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 P oposal and Calib a ion o a Biodynamic Model o Human-S uc u e In e ac ion by he Resolu ion o he In e se Dynamic P oblem. Applica ion o Pedes ian B idges. Ja ie Fe nando Jiménez Alonso Depa men o Con inuum Mechanics and S uc u al Analysis School o Enginee ing UNIVERSITY OF SEVILLE Se ille, Spain 2015 ii  iii  P oposal and calib a ion o a biodynamic model o human-s uc u e in e ac ion by he esolu ion o he in e se dynamic p oblem. Applica ion o pedes ian b idges. PhD’s Thesis in he Dynamic o S uc u es and Ea hquake Enginee ing p og am. Ja ie Fe nando Jiménez Alonso Ad iso : P o . D . And és Sáez Pé ez. Depa men o Con inuum Mechanics and S uc u al Analysis School o Enginee ing Uni e si y o Se ille Abs ac In his hesis a biomechanical c owd-s uc u e in e ac ion model is p oposed and u he implemen ed in o de o adequa ely es ima e he ene gy exchange be ween pedes ians and oo b idge. The p oposed model ocuses on bo h he ib a ions in he e ical and la e al di ec ions and i allows o ake in o accoun he change o he modal p ope ies o he s uc u e due o he p esence o pedes ians, hus imp o ing he nume ical es ima ion o he esponse o he s uc u e unde pedes ian lows. I u he pe mi s o analyze in mo e de ail he la e al lock-in phenomenon. The model in ol es wo sub-models, namely (i) a pedes ian- s uc u e in e ac ion sub-model plus (ii) a c owd sub-model. The i s sub-model ollows om a modal p ojec ion o a wo deg ee o eedom sys em ha simula es he beha io o each pedes ian, on he ib a ion modes o he s uc u e. The pa ame e s o his model a e es ima ed om he accele a ions eco ded on a eal oo b idge by implemen ing an in e se dynamic app oach. Fo he second sub- model, he c owd beha io is simula ed ia a mul i-agen me hod. The pe o mance o he esul ing o e all model is assessed by co ela ing he expe imen al and nume ical dynamic beha io o wo eal oo b idges. In pa icula wo phenomena a e analyzed in de ailed: (i) he change in he i s e ical na u al equency o a eal oo b idge induced by he pedes ian-s uc u e in e ac ion and (ii) he occu ence o he la e al lock-in phenomenon due o he pedes ian ac ion. The p oposed model leads o nume ical esul s ha exhibi good ag eemen wi h he ob ained expe imen al alues. The e o e, i becomes a aluable ool o accoun o he change on he modal p ope ies o a oo b idge induced by he c owd-s uc u e in e ac ion phenomenon. The conside a ion o his ac o allows es ima ing mo e accu a ely he dynamic esponse o he oo b idge unde he pedes ian ac ion, analyzing in mo e de ailed he occu ence o he la e al lock-in phenomenon o imp o ing he e iciency in he design o passi e and ac i e dampe s i hei ins alla ion was necessa y o gua an ee an adequa e com o le el on he oo b idge. Keywo ds: simpli ied biomechanical model, human-s uc u e in e ac ion, c owd dynamics, change o na u al equencies, ope a ional modal analysis, model upda ing, pa ame e iden i ica ion, la e al lock-in phenomenon i   PREFACE This Thesis has been ca ied ou a he Depa men o Con inuum Mechanics and S uc u al Analysis a he Uni e si y o Se ille. The wo k has been supe ised by Full P o esso D . And és Sáez Pe ez. I always owe my deepes g a i ude o And és o encou aging and suppo ing me cons an ly du ing he de elopmen o his wo k. This esea ch would no ha e been possible wi hou his in ini e pa ien and help ul guidance o he o ganiza ion o he pape s. Thank him o his ime, e o and iendship. I eel eally lucky o ha ing had he oppo uni y o de elop his wo k unde his u elage. Addi ionally, I would like o exp ess some lines o g a i ude o hose who ha e con ibu ed o he de elopmen o he esea ch ca ied ou by he au ho o his Thesis. To P o . Al a o Cunha, who p o ided me a nice wo kplace du ing my esea ch s ay a he Labo a o y o Vib a ions and S uc u al Moni o ing (ViBes ) o he Uni e si y o Po o (Po ugal), allowing me o imp o e my knowledge o he ope a ional modal analysis me hodology and pe o m se e al expe imen al es s o g ea impo ance o he de elopmen o his wo k. The esul s I collec ed du ing my esea ch s ay cons i u e a i al pa o my Thesis and would no ha e been possible wi hou he suppo o Associa e P o esso Elsa Cae ano and Assis an P o esso Filipe Magalhães. To P o . Alexande Pa ic, o ecei ing me in o his g oup o he Vib a ion Enginee ing Sec ion o he Uni e si y o Exe e (U.K.) du ing my second esea ch s ay allowing me o in oduce mysel in he in e es ing ield o he con ol o ci il enginee ing s uc u es. To my pa en s, An onio and Ma ia del Ca men, who augh me he alue o he educa ion and who made a ema kable e o o us, hei child en, so ha we could ha e all he oppo uni ies ha hey did no ha e. To my colleagues (pas and p esen ) a he Depa men o Building S uc u es o he Uni e si y o Se ille and a he B idge Enginee ing Fi m, IDES, o hei cons an suppo . This hesis is based on scien i ic pape s which ha e al eady been accep ed o publica ion in ele an scien i ic jou nals o p esen ed a in e na ional con e ence wi h pee - e iew. Finally, wo addi ional pape s, cu en ly unde e iew, ha e been included. Las bu no leas , I am e y g a e ul o he lo ing suppo o my amily, in pa icula my b illian and comp ehensi e wi e, Pa icia, and ou wo lo ely daugh e s, Claudia and Lo ena. I hope ha one day I could compensa e he ime ha I s ool hem o he de elopmen o his wo k. To Ma ibel o he con inuous suppo , o me and my amily, du ing hese di icul yea s. i  My g a i ude goes also o o he ela i es and iends ha ha e helped me o o e ake success ully all he di icul ies o he achie emen o his Thesis. Se ille, Oc obe 2015 Ja ie Fe nando Jiménez Alonso ACKNOWLEDGEMENTS This wo k was pa ially unded by he Spanish Minis y o Science unde esea ch p ojec DPI2014-53947-R. ii  THESIS This Thesis consis s o an ex ended summa y and he ollowing appended pape s: Pape A J.F. Jiménez-Alonso and A. Sáez A di ec -pedes ian s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges In o mes de la Cons ucción, Vol. 66 (Ex a 1). m007 Pape B J.F. Jiménez-Alonso, A. Sáez, E. Cae ano, F. Magalhães Ve ical c owd–s uc u e in e ac ion model o analyze he change o he modal p ope ies o a oo b idge Jou nal o B idge Enginee ing. ASCE (in p ess) Pape C J.F. Jiménez-Alonso and A. Sáez Model upda ing o he selec ion o he e o i me hod o an ancien b idge (Alme ia, Spain). S uc u al Enginee ing In e na ional. IABSE (in p ess). Pape D J.F. Jiménez-Alonso and A. Sáez Con olling he human-induced longi udinal ib a ions o a Nielsen- uss oo b idge ia he modi ica ion o i s na u al equencies Unde e iew Pape E J.F. Jiménez-Alonso, A. Sáez, E. Cae ano and A. Cunha La e al c owd-s uc u e in e ac ion model o analyze he la e al lock-in phenomenon on a eal oo b idge Unde e iew Pape F J.F. Jiménez-Alonso, E. Cae ano and A. Cunha Dynamic es ing o Ca pin ei a oo b idge a Col ihã (Po ugal) 5 h In e na ional Ope a ional Modal Analysis Con e ence (Guima ães, Po ugal) 13-15 May 2013 Pape G J.F. Jiménez-Alonso and A. Sáez Assessmen o he dynamic beha io o Palmas Al as oo b idge a Se ille (Spain) 37 h IABSE Symposium. Mad id (Spain) 3-5 Sep embe 2014 The appended pape s we e p epa ed in collabo a ion wi h co-au ho s. The au ho o his Thesis is esponsible o he majo p og ess o wo k in hese pape s, including he de elopmen /deduc ion o solu ions and nume ical me hods, pe o ming he nume ical simula ions and expe imen al es s and w i ing he main pa s o he pape s. iii  ix  Table o Con en s I. Ex ended summa y ................................................................................... 1 1. In oduc ion. ............................................................................................ 1 1.1 Mo i a ion. .......................................................................................... 3 1.2 Objec i es. ......................................................................................... 3 2. Vib a o y p oblems due o pedes ian lows on oo b idges. ............................ 5 2.1. Load models o a single pedes ian. ...................................................... 5 2.2. Load models o c owds. ...................................................................... 8 2.3. Synch oniza ion and lock-in. ............................................................... 12 2.3.1. Models o he simula ion o he synch oniza ion and lock-in. ............. 13 2.3.2. Synch oniza ion and e ical lock-in. .............................................. 16 2.3.3. Synch oniza ion and la e al lock-in. ............................................... 17 2.4. The pe cep ion o he ib a ion. .......................................................... 18 2.5. Dynamic p ope ies o he s uc u es unde pedes ian ac ion. ................. 19 2.6. The con ol o he ib a o y esponse. .................................................. 20 2.6.1. Modi ica ion o he mass induced by pedes ian ac ion. ..................... 20 2.6.2. Modi ica ion o he s i ness induced by pedes ian ac ion. ................. 21 2.6.3. Modi ica ion o he damping induced by pedes ian ac ion. ................ 22 3. P oposal o a simpli ied biomechanical c owd-s uc u e in e ac ion model. ....... 23 3.1. Modelling he pedes ian-s uc u e in e ac ion. ...................................... 24 3.2. Modelling he c owd-beha io . ............................................................ 28 3.3. C owd-s uc u e in e ac ion. ............................................................... 33 4. In e se dynamic p oblem app oach. .......................................................... 35 4.1. In e se dynamic p oblem: pa ame e iden i ica ion in e ical di ec ion. ... 36 4.2. In e se dynamic p oblem: pa ame e iden i ica ion in la e al di ec ion. ..... 37 5. Expe imen al es ima ion o he pa ame e s o he pedes ian-s uc u e in e ac ion model. ...................................................................................... 40 5.1. Desc ip ion and ini e elemen model o he “labo a o y” oo b idge: Viana oo b idge. ............................................................................................. 40 5.2. Expe imen al iden i ica ion o he modal pa ame e s o he “labo a o y” oo b idge. ............................................................................................. 42 5.3. Model upda ing o he “labo a o y” oo b idge. ...................................... 45 5.4. Expe imen al pedes ian and c owd es s. ............................................. 47 5.5. Es ablishing a sea ch domain o he pa ame e s o he pedes ian-s uc u e model. ................................................................................................... 49 5.6. Pa ame e iden i ica ion o he pedes ian-s uc u e in e ac ion model: e ical di ec ion. .................................................................................... 50 5  2. Vib a o y p oblems due o pedes ian lows on oo b idges. In his sec ion a summa y o he main aspec s o he ib a o y p oblems induced by he c owd-s uc u e in e ac ion is p esen ed. The sec ion includes he di e en models ha ha e mainly in luenced he au ho o he de elopmen o he p oposed c owd-s uc u e in e ac ion model. On he o he hand, his sec ion cons i u es a b ie summa y o he s a e o he a abou his subjec . 2.1. Load models o a single pedes ian. The i s models p oposed in o de o s udy he e ec o a pedes ian c ossing a oo b idge we e based on he assump ion ha he pedes ian’s ac ion can be app oxima ed by a ha monic o ce. F om his app oach a ises he p oposal o he B i ish s anda d (BSI, 2006), he single model, ha la e was adop ed by o he coun ies, as o ins ance, Canada (On a io, 1995) and Spain (RPM-95, 1995). This model conside ed ha he e ec o he passage o a pedes ian on he s uc u e is equi alen o a mo ing e ical sinusoidal o ce p F (in N), wi h a pedes ian s ep equency p (in Hz), a pedes ian s ep eloci y p (m/s) a   F pp  2sin180)(  [1] pp 9.0)(  [2] being he ime a iable (sec.). F om he end o he las cen u y, se e al esea ches ha e ocused hei e o s on he cha ac e iza ion o his o ce mo e p ecisely, including addi ional e ms in he Fou ie se ies and conside ing i s e ec in he h ee spa ial di ec ions. In he ollowing equa ions, [3 and 4], and in Table 2 and Figu e 2 a summa y o he main p oposals epo ed is shown (Se a, 2006, Bu z e al., 2007, Racic e al., 2009).          nh i p e is e ip e p i P F 1 ,,, 2sin1)(  [3]     nh i pla isla ipla p i P F 1 ,,, sin)(  [4] whe e e p F,is he e ical pe iodic o ce due o walking. la p F, is he la e al pe iodic o ce due o walking. 700 p PN is he mean pedes ian’s weigh (Bu z, e al.,2007). 6  e i,  and la i,  a e he Fou ie coe icien s o he i h ha monic o e ical and la e al o ce, dynamic load ac o s (DLFs) s [Hz] is he s ep equency. e i,  and la i,  a e phase shi o he i h ha monic. nh is o al numbe o con ibu ing ha monics. p  is he phase shi among pedes ians. F om he analysis o he esul s p o ided by Table 2, i can be concluded ha a leas wo ha monics a e necessa y o cha ac e ize adequa ely he e ical pedes ian o ce while h ee ha monics a e necessa y o he la e al di ec ion. On he o he hand, he la e al componen o he pedes ian o ce is cha ac e ized by equencies ha a e he hal o he equencies ansmi ed in he e ical di ec ion. The ela ionship be ween he pedes ian eloci y ( p ) and he pedes ian s ep equency ( s ) has been s udied by di e en au ho s. The wo ks epo ed by Bu z e al. (2007), Ricca delli and Pizzimen i (2007), Ricca delli e al. (2007) and Be am and Ruina (2001) may be highligh ed. The ul ima e p oposal, in e na ionally accep ed by he scien i ic communi y, is go e ned by he ollowing equa ion. 32 35.059.193.2 ppps  [5] La ely, his ela ionship will be conside ed in he c owd-s uc u e in e ac ion model p oposed in his wo k. 7  Table 2. Dynamics Load Fac o s (DLFs) acco ding o di e en au ho o walking ac ion in e ical and la e al di ec ion (Se a, 2006; Bu z e al., 2007; Racic e al. 2009). Au ho Fou ie Coe ./Phase Commen a ies Ac ion-Di ec ion Blancha d e al. (1977) 1, e =0.257 Walking-Ve ical Bachmann & Ammann (1987) 1, e =0.40-0.50; 2, e =3, e =0.10 s=2.00-4.00 Hz Walking-Ve ical Schulze (1980) 1, e =0.37;2, e =0.10; 3, e =0.12;4, e =0.04; 5 , e =0.015; s=2.00 Hz Walking-Ve ical Bachmann e al. (1995) 1, e =0.40/0.50; 2, e =3, e =0.10; 1/2,la 1,la =3/2,la =0.10; 2=3=pi/2; s=2.00-2.40 Hz s=2.00 Hz s=2.00 Hz Walking-Ve ical Walking-Ve ical Walking -La e al Walking-Ve ical- La e al Ke (1998) 1, e =0.40/0.50; 2, e =3, e =0.10; 1 acco ding o he equency Walking-Ve ical Young (2001) 1, e =0.37 ( p-0.95) ≤0.50 2, e =0.054+0.0088 s 3, e =0.026+0.015 s 4 , e =0.01+0.0204 s Mean alues o Fou ie Coe . Walking-Ve ical EC5 (2003) 1, e =0.40;2, e =0.20 1 , la =2 , la =0.10 Walking-Ve ical Walking-La e al SETRA (2006) 1, e =0.40 2, e =3, e =0.04 2, e =3, e =pi/2; 1/2,la =3/2,la =0.05 1 , la =2 , la =0.01 Walking-Ve ical Walking-Ve ical Walking-Ve ical Walking-La e al Walking-La e al SYNPEX (2007) 1, e =0.0115 s2+0.2803 s- 0.2902 1, e =0.00 [º] 2, e =0.0669 s2+0.1067 s-0.0417 2, e = -99.76 s2+478.92 s - 387.80 [º] 3, e =0.0247 s2+0.1149 s-0.1518 I s<2.00 Hz 3, e = -150.88 s3+819.65 s2 - 1431.35 s+811.93 [º] I s≥2.00 Hz  3, e = 813.12 s3-5357.60 s2 +11726.00 s -8505.90 [º] 4, e =-0.0039 s2+0.0285 s-0.0082 4 , e = -34.19 s-65.14 [º] Fou ie Coe . and phases o mean pedes ian loads Walking-Ve ical 8  2.2. Load models o c owds. The main limi a ion o he abo e models is ha hey a e no able o p edic he esponse o he oo b idge when he s uc u e is subjec ed o a pedes ian low, being necessa y, he e o e, o de elop models ha accoun o he beha io o he c owd. The me hodology used mo e equen ly in he li e a u e consis s in mul iplying he esponse o a single pedes ian by a ac o ha conside s globally he e ec o he c owd. Among he di e en p oposed models, i is p esen ed below a summa y o he mos his o ically in luen ial, acco ding o he au ho ’s c i e ion. The i s conside ed model was p oposed by Ma sumo o e al. (1978) and akes in o accoun as mul iplica ion ac o he magni ude p n, being p n he numbe o pedes ians on he deck a ce ain ins an . This ac o es ablishes, acco ding o a Poisson dis ibu ion, he p opo ion o pedes ians ha due o he haza d a e mo ing in phase, a oiding he e ec o he es o indi iduals. Howe e , his ac o was used, wi hou success, in o de o p edic he esponse o he T- oo b idge (Tokyo) ha unde wen ib a o y p oblems due o he la e al synch oniza ion o he pedes ians (Zi ano ic e al., 2005). Subsequen ly, he Swiss s anda ds, SIA 260 (2003), p esen ed as no el y he modi ica ion o he mul iplica ion ac o in unc ion o he pedes ian densi y. In ha sense, i p oposes: (i) up o 10 pedes ians, a linea law wi h se e al sec ions and a maximum alue o 3; (ii) up o a pedes ian densi y o 0.30 P/m2 (Pe sons/m2) i accep s he abo e p oposal and (iii) up o abou his poin i es ablishes a pa abolic law wi h a maximum ac o o 20. The e o e, he pedes ian densi y de e mines a egime change o he pedes ian beha io : while alues lowe han 0.30 P/m2 allows he ee mo emen o he pedes ians, as he pedes ian densi y inc eases hei ee mo emen becomes di icul , so ha he pedes ians end o synch onize. The Eu ocode (2002) es ablishes, simila ly o he Swiss s anda ds, h ee load models acco ding o he expec ed pedes ian densi y. The i s model, DLM1, which is he basis o he es , is used o cha ac e ize he ac ion o a single pedes ian, being i s ac ion de ined as a mo ing sinusoidal o ce wi h wo spa ial componen s: one e ical, o alue 280 N and one la e al, wi h magni ude 70 N. Bo h he pedes ian s ep equency and he pedes ian eloci y ollow he c i e ion o he B i ish s anda d (BSI, 2006). The second model, DLM2, cha ac e izes he ac ion o a g oup o up o 15 pedes ians. The esponse o he oo b idge unde he ac ion o he g oup is calcula ed mul iplying he esponse o he model DLM1 by a ac o , wi h a maximum alue o 3, which akes in o accoun he p obabili y ha a esonance phenomenon occu s on he s uc u e due o he ac ion o he g oup o pedes ians. The modi ica ion o he modal p ope ies o he oo b idge due o he pedes ian e ec is quan i ied h ough he addi ion, a he poin wi h a maximum modal de lec ion, o a poin load o 800 kg. The las model, DLM3, applicable o scena ios unde a con inuous pedes ian low, cha ac e izes he ac ion o he c owd h ough a ha monic load equi alen o he weigh o a pedes ian densi y 0.60 P/m2 mul iplied by wo ac o s in o de o accoun o he possible esonance be ween he pedes ians and he s uc u e, as well as he ansi o y cha ac e o he load. The i s ac o , adop s a a iable alue anging be ween 0.05-0.30 and he second ac o is equal o 0.75. The equi alen load is only applied in he pa o he s uc u e whe e he modal de o ma ion has he same sign as he load and i s e ec 9  is un a o able. The pedes ian load is applied o he same equency han he model DLM1. Finally, he model assumes ha he pedes ian low p oduces an inc ease o he modal mass o he s uc u e o 40 kg/m2. Figu e 2. Ve ical and La e al pedes ian walking o ce acco ding o di e en au ho s (Se a, 2006; Bu z e al., 2007 and Racic e al., 2009). The F ench s anda d, Se a (2006), based on bo h he esea ch conduc ed on he Sol e ino oo b idge (Pa is, F ance) and labo a o y es s on eadmills, p esen s a new me hodology ha has been widely adop ed by esea che s and designe s. The me hodology has been accep ed equally by he Eu opean Resea ch P ojec , SYNPEX (Bu z e al., 2007). The p oposed me hod simula es he e ec o he pedes ian lows as an equi alen uni o m dis ibu ed load applied acco ding he conside ed ib a ion mode and whose alue is equal o he e ec o he g oup o pedes ians ha a e synch onized among hem. This magni ude is named as equi alen numbe o pedes ians, 'n. In o de o de e mine he numbe o pedes ians on he 0.00 200.00 400.00 600.00 800.00 1000.00 1200.00 1400.00 0.00 0.20 0.40 Ve ical Load [N] Time [sec.] Blancha d e al. (1977) Bachmann & Ammann (1987) Schulze (1980) Bachmann e al. (1995) Ke (1998) Young (2001) Eu ocode 5 (2003) Se a (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 La e al Load [N] Time [sec.] Bachmann e al. (1995) Eu ocode 5 (2003) Se a (2006) 10  oo b idge, he s uc u e is classi ied acco ding o he expec ed a ic le el. In Table 3, he ou possible a ic ypes a e speci ied (d=pedes ian densi y=Pe sons/m2) which allow gene a ing on he s uc u e he di e en load scena ios. A com o le el will be associa ed wi h each a ic le el. The equi alen uni o m load applied on he deck o he s uc u e is ob ained om he ollowing equa ion.         ')2cos()( p oo n G p [6] whe e G is he componen o he pedes ian load ( 280  G N o he e ical walking and 35G N o he la e al walking). oo is he na u al equency o he s uc u e unde conside a ion. ' p nis he equi alen numbe o pedes ians acco ding o Table 3.  