Proposal and calibration of a biodynamic model of human-structure interaction by the resolution of the inverse dynamic problem: application to pedestrian bridges

 THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY WITH INTERNATIONAL MENTION FOR THE UNIVERSITY OF SEVILLE Proposal and Calibration of a Biodynamic Model of Human-Structure Interaction by the Resolution of the Inverse Dynamic Problem. Application to Pedestrian Bridges. Javier Fernando Jiménez Alonso Department of Continuum Mechanics and Structural Analysis School of Engineering UNIVERSITY OF SEVILLE Seville, Spain 2015 ii  iii  Proposal and calibration of a biodynamic model of human-structure interaction by the resolution of the inverse dynamic problem. Application to pedestrian bridges. PhD’s Thesis in the Dynamic of Structures and Earthquake Engineering program. Javier Fernando Jiménez Alonso Advisor: Prof. Dr. Andrés Sáez Pérez. Department of Continuum Mechanics and Structural Analysis School of Engineering University of Seville Abstract In this thesis a biomechanical crowd-structure interaction model is proposed and further implemented in order to adequately estimate the energy exchange between pedestrians and footbridge. The proposed model focuses on both the vibrations in the vertical and lateral directions and it allows to take into account the change of the modal properties of the structure due to the presence of pedestrians, thus improving the numerical estimation of the response of the structure under pedestrian flows. It further permits to analyze in more detail the lateral lock-in phenomenon. The model involves two sub-models, namely (i) a pedestrianstructure interaction sub-model plus (ii) a crowd sub-model. The first sub-model follows from a modal projection of a two degree of freedom system that simulates the behavior of each pedestrian, on the vibration modes of the structure. The parameters of this model are estimated from the accelerations recorded on a real footbridge by implementing an inverse dynamic approach. For the second submodel, the crowd behavior is simulated via a multi-agent method. The performance of the resulting overall model is assessed by correlating the experimental and numerical dynamic behavior of two real footbridges. In particular two phenomena are analyzed in detailed: (i) the change in the first vertical natural frequency of a real footbridge induced by the pedestrian-structure interaction and (ii) the occurrence of the lateral lock-in phenomenon due to the pedestrian action. The proposed model leads to numerical results that exhibit good agreement with the obtained experimental values. Therefore, it becomes a valuable tool to account for the change on the modal properties of a footbridge induced by the crowd-structure interaction phenomenon. The consideration of this factor allows estimating more accurately the dynamic response of the footbridge under the pedestrian action, analyzing in more detailed the occurrence of the lateral lock-in phenomenon or improving the efficiency in the design of passive and active dampers if their installation was necessary to guarantee an adequate comfort level on the footbridge. Keywords: simplified biomechanical model, human-structure interaction, crowd dynamics, change of natural frequencies, operational modal analysis, model updating, parameter identification, lateral lock-in phenomenon iv  v  PREFACE This Thesis has been carried out at the Department of Continuum Mechanics and Structural Analysis at the University of Seville. The work has been supervised by Full Professor Dr. Andrés Sáez Perez. I always owe my deepest gratitude to Andrés for encouraging and supporting me constantly during the development of this work. This research would not have been possible without his infinite patient and helpful guidance for the organization of the papers. Thank him for his time, effort and friendship. I feel really lucky for having had the opportunity to develop this work under his tutelage. Additionally, I would like to express some lines of gratitude to those who have contributed to the development of the research carried out by the author of this Thesis. To Prof. Alvaro Cunha, who provided me a nice workplace during my research stay at the Laboratory of Vibrations and Structural Monitoring (ViBest) of the University of Porto (Portugal), allowing me to improve my knowledge of the operational modal analysis methodology and perform several experimental tests of great importance for the development of this work. The results I collected during my research stay constitute a vital part of my Thesis and would not have been possible without the support of Associate Professor Elsa Caetano and Assistant Professor Filipe Magalhães. To Prof. Alexander Pavic, for receiving me into his group of the Vibration Engineering Section of the University of Exeter (U.K.) during my second research stay allowing me to introduce myself in the interesting field of the control of civil engineering structures. To my parents, Antonio and Maria del Carmen, who taught me the value of the education and who made a remarkable effort for us, their children, so that we could have all the opportunities that they did not have. To my colleagues (past and present) at the Department of Building Structures of the University of Seville and at the Bridge Engineering Firm, IDES, for their constant support. This thesis is based on scientific papers which have already been accepted for publication in relevant scientific journals or presented at international conference with peer-review. Finally, two additional papers, currently under review, have been included. Last but not least, I am very grateful for the loving support of my family, in particular my brilliant and comprehensive wife, Patricia, and our two lovely daughters, Claudia and Lorena. I hope that one day I could compensate the time that I stool them for the development of this work. To Maribel for her continuous support, for me and my family, during these difficult years. vi  My gratitude goes also to other relatives and friends that have helped me to overtake successfully all the difficulties for the achievement of this Thesis. Seville, October 2015 Javier Fernando Jiménez Alonso ACKNOWLEDGEMENTS This work was partially funded by the Spanish Ministry for Science under research project DPI2014-53947-R. vii  THESIS This Thesis consists of an extended summary and the following appended papers: Paper A J.F. Jiménez-Alonso and A. Sáez A direct-pedestrian structure interaction model to characterize the human induced vibrations on slender footbridges Informes de la Construcción, Vol. 66 (Extra 1). m007 Paper B J.F. Jiménez-Alonso, A. Sáez, E. Caetano, F. Magalhães Vertical crowd–structure interaction model to analyze the change of the modal properties of a footbridge Journal of Bridge Engineering. ASCE (in press) Paper C J.F. Jiménez-Alonso and A. Sáez Model updating for the selection of the retrofit method of an ancient bridge (Almeria, Spain). Structural Engineering International. IABSE (in press). Paper D J.F. Jiménez-Alonso and A. Sáez Controlling the human-induced longitudinal vibrations of a Nielsen-truss footbridge via the modification of its natural frequencies Under review Paper E J.F. Jiménez-Alonso, A. Sáez, E. Caetano and A. Cunha Lateral crowd-structure interaction model to analyze the lateral lock-in phenomenon on a real footbridge Under review Paper F J.F. Jiménez-Alonso, E. Caetano and A. Cunha Dynamic testing of Carpinteira footbridge at Colvihã (Portugal) 5th International Operational Modal Analysis Conference (Guimarães, Portugal) 13-15 May 2013 Paper G J.F. Jiménez-Alonso and A. Sáez Assessment of the dynamic behavior of Palmas Altas footbridge at Seville (Spain) 37th IABSE Symposium. Madrid (Spain) 3-5 September 2014 The appended papers were prepared in collaboration with co-authors. The author of this Thesis is responsible for the major progress of work in these papers, including the development/deduction of solutions and numerical methods, performing the numerical simulations and experimental tests and writing the main parts of the papers. viii  ix  Table of Contents I. Extended summary ................................................................................... 1 1. Introduction. ............................................................................................ 1 1.1 Motivation. .......................................................................................... 3 1.2 Objectives. ......................................................................................... 3 2. Vibratory problems due to pedestrian flows on footbridges. ............................ 5 2.1. Load models for a single pedestrian. ...................................................... 5 2.2. Load models for crowds. ...................................................................... 8 2.3. Synchronization and lock-in. ............................................................... 12 2.3.1. Models for the simulation of the synchronization and lock-in. ............. 13 2.3.2. Synchronization and vertical lock-in. .............................................. 16 2.3.3. Synchronization and lateral lock-in. ............................................... 17 2.4. The perception of the vibration. .......................................................... 18 2.5. Dynamic properties of the structures under pedestrian action. ................. 19 2.6. The control of the vibratory response. .................................................. 20 2.6.1. Modification of the mass induced by pedestrian action. ..................... 20 2.6.2. Modification of the stiffness induced by pedestrian action. ................. 21 2.6.3. Modification of the damping induced by pedestrian action. ................ 22 3. Proposal of a simplified biomechanical crowd-structure interaction model. ....... 23 3.1. Modelling the pedestrian-structure interaction. ...................................... 24 3.2. Modelling the crowd-behavior. ............................................................ 28 3.3. Crowd-structure interaction. ............................................................... 33 4. Inverse dynamic problem approach. .......................................................... 35 4.1. Inverse dynamic problem: parameter identification in vertical direction. ... 36 4.2. Inverse dynamic problem: parameter identification in lateral direction. ..... 37 5. Experimental estimation of the parameters of the pedestrian-structure interaction model. ...................................................................................... 40 5.1. Description and finite element model of the “laboratory” footbridge: Viana footbridge. ............................................................................................. 40 5.2. Experimental identification of the modal parameters of the “laboratory” footbridge. ............................................................................................. 42 5.3. Model updating of the “laboratory” footbridge. ...................................... 45 5.4. Experimental pedestrian and crowd tests. ............................................. 47 5.5. Establishing a search domain for the parameters of the pedestrian-structure model. ................................................................................................... 49 5.6. Parameter identification of the pedestrian-structure interaction model: vertical direction. .................................................................................... 50 5  2. Vibratory problems due to pedestrian flows on footbridges. In this section a summary of the main aspects of the vibratory problems induced by the crowd-structure interaction is presented. The section includes the different models that have mainly influenced the author for the development of the proposed crowd-structure interaction model. On the other hand, this section constitutes a brief summary of the state of the art about this subject. 2.1. Load models for a single pedestrian. The first models proposed in order to study the effect of a pedestrian crossing a footbridge were based on the assumption that the pedestrian’s action can be approximated by a harmonic force. From this approach arises the proposal of the British standard (BSI, 2006), the single model, that later was adopted by other countries, as for instance, Canada (Ontario, 1995) and Spain (RPM-95, 1995). This model considered that the effect of the passage of a pedestrian on the structure is equivalent to a moving vertical sinusoidal force p F (in N), with a pedestrian step frequency p f (in Hz), a pedestrian step velocity p v (m/s) a   tftF pp  2sin180)(  [1] pp ftv 9.0)(  [2] being t the time variable (sec.). From the end of the last century, several researches have focused their efforts on the characterization of this force more precisely, including additional terms in the Fourier series and considering its effect in the three spatial directions. In the following equations, [3 and 4], and in Table 2 and Figure 2 a summary of the main proposals reported is shown (Setra, 2006, Butz et al., 2007, Racic et al., 2009).          nh i pverisveripverp tifPtF 1 ,,, 2sin1)(  [3]     nh i platislatiplatp tifPtF 1 ,,, sin)(  [4] where verp F,is the vertical periodic force due to walking. latp F, is the lateral periodic force due to walking. 700 p PN is the mean pedestrian’s weight (Butz, et al.,2007). 6  veri,  and lati,  are the Fourier coefficients of the ith harmonic for vertical and lateral force, dynamic load factors (DLFs) s f [Hz] is the step frequency. veri,  and lati,  are phase shift of the ith harmonic. nh is total number of contributing harmonics. p  is the phase shift among pedestrians. From the analysis of the results provided by Table 2, it can be concluded that at least two harmonics are necessary to characterize adequately the vertical pedestrian force while three harmonics are necessary for the lateral direction. On the other hand, the lateral component of the pedestrian force is characterized by frequencies that are the half of the frequencies transmitted in the vertical direction. The relationship between the pedestrian velocity ( p v) and the pedestrian step frequency ( s f) has been studied by different authors. The works reported by Butz et al. (2007), Riccardelli and Pizzimenti (2007), Riccardelli et al. (2007) and Bertram and Ruina (2001) may be highlighted. The ultimate proposal, internationally accepted by the scientific community, is governed by the following equation. 32 35.059.193.2 ppps vvvf  [5] Lately, this relationship will be considered in the crowd-structure interaction model proposed in this work. 7  Table 2. Dynamics Load Factors (DLFs) according to different author for walking action in vertical and lateral direction (Setra, 2006; Butz et al., 2007; Racic et al. 2009). Author Fourier Coef./Phase Commentaries Action-Direction Blanchard et al. (1977) 1,ver=0.257 Walking-Vertical Bachmann & Ammann (1987) 1,ver=0.40-0.50; 2,ver=3,ver=0.10 fs=2.00-4.00 Hz Walking-Vertical Schulze (1980) 1,ver=0.37;2,ver=0.10; 3,ver=0.12;4,ver=0.04; 5 , ve r =0.015; fs=2.00 Hz Walking-Vertical Bachmann et al. (1995) 1,ver=0.40/0.50; 2,ver=3,ver=0.10; 1/2,lat1,lat=3/2,lat=0.10; 2=3=pi/2; fs=2.00-2.40 Hz fs=2.00 Hz fs=2.00 Hz Walking-Vertical Walking-Vertical Walking -Lateral Walking-VerticalLateral Kerr (1998) 1,ver =0.40/0.50; 2,ver =3,ver =0.10; 1 according to the frequency Walking-Vertical Young (2001) 1,ver=0.37 (fp-0.95) ≤0.50 2,ver=0.054+0.0088 fs 3,ver=0.026+0.015 fs 4 , ve r =0.01+0.0204 fs Mean values of Fourier Coef. Walking-Vertical EC5 (2003) 1,ver=0.40;2,ver=0.20 1 , lat=2 , lat=0.10 Walking-Vertical Walking-Lateral SETRA (2006) 1,ver=0.40 2,ver=3,ver=0.04 2,ver=3,ver=pi/2; 1/2,lat=3/2,lat =0.05 1 , lat =2 , lat =0.01 Walking-Vertical Walking-Vertical Walking-Vertical Walking-Lateral Walking-Lateral SYNPEX (2007) 1,ver=0.0115 fs2+0.2803 fs0.2902 1,ver =0.00 [º] 2,ver =0.0669 fs2+0.1067 fs-0.0417 2,ver = -99.76 fs2+478.92 fs - 387.80 [º] 3,ver =0.0247 fs2+0.1149 fs-0.1518 If fs<2.00 Hz 3,ver = -150.88 fs3+819.65 fs2 - 1431.35 fs+811.93 [º] If fs≥2.00 Hz  3,ver = 813.12 fs3-5357.60 fs2 +11726.00 fs -8505.90 [º] 4,ver =-0.0039 fs2+0.0285 fs-0.0082 4 , ve r = -34.19 fs-65.14 [º] Fourier Coef. and phases for mean pedestrian loads Walking-Vertical 8  2.2. Load models for crowds. The main limitation of the above models is that they are not able to predict the response of the footbridge when the structure is subjected to a pedestrian flow, being necessary, therefore, to develop models that account for the behavior of the crowd. The methodology used more frequently in the literature consists in multiplying the response of a single pedestrian by a factor that considers globally the effect of the crowd. Among the different proposed models, it is presented below a summary of the most historically influential, according to the author’s criterion. The first considered model was proposed by Matsumoto et al. (1978) and takes into account as multiplication factor the magnitude p n, being p n the number of pedestrians on the deck at certain instant. This factor establishes, according to a Poisson distribution, the proportion of pedestrians that due to the hazard are moving in phase, avoiding the effect of the rest of individuals. However, this factor was used, without success, in order to predict the response of the T-footbridge (Tokyo) that underwent vibratory problems due to the lateral synchronization of the pedestrians (Zivanovic et al., 2005). Subsequently, the Swiss standards, SIA 260 (2003), presented as novelty the modification of the multiplication factor in function of the pedestrian density. In that sense, it proposes: (i) up to 10 pedestrians, a linear law with several sections and a maximum value of 3; (ii) up to a pedestrian density of 0.30 P/m2 (Persons/m2) it accepts the above proposal and (iii) up to about this point it establishes a parabolic law with a maximum factor of 20. Therefore, the pedestrian density determines a regime change of the pedestrian behavior: while values lower than 0.30 P/m2 allows the free movement of the pedestrians, as the pedestrian density increases their free movement becomes difficult, so that the pedestrians tend to synchronize. The Eurocode (2002) establishes, similarly to the Swiss standards, three load models according to the expected pedestrian density. The first model, DLM1, which is the basis of the rest, is used to characterize the action of a single pedestrian, being its action defined as a moving sinusoidal force with two spatial components: one vertical, of value 280 N and one lateral, with magnitude 70 N. Both the pedestrian step frequency and the pedestrian velocity follow the criterion of the British standard (BSI, 2006). The second model, DLM2, characterizes the action of a group of up to 15 pedestrians. The response of the footbridge under the action of the group is calculated multiplying the response of the model DLM1 by a factor, with a maximum value of 3, which takes into account the probability that a resonance phenomenon occurs on the structure due to the action of the group of pedestrians. The modification of the modal properties of the footbridge due to the pedestrian effect is quantified through the addition, at the point with a maximum modal deflection, of a point load of 800 kg. The last model, DLM3, applicable to scenarios under a continuous pedestrian flow, characterizes the action of the crowd through a harmonic load equivalent to the weight of a pedestrian density 0.60 P/m2 multiplied by two factors in order to account for the possible resonance between the pedestrians and the structure, as well as the transitory character of the load. The first factor, adopts a variable value ranging between 0.05-0.30 and the second factor is equal to 0.75. The equivalent load is only applied in the part of the structure where the modal deformation has the same sign as the load and its effect 9  is unfavorable. The pedestrian load is applied to the same frequency than the model DLM1. Finally, the model assumes that the pedestrian flow produces an increase of the modal mass of the structure of 40 kg/m2. Figure 2. Vertical and Lateral pedestrian walking force according to different authors (Setra, 2006; Butz et al., 2007 and Racic et al., 2009). The French standard, Setra (2006), based on both the research conducted on the Solferino footbridge (Paris, France) and laboratory tests on treadmills, presents a new methodology that has been widely adopted by researchers and designers. The methodology has been accepted equally by the European Research Project, SYNPEX (Butz et al., 2007). The proposed method simulates the effect of the pedestrian flows as an equivalent uniform distributed load applied according the considered vibration mode and whose value is equal to the effect of the group of pedestrians that are synchronized among them. This magnitude is named as equivalent number of pedestrians, 'n. In order to determine the number of pedestrians on the 0.00 200.00 400.00 600.00 800.00 1000.00 1200.00 1400.00 0.00 0.20 0.40 Vertical Load [N] Time [sec.] Blanchard et al. (1977) Bachmann & Ammann (1987) Schulze (1980) Bachmann et al. (1995) Kerr (1998) Young (2001) Eurocode 5 (2003) Setra (2006) Synpex (2007) -200.00 -150.00 -100.00 -50.00 0.00 50.00 100.00 150.00 200.00 0.00 0.20 0.40 0.60 0.80 1.00 Lateral Load [N] Time [sec.] Bachmann et al. (1995) Eurocode 5 (2003) Setra (2006) 10  footbridge, the structure is classified according to the expected traffic level. In Table 3, the four possible traffic types are specified (d=pedestrian density=Persons/m2) which allow generating on the structure the different load scenarios. A comfort level will be associated with each traffic level. The equivalent uniform load applied on the deck of the structure is obtained from the following equation.         ')2cos()( pfoot ntfGtp [6] where G is the component of the pedestrian load ( 280  G N for the vertical walking and 35G N for the lateral walking). foot f is the natural frequency of the structure under consideration. ' p nis the equivalent number of pedestrians according to Table 3.  is the reduction coefficient to take into account the probability that the footfall frequency approaches the natural frequency under consideration (Figure 3). Figure 3. Reduction factor  versus the natural frequency of the structure (Butz et al., 2007). The equivalent number of pedestrian depends on the pedestrian density on the structure, the natural frequency under consideration and the ratio between the structural and the critical damping,  . In Table 3, several proposals for its determination are shown. 0.00 0.25 0.50 0.75 1.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Reduction coefficient  Frequency [Hz] Vertical. 1º Harmonic Vertical. 2º Harmonic Lateral 11  Table 3. Equivalent number of pedestrians, ' p n, according to French code (Setra, 2006). Range of natural frequencies [Hz] Traffic Class d [P/m2] 1.70-2.10 1.00-1.70 2.10-2.60 2.60-5.00 <1.00 >5.00 IV <0.20 --- --- --- --- III 0.50  p n8.10 --- --- --- II 0.80  p n8.10  p n8.10  p n8.10 --- I 1.00 p n85.1 p n85.1 p n85.1 --- Meanwhile, the structural damping ratios for different construction types can been obtained from Table 4 (Setra, 2006 and Butz et al., 2007). Table 4. Damping structural ratios for Service and Ultimate Limit states (Setra, 2006 and Butz et al., 2007). Construction type S.L.S. U.L.S. min  [%] med  [%] max  [%] Reinforced concrete 0.80 1.30 5.00 Prestressed concrete 0.50 1.00 2.00 Composite steel-concrete 0.30 0.60 2.00 Welded steel 0.20 0.40 2.00 Screwed steel 0.20 0.40 4.00 Timber 1.00 1.50 4.00 Stress-ribbon 0.70 1.00 2.00 Elastomers ---- ---- 7.00 Thus, the most advanced international standards (Setra, 2006 and Butz et al., 2007) estimate the modification of the dynamic properties of the footbridge due to the pedestrian flows only by considering the modification of the modal mass of the structure, adding directly the passive mass implied by the pedestrians. The effect of the second harmonic that characterizes the pedestrian step is only considered in footbridges under pedestrian densities larger than 0.80 P/m2. However, these international standards present the following limitations: (i) a simplified estimation of the change of the dynamic properties of the footbridge due to the presence of the pedestrians, (ii) the consideration of the synchronization phenomenon from an experimental relationship obtained from tests on only one real footbridge, (iii) DLFs obtained from laboratory test, (iv) a formulation that does not fit well to the case where several vibration modes of the footbridge are in the range that characterizes the pedestrian-structure interaction, and (v) they do not take into account the effect of the non-synchronized pedestrians. In the present Thesis a crowd-structure interaction will be proposed and calibrated in order to overcome the above limitations. 