; it is extracting energy from a crowd that is walking at a completely different rate, and the growth continues until the pedestrians can no longer walk normally and stop contributing.

The critical number, and what it depends on #

Setting the bracket to zero gives

$$N_{\text{crit}} = \frac{8\pi \zeta m f}{k}$$

— structural damping, modal mass and frequency on the top, and the per-person coefficient below.

The factor of eight rather than four is worth a note, because it is the mode shape. Each person contributes at the point they are standing on, weighted by how much the mode moves there, and a crowd spread evenly along a span contributes the mean square of the shape — a half for a half sine. Nobody is at the antinode except the people who are.

The dependencies are all the ones a designer would want and none of them is available cheaply:

Damping is the cheap lever and it is drawn first because it is the only one that can be pulled after the bridge is built. The next two are the ones a designer has while the bridge is still on paper, and each buys the same doubling for a much larger price.

The third lever is the frequency, and it is the one whose behaviour on this problem is least like its behaviour everywhere else on this site.

The linear dependence on frequency deserves a note, because it is not the dependence intuition offers. Raising a structure’s frequency usually helps by moving it away from an excitation; here there is no excitation to move away from, and the frequency helps only because a faster structure needs more velocity-proportional force per cycle to overcome its own damping. There is a second, larger effect that this model does not contain: above about 1.3 Hz people stop synchronising with the deck at all, so the coefficient itself collapses. Stiffening past that point does not raise the threshold — it removes the mechanism.

The crowd is a distribution, not a number #

The factor of eight in the threshold formula was explained as the mode shape: each person contributes weighted by $\phi^2$ at the point they are standing, and a crowd spread evenly along a half-sine mode contributes a mean of one half. That weighting has more in it than a factor of two.

A crowd standing at the antinode contributes twice as much per person. At midspan $\phi^2 = 1$ rather than the spread crowd’s mean of $\tfrac12$, so the same people gathered in the middle halve the critical number. A bridge whose threshold is 160 for a crowd distributed along it has a threshold of 80 for the same crowd gathered at the centre — and people do gather at the centre of a bridge, because that is where the view is.

And a concentrated crowd selects which mode it destabilises. Since the weighting is $\phi^2$, a crowd spread over the whole span contributes a mean of one half to every mode — the average of a sine squared over a whole number of half-waves is a half whatever the mode number. So a uniform crowd drives all the lateral modes equally per unit of modal mass, and the one that goes unstable is simply the one with the smallest product of modal mass and frequency, which is usually the first.

Concentrate the crowd and that stops being true. A group at midspan sits at the antinode of the first mode and at the node of the second, so it drives the first hard and the second not at all. A group on one half of the span drives the second mode as effectively as the first. The crowd’s position decides which mode is at risk, and no count of people contains that information.

Which turns the check into something with a familiar shape. It is not “how many people” but “which arrangement of how many people”, evaluated against each mode in turn — a pattern loading problem, of exactly the kind a continuous beam’s imposed load produces, arriving in a stability calculation instead of a strength one. And as there, no single arrangement is worst for every mode: the crowd position that most endangers the first mode is the one that leaves the second alone.

There is one more consequence of the $\phi^2$ weighting, and it is the reason a multi-span bridge is a different problem from a single one. On a continuous deck the lateral modes are shaped over the whole structure, so a crowd on one span is standing on a shape that has a node in it somewhere else — and the mode it drives is a property of the whole bridge rather than of the span the people are on. A crowd on an approach span can destabilise a mode whose largest motion is over the water.

Two practical residues come out of it.

The first is that a test with a uniformly distributed crowd is not the worst case, and the crowd tests done on the Millennium Bridge deliberately included concentrated groups for that reason.

The second is more useful and is available at design stage. If the crowd’s distribution matters this much, then controlling it is a remedy — and it is the cheapest one on the list. A bridge whose midspan is a place people walk past rather than stop at, because there is nothing to look at and nowhere to lean, is a bridge whose worst crowd distribution never occurs. That is an argument about handrails, viewing platforms and where the wide bit is, made by somebody who is not a structural engineer, and it moves the threshold by a factor of two.

Why nobody had designed against it #

Three reasons, and together they explain how a mechanism this simple stayed hidden.

The frequency is unusual. A lateral mode below about 1.3 Hz is needed, and most footbridges are stiffer than that laterally. Long, slender, shallow bridges are not, and the Millennium Bridge’s central span had a lateral mode at about half a hertz.

The load model was the wrong shape. Crowd loading was specified as a force — so many people times so many newtons, with a correlation factor. A force model cannot represent a term proportional to velocity, so no amount of care with the numbers in it would have found this.

It needs a crowd, and crowds are not used for testing. The bridge had been analysed and load-tested, and neither exercise involved two thousand people. The threshold behaviour makes this worse than it sounds: a test with fifty people on a bridge whose threshold is 160 shows nothing at all, because below the threshold the response is genuinely small.

The third reason is the one that generalises. A threshold mechanism gives no warning on the way to it, so the ordinary engineering habit of extrapolating from a small test says nothing about the large case — and the third figure above is what that looks like drawn: two runs from the same 2 mm nudge, one either side of a crossing, one falling to 1.6 mm and the other at 2.37 mm and still rising, with nothing forcing either of them.

What it has in common with the wind cases #

This mechanism belongs beside two others in this field, and the family resemblance is exact.