is he educ ion coe icien o ake in o accoun he p obabili y ha he oo all equency app oaches he na u al equency unde conside a ion (Figu e 3). Figu e 3. Reduc ion ac o  e sus he na u al equency o he s uc u e (Bu z e al., 2007). The equi alen numbe o pedes ian depends on he pedes ian densi y on he s uc u e, he na u al equency unde conside a ion and he a io be ween he s uc u al and he c i ical damping,  . In Table 3, se e al p oposals o i s de e mina ion a e 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 Reduc ion coe icien  F equency [Hz] Ve ical. 1º Ha monic Ve ical. 2º Ha monic La e al 11  Table 3. Equi alen numbe o pedes ians, ' p n, acco ding o F ench code (Se a, 2006). Range o na u al equencies [Hz] T a ic 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, he s uc u al damping a ios o di e en cons uc ion ypes can been ob ained om Table 4 (Se a, 2006 and Bu z e al., 2007). Table 4. Damping s uc u al a ios o Se ice and Ul ima e Limi s a es (Se a, 2006 and Bu z e al., 2007). Cons uc ion ype S.L.S. U.L.S. min  [%] med  [%] max  [%] Rein o ced conc e e 0.80 1.30 5.00 P es essed conc e e 0.50 1.00 2.00 Composi e s eel-conc e e 0.30 0.60 2.00 Welded s eel 0.20 0.40 2.00 Sc ewed s eel 0.20 0.40 4.00 Timbe 1.00 1.50 4.00 S ess- ibbon 0.70 1.00 2.00 Elas ome s ---- ---- 7.00 Thus, he mos ad anced in e na ional s anda ds (Se a, 2006 and Bu z e al., 2007) es ima e he modi ica ion o he dynamic p ope ies o he oo b idge due o he pedes ian lows only by conside ing he modi ica ion o he modal mass o he s uc u e, adding di ec ly he passi e mass implied by he pedes ians. The e ec o he second ha monic ha cha ac e izes he pedes ian s ep is only conside ed in oo b idges unde pedes ian densi ies la ge han 0.80 P/m2. Howe e , hese in e na ional s anda ds p esen he ollowing limi a ions: (i) a simpli ied es ima ion o he change o he dynamic p ope ies o he oo b idge due o he p esence o he pedes ians, (ii) he conside a ion o he synch oniza ion phenomenon om an expe imen al ela ionship ob ained om es s on only one eal oo b idge, (iii) DLFs ob ained om labo a o y es , (i ) a o mula ion ha does no i well o he case whe e se e al ib a ion modes o he oo b idge a e in he ange ha cha ac e izes he pedes ian-s uc u e in e ac ion, and ( ) hey do no ake in o accoun he e ec o he non-synch onized pedes ians. In he p esen Thesis a c owd-s uc u e in e ac ion will be p oposed and calib a ed in o de o o e come he abo e limi a ions. 12  2.3. Synch oniza ion and lock-in. In he con ex o he pedes ian-s uc u e in e ac ion, he synch oniza ion e lec s he endency o he pedes ians o walk wi h he same spacing and phase among hem, while he lock-in e lec s he endency o he pedes ians in coupling hei s ep wi h he ib a o y mo emen o he s uc u e. This phenomenon can ha e a delibe a e cha ac e ( andalism) o unin en ional. This las case, esponsible o many o he ib a o y p oblems o oo b idges de ec ed du ing he las yea s, can be gene a ed by wo di e en mechanisms, acco ding o he scheme shown in Figu e 4. Figu e 4. Flowcha o unin en ional o pedes ian-s uc u e synch oniza ion (Racic e al., 2009). The i s synch oniza ion mechanism may occu when he pedes ian densi y on he oo b idge, d, is lowe han a c i ical alue, c d (limi densi y o which he mo emen o a pedes ian is in luenced by he es o he g oup), and he alue o he ampli ude o he deck induced by he pedes ians, u, was uppe a limi alue, c u, ha ma ks he limi alue om he pedes ians end o synch onize wi h he mo ion o he deck. On he o he hand, a second mechanism, ha o igina es he synch oniza ion, may occu , unde high pedes ian densi ies, i he mass p o ided by he pedes ians, M , is uppe a c i ical alue, c M, equal o he mass o he pedes ians which ine ial o ce may induce a ib a ion ampli ude, c u. In he ollowing sec ions, a li e a u e e iew o he main exis ing models o he analysis o he lock-in phenomenon in oo b idges is p esen ed. The p ac ical applica ion o hese s udies, adop ed by he cu en s anda ds, is also desc ibed. Pedes ian dens iy d [P/m 2 ] d≤d c d>d c u≤u c u>u c M≤M c M>M c Fo ced Vib a ion Fo ced Vib a ion Synch oniza ion Synch oniza ion 13  2.3.1. Models o he simula ion o he synch oniza ion and lock-in. The s udy o he p oblem o synch oniza ion be ween pedes ians and he oo b idge, he lock-in phenomenon, has been pe o med his o ically independen ly o he simula ion o he beha io o he c owd, o in he bes case, an addi ional checking c i e ion has been es ablished (Se a, 2006 and Bu z e al., 2007). The i s epo ed phenomenon o his ype occu ed in a Ge man oo b idge (1972), du ing i s opening, when a ib a ion mode o 1.10 Hz was exci ed by 300-400 pedes ians, as i is desc ibed by Bachmann and Ammann (1987) in hei book. The com o le el o he oo b idge was gua an eed by he addi ion o se e al uned mass dampe s wi hou gi ing addi ional impo ance o he phenomenon. Nex , a li e a u e e iew o he mos ele an models o he analysis o he phenomenon is p esen ed summa ized. One o he i s models, p oposed by Fujino e al. (1993), was de eloped om he esul s o he s udy o he la e al lock-in phenomenon in he T- oo b idge (Tokyo). By he analysis o ideo images o 2000 pedes ians c ossing he s uc u e, hey es ima ed ha 20% o he pedes ians we e synch onized wi h he oo b idge, wi h a maximum la e al displacemen o he deck o 10 mm and a la e al na u al equency o 0.90 Hz. I was es ablished, o he i s ime, as cause o he ib a o y p oblem he synch oniza ion be ween he pedes ians and he s uc u e. F om hese esul s, a gene al calcula ion ule was p esen ed, es ablishing a ixed alue o he synch oniza ion o 20% (among pedes ian wi h each o he and wi h he s uc u e) and a alue o he mean la e al o ce gene a ed by a pedes ian o 35 N. The model did no conside nei he he inc ease o he synch oniza ion wi h he ampli ude o he mo emen o he deck no he e ec o he pedes ians ha only was synch onized wi h each o he bu no wi h he deck. This model was unsuccess ully applied o he s udy o he dynamic beha io o he Millennium oo b idge (London) du ing i s design phase. A e he ib a o y p oblems de ec ed in i , la ge scale and labo a o y es s we e pe o med in o de o calib a e a new p oposal o simula ing he pedes ian beha io . The pedes ian ac ion, acco ding o Dalla d (Dalla d e al., 2001), may be app oxima ed as a o ce ha depends on he eloci y o he deck and a nega i e coe icien o pedes ian damping. In his way, he inc ease o he pedes ians on he oo b idge p oduces a educ ion o he global damping o he s uc u e, o such an ex en ha a dynamic ins abili y s a e may be eached, which is addi ionally a o ed by he complemen a y synch oniza ion p ocess expe ienced by he pedes ians. The concep o he equi alen numbe o pedes ians is in oduced as he numbe o pedes ians ha elimina e he damping o he sys em. The p ac ical applica ion o his model is e lec ed by he A up o mula. The Millennium oo b idge expe ienced a la e al mo emen o 50 mm wi h a na u al equency o 0.80 Hz in he la e al span and a la e al mo emen o 75 mm wi h a na u al equency o 1.00 Hz in he cen al span, synch onizing he mo emen o 50% o he pedes ians ha c ossed he s uc u e. The esul ing alue o he la e al o ce ansmi ed by each pedes ian, 30 N, is simila o he alue p oposed by Fujino e al. (1993). On he o he hand, he model p o ided, a limi alue o he pedes ian densi y o 1.50 P/m2, om which is so ha d o walk on he deck ha he dynamic e ec s a e negligible. Ne e heless, he model p esen s some limi a ions: (i) he model igno es he ene gy ansmi ed by he no -synch onized pedes ians wi h he s uc u e, (ii) 14  i does no ake in o accoun he change o he modal p ope ies o he s uc u e due o he p esence o he pedes ians, (iii) he change o he beha io o he pedes ians wi h he ib a ion le el o he s uc u e is no conside ed and (i ) he alue o he equi alen pedes ian damping is only es ima ed o one oo b idge and speci ic ange o equencies. In o de o conside in a mo e app op ia e manne he change o he beha io o he pedes ians induced by he ib a ion le el, Nakamu a e al. (2004) modi ied Dalla d’s p oposal (Dalla d e al., 2001) in o de o ake in o accoun ha , acco ding o he obse a ions pe o med on he T and M oo b idges (Tokyo), om ce ain alue o he la e al displacemen , 10 mm, he pedes ians modi ied hei s ep o gua an ee an adequa e com o le el, u he educing he la e al o ce o igina ed by hei s ep. In ha way, a sa u a ion ac o is in oduced in each pedes ian la e al o ce ha a oids ha he o ce inc eases linea ly wi h he eloci y o he deck inde ini ely. This p oposal educes he a io o he inc ease o he eloci y when he eloci y inc eases un il making i null. Al hough his model is an imp o emen as compa ed o he p e ious p oposals, i has as main limi a ion he necessi y o knowing he maximum displacemen o he s uc u e o scale he sa u a ion a io, while sha es he o he limi a ions o he p e ious models. Subsequen ly, he F ench s anda d (Se a, 2006) p oposed a mo e compac model conside ing he esul s o wo ypes o es s, on eadmills a labo a o y and la ge scale pedes ian es s on Sol e ino oo b idge (Pa is). As conclusions o hese wo se s o es s: (i) an expe imen al ela ionship ha allows es ima ing he numbe o synch onized pedes ians on he oo b idge was p oposed, as well as (ii) a new c i e ion in o de o de e mine he sensi i i y o he oo b idge o he la e al lock-in phenomenon. I was e i ied, in his sense, ha a change o egime in he pedes ian beha io occu s, om o ced o synch onized ib a ion, wi h a maximum synch oniza ion a io o 60% o a la e al accele a ion o he deck be ween 0.10- 0.15 m/s2. This limi accele a ion has been adop ed as a c i e ion in o de o de e mine he sensi i i y o he oo b idge o he la e al lock-in. In he las i e yea s, se e al mo e sophis ica ed models, no ye implemen ed in he in e na ional s anda ds, ha e appea ed p o iding a new app oach o he p oblem. Among hese models, he mos in luen ial ones o he de elopmen o his Thesis a e desc ibed, in summa y, in he ollowing pa ag aphs In he i s model, p oposed by Macdonald (Macdonald, 2008) and based on he in e se pendulum model by Bake (Bake , 2002), he pedes ian is modelled as a lumped mass a ached o he oo b idge by an inclined ba (Figu e 5). The pedes ian unde he la e al ib a ions modi ies he il o his legs, sea ching his s abili y, inc easing he la e al componen o he walking pedes ian o ce and being able o a ain a dynamic ins abili y si ua ion. The p oposed model has been accep ed o desc ibe he ini ia ion o he phenomenon, bu he e ec o modi ica ion o he pedes ian s ep due o la ge la e al ib a ions is no included in he p oposal. Despi e i s limi a ions, he e a e e olu ions o he model (Mo bia o e al., 2011) wi h g ea e complexi y and p ecision. 21  On he basis o he spec al design model, Bu z (2006) de eloped an empi ical exp ession o he de e mina ion o he equi ed modal mass, i M (kg), o a gi en pedes ian a ic o ensu e a equi ed com o le el unde he assump ion ha some na u al equency o he s uc u e is in he ange ha cha ac e izes he walking pedes ian ac ion.   lim 31 42 65.1 a kkn M kk p i    [12] whe e 1 k o 4 k a e cons an s, as gi en in Table 8. lim ais he limi accele a ion acco ding o he conside ed com o le el (Table 6). and, p n, is he numbe o pedes ians. Table 8. Cons an s o equi ed modal mass (Bu z, 2006). Ve ical-To sion La e al 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. Modi ica ion o he s i ness induced by pedes ian ac ion. The alue o he na u al equencies o he oo b idge is p opo ional o he squa e oo o he a io be ween he modal s i ness and mass o he s uc u e. In his manne , la ge s uc u al modi ica ions a e necessa y i he na u al equencies o he oo b idge mus be loca ed ou o he pedes ian-s uc u e in e ac ion ange. The cu en end in he design o oo b idges, unde aes he ics, esis an and economic equi emen s causes ha he abo e c i e ion is no always easible in o de o gua an ee an adequa e com o le el (Slaich, 2005). Howe e , he e a e occasions, whe e he i s na u al equency o he s uc u e is inside he walking pedes ian ange, and he second na u al equency is ou side ha ange, whe e i can be easonable o educe he s i ness o he s uc u e so bo h na u al equencies a e ou side he pedes ian-s uc u e in e ac ion ange and checking addi ionally ha he s a ic de lec ion o he oo b idge is compa ible wi h i s use (Se a, 2006 and Bu z e al., 2007). The mos common s a egies in o de o modi y he na u al equencies o he oo b idge, om he iewpoin o he s i ness, a e (Se a, 2006 and Bu z e al., 2007): (i) inc ease he deg ee o s a ically inde e mina ion, (ii) p o ide s uc u al cha ac e is ics o p o ec ion o su ace elemen s, and (iii) use cable sys ems wi h a s abilizing unc ion (Se a, 2006). I is ecommended in oo b idges wi h a wid h la ge han 4.00 m and spans wi h leng hs abo e 50.00 m, o ins all a la e al load ansmission sys em, as an e ec i e me hod o con ol he la e al lock-in phenomenon (Low, 2008). Finally, a high seismici y a eas, he inc ease o he 22  s i ness o he s uc u e in o de o a oid ib a o y p oblems induced by pedes ians may cause, by con as , an inc ease o he seismic ac ion (Slaich, 2005). 2.6.3. Modi ica ion o he damping induced by pedes ian ac ion. The inc ease o he s uc u al damping has been, un il he da e, he mos used me hod o con ol he ib a ions induced by pedes ians on oo b idges (Fujino e al., 1993; Dalla d e al., 2001 and Cae ano e al., 2010). This inc emen can be achie ed ei he by he ac ua ion on in e nal elemen s o he s uc u e, o by he implemen a ion o ex e nal con ol de ices o di e en na u e acco ding o hei pe o mance: ac i e, semi-ac i e, hyb id o passi e de ices (Mou inho e al., 2010). The mos usual is he use o passi e dampe s as: (i) iscous dampe s (Bu z e al., 2007 and Taylo , 2003), (ii) uned mass, liquid o liquid column dampe s (Fujino e al., 1993; Bu z e al, 2007 and Cae ano e al., 2010) and (iii) pendulum dampe s (Bu z e al., 2007). Howe e , he use o hese passi e dampe s mus be limi ed since al hough hey allow achie ing a high le el o damping wi h a easonable cos , hey p esen se e al p oblems ha discou age hei widely use. Among hese p oblems, he mos impo an a e: (i) he necessi y o damping all he na u al equencies o he s uc u e inside he ange o pedes ian in e ac ion, (ii) a bad pe o mance may be he cause o a spli ing o he na u al equency o iginally damped, (iii) hey a e mechanical elemen s ha equi e main enance, and (i ) due o hei weigh i can be non- iable hei placemen on exis ing oo b idges due o s eng h easons (Meinha d , 2009). 23  3. P oposal o a simpli ied biomechanical c owd-s uc u e in e ac ion model. The p oposed c owd-s uc u e in e ac ion has been simula ed using wo indi idual sub-models (Figu e 7): (i) he pedes ian-s uc u e in e ac ion sub-model and (ii) he c owd sub-model. In he i s sub-model, all he dynamics e ec s induced by he pedes ians on he oo b idge a e conside ed. The la e al o e ical accele a ion, a y  o a z  , expe imen ed by each pedes ian is he ou pu ob ained om his model (acco ding o he analysed di ec ion). In he second sub-model, he c owd is simula ed as a beha iou al model, p o iding a desc ip ion o he indi idual pedes ian posi ion, p x, walking pedes ian eloci y, p , s ep pedes ian equency, s , and phase among pedes ians, p  . In o de o ake in o accoun he change o he pedes ian beha iou associa ed wi h he accele a ion le el expe ienced, wo addi ional condi ions ha e been included in his la e sub-model. In e ical di ec ion only a com o h eshold has been included, while in la e al di ec ion a com o and la e al lock-in h esholds ha e been conside ed. The i s condi ion modi ies he pedes ian eloci y, p , acco ding o he com o le el expe ienced by each pedes ian and he second condi ion modi ies he s ep pedes ian equency, s , and he phase among pedes ians, p  , in o de o synch onize he mo emen o he pedes ians and he s uc u e i ce ain limi is exceeded. Figu e 7. Layou o he biomechanical c owd-s uc u e in e ac ion model. Fo each in e ac ion he c owd sub-model de e mines he posi ion, eloci y s ep equency and phase o each pedes ian. Subsequen ly, hese ou pa ame e s a e used as inpu o de ine he walking o ce o each pedes ian-s uc u e model, ob aining as ou pu he la e al o e ical accele a ion o each pedes ian. The pedes ian eloci y and equency o each indi idual is modi ied acco ding o he PEDESTRIAN-STRUCTURE MODEL CROWD MODEL PEDESTRIAN/STRUCTURE PARAMETERS CROWD-STRUCTURE MODEL p x p a y  COMFORT THRESHOLDS LOCK-IN THRESHOLD s p  a z  24  com o le el expe imen ed by each pedes ian and addi ionally his phase shi i a la e al lock-in h eshold is exceeded. Finally, he p ocess is epea ed by he upda ed alues o he posi ion, he pedes ian eloci y, he s ep equency and he phase shi (Figu e 7). 3.1. Modelling he pedes ian-s uc u e in e ac ion. The p oposed pedes ian-s uc u e in e ac ion in each di ec ion ollows om he applica ion o dynamic equilib ium equa ions (Clough and Penzien, 1993; Dominguez, 2001; Xia and Zhang, 2005) o a simpli ied model o in e ac ion (Figu e 8) wi h sp ung ( a m) and unsp ung masses ( s m). This me hodology has been applied sepa a ely o e ical (Pape A and Pape B) and la e al (Pape E) di ec ions and he e i will summa ized o he case o e ical di ec ion. I s implemen a ion in he la e al di ec ion is desc ibed in Pape E. The o mula ion o he p oposed model may be u he gene alized o he h ee di ec ions by acco dingly modi ying bo h he equa ion ha go e ns he conside ed pedes ian load in each di ec ion and he alue o he modal pa ame e s ha de ine he TDOF ( wo deg ees o eedom) pedes ian model. In his way, he esul ing model would be sui able o he mo e gene al 3-D p oblem and i could he e o ake in o accoun he possible in e ac ion in he h ee spa ial di ec ions. Figu e 8. Biomechanical pedes ian-s uc u e in e ac ion model in e ical di ec ion (Pape B). Conside ing he balance o he sys em, s uc u e and pedes ian model, he ollowing coupled equa ions o mo ion may be w i en. in _ )( FxzKzCzM piNUMiiiiii       [13]     0     sapsapaa zzkzzczm  [14]     in , FFzzkzzczm e paspaspss       [15] m a c p k p m s z a z s F in L z xF in F s M i C i K i d p y x p w(x, ) 25  whe e a m is he sp ung mass o he pedes ian in he conside ed di ec ion [kg]. s m is he unsp ung mass o he pedes ian in he conside ed di ec ion [kg]. as mmm  is he o al mass o he pedes ian in he conside ed di ec ion [kg]. i zis he modal displacemen o he ib a ion mode i [m] a z is he absolu e e ical displacemen o he sp ung mass [m]. s z is he absolu e e ical displacemen o he unsp ung mass [m]. p k is he equi alen s i ness o a pedes ian [N/m]. p c is he equi alen damping o a pedes ian [sN/m]. e p F, is he e ical pedes ian o ce due o walking [N]. in F is he in e ac ion o ce be ween he pedes ian and he s uc u e [N]. i M is he modal mass o he ib a ion mode i [kg]. i C is he modal damping o he ib a ion mode i [sN/m] i Kis he modal s i ness o he ib a ion mode i [N/m]. iNUM _  is he e ical componen o he nume ical ib a ion mode i. x pxp  is he longi udinal posi ion o he pedes ian [m]. is he ime [sec.] px is he longi udinal componen o he pedes ian eloci y ec o [m/s]. p dis he dis ance among pedes ians [m]. ),( xw is he de lec ion o he oo b idge a he posi ion x [m]. Lis he leng h o he oo b idge [m]. F om Eq.(15) he ollowing exp ession is ob ained o , in F:     aspaspss e p zzkzzczmFF       ,in [16] and subs i u ing his equa ion in o Eq.(13) yields. 26         aspaspss e ppiNUMiiiiii zzkzzczmFxzKzCzM          ,_  [17] Applying, a he con ac poin be ween he pedes ian and he s uc u e, he equa ions o compa ibili y o displacemen s, eloci y and accele a ion be ween he s uc u e and he simpli ied pedes ian-model o in e ac ion: ),(),( w xwz pxps    [18] ),(),( w xwz pxps      [19] ),(),( w xwz pxps      [20] These quan i ies may be exp essed in e ms o he ampli ude )( zi and he modal shape o he n nume ical conside ed ib a ion modes )( _x iNUM  , neglec ing he e m o a ia ion o he pedes ian eloci y along ime, as:    n i piNUMip x z xw 1 _)()(),(  [21]     n i piNUMpxi n i piNUMip x zx z xw 1 _ 1 _)()()()(),(    [22]       n i piNUMxpip n i iNUMxpi n i piNUMip x zx zx z xw 1 _ 2 , 1 _, 1 _))()()()(2)()(),(    [23] dx xd xiNUM iNUM )( )( _ _     [24] 2 _ 2 _ )( )( dx xd xiNUM iNUM     [25] whe e )( _x iNUM  is he i s spa ial de i a e o he mode o ib a ion i. )( _x iNUM   is he second spa ial de i a e o he mode o ib a ion i. The nume ical ib a ion modes, )( _x iNUM  , a e ob ained in a disc e e way using he co esponding ini e elemen me hod as:   j j j iiNUM xNx )()( _  [26] whe e )(xN ja e he shape unc ions and j i  a e he nodal alues. In he p e ious exp essions, he alue o he nume ical ib a ion modes is se o ze o when he pedes ian emains ou side he s uc u e. 27  0)( _  piNUM x  o L x x p p   )( 0)( [27] The abo e ela ions –Eqs.(18) o (23)- a e hen subs i u ed in he o e all dynamic equilib ium equa ions -Eqs.(13) o (15)- so ha , o ganizing in o ma ion in a ma ix o m, he ollowing model o in e ac ion is ob ained (see Pape B o u he de ails and ma ix o mula ion). )()()()()()()( FzKzCzM       [28] Fo a g oup o k pedes ians (Figu e 8), each o hem will be ep esen ed by he abo e simpli ied in e ac ion model. In he case o a single pedes ian, he p oposed model leads o a sys em o n+1 equa ions, co esponding o he conside ed numbe o ib a ion modes n plus he simpli ied in e ac ion equa ion. Simila ly, when conside ing a g oup o k pedes ians, a sys em o n+k di e en ial equa ions will need o be sol ed. Conside ing he na u e o he esul ing sys em, he use o a me hod o  -Newma k in eg a ion amily is p oposed, wi h pa ame e s 41   and 21   , hus ensu ing an uncondi ionally s able sys em. Fu he mo e, he in eg a ion s ep,  , is es ablished acco ding o he usual ecommenda ions (Clough and Penzien, 1993; Dominguez, 2001) o dynamics models based on modal decomposi ion echnique, as he minimum o he ollowing alues. )01.0, 4 , 200 , 8 1 min( minmin max pp n L L    sec. [29] wi h max [Hz] being he highes conside ed ib a ion equency o he s uc u e and min L[m] he minimum span leng h o he pedes ian b idge. In o de o de e mine he numbe o pedes ians ha c oss he oo b idge in phase, a Poisson dis ibu ion has been adop ed, acco ding o he esul s by Ma sumo o e al. (1978). Thus, when a g oup o p n pedes ians a i e a he oo b idge, he numbe o pedes ians andomly synch onized is p n. This synch oniza ion c i e ion has been adop ed o iginally in he c owd-s uc u e in e ac ion model. I s implemen a ion is achie ed by he phase shi pa ame e , p  . Fo a gi en gene a ion/g oup o pedes ians, he alue o he phase shi o he p n andomly synch onized pedes ians has been se equal o ze o. Fo he emaining pedes ians he phase shi has been assigned andomly, using a Gaussian dis ibu ion in he ange   2,0 . Subsequen ly, in he la e al di ec ion, his pa ame e will be modi ied i he la e al lock-in h eshold is exceeded. 28  3.2. Modelling he c owd-beha io . The pedes ian walking inside a c owd may be modelled using he go e ning equa ions o pa icle dynamics (Rapapo , 2004), conside ing ha di e en social o ces in e ac among pedes ians. This app oach has been success ully applied by se e al au ho s (Helbing and Molná , 1995; Ca oll e al., 2012). In his way, he di e en mo i a ion and in luences expe imen ed by he pedes ians a e desc ibed by se e al o ce e ms. The model is based on New on dynamics and is able o ep esen he ollowing ules in ela ion wi h he na u al pedes ian mo emen (see (Helbing and Molná , 1995) o a mo e in ol ed desc ip ion): (i) pedes ians no mally choose he as es ou e, (ii) each pedes ian has an indi idual speed ha may be de ined by a Gaussian dis ibu ion (iii) and he dis ance be ween pedes ians depends on he pedes ian densi y and he walking pedes ian speed. Figu e 9 ske ches he social o ces ac ing on a pedes ian in a c owd, as desc ibed in de ail in he ollowing pa ag aphs.  Figu e 9. Pedes ian-c owd in e ac ion o ces (Helbing and Molná , 1995). All he pa ame e s o he c owd model conside ed in his s udy ha e been ob ained om he epo ed esul s p o ided by di e en au ho s (Helbing and Molná , 1995; Ca oll e al., 2012), as summa ized in Table 9 and b ie ly desc ibed nex . Bounda y Bounda y Desi ed des ina ion no bou F an bou F d i F Pedes ian j Pedes ian i an_phy ped F p d d e d d zx y p x p  p Aniso opic pedes ian beha iou Legend Fo ce ec o Axis o he s uc u e Geome ic magni ude Coo dina e sys em Mo emen di ec ion an_phy ped F no phy ped soc ped _ FF  p n b b n p p no phy ped soc ped _ FF  b nb an bou F no bou F 0.1 5.0 1.0 p  29  D i ing o ce. Each pedes ian has a ce ain mo i a ion o each his desi ed des ina ion, d d, wi h his desi ed eloci y, d , which is ep esen ed by he d i ing o ce, d i F, as:           p dd d i m e F [30] whe e d e is he desi ed di ec ion ec o , p is he pedes ian s ep eloci y and is he elaxa ion ime o he pedes ian (Helbing and Molná , 1995) (Table 9). The desi ed di ec ion o he mo emen may be ob ained om he posi ion o he pedes ian in each ins an , p x, and i s desi ed des ina ion acco ding o: pd pd dxd xd e   [31] In e ac ions among pedes ians. The in e ac ion among pedes ians o igina es a epulsi e o ce (Helbing and Molná , 1995), ped F, wi h wo componen s, a socio-psychological o ce, soc ped F, and a physical in e ac ion o ce, phy ped F, as: phy ped soc pedped FFF  [32] The socio-psychological o ce e lec s he ac ha he pedes ians y o main ain a ce ain dis ance o o he pedes ians in he c owd. This socio-psychological o ce depends on he dis ance be ween pedes ians, eaching i s maximum alue a he lowes es ablished dis ance and ending o ze o as such dis ance inc eases. The socio-psychological o ce is de ined as: pp p pp p soc ped s B d A          nF 2 exp [33] whe e p A is he in e ac ion s eng h be ween pedes ians (Table 9). p B is he ange o he epulsi e in e ac ion be ween pedes ians (Table 9). p d is he dis ance be ween wo pedes ians. p is he so-called pedes ian adius (Table 9). p n is he no malized ec o poin ing be ween pedes ians. 30  p s is a o m ac o o conside he aniso opic beha iou ( he pedes ian ac ion in on o he pedes ian is mo e impo an han behind him) o he pedes ians, which alue may be ob ained om:   2 cos1 )1( p ppp s     [34] whe e p  (Table 9) is a po en ial ac o ha conside s he in luence on he pedes ian mo emen o o he pedes ians si ua ed in on o him (Figu e 9) and p  is he angle be ween wo pedes ians (Figu e 9). The physical in e ac ion o ce, phy ped F, is only conside ed in si ua ions o physical con ac among pedes ians (i pp d   2), associa ed wi h si ua ions o high pedes ian densi ies (≥0.80 P=Pe son/m2). The physical in e ac ion o ce is de ined by he supe posi ion o wo componen s: (i) he body o ce, no phy ped _ F, ha desc ibes he coun e ac ing body ac ion ha he pedes ians pe o m o a oid physical damage due o hei physical con ac wi h o he indi iduals, (ii) and he sliding o ce, an_phy ped F, ha ep esen s he pedes ians’ endency o a oid passing o he indi iduals wi h a high eloci y a small dis ances (Helbing and Molná , 1995). I is de ined as: an__ phy ped no phy ped phy ped FFF  [35]   pppp no phy ped d HC nF  2 _ [36]   p pppp phy ped d HD F  2 an_ [37] whe e no phy ped _ F is he no mal componen o he physical in e ac ion o ce (body o ce). an_phy ped F is he angen ial componen o he physical in e ac ion o ce (sliding o ce). p C is he body o ce s eng h due o he con ac be ween pedes ians (Table 9). p D is he sliding o ce s eng h due o he con ac be ween pedes ians (Table 9). p is a no malized ec o pe pendicula o p n. pp p  is he angen ial componen o he ela i e pedes ian eloci y, wi h p  being he di e ence o ec o eloci ies be ween wo gi en pedes ians. and unc ion H is de ined as: 37 Figu e 10. Flowcha o he iden i ica ion p ocedu e in e ical di ec ion. An i e a i e p ocess o educe he di e ences be ween he expe imen al and nume ical e ical accele a ions was pe o med, unde he ules o gene ic algo i hms (Koh and Pe y, 2010; Nocen al and W igh , 1999), The es ima ed alues o pa ame e s o he pedes ian-s uc u e in e ac ion model, in e ical di ec ion, we e co ela ed success ully wi h: (i) he walking pedes ian e ical o ce sugges ed by di e en au ho s (summa ized in sec ion 2.1) and (ii) a p e ious es ima ion o he modal pa ame e s (Pape A) ob ained om he analysis o he change o he modal p ope ies o a oo b idge induced by a con olled g oup o pedes ians (Geo gakis and Jo gesen, 2013). On he o he hand, he es ima ed alues o he modal pa ame e s we e inside he ange, es ablished by Shahabpoo e al. (2013). 