Keywords: suspension footbridge, coupled modes, walking, resonance, dynamic response.

Cable supported pedestrian bridges offer an elegant and economical solution for bridging over long spans. Due to the application of high strength and light materials, bridge structures can be designed and constructed very light, flexible and slender with low stiffness, damping ratio and structural mass. As a result, they are prone to vibration induced by pedestrians and other human activities [1].

When walking or running cross a bridge, pedestrians generate dynamic loads in vertical, lateral and longitudinal directions which cover a range of frequencies. If the natural frequencies of the bridge structure are within the range of pacing rates, some pedestrians will change their pacing rates to match the vibrational frequency and the bridge structure will resonate and suffer excessive vibration due to the synchronous excitation. This evidence has been shown by many footbridges with different types of structural forms in the world such as the Millennium Bridge in London [2], the T-Bridge in Japan [3] which experienced excessive vibration induced by synchronous lateral excitation.

An innovative cable supported bridge model with pre-tensioned reverse profiled cables in vertical and horizontal planes is proposed in this conceptual study to investigate the vibration characteristics of shallow suspension pedestrian bridge structures. In this bridge model, the tension force in the supporting cables can be adjusted by introducing pre-tensions to the reverse profiled cables, and therefore the natural frequencies can be altered and cover the range of load frequencies induced by pedestrian walking and running.

In the numerical analysis, SAP2000 [4] is adopted to study the vibration properties and dynamic response under walking loads. The crowd walking loads have been modelled as uniform loads and consist of three parts: vertical dynamic force, lateral dynamic force and vertical static force. It is assumed that 20 percent of pedestrians participated fully in the synchronization process and generated vertical and lateral dynamic forces. For the vertical dynamic force, different force-time functions in the vertical direction are used according to the literature on pacing rates [5]. While for the lateral dynamic force, the force-time functions for one foot take the same form as in vertical direction, but the force magnitude is 4% of its vertical component. A bridge model with fundamental frequency of 0.75 Hz in the lateral direction has been studied to illustrate the dynamic characteristics of shallow suspension footbridges under walking dynamic loads, and the synchronous excitation has been simulated by resonant vibration. Numerical results show that for shallow suspension bridges, the lateral and torsional vibration modes always combine together and form two types of coupled vibration modes: coupled lateral-torsional modes and coupled torsional-lateral modes. It is also found that these coupled modes have lower frequencies compared to other modes and hence are easy to excite. When the bridge resonates under walking loads, the dynamic behaviour in the lateral direction is quite different from that in the vertical direction. In the resonant vibration of coupled modes, the amplitudes of lateral deflection and acceleration increase to the maximum values and decay, and then the vibration tends to be steady. However, in the resonant vibration under vertical modes, the amplitudes of vertical deflection and acceleration increase gradually to a constant value. It is also found that for the proposed pre-tensioned cable supported bridge model, damping has significant effect on the vertical vibration, but only a small effect on the lateral vibration.

1

H. Bachmann, "Lively footbridges - a real challenge", in: Proceedings of the International Conference on the Design and Dynamic Behaviour of Footbridges, Paris, France, November 20-22, 2002.

2

P. Dallard, T. Fitzpatrick, A. Flint, A. Low, R. Ridsdill Smith, M. Willford, "Pedestrian-induced vibration of footbridges", The Structural Engineer, 78(23/24), 13-17, 2000.

3

Y. Fujino, B.M. Pacheco, S.I. Nakamura, P. Warnitchai, "Synchronization of human walking observed during lateral vibration of a congested pedestrian bridge", Earthquake Engineering & Structural Dynamics, 22(9), 741-758, 1993. doi:10.1002/eqe.4290220902

4

Computer and Structures, Inc., "CSI analysis reference manual", Computer and Structures, inc., Berkeley, California, USA, September 2004.

5

J.E. Wheeler, "Prediction and control of pedestrian-induced vibration in footbridge", Journal of Structural Division, 108(9), 2045-2065, 1982.

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