A direct pedestrian-structure interaction model to characterize the human induced vibrations on slender footbridges

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 (Lon- don, 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 (Lond 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 ención. 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 publicados 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 i- cación 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 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.13.1106 ou nume ical p edic ions e sus he expe imen al esul s ob ained by Geo gakis and Jo gesen (7). In hei wo k, hey s udied he change o he dynamic beha iou o a labo a o y oo b idge due o he s ep o con olled pedes ians g oups. The labo a o y oo b idge, conside ed in (7), is a 16 m-long s eel double U simply suppo ed beam. The longi udinal beam p o iles a e UNP-350, wi h UNP-200 c ossbeams placed a 1400.00 mm in e als. Se e al masses we e added o he oo b idge in di e en poin o he s uc u e o inc ease he modal mass and hus dec ease he oo b idge’s equency o a le el close o he expec ed mean s ep equency o he pedes ians (Figu e 5). The o al mass o he s uc u e was 5224.00 kg. A modal analysis was pe o med in o de o iden i y he oo - b idge na u al equencies and damping a io acco ding o he le el o ib a ion. The oo b idge was exci ed by an ac ua- o in he mid-span poin o con ol i s le el o displacemen s. Figu es 6 and 7 co ela e he dependency be ween he i s e ical ib a ion equency ( 1) and he damping a io (ζ oo ) wi h he oo b idge ib a ion le el a mid-span, wi hou ak- ing in o accoun he in luence o he pedes ians (7). Subsequen ly, he s uc u e was subjec ed o nine di e en load scena ios (LS), unde a con inuous low o 4, 7 and 10 pedes ians (leading o mean low a es o app oxima ely 0.35, 0.62 and 0.88 P/s), du ing 180.00 seconds wi h a s ep equency o 1.38 Hz, a s ep eloci y o 1.40 m/s and h ee di e en mid-span le els o displacemen o 1.00, 5.00 and 10.00 mm. Pedes ians we e allowed o walk in bo h di ec- ions along he oo b idge. The es ima ion o he dynamic p ope ies was pe o med in each case om a equency sweep o he ac ua o . The na u al This ange o a ia ion is conside ed as a e e ence in o de o pe o m a alida ion o he in e ac ion model in he e ical di ec ion, bu p o ided hey a e based on a passi e pedes i- ans s a e, sligh a ia ions may be expec ed. In o de , on one hand, o conside he mass dis ibu ion o he human body (3) and on he o he hand, he physical un- de s anding o he p oblem ha sugges s a educ ion o he damping and s i ness o he pedes ian associa ed wi h he mo emen , he abo e limi s a e inc eased, educing he lowe limi s o he alues se in he ollowing anges: • sp ung mass, ma, 80-100 %. • human damping a io, ζh, 10-69 %. • human na u al equency, h, 1.00-10.43 Hz. The es ablishmen o a sea ch domain imp o es he e iciency o he pa ame e iden i ica ion me hods based on a p obabilis- ic sea ch. The alue o he pa ame e s o he pedes ian-s uc- u e in e ac ion model will be de e mined using a p obabilis ic es ima ion me hod applied on he abo e sea ch domain. 6. EXPERIMENTAL PARAMETERS IDENTIFICATION AND MODEL VALIDATION 6.1. Re e ence expe imen al esul s Nex , we will pe o m a mo e e icien es ima ion o he mod- el pa ame e s and a alida ion o i s eliabili y by compa ing Table 5. Es ima ed ange o he pa ame e s o he human-s uc u e in e ac ion model. ma [%] ζh [%] h [Hz] Minimum 91.00 34.00 3.30 Maximum 100.00 69.00 10.43 Figu e 5. (a) Ele a ion and (b) c oss-sec ion o he labo a o y oo b idge s uc u e (7). Figu e 6. Fi s na u al equency e sus ib a ion ampli ude o emp y oo b idge (7). 