ABSTRACT: The evaluation of the serviceability of pedestrian structures requires knowledge on the induced loads and appropriate approaches for describing the pedestrian stream. Basically, both aspects are considered in recent standards and technical guidelines. However, the presented idealized approaches cannot be considered to describe realistic load scenarios and may lead to misinterpretations in regard to serviceability aspects. Therefore, the paper presents a refined load model which is based on experiments with a long walkway. Furthermore, the quality of the Poisson approach for describing the arrival rates in pedestrian streams is discussed. Additionally, the estimation of maximum vibration amplitudes induced by N pedestrians based on the square root of N is evaluated.

KEY WORDS: Human induced vibrations; Pedestrian structures; Intra-individual variability; Simulation; Pedestrian streams.

1 INTRODUCTION As a consequence of sophisticated calculation methods and the introduction of new construction materials, the number of lightweight and slender pedestrian structures has rapidly increased in recent years. However, this development coincides with an increased susceptibility to vibrations. Thus, especially in regard to walking induced vibrations the evaluation of serviceability has gained high importance. One key issue which has to be addressed in this context deals with the realistic description of the induced loads of the single person. Commonly, the loads due to human locomotion are represented by a sinusoidal load time history or the sum of multiple sinusoidal traces [e.g. 1,2,3,4,5]. This idealized approach implies perfect rhythmical walking behaviour with identical locomotion and load parameters for the left and the right leg. Thus, the frequency content of the loads is strictly limited to the integer multiples of the basic step frequency. However, imperfections in natural walking may excite natural frequencies which are clearly outside the frequency content of the idealized load model. A first comprehensive study with the aim of identifying neglected excitation mechanisms has been performed by Kasperski and Sahnaci, using a short walkway [6]. Especially the differences between the step frequencies of the left and right foot lead to sub- and intermediate harmonics. Assuming that there are additionally differences between consecutive double steps, the analysis of the complete frequency content requires a longer sequence of consecutive steps. In order to identify the randomness and the imperfections in the load and the basic walking parameters, therefore, this study is based on experiments with a 28 m walkway which has been constructed at the Ruhr-University Bochum [7]. Section 2 of this paper deals with the setup and the basic layout of the extensive experiments. The analysis of the experimental data and the identification of the different load phases within the walking pattern are presented in section 3. In section 4 a novel approach for modelling the loads of natural walking on a per step basis is presented. The

evaluation of the quality is based on structural response calculations with both the monitored and the modelled load traces. Section 5 deals with a simulation strategy which aims to generate a random step sequence of any length and which considers the individual scatter in the basic walking parameters and the load parameters. Finally, section 6 is about pedestrian streams and the evaluation of the Poisson approach for modelling the arrival rates. Additionally, the quality of the N1/2 - approach for estimating structural responses due to streams composed of N pedestrians is discussed.

2 EXPERIMENTAL SETUP AND BASIC TEST PROCEDURE For the analysis of the randomness within the walking pattern of an individual, i.e. the intra-individual differences, it is required to observe a longer sequence of consecutive steps. The experimental setup which is used in this study is a 28 m walkway which is made up of three sections: a lead-in section, a measuring section and a lead-out section (figure 1).

Figure 1. 28 m walkway.

The measuring section is made up of 4 frame elements each with a length of 4 m. Each element is mounted on 4 vertically oriented unidirectional force cells with a maximum loading capacity of 5 kN. The cells are mounted at the relative positions x/L = 2/9, which mark the optimum of a supported beam in regard to the natural frequency. The sampling frequency for measuring the loads is set to 2400 Hz. The natural frequencies of the frame elements are beyond 40 Hz.

Prediction of the vibrations of pedestrian structures under random pedestrian streams

Ceyhun Sahnaci1, Michael Kasperski2

1,2Department of Civil and Environmental Engineering Sciences, Research Team EKIB, Ruhr-University Bochum, 44780 Bochum, Germany

email: Ceyhun.Sahnaci@rub.de, Michael.Kasperski@rub.de

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

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However, to avoid any distortions from additional resonance effects, the measured time series are digitally low-pass filtered with a cut-off frequency of 25 Hz. The additional load components in both horizontal directions are mechanically separated by a sophisticated arrangement of force cells, steering arms and hinges which connect the upper to the lower frame.

The lead-in and lead-out sections are required to assure stationary and unbiased locomotion conditions while crossing the walkway. The lead-in section is required to prevent that the subject is still in the acceleration phase when entering the measuring section. The lead-out section assures that the test person initiates the deceleration phase only after leaving the measuring section. Both sections require a sufficient length which has been chosen with 4 m for the lead-in and 8 m for the lead-out section. The width of the walkway is 1.50 m.

