Performance of single pedestrian load models in predicting dynamic behaviour of flexible aluminum footbridges

Transcription

1 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 Porto, Portugal, 3 June - 2 July 214 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: ; ISBN: Performance of single pedestrian load models in predicting dynamic behaviour of flexible aluminum footbridges Pampa Dey 1, Ann Sychterz 1, Sriram Narasimhan 1, Scott Walbridge 1 1 Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario, Canada pdey@uwaterloo.ca, asychterz@uwaterloo.ca, sriram.narasimhan@uwaterloo.ca, swalbrid@uwaterloo.ca ABSTRACT: Their high strength-to-weight ratio and slenderness can make aluminium pedestrian bridges lively structures. The large amplitude of vibrations may reduce their serviceability under operational loads and may drive the dynamic design over the static one. In order to properly design aluminium bridges, it is important to define suitable load models to estimate structural response under human induced vibrations. The pedestrian force is commonly modeled in terms of periodic ground reaction forces (GRFs) on rigid surfaces, depending on weight and pacing frequency of the pedestrian, and the GRFs so obtained are approximated through Fourier series with dynamic load factors corresponding to the harmonics. However, most of these models have not been sufficiently verified from experimental and field measurements of bridge response under pedestrian excitations. The scope of this paper is to provide an overview of the periodic load models and their performance in predicting the response of footbridges. Predictions by the existing analytical load models are compared with field measurements conducted on a flexible aluminum bridge, and key observations regarding the applicability of existing models to predict the vibration behaviour of these bridges is reported. KEY WORDS: Pedestrian load models; Dynamic behaviour; Aluminum footbridges. 1 INTRODUCTION Flexible aluminum footbridges are prone to large amplitudes of vibration induced by different human activities like walking, running, jumping etc. Among all the human activities, walking is the most important loading case due to its frequent occurrence. Low-frequency flexible pedestrian bridges with natural frequencies within the normal walking frequency range are expected to generate the highest level of response, which results in exceeding the serviceability limit state of footbridges. Under this situation dynamic design criteria may dominate over the static criteria. In recent years, many studies have been done focusing on vibration serviceability of flexible footbridges subjected to pedestrian induced walking force. Accurately characterizing the walking forces is key to evaluating the performance of footbridges in terms of serviceability-limit state. Many codes of practice employ a deterministic walking load model in terms of ground reaction forces (GRFs) on rigid surfaces, depending on weight and pacing frequency of pedestrian where Fourier series are employed to represent GRFs mathematically as proposed by Blanchard et al. [1]. A number of numerical and experimental investigations have been carried out in the literature to evaluate dynamic load factors (DLFs) and phase angles corresponding to each GRF harmonic based on measurements on very high frequency structures [1-5]. However, the behaviour of pedestrians on a low frequency vibrating footbridge is complicated and quite different that on a rigid surface because of human-structure interaction. Attempts have been made in the past to incorporate the effect of humanstructure interaction with various formulations for the dynamic load factor and phase angles of the Fourier series [6-7]. However, these existing models have been developed with the assumption of perfect periodicity of the walking force. In reality, walking is not perfectly periodic [8] but narrow band in nature [9] and could be altered due to interaction between the pedestrian and the structure. Multiple simplified design methodologies have been developed in the codes of practice [6, 1-13] based on DLFs proposed by different authors. However, validation of these guidelines and application to real structures has been given little attention so far. In order to validate and further develop the design guidelines for flexible footbridges, it is necessary to evaluate of the effectiveness of the existing load models in predicting the vibration behaviour of real structures. With the objective of verifying the performance of existing walking load models for single pedestrians, a field study was conducted on a full-scale aluminium bridge: the Daigneault Creek pedestrian bridge in Brossard, Quebec. Vibration (i.e. acceleration) measurements were obtained along the length of the bridge for various walking scenarios. The response of the bridge under single pedestrian walking was then simulated using multiple periodic moving load models with proposed DLFs [1,6,7]. Finally, to assess the applicability of these models, the estimated response of the bridge is compared with the measured response. 2 VERTICAL PERIODIC LOAD MODELS Generally, most of the design guidelines consider loading from a pedestrian as a periodic point force exerted on the bridge deck. The walking forces of a pedestrian are represented using the following Fourier series [1], P( t) G (1 n i1 sin( i 2 f i s t ) i (1) 187

