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Since the occurrence of pedestrian-induced large lateral vibration of the London Millennium Bridge in 2000, scholars have realized the complexity of pedestrian-induced lateral vibrations. Despite extensive research spanning over two decades, the underlying mechanisms between pedestrians and footbridges remain incompletely understood. Currently, there are two main popular models for explaining the pedestrian-induced lateral vibration of the footbridge: the synchronization lock-in model and the self-excited force model. Among existing studies, the inverted pendulum model (IPM) essentially belongs to the self-excited force model, has gradually gained recognition. This model assumes that pedestrians maintain a constant step frequency and suggests that footbridge lateral vibration divergence can occur through pedestrian-bridge interaction without synchronization. Although the IPM theoretically elucidates the mechanism by which pedestrian-bridge interaction leads to self-excited forces, it still has its shortcomings: it overestimates the critical number of pedestrians required for triggering vibration divergence of the footbridge. The underlying cause of this problem stems from the inverted pendulum model’s inherent limitation as a single-mechanism framework, which fails to consider the adjustments of pedestrian’s step frequency and instead solely relies on the adjustments of pedestrian’s step width. Consequently, this results in an underestimation of the virtual equivalent damping coefficient that is in phase with the vibration velocity of the footbridge. This study proposes a generalized-inverted pendulum model (G-IPM), in which the pedestrian walking phase evolution is effectively combined with the IPM, thereby the adjustments of pedestrian’s step frequency and step width can be considered simultaneously. Compared to the single-mechanism-based IPM, the numerical simulation results indicate that the proposed dual-mechanism-based G-IPM requires fewer pedestrians to trigger the bridge vibration divergence, which is closer to the actual results. Additionally, parameter analysis reveals that varying pedestrian characteristic parameters exert different impacts on the vibration response of bridge. The proposed model pioneers the use of a dual-mechanism-based model, which is of significant theoretical importance in revealing the underlying mechanism of pedestrian-induced lateral vibrations of the footbridge.

Incidents of pedestrian-induced large lateral vibrations of footbridges reveal the occurrence of an instability-type phenomenon in pedestrian-induced vibration. However, the mechanism involved has yet to be clearly explained. In this work, a novel model for predicting the lateral vibration of footbridges is proposed. Under an algebraic framework of nonlinear stochastic vibration, the narrow-band vibration caused by the pedestrian intra-subject randomness and the phase lag between the footbridge motion and the pedestrian load are considered. The critical condition that triggers the large lateral vibration of the footbridge is identified using the stochastic averaging method and the concept of stability based on the Lyapunov exponent. The validity of the proposed method is confirmed through case studies of three bridges. Through a parametric analysis, the effects of several crucial parameters on the stability/instability of vibration are discussed. Finally, conclusions are drawn regarding insights that can be useful to the future designs of footbridges.

A three-dimensional (3D) pedestrian–structure interaction (PSI) system based on the biomechanical bipedal model is presented for general applications. The pedestrian is modeled by a bipedal mobile system with one lump mass and two compliant legs, which comprise damping and spring elements. The continuous gaits of the pedestrian are maintained by a self-driven walking kinetic energy, which is a new driven mechanism for the mobile unit. This self-driven mechanism enables the pedestrian to operate at a varying total energy level, as an important component for further modeling of the crowd-structure dynamic interaction. Numerical studies show that the pedestrian walking on the structure leads to a reduction in the natural frequency, but an increase in the damping ratio of the structure. This model can also reproduce the reaction forces between the feet and structure, similar to those measured in the field. In addition, the proposed model can well describe the 3D pedestrian–structure dynamic interaction. It is recommended for use in further study of more complicated scenarios such as the dynamic interaction between a large scale kinetic crowd and slender footbridge.

Owing to the slenderness and lightness of most modern footbridges, vibration serviceability assessment becomes a crucial issue in the design process. As one of the key factors, the vibration comfort criterion has an important influence on the assessment of the final result. However, there is an obvious lack of experimental studies in this field, especially regarding the pedestrians' perception of the induced vibrations. In this study, an experiment was conducted to investigate the pedestrians' perception of human-induced vibrations of footbridges. During the experiment, the subjects walked on a pathway that was mounted on top of a shaking table. By imposing sinusoidal excitations with different amplitudes and frequencies, the experiment aimed to determine the influence of the two factors on the walking people's perception. Based on the data collected, perception scales were proposed for both the vertical and lateral vibrations of the footbridge. The established scales comprise five levels that depend on the acceleration amplitude and the frequency. Finally, a comparison between the proposed scales, existing comfort criteria in the literature and international codes was carried out.

In this paper, the inverted pendulum model is proposed to describe a pedestrian’s walking motion by considering that the pivot point vibrates periodically up and down. The stability, periodic solutions and oscillations of the inverted pendulum are theoretically investigated, the correctness of which is illustrated by numerical simulations. According to frequency spectrum analysis, the inverted pendulum can exhibit periodically or quasi-periodically stable oscillations. However, we demonstrate that the inverted pendulum will maintain the ratio between the lateral and vertical vibration frequencies near $1∕2$ as an optimizing selection of stability. The theoretical result agrees with the measurement result for a normal pedestrian such that the lateral step frequency is always half the vertical step frequency, which means that it is feasible and reasonable to describe a pedestrian’s walking motion using the inverted pendulum with the pivot vibrating harmonically in the vertical direction. The inverted pendulum model suggested in this paper could contribute to the study of pedestrian–footbridge interaction, which overcomes the difficulty of directly determining the expression of the lateral force induced by pedestrians.

It is essential to reliably predict the human-induced vibrations in serviceability design of footbridges to ensure the vibration levels to be within the acceptable comfort limits. The human-induced structural responses are dependent on the dynamic properties of structures and human-induced excitations. For concrete footbridges, the elastic modulus of concrete is a vital parameter for determining the dynamic structural properties. To this end, a two-stage machine learning (ML)-based method is first proposed for modeling the elastic modulus of concrete. At the first stage, the ensemble algorithm, i.e. the gradient boosting regression tree (GBRT), is used to predict the compressive strength by selecting eight parameters, including concrete ingredients and curing time, as the inputs. At the second stage, the elastic modulus of concrete is modeled by using the GBRT method with the compressive strength as the input. Pedestrian crowd-induced load is the most common and crucial design load for footbridges. To consider the inter- and intra-subject variability in walking parameters and induced forces among persons in a crowd, a load model is developed by associating a modified social force model with a walking force model. By integrating the two submodels of structure and excitation, an intelligent analysis method for human-induced vibration is finally developed. A concrete footbridge with typical box cross-section subjected to human-induced excitation is analysed to illustrate the application of the proposed method.