The experiments have been performed at the halls of the department of mechanical engineering at the Ruhr-University Bochum. Altogether 227 subjects (38 female and 189 male subjects) have participated in the tests. The basic test procedure starts with the determination of the subject’s bodyweight on the first element of the active measuring section. The bodyweight is later used to normalize the measured forces. Then, the subject is asked to go back to the top-end of the lead-in section and to cross the walkway two times by walking and two times by running with a deliberately chosen speed. Each time the end of the lead-out section is reached, the subject is asked to leave the walkway at the bottom-end of the lead-out section and return to the top-end of the lead-in section. The present paper focuses only on the walking tests.

3 ANALYSIS OF THE EXPERIMENTAL DATA A typical vertical load trace measured from a one-way crossing of an individual is shown in figure 2. The locomotion pattern for walking is characterized by alternating double stance phases (dsp) and single stance phases (ssp) (figure 3). The double stance phases are usually indicated by larger peaks. The single stance phase can be found between two adjacent double stance phases and can basically be characterized by two local maxima and one local minimum. The single step is defined to start with the beginning of a double stance phase and to end with the end of the adjacent single stance phase.

Figure 2. Measured load trace for walking.

Figure 3. Identification of the individual phases.

Occasionally, when entering or leaving the measuring section it can be observed that the test person places the foot on the gap between the measuring section and the lead-in or the lead-out section, respectively. Correspondingly, this scenario leads to a bias in the load time histories of the first and the last step. Therefore, for the further analysis only steps are considered which are known to be definitely on the measuring section. The respective load time history is labeled as ‘analyzed range’ in figure 2. The dashed lines mark the first and the last double stance phase of the load pattern leading in this example to a number of 17 clean steps. The total number of observed individual steps is larger than 8000. From the measurements for each individual step the following parameters are obtained:

• step frequency

• step length

• step width

• duration of the double stance and single stance phases

• normalized harmonic load amplitudes

• corresponding phases The step length and the step width are determined from the position of the effective load, which is obtained from equilibrium conditions of the measured forces. Then, the longitudinal distance between the heel contacts for the left and right foot leads to the step length. The step width is obtained from the lateral distance of the centered loads.

Due to anatomic differences between males and females, the respective walking parameters are usually different. Therefore, separate statistics have to be performed. In figure 4, the walking parameters step frequency, step length and step width are plotted in normal probability paper. In several studies it is stated, that in one and the same situation the larger step frequencies can be observed for women if compared to men. Correspondingly, the mean values from the experiments are 1.99 Hz for female subjects and 1.90 Hz for male subjects with a coefficient of variation (cov) of 0.061 and 0.064, respectively. However, the larger step lengths are observed for male persons due to on average larger body heights. The mean

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values for males and females are obtained with 0.81 m and 0.76 m. The corresponding cov-values are 0.089 and 0.105. The largest relative differences are observed for the step width where males have the wider gait. The respective mean value is 0.15 m with a variation coefficient of 0.34 m while the mean value of the step width for females is 0.10 m with a variation coefficient of 0.45. The analysis of the parameters leads to the basic conclusion that a respective load model for walking has to consider gender specific differences.

Figure 4. Statistical analysis of the basic walking parameters.

4 DEVELOPMENT OF A REFINED LOAD MODEL Contrary to the idealized load model where basically the load pattern of a single step is perfectly repeated, the following section introduces a refined model working on a per-step basis. However, due to the complexity of the walking pattern the refined approach requires a higher density of load and time information. In a first analysis step the single step load traces are extracted from the measured sequences (figure 5).

Figure 5. Characteristic points used for the refined model.

The parameters of the double stance phase dspstart, dspend and the maximum dspmax can directly be determined from the measured trace. However, during the single stance phase (ssp) usually muscular tremble effects occur leading to slight disturbances in the load trace. In a first step this load range, therefore, is smoothed by fitting a polynomial of 8th degree by means of method of least-squares. Then, the local extremes sspmax1, sspmin and sspmax2 and the inflection points ip1 and ip2 can be determined. Finally, the end of the single stance phase can be identified with the start of the next double stance phase. Thus, for each individual step 9 load and time parameters are obtained. Based on these characteristic points

the corresponding load shape can be reproduced by using triggered spline functions which consider the basic boundary conditions for the local extremes and the transition points. On the example of a sequence of steps in figure 6 it is shown that the modeled trace based on the spline fitting method is nearly in perfect agreement with the true load time history.

Figure 6. Triggered spline fitting for consecutive steps.

