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ABSTRACT: The realistic evaluation of the serviceability of footbridges in regard to pedestrian-induced vibrations requires a comprehensive load model, a correct model of the random pedestrian stream and finally a thorough understanding of the dynamic behavior of the coupled system structure-users. For passive persons, a probabilistic model is available which allows the prediction of the change in the natural frequency and the effective damping. However, there is no equivalent approach for active persons. Based on field experiments with a shaker, the presented study identifies the induced additional damping for a single person walking on a pedestrian bridge which has a natural frequency of 1.8 Hz. It is shown that the induced damping exceeds that of a passive person and depends on the step frequency. Based on measurements during a mass event, the influence of the number of pedestrians is studied on the change of the natural frequency and the damping ratio.
KEY WORDS: Pedestrian structures, pedestrian-induced damping, full-scale testing.
1 INTRODUCTION
Modern footbridges are light and slender structures which are prone to vibrations induced e.g. by walking pedestrians. While a structural failure due to walking-induced vibrations probably can be excluded, ‘lively’ bridges may cause considerable serviceability problems. Beside basic threshold events in terms of the sensation of discomfort in different degrees, further criteria for the evaluation of the serviceability should consider intrusion, alarm and fear and finally interference with the activities of the users. Low mass and low damping capacities increase the probability of serviceability problems. Especially steel and steel-concrete composite footbridges may have damping values of 0.5% critical damping or lower.
The presence of human occupants may considerably change the dynamic behaviour of structures. While for passive persons these changes can be analysed based on an equivalent mass-spring-damper system for the human body considering random dynamic parameters [1], a consistent model is still missing for the influence of active persons. At least there should be an influence for activities where the person is in permanent contact to the structure. In [2], a respective model has been introduced for the activity bobbing, however, with deterministic describing parameters. Especially, there are no studies of the influence of walking persons in the range of natural frequencies which are excited by the first harmonic of the walking-induced loads.
The actual study is mainly based on field tests. Object of the study is a cable supported bridge which is described in chapter 2 in regard to its basic dynamic characteristics. The basic influence of a single passive person on the dynamic characteristics is discussed in chapter 3. Chapter 4 deals with a test series aiming in identifying the damping effect of a person walking with different step frequencies. Finally, in chapter 5, the bridge vibrations observed during a mass event are analysed in regard to changes in the natural frequency and the effective damping.
2 OBJECT OF THE STUDY
The field tests are performed on a pedestrian bridge with a total span of 66 m which is separated into two fields of 18 m and 48 m. The longer field, which has a width of 4.5 m, is supported at the third points by ropes which are connected to two pylons (fig. 1). The two pylons are connected under the walkway with a horizontal beam which serves as support of the bridge deck and separates the two fields. The width in the shorter field increases from 4.5 m at the pylons to 9.7 m at the end of the bridge.
Figure 1. The OLGA-bridge at Oberhausen, Germany The dynamic characteristics of the bridge are analysed
based on measuring the accelerations at 15 nodes. Beside ambient excitations from wind and traffic underneath the bridge, shaker tests are performed. The bridge has two dominant natural frequencies. The first natural frequency with 1.8 Hz is in the range of the step frequencies of pedestrians. As can be seen in figure 2, which shows the mode shapes of the first two natural frequencies, the first natural frequency is
Damping induced by pedestrians
Michael Kasperski
Research Team EKIB, Department of Civil and Environmental Engineering Sciences, Ruhr-University Bochum, 44780 Bochum, Germany
email: [email protected]
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4
1059
almost exclusively a vibration of the longer field. The corresponding relative vibration amplitudes in the shorter field are below 4% of the maximum vibration amplitude in the longer field. The second natural frequency of 3.9 Hz corresponds to a vibration mode with an additional node in the longer field. The position of the node of the second mode is identical to the position of the anti-node of the first mode. Thus, it is possible to excite the first mode with a shaker without getting influences from the second mode. The second mode shows considerable vibrations of the shorter field with relative vibration amplitudes of 30%.
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Figure 2. Dominant vibration modes of the OLGA-bridge The respective mode shape amplitudes are used to estimate
the modal masses. The modal mass of the first mode is 34 tons; the modal mass of the second mode is slightly larger with 37 tons due to the increase of the width in the shorter field.
