Response Of Footbridges Research Articles

Dynamic behaviour of a GFRP-steel hybrid pedestrian bridge in serviceability conditions. Part 2: Numerical and analytical study

The Part 2 of this two-part paper reports numerical and analytical studies concerning the serviceability dynamic behaviour under dynamic human loads of a typical GFRP-steel hybrid girder system for pedestrian bridges. These studies, which focused on the recent St. Mateus footbridge (Portugal), had two main objectives: (i) to assess the ability of conventional numerical models and analytical formulae to simulate the vibration behaviour of GFRP-steel hybrid footbridges; and (ii) to complement the observations and measurements made in the experimental campaign (Part 1), thus providing in-depth understanding of the vibration performance of this type of footbridges. First, a finite element (FE) model was developed and calibrated with the static and modal experimental data. Then, the FE-model was used to evaluate the vibration response of the footbridge for the same pedestrian activities induced in situ. The design formulae available in the literature for predicting the maximum acceleration of structural systems are also reviewed and were applied to evaluate the resonant response of the footbridge for both single and crowded conditions of pedestrian traffic. The results obtained confirmed the adequate structural behaviour of the St. Mateus footbridge under real dynamic (human) pedestrian load cases and show that this innovative GFRP-steel hybrid structural solution is suitable for pedestrian bridges. The results of the different investigations reported in this paper also showed that both numerical and analytical approaches are able to predict the vibration response of GFRP-steel hybrid footbridges with reasonable accuracy.

The impact of vertical human-structure interaction on the response of footbridges to pedestrian excitation

In the present paper, a detailed crowd model is applied to investigate the impact of vertical human-structure interaction (HSI) on the dynamic response of footbridges to pedestrian excitation. Assuming that the walking behaviour of the pedestrian is not affected by the motion of the supporting structure, the contact force between the pedestrian and the supporting structure is decomposed into the force exerted by the pedestrian on a perfectly rigid floor and an interaction term determined by the mechanical interaction between the person and the structure. The model is also used to evaluate the impact of various simplifying assumptions and inter- and intra-person variabilities.The results show that for the low-frequency dynamic behaviour of footbridges (<6Hz), HSI leads to an effective damping ratio which is (significantly) higher than the inherent damping ratio of the empty structure. In addition, it is demonstrated that the detailed moving crowd model can be well approximated by a time-invariant crowd-structure model. Furthermore, the influence of inter- and intra-person variability in step frequency is observed to decrease as the impact of HSI increases. Finally, it is shown that in many cases accounting for HSI leads to a significant reduction in the response of footbridges to pedestrian excitation which may be considered in practical design.

Pedestrian load models of footbridges

The increase of vibration problems in modern footbridges shows that footbridges should no longer be designed for static loads only. Not only natural frequencies but also damping properties and pedestrian loading determine the dynamic response of footbridges and design tools should consider all of these factors. In this paper the pedestrian load models for serviceability verification of footbridges, which are missing in the current European codes, are presented. For simplicity reasons the proposed pedestrian load models are based on stationary pulsating loads instead of moving pulsating loads. It is shown that simplified procedure can be used in verification of the serviceability limit state related to vibration due to pedestrians. Footbridge vibrations don’t cause usually structural problems, but if the vibration behaviour does not satisfy the comfort criteria, changes in the design or damping devices could be considered. The most popular external damping devices are viscous dampers and tuned mass dampers (TMD). The efficiency of TMD is demonstrated on the example of a footbridge prone to vibrations induced by pedestrians. It is shown that if the TMD is tuned quite precisely the reduction of accelerations can be very significant.

Design of Footbridges against Pedestrian-Induced Vibrations

AbstractIn 1999 and 2000, the two vibration incidents at the Paris Passerelle Solferino Bridge and the London Millennium Bridge triggered a major revision of existing knowledge concerning footbridge response to pedestrian-induced actions. In the last 15 years, an incredibly large amount of research has emerged on the topic. Although researchers have provided many valuable scientific contributions regarding the understanding and modeling of pedestrian-induced vibrations of footbridges, there is still a need to determine what real improvements have been achieved in design procedures. This article provides a critical overview of the methodologies proposed over the last four decades, as well as their code implementations, and summarizes the actual advances available that consultants can use to their advantage in bridge design.

Serviceability Assessment of Composite Footbridge Under Human Walking and Running Loads

Footbridge responses under loads induced by human remain amongst the least explored matters, due to various uncertainties in determining the description of the imposed loadings. To address this gap, serviceability of an existing composite footbridge under human walking and running loadings is analyzed dynamically in this paper employing a finite element approach. The composite footbridge is made-up of a reinforced concrete slab simply supported at two ends on top of two T-section steel beams. To model the walking and running loads, a harmonic force function is applied as the vibration source at the center of the bridge. In the model verification, the computed natural frequency of footbridge exhibits a good agreement with that reported in literature. The vibration responses in terms of peak acceleration and displacement are computed, from which they are then compared with the current design standards for assessment. It is found that the maximum accelerations and displacements of composite footbridge in presence of excitations from one person walking and running satisfy the serviceability limitation recommended by the existing codes of practice. In conclusion, the studied footbridge offers sufficient human safety and comfort against vibration under investigated load prescription.