12  2.3. Synchronization and lock-in. In the context of the pedestrian-structure interaction, the synchronization reflects the tendency of the pedestrians to walk with the same spacing and phase among them, while the lock-in reflects the tendency of the pedestrians in coupling their step with the vibratory movement of the structure. This phenomenon can have a deliberate character (vandalism) or unintentional. This last case, responsible of many of the vibratory problems of footbridges detected during the last years, can be generated by two different mechanisms, according to the scheme shown in Figure 4. Figure 4. Flowchart of unintentional of pedestrian-structure synchronization (Racic et al., 2009). The first synchronization mechanism may occur when the pedestrian density on the footbridge, d, is lower than a critical value, c d (limit density for which the movement of a pedestrian is influenced by the rest of the group), and the value of the amplitude of the deck induced by the pedestrians, u, was upper a limit value, c u, that marks the limit value from the pedestrians tend to synchronize with the motion of the deck. On the other hand, a second mechanism, that originates the synchronization, may occur, under high pedestrian densities, if the mass provided by the pedestrians, M , is upper a critical value, c M, equal to the mass of the pedestrians which inertial force may induce a vibration amplitude, c u. In the following sections, a literature review of the main existing models for the analysis of the lock-in phenomenon in footbridges is presented. The practical application of these studies, adopted by the current standards, is also described. Pedestrian denstiy d [P/m 2 ] d≤d c d>d c u≤u c u>u c M≤M c M>M c Forced Vibration Forced Vibration Synchronization Synchronization 13  2.3.1. Models for the simulation of the synchronization and lock-in. The study of the problem of synchronization between pedestrians and the footbridge, the lock-in phenomenon, has been performed historically independently to the simulation of the behavior of the crowd, or in the best case, an additional checking criterion has been established (Setra, 2006 and Butz et al., 2007). The first reported phenomenon of this type occurred in a German footbridge (1972), during its opening, when a vibration mode of 1.10 Hz was excited by 300-400 pedestrians, as it is described by Bachmann and Ammann (1987) in their book. The comfort level of the footbridge was guaranteed by the addition of several tuned mass dampers without giving additional importance to the phenomenon. Next, a literature review of the most relevant models for the analysis of the phenomenon is presented summarized. One of the first models, proposed by Fujino et al. (1993), was developed from the results of the study of the lateral lock-in phenomenon in the T-footbridge (Tokyo). By the analysis of video images of 2000 pedestrians crossing the structure, they estimated that 20% of the pedestrians were synchronized with the footbridge, with a maximum lateral displacement of the deck of 10 mm and a lateral natural frequency of 0.90 Hz. It was established, for the first time, as cause of the vibratory problem the synchronization between the pedestrians and the structure. From these results, a general calculation rule was presented, establishing a fixed value of the synchronization of 20% (among pedestrian with each other and with the structure) and a value of the mean lateral force generated by a pedestrian of 35 N. The model did not consider neither the increase of the synchronization with the amplitude of the movement of the deck nor the effect of the pedestrians that only was synchronized with each other but not with the deck. This model was unsuccessfully applied for the study of the dynamic behavior of the Millennium footbridge (London) during its design phase. After the vibratory problems detected in it, large scale and laboratory tests were performed in order to calibrate a new proposal for simulating the pedestrian behavior. The pedestrian action, according to Dallard (Dallard et al., 2001), may be approximated as a force that depends on the velocity of the deck and a negative coefficient of pedestrian damping. In this way, the increase of the pedestrians on the footbridge produces a reduction of the global damping of the structure, to such an extent that a dynamic instability state may be reached, which is additionally favored by the complementary synchronization process experienced by the pedestrians. The concept of the equivalent number of pedestrians is introduced as the number of pedestrians that eliminate the damping of the system. The practical application of this model is reflected by the Arup formula. The Millennium footbridge experienced a lateral movement of 50 mm with a natural frequency of 0.80 Hz in the lateral span and a lateral movement of 75 mm with a natural frequency of 1.00 Hz in the central span, synchronizing the movement of 50% of the pedestrians that crossed the structure. The resulting value of the lateral force transmitted by each pedestrian, 30 N, is similar to the value proposed by Fujino et al. (1993). On the other hand, the model provided, a limit value of the pedestrian density of 1.50 P/m2, from which is so hard to walk on the deck that the dynamic effects are negligible. Nevertheless, the model presents some limitations: (i) the model ignores the energy transmitted by the not-synchronized pedestrians with the structure, (ii) 14  it does not take into account the change of the modal properties of the structure due to the presence of the pedestrians, (iii) the change of the behavior of the pedestrians with the vibration level of the structure is not considered and (iv) the value of the equivalent pedestrian damping is only estimated for one footbridge and specific range of frequencies. In order to consider in a more appropriate manner the change of the behavior of the pedestrians induced by the vibration level, Nakamura et al. (2004) modified Dallard’s proposal (Dallard et al., 2001) in order to take into account that, according to the observations performed on the T and M footbridges (Tokyo), from certain value of the lateral displacement, 10 mm, the pedestrians modified their step to guarantee an adequate comfort level, further reducing the lateral force originated by their step. In that way, a saturation factor is introduced in each pedestrian lateral force that avoids that the force increases linearly with the velocity of the deck indefinitely. This proposal reduces the ratio of the increase of the velocity when the velocity increases until making it null. Although this model is an improvement as compared to the previous proposals, it has as main limitation the necessity of knowing the maximum displacement of the structure to scale the saturation ratio, while shares the other limitations of the previous models. Subsequently, the French standard (Setra, 2006) proposed a more compact model considering the results of two types of tests, on treadmills at laboratory and large scale pedestrian tests on Solferino footbridge (Paris). As conclusions of these two sets of tests: (i) an experimental relationship that allows estimating the number of synchronized pedestrians on the footbridge was proposed, as well as (ii) a new criterion in order to determine the sensitivity of the footbridge to the lateral lock-in phenomenon. It was verified, in this sense, that a change of regime in the pedestrian behavior occurs, from forced to synchronized vibration, with a maximum synchronization ratio of 60% for a lateral acceleration of the deck between 0.100.15 m/s2. This limit acceleration has been adopted as a criterion in order to determine the sensitivity of the footbridge to the lateral lock-in. In the last five years, several more sophisticated models, not yet implemented in the international standards, have appeared providing a new approach to the problem. Among these models, the most influential ones for the development of this Thesis are described, in summary, in the following paragraphs In the first model, proposed by Macdonald (Macdonald, 2008) and based on the inverse pendulum model by Baker (Baker, 2002), the pedestrian is modelled as a lumped mass attached to the footbridge by an inclined bar (Figure 5). The pedestrian under the lateral vibrations modifies the tilt of his legs, searching his stability, increasing the lateral component of the walking pedestrian force and being able to attain a dynamic instability situation. The proposed model has been accepted to describe the initiation of the phenomenon, but the effect of modification of the pedestrian step due to large lateral vibrations is not included in the proposal. Despite its limitations, there are evolutions of the model (Morbiato et al., 2011) with greater complexity and precision. 21  On the basis of the spectral design model, Butz (2006) developed an empirical expression for the determination of the required modal mass, i M (kg), for a given pedestrian traffic to ensure a required comfort level under the assumption that some natural frequency of the structure is in the range that characterizes the walking pedestrian action.   lim 31 42 65.1 a kkn M kk p i    [12] where 1 k to 4 k are constants, as given in Table 8. lim ais the limit acceleration according to the considered comfort level (Table 6). and, p n, is the number of pedestrians. Table 8. Constants for required modal mass (Butz, 2006). Vertical-Torsion Lateral d [P/m2] k1 k 2 k 3 k 4 k 1 k 2 k 3 k 4 <0.50 0.7603 0.050 1.00 0.5700 0.4680 0.040 0.675 0.1205 0.4500 0.0120 0.6405 1.50 0.4000 0.035 2.6.2. Modification of the stiffness induced by pedestrian action. The value of the natural frequencies of the footbridge is proportional to the square root of the ratio between the modal stiffness and mass of the structure. In this manner, large structural modifications are necessary if the natural frequencies of the footbridge must be located out of the pedestrian-structure interaction range. The current trend in the design of footbridges, under aesthetics, resistant and economic requirements causes that the above criterion is not always feasible in order to guarantee an adequate comfort level (Slaich, 2005). However, there are occasions, where the first natural frequency of the structure is inside the walking pedestrian range, and the second natural frequency is outside that range, where it can be reasonable to reduce the stiffness of the structure so both natural frequencies are outside the pedestrian-structure interaction range and checking additionally that the static deflection of the footbridge is compatible with its use (Setra, 2006 and Butz et al., 2007). The most common strategies in order to modify the natural frequencies of the footbridge, from the viewpoint of the stiffness, are (Setra, 2006 and Butz et al., 2007): (i) increase the degree of statically indetermination, (ii) provide structural characteristics to protection or surface elements, and (iii) use cable systems with a stabilizing function (Setra, 2006). It is recommended in footbridges with a width larger than 4.00 m and spans with lengths above 50.00 m, to install a lateral load transmission system, as an effective method to control the lateral lock-in phenomenon (Low, 2008). Finally, at high seismicity areas, the increase of the 22  stiffness of the structure in order to avoid vibratory problems induced by pedestrians may cause, by contrast, an increase of the seismic action (Slaich, 2005). 2.6.3. Modification of the damping induced by pedestrian action. The increase of the structural damping has been, until the date, the most used method to control the vibrations induced by pedestrians on footbridges (Fujino et al., 1993; Dallard et al., 2001 and Caetano et al., 2010). This increment can be achieved either by the actuation on internal elements of the structure, or by the implementation of external control devices of different nature according to their performance: active, semi-active, hybrid or passive devices (Moutinho et al., 2010). The most usual is the use of passive dampers as: (i) viscous dampers (Butz et al., 2007 and Taylor, 2003), (ii) tuned mass, liquid or liquid column dampers (Fujino et al., 1993; Butz et al, 2007 and Caetano et al., 2010) and (iii) pendulum dampers (Butz et al., 2007). However, the use of these passive dampers must be limited since although they allow achieving a high level of damping with a reasonable cost, they present several problems that discourage their widely use. Among these problems, the most important are: (i) the necessity of damping all the natural frequencies of the structure inside the range of pedestrian interaction, (ii) a bad performance may be the cause of a splitting of the natural frequency originally damped, (iii) they are mechanical elements that require maintenance, and (iv) due to their weight it can be non-viable their placement on existing footbridges due to strength reasons (Meinhardt, 2009). 23  3. Proposal of a simplified biomechanical crowd-structure interaction model. The proposed crowd-structure interaction has been simulated using two individual sub-models (Figure 7): (i) the pedestrian-structure interaction sub-model and (ii) the crowd sub-model. In the first sub-model, all the dynamics effects induced by the pedestrians on the footbridge are considered. The lateral or vertical acceleration, a y  or a z  , experimented by each pedestrian is the output obtained from this model (according to the analysed direction). In the second sub-model, the crowd is simulated as a behavioural model, providing a description of the individual pedestrian position, p x, walking pedestrian velocity, p v, step pedestrian frequency, s f, and phase among pedestrians, p  . In order to take into account the change of the pedestrian behaviour associated with the acceleration level experienced, two additional conditions have been included in this latter sub-model. In vertical direction only a comfort threshold has been included, while in lateral direction a comfort and lateral lock-in thresholds have been considered. The first condition modifies the pedestrian velocity, p v, according to the comfort level experienced by each pedestrian and the second condition modifies the step pedestrian frequency, s f, and the phase among pedestrians, p  , in order to synchronize the movement of the pedestrians and the structure if certain limit is exceeded. Figure 7. Layout of the biomechanical crowd-structure interaction model. For each interaction the crowd sub-model determines the position, velocity step frequency and phase of each pedestrian. Subsequently, these four parameters are used as input to define the walking force of each pedestrian-structure model, obtaining as output the lateral o vertical acceleration of each pedestrian. The pedestrian velocity and frequency of each individual is modified according to the PEDESTRIAN-STRUCTURE MODEL CROWD MODEL PEDESTRIAN/STRUCTURE PARAMETERS CROWD-STRUCTURE MODEL p x p va y  COMFORT THRESHOLDS LOCK-IN THRESHOLD s f p  a z  24  comfort level experimented by each pedestrian and additionally his phase shift if a lateral lock-in threshold is exceeded. Finally, the process is repeated by the updated values of the position, the pedestrian velocity, the step frequency and the phase shift (Figure 7). 3.1. Modelling the pedestrian-structure interaction. The proposed pedestrian-structure interaction in each direction follows from the application of dynamic equilibrium equations (Clough and Penzien, 1993; Dominguez, 2001; Xia and Zhang, 2005) to a simplified model of interaction (Figure 8) with sprung ( a m) and unsprung masses ( s m). This methodology has been applied separately for vertical (Paper A and Paper B) and lateral (Paper E) directions and here it will summarized for the case of vertical direction. Its implementation in the lateral direction is described in Paper E. The formulation of the proposed model may be further generalized to the three directions by accordingly modifying both the equation that governs the considered pedestrian load in each direction and the value of the modal parameters that define the TDOF (two degrees of freedom) pedestrian model. In this way, the resulting model would be suitable for the more general 3-D problem and it could therefor take into account the possible interaction in the three spatial directions. Figure 8. Biomechanical pedestrian-structure interaction model in vertical direction (Paper B). Considering the balance of the system, structure and pedestrian model, the following coupled equations of motion may be written. int_ )( FxzKzCzM piNUMiiiiii       [13]     0     sapsapaa zzkzzczm  [14]     int, FFzzkzzczm verpaspaspss       [15] m a c p k p m s z a z s F int L z xF int F s M i C i K i d p y x p w(x,t) 25  where a m is the sprung mass of the pedestrian in the considered direction [kg]. s m is the unsprung mass of the pedestrian in the considered direction [kg]. as mmm  is the total mass of the pedestrian in the considered direction [kg]. i zis the modal displacement of the vibration mode i [m] a z is the absolute vertical displacement of the sprung mass [m]. s z is the absolute vertical displacement of the unsprung mass [m]. p k is the equivalent stiffness of a pedestrian [N/m]. p c is the equivalent damping of a pedestrian [sN/m]. verp F, is the vertical pedestrian force due to walking [N]. int F is the interaction force between the pedestrian and the structure [N]. i M is the modal mass of the vibration mode i [kg]. i C is the modal damping of the vibration mode i [sN/m] i Kis the modal stiffness of the vibration mode i [N/m]. iNUM _  is the vertical component of the numerical vibration mode i. tvx pxp  is the longitudinal position of the pedestrian [m]. t is the time [sec.] px v is the longitudinal component of the pedestrian velocity vector [m/s]. p dis the distance among pedestrians [m]. ),( txw is the deflection of the footbridge at the position x [m]. Lis the length of the footbridge [m]. From Eq.(15) the following expression is obtained for, int F:     aspaspssverp zzkzzczmFF       ,int [16] and substituting this equation into Eq.(13) yields. 26         aspaspssverppiNUMiiiiii zzkzzczmFxzKzCzM          ,_  [17] Applying, at the contact point between the pedestrian and the structure, the equations of compatibility of displacements, velocity and acceleration between the structure and the simplified pedestrian-model of interaction: ),(),( ttvwtxwz pxps    [18] ),(),( ttvwtxwz pxps      [19] ),(),( ttvwtxwz pxps      [20] These quantities may be expressed in terms of the amplitude )(tzi and the modal shape of the n numerical considered vibration modes )( _x iNUM  , neglecting the term of variation of the pedestrian velocity along time, as:    n i piNUMip xtztxw 1 _)()(),(  [21]     n i piNUMpxi n i piNUMip xvtzxtztxw 1 _ 1 _)()()()(),(    [22]       n i piNUMxpip n i iNUMxpi n i piNUMip xvtzxvtzxtztxw 1 _ 2 , 1 _, 1 _))()()()(2)()(),(    [23] dx xd xiNUM iNUM )( )( _ _     [24] 2 _ 2 _ )( )( dx xd xiNUM iNUM     [25] where )( _x iNUM  is the first spatial derivate of the mode of vibration i. )( _x iNUM   is the second spatial derivate of the mode of vibration i. The numerical vibration modes, )( _x iNUM  , are obtained in a discrete way using the corresponding finite element method as:   j j j iiNUM xNx )()( _  [26] where )(xN jare the shape functions and j i  are the nodal values. In the previous expressions, the value of the numerical vibration modes is set to zero when the pedestrian remains outside the structure. 27  0)( _  piNUM x  for Ltx tx p p   )( 0)( [27] The above relations –Eqs.(18) to (23)- are then substituted in the overall dynamic equilibrium equations -Eqs.(13) to (15)- so that, organizing information in a matrix form, the following model of interaction is obtained (see Paper B for further details and matrix formulation). )()()()()()()( ttttttt FzKzCzM       [28] For a group of k pedestrians (Figure 8), each of them will be represented by the above simplified interaction model. In the case of a single pedestrian, the proposed model leads to a system of n+1 equations, corresponding to the considered number of vibration modes n plus the simplified interaction equation. Similarly, when considering a group of k pedestrians, a system of n+k differential equations will need to be solved. Considering the nature of the resulting system, the use of a method of  -Newmark integration family is proposed, with parameters 41   and 21   , thus ensuring an unconditionally stable system. Furthermore, the integration step, t  , is established according to the usual recommendations (Clough and Penzien, 1993; Dominguez, 2001) for dynamics models based on modal decomposition technique, as the minimum of the following values. )01.0, 4 , 200 , 8 1 min( minmin max pp vn L v L f t   sec. [29] with max f[Hz] being the highest considered vibration frequency of the structure and min L[m] the minimum span length of the pedestrian bridge. In order to determine the number of pedestrians that cross the footbridge in phase, a Poisson distribution has been adopted, according to the results by Matsumoto et al. (1978). Thus, when a group of p n pedestrians arrive at the footbridge, the number of pedestrians randomly synchronized is p n. This synchronization criterion has been adopted originally in the crowd-structure interaction model. Its implementation is achieved by the phase shift parameter, p  . For a given generation/group of pedestrians, the value of the phase shift of the p n randomly synchronized pedestrians has been set equal to zero. For the remaining pedestrians the phase shift has been assigned randomly, using a Gaussian distribution in the range   2,0 . Subsequently, in the lateral direction, this parameter will be modified if the lateral lock-in threshold is exceeded. 28  3.2. Modelling the crowd-behavior. The pedestrian walking inside a crowd may be modelled using the governing equations of particle dynamics (Rapaport, 2004), considering that different social forces interact among pedestrians. This approach has been successfully applied by several authors (Helbing and Molnár, 1995; Carroll et al., 2012). In this way, the different motivation and influences experimented by the pedestrians are described by several force terms. The model is based on Newton dynamics and is able to represent the following rules in relation with the natural pedestrian movement (see (Helbing and Molnár, 1995) for a more involved description): (i) pedestrians normally choose the fastest route, (ii) each pedestrian has an individual speed that may be defined by a Gaussian distribution (iii) and the distance between pedestrians depends on the pedestrian density and the walking pedestrian speed. Figure 9 sketches the social forces acting on a pedestrian in a crowd, as described in detail in the following paragraphs.  Figure 9. Pedestrian-crowd interaction forces (Helbing and Molnár, 1995). All the parameters for the crowd model considered in this study have been obtained from the reported results provided by different authors (Helbing and Molnár, 1995; Carroll et al., 2012), as summarized in Table 9 and briefly described next. Boundary Boundary Desired destination nor bou F tan bou F dri F Pedestrian j Pedestrian i tan_phy ped F p d d e d d zx y p x p  p r Anisotropic pedestrian behaviour Legend Force vector Axis of the structure Geometric magnitude Coordinate system Movement direction tan_phy ped F norphy ped soc ped _ FF  p n b t b n p t p t norphy ped soc ped _ FF  b nb t tan bou F nor bou F 0.1 5.0 1.0 p  29  Driving force. Each pedestrian has a certain motivation to reach his desired destination, d d, with his desired velocity, d v, which is represented by the driving force, dri F, as:           r p r dd dri tt v mv e F [30] where d e is the desired direction vector, p v is the pedestrian step velocity and r t is the relaxation time of the pedestrian (Helbing and Molnár, 1995) (Table 9). The desired direction of the movement may be obtained from the position of the pedestrian in each instant, p x, and its desired destination according to: pd pd dxd xd e   [31] Interactions among pedestrians. The interaction among pedestrians originates a repulsive force (Helbing and Molnár, 1995), ped F, with two components, a socio-psychological force, soc ped F, and a physical interaction force, phy ped F, as: phy ped soc pedped FFF  [32] The socio-psychological force reflects the fact that the pedestrians try to maintain a certain distance to other pedestrians in the crowd. This socio-psychological force depends on the distance between pedestrians, reaching its maximum value at the lowest established distance and tending to zero as such distance increases. The socio-psychological force is defined as: pp p pp p soc ped s B dr A          nF 2 exp [33] where p A is the interaction strength between pedestrians (Table 9). p B is the range of the repulsive interaction between pedestrians (Table 9). p d is the distance between two pedestrians. p r is the so-called pedestrian radius (Table 9). p n is the normalized vector pointing between pedestrians. 