2 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 where P(t) is the human-induced dynamic load, G is the pedestrian weight, α i is the dynamic load factor (DLF) of the i th harmonic, f s is the pedestrian step-frequency (Hz), and i is the phase shift of the i th harmonic. The Canadian Highway Bridge [12] and the British Standard [13] codes adopt a DLF value of.257 considering the first harmonic only, as proposed by Blanchard et al. [1]. With time, many authors have proposed improved DLF values based on direct or indirect GRF measurements. In 21, [7] proposed the first four harmonics of the vertical force as a function of the walking frequency f s as follows: 1.37( fs.95).5 (2) f s (3) f s (4) f s (5) i (6) The [7] guidelines for bridge design have subsequently considered the DLFs as.4,.1, and.1 for the first three harmonics, respectively, with no phase angles. For the present study, the response of the bridge has been simulated based on all the above studies for comparison and evaluation purposes. 3 FIELD STUDY 3.1 Daigneault Creek Bridge in Brossard, Quebec Daigneault Creek Bridge is an aluminium bridge in Brossard, Quebec (Figure 1), connecting a new subdivision from Rue Claudel to a transportation hub and commercial area on Rue Grande Allée. It has a clear span of 43.7 m and a deck width of approximately 4.4 m. For the current study the first vertical vibration mode of the bridge has been considered. The frequency for this mode is 3.43 Hz with modal mass of 7641 kg and modal damping ratio of.12. These parameters were obtained from a combination of finite modelling of the bridge in the software Sofistik and also from the results of modal testing performed on the bridge. A more detailed description of the footbridge and the modal testing conducted can be found in publications by Sychterz et al. [14]. 3.2 Walking Test The field study was done in two stages. In the first stage, vertical as well as lateral accelerations were measured through 12 accelerometers at the midspan and two quarter span locations of the bridge. The 12 accelerometers were attached to six mounting blocks three on each side of the bridge. Each mounting block carried an accelerometer in the lateral and vertical directions (Figure 2). A detailed description of the first stage of experimentation is provided by Sychterz et al. [14]. 5 test subjects were asked to walk with their natural step frequencies. The physical parameters of the test subjects are summarized in Table 1. Figure 1: Daigneault Creek Bridge in Brossard, Quebec (taken from Sychterz et al. [14]) Figure 2: Accelerometer installed in the mounting block (taken from Sychterz et al. [14]). Table 1 Physical and gait parameters of test subjects Test Subject Mass (kg) Height (m) Leg Length Step Frequency (Hz) (m) P P P P P In the second stage of experiment, 3 sets of walking tests were conducted for test subject P5. Each set corresponds to nine different step frequencies ranging from 1.67 Hz to 2.33 Hz, with a mean of 2 Hz and interval of.83 Hz. To achieve the desired step frequencies, a metronome was used. The footbridge vertical acceleration was recorded at the midspan and one quarter span point along one side of the bridge. In addition, the time instants when a test subject entered and left the bridge in a single crossing were recorded using a stopwatch. In this way it was possible to obtain the average 188