A Review of Human Induced Vibrations on Footbridges

An extensive literature review of human induced vibrations that flexible footbridges experience is addressed in this study. Qualitative information is comprehensively included herein to provide common methods and code recommendations for the practicing engineers. The parameters affecting the dynamic response of footbridges excited by pedestrians are highlighted. In particular, the synchronous lateral excitation is addressed. This investigation can be valuable for the design criteria selection. In addition, this work contributes to the review of numerous case studies correlating important dynamic characteristics for various footbridge structural types, the variability of which confirms the complexity of the issue. Furthermore, vibration upgrading methods are described, focusing on applications of tuned mass dampers and remarkable conclusions are drawn.

Application of the pseudo-excitation method to assessment of walking variability on footbridge vibration

The pseudo-excitation method is applied to determine the non-stationary vibration response of footbridges to variable pedestrian excitation. Excitation forces are described by their spectral densities, the models of which are taken from the literature. In addition a simple spectral model is proposed to encompass both intra- and inter-pedestrian variability. Comparison is made to Monte Carlo simulations of random pedestrian events and a means of estimating extreme statistics of random walking is given. The method is shown to be accurate and efficient. Consequently, this work should find value in explaining differences between observed and modelled responses.

Evaluating the vertical vibration response of footbridges using a response spectrum approach

In this paper, the vertical vibration response of footbridges subjected to dynamic loads induced by walking humans is assessed via a response spectrum approach. The dynamic walking load in the vertical direction is applied on the bridges using two different loading schemes: (1) a stationary load at the mid-span and (2) a moving load across the bridge. The response spectrum analysis is carried out using a Generalized Single Degree of Freedom procedure which has been verified by comparing its predictions with the results of a Multi-Degrees of Freedom modeling.The results obtained indicated that the response spectrum approach is capable of accurately predicting the footbridge vibration response. The results obtained also indicated that the main parameters that affect the induced accelerations in footbridges due to the human walking loads in the vertical direction are the footbridge mass and damping ratio.

Dynamic response of concrete footbridges

AbstractExperience with the analysis and performance of 15 lightweight concrete footbridges is presented from the point of view of their dynamic response caused by moving people. The footbridges are formed either by a stress ribbon or by a slender deck supported by arches or suspended from pylons. The dynamic response was determined according to a procedure given in a 1998 draft of Eurocode 2: Design of Concrete Structures – Part 2: Concrete Bridges. Thirteen footbridges have been built and so far no complaints about their dynamic behaviour have been recorded.

Enhancement factors for the vertical response of footbridges subjected to stochastic crowd loading

The vertical acceleration response of a hypothetical footbridge is predicted for a sample of single pedestrians and a crowd of pedestrians using a probabilistic approach. This approach uses statistical distributions to account for the fact that pedestrian parameters are not identical for all pedestrians. Enhancement factors are proposed for predicting the response due to a crowd based on the predicted accelerations of a single pedestrian. The significant contribution of this work is the generation of response curves identifying enhancement factors for a range of crowd densities and synchronization levels.

Mitigation of Pedestrian-induced Vibrations in Suspension Footbridges via Multiple Tuned Mass Dampers

The dynamic response of suspension footbridges to pedestrian-induced excitations and its passive mitigation, via multiple tuned mass dampers (TMDs), are investigated. First, the nonlinear equations of motion are obtained assuming finite planar motions of the suspension bridge. A suitable approximate version of the equations of motion is shown to be in agreement with existing theories and its linearization is then employed in the structural dynamics analyses. A Galerkin discretization is exploited to calculate both the free and forced dynamic response towards the design of the vibration control system. First, the leading characteristics of the bridge dynamic response are outlined. Resonant vibrations induced by the passage of pedestrians are shown to be effectively reduced using viscoelastic TMDs. As the frequencies of the lowest two modes in suspension footbridges can be very close in the proximity of the crossover phenomenon, three different design scenarios are considered: below, near and above the crossover. In particular, the influence of these scenarios on the passive control architecture is investigated.

