VIBRATION OF THE LONDONM
MILLENNIUM FOOTBRIDGE:
PART 1 - CAUSE

David E Newland

Department of Engineering
University of Cambridge
Trumpington Street
CAMBRIDGE
CB2 1PZ, UK
den@eng.cam.ac.uk

Abstract

When the London Millennium footbridge was opened in June 2000, it swayed alarmingly. This generated huge public interest and the bridge became known as London’s “wobbly bridge”. Part 1 of this paper explains how pedestrians unwittingly caused the bridge to self-excite. Although previously documented, this phenomenon was not well-known.                                                                                                                                           

INTRODUCTION

To mark the Millennium, a new footbridge has been built across the river Thames in London. It is a shallow suspension bridge linking St. Paul’s Cathedral on the north side of the river with the Tate Modern art gallery on the south side. The bridge is over 300 metres long with three spans, the longest being the centre span of 144 metres. To meet the designers’ artistic requirements, the bridge’s suspension cables sag only 2.3 metres, a fraction of the sag of a traditional suspension bridge of the same span. As a result, the cables carry a very high tension force for a bridge of this size, totalling some 2000 tonnes. When the bridge was opened in June 2000, it was found that the it swayed noticeably. With a large number of pedestrians on the bridge, its sideways movement was sufficient to cause people to stop walking and hold on to the hand-rails. Because there was danger of personal injury, it was decided to close the bridge after a few days for remedial work.

Figure 1. The London Millennium footbridge shortly after its completion. St Paul's Cathedral is at the top left

HISTORY

The bridge opened on 10 June 2000. For the opening ceremony, a crowd of over 1000 people had assembled on the south half of the bridge with a band in front. When they started to walk across with the band playing, there was immediately an unexpectedly pronounced lateral movement of the bridge deck. This movement became sufficiently large for people to stop walking to retain their balance and sometimes to hold onto the hand rails for support. Video pictures showed later that the south span had been moving through an amplitude of about 50 mm at 0.8 Hz and the centre span about 75 mm at 1 Hz approximately. Probably higher amplitudes occurred periodically and several modes were involved. It was decided immediately to limit the number of people on the bridge, but even so the deck movement was sufficient to be uncomfortable and to raise concern for public safety so that on 12 June the bridge was closed until the problem could be solved. It was not reopened to the public until 22 February 2002.

There was a significant wind blowing on the opening days (force 3-4) and the bridge had been decorated with large flags, but it was rapidly concluded that wind buffeting had not contributed significantly to vibration of the bridge. Another possible explanation was that coupling between lateral and torsional deck movements was allowing vertical footfall excitation to excite lateral modes, but this was not found to be a significant factor. Evidence in support of this conclusion was that the 1 Hz mode of the centre span was the span’s second lateral mode; with nodes at its centre and at the two bridge piers, this mode had practically no torsional movement.

It was realised very quickly that the problem was one of lateral excitation and although allowance had been made for lateral forces it had not been expected that pedestrians would so easily fall into step or that the lateral force per person would be as great as was apparently proving to be the case.

RESEARCH

An immediate research programme was launched by the bridge's engineering designers Ove Arup, supported by a number of universities and research organisations.

It was found that some similar experiences had been recorded in the literature, although these were not well-known and had not yet been incorporated into the relevant bridge building codes. A German report in 1972 quoted by Bachmann and Ammann in their IABSE book (1987), described how a new steel footbridge had experienced strong lateral vibration during an opening ceremony with 300-400 people.  They explained  how the lateral sway of a person’s centre of gravity occurs at half the walking pace. Since the footbridge had a lowest lateral mode of about 1.1 Hz, the frequency of excitation was very close to the mean pacing rate of walking of about 2 Hz. Thus in this case "an almost resonating vibration occurred. Moreover it could be supposed that in this case the pedestrians synchronised their step with the bridge vibration, thereby enhancing the vibration considerably" (Bachmann, 1992, p. 636). The problem is said to have been solved by the installation of horizontal tuned vibration absorbers.