3 Fourier Amplitude Acceleration (g) Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 walking velocity of the pedestrian which is required for the response simulations. Later on, all the measured acceleration time histories were processed to obtain the first modal acceleration response as well as the response corresponding to the excitation frequency (first harmonic) and the natural frequency of the structure. 3.3 Processing of Field Data Since the response measurements contain contributions from all the harmonics (natural as well as excitation), in the present study, the acceleration were filtered appropriately according to frequencies of interest. To do this, the time domain measurements are converted to the frequency domain through Fourier transform (FFT). Appropriate windowing was undertaken to reduce spectral leakage [14]. Although windows are useful in reducing leakage, they can modify the overall amplitude of the signal. Hence, to compensate for the amplitude distortion in the time domain, the windows were scaled by dividing the windowed array by the coherent gain of the window [14]. Figure 3 plots the acceleration time history and corresponding Fourier spectra using different window functions and it is clear that the effects of leakage is relatively small and different windows yield similar spectral behaviour in terms of frequency and amplitude. After obtaining FFT of the field data, the Fourier amplitudes were made zero outside the desired low and high cut-off frequencies and performed an inverse Fourier transform to obtain the filtered data. 4 SIMULATION OF VERTICAL RESPONSE Moving periodic load models have been adopted to simulate the vertical acceleration response of Daigneault Creek footbridge under single pedestrian excitation. To simulate the first modal response by periodic moving load models, the bridge is assumed to be a single degree-of-freedom (SDOF) linear elastic system with the first mode being the flexural one. The modal equation of motion for the bridge response due to the pedestrian excitation given by: M x Cx Kx f (t) (7) where, x(t) is the modal displacement response of the system at time instant t and its first and second derivative represent the velocity and acceleration of the bridge due to the excitation. M, C and K represent the modal mass, damping and stiffness of the bridge (for natural frequency of 3.43 Hz in this study). The modal load f(t) can be written as: vt f ( t) P( t)sin( ) (8) L with the assumption that the bridge deck acts as a simply supported beam of length L and the first mode shape of the beam (Figure 4) is given by: x ( x) sin( ) (9) L where x = v t is the position of moving load at any time t on the beam and v is walking speed of the pedestrian. Here P(t) represents the ground reaction force due to walking, as given by Equation 1. Response was calculated from the time instant the pedestrian steps on to the bridge to the instant the pedestrian left the bridge at the other end. Hence the total duration required to cross the bridge is T = L/v. The response of the bridge at the mid-span was estimated from the analytical solution proposed by Abu-Hilal and Mohsen [17] for simply supported beam under moving harmonic load in case of different dynamic load factors as described in Section 2. The bridge is assumed to be at rest prior to the application of load. The peak as well as the root mean square accelerations values are calculated and compared with the measured responses in order to evaluate the performances of the existing models in predicting the response of flexible footbridges Time(sec) (a) x 1-3 No windoiw Hann Hamming Blackman-harris (b) Figure 3: (a) Acceleration time history and (b) corresponding Fourier spectra of the data recorded for test subject P5 walking with step frequency of 2.7 Hz. Figure 4: Computational model of a pedestrian crossing a bridge [16]. 189

4 Fourier Amplitude Fourier Amplitude Fourier Amplitude Acceleartion (m/s 2 ) Fourier Amplitude Acceleartion (m/s 2 ) Acceleartion (m/s 2 ) Acceleartion (m/s 2 ) Proceedings of the 9th International Conference on Structural Dynamics, EURODYN COMPARISION OF SIMULATED AND MEASURED RESPONSE Figures 5 and 6 present the measured and simulated vertical acceleration versus time plots and corresponding Fourier spectra of the responses at the midspan for one case where the test subject P5 walked at 2 Hz. Blanchard et al. [1] have included contribution from only the first GRF harmonic, while [7] and [6] have included 2 and 3 higher subharmonics, respectively in their respective models. Figure 6 shows that all of the simulated load models lead to a response that is dominated by the forcing frequency. However, measurements show significant dominance of the natural frequency of the structure. This suggests that the simulated moving load models are unable to replicate the observed frequency content in the dynamic response of a full-scale flexible footbridge under a single pedestrian walking case Blanchard et al.[1] [6] Time (sec) [7] Time (sec).5 Blanchard et al.[1].5 [7] Figure 7: Simulated and measured acceleration time histories of test subject P5 filtered at first harmonic frequency of 2 Hz..15 Blanchard et al.[1].15 [7] [6] Time (sec) Time (sec) Figure 5: Simulated and measured acceleration time histories of test subject P5 walking at 2 Hz step frequency Blanchard et al.[1] [6] [7] Figure 6: Fourier spectra of acceleration time histories [6] Figure 8: Fourier spectra of the filtered time histories. Generally, peak and root mean square (RMS) responses are two different measures of any time series. Peak response represents only the highest amplitude of the series while RMS value takes into account the total data length as well as all the smallest to highest peaks. In the present study both response quantities were estimated to compare the simulated and field measurements. With this objective, recorded time histories are filtered to the forcing frequency contribution of the response. Figures 7 and 8 show the simulated and measured acceleration time histories and their Fourier spectra. The experimental data was filtered to isolate the excitation frequency contribution to the response at 2 Hz. Peak and RMS response of the footbridge under walking of five test subjects at their normal frequency based on single measurement are compared in Figures It can be observed that the simulated model with DLF proposed by Blanchard et al. [1] underestimates the measured response corresponding to the first harmonic frequency except for test subject P5, in which case it predicts better than the other two models. For most cases, the model performs better over others in predicting the peak 19