4.2. In e se dynamic p oblem: pa ame e iden i ica ion in la e al di ec ion. In o de o de ine he objec i e unc ion o he pa ame e iden i ica ion o he TDOF-sys em in la e al di ec ion (Pape E), he eco ded dynamic esponse o he Viana oo b idge du ing he expe imen al pedes ian es was analysed again by i s ans o ma ion o he equency domain. In his case, howe e , i was checked ha he dynamic esponse o he oo b idge was cha ac e ized by a ha monic se ies ha con ained only he i s h ee equencies ha cha ac e izes he pedes ian s ep wi hou a ema kable con ibu ion o he ha monics associa ed wi h he la e al na u al equencies o he oo b idge. In his manne , he accele a ions eco ded du ing he expe imen al pedes ian es con ained in o ma ion mainly o he walking pedes ian la e al o ce, being necessa y o conduc a second expe imen al es , a c owd es , in o de o cha ac e ize he modal pa ame e s o he TDOF-sys em. In his c owd es , he dynamic esponse o Viana oo b idge unde a g oup o i y 38 pedes ians a di e en s ep equencies was eco ded in o de o s udy he change o he i s la e al na u al equency o he oo b idge induced by he pedes ian- s uc u e in e ac ion. Due o his ac , he iden i ica ion p ocess was di ided in wo s eps, by sol ing wo in e se dynamic p oblems. Fi s , as he modal pa ame e s o he pedes ian-s uc u e in e ac ion model ha e a di ec e ec on he modal pa ame e s o he oo b idge (Pape A) as objec i e unc ion o he i s minimiza ion p oblem, he mean squa e e o be ween he expe imen al, exp ,1 la , and nume ical, num la ,1 , i s la e al na u al equency o he Viana oo b idge, ob ained du ing he pe o mance o he expe imen al c owd es and i s nume ical simula ion, was conside ed. Addi ionally, as design a iables o his i s in e se p oblem, he h ee modal pa ame e s ha cha ac e izes he TDOF-sys em, in la e al di ec ion ( he pedes ian sp ung mass, la a m,, he pedes ian damping a io, la p,  , and he pedes ian na u al equency, la p ,), we e conside ed. Second, al hough he e a e e y ecen and comp ehensi e s udies o he la e al o ce induced by pedes ians (Ingól sson and Geo gakis, 2011; Ingól sson e al., 2011), hese esea ch do no include he e ec o he pedes ian-s uc u e in e ac ion. The e o e i was necessa y o es ima e he walking pedes ian la e al o ce unde his assump ion. In his manne a second in e se p oblem was sol ed. As objec i e unc ion o he second minimiza ion p oblem, he mean squa e e o be ween he expe imen al ( exp ,ila psd , whe e i is he conside ed sec ion) and nume ical ( num ila psd ,) powe spec al densi y ob ained om he la e al accele a ions eco ded in he p e iously men ioned expe imen al pedes ian es s and i s nume ical simula ion, was conside ed. As design a iables o his second in e se p oblem, he i s h ee LDLF ( la ,1  , la,2  and la ,3  ) and hei co esponding phase shi s o he second and hi d ha monic ( la,2  and la ,3  ) o he pedes ian walking la e al o ce we e conside ed. In Figu e 11 a lowcha o he iden i ica ion p ocedu e is shown. 39 Figu e 11. Flowcha o he iden i ica ion p ocedu e in la e al di ec ion. Ligh blue ma ks he modal pa ame e iden i ica ion me hodology and da k blue he walking pedes ian o ce iden i ica ion me hodology. As in he abo e case ( e ical di ec ion), an i e a i e p ocess o educe he di e ences be ween he expe imen al and nume ical magni udes was pe o med, unde he ules o gene ic algo i hms (Koh and Pe y, 2010; Nocen al and W igh , 1999), In his case, he iden i ica ion p ocedu e is pe o med in wo s eps. Fi s , he modal pa ame e s o he p oposed TDOF-sys em we e es ima ed by he minimiza ion o he ela i e di e ences be ween he expe imen al and nume ical change o he i s la e al na u al equency o he Viana oo b idge du ing an expe imen al c owd es and i s nume ical simula ion. Second, once es ablished he modal pa ame e s o he p oposed model, he walking pedes ian la e al o ce was es ima ed by he minimiza ion o he ela i e di e ences be ween he expe imen al and nume ical powe spec al densi ies ob ained in ou poin s o he Viana oo b idge du ing an expe imen al pedes ian es and i s nume ical simula ion. The es ima ed alues o he pa ame e s o he pedes ian-s uc u e in e ac ion model, in la e al di ec ion, we e co ela ed success ully wi h: (i) he walking pedes ian la e al o ce sugges ed by di e en au ho s (summa ized in sec ion 2.1) and he ange o pedes ian modal pa ame e s sugges ed by Shahabpoo e al. (2013). 40 5. Expe imen al es ima ion o he pa ame e s o he pedes ian-s uc u e in e ac ion model. In his sec ion he es ima ion o he pa ame e s o he p oposed TDOF-sys em was pe o med in e ical and la e al di ec ions. Fi s , a eal oo b idge, Viana do Cas elo oo b idge (Ba bosa e al., 2012), was con e ed in o a “labo a o y” oo b idge by he upda ing o i s ini e elemen model based on he expe imen al modal pa ame e s o he s uc u e ob ained om an ope a ional modal analysis pe o med on he measu emen s eco ded du ing an ambien es . Second, wo expe imen al es s, a pedes ian and c owd es , we e conduc ed in o de o es ablished a basis o he es ima ion o he pa ame e s o he p oposed TDOF- sys em. Finally, he pa ame e s o he pedes ian-s uc u e in e ac ion model we e es ima ed by he esolu ion o an in e se p oblem app oach. 5.1. Desc ip ion and ini e elemen model o he “labo a o y” oo b idge: Viana oo b idge. The Viana do Cas elo oo b idge (Ba bosa e al., 2012) is a mo eable cable-s ayed b idge. The longi udinal s uc u al scheme o he oo b idge consis s o wo spans o abou 36.50 m and 9.00 m espec i ely suspended by 6 amilies o wo hange s ( wo e aining ones) om an inclined mas . The deck, wi h 2.50 m o wid h, is con igu ed by wo olled s eel beams o a iable dep h b aced by ci cula hollow p o iles. The deck loo is co e ed wi h wood. The compensa ion o he main span weigh is achie ed by placing 11 high densi y blocks (wi h a weigh o 800 kN) placed in he sho e span. The mas is welded in i s base o a cylinde ha is connec ed o a wheel gea bea ing ha allows he o a ional mo emen o he s uc u e. The pylon is connec ed o a deep ounda ion ha balances he o ces ansmi ed by he mas . A pe spec i e o he Viana oo b idge is shown in Figu e 12. Figu e 12. La e al iew o he Viana oo b idge (Ba bosa e al., 2012). A p e o de o d e Fig u pa a The he s 3D- c wi h na u p e es i m p eli asso ou num e limina y n o ha e a e ine a s a u e 13. F me e s. so wa e p s uc u e u c able elem e he hang e al eque iously he m a ing i s mina y F E cia ed nu m nume ical be ). n ume ical i a i s app ing poin F ini e ele m p ackage An u sing 3D-b e e n s (LIN K e s ha e b ncies and s ess le angen s E model l e m e ical na u ib a ion m ni e eleme o xima ion equi ed o m en mod e sys (Ansy s e am elem e K 10) we e i b een consi he ib a el o h e s i ness m e ads o h u al equ e m odes a e 41 e n (Figu e o he dyn a pe o m h e l, ambie n s , 2015) w e n s (BEA M implemen e de ed o a ion mod e e hange s m a ix. Th e h e i s o e ncies gi e n shown ( N U 13) modal a mic beha h e ambien n es g as used b a M 188), exc e e d. The n o he de e m e s o he unde p e e nume ic a u nume i n in T able U M_i, wi h i analysis w iou o h e ib a ion id and m a sed on a d e p o he o nlinea e m ina ion o s uc u e, e manen l a l modal a cal ib a i 10. In Fig being he w as conduc e oo b idg e es . m odel up d d isc e iza i hange s w ec s asso c he num by calcu l oads and a nalysis o on modes u e 14 h e ib a ion m ed in e and d a ing on o w he e c ia ed e ical l a ing hus his and e i s m ode 42 NUM_1=3.426 Hz NUM_2=4.421 Hz NUM_3=6.907 Hz NUM_4=7.431 Hz Figu e 14. Fi s ou nume ical ib a ion modes. Ini ial FE model. 5.2. Expe imen al iden i ica ion o he modal pa ame e s o he “labo a o y” oo b idge. In o de o ob ain expe imen ally he modal pa ame e s (na u al equencies, damping a ios and modal shapes) o he oo b idge, an ambien ib a ion es was pe o med. The measu emen s we e eco ded in ambien condi ions, wi h he oo b idge exci ed by a ligh wind. In o de o acqui e a su icien le el o expe ise in he applica ion o his iden i ica ion echnique, he assessmen o he dynamic beha iou o o he s oo b idges was conduc ed by he au ho du ing he de elopmen o his Thesis. Two ep esen a i e examples ha e been included in he documen (Pape F and Pape G). The modal shape coo dina es we e measu ed along wo g idlines sepa a ed ans e sally 1.68 m. A o al o 2x11 poin s equally dis ibu ed along each longi udinal alignmen we e ins umen ed. Fou high sensi i i y i-axial o ce balanced accele ome e s we e used (Figu e 15). Using wo o hese de ices as e e ences, measu emen s we e successi ely made mo ing he o he wo accele ome e s o he de ined ins umen a ion loca ions and eco ding in each poin 1000 sec. ime se ies o accele a ion sampled a 100 Hz (Figu e 13). 43 Figu e 15. One accele ome e used du ing ambien /expe imen al es s. The expe imen al iden i ica ion o he modal pa ame e s was done in he ime domain using he S ochas ic Subspace Iden i ica ion me hod (Magalhães and Cunha, 2011), implemen ed in he so wa e p og am A emis (A emis, 2015). Figu e 16 illus a es he s abiliza ion diag am o he iden i ica ion algo i hm used. The i s ou ib a ion modes we e iden i ied and subsequen ly used o he FE model upda ing p ocess (see Pape B). The ob ained nume ical and expe imen al na u al equencies and ib a ion modes shapes a e compa ed in Table 10 and he co ela ion be ween he i s ou nume ical and expe imen al ib a ion modes is shown in Figu e 17 (wi h he x axis co esponding o he longi udinal di ec ion o he oo b idge). In o de o alida e he co ela ion be ween he nume ical and expe imen al modal pa ame e s, bo h he ela i e di e ence (  ) be ween he nume ical and expe imen al equencies and he modal assu ance c i e ion (M.A.C.) we e analysed (Zi ano ic e al., 2007). A good co ela ion be ween wo modes is achie ed when he alue o hei M.A.C. a io is g ea e han 0.90. These wo magni udes may be de ined acco ding o Eqs. (46 and 47) as ollows: 100 _ __    iEXP iEXPiNUM [%] [46] whe e iNUM _ is he nume ical na u al equency and iEXP _ is he expe imen al na u al equency o he ib a ion mode i.    iEXP T iEXPiNUM T iNUM iEXP T iNUM i MAC ____ 2 __      [47] whe e iNUM _  and iEXP _  a e he nume ical and expe imen al ib a ion modes o be compa ed and T deno es he anspose. 44 Al hough he shapes o he iden i ied ib a ion modes a e in good ag eemen (wi h M.A.C. a ios g ea e han 0.90 in h ee o he ib a ion modes), he ela i e di e ences, , be ween he i s wo nume ical and expe imen al na u al equencies a e s ill signi ican . The e o e, he ini ial es ima ion made on he physical pa ame e s o he s uc u e is no good enough and i becomes necessa y o pe o m a ini e elemen model upda ing (F iswell and Mo e shead, 1995; Teughels, 2003, Pape C) o he oo b idge in o de o imp o e he co ela ion be ween he nume ical and expe imen al modal pa ame e s. Figu e 16. S abiliza ion diag am o he S ochas ic Subspace Iden i ica ion me hod. Finally, in Table 10 an es ima ion o he damping a ios, i  , associa ed wi h he iden i ied ib a ion modes is also shown (Magalhães e al., 2010). These alues will be adop ed la e in he iden i ica ion p ocess o he pa ame e s o he TDOF-sys em ha models he pedes ian. Table 10. Fi s ou nume ical ( NUM) e sus expe imen al ( EXP) ib a ion modes o he oo b idge. Modes NUM [Hz] EXP [Hz] i  [%]  [%] M.A.C. Desc ip ion 1 3.426 3.138 1.22 9.17 0.963 Fi s e ical mode 2 4.421 4.068 1.21 8.67 0.985 Fi s la e al mode 3 6.907 6.810 1.10 1.42 0.809 Second la e al mode 4 7.431 7.345 1.39 1.17 0.939 Second e ical mode 45 1s Vib a ion mode 2nd Vib a ion mode 3 d Vib a ion mode 4 h Vib a ion mode Figu e 17. Fi s ou nume ical (Num.) e sus expe imen al (Exp.) ib a ion modes. 5.3. Model upda ing o he “labo a o y” oo b idge. As indica ed abo e, in o de o educe he le el o unce ain ies o he nume ical analysis a ini e elemen model upda ing (F iswell and Mo e shead, 1995; Teughels, 2003; Zi ano ic e al., 2007, Pape C) o he s uc u e has been pe o med. The ou iden i ied ib a ion modes we e conside ed in he upda ing p ocess due o he good quali y o he expe imen al da a. Bo h measu ed na u al equencies and modal coo dina e alues we e aken in o accoun . The e o e, in o al 48 esidual componen s we e selec ed o he model upda ing ( he ou iden i ied na u al equencies and he ele en coo dina es o each iden i ied ib a ion mode). A mo e de ailed desc ip ion o he me hodology used o pe o m he model upda ing can be ound in Pape C and Pape G. A sensi i i y analysis was pe o med in o de o adequa ely de e mine he physical pa ame e s o he FE model wi h g ea e in luence on he iden i ied ib a ion modes. In his manne , he modal sensi i i ies wi h espec o some possible physical a iables ha e been ob ained nume ically (Fox and Kapoo (1968)). The esul s o his s udy conclude ha he mos in luen ial physical pa ame e s on he dynamic beha iou o he oo b idge a e he s i ness o he ou amilies o main hange s and he soil-s uc u e in e ac ion, modelled by wo sp ing elemen s (in longi udinal, 5  , and la e al, 6  di ec ions, as Figu e 13 illus a es) si ua ed a he ex eme o he longe span. As he s i ness o he hange s is condi ioned by hei s ess le el, he ini ial s ess s a e o each conside ed amily ( 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 aken in o accoun as physical a iable. In Figu e 13 and Table 11, he selec ed physical a iables a e shown. The model upda ing p ocess has been conduc ed by sol ing an op imiza ion p oblem in he so wa e p og ams Ansys (Ansys, 2015) and Ma lab (Ma lab, 2015). As objec i e unc ion he mean squa e e o be ween he expe imen al and nume ical modal pa ame e s (na u al equencies and ib a ion modes) o he Viana oo b idge has been conside ed. In each i e a ion, a popula ion o 1000 ec o s has been gene a ed ha , using he gene ic algo i hms ules o mu a ion, ep oduc ion and c osso e , has minimized he alue o he p oposed objec i e unc ion. The alues o he selec ed pa ame e s ha e been modi ied in o de o minimize he conside ed objec i e unc ion. Addi ionally, a sea ch domain has been de ined o con ol he a ia ion o each pa ame e , inc easing he e iciency o he op imiza ion algo i hm and ye main aining he physical meaning o he ini e elemen model upda ing. Fo he s i ness o he hange s, as a passi e beha iou is expec ed, hei medium ension le el has been de e mined unde pe manen loads, wi h a alue o 250 kPa. Sligh a ia ions o his alue ha e been conside ed expanding he sea ch domain be ween 0-500 kPa. Fo he s i ness o he sp ings ( 5  and 6  ), gi en he unce ain y associa ed wi h he s i ness o he soil, a wide sea ch domain was conside ed. So conside ing, as i is es ablished by he geo echnical epo , a a ia ion o he Young’s modulus o soil be ween 1001   m E GPa and he geome y o he abu men s, he a ia ion o he equi alen s i ness o he sp ings has been de e mined 5  and   97 61010   N/m. In Table 11 he ange o a ia ion o each pa ame e and i s upda ed alues a e shown Table 11. Upda ed alues o conside ed physical pa ame e s. Pa ame e s Minimum Value Upda ed Value Maximum Value Tension s ess cable 1 ( 1  ) 0.00 kPa 112.38 kPa 500.00 kPa Tension s ess cable 2 ( 2  ) 0.00 kPa 59.62 kPa 500.00 kPa Tension s ess cable 3 ( 3  ) 0.00 kPa 31.62 kPa 500.00 kPa Tension s ess cable 4 ( 4  ) 0.00 kPa 147.43 kPa 500.00 kPa Longi udinal Sp ing ( 5  ) 1.00E7 N/m 6.00E7 N/m 1.00E9 N/m La e al Sp ing ( 6  ) 1.00E7 N/m 1.90E8 N/m 1.00E9 N/m The di e ences be ween he nume ical and expe imen al na u al equencies, a e he ini e elemen model upda ing, a e e y small and he co ela ion be ween he nume ical and expe imen al ib a ion modes a e e en highe . The ela i e di e ences be ween he upda ed nume ical ( UPD ) and expe imen al ( EXP ) modal pa ame e s and he M.A.C. alues achie ed a e he model upda ing p ocess a e summa ized in Table 12, whe e he imp o emen wi h espec o he ini ial FE model is clea (see Table 10). 53 s uc u e in e ac ion model The ollowing Gaussian dis ibu ions ha e been ob ained (),(   N, being  he mean alue and  he s anda d de ia ion).  La e al pedes ian sp ung mass, la a m,,)736.2,216.73(N%.  La e al pedes ian damping a io, la p,  ,)405.5,116.49(N %.  La e al pedes ian na u al equency, la p ,, )178.0,201.1(N Hz. Figu e 22 illus a es he co ela ion be ween expe imen al and nume ical esul s o he change o i s la e al na u al equency o he oo b idge induced by he c owd- s uc u e in e ac ion phenomenon. Good ag eemen be ween bo h se s o esul s is obse ed, wi h di e ences below 0.70 % o all he analysed pedes ian walking equencies. The i s la e al na u al equency co esponding o he emp y oo b idge is included in Figu e 22 o e e ence.  Figu e 22. Change o he i s la e al expe imen al (Exp.) and nume ical (Num.) na u al equency ( 1,la ) e sus he s ep equency [Hz]. F om he p e ious esul s he ollowing conclusions may be ex ac ed: (i) he s abili y ha he es ima ed modal pa ame e s p esen o he di e en s ep equencies, allowing ha he p oposed c owd-in e ac ion model may be used as a ool o he cha ac e iza ion o he e ec o he mo ing pedes ians on he dynamic beha iou o oo b idges in la e al di ec ion and (ii) he good co ela ion be ween he expe imen al and nume ical cu es (Figu e 22) ha show he change o he i s la e al na u al equency o he oo b idge e i ies he abili y o he p oposed model o cha ac e ize he pedes ian-s uc u e in e ac ion phenomenon in la e al di ec ion. 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 1.la [Hz] s [Hz] Exp. Num. Emp y 54 Fo he es ima ion o he walking pedes ian la e al o ce o he TDOF-sys em, a second in e se p oblem was sol ed again. In his case, as objec i e unc ion he mean squa e e o be ween he expe imen al and nume ical powe spec al densi y ob ained om he la e al accele a ions eco ded in he men ioned ou poin s o he Viana oo b idge unde he c ossing o wo pedes ians a con olled s ep equencies was conside ed. The expe imen al la e al accele a ions co espond o he denoised measu emen s o he abo e desc ibed pedes ian es . The nume ical la e al accele a ions ha e been ob ained om he implemen a ion o he p oposed pedes ian-s uc u e in e ac ion model on he upda ed ini e elemen model o he Viana oo b idge. The expe imen al and nume ical powe spec al densi y has been ob ained om hese men ioned accele a ions. Six pa ame e s we e adop ed as design a iables: (i) he i s h ee LDLF ha cha ac e ize he pedes ian walking la e al o ce. (ii) he phase shi s o he second and hi d ha monic ha cha ac e ize he pedes ian walking la e al o ce. (iii) a ime lag ha allows adjus ing he beginning o he c ossing o he pedes ian be ween he expe imen al and nume ical esponse. The es ima ion o he phase shi s has been made in a disc e e way, selec ing in each case he op ion ha minimizes he objec i e unc ion. Figu e 23 illus a es he layou o he iden i ica ion p ocess o he pedes ian walking la e al o ce o he p oposed TDOF-sys em. Figu e 23. Layou o he walking pedes ian la e al o ce iden i ica ion me hodology. 55 In Pape E, he esul s o he es ima ion p ocess a e summa ized, showing he di e en es ima ed pa ame e s e sus he pedes ian s ep equency. Acco ding o hese esul s, i is possible o ob ain a s a is ical es ima ion o he design a iables ha cha ac e ize he pedes ian walking la e al o ce.  Fi s LDLF, la ,1  , )017.0,086.0(N.  Second LDLF, la ,2  , )009.0,094.0(N.  Thi d LDLF, la ,3  , )019.0,040.0(N.  Second la e al phase shi 0 ,2  la  º.  Thi d la e al phase shi 0 ,3  la  º. Figu e 24 illus a es he la e al pedes ian walking o ce ob ained om he p oposed iden i ica ion p ocedu e. The maximum and minimum en eloped alues o he la e al o ces shown in Figu e 2 a e also ep esen ed. Figu e 24. La e al pedes ian walking o ce acco ding o he TDOF-sys em. Fo he gene a ion o he pedes ian lows o he c owd-s uc u e in e ac ion model in bo h di ec ions, he abo e Gaussian dis ibu ions we e conside ed. -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 La e al Load [N] Time [sec.] Minimum Maximum TDOF-sys em 56 6. Valida ion and main esul s o his Thesis. Once ob ained he pa ame e s o he p oposed TDOF-sys em in bo h di ec ions, he de ini ion o he p oposed model is comple e. The p oposed model is alida ed in his sec ion by co ela ing he expe imen al and nume ical dynamic esponse o wo eal oo b idges unde he e ec s induced by he pedes ian ac ion. In e ical di ec ion (Pape B), he p oposed model was implemen ed o analyze he dynamic esponse and he change o he i s e ical na u al equency o he Viana oo b idge unde he p e iously desc ibed c owd es . In la e al di ec ion (Pape E), he p oposed model was implemen ed o analyze he la e al lock-in phenomenon on Ped o e Inês oo b idge (Coimb a, Po ugal), including he es ima ion o i s dynamic esponse and he change o i s i s la e al na u al equency due o he pedes ian ac ion. 6.1. Analysis o he change o he modal p ope ies o he Viana oo b idge. The alidi y and applicabili y o he p oposed c owd-s uc u e in e ac ion model in e ical di ec ion is assessed h ough i s p ac ical applica ion o he ollowing case s udy (Pape B). F om he o ced eco ded esponse o he p e iously men ioned c owd es , he expe imen al analysis o he change o he i s e ical na u al equency o he s uc u e, due o he c ossing o he g oup o pedes ians a di e en s ep equencies, was de e mined. Figu e 26 illus a es he expe imen al analysis o he change o he i s e ical na u al equency o he Viana oo b idge. Subsequen ly, he c owd-s uc u e in e ac ion model has been applied o he upda ed ini e elemen model in o de o ob ain, i s , he e ical nume ical accele a ion a he men ioned sec ions and, la e , o analyse nume ically he change o he i s e ical na u al equencies o he oo b idge due o he p esence o he pedes ians. The assessmen o i s pe o mance has been done by co ela ing he abo e expe imen al esul s wi h he nume ical es ima ions p edic ed by he model. Fo each conside ed s ep equency en gene a ions o g oups wi h 50 pedes ians we e simula ed. The numbe o pedes ians in phase in each new simula ion was de e mined by he e alua ion o he pa ame e p  . The desi ed eloci y, d , o each pedes ian was assigned acco ding o Eq.