Study On Crowd-beam Vertical Interaction And Mechanic Of Lateral Vibration Of Footbridge

Over the last few decades, slender footbridges with low natural frequencies have become increasingly popular because of the emergence of new material, advanced engineering technology and the development of the bridge aesthetics. As a consequence, more of these footbridges show great sensitivity to human-induced vibration, especially excessive vibrations of a footbridge occur if the footbridge has a mode of vibration with a natural frequency within the range of the walking frequencies of the pedestrians on the bridge. Although such type of dynamic vibration of the footbridge has never involved structural failures thus far, it has often caused discomfort, nervous even panic for the pedestrians.Generally, the vertical vibrations of footbridges induced by pedestrians have the most significant impact on bridge structures, which therefore are extensively measured and theoretically analyzed in the past. Since excessive lateral vibrations of the London Millennium Bridge were observed in2000, the lateral vibrations problem of footbridges has been of great concern in civil engineering field worldwide. However, the mechanisms of lateral vibrations of footbridges are still not completely understood because of the complex influencing factors. So far, it is considered that there are two main reasons for the excessive lateral vibrations of footbridges:First, the bridge structures may be unreasonable; Second, dynamic interaction between pedestrians and a footbridge is the root of excessive lateral vibrations of the footbridge. The latter involves complicated dynamic systems with multiple-scale and multi-field coupling, which is very difficult to research because of imperfection of the human dynamic model. Thus, unified and effective mathematics models describing dynamics interaction between pedestrians and footbridges are not available yet, study for which is a research direction in the field of lateral vibrations problem of footbridges. For the first reason given above, the mechanisms can be mainly classified into three classes:direct resonance, parametric excitation and internal resonance. By using the principle of the latter two, the mechanisms of excessive lateral vibrations induced by vertical loading of pedestrians are discussed in this paper. The main researches as follows:A model describing a single person crossing a footbridge is established, by which the influence of the structure stiffness on walking load is theoretically and numerically considered. A spring-mass-damping model with distributed parameters is proposed to simulate the distribution of pedestrians on a footbridge, by which the influence of distributed crowd on structure dynamic parameters and the dynamic properties of transfer functions of the coupled system under the action of walking load induced by pedestrians are investigated systematically.According to the construction features of the London Millennium Bridge, two theoretical models for north span and midspan of the bridge are developed, respectively. The planar bifilar torsion model is used to analyze the reason of instability of lateral vibrations of north and middle spans induced by vertical walking load component action. A formula calculating the critical number of pedestrians on the north span of the bridge is obtained based on a nonsingular perturbation strategy and multi-scale method. Comparing with formulas derived by other authors, our formula related to the frequency of the bridge is more reasonable. Furthermore, the model given by us can be used to explain the phenomenon of instability of the first modal lateral vibration without introducing additional assumptions.By simplifying cables of a cable-stayed footbridge to springs, a mathematic model is proposed to describe vibrations of cable-stayed footbridges induced by pedestrians. Some conclusions can be drawn through analyzing dynamic properties of the model mentioned above using multi-scale method:internal resonance may be possible if a bridge is characterized by a about2:1ratio between vertical and lateral mode frequencies, with the vertically excited mode close to direct resonance conditions concerning the loading. The vertical vibrations of the bridge may reach saturation when the number of pedestrians on the bridge is big enough, therefore, the energy provided by vertical loading will entirely transfer to the modal lateral vibrations of the footbridge in this time, which increases the amplitude of lateral vibrations rapidly. Additionally, the influence of lateral force on internal resonance is discussed numerically, which show that the saturation phenomenon does not occur if the lateral force is not big enough.Finally, an inverted pendulum model is given to simulate the lateral instability of an arch footbridge. By theoretical analysis, the first order simplified model for instability of lateral vibrations of the arch footbridge is derived, i.e. the standard Mathieu equation. The excessive lateral vibrations of the arch footbridge induced by vertical loading are explained by employing the parametric excitation principle.In summary, the mechanisms of instability of lateral vibrations of several types of footbridges, including suspension, cable-stayed and arch footbridges, are investigated in this paper, which show that vertical loading provided by pedestrians can result in excessive lateral vibrations for all these types of footbridges mentioned above and may be one reason for instability of lateral vibrations of footbridges induced by pedestrians.