30  p s is a form factor to consider the anisotropic behaviour (the pedestrian action in front of the pedestrian is more important than behind him) of the pedestrians, which value may be obtained from:   2 cos1 )1( p ppp s     [34] where p  (Table 9) is a potential factor that considers the influence on the pedestrian movement of other pedestrians situated in front of him (Figure 9) and p  is the angle between two pedestrians (Figure 9). The physical interaction force, phy ped F, is only considered in situations of physical contact among pedestrians (if pp rd   2), associated with situations of high pedestrian densities (≥0.80 P=Person/m2). The physical interaction force is defined by the superposition of two components: (i) the body force, norphy ped _ F, that describes the counteracting body action that the pedestrians perform to avoid physical damage due to their physical contact with other individuals, (ii) and the sliding force, tan_phy ped F, that represents the pedestrians’ tendency to avoid passing other individuals with a high velocity at small distances (Helbing and Molnár, 1995). It is defined as: tan__ phy ped norphy ped phy ped FFF  [35]   pppp norphy ped drHC nF  2 _ [36]   p t pppp phy ped vdrHD tF  2 tan_ [37] where norphy ped _ F is the normal component of the physical interaction force (body force). tan_phy ped F is the tangential component of the physical interaction force (sliding force). p C is the body force strength due to the contact between pedestrians (Table 9). p D is the sliding force strength due to the contact between pedestrians (Table 9). p tis a normalized vector perpendicular to p n. pp t p vtv  is the tangential component of the relative pedestrian velocity, with p v being the difference of vector velocities between two given pedestrians. and function H is defined as: 37 Figure 10. Flowchart of the identification procedure in vertical direction. An iterative process to reduce the differences between the experimental and numerical vertical accelerations was performed, under the rules of genetic algorithms (Koh and Perry, 2010; Nocental and Wright, 1999), The estimated values of parameters of the pedestrian-structure interaction model, in vertical direction, were correlated successfully with: (i) the walking pedestrian vertical force suggested by different authors (summarized in section 2.1) and (ii) a previous estimation of the modal parameters (Paper A) obtained from the analysis of the change of the modal properties of a footbridge induced by a controlled group of pedestrians (Georgakis and Jorgesen, 2013). On the other hand, the estimated values of the modal parameters were inside the range, established by Shahabpoor et al. (2013). 4.2. Inverse dynamic problem: parameter identification in lateral direction. In order to define the objective function for the parameter identification of the TDOF-system in lateral direction (Paper E), the recorded dynamic response of the Viana footbridge during the experimental pedestrian test was analysed again by its transformation to the frequency domain. In this case, however, it was checked that the dynamic response of the footbridge was characterized by a harmonic series that contained only the first three frequencies that characterizes the pedestrian step without a remarkable contribution of the harmonics associated with the lateral natural frequencies of the footbridge. In this manner, the accelerations recorded during the experimental pedestrian test contained information mainly of the walking pedestrian lateral force, being necessary to conduct a second experimental test, a crowd test, in order to characterize the modal parameters of the TDOF-system. In this crowd test, the dynamic response of Viana footbridge under a group of fifty 38 pedestrians at different step frequencies was recorded in order to study the change of the first lateral natural frequency of the footbridge induced by the pedestrianstructure interaction. Due to this fact, the identification process was divided in two steps, by solving two inverse dynamic problems. First, as the modal parameters of the pedestrian-structure interaction model have a direct effect on the modal parameters of the footbridge (Paper A) as objective function of the first minimization problem, the mean square error between the experimental, exp ,1 lat f, and numerical, num lat f,1 , first lateral natural frequency of the Viana footbridge, obtained during the performance of the experimental crowd test and its numerical simulation, was considered. Additionally, as design variables of this first inverse problem, the three modal parameters that characterizes the TDOF-system, in lateral direction (the pedestrian sprung mass, lata m,, the pedestrian damping ratio, latp,  , and the pedestrian natural frequency, latp f,), were considered. Second, although there are very recent and comprehensive studies of the lateral force induced by pedestrians (Ingólfsson and Georgakis, 2011; Ingólfsson et al., 2011), these research do not include the effect of the pedestrian-structure interaction. Therefore it was necessary to estimate the walking pedestrian lateral force under this assumption. In this manner a second inverse problem was solved. As objective function of the second minimization problem, the mean square error between the experimental ( exp ,ilat psd , where i is the considered section) and numerical ( num ilat psd ,) power spectral density obtained from the lateral accelerations recorded in the previously mentioned experimental pedestrian tests and its numerical simulation, was considered. As design variables for this second inverse problem, the first three LDLF ( lat,1  , la,2  and lat,3  ) and their corresponding phase shifts of the second and third harmonic ( la,2  and lat,3  ) of the pedestrian walking lateral force were considered. In Figure 11 a flowchart of the identification procedure is shown. 39 Figure 11. Flowchart of the identification procedure in lateral direction. Light blue marks the modal parameter identification methodology and dark blue the walking pedestrian force identification methodology. As in the above case (vertical direction), an iterative process to reduce the differences between the experimental and numerical magnitudes was performed, under the rules of genetic algorithms (Koh and Perry, 2010; Nocental and Wright, 1999), In this case, the identification procedure is performed in two steps. First, the modal parameters of the proposed TDOF-system were estimated by the minimization of the relative differences between the experimental and numerical change of the first lateral natural frequency of the Viana footbridge during an experimental crowd test and its numerical simulation. Second, once established the modal parameters of the proposed model, the walking pedestrian lateral force was estimated by the minimization of the relative differences between the experimental and numerical power spectral densities obtained in four points of the Viana footbridge during an experimental pedestrian test and its numerical simulation. The estimated values of the parameters of the pedestrian-structure interaction model, in lateral direction, were correlated successfully with: (i) the walking pedestrian lateral force suggested by different authors (summarized in section 2.1) and the range of pedestrian modal parameters suggested by Shahabpoor et al. (2013). 40 5. Experimental estimation of the parameters of the pedestrian-structure interaction model. In this section the estimation of the parameters of the proposed TDOF-system was performed in vertical and lateral directions. First, a real footbridge, Viana do Castelo footbridge (Barbosa et al., 2012), was converted into a “laboratory” footbridge by the updating of its finite element model based on the experimental modal parameters of the structure obtained from an operational modal analysis performed on the measurements recorded during an ambient test. Second, two experimental tests, a pedestrian and crowd test, were conducted in order to established a basis for the estimation of the parameters of the proposed TDOFsystem. Finally, the parameters of the pedestrian-structure interaction model were estimated by the resolution of an inverse problem approach. 5.1. Description and finite element model of the “laboratory” footbridge: Viana footbridge. The Viana do Castelo footbridge (Barbosa et al., 2012) is a moveable cable-stayed bridge. The longitudinal structural scheme of the footbridge consists of two spans of about 36.50 m and 9.00 m respectively suspended by 6 families of two hangers (two retaining ones) from an inclined mast. The deck, with 2.50 m of width, is configured by two rolled steel beams of variable depth braced by circular hollow profiles. The deck floor is covered with wood. The compensation of the main span weight is achieved by placing 11 high density blocks (with a weight of 800 kN) placed in the shorter span. The mast is welded in its base to a cylinder that is connected to a wheel gear bearing that allows the rotational movement of the structure. The pylon is connected to a deep foundation that balances the forces transmitted by the mast. A perspective of the Viana footbridge is shown in Figure 12. Figure 12. Lateral view of the Viana footbridge (Barbosa et al., 2012). A pr e orde to d e Fig u para The the s 3Dc with natu prev esti m preli asso four num e liminary n r to have a e fine a sta r u re 13. F meters. software p s tructure u c able elem e the hang e ral freque iously the m ating its minary F E ciated nu m numerical ber). n umerical f i a first appr r ting point r F inite ele m p ackage An u sing 3D-b e e nts (LIN K e rs have b ncies and stress le tangent s E model l e m erical nat u vibration m nite eleme o ximation t r equired to m ent mod e sys (Ansy s e am elem e K 10) were i b een consi the vibr a vel of th e s tiffness m e ads to t h u ral frequ e m odes are 41 e nt (Figure t o the dyn a perform t h e l, ambie n s , 2015) w e nts (BEA M implement e dered for a tion mod e e hangers m atrix. Th e h e first fo e ncies give n shown ( f N U 13) modal a mic beha v h e ambien t n t test g r as used b a M 188), exc e e d. The n o the deter m e s of the under p e e numeric a ur numeri n in T able U M_i, with i analysis w v iour of th e vibration t r id and m a sed on a d e pt for the o nlinear ef f m ination o f structure, e rmanent l a l modal a cal vibrati 10. In Fig being the w as conduc t e footbridg e t est. m odel up d d iscretizati hangers w f ects asso c f the num by calcu l oads and a nalysis o f on modes ure 14 th e vibration m t ed in e and d ating on of w here c iated erical l ating thus f this and e first m ode 42 fNUM_1=3.426 Hz fNUM_2=4.421 Hz fNUM_3=6.907 Hz fNUM_4=7.431 Hz Figure 14. First four numerical vibration modes. Initial FE model. 5.2. Experimental identification of the modal parameters of the “laboratory” footbridge. In order to obtain experimentally the modal parameters (natural frequencies, damping ratios and modal shapes) of the footbridge, an ambient vibration test was performed. The measurements were recorded in ambient conditions, with the footbridge excited by a light wind. In order to acquire a sufficient level of expertise in the application of this identification technique, the assessment of the dynamic behaviour of others footbridges was conducted by the author during the development of this Thesis. Two representative examples have been included in the document (Paper F and Paper G). The modal shape coordinates were measured along two gridlines separated transversally 1.68 m. A total of 2x11 points equally distributed along each longitudinal alignment were instrumented. Four high sensitivity tri-axial force balanced accelerometers were used (Figure 15). Using two of these devices as references, measurements were successively made moving the other two accelerometers to the defined instrumentation locations and recording in each point 1000 sec. time series of acceleration sampled at 100 Hz (Figure 13). 43 Figure 15. One accelerometer used during ambient/experimental tests. The experimental identification of the modal parameters was done in the time domain using the Stochastic Subspace Identification method (Magalhães and Cunha, 2011), implemented in the software program Artemis (Artemis, 2015). Figure 16 illustrates the stabilization diagram of the identification algorithm used. The first four vibration modes were identified and subsequently used for the FE model updating process (see Paper B). The obtained numerical and experimental natural frequencies and vibration modes shapes are compared in Table 10 and the correlation between the first four numerical and experimental vibration modes is shown in Figure 17 (with the x axis corresponding to the longitudinal direction of the footbridge). In order to validate the correlation between the numerical and experimental modal parameters, both the relative difference ( f  ) between the numerical and experimental frequencies and the modal assurance criterion (M.A.C.) were analysed (Zivanovic et al., 2007). A good correlation between two modes is achieved when the value of their M.A.C. ratio is greater than 0.90. These two magnitudes may be defined according to Eqs. (46 and 47) as follows: 100 _ __    iEXP iEXPiNUM f ff f [%] [46] where iNUM f_ is the numerical natural frequency and iEXP f_ is the experimental natural frequency of the vibration mode i.    iEXP T iEXPiNUM T iNUM iEXP T iNUM i MAC ____ 2 __      [47] where iNUM _  and iEXP _  are the numerical and experimental vibration modes to be compared and T denotes the transpose. 44 Although the shapes of the identified vibration modes are in good agreement (with M.A.C. ratios greater than 0.90 in three of the vibration modes), the relative differences, f, between the first two numerical and experimental natural frequencies are still significant. Therefore, the initial estimation made on the physical parameters of the structure is not good enough and it becomes necessary to perform a finite element model updating (Friswell and Mottershead, 1995; Teughels, 2003, Paper C) of the footbridge in order to improve the correlation between the numerical and experimental modal parameters. Figure 16. Stabilization diagram of the Stochastic Subspace Identification method. Finally, in Table 10 an estimation of the damping ratios, i  , associated with the identified vibration modes is also shown (Magalhães et al., 2010). These values will be adopted later in the identification process of the parameters of the TDOF-system that models the pedestrian. Table 10. First four numerical (fNUM) versus experimental (fEXP) vibration modes of the footbridge. Modes fNUM [Hz] fEXP [Hz] i  [%] f  [%] M.A.C. Description 1 3.426 3.138 1.22 9.17 0.963 First vertical mode 2 4.421 4.068 1.21 8.67 0.985 First lateral mode 3 6.907 6.810 1.10 1.42 0.809 Second lateral mode 4 7.431 7.345 1.39 1.17 0.939 Second vertical mode 45 1st Vibration mode 2nd Vibration mode 3rd Vibration mode 4th Vibration mode Figure 17. First four numerical (Num.) versus experimental (Exp.) vibration modes. 5.3. Model updating of the “laboratory” footbridge. As indicated above, in order to reduce the level of uncertainties of the numerical analysis a finite element model updating (Friswell and Mottershead, 1995; Teughels, 2003; Zivanovic et al., 2007, Paper C) of the structure has been performed. The four identified vibration modes were considered in the updating process due to the good quality of the experimental data. Both measured natural frequencies and modal coordinate values were taken into account. Therefore, in total 48 residual components were selected for the model updating (the four identified natural frequencies and the eleven coordinates of each identified vibration mode). A more detailed description of the methodology used to perform the model updating can be found in Paper C and Paper G. A sensitivity analysis was performed in order to adequately determine the physical parameters of the FE model with greater influence on the identified vibration modes. In this manner, the modal sensitivities with respect to some possible physical variables have been obtained numerically (Fox and Kapoor (1968)). The results of this study conclude that the most influential physical parameters on the dynamic behaviour of the footbridge are the stiffness of the four families of main hangers and the soil-structure interaction, modelled by two spring elements (in longitudinal, 5  , and lateral, 6  directions, as Figure 13 illustrates) situated at the extreme of the longer span. As the stiffness of the hangers is conditioned by their stress level, the initial stress state of each considered family ( 1  , 2  , 3  and 4  ) -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 10.00 20.00 30.00 40.00 50.00 X [m] Num. Exp. -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 10.00 20.00 30.00 40.00 50.00 X [m] Num. Exp. -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 0.00 10.00 20.00 30.00 40.00 50.00 X [m] Num. Exp. -1.00 -0.50 0.00 0.50 1.00 1.50 0.00 10.00 20.00 30.00 40.00 50.00 X [m] Num. Exp. 46 has also been taken into account as physical variable. In Figure 13 and Table 11, the selected physical variables are shown. The model updating process has been conducted by solving an optimization problem in the software programs Ansys (Ansys, 2015) and Matlab (Matlab, 2015). As objective function the mean square error between the experimental and numerical modal parameters (natural frequencies and vibration modes) of the Viana footbridge has been considered. In each iteration, a population of 1000 vectors has been generated that, using the genetic algorithms rules of mutation, reproduction and crossover, has minimized the value of the proposed objective function. The values of the selected parameters have been modified in order to minimize the considered objective function. Additionally, a search domain has been defined to control the variation of each parameter, increasing the efficiency of the optimization algorithm and yet maintaining the physical meaning of the finite element model updating. For the stiffness of the hangers, as a passive behaviour is expected, their medium tension level has been determined under permanent loads, with a value of 250 kPa. Slight variations of this value have been considered expanding the search domain between 0-500 kPa. For the stiffness of the springs ( 5  and 6  ), given the uncertainty associated with the stiffness of the soil, a wider search domain was considered. So considering, as it is established by the geotechnical report, a variation of the Young’s modulus of soil between 1001   m E GPa and the geometry of the abutments, the variation of the equivalent stiffness of the springs has been determined 5  and   97 61010   N/m. In Table 11 the range of variation of each parameter and its updated values are shown Table 11. Updated values of considered physical parameters. Parameters Minimum Value Updated Value Maximum Value Tension stress cable 1 ( 1  ) 0.00 kPa 112.38 kPa 500.00 kPa Tension stress cable 2 ( 2  ) 0.00 kPa 59.62 kPa 500.00 kPa Tension stress cable 3 ( 3  ) 0.00 kPa 31.62 kPa 500.00 kPa Tension stress cable 4 ( 4  ) 0.00 kPa 147.43 kPa 500.00 kPa Longitudinal Spring ( 5  ) 1.00E7 N/m 6.00E7 N/m 1.00E9 N/m Lateral Spring ( 6  ) 1.00E7 N/m 1.90E8 N/m 1.00E9 N/m The differences between the numerical and experimental natural frequencies, after the finite element model updating, are very small and the correlation between the numerical and experimental vibration modes are even higher. The relative differences between the updated numerical ( UPD f) and experimental ( EXP f) modal parameters and the M.A.C. values achieved after the model updating process are summarized in Table 12, where the improvement with respect to the initial FE model is clear (see Table 10). 53 structure interaction model The following Gaussian distributions have been obtained (),(   N, being  the mean value and  the standard deviation).  Lateral pedestrian sprung mass, lata m,,)736.2,216.73(N%.  Lateral pedestrian damping ratio, latp,  ,)405.5,116.49(N %.  Lateral pedestrian natural frequency, latp f,, )178.0,201.1(N Hz. Figure 22 illustrates the correlation between experimental and numerical results for the change of first lateral natural frequency of the footbridge induced by the crowdstructure interaction phenomenon. Good agreement between both sets of results is observed, with differences below 0.70 % for all the analysed pedestrian walking frequencies. The first lateral natural frequency corresponding to the empty footbridge is included in Figure 22 for reference.  Figure 22. Change of the first lateral experimental (Exp.) and numerical (Num.) natural frequency (f1,lat) versus the step frequency [Hz]. From the previous results the following conclusions may be extracted: (i) the stability that the estimated modal parameters present for the different step frequencies, allowing that the proposed crowd-interaction model may be used as a tool for the characterization of the effect of the moving pedestrians on the dynamic behaviour of footbridges in lateral direction and (ii) the good correlation between the experimental and numerical curves (Figure 22) that show the change of the first lateral natural frequency of the footbridge verifies the ability of the proposed model to characterize the pedestrian-structure interaction phenomenon in lateral direction. 3.750 3.800 3.850 3.900 3.950 4.000 4.050 4.100 1.30 1.50 1.70 1.90 2.10 2.30 2.50 f 1.lat [Hz] f s [Hz] Exp. Num. Empty 54 For the estimation of the walking pedestrian lateral force of the TDOF-system, a second inverse problem was solved again. In this case, as objective function the mean square error between the experimental and numerical power spectral density obtained from the lateral accelerations recorded in the mentioned four points of the Viana footbridge under the crossing of two pedestrians at controlled step frequencies was considered. The experimental lateral accelerations correspond to the denoised measurements of the above described pedestrian test. The numerical lateral accelerations have been obtained from the implementation of the proposed pedestrian-structure interaction model on the updated finite element model of the Viana footbridge. The experimental and numerical power spectral density has been obtained from these mentioned accelerations. Six parameters were adopted as design variables: (i) the first three LDLF that characterize the pedestrian walking lateral force. (ii) the phase shifts of the second and third harmonic that characterize the pedestrian walking lateral force. (iii) a time lag that allows adjusting the beginning of the crossing of the pedestrian between the experimental and numerical response. The estimation of the phase shifts has been made in a discrete way, selecting in each case the option that minimizes the objective function. Figure 23 illustrates the layout of the identification process of the pedestrian walking lateral force of the proposed TDOF-system. Figure 23. Layout of the walking pedestrian lateral force identification methodology. 55 In Paper E, the results of the estimation process are summarized, showing the different estimated parameters versus the pedestrian step frequency. According to these results, it is possible to obtain a statistical estimation of the design variables that characterize the pedestrian walking lateral force.  First LDLF, lat,1  , )017.0,086.0(N.  Second LDLF, lat,2  , )009.0,094.0(N.  Third LDLF, lat,3  , )019.0,040.0(N.  Second lateral phase shift 0 ,2  lat  º.  