(5). A pedes ian mass o 70 kg has been conside ed acco ding o he F ench code (Se a, 2006). As ini ial spa ial dis ibu ion o he pedes ians, a ec angula g id was selec ed, conside ing an ini ial dis ance among pedes ians 50.0 p d m wi h an equidis an dis ibu ion in he wid h o he deck. The selec ed ime s ep is 01.0   sec. The nume ical e ical accele a ion ( o h ee o he 50 pedes ian gene a ions) a sec ion 2 So he oo b idge, o a s ep equency o 1.60 Hz, is shown in Figu e 25. 57 Figu e 25. Expe imen al e sus nume ical accele a ion ( h ee gene a ions) a sec ion S2 o Viana oo b idge unde a g oup o 50 pedes ians (walking equency o 60.1 s 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.] Ve ical expe imen al accele a ion (S 2 ). 50 Pedes ians a 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.] 1s Gene a ion. Ve ical nume ical accele a ion (S 2 ). 50 Pedes ians a 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 Gene a ion. Ve ical nume ical accele a ion (S 2 ). 50 Pedes ians a 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.] 3 h Gene a ion. Ve ical nume ical accele a ion (S 2 ). 50 Pedes ians a 1.60 Hz 58 As Figu e 25 shows, he co ela ion be ween he expe imen ally eco ded e ical accele a ion and he nume ically es ima ed alues is adequa e, in e ms o bo h he alue o he maximum accele a ion and i s empo al a ia ion. Finally, he nume ically es ima ed e ical accele a ion a sec ion 2 So he oo b idge unde a g oup o 50 pedes ians o di e en s ep equencies was used o iden i y he i s na u al equency o he s uc u e, ollowing he p ocedu e desc ibed in Pape B. Figu e 26 illus a es he co ela ion be ween expe imen al and nume ical esul s o he change o he i s e ical na u al equency o he oo b idge induced by he c owd-s uc u e in e ac ion phenomenon. The nume ical es ima ion o he change o he i s e ical na u al equency was ob ained om he mean alues o en simula ions o each s ep equency. Good ag eemen be ween bo h se s o esul s is obse ed, wi h di e ences below 1.50 % o all he analysed pedes ian walking equencies. The i s e ical na u al equency co esponding o he emp y oo b idge is included in Figu e 26 o e e ence. Figu e 26. Change o he i s e ical, e ,1 [Hz], expe imen al (Exp.) and nume ical (Num.) na u al equency e sus he s ep equency s [Hz]. 6.2. Analysis o he la e al lock-in phenomenon on he Ped o e Inês oo b idge. The Ped o e Inês oo b idge is loca ed a Coimb a (Po ugal). The o al leng h o he s uc u e is 274.5 m, con igu ed by one cen al a ch o 110 m, wo la e al semi- a ches o 64 m and wo ansi ion spans o 30.5 and 6 m (Figu e 27). The main ea u e o he oo b idge is he an i-symme ical con igu a ion o he deck and he a ches wi h espec o he longi udinal axis o he s uc u e. The deck is a conc e e- s eel composi e box-gi de wi h a a iable wid h be ween 4 and 8 m, wha gene a es a pano amic squa e a mid-span o he oo b idge (Figu e 28.a). F om i s design phase, he nume ical s udies de eloped abou he oo b idge indica ed ha he s uc u e was p one o ib a ions induced by pedes ians in la e al di ec ion. This ac mo i a ed he de elopmen o a p ecise and de ailed wo k o he expe imen al assessmen o i s dynamic esponse and he implemen a ion o a con ol sys em in o de o gua an ee an adequa e com o le el o he oo b idge. 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 1, e [Hz] s [Hz] Exp. Num. Emp y 59 This wo k was pe o med and epo ed by Cae ano e al. (2010) and i s esul s ha e been used in his Thesis in o de o alida e he p oposed c owd-s uc u e in e ac ion model in la e al di ec ion. Figu e 27. Ele a ion and plan o he Ped o e Inës oo b idge (Cae ano e al., 2010). The oo b idge p esen ed a i s la e al ib a ion mode wi h an expe imen al na u al equency o 0.91 Hz and an associa ed damping a io o 0.55 % ha was easily exci ed by he pedes ian lows. In o de o de e mine expe imen ally he numbe o pedes ians ha o igina es he la e al lock-in phenomenon an expe imen al es was pe o med. Subsequen ly, in o de o alida e he pe o mance o he p oposed c owd-s uc u e model, an expe imen al and nume ical analysis o he la e al lock- in phenomenon on he Ped o e Inês oo b idge has been co ela ed. The analysis ocused on he beginning o he ins abili y phenomenon, as i is he si ua ion whe e he e ec o he modal pa ame e s o he pedes ians has mo e in luence in he dynamic beha iou o he s uc u e (Dalla d e al., 2001). The nume ical la e al lock-in simula ion is ob ained om he implemen a ion o he p oposed c owd- s uc u e in e ac ion model on an upda ed ini e elemen model o he Ped o e Inês oo b idge epo ed in he li e a u e (Cae ano e al., 2010). Figu e 28. a) Pe spec i e o he oo b idge and b) expe imen al la e al lock-in pedes ian es on his oo b idge (Cae ano e al., 2010). a ) b ) 60 In he expe imen al la e al lock-in es , he la e al accele a ion, la a, a mid-span o he oo b idge unde he c ossing o di e en g oup o pedes ians was eco ded (Figu e 28.b). A g aphical ep esen a ion o he maximum la e al accele a ion a his posi ion e sus he numbe o pedes ians on he oo b idge (Figu e 29) allows iden i ying he ins abili y si ua ion associa ed wi h he la e al lock-in phenomenon. As i is epo ed in he li e a u e (Cae ano e al., 2010) and i is illus a ed in Figu e 29 he numbe o pedes ians ha o igina es he beginning o he la e al lock-in phenomenon is a ound 75. Figu e 29. Expe imen al (Cae ano e al. 2010) and nume ical a ia ion o he maximum la e al accele a ion,   max la a, a mid-span e sus he numbe o pedes ians. Subsequen ly, a nume ical la e al lock-in analysis based on he p oposed c owd- in e ac ion model was pe o med. Each conside ed g oup o pedes ians was simula ed conside ing as ini ial spa ial dis ibu ion a ec angula -shaped g id wi h an ini ial dis ance among pedes ians 50.0  p d m and a equidis an dis ibu ion in he wid h o he deck. The coo dina es o he conside ed la e al ib a ion modes o he s uc u e we e conside ed om he esul s p o ided by he li e a u e (Cae ano e al. 2010). In o de o accoun o he change o he s uc u al damping o he oo b idge acco ding o i s ib a ion le el a pa abolic unc ion has been es ablished based on he esul s ob ained by Geo gakis and Jo gesen (Geo gakis and Jo gesen, 2014) in a labo a o y oo b idge. The ange o a ia ion o he damping a io was comp ised be ween he expe imen al alue ob ained in he p e iously men ioned ee ib a ion es and he limi alue unde s ong ib a ions p oposed by he mo e ecen in e na ional s anda ds (Bu z e al., 2007; Se a, 2006). The maximum nume ical la e al accele a ion a mid-span e sus he numbe o pedes ians on he oo b idge is shown in Figu e 29. As Figu e 29 shows, he co ela ion be ween he expe imen al la e al maximum accele a ions and he nume ically es ima ed maximum alues a e adequa e. Addi ionally, he es ima ion o he nume ical maximum accele a ion ob ained applying he me hodology p oposed by he mo e 0.00 0.10 0.20 0.30 0.40 0.50 0.60 15 25 35 45 55 65 75 85 (a la ) max [m/s 2 ] Numbe o pedes ians Lock-in c i e ion (Se a, 2006) Exp. Num. (Synpex) Num. (TDOF-sys em) 61 ecen in e na ional s anda ds (Bu z e al., 2007; Se a, 2006) is shown in Figu e 29. The p oposed model allows ob aining a mo e accu a e nume ical analysis o he la e al lock-in phenomenon han he conside ed s anda ds. The la e al lock-in c i e ion es ablished by F ench s anda ds (Se a, 2006) is also illus a ed o e e ence in Figu e 29. Figu e 30. Expe imen al (Cae ano e al., 2010) and nume ical a ia ion o he i s la e al, la ,1 , na u al equency o he oo b idge e sus he numbe o pedes ians. Finally, he i s la e al nume ical na u al equency o he oo b idge du ing he occu ence o he la e al lock-in phenomenon was ob ained and i is shown in Figu e 26. The expe imen al i s la e al nume ical na u al equency (Cae ano e al., 2010) is also shown in Figu e 30. Good ag eemen be ween bo h se s o esul s is obse ed, wi h di e ences below 0.35 % in he s udied ange o he numbe o pedes ians. Addi ionally, he alue o he i s la e al na u al equency co esponding o he emp y oo b idge is illus a ed o e e ence (Figu e 30). 0.885 0.890 0.895 0.900 0.905 0.910 0.915 65 70 75 80 85 1,la [Hz] Numbe o pedes ians Exp. Num. Emp y 62 7. Conclusions and u u e esea ch. 7.1 Conclusions. In his wo k, a new c owd-s uc u e in e ac ion model has been p esen ed. The p oposed model has been alida ed h ough he co ela ion be ween he expe imen al and nume ical dynamic esponses o wo eal oo b idges unde he pedes ian ac ion. The p oposed model is o ganized in wo sub-models: (i) a pedes ian-s uc u e in e ac ion and (ii) a c owd sub-model. The pedes ian- s uc u e in e ac ion sub-model is de ined in e ms o a TDOF-sys em, wi h sp ung and unsp ung masses, whose pa ame e s ha e been es ima ed expe imen ally om he esul s o wo expe imen al es s conduc ed on he Viana oo b idge (Viana do Cas elo, Po ugal). As iden i ica ion echnique he solu ion o an in e se dynamic p oblem has been u ilized, minimizing an objec i e unc ion de ined as he mean squa e di e ences be ween an expe imen al and nume ical magni ude. The es ima ion o he pa ame e s o he model has been limi ed o e ical and la e al di ec ion since he e a e ew epo ed cases o ib a o y p oblems in longi udinal di ec ion. In e ical di ec ion, he expe imen al and nume ical accele a ions on ou poin s o he Viana oo b idge unde he c ossing o wo pedes ians a con olled s ep equencies ha e been conside ed as objec i e unc ion. In la e al di ec ion, he iden i ica ion p ocess has been di ided in wo s eps. In he i s s ep, he modal pa ame e s o he TDOF-sys em has been es ima ed conside ing as objec i e unc ion he mean squa e e o be ween he i s expe imen al and nume ical la e al na u al equency o he Viana oo b idge unde he c ossing o a g oup o i y pedes ians a di e en con olled s ep equencies. Subsequen ly, in he second s ep he walking pedes ian la e al o ce o he p oposed model is es ima ed conside ing as objec i e unc ion he mean squa e di e ences be ween he expe imen al and nume ical powe spec al densi y ob ained in ou poin s o he Viana oo b idge unde he c ossing o he wo men ioned pedes ians. Fo he minimiza ion o he abo e objec i e unc ions, as global op imiza ion me hod, he gene ic algo i hms ha e been used in all he cases. The es ima ed pa ame e s a e wi hin he ange ecommended by p e ious wo ks in he li e a u e. The c owd sub-model is de ined in e ms o a mul i-agen model based on he ela ionships es ablished by he social o ce model. The in e ac ion be ween he wo sub-models is achie ed by imposing wo beha iou al condi ions, a com o and la e al lock-in h esholds. I he e ical o la e al accele a ions expe ienced by each pedes ian a e abo e ce ain accele a ion limi s, he a ec ed pedes ian modi ies his s ep eloci y. Addi ionally, i he la e al accele a ions exceed he limi es ablished by he F ench s anda d in o de o cha ac e ize he la e al lock-in phenomenon, he a ec ed pedes ian synch onizes his/he equency s ep and phase shi wi h he mo emen o he deck. The p oposed model is o mula ed unde he ollowing hypo hesis: (i) he pa ame e o he pedes ian-s uc u e in e ac ion model a e assumed cons an , so ha hey do no a y acco ding o he s ep equency o each pedes ian, (ii) he 69 Newland D.E. (2004). Pedes ian exci a ion o b idges. P oceedings o he Ins i u ion o Mechanical Enginee s, Pa C: Jou nal o Mechanical Enginee ing Science Vol. 218: 477-492. Nocen al J., W igh S.J. (1999). Nume ical Op imiza ion. Sp inge , New Yo k, USA. On a io highway b idge design code (1995) The Highway Enginee ing Di ision, On a io (Canada). Picca do, G., Tubino, F. (2008). Pa ame ic esonance o lexible oo b idges unde c owd-induced la e al exci a ion. Jou nal o Sound and Vib a ion, 311(1-2): 353-371. Racic, V., Pa ic, A., B ownjonh, J.M.W. (2009). Expe imen al iden i ica ion and analy ical modelling o human walking o ces: Li e a u e e iew. Jou nal o Sound and Vib a ion, 326 (1): 1-49. doi:10.1016/j.js .2009.04.020. Rapapo , D.C. (2004). The a o molecula dynamic. Camb idge Uni e si y P ess. Ricca delli, F., B ia ico, C., Ingól sson, E.T., Geo gakis, C. (2007). Expe imen al alida ion and calib a ion o pedes ian loading models o oo b idges. P oceedings o he 2 h In e na ional Con e ence on Expe imen al Vib a ion Analysis o Ci il Enginee ing S uc u es. Po o, 24-26 Oc obe . Ricca delli, F., Pizzimen i, D. (2007). La e al Walking-induced o ces on oo b idges. Jou nal o B idge Enginee ing. Vol. 12, nº6, pp.677-688. Ronnquis , A. (2005). .Pedes ian Induced La e al Vib a ions on Slende Foo b idges. PhD Thesis. No wegian Uni e si y o Science and Technoloy. Recomendaciones pa a el p oyec o de puen es me álicos pa a ca e e as (RPM-95) (2003). Minis e io de Fomen o. Mad id (España). Schulze, H. (1980). Dynamic e ec s o he li e load on oo b idges (in Ge man). Signal und Schiene, Vol. 24, (2), pp. 91-93 and (3) pp. 143-147. Se a (2006). Guide mé hodologique passe elles pié onnes (Technical guide oo b idges: Assessmen o ib a ional beha io o oo b idges unde pedes ian loading), Se a. SIA 260 (2003) Basis o s uc u al design. Swiss Socie y o Enginee s and A chi ec s SIA. Slaich, M. (2005). Guidelines o he Design o Foo b idges. Foo b idges 2005. Venice. Shahabpoo , E., Pa ic, A., Racic, V. (2013). Modelling e ec o pedes ians walking on dynamic p ope ies o s uc u es. IMAC XXXI: A Con e ence and Exposi ion on S uc u al Dynamics, 11-14 Feb ua y, O ange Coun y, Cali o nia, USA. S uc u es Design Manual o Highways and Railways (2009). The Go e nmen o Hong Kong. Special Adminis a i e Region. 70 Taylo , D. (2003). Dampe e o i o he London Millennium Foo b idge, a case s udy in biodynamic design. In he P oceedings o he 73 h Shock and Vib a ion Symposium, San Diego, Cali o nia, USA. Teughels, A., (2003). In e se Modelling o Ci il Enginee ing S uc u es Based on Ope a ional Modal Da a. Ph. D. Thesis, Ka holieke Uni e si ei Leu en. Venu i, F., B uno, L., Bellomo, N. (2007). C owd dynamics on a mo ing pla o m: Ma hema ical modelling and applica ion o li ely oo b idges. Ma hema ical and Compu e Modelling, 45(3-4): 252-269. Wolmu h, B., Su ees, J. (2003). C owd- ela ed ailu e o b idges. Ci il Enginee ing, 156(3): 116-123. Xia, H., Zhang, N. (2005). Dynamic analysis o ailway b idge unde high-speed ains. Compu e s and S uc u es, Vol. 83 (23-34), pp. 1891-1901. Young, P. (2001). Imp o ed loo ib a ion p edic ion me hodologies, ARUP Vib a ion Semina . Zheng, X., B ownjohn, J.M.W. (2001). Modelling and simula ion o human- loo sys em unde e ical ib a ion. Sma S uc u es and Ma e ials 2001: Sma S uc u es and In eg a ed Sys ems. SPIE. Zi ano ic, S., Pa ic, A., Ingol sson, E. (2010). Modelling spa ially un es ic ed pedes ian a ic on oo b idges. Jou nal o S uc u al Enginee ing, Vol. 136 (10), pp. 1296-1308. Zi ano ic, S., Pa ic A., Reynolds P. (2007). Fini e elemen modelling and upda ing o a li ely oo b idge: The comple e p ocess. Enginee ing S uc u es, Vol. 301(1-2), pp. 126-145. Zi ano ic, S., Pa ic, A., Reynolds, R. (2005). Vib a ion se iceabili y o oo b idges unde human-induced exci a ion: a li e a u e e iew. Jou nal o Sound and Vib a ion, 279 (1-2): 1-74. PART II APPENDED PAPERS 71 II. Appended pape s. 72 Pape A: A di ec -pedes ian s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges The o iginal e sion o his pape can be ound in doi: 10.3989/ic.2014. 66.iEx a-1 Jou nal name: In o mes de la cons ucción ISI 2014 Classi ica ion: Q4 (53/59) Cons uc ion and Building Enginee ing. Impac Fac o : 0.273 SCIMAGO 2014 Classi ica ion: Q3 (121/215) Ci il Enginee ing SJR: 0.345 ISSN: 0020-0883 In o mes de la Cons ucción Vol. 66, EXTRA 1, m007 diciemb e 2014 ISSN-L: 0020-0883 doi: h p://dx.doi.o g/10.3989/ic.13.110 Recibido/Recei ed: 19/07/2013 Acep ado/Accep ed: 19/11/2013 ABSTRACT Al hough he scien i ic communi y had knowledge o he human induced ib a ion p oblems in s uc u es since he end o he 19 h cen u y, i was no un il he occu ence o he ib a ion phenomenon happened in he Millennium B idge (London, 2000) ha he impo ance o he p oblem e ealed and a highe le el o a en ion de o ed. Despi e he la ge ad- ances achie ed in he de e mina ion o he human-s uc u e in e ac ion o ce, one o he main de iciencies o he exis ing models is he exclusion o he e ec o changes in he oo b idge dynamic p ope ies due o he p esence o pedes ians. In his pape , he o mula ion o a human-s uc u e in e ac ion model, add esses hese limi a ions, is ca ied ou and i s eliabili y is e i ied om p e iously published expe imen al esul s. Keywo ds: Slende oo b idges; human induced ib a ion; pedes ian-s uc u e in e ac ion; dynamic beha iou change. RESUMEN Aunque la comunidad cien í ica enía conocimien o de los p oblemas ib a o ios inducidos po pea ones en es uc u as desde inales del siglo xix, no ue has a la ocu encia de los e en os ib a o ios acon ecidos en la pasa ela del Milenio (Lon- d es, 2000), cuando la impo ancia del p oblema se puso de mani ies o y se le comenzó a dedica un mayo ni el de a en- ción. A pesa de los g andes a ances alcanzados en la ca ac e ización de la ue za de in e acción pea ón-es uc u a una de las p incipales de iciencias de los modelos exis en es es la exclusión del cambio en las p opiedades dinámicas de la pasa ela po la p esencia de pea ones. En es e a ículo, se p esen a la o mulación de un modelo de in e acción pea ón-es uc u a que in en a da espues a a dichas limi aciones, y su alidación a pa i de esul ados expe imen ales p e iamen e publi- cados po o os au o es. Palab as cla e: Pasa elas esbel as; ib aciones inducidas po se es humanos; in e acción pea ón-es uc u a; modi ica- ción de compo amien o dinámico. (*) Uni e si y o Se ille (España). Pe sona de con ac o/Co esponding au ho : [email p o ec ed] (J. F. Jiménez-Alonso) A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges Un modelo di ec o de in e acción pea ón-es uc u a pa a ca ac e iza las ib aciones inducidas po pea ones en pasa elas esbel as J. F. Jiménez-Alonso(*), A. Sáez(*) Cómo ci a es e a ículo/Ci a ion: Jiménez-Alonso, J. F., Sáez, A. (2014). A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges. In o mes de la Cons ucción, 66(ex a-1): m007, doi: h p://dx.doi.o g/10.3989/ic.13.110. Licencia / License: Sal o indicación con a ia, odos los con enidos de la edición elec ónica de In o mes de la Cons ucción se dis ibuyen bajo una licencia de uso y dis ibución C ea i e Commons Reconocimien o no Come cial 3.0. España (cc-by-nc). J. F. Jiménez-Alonso, A. Sáez In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.1102 1. INTRODUCTION The phenomenon o in e ac ion be ween pedes ians and b idges is known since, a he end o he 19 h cen u y (1), a g oup o 60 soldie s exci ed, unde hei s ep, a b idge lo- ca ed in he B i ish own o B ough on. Al hough he scien- i ic communi y did no s op s udying his issue, i was he occu ence o he phenomenon happened in he Millennium B idge (London) ha s essed he impo ance o he p ob- lem and led o a highe le el o a en ion (2). In mos cases, he e ec ha he pedes ians induce on he oo b idge has been idealized like a mo ing a iable o ce on he s uc u e (3). The a iabili y o he abo e men ioned load ies o ha e in conside a ion he a ia ion o he le el o p essu es ha akes place be ween he pedes ian and he deck du ing he phenomenon o he s ep. Howe e , in all hese models, ei he he e ec ha he pedes ians ha e on he dynamic cha ac- e is ics o he s uc u e is neglec ed, o such e ec is consid- e ed by means o e y simpli ied inge ules. Consequen ly, hese models do no inco po a e app op ia ely he ene ge ic exchange ha akes place be ween bo h sys ems du ing he s ep o he pedes ian lows on he s uc u e. Ne e heless, in he exis ing publica ions (3) he e a e clea indica ions abou he impo ance o he dynamic in e ac ion phenomena, wi h e idence ha bo h he equencies and he modes o ib a- ion o he s uc u e a e a ec ed by he s ep o pedes ian g oups. In he case o s uc u es subjec ed o la ge pedes ian lows, he co ec es ima ion o he change o hei dynamic p ope ies due o he pedes ian c ossing is e y impo an du ing he design phase, in o de o adjus as much as possi- ble he na u al equencies o he s uc u e ou side he ange o pedes ian s ep equencies and, in he case o an in e en- ion on an exis ing oo b idge, in o de o imp o e i s com o le el (4) (5). In he p esen wo k, a me hodology o he co ec cha - ac e iza ion o he whole dynamic beha io is p oposed, by implemen ing a human-s uc u e in e ac ion model wi h h ee deg ees o eedom, in o de o cha ac e ize he mo emen o he g a i y cen e o he pedes ian in he h ee spa ial di ec ions. The p oblem o ene ge ic ex- change is add essed in a di ec o m, ealizing he modal p ojec ion o he coo dina es in con ac be ween he pe- des ian and he s uc u e, and main aining he physical coo dina es o he g a i y cen e o he pedes ian. The model conside s, in he same way, he local e ec o he s ep by means o he modal p ojec ion o he co espond- ing in e ac ion o ce. This p ocedu e o esolu ion allows, on he one hand, o un- couple he equa ions o he dynamic sys em ha go e ns he beha io o he s uc u e, hus acili a ing he e ec i e ap- plica ion o he model om he modal cha ac e is ics o he oo b idge, as ob ained om any comme cial so wa e based on he ini e elemen me hod; and on he o he hand, i al- lows o es ima e in a di ec o m bo h he dynamic cha ac- e is ics o he s uc u e du ing he pedes ian s ep, as well as he componen s o he pedes ian cen e o g a i y ac- cele a ion. Fu he mo e, addi ional pa ame e s, such as he sign o pedes ian damping in oduced in o he sys em, may be included in he model. Finally, a alida ion example o he p oposed model is p esen ed, whe e he change o he dynamic beha iou o a eal labo a o y oo b idge du ing a a iable low o pedes ians is a o ably compa ed wi h he model p edic ions. 2. ANALYSIS OF CURRENT STANDARDS Cu en ly, he mos ad anced in e na ional codes abou he dynamic beha iou o slende oo b idges (4) (5) de e mine ha , in a wide way, i he na u al equencies o he s uc u e is in he ange o pedes ian walking s ep equency (1.25- 2.30 Hz o e ical ib a ions and 0.50-1.20 Hz o ho izon- al ib a ions) he accele a ion, in ha di ec ion, needs o be de e mined and checked agains accele a ion limi s (Table 1) o gua an ee an app op ia e com o le el o each design scena io. Fu he mo e, o a oid la e al synch oniza ion he accele a ion in his di ec ion mus be below 0.10-0.15 m/s2. The design scena io is es ablished by he expec ed pedes ian a ic (Table 2) and he si ua ion o impo ance o he s uc- u e. The com o le el is de e mined by he owne o he s uc- u e, and no mally a medium com o le el mus be gua an- eed o all a ic classes, excep o pedes ian densi ies abo e 1.00 P (Pe son)/m2 whe e a minimum com o is accep able. The pedes ian induced ac ion is ep esen ed as an oscilla- o y dis ibu ed load p( ), de ined as: [1] () cos( 2) πψ =⋅ ⋅⋅⋅⋅ ′⋅p G n p whe e: G, is he conside ed componen o he s ep o ce (G=280 N e ical, 140 N longi udinal and 35 N la e al) (4) (5). , is he na u al equency o he s uc u e unde conside a ion. ′ np, is he equi alen pedes ians numbe , de ined by [2] 10.80 ζ ′=⋅⋅ nn pp o a ic classes TC1-TC3 o [3] 1.85 ′=⋅ nn pp o a ic classes TC4-TC5. ψ, is he educ ion coe icien ha akes in o accoun he p obabili y ha he oo all equency app oaches he na u al equency unde conside a ion. ζ, is he s uc u al damping a io. np, is he numbe o he pedes ians on he loaded su ace S (np = S · densi y). S, is he loaded su ace ha depends on he shape o he no - mal mode unde conside a ion. Table 1. De ined com o classes wi h limi accele a ion anges (5). Le el Deg ee Ve ical accele a ion Ho izon al accele a ion 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 Discom o >2.50 m/s2>0.80 m/s2 Table 2. T a ic classes (5). Classes Densi y d [P/m2] Cha ac e is ics TC1 < 15 P 15 single pe sons TC2 < 0.20 P/m2Com o able and ee walking TC3 < 0.50 P/m2Un es ic ed walking, signi ican ly dense a ic TC4 < 1.00 P/m2Uncom o able si ua ion, obs uc ed walking TC5 < 1.50 P/m2Unpleasan walking, e y dense a ic A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges Un modelo di ec o de in e acción pea ón-es uc u a pa a ca ac e iza las ib aciones inducidas po pea ones en pasa elas esbel as In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.110 3 Howe e , his me hodology p esen s some limi a ions: • he equi alen pedes ian numbe (pedes ian mo ing in phase wi h he s uc u e) has been de e mined by he ex- pe imen al esul s o only one oo b idge (5). • he change in he dynamic s uc u al p ope ies ha he pedes- ians low causes is conside ed h ough a inge ule (addi ion o all he pedes ian mass densi y o he s uc u e mass ma ix). • he in e ac ion be ween pedes ians and he s uc u e is only sligh ly conside ed, so he in e na ional s anda ds do no conside adequa ely he synch oniza ion phenomenon be- ween pedes ians o be ween hese ones and he s uc u e. The es ima ions ca ied ou , unde his me hodology, no - mally o e es ima e he eal esul s (5). 