Lateral Parametric Vibration of Footbridge under Pedestrian Excitation considering the Time-Delay Effect

Pedestrian excitation may consequently cause large-scale lateral vibration of the long-span softness of footbridges. Considering the influence of structural geometric nonlinearity, a nonlinear lateral parametric vibration model is established based on the relationship between force and speed. Taking the London Millennium Footbridge as an example, the Galerkin method is applied to formulate parametric vibration equations. In addition, the multi-scale method is used to analyze the parametric vibration of footbridge system theoretically and numerically. The paper aims to find out the reasons for the large-scale vibration of the Millennium Footbridge by calculating the critical number of pedestrians, amplitude-frequency, and phase-frequency characteristics of the Millennium Footbridge during parametric vibration. On the other hand, the paper also studies the influence parameters of the vibration amplitude as well as simulates the dynamic response of the bridge during the whole process of pedestrians on the footbridge. Finally, the paper investigates influences of the time-delay effect on the system parameter vibration. Research shows that: the model established in the paper is reliable; the closer the walking frequency is to two times of the natural frequency, the fewer number of pedestrians are required to excite large vibrations; when the number of pedestrians exceeds the critical number in consideration of nonlinear vibration, the vibration amplitude tends to be stable constant-amplitude vibration, and the amplitude of vibration response is unstable constant-amplitude vibration when only linear vibration is considered; the following factors have an impact on the response amplitude, including the number of pedestrians on footbridge per unit time, damping, initial conditions, and the number of pedestrians in synchronized adjustment. At last, when considering the lag of the pedestrian’s force on the footbridge, the time-lag effect has no effect on the amplitude but has an effect on the time needed to reach a stable amplitude.

Parameter Calibration of a Social Force Model for the Crowd-Induced Vibrations of Footbridges

A reliable prediction of the human-induced vibrations of footbridges relies on an accurate representation of the pedestrian excitation for different loading scenario. Particularly, the modeling of crowd-induced dynamic loading is a critical issue for the serviceability assessment of footbridges. At the design stage, the modeling of crowd loading is often derived from single pedestrian models, neglecting the effect of the structural vibrations as well as the interactions among pedestrians. A detailed description of the crowd behavior can be achieved employing a social force model that describes the different influences affecting individual pedestrian motion. These models are widely adopted to describe the crowd behavior especially in the field of evacuation of public buildings, public safety and transport station management while applications in the serviceability assessment of footbridges are less common. To simulate unidirectional pedestrian flows on footbridges, this paper proposes a parameter calibration of the Helbing’s social force model performed adopting the response surface methodology. Parameters of the social force model are calibrated so as to represent the fundamental relation between mean walking speed and density of the pedestrian crowd. The crowd-induced vibrations are then simulated by modeling each pedestrian in the crowd as a vertical load that crosses the footbridge with time varying trajectory and velocity estimated from the calibrated social force model. Finally, results are compared to those obtained from a multiplication factor approach proposed in literature. This considers the crowd as a uniform distribution of pedestrians with constant speed and given synchronization level and the footbridge response is evaluated as the response to a single pedestrian scaled by a proper enhancement factor.

All-GFRP footbridge under human-induced excitation

The dynamic behavior of lively footbridge is a complex problem. Recently there were numerous publications and recommendations related to the dynamic nature of footbridge. The complicated procedure which was set in a number of instructions and standards says nothing about actions aimed at avoiding critical frequency range in structure. In the paper, results of dynamic in-situ tests of cable-stayed all-GFRP (Glass Fiber Reinforced Polymer) footbridge are presented. Fiberline Footbridge, located in Kolding city in Denmark, was constructed in 1997 using 12 different pultruded profiles all made of GFRP material. The dynamic characteristics as well as vertical response of the tested footbridge under human excitation are given and discussed. Firstly, in order to estimate the dynamic properties of the footbridge, a series of free-decay responses under human jumping were conducted. The fundamental frequency of the analyzed structure was within a critical range. A methodology for footbridge classification with regard to their dynamic sensitivity was worked out and the correlation between the structure's properties and its dynamic response under pedestrian excitation was formulated. It was found that the analyzed footbridge fulfilled vibration comfort criteria elaborated by technical guide Sétra, however, more restricted acceleration limits suggested by Eurocode were not met.

Research on dynamic characteristics of plate under pedestrian excitation based on Newmark-β

Development of micro and ultra-precision machining, precision instruments and equipment, precision assembly and testing has put forward more and more high requirements to vibration isolation on environmental elements, especially the pedestrian excitation generated by workers' normal walking. Therefore, it is very important to study the pedestrian excitation's influence on vibration characteristics of precision instruments and equipment. In this study, dynamic model including mathematical model of pedestrian excitation, interaction model between pedestrian and rectangular plate structure, the human–plate coupled dynamic equation in vertical direction of pedestrian–plate structure was established. And then we use the Newmark-β method to solve the time-domain step-by-step integration of the first four order modes' dynamic equations and study the influence of the linear notion trajectory along the central axis direction on the dynamic characteristics of the rectangular plate. By simulation, we discussed plate structure response under different conditions, including plate structure displacement and acceleration response under the single person excitation with different velocities, under normal walking velocity with different number of pedestrians and under this case of different distance between two pedestrians. The results show that the structural vibration induced by pedestrian excitation has great influence on dynamic characteristics of plate.