Third lateral phase shift 0 ,3  lat  º. Figure 24 illustrates the lateral pedestrian walking force obtained from the proposed identification procedure. The maximum and minimum enveloped values of the lateral forces shown in Figure 2 are also represented. Figure 24. Lateral pedestrian walking force according to the TDOF-system. For the generation of the pedestrian flows of the crowd-structure interaction model in both directions, the above Gaussian distributions were considered. -200.00 -150.00 -100.00 -50.00 0.00 50.00 100.00 150.00 200.00 0.00 0.20 0.40 0.60 0.80 1.00 Lateral Load [N] Time [sec.] Minimum Maximum TDOF-system 56 6. Validation and main results of this Thesis. Once obtained the parameters of the proposed TDOF-system in both directions, the definition of the proposed model is complete. The proposed model is validated in this section by correlating the experimental and numerical dynamic response of two real footbridges under the effects induced by the pedestrian action. In vertical direction (Paper B), the proposed model was implemented to analyze the dynamic response and the change of the first vertical natural frequency of the Viana footbridge under the previously described crowd test. In lateral direction (Paper E), the proposed model was implemented to analyze the lateral lock-in phenomenon on Pedro e Inês footbridge (Coimbra, Portugal), including the estimation of its dynamic response and the change of its first lateral natural frequency due to the pedestrian action. 6.1. Analysis of the change of the modal properties of the Viana footbridge. The validity and applicability of the proposed crowd-structure interaction model in vertical direction is assessed through its practical application to the following case study (Paper B). From the forced recorded response of the previously mentioned crowd test, the experimental analysis of the change of the first vertical natural frequency of the structure, due to the crossing of the group of pedestrians at different step frequencies, was determined. Figure 26 illustrates the experimental analysis of the change of the first vertical natural frequency of the Viana footbridge. Subsequently, the crowd-structure interaction model has been applied to the updated finite element model in order to obtain, first, the vertical numerical acceleration at the mentioned sections and, later, to analyse numerically the change of the first vertical natural frequencies of the footbridge due to the presence of the pedestrians. The assessment of its performance has been done by correlating the above experimental results with the numerical estimations predicted by the model. For each considered step frequency ten generations of groups with 50 pedestrians were simulated. The number of pedestrians in phase in each new simulation was determined by the evaluation of the parameter p  . The desired velocity, d v, of each pedestrian was assigned according to Eq.(5). A pedestrian mass of 70 kg has been considered according to the French code (Setra, 2006). As initial spatial distribution of the pedestrians, a rectangular grid was selected, considering an initial distance among pedestrians 50.0 p d m with an equidistant distribution in the width of the deck. The selected time step is 01.0   tsec. The numerical vertical acceleration (for three of the 50 pedestrian generations) at section 2 Sof the footbridge, for a step frequency of 1.60 Hz, is shown in Figure 25. 57 Figure 25. Experimental versus numerical acceleration (three generations) at section S2 of Viana footbridge under a group of 50 pedestrians (walking frequency of 60.1 s f Hz). -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] Vertical experimental acceleration (S 2 ). 50 Pedestrians at 1.60 Hz -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] 1st Generation. Vertical numerical acceleration (S 2 ). 50 Pedestrians at 1.60 Hz -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] 2nd Generation. Vertical numerical acceleration (S 2 ). 50 Pedestrians at 1.60 Hz -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] 3th Generation. Vertical numerical acceleration (S 2 ). 50 Pedestrians at 1.60 Hz 58 As Figure 25 shows, the correlation between the experimentally recorded vertical acceleration and the numerically estimated values is adequate, in terms of both the value of the maximum acceleration and its temporal variation. Finally, the numerically estimated vertical acceleration at section 2 Sof the footbridge under a group of 50 pedestrians for different step frequencies was used to identify the first natural frequency of the structure, following the procedure described in Paper B. Figure 26 illustrates the correlation between experimental and numerical results for the change of the first vertical natural frequency of the footbridge induced by the crowd-structure interaction phenomenon. The numerical estimation of the change of the first vertical natural frequency was obtained from the mean values of ten simulations for each step frequency. Good agreement between both sets of results is observed, with differences below 1.50 % for all the analysed pedestrian walking frequencies. The first vertical natural frequency corresponding to the empty footbridge is included in Figure 26 for reference. Figure 26. Change of the first vertical, ver f,1 [Hz], experimental (Exp.) and numerical (Num.) natural frequency versus the step frequency s f [Hz]. 6.2. Analysis of the lateral lock-in phenomenon on the Pedro e Inês footbridge. The Pedro e Inês footbridge is located at Coimbra (Portugal). The total length of the structure is 274.5 m, configured by one central arch of 110 m, two lateral semiarches of 64 m and two transition spans of 30.5 and 6 m (Figure 27). The main feature of the footbridge is the anti-symmetrical configuration of the deck and the arches with respect to the longitudinal axis of the structure. The deck is a concretesteel composite box-girder with a variable width between 4 and 8 m, what generates a panoramic square at mid-span of the footbridge (Figure 28.a). From its design phase, the numerical studies developed about the footbridge indicated that the structure was prone to vibrations induced by pedestrians in lateral direction. This fact motivated the development of a precise and detailed work for the experimental assessment of its dynamic response and the implementation of a control system in order to guarantee an adequate comfort level for the footbridge. 2.940 2.960 2.980 3.000 3.020 3.040 3.060 3.080 3.100 3.120 3.140 3.160 1.30 1.50 1.70 1.90 2.10 2.30 2.50 f 1,ver [Hz] f s [Hz] Exp. Num. Empty 59 This work was performed and reported by Caetano et al. (2010) and its results have been used in this Thesis in order to validate the proposed crowd-structure interaction model in lateral direction. Figure 27. Elevation and plan of the Pedro e Inës footbridge (Caetano et al., 2010). The footbridge presented a first lateral vibration mode with an experimental natural frequency of 0.91 Hz and an associated damping ratio of 0.55 % that was easily excited by the pedestrian flows. In order to determine experimentally the number of pedestrians that originates the lateral lock-in phenomenon an experimental test was performed. Subsequently, in order to validate the performance of the proposed crowd-structure model, an experimental and numerical analysis of the lateral lockin phenomenon on the Pedro e Inês footbridge has been correlated. The analysis focused on the beginning of the instability phenomenon, as it is the situation where the effect of the modal parameters of the pedestrians has more influence in the dynamic behaviour of the structure (Dallard et al., 2001). The numerical lateral lock-in simulation is obtained from the implementation of the proposed crowdstructure interaction model on an updated finite element model of the Pedro e Inês footbridge reported in the literature (Caetano et al., 2010). Figure 28. a) Perspective of the footbridge and b) experimental lateral lock-in pedestrian test on this footbridge (Caetano et al., 2010). a ) b ) 60 In the experimental lateral lock-in test, the lateral acceleration, lat a, at mid-span of the footbridge under the crossing of different group of pedestrians was recorded (Figure 28.b). A graphical representation of the maximum lateral acceleration at this position versus the number of pedestrians on the footbridge (Figure 29) allows identifying the instability situation associated with the lateral lock-in phenomenon. As it is reported in the literature (Caetano et al., 2010) and it is illustrated in Figure 29 the number of pedestrians that originates the beginning of the lateral lock-in phenomenon is around 75. Figure 29. Experimental (Caetano et al. 2010) and numerical variation of the maximum lateral acceleration,   max lat a, at mid-span versus the number of pedestrians. Subsequently, a numerical lateral lock-in analysis based on the proposed crowdinteraction model was performed. Each considered group of pedestrians was simulated considering as initial spatial distribution a rectangular-shaped grid with an initial distance among pedestrians 50.0  p d m and a equidistant distribution in the width of the deck. The coordinates of the considered lateral vibration modes of the structure were considered from the results provided by the literature (Caetano et al. 2010). In order to account for the change of the structural damping of the footbridge according to its vibration level a parabolic function has been established based on the results obtained by Georgakis and Jorgesen (Georgakis and Jorgesen, 2014) in a laboratory footbridge. The range of variation of the damping ratio was comprised between the experimental value obtained in the previously mentioned free vibration test and the limit value under strong vibrations proposed by the more recent international standards (Butz et al., 2007; Setra, 2006). The maximum numerical lateral acceleration at mid-span versus the number of pedestrians on the footbridge is shown in Figure 29. As Figure 29 shows, the correlation between the experimental lateral maximum accelerations and the numerically estimated maximum values are adequate. Additionally, the estimation of the numerical maximum acceleration obtained applying the methodology proposed by the more 0.00 0.10 0.20 0.30 0.40 0.50 0.60 15 25 35 45 55 65 75 85 (a lat ) max [m/s 2 ] Number of pedestrians Lock-in criterion (Setra, 2006) Exp. Num. (Synpex) Num. (TDOF-system) 61 recent international standards (Butz et al., 2007; Setra, 2006) is shown in Figure 29. The proposed model allows obtaining a more accurate numerical analysis of the lateral lock-in phenomenon than the considered standards. The lateral lock-in criterion established by French standards (Setra, 2006) is also illustrated for reference in Figure 29. Figure 30. Experimental (Caetano et al., 2010) and numerical variation of the first lateral, lat f,1 , natural frequency of the footbridge versus the number of pedestrians. Finally, the first lateral numerical natural frequency of the footbridge during the occurrence of the lateral lock-in phenomenon was obtained and it is shown in Figure 26. The experimental first lateral numerical natural frequency (Caetano et al., 2010) is also shown in Figure 30. Good agreement between both sets of results is observed, with differences below 0.35 % in the studied range of the number of pedestrians. Additionally, the value of the first lateral natural frequency corresponding to the empty footbridge is illustrated for reference (Figure 30). 0.885 0.890 0.895 0.900 0.905 0.910 0.915 65 70 75 80 85 f 1,lat [Hz] Number of pedestrians Exp. Num. Empty 62 7. Conclusions and future research. 7.1 Conclusions. In this work, a new crowd-structure interaction model has been presented. The proposed model has been validated through the correlation between the experimental and numerical dynamic responses of two real footbridges under the pedestrian action. The proposed model is organized in two sub-models: (i) a pedestrian-structure interaction and (ii) a crowd sub-model. The pedestrianstructure interaction sub-model is defined in terms of a TDOF-system, with sprung and unsprung masses, whose parameters have been estimated experimentally from the results of two experimental tests conducted on the Viana footbridge (Viana do Castelo, Portugal). As identification technique the solution of an inverse dynamic problem has been utilized, minimizing an objective function defined as the mean square differences between an experimental and numerical magnitude. The estimation of the parameters of the model has been limited to vertical and lateral direction since there are few reported cases of vibratory problems in longitudinal direction. In vertical direction, the experimental and numerical accelerations on four points of the Viana footbridge under the crossing of two pedestrians at controlled step frequencies have been considered as objective function. In lateral direction, the identification process has been divided in two steps. In the first step, the modal parameters of the TDOF-system has been estimated considering as objective function the mean square error between the first experimental and numerical lateral natural frequency of the Viana footbridge under the crossing of a group of fifty pedestrians at different controlled step frequencies. Subsequently, in the second step the walking pedestrian lateral force of the proposed model is estimated considering as objective function the mean square differences between the experimental and numerical power spectral density obtained in four points of the Viana footbridge under the crossing of the two mentioned pedestrians. For the minimization of the above objective functions, as global optimization method, the genetic algorithms have been used in all the cases. The estimated parameters are within the range recommended by previous works in the literature. The crowd sub-model is defined in terms of a multi-agent model based on the relationships established by the social force model. The interaction between the two sub-models is achieved by imposing two behavioural conditions, a comfort and lateral lock-in thresholds. If the vertical or lateral accelerations experienced by each pedestrian are above certain acceleration limits, the affected pedestrian modifies his step velocity. Additionally, if the lateral accelerations exceed the limit established by the French standard in order to characterize the lateral lock-in phenomenon, the affected pedestrian synchronizes his/her frequency step and phase shift with the movement of the deck. The proposed model is formulated under the following hypothesis: (i) the parameter of the pedestrian-structure interaction model are assumed constant, so that they do not vary according to the step frequency of each pedestrian, (ii) the 69 Newland D.E. (2004). Pedestrian excitation of bridges. 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Vibration serviceability of footbridges under human-induced excitation: a literature review. Journal of Sound and Vibration, 279 (1-2): 1-74. PART II APPENDED PAPERS 71 II. Appended papers. 72 Paper A: A direct-pedestrian structure interaction model to characterize the human induced vibrations on slender footbridges The original version of this paper can be found in doi: 10.3989/ic.2014.v66.iExtra-1 Journal name: Informes de la construcción ISI 2014 Classification: Q4 (53/59) Construction and Building Engineering. Impact Factor: 0.273 SCIMAGO 2014 Classification: Q3 (121/215) Civil Engineering SJR: 0.345 ISSN: 0020-0883 Informes de la Construcción Vol. 66, EXTRA 1, m007 diciembre 2014 ISSN-L: 0020-0883 doi: http://dx.doi.org/10.3989/ic.13.110 Recibido/Received: 19/07/2013 Aceptado/Accepted: 19/11/2013 ABSTRACT Although the scientific community had knowledge of the human induced vibration problems in structures since the end of the 19th century, it was not until the occurrence of the vibration phenomenon happened in the Millennium Bridge (London, 2000) that the importance of the problem revealed and a higher level of attention devoted. Despite the large advances achieved in the determination of the human-structure interaction force, one of the main deficiencies of the existing models is the exclusion of the effect of changes in the footbridge dynamic properties due to the presence of pedestrians. In this paper, the formulation of a human-structure interaction model, addresses these limitations, is carried out and its reliability is verified from previously published experimental results. Keywords: Slender footbridges; human induced vibration; pedestrian-structure interaction; dynamic behaviour change. RESUMEN Aunque la comunidad científica tenía conocimiento de los problemas vibratorios inducidos por peatones en estructuras desde finales del siglo xix, no fue hasta la ocurrencia de los eventos vibratorios acontecidos en la pasarela del Milenio (Londres, 2000), cuando la importancia del problema se puso de manifiesto y se le comenzó a dedicar un mayor nivel de atención. A pesar de los grandes avances alcanzados en la caracterización de la fuerza de interacción peatón-estructura una de las principales deficiencias de los modelos existentes es la exclusión del cambio en las propiedades dinámicas de la pasarela por la presencia de peatones. En este artículo, se presenta la formulación de un modelo de interacción peatón-estructura que intenta dar respuesta a dichas limitaciones, y su validación a partir de resultados experimentales previamente publicados por otros autores. Palabras clave: Pasarelas esbeltas; vibraciones inducidas por seres humanos; interacción peatón-estructura; modificación de comportamiento dinámico. (*) University of Seville (España). Persona de contacto/Corresponding author: [email protected] (J. F. Jiménez-Alonso) A direct pedestrian-structure interaction model to characterize the human induced vibrations on slender footbridges Un modelo directo de interacción peatón-estructura para caracterizar las vibraciones inducidas por peatones en pasarelas esbeltas J. F. Jiménez-Alonso(*), A. Sáez(*) Cómo citar este artículo/Citation: Jiménez-Alonso, J. F., Sáez, A. (2014). A direct pedestrian-structure interaction model to characterize the human induced vibrations on slender footbridges. Informes de la Construcción, 66(extra-1): m007, doi: http://dx.doi.org/10.3989/ic.13.110. Licencia / License: Salvo indicación contraria, todos los contenidos de la edición electrónica de Informes de la Construcción se distribuyen bajo una licencia de uso y distribución Creative Commons Reconocimiento no Comercial 3.0. España (cc-by-nc). J. F. Jiménez-Alonso, A. Sáez Informes de la Construcción, Vol. 66, EXTRA 1, m007, diciembre 2014. ISSN-L: 0020-0883. doi: http://dx.doi.org/10.3989/ic.13.1102 1. INTRODUCTION The phenomenon of interaction between pedestrians and bridges is known since, at the end of the 19th century (1), a group of 60 soldiers excited, under their step, a bridge located in the British town of Broughton. Although the scientific community did not stop studying this issue, it was the occurrence of the phenomenon happened in the Millennium Bridge (London) that stressed the importance of the problem and led to a higher level of attention (2). In most cases, the effect that the pedestrians induce on the footbridge has been idealized like a moving variable force on the structure (3). The variability of the above mentioned load tries to have in consideration the variation of the level of pressures that takes place between the pedestrian and the deck during the phenomenon of the step. However, in all these models, either the effect that the pedestrians have on the dynamic characteristics of the structure is neglected, or such effect is considered by means of very simplified finger rules. Consequently, these models do not incorporate appropriately the energetic exchange that takes place between both systems during the step of the pedestrian flows on the structure. Nevertheless, in the existing publications (3) there are clear indications about the importance of the dynamic interaction phenomena, with evidence that both the frequencies and the modes of vibration of the structure are affected by the step of pedestrian groups. In the case of structures subjected to large pedestrian flows, the correct estimation of the change of their dynamic properties due to the pedestrian crossing is very important during the design phase, in order to adjust as much as possible the natural frequencies of the structure outside the range of pedestrian step frequencies and, in the case of an intervention on an existing footbridge, in order to improve its comfort level (4) (5). In the present work, a methodology for the correct characterization of the whole dynamic behavior is proposed, by implementing a human-structure interaction model with three degrees of freedom, in order to characterize the movement of the gravity center of the pedestrian in the three spatial directions. The problem of energetic exchange is addressed in a direct form, realizing the modal projection of the coordinates in contact between the pedestrian and the structure, and maintaining the physical coordinates of the gravity center of the pedestrian. The model considers, in the same way, the local effect of the step by means of the modal projection of the corresponding interaction force. This procedure of resolution allows, on the one hand, to uncouple the equations of the dynamic system that governs the behavior of the structure, thus facilitating the effective application of the model from the modal characteristics of the footbridge, as obtained from any commercial software based on the finite element method; and on the other hand, it allows to estimate in a direct form both the dynamic characteristics of the structure during the pedestrian step, as well as the components of the pedestrian center of gravity acceleration. Furthermore, additional parameters, such as the sign of pedestrian damping introduced into the system, may be included in the model. Finally, a validation example of the proposed model is presented, where the change of the dynamic behaviour of a real laboratory footbridge during a variable flow of pedestrians is favorably compared with the model predictions. 2. ANALYSIS OF CURRENT STANDARDS Currently, the most advanced international codes about the dynamic behaviour of slender footbridges (4) (5) determine that, in a wide way, if the natural frequencies of the structure is in the range of pedestrian walking step frequency (1.252.30 Hz for vertical vibrations and 0.50-1.20 Hz for horizontal vibrations) the acceleration, in that direction, needs to be determined and checked against acceleration limits (Table 1) to guarantee an appropriate comfort level for each design scenario. Furthermore, to avoid lateral synchronization the acceleration in this direction must be below 0.10-0.15 m/s2. The design scenario is established by the expected pedestrian traffic (Table 2) and the situation or importance of the structure. The comfort level is determined by the owner of the structure, and normally a medium comfort level must be guaranteed for all traffic classes, except for pedestrian densities above 1.00 P (Person)/m2 where a minimum comfort is acceptable. The pedestrian induced action is represented as an oscillatory distributed load p(t), defined as: [1] () cos( 2) πψ =⋅ ⋅⋅⋅⋅ ′⋅pt Gf tn p where: G, is the considered component of the step force (G=280 N vertical, 140 N longitudinal and 35 N lateral) (4) (5). f, is the natural frequency of the structure under consideration. ′ np, is the equivalent pedestrians number, defined by [2] 10.80 ζ ′=⋅⋅ nn pp for traffic classes TC1-TC3 or [3] 1.85 ′=⋅ nn pp for traffic classes TC4-TC5. ψ, is the reduction coefficient that takes into account the probability that the footfall frequency approaches the natural frequency under consideration. ζ, is the structural damping ratio. np, is the number of the pedestrians on the loaded surface S (np = S · density). S, is the loaded surface that depends on the shape of the normal mode under consideration. Table 1. Defined comfort classes with limit acceleration ranges (5). Level Degree Vertical acceleration Horizontal acceleration CL1 Maximum <0.50 m/s2<0.10 m/s2 CL2 Medium 0.50-1.00 m/s20.10-0.30 m/s2 CL3 Minimum 1.00-2.50 m/s20.30-0.80 m/s2 CL4 Discomfort >2.50 m/s2>0.80 m/s2 Table 2. Traffic classes (5). Classes Density d [P/m2] Characteristics TC1 < 15 P 15 single persons TC2 < 0.20 P/m2Comfortable and free walking TC3 < 0.50 P/m2Unrestricted walking, significantly dense traffic TC4 < 1.00 P/m2Uncomfortable situation, obstructed walking TC5 < 1.50 P/m2Unpleasant walking, very dense traffic A direct pedestrian-structure interaction model to characterize the human induced vibrations on slender footbridges Un modelo directo de interacción peatón-estructura para caracterizar las vibraciones inducidas por peatones en pasarelas esbeltas Informes de la Construcción, Vol. 66, EXTRA 1, m007, diciembre 2014. ISSN-L: 0020-0883. doi: http://dx.doi.org/10.3989/ic.13.110 3 However, this methodology presents some limitations: • the equivalent pedestrian number (pedestrian moving in phase with the structure) has been determined by the experimental results of only one footbridge (5). • the change in the dynamic structural properties that the pedestrians flow causes is considered through a finger rule (addition of all the pedestrian mass density to the structure mass matrix). • the interaction between pedestrians and the structure is only slightly considered, so the international standards do not consider adequately the synchronization phenomenon between pedestrians or between these ones and the structure. The estimations carried out, under this methodology, normally overestimate the real results (5). 