3. PROPOSAL OF A HUMAN-STRUCTURE INTERACTION MODEL In his sec ion a me hod o he simula ion o he in e ac ion be ween he pedes ian and he oo b idge is p oposed. I ol- lows om he applica ion o he dynamic equilib ium equa- ions o a simpli ied model o in e ac ion wi h sp ung and unsp ung masses (Figu e 1). Fo n modes o ib a ion φi (x), he o al esponse o he s uc u e may be decomposed in e ms o he ampli ude o he di e en modes yi ( ) as: [4] w(x, )=yi( )⋅ ϕ i(x) i=1 n ∑ [5]  w(x, )= yi( )⋅ ϕ i(x) i=1 n ∑+yi( )⋅ ⋅′ ϕ i(x) i=1 n ∑ [6]  w(x, )= yi( )⋅ ϕ i(x) i=1 n ∑+2⋅ yi( )⋅ ⋅′ ϕ i i=1 n ∑(x)+yi( )⋅ 2⋅′′ ϕ i(x) i=1 n ∑ whe e [7] ′ ϕ i(x)=d dx ϕ i(x) is he spa ial de i a e o he mode o ib a ion i. [8] ′′ ϕ i(x)=d2 dx2 ϕ i(x) is he second spa ial de i a e o he mode o ib a ion i. and i is neglec ed, due o i s low magni ude, he empo al a ia ion o he s ep speed . Conside ing he equilib ium o he sys em, s uc u e and pe- des ian model, he ollowing coupled equa ion sys em may be ob ained. [9] Mi yi+Ci yi+Kiyi= ϕ i ( ) ⋅Fin [10] 0   ()() +−+−=my cy ykyy aa as as [11] in   ()() +−+−=−my cy ykyy FF ss sa sa s Thus, Fin ollows om he abo e equa ion o yield. [12] in   ()() =− −− −− FFmy cy yk yy ssss as a And subs i u ing his equa ion in o he equilib ium equa ion o he s uc u e. [13] M i  y i +C i  y i +K i y i = ϕ i ( ) ⋅F s −m s  y s −c y s − y a ( ) −k y s −y a ( ) ( ) Applying he equa ions o compa ibili y o displacemen s, e- loci y and accele a ion be ween he s uc u e and he simpli- ied model o in e ac ion. [14] (,)=ywx s [15] (,)  =ywx s [16] (,)   =ywx s Subs i u ing hese ela ions in he o e all dynamic equilib- ium equa ion o he s uc u e and o ganizing in o ma ion in a ma ix o m, he ollowing model o in e ac ion is ob- ained. [17] () () () () () () ()   ⋅+ ⋅+ ⋅= M y C y K y F Conside ing he na u e o he esul ing sys em, he use o a me hod o β-Newma k in eg a ion amily is p oposed, wi h pa ame e s β=1/4 and γ=1/2, hus ensu ing an uncondi ion- ally s able sys em. Figu e 1. Pedes ian-s uc u e in e ac ion model. J. F. Jiménez-Alonso, A. Sáez In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.1104 unde h ee con olled g oup o pedes ians, will be com- pa ed in o de o de e mine he pa ame e s and goodness o he p oposed model. The alida ion will be ca ied ou , o simplici y, in he e - ical di ec ion, al hough he ex ac ed esul s a e easily ex- apola ed o he o he di ec ions. 4. DETERMINATION OF THE WALKING FORCES The mo emen o he body mass and he pu -down, olling and push-o o he ee o one pedes ian gene a e he in- duced h ee-dimensional o ces be ween bo h elemen s, Fs, ha acco ding o he esea ch de eloped by di e en au ho s (4), can be de e mina ed om a Fou ie se ies decomposi ion in he h ee-space componen s. [20] Fp, e ( )=P1+ α i, e sin 2 π i s − φ i ( ) i=1 n ∑ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ [21] Fp,la ( )=P α i,la sin π i s − φ i ( ) i=1 n ∑ [22] Fp,long( )=P α i,long sin 2 π i s − φ i ( ) i = 1 n ∑ whe e Fp, e e ical pe iodic o ce due o walking o unning Fp,la la e al pe iodic o ce due o walking o unning Fp,long longi udinal pe iodic o ce due o walking o unning P [N] medium pedes ian weigh (in e na ionally conside ed as P=700.00 N) αi, e αi,la αi,long Fou ie coe icien o he i h ha monic o e - ical, la e al and longi udinal o ces o dynamic load ac o (DLF). s [Hz] s ep equency φi phase shi o he i h ha monic n o al numbe o con ibu ing ha monics. Among he con ibu ions o he di e en au ho s, o he de elopmen o he p esen documen , he e ical dynamic In he p e ious exp essions, he alue o he ib a ion modes is ze o, when he pedes ian emains ou side he s uc u e. [18] ϕ i(x)=0 o 0≥x≥L o , wi h L being he leng h o he s uc u e In he p oposed me hod, φi (x) is ob ained, in a disc e e way, using he ini e elemen me hod, collec ing he modal dis- placemen s and de i a es in each o he nodes o he s uc- u e. To ob ain a con inuous unc ion o he modes hey a e de e mined om he shape unc ions consis en wi h he i- ni e elemen app oxima ion. Fo he oo b idge, he in e po- la ion unc ions a e cubic adop ing he Be noulli hypo hesis o he beam elemen s. [19] ϕ i(x)= ϕ i j⋅Nj(x) j ∑ Whe e Nj (x) a e he shape unc ions and ϕ i j a e he nodal alues. Fo a g oup o k pedes ians (Figu e 2), we may u he ep esen each one by he abo e simpli ied in e ac ion model. When a g oup o pedes ians is conside ed in he calcula- ions, he numbe o di e en ial equa ions o sol e inc eas- es. In he case o a single pedes ian, he p oposed model leads o a sys em o n+1 equa ions, co esponding o he conside ed numbe o ib a ion modes n plus he app op i- a e simpli ied in e ac ion mechanical elemen sys em. Simi- la ly, when conside ing a g oup o k pedes ians, a sys em o n+k di e en ial equa ions will need o be sol ed. I is im- po an o no e ha he equa ions o he modes o ib a ion o he s uc u e a y in e ms o he posi ion o pedes ians. A e e y ins an , he numbe s o pedes ian on he de o med shape mus be calcula ed, as well as he alue o he ampli- ude, slope and cu a u e co esponding o hei posi ion. As a p elimina y alida ion o he p oposed o mula ion, he p e iously de ined pa ame e s will be es ima ed om he e- sul s a ailable in he li e a u e o compa able s udies (6), as summa ized in nex sec ions. Finally, he expe imen al and nume ical dynamic cha ac e is ics o a labo a o y oo b idge, Figu e 2. Pedes ians g oup acco ding o he pedes ian-s uc u e in e ac ion model. A di ec pedes ian-s uc u e in e ac ion model o cha ac e ize he human induced ib a ions on slende oo b idges Un modelo di ec o de in e acción pea ón-es uc u a pa a ca ac e iza las ib aciones inducidas po pea ones en pasa elas esbel as In o mes de la Cons ucción, Vol. 66, EXTRA 1, m007, diciemb e 2014. ISSN-L: 0020-0883. doi: h p://dx.doi.o g/10.3989/ic.13.110 5 load ac o s p oposed by Se a (5) (see Table 3 and Figu e 3) will be conside ed o cons uc and alida e ou model. This c i e ion is widely accep ed by bo h he scien i ic communi y and he designe s o his ype o s uc u es. These s anda ds ob ained he dynamic coe icien s om expe imen al es s pe o med on mobile pla o ms. The pedes ian load, acco d- ing he esul s o such es s, is adequa ely cha ac e ized by he con ibu ion o he i s h ee ha monics. The ela ion be ween he eloci y magni ude, , and he pac- ing equency, s, is conside ed by he empi ical ela ionship based on he wo k o Be am and Ruina (3). [23] s=0.35⋅ 3−1.59 ⋅ 2+2.93⋅ 5. INITIAL ESTIMATION OF THE DYNAMIC PROPERTIES OF THE HUMAN-STRUCTURE INTERACTION MODEL Fo he es ima ion o he dynamic cha ac e is ics o he SDOF-sys em, as a i s app oxima ion, a wide bibliog aphic s udy has been made. The e a e se e al s udies ha collec he e ec o spec a o s on s adiums s ands in he dynamic beha iou o he s uc u e by a SDOF s a ic sys em (6). In Figu e 4 and Table 4, a scheme o he models used and he es ima ed dynamic pa ame e s a e shown, whe e h and ζh a e he na u al equency and he equi alen damping a io o he human sys em. The abo e esul s allow es ablishing a likely ange o a ia- ion o he sys em pa ame e s. Thus, conside ing he maxi- mum and minimum alues (Figu e 4) o he sp ung mass (ma), he equi alen human damping a io (ζh) and he e i- cal na u al equency ( h) o each pedes ian, i is shown in Table 5 a possible ange o a ia ion o he pa ame e s o he p oposed model. The sp ung mass (ma) is p esen ed as a pe - cen age o he o al mass. A pedes ian ype wi h a mean o al mass o 70.00 kg is conside ed, as es ablished by he Eu o- pean s anda ds (4). Figu e 3. Ve ical componen o walking pedes ian o ce (5). Figu e 4. Simpli ied dynamic ep esen a ions o he s anding human body: (1) SDOF model (2) SDOF model wi h igid suppo . Table 4. Dynamic p ope ies o SDOF equi alen s o s anding humans (6). Human Model Modal P ope ies Human Model Modal P ope ies Foschi e al. (Model 1) h=3.30 Hz Fala i (Model 1) h=10.43 Hz ζh=53.00 % ζh=50.00 % ma=91.00 kg ma=25.00 kg Al-Foqaha’a (Model 1) h=3.50 Hz Zheng and B ownjohn (Model 1) h=5.24 Hz ζh=34.00 % ζh=39.00 % ma=83.00 kg ma=85.00 kg Al-Foqaha’a (Model 2) h=3.70 Hz Ma sumo o and G i in (Model 1) h=5.74 Hz ζh=36.00 % ζh=69.00 % ma=75.00 kg ms=8.00 kg ma=76.10 kg B ownjohn (Model 1) h=4.90 Hz Ma sumo o and G i in (Model 2) h=5.88 Hz ζh=37.00 % ζh=61.00 % ma=80.00 kg ma=70.60 kg ms=7.06 kg Table 3. Fou ie coe icien and phase shi o e ical dynamic load ac o s (5). α1α2α3φ1φ2φ 0.40 0.04 0.04 0.00 90.00 90.00 pedes ian-s uc u e in e ac ion sub-model plus (ii) a c owd sub-model. The i s sub-model ollows om a modal p ojec ion o a sys em wi h wo d.o. , ha simula es he beha io o each pedes ian, on he ib a ion modes o he s uc u e. The pa ame e s o his model ha e been es ima ed om he accele a ions eco ded on a eal oo b idge. Fo he second sub-model, he c owd beha iou is simula ed ia a mul i-agen me hod. The pe o mance o he esul ing o e all model is assessed by co ela ing he expe imen al and nume ical dynamic o a eal oo b idge unde a g oup o pedes ians a di e en con olled s ep equencies. In pa icula , he change in he i s na u al equency induced by he pedes ian- oo b idge in e ac ion is discussed in de ail. The p oposed model leads o nume ical esul s ha exhibi good ag eemen wi h he eco ded expe imen al alues. The e o e i is a aluable ool o es ima e he change on he modal p ope ies o a oo b idge induced by he c owd-s uc u e in e ac ion phenomenon. Keywo ds: simpli ied biomechanical model, human-s uc u e in e ac ion, c owd dynamics, change o na u al equencies, oo b idge. INTRODUCTION. Du ing he las i een yea s, signi ican e o has been made by he scien i ic communi y o cha ac e ize adequa ely he dynamic esponse o oo b idges unde pedes ian lows (Racic e al., 2009; Zi ano ic a al., 2005). Al hough impo an ad ances ha e been achie ed in he de ini ion o he pedes ian walking o ce (Bu z e al., 2007; Se a, 2006), some aspec s o he c owd-s uc u e in e ac ion p oblem ha e no been comple ely sol ed and s ill dese e a en ion. The s udy o he c owd-s uc u e in e ac ion p oblem has been pe o med acco ding o h ee key aspec s in o de o: (i) cha ac e ize he walking o ce ansmi ed by each pedes ian; (ii) cha ac e ize he pedes ian-s uc u e in e ac ion and inally (iii) o cha ac e ize he in e ac ion among pedes ians in he c owd. In his way, esea ch e o s ocused ini ially on he de e mina ion o analy ical exp essions o he walking o ce induced by a pedes ian (Zi ano ic e al., 2005; Bu z e al., 2007). Howe e , as he inc easing sophis ica ion o his p oposed exp essions, cha ac e izing he pedes ian walking o ce, did no lead o signi ican imp o emen s in he nume ical es ima ions o he esponse o he oo b idge unde pedes ian lows, new ac o s we e conside ed in he models. In ha sense, as a esul o he esea ch conduc ed a he Millennium oo b idge (Dalla d e al., 2001), i was concluded ha he e ec o a pedes ian low on he s uc u e in ol ed no only an equi alen pedes ian o ce bu also he modi ica ion o he dynamic p ope ies o he s uc u e. Subsequen ly, his esul was alida ed by o he epo ed wo ks (Ingol sson e al, 2008) whe e pedes ians we e conside ed as ac i e damping o ces ha inc eased he o e all damping o he s uc u e. Following hese wo ks, se e al app oaches ha e been p esen ed in he li e a u e o model de pedes ian-s uc u e in e ac ion p oblem. The i s models main ained he idea o equa ing he pedes ian o an ac i e iscous dampe (Geo gakis and Jo gesen, 2013). Subsequen ly, o he s modal pa ame e s we e conside ed in he in e ac ion phenomenon leading o he appea ance o single deg ee o eedom sys ems o cha ac e ize he beha iou o each pedes ian (Shahabpoo e al., 2013). Acco ding o hese la e models, each pedes ian induced a modi ica ion o bo h he damping and s i ness ma ix o he oo b idge du ing i s c ossing. On he o he hand, he cha ac e iza ion o he dynamic esponse o he s uc u e unde pedes ian lows mo i a ed he s udy o how he pedes ians in e ac in a c owd (Zi ano ic a al., 2010). In ha sense, bo h s a is ical dis ibu ions o he s ep pedes ian equencies in a c owd (Venu i e al., 2007) and ela ions be ween he pedes ian eloci y and s ep equency (B uno and Venu i, 2009) we e es ablished. In o de o cha ac e ize he c owd beha iou , he models ha e e ol ed om a mac oscopic o a mic oscopic app oach. The mo emen o he c owd simula ed o iginally by he luid mechanics laws (Venu i e al., 2007), is cu en ly modelled using pa icle dynamics (Ca oll e al., 2012), by conside ing each pedes ian as an agen whose equilib ium is achie ed h ough he in e ac ion o ces applied by i s en i onmen . Cu en ly, se e al p oposals ha e eme ged (Venu i e al., 2014, Ta a es e al. 2014).in o de o o e a di ec and join esponse o he h ee abo e men ioned key aspec s. In all hese p oposals, he c owd-s uc u e in e ac ion has been simula ed h ough he coupling o wo sub-models: a pedes ian-s uc u e in e ac ion model and a c owd model based on mul i-agen heo y. In his pape , a new c owd-s uc u e in e ac ion model in he e ical di ec ion is p oposed. The model ep esen s an e olu ion o he exis ing p oposals and aims o imp o e some sho comings o he p e ious epo ed wo ks. The p oposed model in ol es wo sub-models as well. A pedes ian-s uc u e in e ac ion sub-model ollows om he modal p ojec ion o a wo deg ee o eedom sys em, whe e he pedes ian mass is di ided in sp ung plus unsp ung componen s, on he ib a ion modes o he s uc u e. The c owd beha iou is simula ed ia a mul i-agen sub-model whe e he mo emen o each pedes ian is go e ned by he expe ienced in e ac ion o ces. A physical in e ac ion o ce has been included o assess mo e accu a ely he c owd beha iou unde high pedes ian densi ies. The in e ac ion be ween he wo sub-models in he e ical di ec ion is achie ed by implemen ing a s op h eshold, so ha i ce ain accele a ion limi is exceeded he a ec ed pedes ians s op. The es ima ion o he pa ame e s o he p oposed pedes ian-s uc u e in e ac ion model has been expe imen ally pe o med based on he esul s o a pedes ian es conduc ed a he Viana oo b idge (Viana do Cas elo, Po ugal). Subsequen ly, he c owd-in e ac ion model has been assessed by co ela ing he expe imen al and nume ical esponse o he Viana oo b idge unde a g oup o 50 pedes ians. Finally, he model has been applied o s udy he nume ical change o he i s e ical ib a ion mode o he Viana oo b idge due o he p esence o he g oup o pedes ians. The p oposed model may u he be used o p edic he occu ence o he la e al lock-in phenomenon in oo b idges o o imp o e he e iciency o he con ol de ices in oduced in a oo b idge when ib a o y p oblems a e de ec ed. The pape is o ganized as ollows: The p oposed c owd-s uc u e in e ac ion model is p esen ed in sec ion 2, by desc ibing (i) he pedes ian-s uc u e in e ac ion sub-model; (ii) he c owd sub-model as well as (iii) he in e ac ion mechanisms be ween bo h sub- models. Sec ion 3 is de o ed o he expe imen al es ima ion o he main pa ame e s ha cha ac e ize he pedes ian-s uc u e in e ac ion sub-model. In sec ion 4, he alidi y and accu acy o he o e all c owd-s uc u e in e ac ion model is success ully assessed by co ela ing bo h he expe imen al and nume ical esul s o a eal oo b idge (Viana do Cas elo, Po ugal). Finally, some concluding ema ks a e d awn o close he pape in sec ion 5. PROPOSAL OF A SIMPLIFIED BIOMECHANICAL CROWD-STRUCTURE INTERACTION MODEL IN VERTICAL DIRECTION. The comple e c owd-s uc u e in e ac ion consis s o wo indi idual submodels (Fig. 1), one o he pedes ian-s uc u e in e ac ion ( ha includes he pedes ian and oo b idge dynamic beha iou ) and ano he o he c owd. The pedes ian-s uc u e in e ac ion model is esponsible o modelling, in a simpli ied way, all he dynamics e ec s (ine ia, damping, s i ness) induced on he oo b idge by he c ossing o a pedes ian. The e ical accele a ion a z  expe imen ed by each pedes ian is ob ained as ou pu om his model. The c owd model is implemen ed as a beha iou al model p o iding a desc ip ion o he indi idual pedes ian posi ion, p x, walking pedes ian eloci y, p , and s ep pedes ian equency, p , wha allows o simula ing he o e all beha iou o he c owd and i s in luence in he dynamic beha iou o he oo b idge. Fig.1. Layou o he biomechanical c owd-s uc u e in e ac ion model. Fo each i e a ion he c owd model de e mines he posi ion and s ep eloci y o each pedes ian. These wo pa ame e s a e used as inpu o de ine he s ep equency and walking o ce o each pedes ian in o he pedes ian-s uc u e model, ob aining as ou pu he pedes ian e ical accele a ion. The pedes ian eloci y o each indi idual is hen modi ied acco ding o he le el o accele a ion expe imen ed by each pedes ian. Finally, he p ocess is epea ed wi h he upda ed alues o he posi ion and he eloci y (Fig. 1). Modelling he pedes ian-s uc u e in e ac ion in he e ical di ec ion. The p oposed pedes ian-s uc u e in e ac ion in he e ical di ec ion ollows om he applica ion o dynamic equilib ium equa ions (Clough and Penzien, 1993; Dominguez, 2001) o a simpli ied model o in e ac ion (Fig. 2) wi h sp ung ( a m) and unsp ung masses ( s m). This me hodology has been applied p e iously by he au ho s success ully (Jiménez-Alonso and Sáez, 2014), and in his pape i is gene alized in o de o ake in o INPUT OUTPUT PEDESTRIAN-STRUCTURE MODEL CROWD MODEL PEDESTRIAN/STRUCTURE PARAMETERS CROWD-STRUCTURE MODEL p x p a z  accoun he modi ica ion o he pedes ian eloci y due o he c owd-s uc u e in e ac ion. Fig.2. Biomechanical pedes ian-s uc u e in e ac ion model. Conside ing he balance o he sys em, s uc u e and pedes ian model, he ollowing coupled equa ions a e ob ained. in _ )( FxzKzCzM piNUMiiiiii       (1)    0     sapsapaa zzkzzczm  (2)    in , FFzzkzzczm e paspaspss       (3) whe e a m is he sp ung mass o he pedes ian [kg]. s m is he unsp ung mass o he pedes ian [kg]. as mmm  is he o al mass o he pedes ian [kg]. a z is he absolu e e ical displacemen o he sp ung mass [m]. s z is he absolu e e ical displacemen o he unsp ung mass [m]. p k is he equi alen s i ness o a pedes ian [N/m]. m a c p k p m s z a z s F in L z xF in F s M i C i K i d p y x p w(x, ) p c is he equi alen damping o a pedes ian [sN/m]. e p F, is he e ical pedes ian o ce due o walking [N]. in F is he in e ac ion o ce be ween he pedes ian and he s uc u e [N]. i M is he modal mass o he ib a ion mode i [kg]. i C is he modal damping o he ib a ion mode i [sN/m] i Kis he modal s i ness o he ib a ion mode i [N/m]. iNUM _  is he e ical componen o he nume ical ib a ion mode i. x pxp  is he longi udinal posi ion o he pedes ian [m]. px is he longi udinal componen o he pedes ian eloci y ec o [m/s]. F om Eq.(3) he ollowing exp ession is ob ained o , in F,     aspaspss e p zzkzzczmFF       ,in ……………………(4) and subs i u ing his equa ion in o Eq.(1) yields.        aspaspss e ppiNUMiiiiii zzkzzczmFxzKzCzM         ,_  …(5) Applying, a he con ac poin he equa ions o compa ibili y o displacemen s, eloci y and accele a ion be ween he s uc u e and he simpli ied pedes ian-model o in e ac ion a e ob ained. ),(),( w xwz pxps    (6) ),(),( w xwz pxps      (7) ),(),( w xwz pxps      (8) These quan i ies may be exp essed in e ms o he ampli ude )( zi and he modal shape o he n nume ical conside ed ib a ion modes )( _x iNUM  , neglec ing he e m o a ia ion o he pedes ian eloci y o e he ime, as:    n i piNUMip x z xw 1 _)()(),(  (9)     n i piNUMpxi n i piNUMip x zx z xw 1 _ 1 _)()()()(),(    (10)       n i piNUMxpip n i iNUMxpi n i piNUMip x zx zx z xw 1 _ 2 , 1 _, 1 _))()()()(2)()(),(    …(11) dx xd xiNUM iNUM )( )( _ _     (12) 2 _ 2 _ )( )( dx xd xiNUM iNUM     (13) whe e )( _x iNUM  is he i s spa ial de i a e o he mode o ib a ion i. )( _x iNUM   is he second spa ial de i a e o he mode o ib a ion i. The abo e ela ions -Eqs.(6) o (11)- a e hen subs i u ed in he o e all dynamic equilib ium equa ions -Eqs.(1) o (3)- so ha , o ganizing in o ma ion in a ma ix o m, he ollowing model o in e ac ion is ob ained (see Appendix I o ma ix o mula ion). )()()()()()()( FzKzCzM       (14) In he p e ious exp essions, he alue o he nume ical ib a ion modes is ze o, when he pedes ian emains ou side he s uc u e. 0)( _  piNUM x  o L x x p p   )( 0)( (15) wi h L being he leng h o he s uc u e The nume ical ib a ion modes, )( _x iNUM  , a e ob ained in a disc e e way using he co esponding ini e elemen me hod as:   j j j iiNUM xNx )()( _  ..