Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5967491, 12 pages

https://doi.org/10.1155/2017/5967491

## Nonlinear Stochastic Analysis for Lateral Vibration of Footbridge under Pedestrian Narrowband Excitation

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, China

Correspondence should be addressed to Yu Xiao-lin

Received 11 March 2017; Revised 14 May 2017; Accepted 7 June 2017; Published 9 July 2017

Academic Editor: Roman Lewandowski

Copyright © 2017 Jia Bu-yu 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

During the lateral vibration of footbridge, the pedestrian lateral load shows two distinct features: one is the vibration-dependency; another is the narrowband randomness caused by the variability between two subsequent walking steps. In this case, the lateral vibration of footbridge is actually a complicated, nonlinear stochastic system. In this paper, a novel nonlinear stochastic model for lateral vibration of footbridge is proposed, in which a velocity-dependent load model developed from Nakamura model is adopted to represent the pedestrian-bridge interaction and the narrowband stochastic characteristic is considered. The amplitude and phase involved Itô equations are established using the multiscale method. Based on the maximal Lyapunov exponent derived from these equations, the critical condition for triggering a large lateral vibration can be obtained by solving the stability problem. The validity of the proposed method is confirmed, based on performing the case studies of two bridges. Meanwhile, through parameter analysis, the influences of several crucial parameters on the stability of vibration are discussed.

#### 1. Introduction

The infamous incidents of large lateral vibrations of the Solferino Bridge in 1999 and the London Millennium Bridge in 2000 highlight the divergence instability in pedestrian-induced vibration; that is, a slight increase in the number of people will cause the finite vibration of footbridge to diverge with large amplitude. Since these infamous incidents, scholars have begun to study the mechanism of these vibrations and gradually realized that the large lateral vibration of footbridge is a complicated process that may be governed by several coexisting mechanisms rather than a single one. To explain such vibration, several models have been proposed and classified into deterministic and stochastic models.

The representative deterministic models include the Fujino [1], Dallard [2], Nakamura [3], Ingólfsson [4, 5], Macdonald [6, 7], Roberts [8], Newland [9], Piccardo [10], Blekherman [11, 12], and Strogatz [13] models. The Fujino model can be regarded as a linear direct resonance model where the lateral vibrations are caused by direct resonance; that is, the pedestrian walking frequency is in resonance with the natural frequency of one or more lateral vibration modes. This model assumes that the vibration amplitude continues to increase along with the number of pedestrians, but such assumption contradicts the observed sudden divergent vibration on the Millennium Bridge. The Dallard, Nakamura, and Ingólfsson models are velocity-dependent nonlinear models where the pedestrian lateral load is assumed to be influenced by the footbridge velocity response and such influence can be described using a known model. The velocity-related terms in a velocity-dependent nonlinear model represent the external addition of negative damping to the bridge. The critical number of pedestrians needed to trigger the divergence can be obtained when the overall modal damping becomes negative. The Macdonald, Roberts, and Newland models couple pedestrian motion with bridge vibration as well as considering the influence of footbridge vibration on the pedestrian lateral load. However, such influence is not described by a known empirical parameter model but rather by coupled equations that include pedestrian motion and bridge vibration. The Piccardo model is a parametric resonance model that attributes the excessive pedestrian-induced lateral vibration in flexible footbridges to a parametric resonance in which the lateral natural frequency is equivalent to half of the pedestrian lateral walking frequency. The Blekherman model is an autoparametric resonance model that assumes that the vertical and lateral modes are coupled and that the vertical excitation energy is transferred to the lateral direction when the vertical and lateral frequencies of the footbridge show multiple relationships. The Strogatz model is a pedestrian phase synchronization model that is originally used to analyze the onset of synchronization in populations of coupled oscillators. According to this model, the initially randomly distributed walking frequencies of pedestrians will be synchronized with that of bridge if an external stimulus (e.g., vibration amplitude) is strong enough or if the step frequency is close to the vibration frequency of the bridge.