3. PROPOSAL OF A HUMAN-STRUCTURE INTERACTION MODEL In this section a method for the simulation of the interaction between the pedestrian and the footbridge is proposed. It follows from the application of the dynamic equilibrium equations to a simplified model of interaction with sprung and unsprung masses (Figure 1). For n modes of vibration φi (x), the total response of the structure may be decomposed in terms of the amplitude of the different modes yi (t) as: [4] w(x,t)=yi(t)⋅ ϕ i(x) i=1 n ∑ [5]  w(x,t)= yi(t)⋅ ϕ i(x) i=1 n ∑+yi(t)⋅v⋅′ ϕ i(x) i=1 n ∑ [6]  w(x,t)= yi(t)⋅ ϕ i(x) i=1 n ∑+2⋅ yi(t)⋅v⋅′ ϕ i i=1 n ∑(x)+yi(t)⋅v2⋅′′ ϕ i(x) i=1 n ∑ where [7] ′ ϕ i(x)=d dx ϕ i(x) is the spatial derivate of the mode of vibration i. [8] ′′ ϕ i(x)=d2 dx2 ϕ i(x) is the second spatial derivate of the mode of vibration i. and it is neglected, due to its low magnitude, the temporal variation of the step speed v. Considering the equilibrium of the system, structure and pedestrian model, the following coupled equation system may be obtained. [9] Mi yi+Ci yi+Kiyi= ϕ ivt ( ) ⋅Fint [10] 0   ()() +−+−=my cy ykyy aa as as [11] int   ()() +−+−=−my cy ykyy FF ss sa sa s Thus, Fint follows from the above equation to yield. [12] int  ()() =− −− −− FFmy cy yk yy ssss as a And substituting this equation into the equilibrium equation of the structure. [13] M i  y i +C i  y i +K i y i = ϕ i vt ( ) ⋅F s −m s  y s −c y s − y a ( ) −k y s −y a ( ) ( ) Applying the equations of compatibility of displacements, velocity and acceleration between the structure and the simplified model of interaction. [14] (,)=ywxt s [15] (,)  =ywxt s [16] (,)   =ywxt s Substituting these relations in the overall dynamic equilibrium equation of the structure and organizing information in a matrix form, the following model of interaction is obtained. [17] () () () () () () ()   ⋅+ ⋅+ ⋅= Mt yt Ct yt Kt yt Ft Considering the nature of the resulting system, the use of a method of β-Newmark integration family is proposed, with parameters β=1/4 and γ=1/2, thus ensuring an unconditionally stable system. Figure 1. Pedestrian-structure interaction model. J. F. Jiménez-Alonso, A. Sáez Informes de la Construcción, Vol. 66, EXTRA 1, m007, diciembre 2014. ISSN-L: 0020-0883. doi: http://dx.doi.org/10.3989/ic.13.1104 under three controlled group of pedestrians, will be compared in order to determine the parameters and goodness of the proposed model. The validation will be carried out, for simplicity, in the vertical direction, although the extracted results are easily extrapolated to the other directions. 4. DETERMINATION OF THE WALKING FORCES The movement of the body mass and the put-down, rolling and push-off of the feet of one pedestrian generate the induced three-dimensional forces between both elements, Fs, that according to the research developed by different authors (4), can be determinated from a Fourier series decomposition in the three-space components. [20] Fp,vert (t)=P1+ α i,vert sin 2 π ifst− φ i ( ) i=1 nf ∑ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ [21] Fp,lat (t)=P α i,lat sin π ifst− φ i ( ) i=1 nf ∑ [22] Fp,long(t)=P α i,long sin 2 π ifst− φ i ( ) i = 1 nf ∑ where Fp,vert vertical periodic force due to walking or running Fp,lat lateral periodic force due to walking or running Fp,long longitudinal periodic force due to walking or running P [N] medium pedestrian weight (internationally considered as P=700.00 N) αi,vert αi,lat αi,long Fourier coefficient of the ith harmonic for vertical, lateral and longitudinal forces or dynamic load factor (DLF). fs [Hz] step frequency φi phase shift of the ith harmonic nf total number of contributing harmonics. Among the contributions of the different authors, for the development of the present document, the vertical dynamic In the previous expressions, the value of the vibration modes is zero, when the pedestrian remains outside the structure. [18] ϕ i(x)=0 for 0≥x≥L for , with L being the length of the structure In the proposed method, φi (x) is obtained, in a discrete way, using the finite element method, collecting the modal displacements and derivates in each of the nodes of the structure. To obtain a continuous function of the modes they are determined from the shape functions consistent with the finite element approximation. For the footbridge, the interpolation functions are cubic adopting the Bernoulli hypothesis for the beam elements. [19] ϕ i(x)= ϕ i j⋅Nj(x) j ∑ Where Nj (x) are the shape functions and ϕ i j are the nodal values. For a group of k pedestrians (Figure 2), we may further represent each one by the above simplified interaction model. When a group of pedestrians is considered in the calculations, the number of differential equations to solve increases. In the case of a single pedestrian, the proposed model leads to a system of n+1 equations, corresponding to the considered number of vibration modes n plus the appropriate simplified interaction mechanical element system. Similarly, when considering a group of k pedestrians, a system of n+k differential equations will need to be solved. It is important to note that the equations for the modes of vibration of the structure vary in terms of the position of pedestrians. At every instant, the numbers of pedestrian on the deformed shape must be calculated, as well as the value of the amplitude, slope and curvature corresponding to their position. As a preliminary validation of the proposed formulation, the previously defined parameters will be estimated from the results available in the literature for comparable studies (6), as summarized in next sections. Finally, the experimental and numerical dynamic characteristics of a laboratory footbridge, Figure 2. Pedestrians group according to the pedestrian-structure interaction model. A direct pedestrian-structure interaction model to characterize the human induced vibrations on slender footbridges Un modelo directo de interacción peatón-estructura para caracterizar las vibraciones inducidas por peatones en pasarelas esbeltas Informes de la Construcción, Vol. 66, EXTRA 1, m007, diciembre 2014. ISSN-L: 0020-0883. doi: http://dx.doi.org/10.3989/ic.13.110 5 load factors proposed by Setra (5) (see Table 3 and Figure 3) will be considered to construct and validate our model. This criterion is widely accepted by both the scientific community and the designers of this type of structures. These standards obtained the dynamic coefficients from experimental tests performed on mobile platforms. The pedestrian load, according the results of such tests, is adequately characterized by the contribution of the first three harmonics. The relation between the velocity magnitude, v, and the pacing frequency, fs, is considered by the empirical relationship based on the work of Bertram and Ruina (3). [23] fs=0.35⋅v3−1.59 ⋅v2+2.93⋅v 5. INITIAL ESTIMATION OF THE DYNAMIC PROPERTIES OF THE HUMAN-STRUCTURE INTERACTION MODEL For the estimation of the dynamic characteristics of the SDOF-system, as a first approximation, a wide bibliographic study has been made. There are several studies that collect the effect of spectators on stadiums stands in the dynamic behaviour of the structure by a SDOF static system (6). In Figure 4 and Table 4, a scheme of the models used and the estimated dynamic parameters are shown, where fh and ζh are the natural frequency and the equivalent damping ratio of the human system. The above results allow establishing a likely range of variation of the system parameters. Thus, considering the maximum and minimum values (Figure 4) of the sprung mass (ma), the equivalent human damping ratio (ζh) and the vertical natural frequency (fh) of each pedestrian, it is shown in Table 5 a possible range of variation of the parameters of the proposed model. The sprung mass (ma) is presented as a percentage of the total mass. A pedestrian type with a mean total mass of 70.00 kg is considered, as established by the European standards (4). Figure 3. Vertical component of walking pedestrian force (5). Figure 4. Simplified dynamic representations of the standing human body: (1) SDOF model (2) SDOF model with rigid support. Table 4. Dynamic properties of SDOF equivalents to standing humans (6). Human Model Modal Properties Human Model Modal Properties Foschi et al. (Model 1) fh=3.30 Hz Falati (Model 1) fh=10.43 Hz ζh=53.00 % ζh=50.00 % ma=91.00 kg ma=25.00 kg Al-Foqaha’a (Model 1) fh=3.50 Hz Zheng and Brownjohn (Model 1) fh=5.24 Hz ζh=34.00 % ζh=39.00 % ma=83.00 kg ma=85.00 kg Al-Foqaha’a (Model 2) fh=3.70 Hz Matsumoto and Griffin (Model 1) fh=5.74 Hz ζh=36.00 % ζh=69.00 % ma=75.00 kg ms=8.00 kg ma=76.10 kg Brownjohn (Model 1) fh=4.90 Hz Matsumoto and Griffin (Model 2) fh=5.88 Hz ζh=37.00 % ζh=61.00 % ma=80.00 kg ma=70.60 kg ms=7.06 kg Table 3. Fourier coefficient and phase shift for vertical dynamic load factors (5). α1α2α3φ1φ2φ 0.40 0.04 0.04 0.00 90.00 90.00 pedestrian-structure interaction sub-model plus (ii) a crowd sub-model. The first sub-model follows from a modal projection of a system with two d.o.f, that simulates the behavior of each pedestrian, on the vibration modes of the structure. The parameters of this model have been estimated from the accelerations recorded on a real footbridge. For the second sub-model, the crowd behaviour is simulated via a multi-agent method. The performance of the resulting overall model is assessed by correlating the experimental and numerical dynamic of a real footbridge under a group of pedestrians at different controlled step frequencies. In particular, the change in the first natural frequency induced by the pedestrian-footbridge interaction is discussed in detail. The proposed model leads to numerical results that exhibit good agreement with the recorded experimental values. Therefore it is a valuable tool to estimate the change on the modal properties of a footbridge induced by the crowd-structure interaction phenomenon. Keywords: simplified biomechanical model, human-structure interaction, crowd dynamics, change of natural frequencies, footbridge. INTRODUCTION. During the last fifteen years, significant effort has been made by the scientific community to characterize adequately the dynamic response of footbridges under pedestrian flows (Racic et al., 2009; Zivanovic at al., 2005). Although important advances have been achieved in the definition of the pedestrian walking force (Butz et al., 2007; Setra, 2006), some aspects of the crowd-structure interaction problem have not been completely solved and still deserve attention. The study of the crowd-structure interaction problem has been performed according to three key aspects in order to: (i) characterize the walking force transmitted by each pedestrian; (ii) characterize the pedestrian-structure interaction and finally (iii) to characterize the interaction among pedestrians in the crowd. In this way, research efforst focused initially on the determination of analytical expressions for the walking force induced by a pedestrian (Zivanovic et al., 2005; Butz et al., 2007). However, as the increasing sophistication of this proposed expressions, characterizing the pedestrian walking force, did not lead to significant improvements in the numerical estimations of the response of the footbridge under pedestrian flows, new factors were considered in the models. In that sense, as a result of the research conducted at the Millennium footbridge (Dallard et al., 2001), it was concluded that the effect of a pedestrian flow on the structure involved not only an equivalent pedestrian force but also the modification of the dynamic properties of the structure. Subsequently, this result was validated by other reported works (Ingolfsson et al, 2008) where pedestrians were considered as active damping forces that increased the overall damping of the structure. Following these works, several approaches have been presented in the literature to model de pedestrian-structure interaction problem. The first models maintained the idea of equating the pedestrian to an active viscous damper (Georgakis and Jorgesen, 2013). Subsequently, others modal parameters were considered in the interaction phenomenon leading to the appearance of single degree of freedom systems to characterize the behaviour of each pedestrian (Shahabpoor et al., 2013). According to these latter models, each pedestrian induced a modification of both the damping and stiffness matrix of the footbridge during its crossing. On the other hand, the characterization of the dynamic response of the structure under pedestrian flows motivated the study of how the pedestrians interact in a crowd (Zivanovic at al., 2010). In that sense, both statistical distributions of the step pedestrian frequencies in a crowd (Venuti et al., 2007) and relations between the pedestrian velocity and step frequency (Bruno and Venuti, 2009) were established. In order to characterize the crowd behaviour, the models have evolved from a macroscopic to a microscopic approach. The movement of the crowd simulated originally by the fluid mechanics laws (Venuti et al., 2007), is currently modelled using particle dynamics (Carroll et al., 2012), by considering each pedestrian as an agent whose equilibrium is achieved through the interaction forces applied by its environment. Currently, several proposals have emerged (Venuti et al., 2014, Tavares et al. 2014).in order to offer a direct and joint response to the three above mentioned key aspects. In all these proposals, the crowd-structure interaction has been simulated through the coupling of two sub-models: a pedestrian-structure interaction model and a crowd model based on multi-agent theory. In this paper, a new crowd-structure interaction model in the vertical direction is proposed. The model represents an evolution of the existing proposals and aims to improve some shortcomings of the previous reported works. The proposed model involves two sub-models as well. A pedestrian-structure interaction sub-model follows from the modal projection of a two degree of freedom system, where the pedestrian mass is divided in sprung plus unsprung components, on the vibration modes of the structure. The crowd behaviour is simulated via a multi-agent sub-model where the movement of each pedestrian is governed by the experienced interaction forces. A physical interaction force has been included to assess more accurately the crowd behaviour under high pedestrian densities. The interaction between the two sub-models in the vertical direction is achieved by implementing a stop threshold, so that if certain acceleration limit is exceeded the affected pedestrians stop. The estimation of the parameters of the proposed pedestrian-structure interaction model has been experimentally performed based on the results of a pedestrian test conducted at the Viana footbridge (Viana do Castelo, Portugal). Subsequently, the crowd-interaction model has been assessed by correlating the experimental and numerical response of the Viana footbridge under a group of 50 pedestrians. Finally, the model has been applied to study the numerical change of the first vertical vibration mode of the Viana footbridge due to the presence of the group of pedestrians. The proposed model may further be used to predict the occurrence of the lateral lock-in phenomenon in footbridges or to improve the efficiency of the control devices introduced in a footbridge when vibratory problems are detected. The paper is organized as follows: The proposed crowd-structure interaction model is presented in section 2, by describing (i) the pedestrian-structure interaction sub-model; (ii) the crowd sub-model as well as (iii) the interaction mechanisms between both submodels. Section 3 is devoted to the experimental estimation of the main parameters that characterize the pedestrian-structure interaction sub-model. In section 4, the validity and accuracy of the overall crowd-structure interaction model is successfully assessed by correlating both the experimental and numerical results for a real footbridge (Viana do Castelo, Portugal). Finally, some concluding remarks are drawn to close the paper in section 5. PROPOSAL OF A SIMPLIFIED BIOMECHANICAL CROWD-STRUCTURE INTERACTION MODEL IN VERTICAL DIRECTION. The complete crowd-structure interaction consists of two individual submodels (Fig. 1), one for the pedestrian-structure interaction (that includes the pedestrian and footbridge dynamic behaviour) and another for the crowd. The pedestrian-structure interaction model is responsible for modelling, in a simplified way, all the dynamics effects (inertia, damping, stiffness) induced on the footbridge by the crossing of a pedestrian. The vertical acceleration a z  experimented by each pedestrian is obtained as output from this model. The crowd model is implemented as a behavioural model providing a description of the individual pedestrian position, p x, walking pedestrian velocity, p v, and step pedestrian frequency, p f, what allows for simulating the overall behaviour of the crowd and its influence in the dynamic behaviour of the footbridge. Fig.1. Layout of the biomechanical crowd-structure interaction model. For each iteration the crowd model determines the position and step velocity of each pedestrian. These two parameters are used as input to define the step frequency and walking force of each pedestrian into the pedestrian-structure model, obtaining as output the pedestrian vertical acceleration. The pedestrian velocity of each individual is then modified according to the level of acceleration experimented by each pedestrian. Finally, the process is repeated with the updated values of the position and the velocity (Fig. 1). Modelling the pedestrian-structure interaction in the vertical direction. The proposed pedestrian-structure interaction in the vertical direction follows from the application of dynamic equilibrium equations (Clough and Penzien, 1993; Dominguez, 2001) to a simplified model of interaction (Fig. 2) with sprung ( a m) and unsprung masses ( s m). This methodology has been applied previously by the authors successfully (Jiménez-Alonso and Sáez, 2014), and in this paper it is generalized in order to take into INPUT OUTPUT PEDESTRIAN-STRUCTURE MODEL CROWD MODEL PEDESTRIAN/STRUCTURE PARAMETERS CROWD-STRUCTURE MODEL p x p va z  account the modification of the pedestrian velocity due to the crowd-structure interaction. Fig.2. Biomechanical pedestrian-structure interaction model. Considering the balance of the system, structure and pedestrian model, the following coupled equations are obtained. int_ )( FxzKzCzM piNUMiiiiii       (1)    0     sapsapaa zzkzzczm  (2)    int, FFzzkzzczm verpaspaspss       (3) where a m is the sprung mass of the pedestrian [kg]. s m is the unsprung mass of the pedestrian [kg]. as mmm  is the total mass of the pedestrian [kg]. a z is the absolute vertical displacement of the sprung mass [m]. s z is the absolute vertical displacement of the unsprung mass [m]. p k is the equivalent stiffness of a pedestrian [N/m]. m a c p k p m s z a z s F int L z xF int F s M i C i K i d p y x p w(x,t) p c is the equivalent damping of a pedestrian [sN/m]. verp F, is the vertical pedestrian force due to walking [N]. int F is the interaction force between the pedestrian and the structure [N]. i M is the modal mass of the vibration mode i [kg]. i C is the modal damping of the vibration mode i [sN/m] i Kis the modal stiffness of the vibration mode i [N/m]. iNUM _  is the vertical component of the numerical vibration mode i. tvx pxp  is the longitudinal position of the pedestrian [m]. px v is the longitudinal component of the pedestrian velocity vector [m/s]. From Eq.(3) the following expression is obtained for, int F,     aspaspssverp zzkzzczmFF       ,int ……………………(4) and substituting this equation into Eq.(1) yields.        aspaspssverppiNUMiiiiii zzkzzczmFxzKzCzM         ,_  …(5) Applying, at the contact point the equations of compatibility of displacements, velocity and acceleration between the structure and the simplified pedestrian-model of interaction are obtained. ),(),( ttvwtxwz pxps    (6) ),(),( ttvwtxwz pxps      (7) ),(),( ttvwtxwz pxps      (8) These quantities may be expressed in terms of the amplitude )(tzi and the modal shape of the n numerical considered vibration modes )( _x iNUM  , neglecting the term of variation of the pedestrian velocity over the time, as:    n i piNUMip xtztxw 1 _)()(),(  (9)     n i piNUMpxi n i piNUMip xvtzxtztxw 1 _ 1 _)()()()(),(    (10)       n i piNUMxpip n i iNUMxpi n i piNUMip xvtzxvtzxtztxw 1 _ 2 , 1 _, 1 _))()()()(2)()(),(    …(11) dx xd xiNUM iNUM )( )( _ _     (12) 2 _ 2 _ )( )( dx xd xiNUM iNUM     (13) where )( _x iNUM  is the first spatial derivate of the mode of vibration i. )( _x iNUM   is the second spatial derivate of the mode of vibration i. The above relations -Eqs.(6) to (11)- are then substituted in the overall dynamic equilibrium equations -Eqs.(1) to (3)- so that, organizing information in a matrix form, the following model of interaction is obtained (see Appendix I for matrix formulation). )()()()()()()( ttttttt FzKzCzM       (14) In the previous expressions, the value of the numerical vibration modes is zero, when the pedestrian remains outside the structure. 0)( _  piNUM x  for Ltx tx p p   )( 0)( (15) with L being the length of the structure The numerical vibration modes, )( _x iNUM  , are obtained in a discrete way using the corresponding finite element method as:   j j j iiNUM xNx )()( _  ..