(16) whe e )(xN ja e he shape unc ions and j i  a e he nodal alues. Al hough he pape ocuses on ib a ions in e ical di ec ion. The o mula ion o he p oposed model may be u he gene alized o he o he wo di ec ions, longi udinal and la e al, by acco dingly modi ying bo h he equa ion ha go e ns he conside ed pedes ian load in each di ec ion and he alue o he modal pa ame e s ha de ine he TDOF ( wo deg ees o eedom) pedes ian model. In his way, he esul ing model would be sui able o he mo e gene al 3-D p oblem and i could he e o ake in o accoun he possible in e ac ion in he h ee spa ial di ec ions. Fo a g oup o k pedes ians (Fig. 2), each o hem will be ep esen ed by he abo e simpli ied in e ac ion model. In he case o a single pedes ian, he p oposed model leads o a sys em o n+1 equa ions, co esponding o he conside ed numbe o ib a ion modes n plus he simpli ied in e ac ion equa ion. Simila ly, when conside ing a g oup o k pedes ians, a sys em o n+k di e en ial equa ions will need o be sol ed. Conside ing he na u e o he esul ing sys em, he use o a me hod o  -Newma k in eg a ion amily is p oposed, wi h pa ame e s 41   and 21   , hus ensu ing an uncondi ionally s able sys em. Fu he mo e, he in eg a ion s ep,  , is es ablished acco ding o he usual ecommenda ions (Clough and Penzien, 1993; Dominguez, 2001) o dynamics models based on modal decomposi ion echnique, as he minimum o he ollowing alues. )01.0, 4 , 200 , 8 1 min( minmin max pp n L L    sec. (17) wi h max [Hz] being he highes conside ed ib a ion equency o he s uc u e (30 Hz acco ding o Dominguez (2001)) and min L[m] he minimum span leng h o he pedes ian b idge. Pedes ian e ical walking o ce. The mo emen o he body mass and he pu -down, olling and push-o o he ee o one pedes ian gene a e he induced e ical o ces be ween he pedes ian and he s uc u e, e p F,. Acco ding o di e en au ho s (Bu z e al., 2007; Se a, 2006), his o ce can be de e mined om a Fou ie se ies decomposi ion as:          n i pis e i e p iPF 1 ,, 2sin1  (18) whe e gmP  [N] is he medium pedes ian weigh , g being he accele a ion o he g a i y. e i,  is he Fou ie coe icien o he i h ha monic o e ical o ces o e ical dynamic load ac o (VDLF). s [Hz] is he s ep equency o he pedes ian. i  is he phase shi o he i h ha monic o he pedes ian o ce. p  is he phase shi among pedes ians. n is he o al numbe o con ibu ing ha monics. In o de o de e mine he numbe o pedes ians ha c oss he oo b idge in phase, a Poisson dis ibu ion has been adop ed, acco ding o he esul s by Ma sumo o e al. (1978). Fu he es s pe o med on Sol e ino b idge (Se a, 2006), as well as o he s udies, sugges ha lock-in in e ical di ec ion does no seem p obable due o he low sensi i i y o he pedes ians o e ical ib a ions. Thus, when a g oup o p n pedes ians a i e a he oo b idge, he numbe o pedes ians andomly synch onized is p n. This synch oniza ion c i e ion has been adop ed in ou c owd-s uc u e in e ac ion model. I s implemen a ion in he model is achie ed by he phase shi pa ame e , p  . Fo a gi en gene a ion/g oup o pedes ians, he alue o he phase shi p D is he sliding o ce s eng h due o he con ac be ween pedes ians (Table 3). p is a no malized ec o pe pendicula o p n. pp p  is he angen ial componen o he ela i e pedes ian eloci y, wi h p  being he di e ence o eloci ies be ween wo gi en pedes ians. and unc ion H is de ined as: 00 0 )(      i i H (27) In e ac ions wi h bounda ies. The in e ac ion wi h he bounda ies gi es ise o o ces, bou F. These o ces a e equi alen o he ones esul ing om he in e ac ion wi h o he pedes ian, so hey can be o mula ed in a simila ashion. an bou no boubou FFF  (28)  bbpb b bp b no bou d HC B d AnF                     exp (29)   bbpbpbbou d HD F  , an (30) whe e no bou F is he no mal componen o he bounda y in e ac ion o ce. an bou F is he angen ial componen o he bounda y in e ac ion o ce. b A is he in e ac ion s eng h be ween he pedes ian and he bounda y (Table 3).. b B is he ange o he epulsi e in e ac ion be ween he pedes ian and he bounda y (Table 3).. b d is he dis ance be ween he pedes ian and he bounda y. b C is he body o ce s eng h due o he con ac wi h he bounda y (Table 3). b D is he sliding o ce s eng h due o he con ac wi h he bounda y (Table 3). b n is he no malized ec o de ined pe pendicula ly om he pedes ian o he bounda y. b is he ec o pe pendicula o b n. deno es scala p oduc . Resul an o ce. Finally, he p oposed mul i-agen model ha simula es he beha iou o he c owd consis s in he sum o all hese pa ial o ces ha ep esen he di e en in luences ha he pedes ians su e when in e ac ing in a c owd. The e o e, he esul an o ce, pci F, desc ibes he mo emen and di ec ion o each pedes ian in he c owd as: boupedd ipci FFFF    (31) Table 3. C owd model pa ame e s conside ed (Helbing and Molná , 1995; Ca oll e al., 2012). Pa ame e Elemen Value Relaxa ion ime 0.50 sec. In e ac ion s eng h pedes ians p A 2000 N In e ac ion ange pedes ians p B 0.30 m Po en ial ac o p  0.20 Con ac s eng h pedes ians p C 2000 N Sliding s eng h pedes ians p D 4800 N In e ac ion s eng h bounda ies b A 5100 N In e ac ion ange bounda ies b B 0.50 m Con ac s eng h bounda ies b C 2000 N Sliding s eng h bounda ies b D 4800 N Radius o pedes ian p 0.20 m Fo he gene a ion o pedes ians lows h ee pa ame e s ha e been conside ed, he pedes ian densi y es ablished by in e na ional s anda ds (Bu z e al., 2007; Se a, 2006) acco ding o he expec ed pedes ian a ic on he oo b idge, he alue o he desi ed eloci y, d , o he pedes ians and he dis ance be ween pedes ians, p d. In his pape a one-way a ic has been conside ed o simplici y in he gene a ion o he pedes ian lows, al hough he model may be easily gene alized o wo-way a ic. The alues o he desi ed eloci y o each pedes ian ha e been ob ained om he pedes ian s ep equencies, s . Fo he p esen c owd-s uc u e in e ac ion model he Gaussian dis ibu ion o he pedes ian s ep equency p o ided by Zi ano ic e al (2010) has been adop ed, )186.0,87.1(NHz (whe e ),(   N is he Gaussian dis ibu ion,  is he mean alue and  is he s anda d de ia ion). A e assigning a s ep equency o each pedes ian, i s desi ed eloci y is de e mined om he empi ical ela ion gi en by Be am and Ruina (2001) (B uno and Venu i (2009)), ppp  93.259.135.0 23 s (32) so ha he ini ial condi ions o each pedes ian assume ha he pedes ian eloci y, p , is equals o he desi ed eloci y, d . Finally, once he pedes ian densi y and he desi ed eloci y o each pedes ian a e es ablished, he o iginal dis ance among pedes ians is calcula ed conside ing he wid h o he oo b idge and assuming a ec angula -shaped mesh o pedes ians. Solu ion p ocedu e. The esul an c owd-s uc u e in e ac ion o ce, pci F, ac s on each pedes ian du ing each ime i e a ion j. The accele a ion ec o , j p a, ollows om m j pci j p F a (33) conside ing a pedes ian mass, m(as mm   ). The e alua ion o he emaining a iables ha go e n he c owd model, 1j p and 1j p x, is hen pe o med using a mul i-s ep me hod based on a p edic i e-co ec i e me hod, namely he Gea ’s algo i hm (Hee mann, 1986), due o he ac ha he social o ces depend on he eloci y and he posi ion o he pedes ians. The algo i hm calcula es i s an app oxima e alue, called a p edic o ha subsequen ly is co ec ed wi h a co ec o alue. The algo i hm applied in his case is o i h o de . Fi s , he new loca ions, eloci ies, accele a ions and highe de i a i es a e p edic ed acco ding o -Eqs.(34) o (37)-, whe e he supe sc ip )( p indica es a p edic ed alue. j p j p j p j p j p jp p βαa xx         2462 432 1)( (34) j p j p j p j p jp p βαa       62 32 1)( (35) j p j p j p jp p βαaa     2 2 1)( (36) j p j p jp p βαα  1)( (37) whe e p α is he i s de i a i e o p a and p β is he second de i a i e o p a. F om hese p edic ed loca ions, he di e ence be ween he accele a ion a ime s ep j+1 and he p edic ed accele a ion 1)( jp p ais ob ained (Eq.(38)) om 1)(1   jp p j pco aaΔ (38) This co ec ion ac o ec o , co Δ, allows o ob aining he co ec ed loca ions, eloci ies and highe de i a ions acco ding o 120 19 2 2 1)(1     co jp p j pΔxx (39) 4 3 2 1)(1     co jp p j pΔ (40) 2 1 3 1 1)(1     co jp p j pΔαα (41) 12 1 12 1 2 1)(1     co jp p j pΔββ (42) C owd-s uc u e in e ac ion. The maximum e ical accele a ion expe ienced by each pedes ian c ossing he s uc u e,  max a z  may be compa ed agains he accele a ion h eshold alues es ablished by in e na ional codes (Bu z e al., 2007; Se a, 2006) in o de o modi y he indi idual pedes ian beha iou due o he esponse o he s uc u e. Due o he good ole ance o pedes ians o e ical ib a ions (Racic e al., 2009; Zi ano ic a al., 2005) only a s op h eshold has been es ablished. In his way, when he e ical accele a ion expe ienced by a pedes ian exceeds he accele a ion limi e, 50.2 lim  z  m/s2 (Se a, 2006), pedes ians s op walking o main ain balance, and hey emain s opped un il he accele a ion le el educes again, conside ing o bo h ac ions a eac ion ime, 00.2 ea sec. A linea a ia ion o he pedes ian eloci y du ing he eac ion ime has been assumed. In o de o a oid meaningless small walking eloci ies, a p ac ical lowe limi on walking eloci y magni ude has been imposed as sugges ed by Ca ol e al. (2012).    0 1.0 d p i i   lim max lim max 1.0 zz zz a dpa     (43) EXPERIMENTAL ESTIMATION OF THE PARAMETERS OF THE PEDESTRIAN STRUCTURE INTERACTION MODEL. In e se dynamic p oblem me hodology implemen ed. The es ima ion o he pa ame e s ha cha ac e ize he dynamic beha iou o he p oposed pedes ian-s uc u e in e ac ion model in e ical di ec ion has been pe o med om he esponse o a eal oo b idge unde he c ossing o wo pedes ians by sol ing he co esponding in e se dynamic p oblem me hodology. In Fig. 5 he lowcha o he iden i ica ion p ocedu e is shown. Fig.5. Flowcha o he iden i ica ion p ocedu e. INITIALIZATION VARIABLES (1, e 2, e mah h)k NUMERICAL ANALYSIS NUMERICAL ACCELERATIONS EVALUATION OF OBJECTIVE FUNCTION 2 exp ,, 2 1 i num i aa  MINIMIZATION STEP UPDATED VALUES (1, e 2, e mah h)k+1 RESULT: IDENTIFIED VARIABLES (1, e 2, e mazh h)=(1, e 2, e mah h)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 h  1, e  2, e UPDATED F.E.M. EXPERIMENTAL PEDESTRIAN TEST TWO PEDESTRIANS p =1.50, 2.00 AND 2.50 Hz exp ,i a EXPERIMENTAL RESPONSE num i a, As iden i ica ion me hod he minimiza ion o a leas squa es p oblem has been adop ed (Koh and Pe y, 2010). The objec i e unc ion has been de ined as he mean squa e e o be ween he expe imen al ( exp ,i a, whe e i is he conside ed sec ion) and nume ical ( num i a,) e ical accele a ions ob ained, in ou poin s o he Viana oo b idge (Ba bosa e al., 2012), unde he c ossing o wo pedes ians a con olled s ep equencies. Gene ic algo i hms ha e been used o ensu e a global op imiza ion and a sea ch domain o each pa ame e has been es ablished. The cha ac e iza ion o he dynamic beha iou o he oo b idge has been pe o med by he ini e elemen model upda ing (Teughels, 2003; Zi ano ic e al., 2007) based on he modal pa ame e s o he s uc u e es ima ed om he applica ion o an ope a ional modal analysis (Magalhães and Cunha, 2011). An ambien es has been pe o med on Viana oo b idge and he measu ed signals ha e been p ocessed by an ou pu -only iden i ica ion me hod in he ime domain, which p o ided es ima es o he i s i s ou na u al equencies, he co esponding modal shapes and he associa ed damping a ios (Magalhães e al., 2010). (Fig.6). La e , his es has been ep oduced nume ically by he implemen a ion o he p oposed pedes ian- s uc u e in e ac ion model. An i e a i e p ocess o educe he di e ences be ween he expe imen al and nume ical e ical accele a ions has been pe o med, unde he ules o gene ic algo i hms (Koh and Pe y, 2010; Nocen al and W igh , 1999), and conside ing as design a iables he i s wo VDLF ( e ,1  and e ,2  ) o he pedes ian walking o ce and he h ee modal pa ame e s ha cha ac e izes he TDOF pedes ian- s uc u e in e ac ion model; he pedes ian sp ung mass, a m, he pedes ian damping a io, p  , and he pedes ian na u al equency, p . Fig.6. Layou o he iden i ica ion me hodology. Es ablishing a p elimina y sea ch domain o he pa ame e s o he pedes ian- s uc u e model. In o de o educe he unce ain y o he es ima ed alues o he i e conside ed pa ame e s and hus p e en an ill-condi ioned in e se p oblem, a sea ch domain has been es ablished. Acco ding o Table 1 he minimum and maximum alues o each VDLF allow es ablishing a sea ch domain o hei expe imen al es ima ion. In o de o ake in o accoun ha he es ima ion o hese pa ame e s will be pe o med om measu emen s ca ied ou on a eal oo b idge, he abo e men ioned sea ch domains ha e been ex ended. As he alue o he hi d ha monic o he walking pedes ian o ce p oposed by he di e en au ho s has a lowe magni ude i s con ibu ion has been neglec ed. S1 S2 S3 S4 UPDATED F.E.M. OF THE FOOTBRIDGE p a ,1num a ,2exp a ,1exp a ,3exp a ,4exp a ,2num a ,3num a ,4num 2 exp ,, 2 1 i num i aa  EVALUATION OF OBJECTIVE FUNCTION  1, e  2, e m a  h h IDENTIFIED VARIABLES m a  h h  1, e ,  2, e X Z Y Z Y X TRIAXIAL ACCELEROMETER PEDESTRIAN MODAL PARAMETERS VERTICAL WALKING FORCE TDOF MODEL PARAMETERS IDENTIFICATION EXPERIMENTAL NUMERICAL Simila ly, he phase shi s o he second and hi d ha monic o he e ical walking o ce ha e no been conside ed. Simila ly, acco ding o Table 2, he minimum and maximum alues o he pa ame e s o he passi e pedes ian models allow de ining he sea ch domains o hei es ima ion. The knowledge o he physical p oblem sugges s: (i) a educ ion o he damping and s i ness o he pedes ian associa ed wi h i s mo emen and (ii) he change in he human body mass dis ibu ion be ween he ac i e and passi e s a es. The sea ch domains o hese pa ame e s ha e been inc eased o ake in o accoun o hese ac s. Thus he ollowing sea ch domains ha e been adop ed:  Fi s VDLF,  45.000.0 ,1   e  .  Second VDLF,  20.000.0 ,2   e  .  Pedes ian sp ung mass,   10080   a m%.  Pedes ian damping a io,   6910   p  %.  Pedes ian na u al equency,   43.101   p Hz. Desc ip ion and ini e elemen model o he “labo a o y” oo b idge: Viana oo b idge. The Viana do Cas elo oo b idge (Ba bosa e al., 2012) is a mo eable cable-s ayed b idge. The longi udinal s uc u al scheme o he oo b idge consis s o wo spans o abou 36.50 m and 9.00 m espec i ely suspended by 6 amilies o wo hange s ( wo e aining ones) om an inclined mas . The deck, wi h 2.50 m o wid h, is con igu ed by wo olled s eel beams o a iable dep h b aced by ci cula hollow p o iles. The deck loo is co e ed wi h wood. The compensa ion o he main span weigh is achie ed by plac i mas allo w oun d oo b A p o h a poin (An s (BE A impl e o h he s load s his asso c i ng 11 high is welded w s he o a d a ion ha b idge is sh o Fig.7. Fini elimina y n a e a i s a equi ed s ys, 2014) w A M188) e x emen e d . T h e de e mi n s uc u e, by s and hus e p elimina y c ia ed nu m densi y bl o in i s base ional mo balances h o wn in Fig. e elemen m n ume ical i a pp oxima i o pe o m w as used, b a x cep o T he nonline a n a ion o h y calcula in e s ima ing i y FE mod e m e ical na u o cks (wi h a o a cylin d emen o h e o ces a 7 . m odel, amb i ni e eleme n o n o he d y m he ambi e a sed on a d i he hang e a e ec s a h e nume ic a g p e ious l i s angen s e l leads o u al eque n weigh o 8 d e ha is c he s uc u a nsmi ed b y b ien es g i n (Fig.7) m y namic be h e n ib a i o i sc e iza io n e s whe e a ssocia ed w a l na u al l y he s es s s i ness m a o he i s n cies gi e n 8 00 kN) pl a c onnec ed o e. The pyl y he mas . i d and mod e m odal anal y h a iou o h o n es . Th n o he s u 3D-cable w i h he ha n equencies s le el o h a ix. The n ou num e n in Table a ced in he s o a wheel g o n is con n A pe spec e l upda ing y sis was co n h e oo b id g e so wa e u c u e in 3 D elemen s ( n ge s ha e and he i b h e hange s u n ume ical m ical ib a 4. In Fig . s ho e spa n g ea bea in g n ec ed o a i e o he V pa ame e s n duc e d in g e and a s a package A D -beam ele m (LINK10) been consi b a ion mo d u nde pe m m odal anal y ion mode s .8 he i s n . The g ha deep V iana . o de a ing A nsys m en s we e de ed d es o m anen sis o s and wo Fig.10. Fi s wo e ical upda ed nume ical (Upd.) and expe imen al (Exp.) ib a ion modes. A e he de elopmen o he model upda ing, he nume ical dynamic beha iou o he oo b idge simula es accu a ely he eal esponse o he oo b idge. This de ailed knowledge o he dynamic beha iou o he oo b idge allows o adop ing his eal oo b idge as a benchma k o “labo a o y” oo b idge. Expe imen al pedes ian es . The es ima ion o he pa ame e s ha cha ac e ize he beha iou o he p oposed pedes ian-s uc u e in e ac ion model was made h ough he esul s o a pedes ian es . In he es , wo pedes ians (A and B) we e selec ed. The mass o each pedes ian was 70.61 A mkg and 50.100 B mkg. On he deck o he oo b idge ou iaxial accele ome e s we e placed a he in e sec ion poin be ween he hange s and he deck (Fig.6). Th ee se ies we e eco ded o each pedes ian, c ossing he oo b idge h ee imes a di e en s ep equencies, s (1.50, 2.00, 2.50 Hz), con olled by a me onome. In he ou moni o ed sec ions ( 1 S,2 S,3 S and 4 S in Fig.6) he dynamic esponse o he -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 upda ed e sus expe imen al e ical ib a ion mode Upd. Exp. s uc u e was eco ded. In each passage, he ini ial and end ime, ha ma ks he c ossing o he pedes ian on he s uc u e, was eco ded as well, in o de o bo h localize he o ced ib a ion esponse co esponding o he passage o each pedes ian and es ima e he pedes ian eloci y, p . Pa ame e s iden i ica ion in ime domain. Fo he es ima ion o he pa ame e s o he TDOF-sys em (Koh and Pe y, 2010), an in e se dynamic p oblem was sol ed, as i has been desc ibed p e iously. As objec i e unc ion he ela i e di e ences be ween he expe imen al and nume ical e ical accele a ions in ou sec ions o he oo b idge ( 1 S,2 S,3 S and 4 S) has been conside ed. As op imiza ion me hod, gene ic algo i hms ha e been used again. The expe imen al e ical accele a ions co espond o he measu emen s o he abo e desc ibed pedes ian es . The nume ical e ical accele a ions ha e been ob ained om he implemen a ion o he p oposed pedes ian-s uc u e in e ac ion model on he upda ed ini e elemen model o he Viana oo b idge. Six pa ame e s we e adop ed as design a iables: (i) he i s wo VDLF ha cha ac e ize he pedes ian walking o ce. (ii) he h ee modal pa ame e s (pedes ian sp ung mass, pedes ian damping a io and pedes ian na u al equency) ha go e n he beha iou o he TDOF-sys em. (iii) a ime lag ha allows adjus ing he beginning o he c ossing o he pedes ian be ween he expe imen al and nume ical accele a ions. In o de o imp o e he eliabili y o he pa ame e s es ima ion he le el o noise o he signal has been educed (Koh and Pe y, 2010). In ha way, each measu emen eco d has been decomposed using he Wa ele ans o m (Gopalak ishnan and Mi a, 2014), choosing as wa ele amily, he Daubechies. A le el 7 o decomposi ion has been applied o each signal. As h eshold selec ion ule, he p inciple o S ein's Unbiased Risk Es ima e has been conside ed (Gopalak ishnan and Mi a, 2014). Subsequen ly, he econs uc ion o he signal has been ca ied ou using he o iginal app oxima ion coe icien s o le el 7 and he modi ied de ail coe icien s o le els om 1 o 7. Fo each se ies, a gene a ion o 1000 indi iduals has been de ined. Each indi idual modi ies, acco ding o he ules o gene ic algo i hms, he alues o i s componen s in o de o minimize he de ined objec i e unc ion. In Table 7, he esul s o he es ima ion p ocess a e summa ized, showing he di e en es ima ed pa ame e s e sus he pedes ian s ep equency. Table 7. Es ima ion o he pa ame e s o he pedes ian-s uc u al in e ac ion model. s [Hz] Pedes ian sp ung 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 s [Hz] Pedes ian damping a io  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 s [Hz] Pedes ian equency 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 s [Hz] Ve ical dynamic load ac o 1 , e 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 s [Hz] Ve ical dynamic load ac o 2 , e 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 Acco ding o he esul s o Table 7 he pa ame e s o p oposed pedes ian-s uc u e in e ac ion model show some dependence on he s ep pedes ian equency. Howe e due o he numbe o pedes ians used du ing he pedes ian es , only a global mean alue,  , and a s anda d de ia ion,  , ha e been de e mined ),(   N. Fo he p oposed pedes ian-s uc u e model, he ollowing Gaussian dis ibu ions ha e been conside ed. - Fi s VDLF, e ,1  , )04.0,237.0(N. - Second VDLF, e ,2  , )004.0,043.0(N. - Pedes ian sp ung mass, a m, )809.3,139.87(N%. - Pedes ian damping a io, p  , )776.9,44.41(N %. - Pedes ian na u al equency, p , )728.0,933.2(N Hz. The p oposed alues a e inside he ange, es ablished by Shahabpoo e al. (2013), ha cha ac e izes he modal p ope ies o TDOF pedes ian-s uc u e in e ac ion model. Fo he gene a ion o he pedes ian lows o he c owd-s uc u e in e ac ion model, he abo e Gaussian dis ibu ions ha e been conside ed. Once ob ained he pa ame e s, he de ini ion o he p oposed model is comple e. This model will be alida ed in he nex sec ion by co ela ing he nume ical and expe imen al dynamic esponse o he Viana oo b idge unde he c ossing o a g oup o pedes ians and he expe imen al and nume ical analysis o he change o he i s e ical na u al equency o he s uc u e induced by he pedes ian low. MODEL VALIDATION. The alidi y and applicabili y o he p oposed c owd-s uc u e in e ac ion model is nex assessed h ough i s p ac ical applica ion o a case s udy. The e ical accele a ion a h ee sec ions ( 1 S,2 S, and 3 S (Fig.6)) o he Viana oo b idge has been measu ed unde he c ossing o a g oup o 50 pedes ians a di e en s ep equencies (1.30-2.50 Hz). F om he o ced eco ded esponse, he expe imen al s udy o he change o he i s e ical na u al equency o he s uc u e, due o he c ossing o he g oup o pedes ians a di e en s ep equencies, has been de e mined. Subsequen ly, he c owd- s uc u e in e ac ion model has been applied o he upda ed FE model in o de o ob ain i s he e ical nume ical accele a ion a he men ioned sec ions and la e o s udy nume ically he change o he i s e ical na u al equencies o he oo b idge due o he p esence o he pedes ians. Expe imen al c owd es : dynamic esponse and change o he i s na u al equency unde pedes ian low. In he c owd es a g oup o 50 pedes ians has c ossed he oo b idge a di e en s ep equencies (1.30-1.40-1.60-1.75-2-00-2.50 Hz) con olled by a me onome, measu ing he e ical dynamic esponse o he oo b idge a h ee sec ions 1 S, 2 S and 3 S wi h a i-axial accele ome e (Fig.6). Du ing he c owd es he g oup o 50 pedes ians has been dis ibu ed in h ee alignmen s, main aining a la e al sepa a ion among pedes ians a ound 0.85 m and a longi udinal dis ance be ween pedes ians a ound 0.50 m. A pedes ian wi h a me onome has led he g oup in each c ossing. A scheme o he pedes ian dis ibu ion du ing he c owd es is shown in Fig.11. In o he c ib a (hig h and Mi a coe na u na u ob a i sign a me h de o de e c ossing o a ion espo n h e modal d p ocessed a , 2014) b icien s (Fi g al equen c al eque n i ned acco d a l used o h odology p Fig.11. E e mine he c he g oup n se o he s d e lec ion) by he Co b ased on D g .12), in h e c y o he s n cy h oug h d ing o he he es ima i e iously d e E xpe imen a c hange o h o pedes i uc u e ha s h a e b een n inuous W D aubechies e il e ed a n u c u e. Thi s h he pow e Pea k -Pick i i on o he p e sc ibed. a l c owd es h e i s na u i ans a di e s been con s conside ed . W a ele T a amily. T n ge o eq u s esul has e spec al ing me ho d owe spec a Viana u al eque n e en s ep s ide e d . On l . The selec a ns o m ( C T he maxi m u encies, is been alid a densi y o d (Magalh ã al densi y h o o b idge. n cy o he equencies l y he eco ed signal h C WT) (Go p m um alue hen co el a a ed by he e he abo e ã es and Cu n h as been d e oo b idge u , jus he ds a sec i o h as been i l p alak ishna n o he w a a ed wi h h e es ima ion o e signal (F i n ha, 2011) e noised usi n u nde o ced o n 2 S l e ed n and a ele e i s o he i g.12) . The n g he Fi To i l sec i o Fig. 1 The oo b Nu m i s The expe cons i g .12. Es i m l lus a e h e on 2 S und e 1 3. expe ime n b idge is su m m e ical c o w na u al e assessmen imen al e i de ed s e p m a ing he c h e eco ded e e a g oup o n al s udy o m ma ized i n w d es : e s e quenc y u n o he m e sul s wi h p equenc y h ange o h e e sul s, he m o 50 pede s o he ch a n Fig.14. s ima ion o n de pedes m odel pe o he nume i y en gene a e i s e i c m easu ed e s ians a a a nge o i o he d y n a ian low. o mance h cal es ima a ions o g c al na u al e ical acce s ep eque s e ical a mic espo n as been d o ion p edic g oups wi h equency o l e a ion o n cy o 1.6 0 na u al n se and h o ne co el a e d by he m h 50 pedes he s uc u he oo b i d 0 Hz is sho w equency o e chan g e o a ing he a m odel. Fo ians ha e u e. d ge in w n in o he o he a bo e each been simula ed. The numbe o pedes ians in phase in each new simula ion has been de e mined hough he e alua ion o he pa ame e p  . The desi ed eloci y, d , o each pedes ian has been assigned acco ding o Eq.