Dynamics: Resonance

In certain situations resonance should be avoided in civil engineering structures. This requires the knowledge of the natural frequencies of the structure and the frequencies of the dynamic loading applied to it.

The London Millennium Footbridge

Fig. 16-11: The London Millennium Footbridge

The London Millennium Footbridge is the first new pedestrian bridge crossing over the Thames in Central London for more than a century. The bridge is located between Peter’s Hall on the north bank leading to Saint Paul’s Cathedral and the new Tate Modern Art Gallery on the south bank. The bridge has three spans over a total length of 325 metres. The lengths of the three spans are 81 metres for the north span, 144 metres for the main span and 100 metres for the south span.

The London Millennium Footbridge, Figure 16-11, opened on 10 June 2000. It was estimated that between 80,000 and 100,000 people crossed the bridge during the opening day with a maximum of 2000 people on the bridge at any one time. During the opening day unexpected lateral movements, or ‘wobbling’, occurred when people walked across the bridge. Two days later the bridge was officially closed in order to investigate the causes of the vibration. After investigation and modification, the bridge reopened in December 2001 [16.4]. Since then millions of people have visited the bridge and there has been no reoccurrence of ‘wobbling’ problems.

As constructed, the bridge had lateral natural frequencies as follows:

  • South span: first lateral natural frequency was about 0.8Hz;
  • Central span: first two lateral natural frequencies were about 0.5 and 0.95 Hz; and
  • North span: first lateral natural frequency was about 1.0 Hz.

It was reported that excessive vibrations of the bridge in the lateral direction did not occur continuously but built up when a large number of pedestrians were on the south and central spans of the bridge and then died down if the number of people on the bridge reduced or if the people stopped walking [16.4].

Fig. 16-12: Periods of walking forces

Resonance is related to forcing frequencies. Walking is a periodic movement on a flat surface in which two feet move alternately from one position to another and do not leave the surface simultaneously. When people walk in a normal way, walking has its own frequency. This is illustrated in Fig. 16-12.

When people walk, they produce vertical loading on the walking surface, such as floors and footbridges. The time from one vertical load generated by one foot to the next vertical load induced by the other foot is the period of the walking load in the vertical direction, denoted as . People walking generate not only vertical forces, but also lateral forces. The force generated laterally by the right foot is normally in the opposite direction to the force induced by the left foot. Therefore the time required for reproducing the same pattern of force can be counted from the left (or right) foot to the next step of the left (or right) foot. In other words, the period of lateral forces ( ) is just twice the period of the vertical force induced by people walking, or the frequency of the lateral walking loads is a half of that of the vertical walking loads [16.5].

Click to view the enlarged figure
Fig.16-13: Distribution of frequency of walking loads in the vertical direction [16.6]

Fig. 16-13 shows the distribution of walking frequencies obtained from 400 people walking on two footbridges in Manchester [16.6]. The horizontal axis indicates the walking frequency in the vertical direction while the vertical axis shows the number of observations of particular frequencies. It can be seen from Fig. 16-13 that most of the 400 frequencies of people walking are between 1.6 Hz and 2.0 Hz in the vertical direction. As the walking frequency in the lateral direction is just half that in the vertical direction, the corresponding frequencies of people walking in the lateral direction are between 0.8 Hz and 1.0 Hz.

It can be noted that the frequencies of walking loads in the lateral directions were close to the lateral natural frequencies of the south and central spans of the London Millennium Footbridge where excessive vibrations occurred. It was also observed that people walking in large groups tended to synchronise their walking paces. When the footbridge started wobbling, more people would walk at the frequency of the wobbling, which enhanced the synchronisation. This synchronisation magnified the effect of the lateral footfall forces on the footbridge. The wobbling of the footbridge was caused by resonance that synchronisated the walking loads.

(Unknown author)
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