(16) where )(xN jare the shape functions and j i  are the nodal values. Although the paper focuses on vibrations in vertical direction. The formulation of the proposed model may be further generalized to the other two directions, longitudinal and lateral, by accordingly modifying both the equation that governs the considered pedestrian load in each direction and the value of the modal parameters that define the TDOF (two degrees of freedom) pedestrian model. In this way, the resulting model would be suitable for the more general 3-D problem and it could therefor take into account the possible interaction in the three spatial directions. For a group of k pedestrians (Fig. 2), each of them will be represented by the above simplified interaction model. In the case of a single pedestrian, the proposed model leads to a system of n+1 equations, corresponding to the considered number of vibration modes n plus the simplified interaction equation. Similarly, when considering a group of k pedestrians, a system of n+k differential equations will need to be solved. Considering the nature of the resulting system, the use of a method of  -Newmark integration family is proposed, with parameters 41   and 21   , thus ensuring an unconditionally stable system. Furthermore, the integration step, t  , is established according to the usual recommendations (Clough and Penzien, 1993; Dominguez, 2001) for dynamics models based on modal decomposition technique, as the minimum of the following values. )01.0, 4 , 200 , 8 1 min( minmin max pp vn L v L f t   sec. (17) with max f[Hz] being the highest considered vibration frequency of the structure (30 Hz according to Dominguez (2001)) and min L[m] the minimum span length of the pedestrian bridge. Pedestrian vertical walking force. The movement of the body mass and the put-down, rolling and push-off of the feet of one pedestrian generate the induced vertical forces between the pedestrian and the structure, verp F,. According to different authors (Butz et al., 2007; Setra, 2006), this force can be determined from a Fourier series decomposition as:          f n i pisveriverp tfiPF 1 ,, 2sin1  (18) where gmP  [N] is the medium pedestrian weight, g being the acceleration of the gravity. veri,  is the Fourier coefficient of the ith harmonic for vertical forces or vertical dynamic load factor (VDLF). s f [Hz] is the step frequency of the pedestrian. i  is the phase shift of the ith harmonic of the pedestrian force. p  is the phase shift among pedestrians. f n is the total number of contributing harmonics. In order to determine the number of pedestrians that cross the footbridge in phase, a Poisson distribution has been adopted, according to the results by Matsumoto et al. (1978). Further tests performed on Solferino bridge (Setra, 2006), as well as other studies, suggest that lock-in in vertical direction does not seem probable due to the low sensitivity of the pedestrians to vertical vibrations. Thus, when a group of p n pedestrians arrive at the footbridge, the number of pedestrians randomly synchronized is p n. This synchronization criterion has been adopted in our crowd-structure interaction model. Its implementation in the model is achieved by the phase shift parameter, p  . For a given generation/group of pedestrians, the value of the phase shift p D is the sliding force strength due to the contact between pedestrians (Table 3). p tis a normalized vector perpendicular to p n. pp t p vtv  is the tangential component of the relative pedestrian velocity, with p v being the difference of velocities between two given pedestrians. and function H is defined as: 00 0 )(      if if H (27) Interactions with boundaries. The interaction with the boundaries gives rise to forces, bou F. These forces are equivalent to the ones resulting from the interaction with other pedestrian, so they can be formulated in a similar fashion. tan bou nor boubou FFF  (28)  bbpb b bp b nor bou drHC B dr AnF                     exp (29)   bbpbpbbou drHD ttvF  , tan (30) where nor bou F is the normal component of the boundary interaction force. tan bou F is the tangential component of the boundary interaction force. b A is the interaction strength between the pedestrian and the boundary (Table 3).. b B is the range of the repulsive interaction between the pedestrian and the boundary (Table 3).. b d is the distance between the pedestrian and the boundary. b C is the body force strength due to the contact with the boundary (Table 3). b D is the sliding force strength due to the contact with the boundary (Table 3). b n is the normalized vector defined perpendicularly from the pedestrian to the boundary. b tis the vector perpendicular to b n. denotes scalar product. Resultant force. Finally, the proposed multi-agent model that simulates the behaviour of the crowd consists in the sum of all these partial forces that represent the different influences that the pedestrians suffer when interacting in a crowd. Therefore, the resultant force, pci F, describes the movement and direction of each pedestrian in the crowd as: boupeddripci FFFF    (31) Table 3. Crowd model parameters considered (Helbing and Molnár, 1995; Carroll et al., 2012). Parameter Element Value Relaxation time r t 0.50 sec. Interaction strength pedestrians p A 2000 N Interaction range pedestrians p B 0.30 m Potential factor p  0.20 Contact strength pedestrians p C 2000 N Sliding strength pedestrians p D 4800 N Interaction strength boundaries b A 5100 N Interaction range boundaries b B 0.50 m Contact strength boundaries b C 2000 N Sliding strength boundaries b D 4800 N Radius of pedestrian p r 0.20 m For the generation of pedestrians flows three parameters have been considered, the pedestrian density established by international standards (Butz et al., 2007; Setra, 2006) according to the expected pedestrian traffic on the footbridge, the value of the desired velocity, d v, of the pedestrians and the distance between pedestrians, p d. In this paper a one-way traffic has been considered for simplicity in the generation of the pedestrian flows, although the model may be easily generalized for two-way traffic. The values of the desired velocity of each pedestrian have been obtained from the pedestrian step frequencies, s f. For the present crowd-structure interaction model the Gaussian distribution of the pedestrian step frequency provided by Zivanovic et al (2010) has been adopted, )186.0,87.1(NHz (where ),(   N is the Gaussian distribution,  is the mean value and  is the standard deviation). After assigning a step frequency to each pedestrian, its desired velocity is determined from the empirical relation given by Bertram and Ruina (2001) (Bruno and Venuti (2009)), ppp fvvv  93.259.135.0 23 s (32) so that the initial conditions for each pedestrian assume that the pedestrian velocity, p v, is equals to the desired velocity, d v. Finally, once the pedestrian density and the desired velocity of each pedestrian are established, the original distance among pedestrians is calculated considering the width of the footbridge and assuming a rectangular-shaped mesh of pedestrians. Solution procedure. The resultant crowd-structure interaction force, pci F, acts on each pedestrian during each time iteration j. The acceleration vector, j p a, follows from m j pci j p F a (33) considering a pedestrian mass, m(as mm   ). The evaluation of the remaining variables that govern the crowd model, 1j p v and 1j p x, is then performed using a multi-step method based on a predictive-corrective method, namely the Gear’s algorithm (Heermann, 1986), due to the fact that the social forces depend on the velocity and the position of the pedestrians. The algorithm calculates first an approximate value, called a predictor that subsequently is corrected with a corrector value. The algorithm applied in this case is of fifth order. First, the new locations, velocities, accelerations and higher derivatives are predicted according to -Eqs.(34) to (37)-, where the superscript )( p indicates a predicted value. j p j p j p j p j p jp p ttt tβαavxx         2462 432 1)( (34) j p j p j p j p jp p tt tβαavv       62 32 1)( (35) j p j p j p jp p t tβαaa     2 2 1)( (36) j p j p jp ptβαα  1)( (37) where p α is the first derivative of p a and p β is the second derivative of p a. From these predicted locations, the difference between the acceleration at time step j+1 and the predicted acceleration 1)( jp p ais obtained (Eq.(38)) from 1)(1   jp p j pcor aaΔ (38) This correction factor vector, cor Δ, allows for obtaining the corrected locations, velocities and higher derivations according to 120 19 2 2 1)(1     t cor jp p j pΔxx (39) 4 3 2 1)(1     t cor jp p j pΔvv (40) 2 1 3 1 1)(1     t cor jp p j pΔαα (41) 12 1 12 1 2 1)(1     t cor jp p j pΔββ (42) Crowd-structure interaction. The maximum vertical acceleration experienced by each pedestrian crossing the structure,  max a z  may be compared against the acceleration threshold values established by international codes (Butz et al., 2007; Setra, 2006) in order to modify the individual pedestrian behaviour due to the response of the structure. Due to the good tolerance of pedestrians to vertical vibrations (Racic et al., 2009; Zivanovic at al., 2005) only a stop threshold has been established. In this way, when the vertical acceleration experienced by a pedestrian exceeds the acceleration limite, 50.2 lim  z  m/s2 (Setra, 2006), pedestrians stop walking to maintain balance, and they remain stopped until the acceleration level reduces again, considering for both actions a reaction time, 00.2 rea t sec. A linear variation of the pedestrian velocity during the reaction time has been assumed. In order to avoid meaningless small walking velocities, a practical lower limit on walking velocity magnitude has been imposed as suggested by Carrol et al. (2012).    0 1.0 d p v v if if   lim max lim max 1.0 zz vvzz a dpa     (43) EXPERIMENTAL ESTIMATION OF THE PARAMETERS OF THE PEDESTRIAN STRUCTURE INTERACTION MODEL. Inverse dynamic problem methodology implemented. The estimation of the parameters that characterize the dynamic behaviour of the proposed pedestrian-structure interaction model in vertical direction has been performed from the response of a real footbridge under the crossing of two pedestrians by solving the corresponding inverse dynamic problem methodology. In Fig. 5 the flowchart of the identification procedure is shown. Fig.5. Flowchart of the identification procedure. INITIALIZATION VARIABLES (1,ver 2,ver mahfh)k NUMERICAL ANALYSIS NUMERICAL ACCELERATIONS EVALUATION OF OBJECTIVE FUNCTION 2 exp ,, 2 1 iv num iv aa  MINIMIZATION STEP UPDATED VALUES (1,ver 2,ver mahfh)k+1 RESULT: IDENTIFIED VARIABLES (1,ver 2,ver mazhfh)=(1,ver 2,ver mahfh)k+1 YES NO k=k+1 CONVERGENCE? TDOF MODEL PARAMETERS IDENTIFICATION F.E.M. MODEL UPDATING PRELIMINARY F.E.M. AMBIENT VIBRATION TEST O.M.A. ESTABLISHING SEARCH DOMAIN VERTICAL PEDESTRIAN WALKING FORCE PEDESTRIAN MODAL PARAMETERS m a  h f h  1,ver  2,ver UPDATED F.E.M. EXPERIMENTAL PEDESTRIAN TEST TWO PEDESTRIANS f p =1.50, 2.00 AND 2.50 Hz exp ,iv a EXPERIMENTAL RESPONSE num iv a, As identification method the minimization of a least squares problem has been adopted (Koh and Perry, 2010). The objective function has been defined as the mean square error between the experimental ( exp ,iv a, where i is the considered section) and numerical ( num iv a,) vertical accelerations obtained, in four points of the Viana footbridge (Barbosa et al., 2012), under the crossing of two pedestrians at controlled step frequencies. Genetic algorithms have been used to ensure a global optimization and a search domain for each parameter has been established. The characterization of the dynamic behaviour of the footbridge has been performed by the finite element model updating (Teughels, 2003; Zivanovic et al., 2007) based on the modal parameters of the structure estimated from the application of an operational modal analysis (Magalhães and Cunha, 2011). An ambient test has been performed on Viana footbridge and the measured signals have been processed by an output-only identification method in the time domain, which provided estimates of the its first four natural frequencies, the corresponding modal shapes and the associated damping ratios (Magalhães et al., 2010). (Fig.6). Later, this test has been reproduced numerically by the implementation of the proposed pedestrianstructure interaction model. An iterative process to reduce the differences between the experimental and numerical vertical accelerations has been performed, under the rules of genetic algorithms (Koh and Perry, 2010; Nocental and Wright, 1999), and considering as design variables the first two VDLF ( ver,1  and ver,2  ) of the pedestrian walking force and the three modal parameters that characterizes the TDOF pedestrianstructure interaction model; the pedestrian sprung mass, a m, the pedestrian damping ratio, p  , and the pedestrian natural frequency, p f. Fig.6. Layout of the identification methodology. Establishing a preliminary search domain for the parameters of the pedestrianstructure model. In order to reduce the uncertainty of the estimated values of the five considered parameters and thus prevent an ill-conditioned inverse problem, a search domain has been established. According to Table 1 the minimum and maximum values of each VDLF allow establishing a search domain for their experimental estimation. In order to take into account that the estimation of these parameters will be performed from measurements carried out on a real footbridge, the above mentioned search domains have been extended. As the value of the third harmonic of the walking pedestrian force proposed by the different authors has a lower magnitude its contribution has been neglected. S1 S2 S3 S4 UPDATED F.E.M. OF THE FOOTBRIDGE v p a v,1num a v,2exp a v,1exp a v,3exp a v,4exp a v,2num a v,3num a v,4num 2 exp ,, 2 1 iv num iv aa  EVALUATION OF OBJECTIVE FUNCTION  1,ver  2,ver m a  h f h IDENTIFIED VARIABLES m a  h f h  1,ver,  2,ver X Z Y Z Y X TRIAXIAL ACCELEROMETER PEDESTRIAN MODAL PARAMETERS VERTICAL WALKING FORCE TDOF MODEL PARAMETERS IDENTIFICATION EXPERIMENTAL NUMERICAL Similarly, the phase shifts of the second and third harmonic of the vertical walking force have not been considered. Similarly, according to Table 2, the minimum and maximum values of the parameters of the passive pedestrian models allow defining the search domains for their estimation. The knowledge of the physical problem suggests: (i) a reduction of the damping and stiffness of the pedestrian associated with its movement and (ii) the change in the human body mass distribution between the active and passive states. The search domains of these parameters have been increased to take into account of these facts. Thus the following search domains have been adopted:  First VDLF,  45.000.0 ,1   ver  .  Second VDLF,  20.000.0 ,2   ver  .  Pedestrian sprung mass,   10080   a m%.  Pedestrian damping ratio,   6910   p  %.  Pedestrian natural frequency,   43.101   p f Hz. Description and finite element model of the “laboratory” footbridge: Viana footbridge. The Viana do Castelo footbridge (Barbosa et al., 2012) is a moveable cable-stayed bridge. The longitudinal structural scheme of the footbridge consists of two spans of about 36.50 m and 9.00 m respectively suspended by 6 families of two hangers (two retaining ones) from an inclined mast. The deck, with 2.50 m of width, is configured by two rolled steel beams of variable depth braced by circular hollow profiles. The deck floor is covered with wood. The compensation of the main span weight is achieved by plac i mas t allo w foun d foot b A p r to h a poin t (An s (BE A impl e for t h the s load s this asso c i ng 11 high t is welded w s the rota t d ation that b ridge is sh o Fig.7. Fini t r eliminary n a ve a first a t required s ys, 2014) w A M188) e x emente d . T h e determi n s tructure, by s and thus e preliminar y c iated nu m density bl o in its base t ional mo v balances t h o wn in Fig. t e element m n umerical fi a pproximati to perfor m w as used, b a x cept for T he nonline a n ation of t h y calculatin e stimating i y FE mod e m erical nat u o cks (with a to a cylin d ement of t h e forces tr a 7 . m odel, amb i nite eleme n o n to the d y m the ambi e a sed on a d i the hang e a r effects a h e numeric a g previous l i ts tangent s e l leads t o u ral freque n weight of 8 d er that is c t he structu r a nsmitted b y b ient test gr i n t (Fig.7) m y namic be h e nt vibrati o i scretizatio n e rs where a ssociated w a l natural f r l y the stres s s tiffness m a o the first n cies give n 8 00 kN) pl a c onnected t o r e. The pyl y the mast. i d and mod e m odal anal y h aviour of t h o n test. Th n of the str u 3D-cable w ith the ha n r equencies s level of t h a trix. The n four num e n in Table a ced in the s o a wheel g o n is con n A perspec t e l updating y sis was co n h e footbrid g e software u cture in 3 D elements ( n gers have and the vi b h e hangers u n umerical m rical vibra t 4. In Fig . s horter spa n g ear bearin g n ected to a t ive of the V parameters n ducte d in g e and a st a package A D -beam ele m (LINK10) been consi b ration mo d u nder per m m odal anal y tion mode s .8 the firs t n . The g that deep V iana . order a rting A nsys m ents were dered d es of m anent sis of s and t two Fig.10. First two vertical updated numerical (Upd.) and experimental (Exp.) vibration modes. After the development of the model updating, the numerical dynamic behaviour of the footbridge simulates accurately the real response of the footbridge. This detailed knowledge of the dynamic behaviour of the footbridge allows for adopting this real footbridge as a benchmark or “laboratory” footbridge. Experimental pedestrian test. The estimation of the parameters that characterize the behaviour of the proposed pedestrian-structure interaction model was made through the results of a pedestrian test. In the test, two pedestrians (A and B) were selected. The mass of each pedestrian was 70.61 A mkg and 50.100 B mkg. On the deck of the footbridge four triaxial accelerometers were placed at the intersection point between the hangers and the deck (Fig.6). Three series were recorded for each pedestrian, crossing the footbridge three times at different step frequencies, s f (1.50, 2.00, 2.50 Hz), controlled by a metronome. In the four monitored sections ( 1 S,2 S,3 S and 4 S in Fig.6) the dynamic response of the -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 10.00 20.00 30.00 40.00 50.00 X [m] Second updated versus experimental vertical vibration mode Upd. Exp. structure was recorded. In each passage, the initial and end time, that marks the crossing of the pedestrian on the structure, was recorded as well, in order to both localize the forced vibration response corresponding to the passage of each pedestrian and estimate the pedestrian velocity, p v. Parameters identification in time domain. For the estimation of the parameters of the TDOF-system (Koh and Perry, 2010), an inverse dynamic problem was solved, as it has been described previously. As objective function the relative differences between the experimental and numerical vertical accelerations in four sections of the footbridge ( 1 S,2 S,3 S and 4 S) has been considered. As optimization method, genetic algorithms have been used again. The experimental vertical accelerations correspond to the measurements of the above described pedestrian test. The numerical vertical accelerations have been obtained from the implementation of the proposed pedestrian-structure interaction model on the updated finite element model of the Viana footbridge. Six parameters were adopted as design variables: (i) the first two VDLF that characterize the pedestrian walking force. (ii) the three modal parameters (pedestrian sprung mass, pedestrian damping ratio and pedestrian natural frequency) that govern the behaviour of the TDOF-system. (iii) a time lag that allows adjusting the beginning of the crossing of the pedestrian between the experimental and numerical accelerations. In order to improve the reliability of the parameters estimation the level of noise of the signal has been reduced (Koh and Perry, 2010). In that way, each measurement record has been decomposed using the Wavelet transform (Gopalakrishnan and Mitra, 2014), choosing as wavelet family, the Daubechies. A level 7 of decomposition has been applied to each signal. As threshold selection rule, the principle of Stein's Unbiased Risk Estimate has been considered (Gopalakrishnan and Mitra, 2014). Subsequently, the reconstruction of the signal has been carried out using the original approximation coefficients of level 7 and the modified detail coefficients of levels from 1 to 7. For each series, a generation of 1000 individuals has been defined. Each individual modifies, according to the rules of genetic algorithms, the values of its components in order to minimize the defined objective function. In Table 7, the results of the estimation process are summarized, showing the different estimated parameters versus the pedestrian step frequency. Table 7. Estimation of the parameters of the pedestrian-structural interaction model. fs [Hz] Pedestrian sprung mass ma [%] Minimum Medium Maximum 1.50 84.995 88.367 91.738 2.00 81.636 86.467 91.297 2.50 82.119 86.583 91.048 fs [Hz] Pedestrian damping ratio  p [%] Minimum Medium Maximum 1.50 22.143 32.115 42.087 2.00 38.908 45.992 53.076 2.50 40.132 46.214 52.295 fs [Hz] Pedestrian frequency f p [Hz] Minimum Medium Maximum 1.50 1.923 2.923 3.924 2.00 2.094 2.915 3.736 2.50 2.295 2.962 3.629 fs [Hz] Vertical dynamic load factor 1 , ver Minimum Medium Maximum 1.50 0.201 0.203 0.206 2.00 0.214 0.235 0.255 2.50 0.224 0.273 0.322 fs [Hz] Vertical dynamic load factor 2 , ver Minimum Medium Maximum 1.50 0.039 0.040 0.041 2.00 0.040 0.042 0.043 2.50 0.043 0.047 0.051 According to the results of Table 7 the parameters of proposed pedestrian-structure interaction model show some dependence on the step pedestrian frequency. However due to the number of pedestrians used during the pedestrian test, only a global mean value,  , and a standard deviation,  , have been determined ),(   N. For the proposed pedestrian-structure model, the following Gaussian distributions have been considered. - First VDLF, ver,1  , )04.0,237.0(N. - Second VDLF, ver,2  , )004.0,043.0(N. - Pedestrian sprung mass, a m, )809.3,139.87(N%. - Pedestrian damping ratio, p  , )776.9,44.41(N %. - Pedestrian natural frequency, p f, )728.0,933.2(N Hz. The proposed values are inside the range, established by Shahabpoor et al. (2013), that characterizes the modal properties of TDOF pedestrian-structure interaction model. For the generation of the pedestrian flows of the crowd-structure interaction model, the above Gaussian distributions have been considered. Once obtained the parameters, the definition of the proposed model is complete. This model will be validated in the next section by correlating the numerical and experimental dynamic response of the Viana footbridge under the crossing of a group of pedestrians and the experimental and numerical analysis of the change of the first vertical natural frequency of the structure induced by the pedestrian flow. MODEL VALIDATION. The validity and applicability of the proposed crowd-structure interaction model is next assessed through its practical application to a case study. The vertical acceleration at three sections ( 1 S,2 S, and 3 S (Fig.6)) of the Viana footbridge has been measured under the crossing of a group of 50 pedestrians at different step frequencies (1.30-2.50 Hz). From the forced recorded response, the experimental study of the change of the first vertical natural frequency of the structure, due to the crossing of the group of pedestrians at different step frequencies, has been determined. Subsequently, the crowdstructure interaction model has been applied to the updated FE model in order to obtain first the vertical numerical acceleration at the mentioned sections and later to study numerically the change of the first vertical natural frequencies of the footbridge due to the presence of the pedestrians. Experimental crowd test: dynamic response and change of the first natural frequency under pedestrian flow. In the crowd test a group of 50 pedestrians has crossed the footbridge at different step frequencies (1.30-1.40-1.60-1.75-2-00-2.50 Hz) controlled by a metronome, measuring the vertical dynamic response of the footbridge at three sections 1 S, 2 S and 3 S with a tri-axial accelerometer (Fig.6). During the crowd test the group of 50 pedestrians has been distributed in three alignments, maintaining a lateral separation among pedestrians around 0.85 m and a longitudinal distance between pedestrians around 0.50 m. A pedestrian with a metronome has led the group in each crossing. A scheme of the pedestrian distribution during the crowd test is shown in Fig.11. In o r the c vibr a (hig h and Mitr a coef f natu r natu r obta i sign a met h r der to det e c rossing of a tion respo n h er modal d processed a , 2014) b f icients (Fi g r al frequen c r al freque n i ned accor d a l used for t h odology p r Fig.11. E e rmine the c the group n se of the s t d eflection) by the Co b ased on D g .12), in th e c y of the st r n cy throug h d ing to the t he estimat i r eviously d e E xperiment a c hange of t h of pedestr i t ructure ha s h ave b een ntinuous W D aubechies e filtered ra n u cture. Thi s h the pow e Pea k -Pick i i on of the p e scribed. a l crowd tes h e first nat u i ans at dif fe s been con s considered . W avelet Tr a family. T n ge of freq u s result has e r spectral ing metho d ower spect r t at Viana f u ral freque n fe rent step f s idere d . On l . The selec t a nsform ( C T he maxi m u encies, is t been valid a density of d (Magalh ã r al density h fo otbridge. n cy of the f f requencies l y the reco r t ed signal h C WT) (Go p m um value t hen correl a a ted by the e the above ã es and Cu n h as been d e f ootbridge u , just the f r ds at secti o h as been fi l p alakrishna n of the w a a ted with th e estimation o e signal (F i n ha, 2011) e noised usi n u nder f orced o n 2 S l tered n and a velet e first o f the i g.12) . The n g the Fi To i l secti o Fig. 1 The foot b Nu m first The expe cons i g .12. Esti m l lustrate th e on 2 S und e 1 3. experime n b ridge is su m m erical cro w natural fr e assessmen t rimental r e i dered ste p m ating the c h e recorded r e e r a group o n tal study o m marized i n w d test: e s e quenc y u n t of the m e sults with p frequenc y h ange of th e e sults, the m o f 50 pede s o f the ch a n Fig.14. s timation o n der pedes t m odel perf o the numeri y ten gener a e first verti c m easured v e s trians at a a nge of fi r o f the d y n a t rian flow. fo rmance h cal estima t a tions of g c al natural f e rtical acce step freque r st vertical a mic respo n as been d o ion predict g roups wit h fr equency o f l eration of t n cy of 1.6 0 natural f r n se and th o ne correl a e d by the m h 50 pedes t f the struct u t he footbri d 0 Hz is sho w r equency o e chan g e o a ting the a m odel. For t rians have u re. d ge in w n in o f the o f the a bove each been simulated. The number of pedestrians in phase in each new simulation has been determined though the evaluation of the parameter p  . The desired velocity, d v, of each pedestrian has been assigned according to Eq.