(32). A pedes ian mass o 70 kg has been conside ed acco ding o he F ench code (Se a, 2006). As ini ial spa ial dis ibu ion o he pedes ians, a ec angula g id has been selec ed, conside ing an ini ial dis ance among pedes ians 50.0 p d m wi h an equidis an dis ibu ion in he wid h o he deck. The selec ed ime s ep is 01.0   sec. The nume ical e ical accele a ion ( o h ee o he 50 pedes ians gene a ions) a sec ion 2 So he oo b idge, o a s ep equency o 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.] Ve ical expe imen al accele a ion (S2). 50 Pedes ians a 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.] 1s Gene a ion. Ve ical nume ical accele a ion (S 2 ). 50 Pedes ians a 1.60 Hz Fig.13. Expe imen al e sus nume ical accele a ion ( h ee gene a ions) a sec ion S2 o Viana oo b idge unde a g oup o 50 pedes ians (walking equency o 60.1 p Hz). As Fig.13 shows, he co ela ion be ween he expe imen ally eco ded e ical accele a ion and he nume ically es ima ed alues is adequa e, in e ms o bo h he alue o he maximum accele a ion and i s empo al a ia ion. Finally, he nume ically es ima ed e ical accele a ion a sec ion 2 So he oo b idge unde a g oup o 50 pedes ians o di e en s ep equencies has been used o iden i y he i s na u al equency o he s uc u e, ollowing he p ocedu e desc ibed in he p e ious sec ion. Fig.14 illus a es he co ela ion be ween expe imen al and nume ical esul s o he change o i s e ical na u al equency o he oo b idge induced by he -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 Gene a ion. Ve ical nume ical accele a ion (S 2 ). 50 Pedes ians a 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.] 3 h Gene a ion. Ve ical nume ical accele a ion (S 2 ). 50 Pedes ians a 1.60 Hz c owd-s uc u e in e ac ion phenomenon. The nume ical es ima ion o he change o he i s e ical na u al equency has been ob ained om he mean alues o en simula ions o each s ep equency. Good ag eemen be ween bo h se s o esul s is obse ed, wi h di e ences below 1.50 % o all he analysed pedes ian walking equencies. The i s e ical na u al equency co esponding o he emp y oo b idge is included in Fig.14 o e e ence. Fig.14. Change o he i s e ical, e ,1 [Hz], expe imen al (Exp.) and nume ical (Num.) na u al equency e sus he s ep equency p [Hz]. CONCLUSIONS. In his pape , a new c owd-s uc u e in e ac ion model in he e ical di ec ion has been p esen ed and u he alida ed h ough he co ela ion be ween he expe imen al and nume ical dynamic esponse o a eal oo b idge (Viana) adop ed as benchma k. The p oposed model has been o ganized in wo sub-models: a pedes ian-s uc u e in e ac ion and a c owd sub-model. The pedes ian-s uc u e in e ac ion sub-model is con igu ed by a TDOF sys em, wi h sp ung and unsp ung masses, whose pa ame e s ha e been es ima ed expe imen ally om he solu ion o an in e se p oblem on he 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 1, e [Hz] p [Hz] Exp. Num. Emp y Koh Ghee C., Pe y M.C.. S uc u al Iden i ica ion and Damage De ec ion using Gene ic Algo i hms. CRC P ess, Taylo &F ancis G oup, 2010. Magalhães, F., Cunha, A.. Explaining Ope a ional Modal Analysis wi h da a om an a ch b idge. Mechanical Sys ems and Signal P ocessing, In i ed Tu o ial Pape , Volume 25, Issue 5, pp. 1431-1450 , 2011. Magalhães, F., Cunha, A., Cae ano, E., B incke , R. Damping es ima ion using ee decays and ambien ib a ion es s. Mechanical Sys ems and Signal P ocessing, Volume 24, Issue 5, pp. 1274–1290, July 2010. Ma lab R2011a. . h p://www.ma hwo ks.com/. Ma sumo o, Y., Nishioka, T., Shioji i, H., Ma suzaki, K.. Dynamic design o oo b idges. IABSE P oceedings, No. P-17/78, pp. 1-15, 1978. Nocen al J., W igh S.J.. Nume ical Op imiza ion. Sp inge , New Yo k, USA, 1999. Racic, V., Pa ic, A., B ownjohn, J.M.W. Expe imen al iden i ica ion and analy ical modelling o human walking o ces: Li e a u e e iew. Jou nal o Sound and Vib a ion, Vol. 326, pp. 1-49, Ap il 2009. Rapapo , D.C.. The a o molecula dynamic. Camb idge Uni e si y P ess; 2004. SETRA/AFGC. Guide mé hodologique passe elles pié onnes (Technical Guide Foo b idges: Assessmen o ib a ion beha iou o oo b idge unde pedes ian loading). SETRA, 2006. Shahabpoo , E., Pa ic, A., Racic, V.. Modelling e ec o pedes ians walking on dynamic p ope ies o s uc u es. IMAC XXXI: A Con e ence and Exposi ion on S uc u al Dynamics, 11-14 Feb ua y, O ange Coun y, Cali o nia, USA (2013). Ta a es, F., Leal, R., Shube M. Biodynamic single pe son and c owd pedes ian model: a wo ked es case and simula ion. Foo b idge 2014. London, Uni ed Kingdom (2014). Teughels, A., In e se Modelling o Ci il Enginee ing S uc u es Based on Ope a ional Modal Da a. Ph. D. Thesis, Ka holieke Uni e si ei Leu en, 2003. Venu i, F., B uno, L., Bellomo, N. C owd dynamics on a mo ing pla o m: ma hema ical modelling and applica ion o li ely oo b idges. Ma hema ical and Compu e Modelling 45 (3-4), 252-269, Feb ua y 2007. Venu i, F., Racic, V., Co be a, A. Pedes ian-s uc u e in e ac ion in he e ical di ec ion: coupled oscilla o - o ce model o ib a ion se iceabili y assessmen . P oceedings o he 9 h In e na ional Con e ence on S uc u al Dynamics, EURODYN 2014. Po o, Po ugal, 30 June - 2 July 2014. Zi ano ic, S., Pa ic, A., Ingol sson, E., Modelling spa ially un es ic ed pedes ian a ic on oo b idges. Jou nal o S uc u al Enginee ing, Vol. 136 (10), pp. 1296-1308. 2010. Zi ano ic, S., Pa ic A., Reynolds P.. Fini e elemen modelling and upda ing o a li ely oo b idge: The comple e p ocess. Enginee ing S uc u es, Vol. 301(1-2), pp. 126-145, Ma ch 2007. Zi ano ic, S., Pa ic A., Reynolds P.. Vib a ion se iceabili y o oo b idges unde human-induced exci a ion: a li e a u e e iew. Jou nal o Sound and Vib a ion, Vol. 279, Issue 1-2, pp. 1-74, Janua y 2005. 74 Pape C: Model upda ing o he selec ion o he e o i me hod o an ancien b idge (Alme ia, Spain). The o iginal e sion o his pape can be ound in doi: 10.2749/101686615X14355644771333 S uc u al Enginee ing In e na ional (in p ess) ISI 2014 Classi ica ion: Q4 (105/124) Ci il Enginee ing. Impac Fac o : 0.414 SCIMAGO 2014 Classi ica ion: Q2 (106/215) Ci il Enginee ing SJR: 0.417 ISSN: 1683-0350 Model upda ing o he selec ion o he e o i me hod o an ancien b idge (Alme ia, Spain). ABSTRACT In his pape we analyse a case s udy whe e a ini e elemen model upda ing is conduc ed on he basis o he expe imen al modal pa ame e s eco ded o a ein o ced conc e e uss b idge buil in Alme ia (Spain) in 1927. The inal aim o his s udy is o help unde s anding he ac ual s a e o s uc u al conse a ion o he b idge, in o de o selec an app op ia e e o i echnique o ein o ce i be o e he planned widening o i s deck. A p esen , he wo mos widely used me hods o e o i ing consis on ei he ein o cing he s uc u e wi h ex e nal p es essing o inc easing i s lexu al s eng h by adhe ing CFRP lamina es. The i s me hod is especially ad an ageous i he s uc u e is de e io a ed o such an ex en ha he s eng hening needs o ocus no only on inc easing he lexu al s eng h o he b idge, bu also on a oiding he de lec ion p oblems associa ed wi h he educed ine ia o he s uc u e. I his we e no he case, s eng hening wi h CFRP would be an adequa e op ion. To his end, and gi en he monumen al cha ac e o his ancien cons uc ion, i is no possible o apply di ec ly des uc i e o s a ic load es s o de e mine he p ope ies o i s cons i uen ma e ials, so ha i becomes necessa y o es ima e i s de e io a ion s a e indi ec ly wi h he use o non-des uc i e echniques. Fo his pu pose, in he p esen pape we ca y ou a ini e elemen model upda ing o he s uc u e, based on expe imen al modal pa ame e s, which allows o checking i s se ice condi ion h ough he es ima ion o he alue o se e al physical pa ame e s o he b idge. Subsequen ly, he esul ing upda ed model cons i u es a aluable ool o help es ablishing which is he mos adequa e e o i me hod. Keywo ds: Re o i me hods, ope a ional modal analysis, model upda ing, ancien ein o ced conc e e b idge, non-des uc i e es ing. 1. In oduc ion. The Molinos B idge is a ein o ced conc e e uss gi de b idge (Fig. 1) loca ed o e he Anda ax Ri e a he ou ski s o he own o Alme ía (Spain). I was buil in 1927, being a p esen one o he bes p ese ed achie emen s o he i s ein o ced conc e e b idges buil in Spain. The design o he s uc u e co esponds o one o he schemes p oposed by he Spanish enginee Juan Manuel Za a in he o icial model o oad b idges o he collec ion o ein o ced conc e e s aigh b idge o spans o 32 m, published in 1920 [1-6]. Recen ly, he Planning Depa men o he Ci y o Alme ia called o a public compe i ion in o de o p oceed wi h he epai , ein o cemen and widening o his ancien conc e e b idge. The g ow h o he ci y owa ds he Eas and he a emp o p omo ing he ou ism on he Eas Coas o he p o ince made necessa y o imp o e he oad access o hose neighbou hoods o he ci y. Since he cons uc ion o a new s uc u e was dismissed due o i s high cos , he p oposal ocused on adap ing he exis ing b idge o he new se ice condi ions. Acco ding o he echnical p o isions o he compe i ion published by he Planning Depa men , he wo ks o he ehabili a ion and unc ional adap a ion o he Molinos b idge should sa is y he ollowing condi ions: (i) widening he exis ing lanes o ehicles, inc easing i s wid h un il 3.25 m pe lane; (ii) c ea ing a pa emen and a cycle lane wi h a minimum wid h o 1.50 m; and (iii) he p ojec should main ain he cu en ypology o he b idge and s ill ensu e i s adequa e bea ing capaci y unde he new loading scena io. All he p esen ed p ojec p oposals add essed h ee main aspec s, namely: (i) design o a s uc u al sys em o inc ease he wid h o he deck; (ii) e o i o he main uss gi de s; and (iii) ea men o he exis ing c acking on he ein o ced conc e e elemen s. Rega ding he e o i me hod o he main gi de s, he al e na i es p esen ed by he di e en enginee ing i ms may be g ouped in wo ca ego ies based on: (i) ei he use ex e nal p es essing [7] o (ii) use ca bon ibe ein o ced polyme s (CFRP) lamina es [8] o e o i he ein o ced conc e e uss gi de s. The main ac o ha condi ions he choice o he e o i me hod is he ac ual de e io a ion le el o he s uc u e and i s in luence in he momen o ine ia o he gi de . In his way, i he s uc u e was se iously damaged, i would be necessa y o ocus no only on con olling he le el o s esses bu on educing he de lec ion o he gi de as well. The e o e, in o de o p ope ly e alua e he di e en p ojec s and selec he mos adequa e e o i me hod, he Planning Depa men o he Ci y o Alme ia decided o conduc a de ailed s udy o he s a e o de e io a ion o he s uc u e using non- des uc i e echniques. The use o he b idge, be o e i s e o i , was jus limi ed o ligh a ic, so he Planning Depa men u he es ablished as a equi emen he impossibili y o pe o ming a s a ic load es , conduc ed wi h hea y loads, in o de o assess he s uc u al condi ion o he b idge. The p esen pape summa izes he wo k de eloped by he au ho s o e alua e he damage le el o he b idge. This s udy desc ibes an inno a i e applica ion o he FE model upda ing me hod, using i as a aluable ool in o de o ind he op imum ein o cemen me hod o a ancien b idge. The wo k mainly ocuses on pe o ming a FE model upda ing [9] o he b idge based on he expe imen al modal pa ame e s. These pa ame e s a e de e mined om an ambien es , using ope a ional modal analysis [10]. This s udy ocuses on he e ical di ec ion in o de o es ima e he de e io a ion s a e o he ancien b idge compa ing wo scena ios: (i) one de ined by he cu en s uc u e, om a nume ical load es pe o med on he upda ed model; and (ii) a second scena io de ined by he o iginal s uc u e, om he o iginal expe imen al load es a he ime o comple ion o he b idge [4]. The esul s o he s udy, and in pa icula he esul ing upda ed model, a e hen used as e e ence o he e alua ion o he di e en p ojec p oposals and he selec ion o he mos app op ia e e o i me hod. The pape is o ganized as ollows: In sec ion 2, a p elimina y s udy o he s uc u al beha iou o he s uc u e is p esen ed, desc ibing he main s uc u al elemen s and he le el o c acking obse ed on he s uc u e om isual inspec ion. Fu he mo e, a ough es ima ion o he ange o a ia ion o he Young’s modulus o he conc e e is ob ained om a hamme ebound es . A he end o his sec ion, a FE model o he s uc u e is pe o med in o de o ob ain an ini ial es ima ion o he dis ibu ion o he p incipal s esses o he s uc u e unde sel -weigh and dead load. La e , he esul s om he hamme ebound es and he s a ic analysis we e used o es ablish bo h he physical pa ame e s used in he upda ing p ocess and i s ange o a ia ion. To make he pape as sel -con ained as possible, a desc ip ion and compa ison o he wo p oposed e o i me hod is b ie ly p esen ed in sec ion 3, and u he a c i e ion o i s selec ion is es ablished. In sec ion 4, he modal pa ame e s o a p e iously selec ed ex eme span a e es ima ed expe imen ally h ough he applica ion o ope a ional modal analysis in he equency domain. In sec ion 5, he FE model upda ing o he selec ed span is pe o med. In sec ion 6, he upda ed model is employed o assess he s uc u al condi ion o he b idge, by compa ing he esul s o he o iginal load es o he b idge [4] wi h he nume ical simula ion o such load scena io using he upda ed model. Finally, om he ob ained esul s, he bes e o i me hod is selec ed and se e al conclusions a e d awn in sec ion 7. 2. P elimina y s udy o he s uc u al beha iou o he b idge. As a p elimina y s ep o he selec ion o any e o i sys em i is o key impo ance o unde s and he ac ual s uc u al beha iou o he di e en elemen s ha con igu e he b idge, as well as o de e mine he mechanical cha ac e is ics o he ma e ial and o conduc an in-dep h isual inspec ion o he cu en s a e o he b idge. 2.1. Desc ip ion o he o iginal s uc u e. The Molinos B idge o e he Anda ax Ri e is loca ed along he oad be ween Alme ía and Níja (Spain). I is an isos a ic s uc u e wi h i e spans (32.64 - 32.72 - 32.72 - 32.68 and 32.78 m) wi h a o al leng h o 163.54 m (Fig. 1). The wid h o he o iginal b idge is a ound 6.20 m, composed by wo lanes o 2.20 m and wo pa emen s o 0.20 m. Fig. 1. Ele a ion o he Molinos B idge om he Eas abu men . Each span [2, 3 and 11] is con igu ed by wo ein o ced conc e e uss gi de s sepa a ed 2.70 m and connec ed a he op by a ein o ced conc e e slab o a iable dep h (0.18- 0.54 m) wi h a o al wid h o 6.20 m. The o al dep h o he b idge is abou 2.66 m. The wid h o he di e en elemen s ha con igu e he uss gi de s is 0.40 m. The dep h o he lowe cho d is 0.40 m, bu he dep h o s u s and diagonal a ies be ween 0.18 m in he mid-span and 0.64 m o e he suppo s. Inside he lowe cho d o he uss gi de s he e a e se e al ec angula s eel pla es 300x12 mm, in a numbe a ying om 10 pla es in each lowe cho d a he mid span o 4 pla es a he suppo s [2, 3 and 11]. The o ce ansmission be ween he lowe cho ds and he s u s and diagonals is achie ed by he placemen o pins o diame e 40-50 mm (Fig. 2). The usses a e join ed a he in e sec ion be ween he lowe cho d and he s u s by se e al ein o ced conc e e ec angula c ossba s wi h 0.40 m o dep h and a iable wid h (Fig. 3.a). Due o he geome ic con igu a ion o he b idge he diagonal elemen s wo k in comp ession and a e ein o ced wi h longi udinal ba s and a high densi y o s i ups. The s u s o he uss gi de s, always in ension, a e ein o ced wi h longi udinal and ans e sal ba s which numbe inc eases wi h he p oximi y o he suppo s. Fig. 2. O iginal de ails o he c oss sec ion o he b idge [11]. The pie s and he abu men s ha e a ec angula c oss-sec ion wi h dimensions 1.60x5.60 m and a heigh o 6.00 m. Each o hese elemen s es on a mason y block wi h ec angula c oss-sec ion 2.00x6.00 m and heigh a ound 5.70 m [3]. The ounda ions o hese elemen s a e ein o ced conc e e slabs wi h 1.50 m o dep h and ho izon al dimensions 4.00x8.00 m. The linking be ween he deck and he pie s is achie ed by ixed (one hinge) and sliding ( wo hinges) bea ings (Fig. 3.b). To a oid es ic ing he longi udinal mo emen s o he deck due o he heological and he mal e ec s, in each pie hese wo ypes o bea ings ha e been ins alled [3]. Fig. 3. a) Spa ial con igu a ion o he uss gi de . b) Suppo on abu men s 2.2. Visual inspec ion o he o iginal s uc u e. The de e io a ion s a e o he b idge was s udied, as a i s app oxima ion, h ough i s isual inspec ion. The main damages obse ed in he s uc u e we e he ollowing: (i) De achmen o he ein o ced conc e e along o he uss gi de s. The in il a ions o wa e and sal s, due o he impai men o he wa e p oo ing, in he uss gi de s ha e caused he oxida ion o he eba s and he appea ance o he de achmen o he conc e e. De achmen s a ound 3.50 cm we e measu ed. The oo b idge. Jou nal o B idge Enginee ing, ASCE (in p ess). doi: 10.1061/(ASCE)BE.1943-5592.0000828 Jones, C.A., Reynolds, P., Pa ic, A. (2010). Vib a ion se iceabili y o s adia s uc u es subjec ed o dynamic c owd loads: A li e a u e e iew. Jou nal o Sound and Vib a ion, Vol. 330, pp. 1531-1566. Koh Ghee C., Pe y M.C. (2010). S uc u al Iden i ica ion and Damage De ec ion using Gene ic Algo i hms. CRC P ess, Taylo &F ancis G oup. Macdonald, J.H.C. (2008). Pedes ian-induced ib a ions o he Cli on Suspension B idge, UK. P oceedings o he ICE-B idge Enginee ing 161 (2), pp. 69-77. Macdonald, J.H.C. (2008). La e al exci a ion o b idges by balancing pedes ians. P oceedings o he Royal Socie y. Magalhães, F., Cunha, A. (2011). Explaining Ope a ional Modal Analysis wi h da a om an a ch b idge. Mechanical Sys ems and Signal P ocessing, In i ed Tu o ial Pape , Volume 25, Issue 5, pp. 1431-1450. Magalhães, F., Cunha, A., Cae ano, E., B incke , R. (2010). Damping es ima ion using ee decays and ambien ib a ion es s. Mechanical Sys ems and Signal P ocessing, Volume 24, Issue 5, pp. 1274–1290. Ma lab R2015a. . h p://www.ma hwo ks.com/. Ma sumo o, Y., G i in, M.J. (2003). Ma hema ical models o he appa en masses o s anding subjec s exposed o e ical whole-body ib a ion. Jou nal o Sound and Vib a ion Vol. 260 (3) pp. 431-451. Ma sumo o, Y., Nishioka, T., Shioji i, H., Ma suzaki, K. (1978). Dynamic design o oo b idges. IABSE P oceedings, No. P-17/78, pp. 1-15. Nocen al J., W igh S.J. (1999). Nume ical Op imiza ion. Sp inge , New Yo k, USA. Racic, V., Pa ic, A., B ownjohn, J.M.W. (2009). Expe imen al iden i ica ion and analy ical modelling o human walking o ces: Li e a u e e iew. Jou nal o Sound and Vib a ion, Vol. 326, pp. 1-49. Rapapo , D.C. (2004). The a o molecula dynamic. Camb idge Uni e si y P ess. Ronnquis , A. (2005). .Pedes ian Induced La e al Vib a ions on Slende Foo b idges. PhD Thesis. No wegian Uni e si y o Science and Technoloy. SETRA/AFGC. (2006). Guide mé hodologique passe elles pié onnes (Technical Guide Foo b idges: Assessmen o ib a ion beha iou o oo b idge unde pedes ian loading). SETRA. Shahabpoo , E., Pa ic, A., Racic, V. (2013). Modelling e ec o pedes ians walking on dynamic p ope ies o s uc u es. IMAC XXXI: A Con e ence and Exposi ion on S uc u al Dynamics, 11-14 Feb ua y, O ange Coun y, Cali o nia, USA. Teughels, A. (2003). In e se Modelling o Ci il Enginee ing S uc u es Based on Ope a ional Modal Da a. Ph. D. Thesis, Ka holieke Uni e si ei Leu en. Venu i, F., B uno, L., Bellomo, N. (2007). C owd dynamics on a mo ing pla o m: ma hema ical modelling and applica ion o li ely oo b idges. Ma hema ical and Compu e Modelling 45 (3-4), 252-269. Zheng, X., B ownjohn, J.M.W. (2001). Modelling and simula ion o human- loo sys em unde e ical ib a ion. Sma S uc u es and Ma e ials 2001: Sma S uc u es and In eg a ed Sys ems. SPIE. Zi ano ic, S., Pa ic, A., Ingol sson, E. (2010). Modelling spa ially un es ic ed pedes ian a ic on oo b idges. Jou nal o S uc u al Enginee ing, Vol. 136 (10), pp. 1296-1308. Zi ano ic, S., Pa ic A., Reynolds P. (2007). Fini e elemen modelling and upda ing o a li ely oo b idge: The comple e p ocess. Enginee ing S uc u es, Vol. 301(1-2), pp. 126-145. Zi ano ic, S., Pa ic A., Reynolds P.(2005). Vib a ion se iceabili y o oo b idges unde human-induced exci a ion: a li e a u e e iew. Jou nal o Sound and Vib a ion, Vol. 279, Issue 1-2, pp. 1-74, Janua y 2005. 77 Pape F: Dynamic es ing o Ca pin ei a oo b idge a Col ihã (Po ugal). Con e ence name: 5 h In e na ional Ope a ional Modal Analysis Con e ence Loca ion and da e: Guima ães (Po ugal). 