Lateral Vibration Of Footbridges Research Articles

Dynamic Analysis for Lateral Vibrations of Footbridges under Crowd Based on a Two-Degrees-of-Freedom Model

Nakamura’s model is widely used to describe lateral vibrations of a footbridge induced by crowd. The predicted responses of Nakamura’s model were compared with measured data of T-bridge and M-bridge in Japan to demonstrate the validity. However, the predicted responses based on Nakamura’s model almost always were stronger than measured data. Considering that both T-bridge and M-bridge are cable-stayed bridges, it seems to be not precise enough to simplify a cable-stayed bridge as a single degree of freedom system in Nakamura’s model. In this paper, we establish a two-degrees-of-freedom model to describe lateral vibrations of a cable-stayed bridge. The cables have one degree, and the bridge deck has the other. Additionally, in this model we introduce a time delay in interaction between the bridge and pedestrians. By employing the center manifold theory, we find that a subcritical Hopf bifurcation occurs in the two-degrees-of-freedom model. We theoretically and numerically illustrate that the cables and time delay have significant influence on the lateral vibration amplitude of a footbridge under crowd. The appropriate increases of tension in the cables and time delay both can decrease the lateral vibration amplitude. The analysis for the proposed two-degrees-of-freedom model shows that the predicted responses of Nakamura’s model can better agree with the measured date if we take the influence of cables and time delay into account.

Effects of Stochastic Excitation and Phase Lag of Pedestrians on Lateral Vibration of Footbridges

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.

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

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.

Excessive Pedestrian-Induced Swaying in Code-Compliant Walkways

Human-induced dynamic loads may trigger noisy vibrations on slender structures, affecting even their serviceability and compromising their use. This paper shows that current code-based design may not be effective enough in preventing excessive lateral vibrations of footbridges. By referring to the case history of the Assago-Forum walkway (Italy), it is shown that the widely adopted ES-ISO-10137 International Standard may, in fact, lead to underestimating—even significantly—the actual peak lateral accelerations in walkways and, consequently, it may erroneously consider structures safe against human-induced lateral vibrations when they are not actually so. The reason can be found in the extent to which the horizontal load code-based model is affected by the design scenario. By considering the horizontal actions relating to different code-compliant scenarios, a numerical dynamic analysis of the Assago-Forum walkway has been carried out. The results have then been compared with those obtained through a recently published, more advanced study. This investigation provides evidence that the design guidance given by ES-ISO-10137 against pedestrian-induced lateral vibrations of walkways may not be adequate. The paper also advances some suggestions on how code-based design might be improved.

Calculation for the Critical Number of Pedestrians on a Footbridge by Using the Modified Nakamura’s Model

The problem of the critical number of pedestrians is significant in the study of lateral vibrations of footbridges since it can cause serviceability for the footbridges. By considering the pedestrians mass on the footbridge increases slowly, the modified Nakamuras model can be regarded as a fast-slow system which can be used to calculate the critical number of pedestrians. The influence of the time delay of interaction between pedestrians and the footbridge on the critical number of pedestrians is analyzed. Comparison with other results indicates the correctness of our formula predicting the critical number of pedestrians. Furthermore, our research suggests that the existence of the time delay can make the critical number of pedestrians less, which means that the time delay helps to improve the safety of the footbridge.

Application of Nakamura’s Model to Describe the Delayed Increase in Lateral Vibration of Footbridges

AbstractNakamura’s model is proposed to describe lateral vibrations of a footbridge induced by pedestrians. The model is found to be a slow-fast system if the number of pedestrians on a footbridge increases slowly. Then, based on the delayed bifurcation theory, Nakamura’s model can be used to describe the delayed increase in the lateral vibration of the footbridge. The critical number of pedestrians on the footbridge is calculated by using the delayed bifurcation theory to quantitatively demonstrate our analysis. By comparison with a previously obtained formula, the rationality of the authors’ formula for predicting the critical number of pedestrians is illustrated. Based on the authors’ formula, the experimental parameter k=300 used in the previously obtained formula does not seem to be always applicable to other bridges. Based on Nakamura’s model, the value of the parameter k seems to be affected mainly by the properties of the pedestrians on the bridge rather than those of the bridge.

Calculation of the Periodic Solution in Nakamura’s Model

Nakamura’s model is one of the most practical models describing the lateral vibrations of footbridges induced by pedestrians. This paper presents a calculation of the periodic solution in Nakamura’s model. After a Hopf bifurcation analysis of Nakamura’s model, the amplitude of the lateral vibration is computed by using the energy method. The correctness and accuracy of the calculation is demonstrated by numerical simulations. Then, how the factors and variables in Nakamura’s model effect the amplitude is investigated based on the analytic solution. Our analysis results may be used to explain why the most of the predict results of Nakamura’s model are larger than the measure data.

Nonlinear Analysis for the Lateral Vibration of Footbridges Induced by Pedestrians

AbstractIn this paper, the factors and variables that affect the amplitude of the lateral vibration of the footbridge induced by pedestrians in Nakamura’s model are investigated in the aspect of nonlinear dynamics. Nakamura’s model is based on the motion of equations including coefficients of the rate of a pedestrian’s lateral force, the pedestrian density, the rate of synchronized pedestrians, the function G(fB) describing how pedestrians synchronize with the bridge natural frequency, and the pedestrians’ attitude to large vibration amplitude. The important coefficient G(fB) cannot be decided in this model because of the lack of data available on the lateral load behaviors of walking people. Nakamura assumed that G(fB)=1.0 in his model by considering that pedestrians are most likely to synchronize at the bridge frequency around 1.0 Hz. The analysis results presented in this paper demonstrate that the amplitude of the lateral vibration increases when the function G(fB) increases. This suggests G(fB)=1.0 m...

Lateral Vibration of Footbridges under Crowd-Loading: Continuous Crowd Modeling Approach

In this paper, a simple 1D crowd model is proposed, which aim is to properly describe the crowd-flow phenomena occurring when pedestrians walk on a flexible footbridge. The crowd is assumed to behave like a continuous compressible fluid and the pedestrian flow is modeled in a 1-D framework using the (total) mass (of pedestrians) conservation equation. This crowd model is then coupled with a simple model for the dynamical behavior of the footbridge and an optimized modeling of synchronization effects is performed. Numerical simulations are presented to show some preliminary results.

Lateral vibration of footbridges under crowd-loading : Continuous crowd modeling approach

Lateral vibration of footbridges under crowd-loading : Continuous crowd modeling approach

Lateral vibration of footbridges by synchronous walking

On the T-bridge (a cable-stayed footbridge) the girder vibrated laterally with a frequency of 0.93 Hz, the natural frequency of the first lateral mode, when a large number of people crossed the bridge. Some pedestrians felt uncomfortable and unsafe. The girder was excited by the lateral dynamic force which was produced by the zigzag movement of pedestrians. Once the bridge started to vibrate, some of the pedestrians synchronized with the girder vibration, which further increased the girder response. The same vibration occurred on the London Millennium Bridge and the M-bridge (a suspension footbridge), where field measurements were conducted. A dynamic model was then proposed to evaluate the pedestrians’ synchronized dynamic forces. Overall behavior of the girder response calculated by this dynamic model agreed with the measured values on the bridges. It was found by parametric studies that smaller bridge damping and smaller bridge mass produced larger girder response. Higher pedestrian density also increased the girder response unless they were too crowded to walk normally.

A briefing on pedestrian-induced lateral vibration of footbridges

A briefing on pedestrian-induced lateral vibration of footbridges