(32). A pedestrian mass of 70 kg has been considered according to the French code (Setra, 2006). As initial spatial distribution of the pedestrians, a rectangular grid has been selected, considering an initial distance among pedestrians 50.0 p d m with an equidistant distribution in the width of the deck. The selected time step is 01.0  tsec. The numerical vertical acceleration (for three of the 50 pedestrians generations) at section 2 Sof the footbridge, for a step frequency of 1.60 Hz, is shown in Fig.13. -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] Vertical experimental acceleration (S2). 50 Pedestrians at 1.60 Hz -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] 1st Generation. Vertical numerical acceleration (S 2 ). 50 Pedestrians at 1.60 Hz Fig.13. Experimental versus numerical acceleration (three generations) at section S2 of Viana footbridge under a group of 50 pedestrians (walking frequency of 60.1 p f Hz). As Fig.13 shows, the correlation between the experimentally recorded vertical acceleration and the numerically estimated values is adequate, in terms of both the value of the maximum acceleration and its temporal variation. Finally, the numerically estimated vertical acceleration at section 2 Sof the footbridge under a group of 50 pedestrians for different step frequencies has been used to identify the first natural frequency of the structure, following the procedure described in the previous section. Fig.14 illustrates the correlation between experimental and numerical results for the change of first vertical natural frequency of the footbridge induced by the -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] 2nd Generation. Vertical numerical acceleration (S 2 ). 50 Pedestrians at 1.60 Hz -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.00 10.00 20.00 30.00 40.00 50.00 60.00 m/s 2 Time [sec.] 3th Generation. Vertical numerical acceleration (S 2 ). 50 Pedestrians at 1.60 Hz crowd-structure interaction phenomenon. The numerical estimation of the change of the first vertical natural frequency has been obtained from the mean values of ten simulations for each step frequency. Good agreement between both sets of results is observed, with differences below 1.50 % for all the analysed pedestrian walking frequencies. The first vertical natural frequency corresponding to the empty footbridge is included in Fig.14 for reference. Fig.14. Change of the first vertical, ver f,1 [Hz], experimental (Exp.) and numerical (Num.) natural frequency versus the step frequency p f [Hz]. CONCLUSIONS. In this paper, a new crowd-structure interaction model in the vertical direction has been presented and further validated through the correlation between the experimental and numerical dynamic response of a real footbridge (Viana) adopted as benchmark. The proposed model has been organized in two sub-models: a pedestrian-structure interaction and a crowd sub-model. The pedestrian-structure interaction sub-model is configured by a TDOF system, with sprung and unsprung masses, whose parameters have been estimated experimentally from the solution of an inverse problem on the 2.940 2.960 2.980 3.000 3.020 3.040 3.060 3.080 3.100 3.120 3.140 3.160 1.30 1.50 1.70 1.90 2.10 2.30 2.50 f 1,ver [Hz] f p [Hz] Exp. Num. Empty Koh Ghee C., Perry M.C.. Structural Identification and Damage Detection using Genetic Algorithms. CRC Press, Taylor&Francis Group, 2010. Magalhães, F., Cunha, A.. Explaining Operational Modal Analysis with data from an arch bridge. Mechanical Systems and Signal Processing, Invited Tutorial Paper, Volume 25, Issue 5, pp. 1431-1450 , 2011. Magalhães, F., Cunha, A., Caetano, E., Brincker, R. Damping estimation using free decays and ambient vibration tests. Mechanical Systems and Signal Processing, Volume 24, Issue 5, pp. 1274–1290, July 2010. Matlab R2011a. . http://www.mathworks.com/. Matsumoto, Y., Nishioka, T., Shiojiri, H., Matsuzaki, K.. Dynamic design of footbridges. IABSE Proceedings, No. P-17/78, pp. 1-15, 1978. Nocental J., Wright S.J.. Numerical Optimization. Springer, New York, USA, 1999. Racic, V., Pavic, A., Brownjohn, J.M.W. Experimental identification and analytical modelling of human walking forces: Literature review. Journal of Sound and Vibration, Vol. 326, pp. 1-49, April 2009. Rapaport, D.C.. The art of molecular dynamic. Cambridge University Press; 2004. SETRA/AFGC. Guide méthodologique passerelles piétonnes (Technical Guide Footbridges: Assessment of vibration behaviour of footbridge under pedestrian loading). SETRA, 2006. Shahabpoor, E., Pavic, A., Racic, V.. Modelling effect of pedestrians walking on dynamic properties of structures. IMAC XXXI: A Conference and Exposition on Structural Dynamics, 11-14 February, Orange County, California, USA (2013). Tavares, F., Leal, R., Shubert M. Biodynamic single person and crowd pedestrian model: a worked test case and simulation. Footbridge 2014. London, United Kingdom (2014). Teughels, A., Inverse Modelling of Civil Engineering Structures Based on Operational Modal Data. Ph. D. Thesis, Katholieke Universiteit Leuven, 2003. Venuti, F., Bruno, L., Bellomo, N. Crowd dynamics on a moving platform: mathematical modelling and application to lively footbridges. Mathematical and Computer Modelling 45 (3-4), 252-269, February 2007. Venuti, F., Racic, V., Corbetta, A. Pedestrian-structure interaction in the vertical direction: coupled oscillator-force model for vibration serviceability assessment. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014. Porto, Portugal, 30 June - 2 July 2014. Zivanovic, S., Pavic, A., Ingolfsson, E., Modelling spatially unrestricted pedestrian traffic on footbridges. Journal of Structural Engineering, Vol. 136 (10), pp. 1296-1308. 2010. Zivanovic, S., Pavic A., Reynolds P.. Finite element modelling and updating of a lively footbridge: The complete process. Engineering Structures, Vol. 301(1-2), pp. 126-145, March 2007. Zivanovic, S., Pavic A., Reynolds P.. Vibration serviceability of footbridges under human-induced excitation: a literature review. Journal of Sound and Vibration, Vol. 279, Issue 1-2, pp. 1-74, January 2005. 74 Paper C: Model updating for the selection of the retrofit method of an ancient bridge (Almeria, Spain). The original version of this paper can be found in doi: 10.2749/101686615X14355644771333 Structural Engineering International (in press) ISI 2014 Classification: Q4 (105/124) Civil Engineering. Impact Factor: 0.414 SCIMAGO 2014 Classification: Q2 (106/215) Civil Engineering SJR: 0.417 ISSN: 1683-0350 Model updating for the selection of the retrofit method of an ancient bridge (Almeria, Spain). ABSTRACT In this paper we analyse a case study where a finite element model updating is conducted on the basis of the experimental modal parameters recorded for a reinforced concrete truss bridge built in Almeria (Spain) in 1927. The final aim of this study is to help understanding the actual state of structural conservation of the bridge, in order to select an appropriate retrofit technique to reinforce it before the planned widening of its deck. At present, the two most widely used methods for retrofitting consist on either reinforcing the structure with external prestressing or increasing its flexural strength by adhering CFRP laminates. The first method is especially advantageous if the structure is deteriorated to such an extent that the strengthening needs to focus not only on increasing the flexural strength of the bridge, but also on avoiding the deflection problems associated with the reduced inertia of the structure. If this were not the case, strengthening with CFRP would be an adequate option. To this end, and given the monumental character of this ancient construction, it is not possible to apply directly destructive or static load tests to determine the properties of its constituent materials, so that it becomes necessary to estimate its deterioration state indirectly with the use of non-destructive techniques. For this purpose, in the present paper we carry out a finite element model updating of the structure, based on experimental modal parameters, which allows for checking its service condition through the estimation of the value of several physical parameters of the bridge. Subsequently, the resulting updated model constitutes a valuable tool to help establishing which is the most adequate retrofit method. Keywords: Retrofit methods, operational modal analysis, model updating, ancient reinforced concrete bridge, non-destructive testing. 1. Introduction. The Molinos Bridge is a reinforced concrete truss girder bridge (Fig. 1) located over the Andarax River at the outskirts of the town of Almería (Spain). It was built in 1927, being at present one of the best preserved achievements of the first reinforced concrete bridges built in Spain. The design of the structure corresponds to one of the schemes proposed by the Spanish engineer Juan Manuel Zafra in the official model of road bridges of the collection of reinforced concrete straight bridge for spans of 32 m, published in 1920 [1-6]. Recently, the Planning Department of the City of Almeria called for a public competition in order to proceed with the repair, reinforcement and widening of this ancient concrete bridge. The growth of the city towards the East and the attempt of promoting the tourism on the East Coast of the province made necessary to improve the road access to those neighbourhoods of the city. Since the construction of a new structure was dismissed due to its high cost, the proposal focused on adapting the existing bridge to the new service conditions. According to the technical provisions of the competition published by the Planning Department, the works for the rehabilitation and functional adaptation of the Molinos bridge should satisfy the following conditions: (i) widening the existing lanes for vehicles, increasing its width until 3.25 m per lane; (ii) creating a pavement and a cycle lane with a minimum width of 1.50 m; and (iii) the project should maintain the current typology of the bridge and still ensure its adequate bearing capacity under the new loading scenario. All the presented project proposals addressed three main aspects, namely: (i) design of a structural system to increase the width of the deck; (ii) retrofit of the main truss girders; and (iii) treatment of the existing cracking on the reinforced concrete elements. Regarding the retrofit method of the main girders, the alternatives presented by the different engineering firms may be grouped in two categories based on: (i) either use external prestressing [7] or (ii) use carbon fiber reinforced polymers (CFRP) laminates [8] to retrofit the reinforced concrete truss girders. The main factor that conditions the choice of the retrofit method is the actual deterioration level of the structure and its influence in the moment of inertia of the girder. In this way, if the structure was seriously damaged, it would be necessary to focus not only on controlling the level of stresses but on reducing the deflection of the girder as well. Therefore, in order to properly evaluate the different projects and select the most adequate retrofit method, the Planning Department of the City of Almeria decided to conduct a detailed study of the state of deterioration of the structure using nondestructive techniques. The use of the bridge, before its retrofit, was just limited to light traffic, so the Planning Department further established as a requirement the impossibility of performing a static load test, conducted with heavy loads, in order to assess the structural condition of the bridge. The present paper summarizes the work developed by the authors to evaluate the damage level of the bridge. This study describes an innovative application of the FE model updating method, using it as a valuable tool in order to find the optimum reinforcement method for a ancient bridge. The work mainly focuses on performing a FE model updating [9] of the bridge based on the experimental modal parameters. These parameters are determined from an ambient test, using operational modal analysis [10]. This study focuses on the vertical direction in order to estimate the deterioration state of the ancient bridge comparing two scenarios: (i) one defined by the current structure, from a numerical load test performed on the updated model; and (ii) a second scenario defined by the original structure, from the original experimental load test at the time of completion of the bridge [4]. The results of the study, and in particular the resulting updated model, are then used as reference for the evaluation of the different project proposals and the selection of the most appropriate retrofit method. The paper is organized as follows: In section 2, a preliminary study of the structural behaviour of the structure is presented, describing the main structural elements and the level of cracking observed on the structure from visual inspection. Furthermore, a rough estimation of the range of variation of the Young’s modulus of the concrete is obtained from a hammer rebound test. At the end of this section, a FE model of the structure is performed in order to obtain an initial estimation of the distribution of the principal stresses of the structure under self-weight and dead load. Later, the results from the hammer rebound test and the static analysis were used to establish both the physical parameters used in the updating process and its range of variation. To make the paper as self-contained as possible, a description and comparison of the two proposed retrofit method is briefly presented in section 3, and further a criterion for its selection is established. In section 4, the modal parameters of a previously selected extreme span are estimated experimentally through the application of operational modal analysis in the frequency domain. In section 5, the FE model updating of the selected span is performed. In section 6, the updated model is employed to assess the structural condition of the bridge, by comparing the results of the original load test of the bridge [4] with the numerical simulation of such load scenario using the updated model. Finally, from the obtained results, the best retrofit method is selected and several conclusions are drawn in section 7. 2. Preliminary study of the structural behaviour of the bridge. As a preliminary step for the selection of any retrofit system it is of key importance to understand the actual structural behaviour of the different elements that configure the bridge, as well as to determine the mechanical characteristics of the material and to conduct an in-depth visual inspection of the current state of the bridge. 2.1. Description of the original structure. The Molinos Bridge over the Andarax River is located along the road between Almería and Níjar (Spain). It is an isostatic structure with five spans (32.64 - 32.72 - 32.72 - 32.68 and 32.78 m) with a total length of 163.54 m (Fig. 1). The width of the original bridge is around 6.20 m, composed by two lanes of 2.20 m and two pavements of 0.20 m. Fig. 1. Elevation of the Molinos Bridge from the East abutment. Each span [2, 3 and 11] is configured by two reinforced concrete truss girders separated 2.70 m and connected at the top by a reinforced concrete slab of variable depth (0.180.54 m) with a total width of 6.20 m. The total depth of the bridge is about 2.66 m. The width of the different elements that configure the truss girders is 0.40 m. The depth of the lower chord is 0.40 m, but the depth of struts and diagonal varies between 0.18 m in the mid-span and 0.64 m over the supports. Inside the lower chord of the truss girders there are several rectangular steel plates 300x12 mm, in a number varying from 10 plates in each lower chord at the mid span to 4 plates at the supports [2, 3 and 11]. The force transmission between the lower chords and the struts and diagonals is achieved by the placement of pins of diameter 40-50 mm (Fig. 2). The trusses are jointed at the intersection between the lower chord and the struts by several reinforced concrete rectangular crossbars with 0.40 m of depth and variable width (Fig. 3.a). Due to the geometric configuration of the bridge the diagonal elements work in compression and are reinforced with longitudinal bars and a high density of stirrups. The struts of the truss girders, always in tension, are reinforced with longitudinal and transversal bars which number increases with the proximity to the supports. Fig. 2. Original details of the cross section of the bridge [11]. The piers and the abutments have a rectangular cross-section with dimensions 1.60x5.60 m and a height of 6.00 m. Each of these elements rest on a masonry block with rectangular cross-section 2.00x6.00 m and height around 5.70 m [3]. The foundations of these elements are reinforced concrete slabs with 1.50 m of depth and horizontal dimensions 4.00x8.00 m. The linking between the deck and the piers is achieved by fixed (one hinge) and sliding (two hinges) bearings (Fig. 3.b). To avoid restricting the longitudinal movements of the deck due to the rheological and thermal effects, in each pier these two types of bearings have been installed [3]. Fig. 3. a) Spatial configuration of the truss girder. b) Support on abutments 2.2. Visual inspection of the original structure. The deterioration state of the bridge was studied, as a first approximation, through its visual inspection. The main damages observed in the structure were the following: (i) Detachment of the reinforced concrete along of the truss girders. The infiltrations of water and salts, due to the impairment of the waterproofing, in the truss girders have caused the oxidation of the rebars and the appearance of the detachment of the concrete. Detachments around 3.50 cm were measured. The footbridge. Journal of Bridge Engineering, ASCE (in press). doi: 10.1061/(ASCE)BE.1943-5592.0000828 Jones, C.A., Reynolds, P., Pavic, A. (2010). Vibration serviceability of stadia structures subjected to dynamic crowd loads: A literature review. Journal of Sound and Vibration, Vol. 330, pp. 1531-1566. Koh Ghee C., Perry M.C. (2010). Structural Identification and Damage Detection using Genetic Algorithms. CRC Press, Taylor&Francis Group. Macdonald, J.H.C. (2008). Pedestrian-induced vibrations of the Clifton Suspension Bridge, UK. Proceedings of the ICE-Bridge Engineering 161 (2), pp. 69-77. Macdonald, J.H.C. (2008). Lateral excitation of bridges by balancing pedestrians. Proceedings of the Royal Society. Magalhães, F., Cunha, A. (2011). Explaining Operational Modal Analysis with data from an arch bridge. Mechanical Systems and Signal Processing, Invited Tutorial Paper, Volume 25, Issue 5, pp. 1431-1450. Magalhães, F., Cunha, A., Caetano, E., Brincker, R. (2010). Damping estimation using free decays and ambient vibration tests. Mechanical Systems and Signal Processing, Volume 24, Issue 5, pp. 1274–1290. Matlab R2015a. . http://www.mathworks.com/. Matsumoto, Y., Griffin, M.J. (2003). Mathematical models for the apparent masses of standing subjects exposed to vertical whole-body vibration. Journal of Sound and Vibration Vol. 260 (3) pp. 431-451. Matsumoto, Y., Nishioka, T., Shiojiri, H., Matsuzaki, K. (1978). Dynamic design of footbridges. IABSE Proceedings, No. P-17/78, pp. 1-15. Nocental J., Wright S.J. (1999). Numerical Optimization. Springer, New York, USA. Racic, V., Pavic, A., Brownjohn, J.M.W. (2009). Experimental identification and analytical modelling of human walking forces: Literature review. Journal of Sound and Vibration, Vol. 326, pp. 1-49. Rapaport, D.C. (2004). The art of molecular dynamic. Cambridge University Press. Ronnquist, A. (2005). .Pedestrian Induced Lateral Vibrations on Slender Footbridges. PhD Thesis. Norwegian University of Science and Technoloy. SETRA/AFGC. (2006). Guide méthodologique passerelles piétonnes (Technical Guide Footbridges: Assessment of vibration behaviour of footbridge under pedestrian loading). SETRA. Shahabpoor, E., Pavic, A., Racic, V. (2013). Modelling effect of pedestrians walking on dynamic properties of structures. IMAC XXXI: A Conference and Exposition on Structural Dynamics, 11-14 February, Orange County, California, USA. Teughels, A. (2003). Inverse Modelling of Civil Engineering Structures Based on Operational Modal Data. Ph. D. Thesis, Katholieke Universiteit Leuven. Venuti, F., Bruno, L., Bellomo, N. (2007). Crowd dynamics on a moving platform: mathematical modelling and application to lively footbridges. Mathematical and Computer Modelling 45 (3-4), 252-269. Zheng, X., Brownjohn, J.M.W. (2001). Modelling and simulation of human-floor system under vertical vibration. Smart Structures and Materials 2001: Smart Structures and Integrated Systems. SPIE. Zivanovic, S., Pavic, A., Ingolfsson, E. (2010). Modelling spatially unrestricted pedestrian traffic on footbridges. Journal of Structural Engineering, Vol. 136 (10), pp. 1296-1308. Zivanovic, S., Pavic A., Reynolds P. (2007). Finite element modelling and updating of a lively footbridge: The complete process. Engineering Structures, Vol. 301(1-2), pp. 126-145. Zivanovic, S., Pavic A., Reynolds P.(2005). Vibration serviceability of footbridges under human-induced excitation: a literature review. Journal of Sound and Vibration, Vol. 279, Issue 1-2, pp. 1-74, January 2005. 77 Paper F: Dynamic testing of Carpinteira footbridge at Colvihã (Portugal). Conference name: 5th International Operational Modal Analysis Conference Location and date: Guimarães (Portugal). 13-15 May 2013. Paper ID: 224 (pp. 1-10). ISBN: 978-972-8692-83-4. Scopus: http://0-www.scopus.com.fama.us.es/inward/record.url?eid=2-s2.084906259319&partnerID=40&md5=3cd5fcc0518066dbff05cdacdea01d34 IOMAC'13 5th International Operational Modal Analysis Conference 2013 May 13-15 Guimarães - Portugal DYNAMIC TESTING OF CARPINTEIRA FOOTBRIDGE AT COVILHÃ, PORTUGAL Javier Fdo. Jiménez-Alonso 1, Elsa Caetano2, Álvaro Cunha3 ABSTRACT The footbridge over the Carpinteira stream establishes a connection between two steep cliffs, at a height of 52.00 m above the water, having a length of about 220.00 m and being composed of three straight sections with different orientations in plan view, joined by circular curves and supported on four columns. In order to describe the dynamic behaviour of the footbridge, an ambient vibration test was performed, complemented by characterisation tests of the vibrations induced by pedestrians in the loading scenarios assumed as critical for the structure. A detailed finite element model of the structure has also been developed for correlation analysis with the measured values. This paper shows the results of these tests and the main conclusions about the comfort level provided by the footbridge under the service conditions. Keywords: Footbridge, Operational modal analysis, Human induced vibrations, Ambient dynamic testing. 1. INTRODUCTION Inserted in the Urbanization Plan of Carpinteira Valley, promoted by the Polis Program, the footbridge over the Carpinteira stream establishes a link between the two steep cliffs of Carpinteira Valley at a height of 52.00 m over the water. The design has been developed by Afassociados, in collaboration with the architect Carrilho da Graça [1], and the construction, performed by the company CERTAR, was finished in September 2009. The dynamic characteristics of the footbridge, predicted at the design stage, motivated some concern owing to the possibility of occurrence of significant lateral and vertical vibrations, since several of the natural frequencies of the calculated vibration modes, both in lateral and vertical directions, would be apparently located at critical intervals from the point of view of the excitation induced by pedestrians [2]. Accordingly, and as it wouldn’t be possible to act on the stiffness or mass of the footbridge in order to achieve significant changes in terms of modifying the dynamic characteristics of the structure, the Designer has foreseen the need to install tuned mass dampers, if after the construction of the structure, there was evidence that they would be really required. 1 Assistant Professor, University of Seville, Higher Technical School of Building Engineering, [email protected] 2 Associate Aggregate Professor, University of Porto, Faculty of Engineering, [email protected] 3 Full Professor, University of Porto, Faculty of Engineering, [email protected] Session 1, J.F. Jiménez-Alonso, Elsa Caetano, Álvaro Cunha 2 The footbridge was constructed in September 2009 and placed at the service of the population without introducing any device to mitigate vibrations, which aroused the interest in conducting the present research. Thus, using the experimental resources available at the Laboratory of Vibrations and Monitoring (ViBest, www.fe.up.pt) of FEUP, an ambient vibration test was performed with the aim of identifying the dynamic characteristics of the footbridge. Additionally, measurements of the response of the structure under service conditions were made, and also under conditions of use potentially hazardous for the footbridge. This paper presents the main results achieved, which show some differences in the dynamic behavior of the structure in relation to the predicted at the design phase and, in particular, allow the characterization of the comfort level of the footbridge under normal use conditions. 