Shock and Vibration

Volume 2019, Article ID 8058191, 11 pages

https://doi.org/10.1155/2019/8058191

## A Suspension Footbridge Model under Crowd-Induced Lateral Excitation

^{1}School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 200093, China^{2}Department of Civil Engineering, Wenzhou Vocation and Technical College, Wenzhou 325035, China

Correspondence should be addressed to Bin Zhen; moc.361@08nibnehz

Received 17 April 2018; Accepted 19 November 2018; Published 2 January 2019

Academic Editor: Salvatore Caddemi

Copyright © 2019 Lijun Ouyang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, a plane pendulum model is proposed to investigate the lateral vibration of a suspension bridge under crowd excitation. The plane model consists of two strings and a rigid body, which represent cables and the bridge deck, respectively. The lateral force induced by crowd is expressed as a cosine function with random phase. Comparing with other existing pedestrian-footbridge interaction models, the proposed model has two features: one is that the structural characteristics of the suspension bridge are taken into account. The other is that the expression of the lateral force induced by crowd has a unified form for different lateral bridge amplitudes. By numerically analyzing the solution stability of the plane model, we exhibit the whole changing process how a suspension bridge increases its lateral amplitude from small to large. It is shown that the worst case occurs when the lateral natural frequency of the bridge is half the lateral step frequency of the pedestrians. Based on the analysis results, the plane pendulum model can be easily used to explain why the central span of the London Millennium Bridge has large lateral oscillations at about 0.48 Hz.

#### 1. Introduction

The excessive lateral vibration problem of footbridges induced by pedestrians in recent years has drawn particular attention in design and research communities around the world [1]. So far, such kind of lateral vibration problem in bridges has never involved structural failures but caused discomfort for pedestrians. Excessive lateral vibrations have been observed in almost all types of footbridges, such as suspension bridges [2, 3], arch bridges, truss bridges [4], and so on. The mechanism of excessive lateral vibrations for footbridges has been studied theoretically and experimentally in the last decade. The key to resolving the problem involves two aspects: one is how to determine the lateral force induced by pedestrians; the other is how to describe the motion of the bridge. For the former, the expression of the lateral force generally is obtained by using an empirical approach because of complexity. For the latter, a bridge usually is viewed as a single degree-of-freedom (SDOF) system for simplicity.

According to experiments carried out on rigid surfaces [5], the ground reaction force (GRF) occurs due to acceleration (and deceleration) of the center of mass of a pedestrian’s body. The GRF is a three-dimensional vector which varies in time and space due to the forward movement of the pedestrian. Early studies on GRFs from Harper et al. [6] revealed that horizontal lateral component of the force is generally very small. Chao et al. [7] measured single footstep forces from several persons and found that the lateral component of the force is about of the body weight for men and little less for women. Both Andriacchi et al. [8] and Masani et al. [9] showed that the peak value of lateral force increases with the walking speed. The crowd excitation usually is estimated by adding together the GRF of every pedestrian on the bridge. A notion of “dynamics loading factor” was introduced [10] in the expression of lateral force induced by crowd suggested by Matsumoto et al. [11]. An analogous model was also given by Roberts [12].

The above force models are mainly proposed based on experimental measurements carried out on stationary platforms. Through video analysis for T-Bridge in Japan, Fujino et al. [2] concluded that pedestrians’ gait increases with the lateral bridge amplitude. Kay and Warren Jr. [13] experimentally showed that the pedestrian gait cycle frequency locks to the driving frequency of a moving surface in a large range of frequencies. Comparing with the case for stationary platforms, McAndrew et al. [14] reported that the step width and the frequency increase, whereas the step length decreases. The experiments of Brady et al. [15] were carried out for the case of low frequencies (0.2–0.3 Hz) and a large amplitude (127 mm), which also indicated that the step width generally increases. McRobie et al. [16] constructed a suspended platform equipped with a treadmill having an adjustable lateral frequency in the range of 0.7 to 0.9 Hz. It was observed that people tended to spread their feet further apart and walked at the same frequency (with a constant phase) as that of the platform. In case of great lateral oscillations of the footbridges, an additional nonnegligible force component is potentially generated due to the interaction between the movement of the center of mass of the pedestrian’s body and that of the structure. For great lateral oscillations of the footbridge, Piccardo and Tubino [17] considered that the lateral force exerted by crowd is harmonic, and its amplitude depends on the bridge displacement. Based on such force model, the parametric excitation mechanism was proposed to explain the causes of excessive lateral vibrations of the Millennium Bridge.