The quality of the model is evaluated based on structural response calculations by means of the Newmark integration method for both the true load sequence and the spline model. Thereby, for varying natural frequencies from 0.5 to 20 Hz the maximum accelerations of a simple beam structure are determined. The length of the structure is chosen with 16 m which corresponds to the length of the measuring section of the walkway. The damping capacity is 1% of critical damping and the effective mass of the first vibration mode is 10 tons. From the calculations for each of the 2 times 227 load time histories the maximum accelerations of the structure are obtained. In figure 7 the non-exceedance probabilities of the respective results for four chosen frequency positions according to multiples of the central step frequency fs,c are plotted in Gumbel probability paper. The central step frequency corresponds to the mean value of the observed step frequencies. The traces of the non-exceedance probabilities for the resonance case central step frequency equals natural frequency (i.e. f0 = fs,c) are in good agreement. Thus, both modeling methods are appropriate to identify resonance effects with the first harmonic. However, the idealized model leads to underestimations in the maximum accelerations for the higher frequency positions. The degree of underestimation is obtained from the ratio of the accelerations amax,true / amax,model and remains small for resonance effects with the second harmonic i.e. f0 = 2·fs,c, but becomes considerable for resonance effects with the first and the second intermediate harmonic, i.e. for f0 = 1.5·fs,c and f0 = 2.5·fs,c, respectively. The traces of the non-exceedance probabilities obtained from the spline model in chapter 4, however, are nearly in perfect agreement with the traces obtained from the true loads.

In figure 8, the contour lines of the acceleration ratios obtained from the spline model are summarized for the range of natural frequencies up to 20 Hz. The color-change from blue to red marks the increasing degree of underestimation. The plot results in a large, blue-colored area up to a range of natural frequencies of approximately 15 Hz where the model leads to only slight over- or underestimations in the accelerations with ratio values around 1.0. Correspondingly, the amplitude and frequency content of the true loads for both

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the integer and intermediate harmonic range is completely identified within this range, and thus, the model can even be applied to high frequency floors. The respective plot for the idealized model in figure 9 reveals that considerable

underestimations for large ranges of non-exceedance probabilities are not only obtained in case of resonance effects with the intermediate harmonics but also for higher integer harmonics.

Figure 7. Maximum structural accelerations obtained from the true loads and the modelled loads.

Figure 8. Contour plot of the relations of the maximum accelerations obtained by the true loads and the spline model.

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Figure 9. Contour plot of the relations of the maximum accelerations obtained by the true loads and the idealized model.

5 SIMULATION OF A RANDOM SEQUENCE OF STEPS In order to generate a random load sequence for consecutive steps of any length, in the following section a simulation strategy is presented which considers the intraindividual variability of the load and walking parameters. The simulations are based on the introduced spline model. Due to the basic connection between the dynamical behavior of a structure and the step frequency, it seems to be reasonable to consider this walking parameter as key parameter for the simulations.

In a first step it is required to develop probabilistic models for the mean values and the standard deviations of the basic walking parameters and the corresponding deviations between left and right. The respective results for the step frequency of the male subjects are shown in figure 10, where the non-exceedance probabilities are plotted in normal probability paper. The traces can approximately be described by a straight line leading to the conclusion that the normal distribution can be assumed to be an appropriate model. The same conclusion is obtained for the walking parameter step length and the respective standard deviations.

Figure 10. Non-exceedance probabilities of the deviations between left and right Δfsm and the mean values of the step

frequencies fsm.

In a similar manner probabilistic models are also developed for the 9 amplitude and time parameters describing the shape of the single step. However, the major task is the description of the correlation structure of the amplitude and time information including the dependencies to the key parameter step frequency. It has to be noted, that the relative time information marking the beginning and the end of the step remain constant with values of 0 and 1 and therefore are not considered in the correlation analysis. Thus, the correlation structure can be described by introducing a 33x33 matrix which is composed of the respective mean values and the standard deviations of the shape parameters and the mean values of the step frequencies.

On the example of the correlations between the shape amplitudes 1 to 9 and the step frequency in figure 11 it can be shown that significant correlations are obtained for the extremes during the single stance phase (amplitudes 4,5,6 or sspmax1, sspmin, sspmax2, respectively). However, for the female subjects the maximum during the double stance phase dspmax (amplitude 2) shows no dependency with the step frequency whereas for the male subjects a strong correlation is obtained.

Figure 11. Correlation coefficients between shape amplitudes and the step frequency.