Sensitivity of footbridge vibrations to stochastic walking parameters

Some footbridges are so slender that pedestrian traffic can cause excessive vibrations and serviceability problems. Design guidelines outline procedures for vibration serviceability checks, but it is noticeable that they rely on the assumption that the action is deterministic, although in fact it is stochastic as different pedestrians generate different dynamic forces. For serviceability checks of footbridge designs it would seem reasonable to consider modelling the stochastic nature of the main parameters describing the excitation, such as for instance the load amplitude and the step frequency of the pedestrian. A stochastic modelling approach is adopted for this paper and it facilitates quantifying the probability of exceeding various vibration levels, which is useful in a discussion of serviceability of a footbridge design. However, estimates of statistical distributions of footbridge vibration levels to walking loads might be influenced by the models assumed for the parameters of the load model (the walking parameters). The paper explores how sensitive estimates of the statistical distribution of vertical footbridge response are to various stochastic assumptions for the walking parameters. The basis for the study is a literature review identifying different suggestions as to how the stochastic nature of these parameters may be modelled, and a parameter study examines how the different models influence estimates of the statistical distribution of footbridge vibrations. By neglecting scatter in some of the walking parameters, the significance of modelling the various walking parameters stochastically rather than deterministically is also investigated providing insight into which modelling efforts need to be made for arriving at reliable estimates of statistical distributions of footbridge vibrations. The studies for the paper are based on numerical simulations of footbridge responses and on the use of Monte Carlo simulations for modelling the stochastic nature of actions of a single pedestrian traversing various pin-supported single-span footbridges with frequencies in vertical bending in the range of 1.6–2.4 Hz.

A method for predicting the lateral girder response of footbridges induced by pedestrians

A numerical method is proposed to predict the lateral girder response induced by pedestrians on footbridges. The method is based on the motion of equations including the coefficients of the rate of a pedestrian’s lateral force, pedestrian density, rate of synchronized pedestrians, and pedestrians’ attitude to large vibration amplitude. These coefficients were determined by the field measured data of two slender footbridges and the experimental data of pedestrian induced forces. The lateral girder responses were then predicted for these bridges with different pedestrian densities by the proposed numerical method. They agreed reasonably well with the field measured girder responses of these bridges, which verified the proposed prediction method.

A parametric study of composite footbridges under pedestrian walking loads

This paper presents a realistic load model developed to incorporate the effects induced by people walking in the dynamical response of footbridges. The effect of the human heel was also incorporated in the dynamical load representation. The investigated structural model was based on several footbridges, with main spans varying from 10 to 35 m. The proposed computational model adopted the usual mesh refinement techniques present in finite element method simulations. The results obtained throughout the investigation showed that the design standards could produce unsafe values since they are based on excessively simplified load models. Hence it was detected that this type of structure can reach high vibration levels that can compromise the footbridge user’s comfort and especially its safety.

Vibration analysis of footbridges due to vertical human loads

Pedestrian footbridges have been constructed with increasingly daring structures encompassing the experience and knowledge of structural designers by using newly developed materials and technologies. This fact has generated very slender structural footbridges and, consequently, changed their associated serviceability and ultimate limit states. A direct consequence of this design trend was a considerable increase of structural vibration problems. In the particular case of pedestrian footbridges this phenomenon occurs when the structural fundamental frequency is near the load excitation frequencies, or higher frequencies multiples. This was the main motivation for the development of a design methodology to better evaluate the footbridge user’s comfort and safety. Considering all these aspects a linear elastic finite element analysis contemplating the dynamical response of pedestrian footbridges focusing on critical acceleration values was conducted. The investigated model was based on an existing footbridge located at the city of Rio de Janeiro, Brazil. Four different loading models were developed to incorporate the dynamical effects, induced by people walking, in the dynamical response of pedestrian footbridges. The results indicated that this footbridge can reach high vibration levels hat could compromise the user’s comfort limit state.

Experimental methods for estimating modal mass in footbridges using human-induced dynamic excitation

Abstract To assess human-induced response of footbridges and, if necessary, design the suppression systems such as tuned-mass dampers, it is vital to have reliable estimates of modal mass for lively vibration modes. Traditional experimental methods for estimating modal mass of existing footbridges require measurement of frequency response functions, implying direct measurement of controllable excitation. This requires expensive experimental facilities which often may not be financially or practically feasible. Out of necessity, a method has been developed for estimating the modal masses for vibration modes of footbridges known to respond perceptibly to pedestrian excitation. The method has the advantage that, in its simplest form, only a single channel measurement of acceleration response of the footbridge to a single person jumping, walking or swaying, needs to be recorded. Rather than applying and directly measuring artificial excitation, the method is based on quantifying the forces that generate the lively response, and characterising the initial part of the resonant build-up due to these forces. The method makes use of a database of laboratory-recorded time series of single-person jumping, walking or swaying with the reasonable assumption that the fundamental characteristics of human-induced forcing change little between laboratory and field testing. This also works because the early resonant build-up is relatively insensitive to the imperfect timing of human jumping, walking or swaying. The method has been evaluated on four full-scale footbridges so far and found to give acceptable accuracy with the error margin of about 15% for modal mass estimation for modes of vibration responding to vertical forcing. The method can be used in general for cases where resonant excitation of pedestrian structures is a concern, hence it can be used for vertical, lateral and torsional footbridge vibration modes and even for floors having low and well separated natural frequencies.