The damping of the empty structure is low with only 0.5% critical damping for vibration amplitudes up to 0.25 m/s². Basically, large vibrations levels may lead to considerably higher damping. Therefore, the bridge is excited by a group of pedestrians walking in step. Once the group has left the structure, the sub-sequent free vibrations can be used to determine the damping capacity for larger vibration amplitudes. Figure 3 shows the damping behaviour for vibration amplitudes up to 0.78 m/s². It is important to note, that during the free vibration a technician is standing at the anti-node of mode one. For a least square fit over 23 cycles, the damping ratio is 0.0054, i.e. there is an increase of 8%.
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Figure 3. Damping behavior for vibration amplitudes
between 0.8 and 0.35 m/s² with a person standing at the anti-node of mode 1
3 INFLUENCE OF A PASSIVE PERSON
With the random model of the dynamic characteristics of the human body in [1], it becomes possible to predict the range of changes of the natural frequency and the effective damping of the bridge due to a single person standing in the anti-node of mode 1. Basically, a passive person is modelled as a two-degree of freedom system with two random masses, two random natural frequencies and corresponding random damping ratios.
The influence of persons on the natural frequency often is estimated based on the mass-only model [3, 4], i.e. the change of the natural frequency is determined based on the increase of the modal mass m of the empty bridge with the mass of the person mp. Introducing the relative mass ratio p = mp / m leads to a reduction of the natural frequency with 1/(1+p)
1/2. As has been shown in [1], the mass-only model tends to overestimate the natural frequency of the coupled system. However, for natural frequencies below 2.5 Hz, this effect is small. Due to the large mass of the empty bridge, there is almost no influence of the random person on the natural frequency of the coupled system. However, there is an influence on the effective damping Deff. The trace of the non-exceedance probability of Deff / Dempty is shown in figure 4. The additional damping Dadd = Deff -1 has a mean value of 0.0328 critical damping, the corresponding variation coefficient is large with 36%. The two sided 95% confidence ranges from 0.0156 to 0.0611.
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Figure 4. Influence of a passive person on the natural frequency and the damping ratio of the OLGA bridge;
simulations based on [1]
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4 INFLUENCE OF A SINGLE WALKING PERSON
The next test series aims in identifying the influence of a walking person on the effective damping. The tests are performed with a shaker placed at the anti-node of mode 1. The shaker provides a constant background excitation at the first natural frequency of 1.8 Hz, leading to vibration amplitudes of about 0.25 m/s². For the first run, the bridge is completely empty. Then, a person is standing still at the anti-node. It is important to note, that the person for these test is different to the person standing in the center of the bridge for the free vibration tests in section 2. The effective damping is increased by a factor of 1.032, which is obviously in the range of the predicted increases in figure 4. The following runs are performed with this person walking at step frequencies considerably different to the natural frequency, which allows separating the effect of the two excitation sources in the frequency range. The step frequency is triggered with an electronic metronome.
Since the control panel of the shaker is close to the shaker position, the technician who starts the measurements has to leave the bridge before the actual test procedure starts. The test person enters the bridge once the shaker-excitations have reached a stationary level. During this starting phase, the distortions induced by the leaving technician have decayed. Since the crossing of the bridge by a single person leads to a transient excitation, which makes it difficult to identify the damping effects, the person walks in circles in a marked bridge section which is given by mode shape amplitudes larger than 0.8. When the shaker excitation stops after about six minutes, the test person leaves the bridge. Finally, the technician enters the bridge to stop the measurements.
The further processing of the data is based on the Discrete Fourier Transformation DFT. This allows considering for
each analyzed frequency a time window which is based on an integer number of cycles. Thus, all biasing effects of the Fast Fourier Transformation FFT are avoided. All DFT-amplitudes are based on a time window with a length of 10 s, allowing a resolution of the frequencies with a step width of 0.1 Hz. During the circling phase, the moving DFT-amplitudes will show some fluctuations, which is to be expected since the weighting factor of the load changes with the position between 0.8 and 1.0. Therefore in the following, the estimation of the effective damping is based on the mean value of the DFT-amplitudes in the respective time window.
Figure 5 shows the resulting time series for a step frequency of 1.7 Hz. Beside the acceleration of the bridge deck, three DFT-amplitudes are shown, namely for 1.7, 1.8 and 1.9 Hz. Before the actual test person enters the bridge, the DFT-amplitudes for 1.7 and 1.9 Hz show some fluctuations which are due to the leaving technician. Once the actual test person has entered the bridge, the DFT-amplitude for 1.7 Hz clearly increases. During the circling phase, the DFT-amplitude for 1.7 Hz fluctuates around a mean value. The same is true for the DFT-amplitude for 1.8 Hz, indicating a change of the effective damping. The control DFT-amplitude for 1.9 Hz also shows clear non-zero amplitudes. This behavior has to be expected since the person is not able to perfectly keep the triggering frequency during walking, especially during the turning phases. Therefore, some excitations at other frequencies are induced, which obviously leads to distortions since these contributions do not perfectly fit into the 10s time window with an integer number of cycles. However, for the identification of the effective damping, only the mean of the DFT-amplitude for 1.8 Hz is used, and the above distortions play only a minor role.