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The contour line plot in figure 12 reveals the inherent dependencies of the shape amplitudes 1 to 9 on the example of the male subjects. For the ranges of the double stance and single stance phases the correlation process shows a clearly defined structure which has to be covered by the simulation. The range which is marked as swing phase corresponds to the load phase between the two inflection points ip1 and ip2. The plot reveals that there is a strong negative correlation between the swing phase and the maxima during the single stance phase sspmax1 and sspmax2 leading to red colored areas. The respective correlation coefficients are between -0.6 and -0.8. Negative correlations of same order can also be found between the swing phase and the maximum during the double stance phase. However, strong positive correlations with correlation coefficients between 0.6 and 0.8 can be found between dspmax and sspmax2.

Figure 12. Inherent correlation structure of the shape amplitudes.

The final simulation routine first generates the gender of a person in accordance to the preset ratio of male to female persons. Next, the gender depending body weight and the gender depending basic walking parameters are generated in terms of the mean values and the standard deviations of the step frequencies and step lengths for the left and right foot. In the third step, the basic load parameters are generated in terms of the mean values and standard deviations of the amplitudes and times for the nine control points, again separately for left and right. Note that the ninth control point of step n equals the first control point of step n+1. After step three, all required information is available to generate a sequence of random steps of any length. A simulated load time history for a male person is shown in figure 13.

If the load time history is going to be used to calculate the excitations of e.g. a pedestrian structure, the generation ends as soon as the crossing time of the actual person is exceeded. Since the duration of the double stance phase is one of the generated information, the routine allows modelling the position of the effective load on the basis of the real walking behaviour, i.e. during the single stance phase the position remains constant, and the forward motion takes place during the two foot stance phase. It is important to note, that usually

this detail is not important, i.e. it is sufficient to assume for the effective position of the load a constant walking speed. However, there are some natural frequencies where the jerky changes of the position of the load lead to some extra excitations. The generated load time history can be weighted in accordance to a desired mode shape, e.g. a half sine wave for a simple beam structure.

Figure 13. Simulated load time history of the vertical loads.

6 SIMULATION OF A RANDOM PEDESTRIAN FLOW For modelling random pedestrian streams, usually a Poisson distribution is assumed for the arrival rate, which is the number of persons per second entering the pedestrian structure. Then, the inter-arrival times follow an exponential distribution. However, observing real random pedestrian streams leads to the conclusion that this approach is less appropriate (figure 14). The main reason for the mismatch is that each individual is assumed to be independent, and thus, pairs or smaller groups cannot be modelled. This also leads to an underestimation of the total number of persons on the structure. Therefore, it is recommended to additionally consider groups of different sizes in a separate Poisson approach.

Figure 14. Stationary pedestrian stream during an observation period of 1h.

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Furthermore, it cannot always be assured, that the process is stationary. This especially has to be considered when pedestrian structures are situated in the environment of e.g. bus stops or train stations. The typical change of the arrival rate over time of a pedestrian bridge in such a situation is shown in figure 15. In this example the bus arrives each 30 minutes which leads to a sudden increase of the arrival rate for a short period of time. Therefore, for the modelling process a variable arrival rate has to be applied considering higher values when the vehicle arrives.

Figure 15. Instationary pedestrian stream during an observation period of 1h.

In standards like ISO 10137 [3] or guidelines like Hivoss [4] the influence of N pedestrians on the maximum vibration amplitudes of a pedestrian bridge is usually considered with an N1/2 - approach. However, in the context of comfort and safety criteria for the active user it is more reasonable to focus on the maximum experienced vibrations of the individual instead of identifying maximum structural responses. Therefore, based on the simulation strategy in chapter 5, in this study structural response calculations for a random pedestrian stream have been performed with determining the maximum experienced acceleration amplitude of each individual. The structure corresponds to a simple beam pedestrian bridge with a span of 65 m and an effective mass of 33 tons. The damping capacity corresponds to 1 % of critical damping. In a first approach the arrival rates are described by means of the Poisson distribution, i.e. a stationary flow is assumed and pairs or groups are not considered. The average ratio of female to male persons is set to 50%. However, the gender of a random person is a basic variable. The step frequencies are assumed to follow the normal distribution with a central value of 1.9 Hz. In order to analyse resonance effects for the first or second harmonic or the first inter-harmonic the calculations are performed for three natural frequencies 1.9 Hz, 2.85 Hz and 3.90 Hz.

The focus in simulations usually lies on the identification of upper limits in terms of fractile values or non-exceedance probabilities, respectively. However, without giving further information on the fundamental reference period i.e. the duration of observation the specification of fractiles can be considered as meaningless. On the one hand those fractiles cannot be compared with results from other studies, on the other hand it is not possible to translate respective results to other reference periods of interest, e.g. one year or lifetime. The connection between probabilities based on different reference periods tref1 and tref2 can be described with equation (1).

