اثر متقابل جمعیت - ساختار در پل عابر پیاده: مدلسازی، کاربرد برای مطالعه موردی واقعی و تجزیه و تحلیل حساسیت

A mathematical and computational model used to simulate crowd–structure interaction in lively footbridges is presented in this work. The model is based on the mathematical and numerical decomposition of the coupled multiphysical nonlinear system into two interacting subsystems. The model was conceived to simulate the synchronous lateral excitation phenomenon caused by pedestrians walking on footbridges. The model was first applied to simulate a crowd event on an actual footbridge, the T-bridge in Japan. Three sensitivity analyses were then performed on the same benchmark to evaluate the properties of the model. The simulation results show good agreement with the experimental data found in literature and the model could be considered a useful tool for designers and engineers in the different phases of footbridge design.

Over the last few decades, several footbridges have shown great sensitivity to human induced vibrations in the lateral direction (e.g. Refs. [1] and [2]). This phenomenon, known as synchronous lateral excitation, can take place any time pedestrians walk on a surface that oscillates laterally with a frequency that is close to the mean lateral walking frequency (around 1 Hz). When a pedestrian walks on a laterally moving surface, in an attempt to maintain balance, he walks with his legs more widespread and adapts his frequency to that of the moving surface, that is, he synchronises to the structure. Hence, the lateral motion of the upper part of the torso increases and the resulting lateral force increases in turn. This phenomenon is amplified if the pedestrian walks in a crowd, since synchronisation among pedestrians increases the effects of pedestrian–structure synchronisation. The synchronous lateral excitation phenomenon has never caused structural failure since it is self-limited, that is, when the vibrations exceed a limit value, pedestrians stop walking or touch the handrails, and this causes the vibration to decay. Nevertheless, the resulting reduced comfort for the users has often led to a temporary closure of the footbridge, with consequent economic and social repercussions. In order to avoid this kind of problem, an intense research activity was begun after the Millennium Bridge in London was closed because of excessive lateral vibration. The results of these studies, which are reviewed in Ref. [3], represent the scientific background of some recently published design guidelines [4] and [5]. The most relevant data concerning pedestrian behaviour have been obtained using an empirical approach. Laboratory tests involving a pedestrian walking on both a motionless platform [6] and a laterally moving treadmill [2] and [7], as well as tests performed on actual footbridges [8], have been carried out to measure the lateral force exerted by one pedestrian and interesting information about the synchronisation between the pedestrian and the structure has been obtained. Moreover, the behaviour of a pedestrian in a crowd has been investigated by means of in situ experiments [2] and through the observation of videos recorded during crowd events [1] and [2]. Several semi-empirical load models have been developed on the basis of the aforementioned experimental data, e.g. in Refs. [4], [9] and [10]. Generally, the pedestrians are considered as a load that has to be applied to the structural dynamic system. To the authors’ knowledge, the crowd was first modelled as part of a complex dynamical system in Refs. [11] and [12], where the modelling framework was presented. This framework is based on the decomposition of the coupled multiphysical crowd–structure dynamical system into two subsystems, the crowd and the structure, which interact with each other through forcing terms. The resulting, very simple model is capable of taking into account for some key features of the phenomenon, such as the self-limited nature of the structural vibration and the effects of various pedestrian traffic conditions [12]. The authors devoted their subsequent work to the development of each single model component. The effects of the structure vibrations on the crowd behaviour have been modelled in Ref. [13], where a relation between the crowd density, the walking velocity and the deck motion has been derived. The crowd-to-structure action has been developed in Ref. [14] using a new lateral force model, referring to pedestrian clusters, which is able to describe both pedestrian-to-pedestrian and pedestrian-to-structure synchronisation effects in each cluster. In the present work, the updated components have been collected in the initial modelling framework. The improved model has been implemented in an ad hoc developed multiphysics numerical code. The model has been applied to an actual crowd event on a real footbridge, the T-bridge (Japan). Detailed in situ measurements of both the crowd conditions and structural response [1] allow a complete comparison with the computational results. The coupled system sensitivity to both structural and pedestrian design parameters has been evaluated through three parametrical studies. The paper is developed in six more sections. Section 2 briefly recalls the proposed model and describes its upgraded components. Section 3 is devoted to the computational approach. The model is applied to a case study, the T-bridge in Japan, which is described in Section 4. In Section 5 the model is validated by simulating a real event occurred on the T-bridge. Sensitivity studies on the pedestrian biometrics and travel purposes, the incoming crowd density and the structural properties are then performed in Section 6. The concluding remarks are outlined in Section 7.

A complete model has been proposed to simulate the phenomenon of synchronous lateral excitation on lively footbridges. The model is based on the partitioning of the coupled system into two interacting subsystems. The crowd is not intended as just a load, but as a part of a complex dynamical system. The model was first applied to simulate a real event that occurred on the T-bridge in Japan, and then subjected to several sensitivity studies on different crowd and structural parameters. The results obtained from the simulation of a real event show an excellent agreement with the recorded data, both for the evolution in time of the crowd condition along the span and for the maximum value of the lateral displacement of the deck. Generally speaking, the sensitivity studies highlight the capabilities of the proposed approach to evaluate the effects of various physical parameters on the crowd dynamics and structural response. The structural response is particularly sensitive to the crowd travel purpose and geographical area. Hence, in the conceptual design phase, it is important to plan the kind of pedestrian traffic that the footbridge is most likely to incur during its lifetime. The sensitivity study on the crowd density has shown that a more crowded condition does not always correspond to higher deck vibrations and confirmed the complexity of the coupled dynamical system. This conclusion has obviously been derived from the assumptions that were made for the proposed model. Simplified comfort criteria, based on the limitation of the number of pedestrians crossing the bridge, might not always be effective in preventing the synchronous lateral excitation phenomenon. However, the complexity of the phenomenon makes it difficult to conceive compact comfort criteria which can take into account all the features involved. Finally, the sensitivity study on the structural stiffness has allowed the most relevant effects of deck acceleration to be considered in the neighbourhood of the acceleration critical value. A limit cycle in the force–acceleration envelope plane can be observed. This shows a self-limiting mechanism of the structural response around the lock-in triggering threshold region, which gives rise to intermittent lock–delock stages. These simulated features strengthen the analogy with the lock-in phenomenon that occurs in fluid–structure interaction and which has been suggested in literature. A structural benchmark, characterised by lower lateral stiffness, could provide more information on the effects of very high amplitudes of deck acceleration on crowd–structure interaction.