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bridge vibration DFT 1.8 Hz bridge vibration DFT 1.7 Hz bridge vibration DFT 1.9 Hz bridge vibration
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Figure 5. Change of the bridge vibration during walking of test person with a step frequency of 1.7 Hz
The above walking test is repeated for step frequencies of 1.6, 1.9, 2.0 and 2.1 Hz. The final results in terms of changes in the effective damping are shown in figure 6. Basically, the test person induces more damping while walking than for standing still. The additionally induced damping by the walking person increases from 5% for fstep = 1.6 Hz to 13% for fstep = 1.9 Hz.
Then, the additional damping decreases again to a value of 6% for fstep = 2.1 Hz. It is worth mentioning that the model for a passive person in [1] can be used as a conservative lower bound for considering the damping induced by pedestrians, since the damping effect induced by an active person is larger than the damping induced by a passive person.
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empty passive w16 w17 w18 w19 w20 w210.90
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Figure 6. Change of the effective damping by a single person
5 INFLUENCE OF A CROWD OF PEDESTRIANS
Basically, the influence of groups or even crowds of pedestrians on the dynamic characteristics of the bridge have to be understood as a random process, since the step frequencies of the individual will be different. Additionally, the step lengths and therefore the walking speeds and the crossing times differ from person to person. Therefore, even for a stationary pedestrian flow, the actual number of persons on the bridge will show some scatter. Simulations allow studying these effects. In the following, it is assumed that both the step frequency and the step length follow a normal distribution with fs, mean = 1.9 Hz and ls, mean = 0.82 m. The corresponding variation coefficients are 6% and 7%. The inter-arrival times are assumed to follow an exponential distribution with the characteristic parameter arrival rate in number of persons per second. Figure 7 shows the probability density of the random number of persons on the longer field for an arrival rate of 0.5 persons per second. The scatter ranges from 3 to 29 persons. In 90% of all cases, the random number of persons on the longer field is between 8 and 20. Therefore, both the natural frequency and the damping capacity of the coupled system structure-users will be random.
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Figure 7. Example for the randomness in the number of persons on the bridge for a stationary pedestrian flow
The question arises, how transient dynamic parameters can be identified. In this study the approach is based on moving DFT-amplitudes over a time window of 40 s for frequencies from 1.4 to 2.2 Hz in 0.05 Hz steps. For each time step, the
four amplitudes around the largest amplitude are used to identify the natural frequency and the damping ratio of an equivalent single degree of freedom system, supposing that the amplitudes to both sides of the maximum are monotonically decreasing. The dynamic amplification function is given as:
222 D2-1
1 )V(
(1)
η – frequency ratio f / f0
f –excitation frequency
f0 – natural frequency
D – damping ratio
The fitting procedure is based on a least square fit, demanding that the largest amplitude is exactly on the theoretical curve. Thus, only the natural frequency and the damping ratio have to be varied. The basic strategy is illustrated in figure 8.
The above estimation method is evaluated based on simulations using the identified load model in [5]. The structural behavior of the analyzed bridge under random pedestrian flow corresponds to the OLGA-bridge, i.e. the natural frequency is 1.8 Hz and the damping ratio is 0.005. The hit rate for the estimation of the natural frequency with an accuracy of ±0.01 Hz is 94.1%, i.e. the method is able to detect a change in the natural frequency. Therefore, it seems to be justified to analyze the randomness in the change of the natural frequency as well. The refinement in the frequency range is 0.01 Hz since wrong entries are only to be expected with a probability of 6%. For the estimation of the damping ratio, the hit rate is, however, considerable smaller. For a range of ±40% of the true damping ratio the successful estimation rate is only 60%. The hit ratio increases, if the fitting is only performed for large amplitudes of the DFTs.
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Figure 8. Basic strategy for the identification of the natural frequency and the damping ratio based on fitting of the
dynamic amplification function to the largest DFT-amplitude and four neighboring values
For a first practical application, bridge measurements during
a techno festival in the nearby OLGA park are used. The accelerations of the bridge are monitored at several positions.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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Additionally, the number of pedestrians is counted for each 10 minutes. Altogether, 17 runs are available [6]. The arrival rate ranges from 0.2 to 0.6 persons per seconds.