The concept of synchronisation turned out to be very important, and a later paper by Fujino et al. (1993) was discovered which described observations of  pedestrian-induced lateral vibration of a cable-stayed steel box girder bridge of similar size to the Millennium Bridge. It was found that when a large number of people were crossing the bridge (2,000 people on the bridge), lateral vibration of the bridge deck at 0.9 Hz could build up to an amplitude of 10 mm with some of the supporting cables whose natural frequencies were close to 0.9 Hz vibrating with an amplitude of up to 300 mm. By analysing video recordings of pedestrians’ head movement, Fujino concluded that lateral deck movement encourages pedestrians to walk in step and that synchronisation increases the human force and makes it resonant with the bridge deck. He summarised his findings as follows: "The growth process of the lateral vibration of the girder under the congested pedestrians can be explained as follows. First a small lateral motion is induced by the random lateral human walking forces, and walking of some pedestrians is synchronised to the girder motion. Then resonant force acts on the girder, consequently the girder motion is increased. Walking of more pedestrians are synchronised, increasing the lateral girder motion. In this sense, this vibration was a self-excited nature. Of course, because of adaptive nature of human being, the girder amplitude will not go to infinity and will reach a steady state."

Although Fujino records the damping ratio of the 0.9Hz lateral mode as , he found that only 20% of the pedestrians on the main span of the bridge were completely synchronised to the girder vibration and the amplitude of vibration was only 10 mm (compared with 75 mm for the Millennium Bridge). Impressions from video clips of the Millennium bridge are that a good deal more than 20% of walkers had synchronised their step. Also in Fujino’s example, the very large movement of the suspension cables (300 mm amplitude) may have made these act as dynamic vibration absorbers and so limit the extent and consequences of synchronisation.

It was clear that data specific to the Millennium bridge was urgently required and Arup undertook an extensive programme of testing to obtain this. In addition to commissioning tests on human gait and how this is affected by movement of the walking surface, the main tests were carried out on the bridge itself. These included artificially shaking the bridge to confirm mode shapes and damping and a comprehensive series of crowd tests. Detailed vibration measurements and video records were made with pedestrians walking at different speeds and densities on each span. These allowed reliable quantitative data on the synchronous lateral excitation phenomenon to be established and a self-excitation model to be developed which could give a reliable prediction of structural response.

ARUP’S PEDESTRIAN LATERAL LOADING MODEL

Arup's loading model is described in Fitzpatrick et al (2001).  Using experimental data from controlled pedestrian loading tests, with an approximately constant density of pedestrians walking at steady speed, Arup found that there was strong correlation between the amplitude of the pedestrians’ (modal) excitation force and the amplitude of the bridge deck’s (modal) lateral velocity. Measurement of deck velocity is straightforward, but the excitation force was calculated from a power flow analysis based on the concept that the work done by the net excitation force (foot-fall force less damping force) is equal to the gain of kinetic energy per cycle. This led to the conclusion that, when synchronisation has occurred, the amplitude of energy-transferring force per pedestrian is linearly proportional to velocity and acts as a negative damping force. This allows the limiting number of people for stability  to be calculated and the effective damping to be calculated for

NEW LINEAR FEEDBACK MODEL

These results can be reached by a completely different approach using a feedback model of synchronous lateral excitation. It is based on the idea that people will naturally fall into step with each other and that they will unconsciously adjust their group phasing so that the bridge vibration increases to a maximum. This is important. As the assumed phase of people’s walking and the bridge’s movement changes, bridge amplitude changes enormously. From this feedback model, the phase to give peak response can be calculated theoretically. It turns out that the answer agrees with the phase found by Arup from their experimental studies.

The physical mechanism of synchronous lateral excitation is well described by Arup  (Dallard et al, 2001) in the following way:
"Chance footfall correlation, combined with the synchronization that occurs naturally within a crowd, may cause the bridge to start to sway horizontally. If the sway is perceptible, a further effect can start to take hold. It becomes more comfortable for the pedestrians to walk in synchronization with the swaying of the bridge. The pedestrians find this makes their interaction with the bridge more predicable and helps them maintain their lateral balance. This instinctive behaviour ensures that the footfall forces are applied at a resonant frequency of the bridge, and with a phase such as to increase the motion of the bridge. As the amplitude of the motion increases, the lateral force imparted by individuals increases, as does the degree of correlation between individuals. The frequency "lock-in" and positive force feedback caused the excessive motions observed at the Millennium Bridge."

