Abstract

INTRODUCTION

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

HISTORY

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

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

NEW LINEAR FEEDBACK MODEL

"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

(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