Figure 9 shows the probability densities of the identified natural frequency and the identified damping ratio for three 10
minute windows with different arrival rates. While the natural frequency drops with increasing arrival rate, the damping ratio increases. The probability distribution of the effective damping becomes broader for increasing arrival rate.
1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.810
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Figure 9. Probability densities of the natural frequency and the damping ratio for the coupled system structure-user
or the OLGA-bridge under different random flow intensities [persons / s]
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The results of all 17 runs are shown in figure 10 in terms of the mean value of the identified natural frequency and the effective damping ratio. With increasing arrival rate in persons per second the mean value of the natural frequency shows a clearly decreasing trend. However, the effects remain small with an average drop of only 1.2 % for the largest observed arrival rate. The effective damping ratio shows a clear increasing trend. Compared to the empty structure, the effective damping under the influence of a random pedestrian stream increases by a factor of 2 to 2.6.
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Figure 10. Influence of the arrival rate on the mean values of the natural frequency and the effective damping ratio for the
OLGA-bridge at Oberhausen
6 SUMMARY AND CONCLUSIONS
The presence of human occupants may considerably change the dynamic behavior of structures. For passive persons, a probabilistic model exists, which describes the human body as a two-degree of freedom system. The corresponding basic dynamic characteristics of the human body are assumed to be random. For an existing bridge with a natural frequency of 1.8 Hz and a mass of 33 tons, the change in the damping ratio is predicted in terms of the non-exceedance probability of the increase of the effective damping, considering a random person standing in the node of the dominant vibration mode. These results are used as anchor points for a field experiment which aims in the identification of the damping induced by a single pedestrian. The field experiment uses a known background excitation which is induced by a shaker. The structural response is observed for the empty structure, for the case of a person standing in the node of the dominant
vibration mode, and for the case of a walking person with varying step frequencies. Since the influence of the person on the natural frequency is negligibly small, the change of the DFT-amplitude for 1.8 Hz can be taken directly as the change of the effective damping. The paper shows that
the refined analysis-method allows identifying very small changes in the damping ratio, e.g. a change of only 3% induced by a passive person
the influence of a walking person on the effective damping ratio is larger than that of a passive person
the increase of the effective damping depends on the step frequency
Similar to the situation for passive persons, the effect of a walking person has to be understood as a random process. Therefore, effective natural frequencies and effective damping ratios will vary over time if the bridge vibration is excited by a random pedestrian flow. This is shown on the example of a mass-event based on 17 individual 10-minute runs. The mean values of the natural frequency and the effect damping show consistent trends; however, the analysis of the dynamic characteristics within the 10-minute time windows shows larger scatter.
Although for the chosen example bridge the observed degree of influence on the damping ratio is small for a single person, the increase becomes substantial when larger groups cross the bridge. Further research is required to exploit these beneficial effects of the pedestrians.
ACKNOWLEDGMENTS
Part of this work has been sponsored by the German Research Foundation under the contract number KA675/13-1 and 13-2. This support is gratefully acknowledged.
REFERENCES
[1] Agu, E. & Kasperski, M. (2011) Influence of crowds on the dynamic behaviour of floors and stand structures, Journal of Sound and Vibration, Vol. 330, No. 3, pp 431-444
[2] Dougill, J.W., Wright, J.R., Parkhouse, J.G. & Harrison, R.E. (2006) Human structure interaction during rhythmic bobbing, The Structural Engineer, 84(22), pp 32-39
[3] HiVoss - Human Induced Vibration of Steel Structures Pedestrian Bridges: Guideline and Background http://www.stb.rwth-aachen.de/projekte/2007/HIVOSS/docs/ Footbridge_Guidelines_EN03.pdf http://www.stb.rwth-aachen.de/projekte/2007/HIVOSS/docs/ Footbridge_Background_EN02.pdf
[4] SETRA – Service d’études technique des routes et autoroutes (2006) Technical guide footbridges http://www.setra.equipement.gouv.fr/IMG/pdf/US_0644A_Footbridges.pdf
[5] Sahnaci, C. (2014) Menscheninduzierte Einwirkungen auf Tragwerke infolge der Lokomotionsformen Gehen und Rennen: Analyse und Modellierung, Dissertation Ruhr-Universität Bochum, Department of Civil and Environmental Engineering Sciences http://www-brs.ub.ruhr-uni-bochum.de/netahtml/HSS/Diss/ SahnaciCeyhun/
[6] Kasperski, M. (2006 ) Vibration serviceability for pedestrian bridges Structures and Buildings, 159, Issue SB5, pp 273-282
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