Figure 1. Feedback system to represent synchronous lateral excitation

The frequency-domain feedback model shown in the diagram above represents this behaviour. Each mode is treated separately. Here   is the Fourier transform of the modal excitation force x(t) with no bridge movement,   is the Fourier transform of the modal displacement response y(t). is the modal frequency response function at frequency   and the complex quantity describes the positive force feedback by which the pedestrians’ modal input force is modified by movement of the bridge.  These are all complex quantities representing amplitude and phase at frequency , using the functional notation where . In this notation, is a complex quantity (amplitude and phase) and   is a real quantity (amplitude only). The control equation is

 
(1)
 
 
giving
 
 
(2)
 
 
The feedback function  is a complex function which we now write as a real argument and complex exponential phase function:
 
 
(3)
 

As noted above, the observed motion of pedestrians is that their phase adjusts itself so as to increase the motion of the bridge. Therefore it is natural to choose the phase angle   so that the bridge’s response is a maximum, that is that   is a maximum. This can be done by substituting (3) into (2) and then differentiating the denominator with respect to   to seach for a minimum to obtain the result that

 
(4)
 
 
For a resonant mode with modal stiffness, mass and damping given by k, m and c,
 
 
(5)
 
 
with which (4) becomes
 
 
(6)
 
 
When the footfall frequency  coincides with the mode’s natural frequency so that
 
 
(7)
 
 
then (6) gives
 
 
(8)
 
 
or, in terms of the modal loss factor  of the bridge structure without pedestrians,
 
  .
(9)
 
 
the maximum non-dimensional response ratio is
 
 
(10)
 

In this formula,  is the amplitude of the modal force exerted on the bridge by the pedestrians walking on it, when their pacing rate coincides with twice , with the phasing of their movement having naturally adjusted itself to give maximum response.

The important conclusion from this analysis is that the walking pedestrians act as negative damping and the effective modal loss factor  is reduced when pedestrians walk over the bridge. This conclusion confirms that obtained by Arup from a purely experimental approach (Fitzpatrick et al. 2001, s. 4.11).

PHASE FOR PEAK RESPONSE

 
If we write
 
 
(11)
 
 
then the value of  for which the maximum response (4) is obtained is
 
 
(12)
 
 
which from (5) is
 
  .
(13)
 

So, from (3), we see that phase of the pedestrians’ feedback force is leading the output displacement of the bridge deck by an angle which becomes exactly  at the resonant frequency defined by (7). This of course is what we expect for a negative damping force.

DETERMINING THE PROPORTIONALITY CONSTANT

Referring to figure 2, the modal feedback force  is generated by all the pedestrians walking on the bridge. If there are N people unformly distributed along a span of length   with  mode shape  normalised so that

 
(14)
 

and if each person contributes an actual force per unit deck displacement of  and per unit modal displacement of   so that the modal force from pedestrians per unit modal displacement is

 
(15)
 
 
Hence the net modal loss factor from (10) is
 
 
(16)
 
where
 
=  nNet modal loss factor
 
=  modal loss factor for structure alone
  N
=  number of people on span
  L
=  length of span
 
=  natural frequency (rad/s)
  k
=  modal stiffness
  m
=  modal mass
 
=  amplitude of feedback force per person and per unit displacement of the bridge deck at frequency

By measuring the net modal loss factor with N/L people per unit length of deck walking steadily at the synchronous speed (footfall frequency twice the natural frequency), the feedback force per person and per unit displacement, , can be calculated from (16) if the loss factor of bare structure has been measured previously. Alternatively, if   and   are known,  can be computed. The application of this formula (16) is considered in Part 2.

References are at the end of Part 2 of this paper