13-15 May 2013. Pape ID: 224 (pp. 1-10). ISBN: 978-972-8692-83-4. Scopus: h p://0-www.scopus.com. ama.us.es/inwa d/ eco d.u l?eid=2-s2.0- 84906259319&pa ne ID=40&md5=3cd5 cc0518066db 05cdacdea01d34 IOMAC'13 5 h In e na ional Ope a ional Modal Analysis Con e ence 2013 May 13-15 Guima ães - Po ugal DYNAMIC TESTING OF CARPINTEIRA FOOTBRIDGE AT COVILHÃ, PORTUGAL Ja ie Fdo. Jiménez-Alonso 1, Elsa Cae ano2, Ál a o Cunha3 ABSTRACT The oo b idge o e he Ca pin ei a s eam es ablishes a connec ion be ween wo s eep cli s, a a heigh o 52.00 m abo e he wa e , ha ing a leng h o abou 220.00 m and being composed o h ee s aigh sec ions wi h di e en o ien a ions in plan iew, joined by ci cula cu es and suppo ed on ou columns. In o de o desc ibe he dynamic beha iou o he oo b idge, an ambien ib a ion es was pe o med, complemen ed by cha ac e isa ion es s o he ib a ions induced by pedes ians in he loading scena ios assumed as c i ical o he s uc u e. A de ailed ini e elemen model o he s uc u e has also been de eloped o co ela ion analysis wi h he measu ed alues. This pape shows he esul s o hese es s and he main conclusions abou he com o le el p o ided by he oo b idge unde he se ice condi ions. Keywo ds: Foo b idge, Ope a ional modal analysis, Human induced ib a ions, Ambien dynamic es ing. 1. INTRODUCTION Inse ed in he U baniza ion Plan o Ca pin ei a Valley, p omo ed by he Polis P og am, he oo b idge o e he Ca pin ei a s eam es ablishes a link be ween he wo s eep cli s o Ca pin ei a Valley a a heigh o 52.00 m o e he wa e . The design has been de eloped by A associados, in collabo a ion wi h he a chi ec Ca ilho da G aça [1], and he cons uc ion, pe o med by he company CERTAR, was inished in Sep embe 2009. The dynamic cha ac e is ics o he oo b idge, p edic ed a he design s age, mo i a ed some conce n owing o he possibili y o occu ence o signi ican la e al and e ical ib a ions, since se e al o he na u al equencies o he calcula ed ib a ion modes, bo h in la e al and e ical di ec ions, would be appa en ly loca ed a c i ical in e als om he poin o iew o he exci a ion induced by pedes ians [2]. Acco dingly, and as i wouldn’ be possible o ac on he s i ness o mass o he oo b idge in o de o achie e signi ican changes in e ms o modi ying he dynamic cha ac e is ics o he s uc u e, he Designe has o eseen he need o ins all uned mass dampe s, i a e he cons uc ion o he s uc u e, he e was e idence ha hey would be eally equi ed. 1 Assis an P o esso , Uni e si y o Se ille, Highe Technical School o Building Enginee ing, j [email protected] 2 Associa e Agg ega e P o esso , Uni e si y o Po o, Facul y o Enginee ing, [email p o ec ed] 3 Full P o esso , Uni e si y o Po o, Facul y o Enginee ing, [email p o ec ed] Session 1, J.F. Jiménez-Alonso, Elsa Cae ano, Ál a o Cunha 2 The oo b idge was cons uc ed in Sep embe 2009 and placed a he se ice o he popula ion wi hou in oducing any de ice o mi iga e ib a ions, which a oused he in e es in conduc ing he p esen esea ch. Thus, using he expe imen al esou ces a ailable a he Labo a o y o Vib a ions and Moni o ing (ViBes , www. e.up.p ) o FEUP, an ambien ib a ion es was pe o med wi h he aim o iden i ying he dynamic cha ac e is ics o he oo b idge. Addi ionally, measu emen s o he esponse o he s uc u e unde se ice condi ions we e made, and also unde condi ions o use po en ially haza dous o he oo b idge. This pape p esen s he main esul s achie ed, which show some di e ences in he dynamic beha io o he s uc u e in ela ion o he p edic ed a he design phase and, in pa icula , allow he cha ac e iza ion o he com o le el o he oo b idge unde no mal use condi ions. 2. DESCRIPTION OF THE STRUCTURE: CARPINTEIRA FOOTBRIDGE Figu e 1 Foo b idge o e he Ca pin ei a s eam, iew om he sou h side. No a: As co as ap e sen adas são med idas ao long o do eix o lo ngi udin al d a pon e Escala 1/400 Plan a 42.267 48.406 49.000 49.302 31.769 Figu e 2 Ca pin ei a oo b idge, la e al and plan iews. 42.267 48.406 49.000 49.302 31.769 5 h In e na ional Ope a ional Modal Analysis Con e ence, Guima ães 13-15 May 2013 3 The oo b idge (Figu e 1) is composed o a s eel deck ha uns a a cons an le el, being o med by h ee linea segmen s wi h di e en o ien a ion in plan , connec ed by ci cula cu es. The o al leng h o he deck is abou 220.00 m and i is suppo ed by ou composi e s eel-conc e e columns, wi h a a iable heigh be ween abou 18.00 and 40.00 m. The oo b idge is hus di ided in i e spans, wi h leng hs om abou 32.00 o 50.00 m (Figu e 2). The c oss-sec ion o he oo b idge, wi h o e all dimensions o 4.40 x 1.75 m2 and 3.50 m o e ec i e wid h, is o med by wo longi udinal s eel welded gi de s and a wooden suppo ing loo s uc u e ha con igu e he cha ac e is ics o he U-shaped sec ion o he oo b idge. 3. NUMERICAL MODELLING OF THE FOOTBRIDGE Due o he complexi y o bo h he geome y and he beha iou o he s uc u e and, wi h he aim o suppo ing he de elopmen o he dynamic es s and subsequen in e p e a ion o he ob ained esul s, a ini e elemen model o he oo b idge was de eloped using 3-D beam elemen s [3, 4 and 5]. In Figu e 3, he ini e elemen mesh used is p esen ed. . Figu e 3 Ca pin ei a oo b idge, ini e elemen mesh used in he nume ical model and de ail o disc e iza ion. The o e all mass o he s eel deck is abou 300 on, co esponding o a linea mass o 1250 kg/m. In he design o he oo b idge, he associa ed dynamic e ec s o a pedes ian low, wi h a densi y o 0.60 pedes ian/m2, ha e been conside ed, which co esponds o an inc ease o he mass o he deck o abou 10 %. Though he whole pedes ian mass is no usually conside ed in he dynamic cha ac e iza ion o he oo b idge, he calcula ion o he na u al equencies has been made assuming ei he an emp y s uc u e o a loaded s a e wi h 0.60 Pedes ian/m2, in o de o ame hei na u al equencies. On he o he hand, gi en he ela i ely low loading le els, i was assumed ha du ing se ice, he elas ic suppo s a he ex emes wouldn’ be ac i a ed. Acco dingly, al e na i e condi ions Session 1, J.F. Jiménez-Alonso, Elsa Cae ano, Ál a o Cunha 4 o connec ion we e simula ed: elas ic beha iou associa ed o neop ene bea ings, o cons ain mo emen in longi udinal di ec ion. Table 1 p esen s he alues o se e al na u al equencies co esponding o ou nume ical simula ions de eloped in o de o ame he b idge na u al equencies: M1 - emp y oo b idge, elas ic suppo s; M2 - ull oo b idge wi h 0.60 pedes ian/m2, elas ic suppo s; M3 - emp y oo b idge, ixed suppo s; M4 - ull oo b idge wi h 0.60 pedes ian/m2, elas ic suppo s. I is desc ibed in Table 1 he main cha ac e is ics o he mos impo an ib a ion modes (L: longi udinal; T: ans e se; V: e ical; To: o sion). On he o he hand, Figu e 4 shows he ou mos ele an ib a ion modes, based on he M3 model. Table 1 Nume ical na u al equencies ob ained unde di e en assump ions in ela ion o mass (emp y/ ull deck wi h a pedes ian densi y o 0.60 pedes ian/m2) and he s i ness o he ex eme suppo s (elas ic/ ixed). Modes M1 (elas ic/ emp y) M2 (elas ic/ ull) M3 ( ixed/emp y) M4 ( ixed/ ull) 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) F eq. 1= 1.21 Hz F eq. 3= 1.97 Hz F eq. 7= 2.44 Hz F eq. 13= 3.83 Hz Figu e 4 Se e al modal shapes ob ained om ini e elemen model M3 (emp y oo b idge wi h ixed longi udinal suppo a abu men s). 4. IDENTIFICATION OF THE DYNAMIC PROPERTIES The iden i ica ion o he na u al equencies, ib a ion modes and modal damping a ios was made pe o ming an ambien ib a ion es , conduc ed in July 2011. In his es , 5 seismog aphs p o ided wi h iaxial accele ome e s, we e used. These elemen s we e successi ely placed along he posi ions indica ed in Figu e 5, keeping wo o he de ices in sec ions 6 and 15, in he wes side o he oo b idge. In each posi ion, eco ds o ambien accele a ion ha e been collec ed in 15 channels wi h 13- minu e du a ion, sampled a 100 Hz. 5 h In e na ional Ope a ional Modal Analysis Con e ence, Guima ães 13-15 May 2013 5 Figu e 6 shows wo images o he es s pe o med, held du ing a no mal day and in ol ing he occasional passage o pedes ians, unde no mal condi ions o use. Figu e 5 Ins umen ed poin s in he ambien ib a ion es . Figu e 6 Images o he ambien ib a ion es s. Acco ding o he image o Figu e 6, and gi en he peculia cha ac e is ics o he oo b idge, i was decided o dispose sys ema ically he senso s, so ha he ans e se axis would coincide wi h he No h di ec ion. This pe mi ed he use o a common e e ence and acili a ed he signal p ocessing. The iden i ica ion o he modal pa ame e s was made using he so wa e ARTEMIS [6]. Table 2 shows a lis o he mos ele an iden i ied na u al equencies and modal damping a ios, whe eas Figu e 7 cha ac e izes se e al o he co esponding ib a ion modes. I is se led in Table 2 a co espondence be ween some iden i ied and calcula ed ib a ion modes, conside ing as basis he p e iously p esen ed M3 model. The analysis o Table 2 and Figu es 4 and 7 shows ha , al hough he main aspec s o he dynamic beha io o he oo b idge a e cha ac e ized by he nume ical ini e elemen model, in e ms o he na u al equencies alues o he ib a ion modes ( ans e se, e ical and o sion), signi ican di e ences exis be ween he modes calcula ed and iden i ied. Indeed, we can no ice ha he i s iden i ied ans e se ib a ion mode has a na u al equency sligh ly highe han he calcula ed one, which is indica i e o he beha io o he ex eme suppo s, almos ixed o he abu men s. Al hough he nume ical modeling has jus in oduced longi udinal cons ic ion, he na u al equency o he second mode o ib a ion, o local cha ac e , is also highe han he calcula ed, which ce ainly also in ol es he cons ic ion o he ans e se mo ion in he ex eme suppo s a he abu men s. Mo eo e , i is also obse ed ha he eal ans e se lexibili y o he columns was in e io o he calcula ed one, since he e ical ib a ion modes in ol e gene ally he o sion o he deck, bu no he ans e se bending o he columns, con a y o he esul s o he calcula ion. Finally, i is no iced some p oximi y be ween he na u al equencies o he modes wi h e ical and o sional componen s, which is ce ainly de e mined by he mechanical cha ac e is ics o he deck. Session 1, J.F. Jiménez-Alonso, Elsa Cae ano, Ál a o Cunha 6 Table 2 Nume ical na u al equencies Modes Iden i ied na u al equency (Hz) Nume ical na u al equency (Hz) Damping a io (%) 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 * Se e al nume ical modes wi h close equencies. F1= 1.37 Hz F2= 2.20 Hz F3= 2.47 Hz F4= 2.76 Hz F5= 2.95 Hz F6= 3.59 Hz Figu e 7 Iden i ied ib a ion modes using ARTEMIS so wa e. 5. CHARACTERIZATION OF THE DYNAMIC BEHAVIOUR F om inspec ion o he iden i ied na u al equencies o he oo b idge, i can be concluded, i s ly, ha he oo b idge is no ulne able o he la e al synch oniza ion phenomenon. Indeed, his phenomenon ypically occu s wi h na u al equencies close o 1 Hz. The in e na ional codes and ecommenda ions [7, 8] de ine his synch oniza ion as possible in he ange o equencies om 0.5 o 1.2 Hz, wi h a c i ical scena io when he undamen al equencies a e si ua ed be ween 0.7 and 1.0 Hz. The ini ial nume ical s udies [2] sugges ed a undamen al equency o 1.14 Hz, so he isk o occu ence o his phenomenon was eal. No e ha , despi e he possible sophis ica ion o he cu en nume ical models, he eal bounda y condi ions o he oo b idge and, some imes, he addi ion o elemen s assumed as non-s uc u al, can g ea ly in luence i s dynamic beha iou , and he e a e o en di e ences in he undamen al equencies o he buil s uc u e agains he nume ical esul s o he design. In his case, he na u al equency o 1.37 Hz measu ed in he ans e se di ec ion educes he isk o la e al 5 h In e na ional Ope a ional Modal Analysis Con e ence, Guima ães 13-15 May 2013 7 synch oniza ion, no a oiding howe e ques ions abou he com o le el p o ided by he s uc u e, bo h in e ms o he e ical and ho izon al ib a ions. This p oblem is pa icula ly ele an aking in mind he loca ion o he b idge, a a e y high le el, and also aking in o accoun he cha ac e is ics o he pa emen , wi h a sla ed wooden ela i ely spa se. I is e e ed, on he o he hand, he e y low damping iden i ied by he ambien ib a ion es . I is impo an o no e ha he quali y o he damping es ima es ob ained in his way is ques ionable, pa icula ly since he e was no oppo uni y o alida e hem using o he me hod. I is no ed, howe e , ha he modal damping a ios iden i ied a e lowe han expec ed. In ela ion o he e ical ib a ions, i was also ound a he design s age [2] ha he e we e se e al ib a ion modes wi h na u al equencies close o 2 Hz, which could lead o esonan phenomena. In he buil oo b idge, i was obse ed ha he equencies o he e ical modes a e gene ally a li le highe han calcula ed, being se led ypically abo e 2.5 Hz. In con as , he lexibili y o he columns is lowe han he modelled one, which educes hei pa icipa ion in e ms o ans e se bending in he main ib a ion modes. This ac leads o a mo e signi ican o sional beha iou o he deck. Also hese cha ac e is ics become bene icial o he s uc u e, since he esonan phenomena a equencies a ound 2 Hz don’ occu . In con as , he dynamic e ec s induced by pedes ians jogging become mo e ele an . Mo eo e , and gi en ha he o sional beha iou o he deck u ns ou o be e iden , also he la e al modes o ib a ion a e associa ed gene ally o he componen s o e ical ib a ion modes, which may esul in mo e se e e ib a ion le els. Taking in o accoun he obse ed dynamic cha ac e is ics o he oo b idge, p e iously discussed, i is assumed ha he dynamic esponse may be c i ical in he ollowing si ua ions: 1) Slow walking o la ge lows o pedes ians, wi h equencies om 1.4 o 1.5 Hz (exci a ion o ib a ion mode 1 and modes 3 o 6 (Figu e 7) h ough he second ha monic); 2) Jogging by a pedes ian o a g oup o pedes ians, wi h na u al equencies o 2.47 Hz, 2.76 Hz o 2.95 Hz (exci a ion o ib a ion modes 3, 4 o 5 (Figu e 7)). Al hough i has been obse ed ha he use o he oo b idge is no in ense, i was possible o es a no mal ope a ing si ua ion ela i ely close o he desc ibed condi ion (1) o he abo e pa ag aph, al hough mobilizing a educed densi y o pedes ians. The dynamic es s we e pe o med in wo days o he summe o 2011. I was ound ha he oo b idge was used me ely occasionally du ing he day. Con e sely, and gi en he wa m empe a u e obse ed in he la e e ening, om 19:00 h, a con inuous use by pedes ians was de ec ed, cha ac e ized by slow walking. Al hough an accu a e quan i ica ion o he densi y o pedes ians has no been made, i is possible howe e o say ha a densi y o a leas o 0.1 pedes ian/m2 has been eached, since mo e han 80 pedes ians we e o e he s uc u e. Unde hese condi ions, he la e al and e ical ib a ions we e clea ly pe cep ible, also mobilizing high and low equencies. Taking as basis he ange o 0-8 Hz, bo h le els o ans e se and e ical accele a ions we e collec ed, wi h alues o 0.08 m/s2 and 0.17 m/s2, espec i ely, cha ac e is ics o he maximum com o le el o he oo b idge, in line wi h he ecommenda ions men ioned abo e [7, 8]. Figu e 8 shows examples o he eco ds o he ans e se and e ical accele a ion collec ed in he ma k numbe 16 (see Figu e 5) unde hese condi ions, oge he wi h hei co esponding spec al con en . Due o he slende ness o he s uc u e, he join be ween he deck and he pie s has been specially s i ened. Fig. 2: P elimina y Fini e Elemen Model. 3. P elimina y nume ical modal analysis. Fi s ly, a nume ical modal analysis was de eloped (Figu e 2), whe e he e ec o he s eel oo was conside ed as a passi e mass uni o mly dis ibu ed on he deck (app oxima ely 500 kg/m). The model o he s uc u e was ca ied ou by he ini e elemen so wa e Au odesk Robo S uc u al Analysis P o essional [4]. Unde his hypo hesis, he nume ical ib a ion modes (Figu e 3) o he oo b idge has been de e mined in wo scena ios, absence o pedes ians ( emp) and he si ua ion whe e a pedes ian low o 1.00 P/m2 (Pedes ians/m2) c oss he s uc u e ( ul) [2]. Nume ical e ical mode 1 ul/emp=2.34/2.43 Hz Nume ical la e al mode 1 ul/em p =2.13/2.20 Hz Nume ical e ical mode 2 ul/em p =2.56/2.66 Hz Nume ical la e al mode 2 ul/em p =5.08/5.25 Hz Fig. 3: Fi s ou ib a ion modes. Emp y (emp) and ull( ul) oo b idge. Gi en he si ua ion o he oo b idge and assuming ha on he s uc u e a e no expec ed pedes ians densi ies abo e 0.80 P/m2, he nume ical es ima ed na u al equencies a e ou side o he no mal anges ha cha ac e ize he pedes ian walking s ep. On he o he hand, he na u al equencies o he s uc u e a e in he ange ha cha ac e izes he ac ion o jogging o unning. The nume ical esponse ha p oduces he c ossing o he p e iously p ede ined ha monic load on he s uc u e (1) eaches i s maximum alue unde a c ossing jogge a 2.56 Hz. In Figu e 4 he e ical dynamic esponse (accele a ion) o he s uc u e unde he passage o his pedes ian is shown. The maximum alue is less han he limi es ablished by he com o le el (1.00 m/s2). Nume ical e ical acele a ion. 1 Jogge =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 ime [sec] [m/s2] Fig. 4: Nume ical e ical accele a ion a mid-span due o a jogge s ep equency =2.56 Hz. 4. Ambien and pedes ian es s. 4.1 Ambien es . The dynamic pa ame e s o he s uc u e ha e been de e mined by he measu es ob ained om an ambien es . The deck o he s uc u e has been di ided in o a 2x15 g id, being he poin s sepa a ed longi udinally 5.65 m and ans e sally 3.15 m (see Figu e 5, blue a ows a e e e ence accele ome e s). Two se ies o measu emen s we e ca ied ou , each one consis ing on 14 se -up, he i s one co esponds o he de e mina ion o e ical dynamic pa ame e s and he second o es ima e he la e al dynamic pa ame e s. The measu es we e made wi h 4 uniaxial accele ome e s, sensi i i y 10 V/g, ype Episenso and p oduced by he company Kineme ics. The du a ion o each se -up was 900 seconds and he sampling equency was 100.00 Hz [5]. Ve ical Ambien Tes Layou La e al Ambien Tes Layou Fig. 5: Measu emen g id o he ambien es . F om he abo e se ies, he dynamic pa ame e s o he s uc u e ha e been de e mined, p ocessing he signals by wo di e en me hods, one in he equency domain, Enhanced F equency Domain Decomposi ion (E.F.D.D.), and ano he in he ime domain, S ochas ic Subspace Iden i ica ion (S.S.I.). Fo he alida ion o he esul s [5], he M.A.C. a io (Modal Assu ance C i e ion) o ce ain ib a ion modes, has been calcula ed (Table 1) p esen ing all he iden i ied ib a ion modes a M.A.C. g ea e han 0.90. In Figu e 6 he g aphical ep esen a ion o he i s ou de e mined ib a ion modes is shown. The p ac ical applica ion o he abo e algo i hms was pe o med using he ARTeMIS Ex ac o P o 2012 so wa e de eloped by SVS A/S [6]. Table 1: Expe imen al na u al equencies Mode EFDD [Hz] SSI [Hz] Desc ip ion M.A.C. 1 2.372 2.370 La e al 0.999 2 3.026 3.024 Ve ical 1.000 3 3.830 3.844 Ve ical 0.890 4 5.508 5.601 La e al 0.901 Expe imen al e ical mode 1 =3.026 Hz Expe imen al la e al mode 1 =2.372 Hz Expe imen al e ical mode 2 =3.844 Hz Expe imen al la e al mode 2 =5.508 Hz Fig. 6: Expe imen al i s ou modes o ib a ion (E.F.D.D.). 4.2 Pedes ian es . Finally, i is pe o med a es wi h i e di e en pedes ians, measu ing he dynamic esponse o he oo b idge unde di e en s ep equencies ( = 1.50, 2.00, 2.50, 3.00, 3.50 and 4.00 Hz). In Figu e 7, he measu ed maximum e ical accele a ion in he cen al mid-span o a pedes ian wi h a weigh o 114.00 kg and a s ep equency o 3.00 Hz is shown. The maximum measu ed accele a ion is less han he limi es ablished by he medium com o le el and he nume ical one es ima ed p e iously. Expe imen al e ical acele a ion. 1 Jogge -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 ime [sec] [m/s 2 ] Fig. 7: Expe imen al e ical accele a ion a mid-span due o a jogge s ep equency =3.00 Hz The s i ening o he s uc u e caused by he p esence o he s eel oo o igina es an imp o emen o he com o le el o he s uc u e. 5. Model upda ing. 5.1 De ailed ini e elemen model o he whole s uc u e. In o de o ha e a mo e accu a e unde s anding o he dynamic beha iou o he s uc u e a de ailed ini e elemen model o he whole s uc u e has been de eloped. Nume ical modal analysis has been de eloped h ough he applica ion o he ini e elemen me hod [7]. In he ini e elemen model o he s uc u e has been necessa y o model all he elemen o he oo b idge and he co e o cha ac e ize as p ecisely as possible he mass and s i ness ma ices. The model has been ca ied ou using 3D-beam (BEAM188) elemen s excep in he case o he s eel co e we e 2D-shell (SHELL63) elemen s has been conside ed. 5.2 Model upda ing o he de ailed ini e elemen model. A model upda ing o he abo e de ailed ini e elemen model has been de eloped [8] om he esul s o he abo e ope a ional modal analysis in o de o cha ac e ize mo e adequa ely he dynamic beha iou o he oo b idge. In his sense, 7 physical pa ame e s o he s uc u e (acco ding o Table 2) ha e been modi ied in o de o minimize he mean squa e e o be ween he expe imen al and nume ical pa ame e s, conside ing he iden i ied na u al equencies and hei co esponding modal coo dina es. A e a sensi i i y s udy o he main physical pa ame e s o he ini e elemen model, i was ound ha he physical pa ame e s wi h g ea e in luence on he dynamic beha iou o he oo b idge a e he s i ness o bea ings. The s i ness o hese elemen s has been simula ed h ough h ee sp ings, one in each di ec ion (longi udinal, la e al and e ical). The objec i e unc ion, in his case, was de ined as he sum o he ela i e di e ences be ween he na u al equencies and he modal coo dina es ob ained expe imen ally and nume ically. As op imiza ion me hod he gene ic algo i hms ha e been chosen. In Figu e 8 he esul s o he adjus men made on he i s ou ib a ion modes a e shown. A e he adjus men o he selec ed physical pa ame e s, high co ela ions be ween expe imen al and nume ical modal shapes (M.A.C. abo e 95 %) ha e been eached in he ou modes iden i ied. -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 Expe imen al&nume ical 1s e ical 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 Expe imen al&nume ical 1s la e al 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 Expe imen al&nume ical 2nd e ical 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 Expe imen al&nume ical 2nd la e al mode Fig. 8: Compa ison be ween expe imen al (Exp.) and nume ical (Num.) ib a ion modes Table 2: Upda ed alues o conside ed physical pa ame e s Pa ame e s Ini ial Value Upda ed Value E ec i e co e hickness 0.003 mm 0.0015 mm E ec i e slab conc e e hickness 0.15 m 0.10 m E ec i e abu men s i ness 30000 MPa 33240 MPa Soil s i ness 5.00E8 kN/m 4.64E8 kN/m Longi udinal bea ing s i ness 1.00E9 kN/m 2.00E8 kN/m La e al bea ing s i ness 1.00E10 kN/m 2.64E9 kN/m Ve ical bea ing s i ness 1.00E11 kN/m 1.28E11 kN/m Finally, in Figu e 9 he upda ed ou i s ib a ion modes om he de ailed ini e elemen me hod a e shown. Upda ed e ical mode 1 =3.026 Hz Upda ed la e al mode 1 =2.372 Hz Upda ed e ical mode 2 =3.844 Hz Upda ed la e al mode 2 =5.508 Hz Fig. 9: Upda ed ou i s ib a ion modes. 6. Nume ical es ima ion o he e ec o he s eel co e cons uc ion. In Table 3 he a ia ion in he na u al equencies o he oo b idge due o he cons uc ion o he oo b idge has been es ima ed. F om he upda ed ini e elemen model, i has been possible o simula e he beha iou he dynamic beha iou o he oo b idge wi hou he s eel oo ( NUM_INI) and he cu en si ua ion ( NUM_COV). Bo h alues ha e been ob ained nume ically. The pe cen age alues a e ep esen a i es o he s i ening e ec ha he co e p esen s in each di ec ion. The s eel co e inc ease he s i ness o he oo b idge in he e ical di ec ion, howe e , in he la e al di ec ion he cons uc ion o he s eel co e educes he alue o he na u al equencies in ha di ec ion. F om he poin o iew o he maximum accele a ion alues achie ed, he e is a sligh imp o emen in he com o le el due o he s i ening o he s uc u e. I p esen s ce ain sa e y ma gin, ensu ing ha he oo b idge eaches a medium com o le el, h ough e en, unde a e y a e load case as he Table 3: Es ima ion o he change in he na u al equencies o he oo b idge Mode NUM_INI [Hz] NUM_COV [Hz] Desc ip ion  [%]. 1 2.569 2.372 La e al -7.69 2 2.839 3.026 Ve ical 6.55 3 3.863 3.844 Ve ical -0.51 4 5.722 5.508 La e al -3.74 ci cula ion on he oo b idge o se e al jogge s in pa allel (Figu e 10). Fig. 10: Change in he i s wo e ical ib a ion modes. – wi hou co e -- wi h co e 7. Conclusions. In his pape , i has been es ima ed expe imen ally and nume ically, he change o he dynamic beha iou o a slende oo b idge due o he cons uc ion o a s eel oo o e he o iginal s uc u e. The s eel oo inc eases he s i ness o he s uc u e in e ical di ec ion bu educes he alue o he na u al equencies o he s uc u e in he la e al di ec ion. This e ec is especially ele an in he i s wo na u al equencies o he s uc u e. Howe e , he alues o he cu en na u al equencies ensu e ha he s uc u e will no su e om com o p oblems due o pedes ians low walking. In ela ion o jogging o unning, i has been shown ha he s uc u al s i ening imp o es i s beha iou unde hese ypes o human ac ion. 8. Re e ences. [1] SETRA, Guide mé hodologique passe elles pié onnes (Technical guide oo b idges: Assemen o ib a ional beha iou o oo b idges unde pedes ian loading), Se a, 2006. [2] SYNPEX Guidelines, Eu opean P ojec on Ad anced Load Models o synch onous Pedes ian Exci a ion and Op imized Design Guidelines o S eel Foo b idges, 2007. [3] CLOUGH, R and PENZIEN, J. Dynamics o S uc u es, 2nd. Edi ion, Mc G aw-Hill, 1993. [4] AUTODESK ROBOT STRUCTURAL ANALYSIS PROFESSIONAL 2011. [5] MAGALHÃES, F., CUNHA, A. "Explaining Ope a ional Modal Analysis wi h da a om an a ch b idge", Mechanical Sys ems and Signal P ocessing, In i ed Tu o ial Pape , Volume 25, Issue 5, pp. 1431-1450 , 2011. [6] ARTeMIS Ex ac o P o 2012. [7] ANSYS Mechanical Release 11.0. [8] ZIVANOVIC, S., PAVIC, A. REYNOLD, P., “Fini e elemen modelling and upda ing o a li ely oo b idge: The comple e p ocess”, Jou nal o Sound and Vib a ion, Vol. 301,. nº 1-2, pp. 126-145,2007.