Development of Human Body Model for the Dynamic Analysis of Footbridges under Pedestrian Induced Excitation
Abstract
Human-induced excitations are the most dominant source in the vibration of footbridges. Such vibration effects are modeled as a mathematical time-domain force model in currently using softwares. However, such models are not capable of considering the dynamic effects of footbridges as they exclude the effects from precise human body model and human-structure interaction. Also, it is so complicated to analyze the dynamic effects from crowded walking load. In this study, a human body model for generating pedestrian excitation is developed which is adequate for considering the interaction between bridge and pedestrians on the footbridge. By using this model, the dynamic behaviors of footbridge are analyzed. It is also compared with the dynamic responses from the analysis using the time-domain force model and an experimental data. The dynamic responses from the analyses using the human body model are larger than the time-domain force model because of the human-structure interaction effects. Also, the proposed model shows good agreement with experimental result. Through a dynamic analysis for the footbridge under crowd-induced excitation, the developed numerical human body model is found to simulate the people walking in crowds effectively. From the analysis results, it is found that the partially grouped randomly walking stream could make even larger response than the synchronized crowded loads.
Keywords: footbridge, crowded pedestrian load, human body model, bridge-human interaction
Introduction
In recent decades there has been a trend towards to improve mechanical characteristics of materials used in footbridge construction. Engineers are now designing lighter, more slender and more aesthetic structures. As a result of these construction trends, many of these footbridges are becoming more susceptible to vibration when subjected to dynamic loads. In most cases the vibrations of footbridges lead to serviceability problems, i.e. the inconvenience of the pedestrians or in some extreme cases a bridge may no longer be used and has to be closed. In rare cases, safety problems may also arise due to overstressing and/or fatigue (Bachmann, 2002).
Note.-Discussion open until May 1, 2009. This manuscript for this paper was submitted for review and possible publication on October 20, 2008; approved on December 15, 2008
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There have been many cases reported to have these problems, and they were especially emphasized to both public and professional researchers after the infamous swaying of the new and attractive Millennium Bridge in London during its opening day on June 10, 2000 (Dallard et al., 2001).
The human-induced dynamic loading occurs frequently and it is often regarded as dominant load for footbridges because it sometimes obstructs pedestrians to walk conveniently. There are two means of investigating footbridge vibration problem, namely experimental and numerical approaches. An experimental method requires considerable time and cost while the numerical approach represents an economical way to examine the dynamic behavior of a footbridge. To apply the appropriate force for the dynamic analysis of footbridges, various mathematical models of human-induced dynamic walking loads are proposed (Zivanovic et al., 2005), such as time and frequency domain force models. Although mathematical time and frequency domain force models are used in contemporary footbridge design, the dynamic analysis
using these models is incapable of considering dynamic effects from the human-structure interaction. This is because these models are based on the data measured on a rigid surface. Also, it is so hard to apply time and frequency domain force model in analyzing dynamic responses from walking crowds. Therefore, a three dimensional analysis program is developed in this study using the human body model. This program can effectively generate pedestrian excitation for the dynamic behavior of a footbridge under human-induced lateral and gravitational load.
1.Dynamic Analysis Methods for Footbridges under Human-induced Load
A time history dynamic analysis method using a commercial FE (finite element) analysis program with the application of the mathematical time-domain force model is the most common method to analyze the dynamic behavior of footbridges. However, this model is not suitable for dynamic analysis of footbridge since it can not consider the effects from human-structure interaction that human makes to maintain his balance. To overcome this problem, a new human body model is proposed in this chapter. It is comparatively easier to consider these effects in the human body model. Also, it has advantages to apply complicated loading conditions such as loads from crowded pedestrians. In this chapter, the analysis methods using the mathematical time-domain model and the proposed mechanical human body model are briefly explained with their theoretical bases.
The dynamic analysis method using the mathematical time-domain force model
A time-domain force model is based on the assumption that both feet produce exactly the same periodic force. It is well-known that each periodic force Fp(t) with a period T can be represented by a Fourier series:
where G is the person’s weight (N), αi is the Fourier’s coefficient (dynamic load factors) of the ith harmonic, i.e. dynamic load factor (DLF), fp is the pacing frequency (Hz), φi is the phase angle of the ith harmonic, i is the order number of the harmonic, and n is the total number of contributing harmonics.
As shown in Eq. (1), the vertical force can be divided into static and dynamic components. The static component corresponds to a person’s weight and the dynamic component is the sum of harmonic functions with frequencies that are an integer multiple of the pacing frequency. The first to third harmonics are dominant; Eq. (1) can be simply rewritten in four terms as follows (Bachmann and Ammann, 1987; Bachmann et al., 1995):
A pedestrian on a footbridge produces not only vertical force but also lateral force. Although the lateral force is relatively small compared to the vertical force, it is sufficient to produce strong vibrations in the case of laterally soft and hence low frequency structures. Fujino et al. (1993) noticed that the frequency of a person’s lateral head movement is 1 Hz which is twice of walking frequency. Lateral movement is ±10mm when crossing a footbridge as shown in Fig. 1. As the frequency of lateral movement is half the vertical pacing frequency, the harmonic motion in lateral direction is half of that in vertical direction.
Table 1 shows the correlation of pacing frequency, forward speed and stride length according to human motion forms (Bachmann and Ammann, 1987). Normal walking is considered as the human motion for the mathematic time-domain force model in this study, hence
the forward speed and the stride length is selected as
1.5 m/s and 0.75 m respectively. Bachmann and Ammann (1987) also proposed the dynamic load factors and the phase angles for normal walking as summarized in Table 2. By using the values on this table, the time-domain force models in vertical and lateral directions can be calculated using Eq. (2) as shown in Fig. 2. The resultant vertical and lateral time-domain force model is then applied to a FE model of a footbridge model.
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