In vortex lock-in, the structure’s motion organises the wake, so the excitation follows the response. In galloping, the section’s motion changes its own angle of attack, so the aerodynamic force acts with the velocity. Here, the deck’s motion changes how people balance, so the crowd’s force acts with the velocity.

All three are negative damping, all three have a threshold rather than a magnification, and all three are invisible to any analysis that treats the load as given. The only difference is what supplies the feedback: a fluid, a fluid, and a nervous system.

The resemblance is close enough that the same family draws the wind case, on the same axes, with the crowd axis replaced by a wind speed.

What is worth noticing is that the two thresholds are found by the same arithmetic and neither of them is a strength. In one case the parameter that is being accumulated is people and in the other it is metres per second, and in both the design question is whether the environment can supply enough of it.

That is the strongest argument for treating dynamics as a field rather than as a collection of load cases. The three problems arise in different industries, are taught in different books and are named after different phenomena, and they are one equation with different words for $k$.

The fix, and why it was dampers #

Stiffening the bridge would have worked too — a higher lateral frequency raises the threshold and eventually moves the bridge out of the range where people synchronise at all — and it was not chosen, because the bridge’s appearance was the point of it. Adding mass would have worked and would have needed the foundations revisited. Adding damping was the only remedy that could be retrofitted to a finished structure without changing what it looked like.

That is worth stating because it is unusual. In most of this subject a remedy is a member size; here the remedy is a device, bolted on afterwards, and the design calculation for it is a calculation about the total damping rather than about any strength.

The measurement that settled it #

The mechanism was not deduced from the drawings. It was measured on the closed bridge, by walking crowds of increasing size across the deck while recording what it did — which is the only experiment that could have distinguished the explanations on offer.

Three things came out of it that no analysis would have produced.

The response has a threshold rather than a slope. Crowds below the critical number produced almost nothing; the same bridge with a few more people produced growing motion. That shape is the signature of an instability and it ruled out every load-based explanation on its own.

The coefficient could be extracted. Fitting the observed growth rates gave the per-person velocity coefficient, and it turned out to be roughly the same across the tests — a number that could then be written into a design rule for other bridges.

The bridge’s own damping was measured at the same time, which was necessary because the threshold is proportional to it and no drawing supplies it.

That sequence — a structure that misbehaves, a full-scale test on the finished thing, a coefficient fitted, a design rule written — is how most of this field’s empirical content was obtained. It is worth comparing with the way the impact factor entered bridge design after the Dee Bridge collapse: a failure, an apparatus, a measurement, a rule. A century and a half apart, and the same shape.

What the picture cannot show #

The coefficient is not a constant. The 300 N·s/m is a fitted average. It varies with the amplitude of the deck’s motion, with the density of the crowd, and probably with what people are doing — a crowd that has noticed the sway behaves differently from one that has not.

The threshold is not sharp in reality. The linear model gives a clean crossing; real bridges show a rapid but finite build-up over a range of crowd sizes.

The crowd is drawn as a number. In the model, N people contribute N times one coefficient. Real crowds have a density, and above about one person per square metre they interfere with each other’s gait, which changes both the pacing rate and the ability to synchronise. Very dense crowds may contribute less per person than sparse ones.

It stops. The model has the amplitude growing without bound. What actually happens is that at 50 or 70 mm of lateral movement, people stop walking normally — they stop, grab the rail, or spread their feet — and the mechanism switches itself off. The bridge is therefore self-limiting and was never in danger of collapse: the failure mode is that it becomes unusable, which is a serviceability limit reached by a stability mechanism.

Vertical, and why it is not the same problem #

Footbridges also bounce vertically, and it is worth separating the two because the same bridge does both and only one of them is an instability.

The vertical case is ordinary resonance: a crowd walking at 2 Hz on a bridge with a 2 Hz vertical mode drives it at that frequency, the response builds up, and it is bounded by the usual one over twice the damping. More people means more force means more response, in proportion, and a load model works. What complicates it is that people on a bouncing bridge do tend to fall into step, so the correlation between them rises with the amplitude — but the excitation still exists with the bridge held still, which the lateral one does not.

The lateral case has no such force. Hold the deck rigid and the net lateral force from a crowd is very nearly zero however many people are on it, because the phases are random and stay random. The excitation is manufactured by the response, and that is the distinction the whole essay turns on.

The practical consequence is that the two checks are different in kind. The vertical check compares a response with a comfort limit and is a serviceability calculation. The lateral check compares a crowd size with a threshold and is a stability calculation, with the same character as asking whether a column will stay straight.

Where the ladder goes #

The immediate rung is the design response, now standard: check the lateral modes of any footbridge below about 1.3 Hz, compute the threshold, and if the expected crowd exceeds it, install damping. That check exists in every footbridge guide written after 2002 and in none written before.

The deeper rung is the class of problem. A structure whose loading depends on its own motion is not a structure with a difficult load case; it is a stability problem, and it belongs with buckling rather than with strength. The question is not how large the response is but whether the equilibrium is stable, and the answer is a threshold in a parameter — which is exactly the form of a buckling load, with damping in the place of stiffness.

Read that way, the field’s three self-excited problems stop being curiosities and become the dynamic half of a subject this site has already spent a whole field on. A column loses stiffness as the load rises and buckles when there is none left; a footbridge loses damping as the crowd grows and sways when there is none left. Both have a critical value, both are indifferent to the size of the disturbance that starts them, and neither is predicted by any calculation of how large the response to a given load would be.