Lateral Vibration Analysis Of Footbridges

The excessive lateral resonance problem of the London Millennium Bridge in 2000 attracted great public and professional attention. It is currently accepted that flexible footbridges with lateral frequencies below 1.3 Hz are potentially the most vulnerable to exhibit excessive lateral vibration,when excited by large crowds. There exist three solutions to this problem. The first is to increase the lateral stiffness of the bridge to move all the lateral frequencies out of the sensitive frequency range, but it would significantly affect the aesthetics of the bridge,and usually it is not feasible in practice for especially flexible footbridge; the second option is to increase the damping of the structure to reduce the resonant response, but this would be a very costly work, such as the retrofit work of the Millennium Bridge,7million dollar were added; another way is to restrict the number of pedestrians across the footbridge. Because the excessive lateral resonance are induced only when the walker number on the bridge is large enough, it is necessary to study the critical number of pedestrians likely to induce the lateral resonance of flexible footbridges.The equation of motion for single degree of freedom model of pedestrian is first simplified based on the method presented by Professor Roberts in British. The estimated formula of the critical people on the footbridge that can result the significant lateral vibration of bridge is derived on the assumption that the displacement response of pedestrian and bridge are equal. The statistic characteristics for the walking frequency and step length of Chinese pedestrians are derived by filed testing. It is found that walking frequency and walking step length obey the normal distributions with N(1.82Hz,0.22) and with N(0.72m,0.078), respectively, which has shown a little bit different form the statistic results in other countries due to different race, and there is also a linear relationship between walking velocity and walking frequency.The finite element (FE) model of three simply supported footbridges with different spans is established to check the accuracy of the estimated formula presented above. As the sensitive frequency for lateral vibration falls in the range of 0.5～1.2 Hz, the natural frequencies for the three bridges are designed as 1.2 Hz,0.887 Hz and 0.6 Hz. The number of critical pedestrian is calculated based on the natural frequency computed form modal analysis of the developed FE model and the estimated formula. By assuming that the number of pedestrian is distributed evenly along the bridge, the maximum amplitude of lateral acceleration response can be obtained from the FE method. Finally, the calculated acceleration responses for these three bridges are compared with the limit acceleration specified by the four categories of user comfort in EN03. It is shown that the maximum accelerations for each of the three cases exceed the lower limit value of acceleration of "uncomfortable category". Therefore, the estimated formula for critical number of people is reasonable. The present formula provides a conservative value for critical number of pedestrians, and it should be also to provide useful reference for traffic control of footbridges.