5 RMS Acceleration (m/s 2 ) RMS Acceleration (m/s 2 ) Peak Acceleration (m/s 2 ) Peak Acceleration (m/s 2 ) Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 response (Figure 9) while the DLF proposed by Blanchard et al. gives a close estimate of the RMS response except test subject P1. It must be emphasized, however, that all these models significantly underestimate the contribution of response arising from the natural frequency of the structure (Figures 11 and 12) Blanchard et al. P1 P2 P3 P4 P5 Test Subjects Figure 9: Comparison of peak responses corresponding to different test subjects walking at their normal frequency (filtered at first harmonic of forcing function). 7.5 x P1 P2 P3 P4 P5 Test Subjects Blanchard et al. Figure 1: Comparison of RMS responses corresponding to different test subjects walking at their normal frequency (filtered at first harmonic of forcing function). The performance of the simulated models for different step frequencies ranging from slow to fast is evaluated next. For this purpose, simulated responses of test subject P5 under different walking frequencies ranging from 1.67 Hz to 2.33 Hz are compared with the statistical values of the observed responses in error bar plots (Figures 13-16). Comparison of peak and RMS responses shows that the DLF value, proposed by Blanchard et al. [1], results in the best estimate with larger deviation for slows pacing rates (Figures 13 and 14). However, all of the models overestimate the measured response. Also the models are unable to capture the contribution of response from transients, with the exception of slow step frequencies of 1.67 and 1.76 Hz (Figures 15 and 16). In these specific cases, the second harmonic of the step frequencies are close to the resonant frequency of the first vertical mode of the footbridge (3.43 Hz). As a result, the second harmonic dominates the vibration of the bridge along with its natural frequency and hence contributes to the total response. However, this is a special case and results have to be interpreted with caution Blanchard et al. P1 P2 P3 P4 P5 Test Subjects Figure 11: Comparison of peak responses corresponding to different test subjects walking at their normal frequency (filtered at first natural frequency of footbridge) Blanchard et al. P1 P2 P3 P4 P5 Test Subjects Figure 12: Comparison of RMS responses corresponding to different test subjects walking at their normal frequency (filtered at first natural frequency of footbridge). 191