Probability-based prediction of multi-mode vibration response to walking excitation

In vibration serviceability checks of footbridges, a force induced by a single person walking is usually modelled as a harmonic force having a frequency that matches one of the footbridge natural frequencies. This approach assumes that, among the infinite number of harmonics a walking force is composed of, only a single harmonic is important for a vibration serviceability check. Another usual assumption is that the footbridge can be modelled as an SDOF system, implying that only vibration in a single mode is of interest. In addition, due to the deterministic nature of this approach, it cannot take into account inter- and intra-subject variabilities in the walking force that are now well documented in the literature. To account for these variabilities, a novel probabilistic approach to carry out a vibration serviceability check is developed in this paper. Factors such as the probability distribution of walking frequencies, step lengths and amplitude of walking force for its five lowest harmonics and subharmonics are taken into account. Using walking force time histories measured on a treadmill, the frequency content of the force was investigated, resulting in the formulation of a multi-harmonic force model. This model can be used to estimate the multi-mode response in footbridges. This was verified successfully on an as-built catenary footbridge structure. Although only the vibration response of footbridges was analysed in this paper, the force model proposed has the potential to be implemented in the estimation of floor vibration as well, where multi-mode response occurs more frequently. The model is easily programmable and as such could present a powerful tool for estimating efficiently the probability of various levels of vibration response due to single person walking. Therefore, the proposed probability-based methodology has the potential to revolutionise the philosophy of the current codes of practice dealing with vibration serviceability of structures under human-induced vibration.

Human–structure dynamic interaction in footbridges

Dynamic load induced in the vertical direction by a single person walking across a footbridge structure is usually modelled as a resonant harmonic force. A comparison between modal responses generated by this harmonic force and a walking force measured on a treadmill confirmed that using the harmonic force is justifiable in the case of walking across a structure that does not vibrate perceptibly. It is further shown that using this harmonic force for predicting the response of a perceptibly vibrating footbridge can significantly overestimate the real footbridge response and therefore its degree of liveliness. This is because of the inability of pedestrians to keep pacing steadily when perceiving strong vibrations. A methodology for systematic comparison of the measured and simulated responses is proposed with the aim of identifying vibration levels which disturb normal walking. On two footbridges investigated this level was 0·33 m/s2 and 0·37 m/s2, in both cases being lower than the levels allowed by the current British footbridge design code. Finally, the observed effect of losing steady walking step can be expressed via either increased damping or modified harmonic dynamic force over time periods when test subjects lose their step. The average increase in the damping over the analysed time periods of imperfect walking was up to 10 times higher than the damping of an empty footbridge.

Vibration Excitation and Control of a Pedestrian Walkway by Individuals and Crowds

As part of a continuing study on effects of humans on loading and dynamic response of footbridges, a steel frame walkway has been the subject of studies on the effects of multiple pedestrians with respect to loading and response mitigation. Following finite element modeling and experimental modal analysis to identify the low frequency vibration modes likely to be excited by normal walking, the variation of response with pedestrian density and of system damping and natural frequency with occupancy by stationary pedestrians were both studied. The potentially mitigating effect of stationary occupants is still not well understood and the study included direct measurement of damping forces and absorbed energy using a force plate. The various tests showed that energy dissipation measured directly was consistent with the observed change in damping, that vertical and lateral response both varied approximately with square root of number of pedestrians, and that the simple model of a human as a single mass‐spring‐damper system may need to be refined to fit observed changes in modal parameters with a crowd of humans present. Modal parameter changes with moving pedestrians were small compared to those with stationary pedestrians indicating that within limits, modal parameters for the empty structure could be used in analysis.

Dynamic loading and response of footbridges

Dynamic forces were measured during walking, running, and jumping, using an instrumented platform. The results are expressed as sinusoidal force amplitudes normalized by the subject's weight and are plotted versus frequency. The maximum dynamic loads for walking were found to be nearly twice as large as those recommended in the 1983 Ontario Highway Bridge Design Code or the British Standard BS5400, and those for running or jumping, more than six times as large.Responses of footbridges are calculated using a simple formula based on the dynamic loading due to one person, the response of a simple span at resonance, and limited duration of excitation. Good agreement was obtained with the measured response of two 17 m experimental spans subjected to human excitation, for both the first and second harmonics of the step rate. The resonant vibrations of the spans can be substantially reduced by resonant dampers. Key words: footbridges, dynamic loads, dynamic response, design criteria, resonant dampers.