The major existent models are almost based on deterministic methods yet ignore the obvious randomness in pedestrian load. Actually, the pedestrian load is a complex stochastic process that involves large intrasubject variability from the same person and intersubject variability among different people, thereby making the pedestrian load in actual cases become significantly different from that in a deterministic case. This paper focuses on intrasubject variability. In the deterministic time domain analysis, the pedestrian load is always assumed to be a perfect periodic load that can be transformed into multiorder harmonic loads by Fourier series. However, the real consecutive pedestrian load is not a perfect periodic load but rather a narrowband stochastic process caused by intrasubject variability. If the presented load is transformed into the frequency domain, then the Fourier spectra are not distributed in discrete frequency points as a perfect periodic load but within a certain distributed width around the main harmonics, thereby leading to a reduced response. Only few stochastic models exist today, such as the Brownjohn [14], Živanović [15], Ingólfsson [4], and Racic models [16, 17]. Although the Brownjohn and Živanović models aim at the pedestrian vertical load, their modeling approaches for intrasubject randomness can also be applied to the pedestrian lateral load. These two models, which can express the pedestrian load through their Power Spectral Density (PSD), allow for the evaluation of the variance of the structural displacement and acceleration response. However, these models do not account for the influence of bridge vibration on pedestrian load. The Ingólfsson model aims at the pedestrian lateral load, in which motion-induced forces, including equivalent damping and inertia forces, are quantified through random coefficients that are generated from a discrete-time Gaussian Markov process. This model also considers the randomness of body weight, walking frequency, step length, walking speed, and arrival time among different people to represent the intersubject randomness. The critical number of pedestrians can be predicted through the criteria of the initial zero-crossing of overall damping and the first exceedance of the acceleration threshold, although this model only generates numerical results instead of an algebraic solution. The Racic model focuses on the multiple Gaussian fitting of the Autospectral Densities (ASD) of pedestrian lateral excitation, by which the variations of amplitude and phase during the real pedestrian lateral excitation process can be effectively reproduced. However, this model employs too many empirical parameters to fit numerous Gaussian functions.

The tests performed on the Millennium Bridge show that the pedestrian lateral load depends on the bridge velocity. In the Dallard model, the pedestrian lateral load is thought to be proportional to bridge velocity, thereby indicating that the bridge vibration will reach infinity as the lateral load increases. This result contradicts the fact that pedestrians will either reduce their walking speed or completely stop when the bridge velocity becomes large. The Nakamura model starts from the Dallard model but introduces a nonlinear velocity-dependent function to represent the self-limiting nature of pedestrian action (i.e., the pedestrian response to bridge vibration will saturate under a large bridge velocity). Given that the nonlinear velocity-dependent function adopts a fraction form and includes an absolute value, this model cannot be used for conveniently implementing an algebraic analysis. Another downside to this model is that the pedestrian lateral load does not contain harmonic terms, thereby indicating that it is independent of walking frequency (i.e., the effect of the variation of pedestrian walking frequency on vibration is ignored). Inspired by the Nakamura model, this paper adopts a nonlinear velocity-dependent model to represent the pedestrian-bridge interaction but replaces the fraction function with a hyperbolic tangent function, like the way that is proposed by Bin and Weiping [18]. Meanwhile, given that the pedestrian gait varies with each step (i.e., intrasubject variability), the pedestrian lateral load is treated as a narrowband stochastic excitation process to take into account the intrasubject variability. Unlike the numerical simulation method that requires a large number of calculations, the proposed model is established in a theoretic framework of nonlinear stochastic vibration. Based on the Itô equations that are derived using the multiscale method, the critical condition for triggering a large lateral vibration is obtained according to the maximal Lyapunov exponent, whose sign is used to indicate stability or instability. The effectiveness of the proposed model is then confirmed through its applications on the Millennium Bridge (M-Bridge) and the Passerelle Simone de Beauvoir Bridge (P-Bridge), and the influences of some crucial parameters on the stability of vibration are discussed.

#### 2. Model for the Lateral Vibration of Footbridge

##### 2.1. Stochastic Model of Pedestrian Lateral Load

The Nakamura model is taken as the basic model to represent the pedestrian lateral load. However, due to the lack of harmonic term, the Nakamura model can only consider the case of direct resonance, while it fails to consider the effect of the pedestrian frequency distribution on vibration. Obviously, such model cannot give a justified explanation for the large lateral vibration of lower frequency bridge modes around 0.5 Hz. To address this concern, a harmonic term (it becomes a stochastic excitation process when the intrasubject randomness is considered, more on this later) is introduced into the load model to refine the Nakamura model.

Then the lateral load per unit length exerted by pedestrians can be defined as follows:where with being the number of pedestrians and being the single pedestrian mass, based on the assumption that the pedestrians are assumed to be uniformly distributed along the bridge length . is the acceleration of gravity, and is the dynamic loading factor of the first harmonic that takes a value of 0.04 according to [5, 14, 19] (this paper considers only the first harmonic and ignores the contribution from other high harmonics [20]). denotes the synchronization coefficient that takes a value of 0.2 according to [1, 3], is the vibration-dependent function that describes the interaction between pedestrian and footbridge vibration response (i.e., the displacement , velocity , and acceleration ), and denotes the stochastic excitation process (or the harmonic function if the deterministic periodic load is considered).

As mentioned previously, it is not convenient to implement an algebraic analysis for the Nakamura model due to its assumption that the velocity-dependent function has a fraction form including an absolute value. Therefore, the function of is expressed by the following velocity-dependent hyperbolic tangent function proposed by [18]:

Figure 1 shows the comparison result of between the hyperbolic tangent function ( s/m) and the fraction function in the Nakamura model.