6 Peak Acceleration (m/s 2 ) RMS Acceleration (m/s 2 ) Peak Acceleration (m/s 2 ) RMS Acceleration (m/s 2 ) Proceedings of the 9th International Conference on Structural Dynamics, EURODYN Blanchard Blanchard Step Figure 13: Comparison of peak responses of test subject P5 walking at different step frequencies with error bar plot of the filed response (filtered at first forcing harmonic) Blanchard Step Figure 14: Comparison of RMS responses of test subject P5 walking at different step frequencies with error bar plot of the filed response (filtered at first forcing harmonic) Blanchard Step Figure 15: Comparison of peak responses of test subject P5 walking at different step frequencies with error bar plot of the filed response (filtered at first natural frequency of the footbridge) Step Figure 16: Comparison of RMS responses of test subject P5 walking at different step frequencies with error bar plot of the filed response (filtered at first natural frequency of the footbridge). 6 CONCLUSION The current study starts with measuring the response of a fullscale footbridge in terms of acceleration due to a single person crossing the bridge in different step frequencies. The study then focuses on estimation of the footbridge accelerations by simulating moving periodic load models of walking using DLFs proposed by different authors. Simulated peak and RMS accelerations are then compared to the measured values. Overall the DLFs proposed by Blanchard et al. seem to capture the responses reasonably well compared to other models studied here. However, none of the models explain the relatively large contributions from the transients in the experimental data. Further investigation is necessary to study this issue in a greater detail and efforts along these lines are currently underway at the University of Waterloo. Finally, if required, the load models need to be improved so that they can better predict observations from full-scale footbridges. ACKNOWLEDGMENTS The support for this research provided by the Aluminum Association of Canada and MAADI Group is gratefully acknowledged. The City of Brossard is thanked for making the field study possible. The authors thank Richard Morrison from the University of Waterloo and Patrick Simon from TU Berlin for assistance with the field study. REFERENCES [1] J. Blanchard, B. L. Davies, J. W Smith, Design criteria and analysis for dynamic loading of footbridges, 1977 [2] H. Bachmann, Vibration Problems in Structures: Practical Guidelines, Springer Verlag, [3] H. Bachmann, W. Ammann, Vibrations in Structures Induced by Man and Machines, Structural Engineering Documents, Vol.3e,International Association of Bridges and Structural Engineering(IABSE),Zurich, [4] J. H. Rainer, G. Pernica, D. E. Allen, Dynamic loading and response of footbridges, Canadian Journal of Civil Engineering 15 (1) (1988) [5] S. C. Kerr, Human Induced Loading on Staircases, PhD Thesis, Mechanical Engineering Department, University College London, UK,

7 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 [6] P., Improved Floor Vibration Prediction Methodologies, ARUP Vibration Seminar, October 4, 21. [7] SETRA, Footbridges, Assessment of vibrational behaviour of footbridges under pedestrian loading, Technical guide SETRA, Paris, France 26. [8] Eriksson, P. E. Vibration of Low-Frequency Floors Dynamic Forces and Response Prediction, PhD Thesis, Unit for Dynamics in Design, Chalmers University of Technology, Goteborg, Sweden, 1994 [9] S. V. Ohlsson, Floor Vibrations and Human Discomfort, PhD Thesis, Chalmers University of Technology, 1982 [1] EC (1997) Eurocode 5, Design of Timber Structures Part 2: Bridges, ENV : 1997, European Committee for [11] ISO, Bases for Design of Structures Serviceability of Buildings and Pedestrian Walkways Against Vibration, ISO/CD 1137, International Standardization Organization, Geneva, Switzerland, 25. [12] CAN/CSA-S6-6. Canadian Highway Bridge Design Code. 26. Canadian Standards Association. [13] BS 54- Steel, Concrete & Composite Bridges, Part Appendix B, BSI, UK. [14] A. Sychterz, A. Sadhu, S. Narasimhan, S. Walbridge, Results from Modal Testing of the Daigneault Creek Bridge, CSCE 213 General Conference, Montréal, Québec, May 29 to June 1, 213 [15] F. J. Harris, On The Use of Windows for Harmonic Analysis with The Discrete Fourier Transform, Proceeding of the I, Vol. 66, No. 1, Taneeary 1978 [16] L. Pendersen, C. Frier, 21. Sensitivity of Footbridge Vibrations to Stochastic Walking Parameters, Journal of Sound and Vibration, 329(13), [17] M. Abu-Hilal, M. Mohsen, Vibration Of Beams with General Boundary Conditions Due to A Moving Harmonic Load, Journal of Sound and Vibration, (2):

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