2. DESCRIPTION OF THE STRUCTURE: CARPINTEIRA FOOTBRIDGE Figure 1 Footbridge over the Carpinteira stream, view from the south side. Nota: As co tas apre sent adas são med idas ao long o do eix o lo ngit udin al d a ponte Escala 1/400 Planta 42.267 48.406 49.000 49.302 31.769 Figure 2 Carpinteira footbridge, lateral and plan views. 42.267 48.406 49.000 49.302 31.769 5th International Operational Modal Analysis Conference, Guimarães 13-15 May 2013 3 The footbridge (Figure 1) is composed of a steel deck that runs at a constant level, being formed by three linear segments with different orientation in plant, connected by circular curves. The total length of the deck is about 220.00 m and it is supported by four composite steel-concrete columns, with a variable height between about 18.00 and 40.00 m. The footbridge is thus divided in five spans, with lengths from about 32.00 to 50.00 m (Figure 2). The cross-section of the footbridge, with overall dimensions of 4.40 x 1.75 m2 and 3.50 m of effective width, is formed by two longitudinal steel welded girders and a wooden supporting floor structure that configure the characteristics of the U-shaped section of the footbridge. 3. NUMERICAL MODELLING OF THE FOOTBRIDGE Due to the complexity of both the geometry and the behaviour of the structure and, with the aim of supporting the development of the dynamic tests and subsequent interpretation of the obtained results, a finite element model of the footbridge was developed using 3-D beam elements [3, 4 and 5]. In Figure 3, the finite element mesh used is presented. . Figure 3 Carpinteira footbridge, finite element mesh used in the numerical model and detail of discretization. The overall mass of the steel deck is about 300 ton, corresponding to a linear mass of 1250 kg/m. In the design of the footbridge, the associated dynamic effects of a pedestrian flow, with a density of 0.60 pedestrian/m2, have been considered, which corresponds to an increase of the mass of the deck of about 10 %. Though the whole pedestrian mass is not usually considered in the dynamic characterization of the footbridge, the calculation of the natural frequencies has been made assuming either an empty structure or a loaded state with 0.60 Pedestrian/m2, in order to frame their natural frequencies. On the other hand, given the relatively low loading levels, it was assumed that during service, the elastic supports at the extremes wouldn’t be activated. Accordingly, alternative conditions Session 1, J.F. Jiménez-Alonso, Elsa Caetano, Álvaro Cunha 4 of connection were simulated: elastic behaviour associated to neoprene bearings, or constraint movement in longitudinal direction. Table 1 presents the values of several natural frequencies corresponding to four numerical simulations developed in order to frame the bridge natural frequencies: M1 - empty footbridge, elastic supports; M2 - full footbridge with 0.60 pedestrian/m2, elastic supports; M3 - empty footbridge, fixed supports; M4 - full footbridge with 0.60 pedestrian/m2, elastic supports. It is described in Table 1 the main characteristics of the most important vibration modes (L: longitudinal; T: transverse; V: vertical; To: torsion). On the other hand, Figure 4 shows the four most relevant vibration modes, based on the M3 model. Table 1 Numerical natural frequencies obtained under different assumptions in relation to mass (empty/full deck with a pedestrian density of 0.60 pedestrian/m2) and the stiffness of the extreme supports (elastic/fixed). Modes M1 (elastic/ empty) M2 (elastic/full) M3 (fixed/empty) M4 (fixed/full) 1 1.15 (T) 1.11 (T) 1.21 (T) 1.16 (T) 2 1.34 (L) 1.29 (L) 1.6 (T local) 1.52 (T local) 3 1.38 (T local) 1.3 (T local) 1.97 (To) 1.88 (To) 4 1.79 (T) 1.71 (T) 2.02 (To+ T) 1.93 (To+T) 5/ 6(M3+M4) 2.01 (V+T) 1.9 (V+T) 2.33 (To+T) 2.20 (T+To) 7 (M3+ M4) 2.44 (To+T) 2.32 (T+To) … … … … … 14/ 13 (M3+ M4) 3.78 (V) 3.54 (V) 3.83 (V) 3.58 (V) Freq. 1= 1.21 Hz Freq. 3= 1.97 Hz Freq. 7= 2.44 Hz Freq. 13= 3.83 Hz Figure 4 Several modal shapes obtained from finite element model M3 (empty footbridge with fixed longitudinal support at abutments). 4. IDENTIFICATION OF THE DYNAMIC PROPERTIES The identification of the natural frequencies, vibration modes and modal damping ratios was made performing an ambient vibration test, conducted in July 2011. In this test, 5 seismographs provided with triaxial accelerometers, were used. These elements were successively placed along the positions indicated in Figure 5, keeping two of the devices in sections 6 and 15, in the west side of the footbridge. In each position, records of ambient acceleration have been collected in 15 channels with 13minute duration, sampled at 100 Hz. 5th International Operational Modal Analysis Conference, Guimarães 13-15 May 2013 5 Figure 6 shows two images of the tests performed, held during a normal day and involving the occasional passage of pedestrians, under normal conditions of use. Figure 5 Instrumented points in the ambient vibration test. Figure 6 Images of the ambient vibration tests. According to the image of Figure 6, and given the peculiar characteristics of the footbridge, it was decided to dispose systematically the sensors, so that the transverse axis would coincide with the North direction. This permitted the use of a common reference and facilitated the signal processing. The identification of the modal parameters was made using the software ARTEMIS [6]. Table 2 shows a list of the most relevant identified natural frequencies and modal damping ratios, whereas Figure 7 characterizes several of the corresponding vibration modes. It is settled in Table 2 a correspondence between some identified and calculated vibration modes, considering as basis the previously presented M3 model. The analysis of Table 2 and Figures 4 and 7 shows that, although the main aspects of the dynamic behavior of the footbridge are characterized by the numerical finite element model, in terms of the natural frequencies values of the vibration modes (transverse, vertical and torsion), significant differences exist between the modes calculated and identified. Indeed, we can notice that the first identified transverse vibration mode has a natural frequency slightly higher than the calculated one, which is indicative of the behavior of the extreme supports, almost fixed to the abutments. Although the numerical modeling has just introduced longitudinal constriction, the natural frequency of the second mode of vibration, of local character, is also higher than the calculated, which certainly also involves the constriction of the transverse motion in the extreme supports at the abutments. Moreover, it is also observed that the real transverse flexibility of the columns was inferior to the calculated one, since the vertical vibration modes involve generally the torsion of the deck, but not the transverse bending of the columns, contrary to the results of the calculation. Finally, it is noticed some proximity between the natural frequencies of the modes with vertical and torsional components, which is certainly determined by the mechanical characteristics of the deck. Session 1, J.F. Jiménez-Alonso, Elsa Caetano, Álvaro Cunha 6 Table 2 Numerical natural frequencies Modes Identified natural frequency (Hz) Numerical natural frequency (Hz) Damping ratio (%) 1 1.37 (T) 1.21 (T) 0.28 2 2.20 (T local) 1.6 (T local) 0.21 3 2.47* (V+ To) 0.17 4 2.76 (V+To) 0.13 5 2.95 (V+To) 0.33 6 3.59 (V+T) 3.83 (V) 0.07 * Several numerical modes with close frequencies. F1= 1.37 Hz F2= 2.20 Hz F3= 2.47 Hz F4= 2.76 Hz F5= 2.95 Hz F6= 3.59 Hz Figure 7 Identified vibration modes using ARTEMIS software. 5. CHARACTERIZATION OF THE DYNAMIC BEHAVIOUR From inspection of the identified natural frequencies of the footbridge, it can be concluded, firstly, that the footbridge is not vulnerable to the lateral synchronization phenomenon. Indeed, this phenomenon typically occurs with natural frequencies close to 1 Hz. The international codes and recommendations [7, 8] define this synchronization as possible in the range of frequencies from 0.5 to 1.2 Hz, with a critical scenario when the fundamental frequencies are situated between 0.7 and 1.0 Hz. The initial numerical studies [2] suggested a fundamental frequency of 1.14 Hz, so the risk of occurrence of this phenomenon was real. Note that, despite the possible sophistication of the current numerical models, the real boundary conditions of the footbridge and, sometimes, the addition of elements assumed as non-structural, can greatly influence its dynamic behaviour, and there are often differences in the fundamental frequencies of the built structure against the numerical results of the design. In this case, the natural frequency of 1.37 Hz measured in the transverse direction reduces the risk of lateral 5th International Operational Modal Analysis Conference, Guimarães 13-15 May 2013 7 synchronization, not avoiding however questions about the comfort level provided by the structure, both in terms of the vertical and horizontal vibrations. This problem is particularly relevant taking in mind the location of the bridge, at a very high level, and also taking into account the characteristics of the pavement, with a slatted wooden relatively sparse. It is referred, on the other hand, the very low damping identified by the ambient vibration test. It is important to note that the quality of the damping estimates obtained in this way is questionable, particularly since there was no opportunity to validate them using other method. It is noted, however, that the modal damping ratios identified are lower than expected. In relation to the vertical vibrations, it was also found at the design stage [2] that there were several vibration modes with natural frequencies close to 2 Hz, which could lead to resonant phenomena. In the built footbridge, it was observed that the frequencies of the vertical modes are generally a little higher than calculated, being settled typically above 2.5 Hz. In contrast, the flexibility of the columns is lower than the modelled one, which reduces their participation in terms of transverse bending in the main vibration modes. This fact leads to a more significant torsional behaviour of the deck. Also these characteristics become beneficial to the structure, since the resonant phenomena at frequencies around 2 Hz don’t occur. In contrast, the dynamic effects induced by pedestrians jogging become more relevant. Moreover, and given that the torsional behaviour of the deck turns out to be evident, also the lateral modes of vibration are associated generally to the components of vertical vibration modes, which may result in more severe vibration levels. Taking into account the observed dynamic characteristics of the footbridge, previously discussed, it is assumed that the dynamic response may be critical in the following situations: 1) Slow walking of large flows of pedestrians, with frequencies from 1.4 to 1.5 Hz (excitation of vibration mode 1 and modes 3 to 6 (Figure 7) through the second harmonic); 2) Jogging by a pedestrian or a group of pedestrians, with natural frequencies of 2.47 Hz, 2.76 Hz or 2.95 Hz (excitation of vibration modes 3, 4 or 5 (Figure 7)). Although it has been observed that the use of the footbridge is not intense, it was possible to test a normal operating situation relatively close to the described condition (1) of the above paragraph, although mobilizing a reduced density of pedestrians. The dynamic tests were performed in two days of the summer of 2011. It was found that the footbridge was used merely occasionally during the day. Conversely, and given the warm temperature observed in the late evening, from 19:00 h, a continuous use by pedestrians was detected, characterized by slow walking. Although an accurate quantification of the density of pedestrians has not been made, it is possible however to say that a density of at least of 0.1 pedestrian/m2 has been reached, since more than 80 pedestrians were over the structure. Under these conditions, the lateral and vertical vibrations were clearly perceptible, also mobilizing high and low frequencies. Taking as basis the range of 0-8 Hz, both levels of transverse and vertical accelerations were collected, with values of 0.08 m/s2 and 0.17 m/s2, respectively, characteristics of the maximum comfort level of the footbridge, in line with the recommendations mentioned above [7, 8]. Figure 8 shows examples of the records of the transverse and vertical acceleration collected in the mark number 16 (see Figure 5) under these conditions, together with their corresponding spectral content. Due to the slenderness of the structure, the joint between the deck and the piers has been specially stiffened. Fig. 2: Preliminary Finite Element Model. 3. Preliminary numerical modal analysis. Firstly, a numerical modal analysis was developed (Figure 2), where the effect of the steel roof was considered as a passive mass uniformly distributed on the deck (approximately 500 kg/m). The model of the structure was carried out by the finite element software Autodesk Robot Structural Analysis Professional [4]. Under this hypothesis, the numerical vibration modes (Figure 3) of the footbridge has been determined in two scenarios, absence of pedestrians (femp) and the situation where a pedestrian flow of 1.00 P/m2 (Pedestrians/m2) cross the structure (fful) [2]. Numerical vertical mode 1 fful/emp=2.34/2.43 Hz Numerical lateral mode 1 fful/em p =2.13/2.20 Hz Numerical vertical mode 2 fful/em p =2.56/2.66 Hz Numerical lateral mode 2 fful/em p =5.08/5.25 Hz Fig. 3: First four vibration modes. Empty (emp) and full(ful) footbridge. Given the situation of the footbridge and assuming that on the structure are not expected pedestrians densities above 0.80 P/m2, the numerical estimated natural frequencies are outside of the normal ranges that characterize the pedestrian walking step. On the other hand, the natural frequencies of the structure are in the range that characterizes the action of jogging or running. The numerical response that produces the crossing of the previously predefined harmonic load on the structure (1) reaches its maximum value under a crossing jogger at 2.56 Hz. In Figure 4 the vertical dynamic response (acceleration) of the structure under the passage of this pedestrian is shown. The maximum value is less than the limit established by the comfort level (1.00 m/s2). Numerical vertical aceleration. 1 Jogger f=2.56 Hz -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0 5 10 15 20 25 30 35 40 45 50 time [sec] [m/s2] Fig. 4: Numerical vertical acceleration at mid-span due to a jogger step frequency f=2.56 Hz. 4. Ambient and pedestrian tests. 4.1 Ambient test. The dynamic parameters of the structure have been determined by the measures obtained from an ambient test. The deck of the structure has been divided into a 2x15 grid, being the points separated longitudinally 5.65 m and transversally 3.15 m (see Figure 5, blue arrows are reference accelerometers). Two series of measurements were carried out, each one consisting on 14 set-up, the first one corresponds to the determination of vertical dynamic parameters and the second to estimate the lateral dynamic parameters. The measures were made with 4 uniaxial accelerometers, sensitivity 10 V/g, type Episensor and produced by the company Kinemetrics. The duration of each set-up was 900 seconds and the sampling frequency was 100.00 Hz [5]. Vertical Ambient Test Layout Lateral Ambient Test Layout Fig. 5: Measurement grid for the ambient test. From the above series, the dynamic parameters of the structure have been determined, processing the signals by two different methods, one in the frequency domain, Enhanced Frequency Domain Decomposition (E.F.D.D.), and another in the time domain, Stochastic Subspace Identification (S.S.I.). For the validation of the results [5], the M.A.C. ratio (Modal Assurance Criterion) of certain vibration modes, has been calculated (Table 1) presenting all the identified vibration modes a M.A.C. greater than 0.90. In Figure 6 the graphical representation of the first four determined vibration modes is shown. The practical application of the above algorithms was performed using the ARTeMIS Extractor Pro 2012 software developed by SVS A/S [6]. Table 1: Experimental natural frequencies Mode fEFDD [Hz] fSSI [Hz] Description M.A.C. 1 2.372 2.370 Lateral 0.999 2 3.026 3.024 Vertical 1.000 3 3.830 3.844 Vertical 0.890 4 5.508 5.601 Lateral 0.901 Experimental vertical mode 1 f=3.026 Hz Experimental lateral mode 1 f=2.372 Hz Experimental vertical mode 2 f=3.844 Hz Experimental lateral mode 2 f=5.508 Hz Fig. 6: Experimental first four modes of vibration (E.F.D.D.). 4.2 Pedestrian test. Finally, it is performed a test with five different pedestrians, measuring the dynamic response of the footbridge under different step frequencies (f = 1.50, 2.00, 2.50, 3.00, 3.50 and 4.00 Hz). In Figure 7, the measured maximum vertical acceleration in the central mid-span for a pedestrian with a weight of 114.00 kg and a step frequency of 3.00 Hz is shown. The maximum measured acceleration is less than the limit established by the medium comfort level and the numerical one estimated previously. Experimental vertical aceleration. 1 Jogger -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0 5 10 15 20 25 30 35 40 45 50 time [sec] [m/s 2 ] Fig. 7: Experimental vertical acceleration at mid-span due to a jogger step frequency f=3.00 Hz The stiffening of the structure caused by the presence of the steel roof originates an improvement of the comfort level of the structure. 5. Model updating. 5.1 Detailed finite element model of the whole structure. In order to have a more accurate understanding of the dynamic behaviour of the structure a detailed finite element model of the whole structure has been developed. Numerical modal analysis has been developed through the application of the finite element method [7]. In the finite element model of the structure has been necessary to model all the element of the footbridge and the cover to characterize as precisely as possible the mass and stiffness matrices. The model has been carried out using 3D-beam (BEAM188) elements except in the case of the steel cover were 2D-shell (SHELL63) elements has been considered. 5.2 Model updating of the detailed finite element model. A model updating of the above detailed finite element model has been developed [8] from the results of the above operational modal analysis in order to characterize more adequately the dynamic behaviour of the footbridge. In this sense, 7 physical parameters of the structure (according to Table 2) have been modified in order to minimize the mean square error between the experimental and numerical parameters, considering the identified natural frequencies and their corresponding modal coordinates. After a sensitivity study of the main physical parameters of the finite element model, it was found that the physical parameters with greater influence on the dynamic behaviour of the footbridge are the stiffness of bearings. The stiffness of these elements has been simulated through three springs, one in each direction (longitudinal, lateral and vertical). The objective function, in this case, was defined as the sum of the relative differences between the natural frequencies and the modal coordinates obtained experimentally and numerically. As optimization method the genetic algorithms have been chosen. In Figure 8 the results of the adjustment made on the first four vibration modes are shown. After the adjustment of the selected physical parameters, high correlations between experimental and numerical modal shapes (M.A.C. above 95 %) have been reached in the four modes identified. -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 x [m] Exp Num Experimental&numerical 1st vertical mode 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 x [m] Exp Num c Experimental&numerical 1st lateral mode -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 x [m] Exp Num Experimental&numerical 2nd vertical mode -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 x [m] Exp Num Experimental&numerical 2nd lateral mode Fig. 8: Comparison between experimental (Exp.) and numerical (Num.) vibration modes Table 2: Updated values of considered physical parameters Parameters Initial Value Updated Value Effective cover thickness 0.003 mm 0.0015 mm Effective slab concrete thickness 0.15 m 0.10 m Effective abutment stiffness 30000 MPa 33240 MPa Soil stiffness 5.00E8 kN/m 4.64E8 kN/m Longitudinal bearing stiffness 1.00E9 kN/m 2.00E8 kN/m Lateral bearing stiffness 1.00E10 kN/m 2.64E9 kN/m Vertical bearing stiffness 1.00E11 kN/m 1.28E11 kN/m Finally, in Figure 9 the updated four first vibration modes from the detailed finite element method are shown. Updated vertical mode 1 f=3.026 Hz Updated lateral mode 1 f=2.372 Hz Updated vertical mode 2 f=3.844 Hz Updated lateral mode 2 f=5.508 Hz Fig. 9: Updated four first vibration modes. 6. Numerical estimation of the effect of the steel cover construction. In Table 3 the variation in the natural frequencies of the footbridge due to the construction of the footbridge has been estimated. From the updated finite element model, it has been possible to simulate the behaviour the dynamic behaviour of the footbridge without the steel roof (fNUM_INI) and the current situation (fNUM_COV). Both values have been obtained numerically. The percentage values are representatives of the stiffening effect that the cover presents in each direction. The steel cover increase the stiffness of the footbridge in the vertical direction, however, in the lateral direction the construction of the steel cover reduces the value of the natural frequencies in that direction. From the point of view of the maximum acceleration values achieved, there is a slight improvement in the comfort level due to the stiffening of the structure. It presents certain safety margin, ensuring that the footbridge reaches a medium comfort level, through even, under a very rare load case as the Table 3: Estimation of the change in the natural frequencies of the footbridge Mode fNUM_INI [Hz] fNUM_COV [Hz] Description f [%]. 1 2.569 2.372 Lateral -7.69 2 2.839 3.026 Vertical 6.55 3 3.863 3.844 Vertical -0.51 4 5.722 5.508 Lateral -3.74 circulation on the footbridge of several joggers in parallel (Figure 10). Fig. 10: Change in the first two vertical vibration modes. – without cover -- with cover 7. Conclusions. In this paper, it has been estimated experimentally and numerically, the change of the dynamic behaviour of a slender footbridge due to the construction of a steel roof over the original structure. The steel roof increases the stiffness of the structure in vertical direction but reduces the value of the natural frequencies of the structure in the lateral direction. This effect is especially relevant in the first two natural frequencies of the structure. However, the values of the current natural frequencies ensure that the structure will not suffer from comfort problems due to pedestrians flow walking. In relation to jogging or running, it has been shown that the structural stiffening improves its behaviour under these types of human action. 8. References. [1] SETRA, Guide méthodologique passerelles piétonnes (Technical guide footbridges: Assement of vibrational behaviour of footbridges under pedestrian loading), Setra, 2006. [2] SYNPEX Guidelines, European Project on Advanced Load Models for synchronous Pedestrian Excitation and Optimized Design Guidelines for Steel Footbridges, 2007. [3] CLOUGH, R and PENZIEN, J. Dynamics of Structures, 2nd. Edition, Mc Graw-Hill, 1993. [4] AUTODESK ROBOT STRUCTURAL ANALYSIS PROFESSIONAL 2011. [5] MAGALHÃES, F., CUNHA, A. "Explaining Operational Modal Analysis with data from an arch bridge", Mechanical Systems and Signal Processing, Invited Tutorial Paper, Volume 25, Issue 5, pp. 1431-1450 , 2011. [6] ARTeMIS Extractor Pro 2012. [7] ANSYS Mechanical Release 11.0. [8] ZIVANOVIC, S., PAVIC, A. REYNOLD, P., “Finite element modelling and updating of a lively footbridge: The complete process”, Journal of Sound and Vibration, Vol. 301,. nº 1-2, pp. 126-145,2007.