1 (1)

For the analysis of the structural vibrations the reference period is set to 30 minutes. For each arrival rate 5000 consecutive hours are simulated. However, the main focus of this study lies on the experienced accelerations which have as implicit reference period simply the crossing time of each individual. The respective results for the mean values and the 95% and 99.9% fractile values of the maximum experienced accelerations are shown in figure 16. As can be seen, the N1/2

– approach only leads to reasonable results for the mean values of the maximum experienced accelerations in the case of central step frequency equals natural frequency of the structure (f0 = 1.9 Hz). For higher fractile values and for the remaining resonance scenarios the square root of N can be considered as a conservative approach assuming that the amplitude for one person is estimated correctly.

Figure 16. Influence of number of persons N on maximum experienced accelerations.

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7 SUMMARY AND CONCLUSIONS In recent years the number of lightweight and slender pedestrian structures has rapidly increased leading to an increased susceptibility to vibrations. Current regulations demand the evaluation of serviceability, however, scientific sound approaches for load models and serviceability criteria are not available. The main objective of this study is to develop a realistic load for the locomotion form walking based on experiments. The basic experimental setup is made up of a 28 m instrumented walkway which allows to monitor the vertical loads of a sufficiently long sequence of steps. Especially, with this setup intermediate harmonic load contributions can be identified which are induced due to intraindividual deviations in the process of natural walking. The analysis of the monitored load traces leads to the basic walking parameters and the load parameters on a per step basis. For both parameter sets significant differences between female and male persons are obtained. Therefore, the development of the load model is based on gender specific approaches.

A sophisticated modelling approach is presented which is based on 9 characteristic load and time information per step and which allows to reproduce the monitored load traces of consecutive steps. The evaluation of the load model is performed by means of structural response calculations leading to the conclusion that the model meets the demand of preciseness and accuracy for the prediction of vibration amplitudes for a very wide range of natural frequencies.

In a next step a simulation strategy is presented which enables to generate a gender dependent random load sequence of consecutive steps. The strategy requires the appropriate probabilistic description of the mean values and the standard deviations of the basic walking parameters and the corresponding deviations between left and right. Especially, the correlation structure of the amplitude and time information is determined including the dependencies to the key parameter step frequency.

Finally, it is shown that the Poisson approach for modelling the arrival rates of a pedestrian stream usually leads to inappropriate conclusions in regard to the total number of persons on the structure. Pairs or groups cannot be modelled and therefore have to be considered in a separate Poisson approach. Instationary pedestrian streams can be considered with variable arrival rates over time.

In recent standards and guidelines it is usually assumed that the influence of N pedestrians on the maximum structural vibrations follows the square root of N. However, in the context of comfort and safety criteria for the user it is more reasonable to focus on the individually experienced accelerations. By means of simulations it is shown that the estimation of the accelerations with an N1/2 – approach is only appropriate for the mean values of the maximum experienced accelerations in case of resonance with the first harmonic. However, in the cases resonance with the first inter-harmonic or the second harmonic this approach leads to conservative results.

ACKNOWLEDGMENTS Part of this work was sponsored by the German Research Foundation under the contract numbers KA675/13-1 and KA675/13-2. This support is gratefully acknowledged. The authors also would like to express their special thanks to the many colleagues from 'foreign' departments who supported the above research efforts.

REFERENCES [1] British Standards Institution BSI. BS5400 - Steel, Concrete and

Composite Bridges - Specification for loads, Part 2 (BD37/01, Appendix B), United Kingdom, 2001.

[2] French association of civil engineering, Assessment of vibrational behaviour of footbridges under pedestrian loading. SETRA - Service d'études techniques des routes et autoroutes, Technical Guide – Footbridges, 2006.

[3] International Organization for Standardization ISO, ISO 10137 - Basis for design of structures - Serviceability of buildings and pedestrian structures against vibration, 2007.

[4] Research Fund for Coal and Steel, Hivoss, Design of Footbridges, 2008.

[5] VDI Guidelines 2038, Serviceability of structures under dynamic loads – Methods of analysis and evaluation in structural dynamics, Germany, 2012.

[6] C. Sahnaci and M. Kasperski, Random loads induced by walking, Proceedings of sixth European conference on structural dynamics (EURODYN), Paris, September 2005.

[7] C. Sahnaci, Menscheninduzierte Einwirkungen auf Tragwerke infolge der Lokomotionsformen Gehen und Rennen: Analyse und Modellierung, PhD Thesis at Ruhr-University Bochum (in German), Department of Civil and Environmental Engineering, Germany, 2013.

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