It is observed that even if a footbridge is static at the beginning, excessive lateral vibrations still occur under crowd. However, we have to choose different models to analyze oscillations of the bridge in small or large amplitude. No force models mentioned above allow describing the whole process how the amplitude increases from small to large. When the lateral bridge amplitude is small, the distribution of pedestrians’ walking phase is random. According to the existing experiments, a moving surface always makes pedestrians synchronize with the surface over a large frequency range. The randomness of pedestrians’ walking phase reduces with the increase of the lateral bridge amplitude. Then, the randomness of pedestrians’ walking phase can be used to evaluate the influence of the moving deck. In this paper, we try to consider a unified expression of the lateral force induced by pedestrians for different bridge amplitudes. The details about the lateral force model will be given in Section 3.

In many dynamic interaction models [3, 4, 18, 19], a footbridge always was simplified as a SDOF system no matter what its structural characteristics. Roberts [12] modelled the Millennium Bridge as a beam but also eventually converted it to a SDOF system. Blekherman [20] used a pendulum system with two degrees of freedom to simulate the lateral vibrations of a footbridge under crowd excitation. Although there is no evidence that the structural characteristics of a footbridge have impact on dynamic interaction between pedestrians and the footbridge, McRobie et al. [21] experimentally investigated the phenomenon of human-structure lock in by using a section model consisting of two strings and a rigid body; Zhou and Ji [22] theoretically and experimentally analyzed dynamic characteristics of a generalized suspension system developed on the basis of the section model. Their research studies showed the parameters about strings have great influence on dynamic behaviour of the suspension system. This implies that we should not ignore structural characteristics in the analysis for mechanism of excessive lateral vibrations of suspension footbridges. The other motivation of our paper is to involve structural characteristic of the suspension bridge in the proposed model.

In this paper, we attempt to use a simple model to exhibit the whole process how a suspension footbridge increases its lateral amplitude from small to large under crowd. According to the research studies of McRobie et al. [21] and Zhou and Ji [22], we study a suspension footbridge by using the section model consisting of two strings and a rigid body. The strings and the rigid body represent cables and bridge deck, respectively, as shown in Figure 1. The lateral force induced by crowd on the bridge here is expressed by a cosine function with random phase evaluating the effect of the moving bridge deck. The randomness of the phase reduces when the lateral bridge amplitude increases, because the swaying bridge always makes pedestrians synchronize with the moving deck. By referring to some criteria on stability of nonlinear and stochastic equations [23, 24, 25, 26], we will show the influence of randomness on stability of a suspension footbridge to understand the whole process that the lateral bridge amplitude increases from small to large. The most advantage of the proposed model is the unified form on description of the lateral force induced by pedestrians, which usually is discontinuously expressed for different lateral bridge amplitudes in other models. Some new explanations might be given based on the plane pendulum model to understand the mechanism of the pedestrian-footbridge interaction. However, it should be pointed out that the section model is appropriate if the distribution of the pedestrian mass is uniform along both the bridge length and the direction orthogonal bridge axis. If an uncertainty of the lateral force induced by the pedestrian occurs, the result based on the model may not be accurate. However, we focus on excessive lateral vibrations of a footbridge, and under such condition, there always are a lot of pedestrians on the bridge. Usually, pedestrians do not cluster together in any place on the bridge. Instead, they always trend to distribute uniformly on the deck. Once the pedestrian flows are steady, the distribution of pedestrians on the deck will be approximately uniform. In addition, only the first-order lateral modal vibration of the bridge is concerned in this model. Since we mainly intend to find a unified form to describe the lateral force induced by pedestrians for different amplitudes, some important parameters, such as walking speed and step frequency, have not been involved in this model. Furthermore, we consider the structural characteristics of the suspension bridge in our model, and then the analysis results are only valid for the suspension bridge.