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 7 addi ional mass dis ibu ion so ha he alue o he i s ex- pe imen al ib a ion equency o he oo b idge is equal o he one ob ained om he nume ical modal analysis, consid- e ing an emp y s uc u e. Then in o de o es ima e he h ee main pa ame e s o he p oposed pedes ian-s uc u e in e ac ion model, sp ung mass (ma), equi alen human damping a io (ζh) and human na u al equency ( h), he mean squa e e o be ween he expe imen- al and nume ical i s e ical na u al equency o he oo - b idge is minimized (8) o each load scena io (LS). Gi en he singula i y o he minimiza ion unc ion chosen, and in o de o a oid local minimums, he gene ic algo i hms me hod (a global sea ch me hod) has been conside ed (9). The op imi- za ion me hod chosen p e en ed he solu ion o he p oblem om being condi ioned by he selec ed s a ing poin o he sea ch. To a oid ill-condi ioning p oblems, he educ ion o he sea ch domain has been de eloped, cons aining he p ob- lem. The sea ch ange p e iously p oposed has been used. Fo each load scena io, a popula ion o 1000 indi iduals ( ec- o s wi h h ee elemen s) is de ined. Each indi idual modi ies h ough he ope a ions o ep oduc ion, c osso e and mu a- ion he alues o i s elemen s in o de o minimize he alue o he objec i e unc ion. In all he cases, he op imiza ion p ocess eached alues o he objec i e unc ion lowe han 10–12 Hz. The maximum numbe o i e a ions is se o 100. The nume ical alue o he damped equency was ob ained om he esponse unc ion o he sys em in he equency do- main. The expe imen al alue o he damped equency was de e mined o each load scena ios acco ding o he pedes- ian low and he le el o ib a ion eached. I is shown in Figu e 9 he lowcha o he human-s uc u e pe- des ian model i ing p ocedu e based on gene ic algo i hms. equency was cha ac e ized by he equency whe e he o ce exe ed by he ac ua o eaches a minimum alue. The main esul s o his esea ch, summa ized in he Table 6, show he shi o he i s na u al equency o he s uc u e, om 1,emp (emp y oo b idge) o 1, ull ( ull oo b idge), e sus he le el o displacemen a he mid-span (A) and he numbe o pedes ians (Np). 6.2. Model alida ion The abo e esul s allow ob aining an expe imen al es ima- ion o he change in he dynamic beha iou o a labo a o y oo b idge depending on he pedes ian densi y. F om hese da a, he di e en pa ame e s o he p oposed human-s uc- u e in e ac ion model will be es ima ed by sol ing he in- e se dynamic p oblem. The solu ion o his p oblem will be made in wo phases. Fi s , a ini e elemen model o he oo b idge was de eloped (see Figu e 8), using 3-D beam elemen s and, adjus ing an Figu e 7. Damping a io e sus ib a ion ampli ude o emp y oo b idge (7). Table 6. Change o i s na u al equency unde he di e en scena ios. LS NpA [mm] 1,emp [Hz] 1, ull [Hz] 1.1 4 1.00 2.280 2.086 1.2 4 5.00 2.270 2.073 1.3 4 10.00 2.220 2.029 2.1 7 1.00 2.280 2.022 2.2 7 5.00 2.270 2.009 2.3 7 10.00 2.220 1.965 3.1 10 1.00 2.300 1.980 3.2 10.00 5.00 2.290 1.970 3.3 10.00 10.00 2.270 1.950 Figu e 8. Fini e elemen model and i s ib a ion mode ( =2.27 Hz) o he oo b idge. 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.13.1108 sligh ly cons an ega dless o he le el o he pedes ian low, he le el o ib a ion eached and he damping a io o he oo b idge. The es ima ed s anda d de ia ions o each model pa ame e (sp ung mass, equi alen human damping a io and na u al equency) a e small and he es ima ed alues a e conside ed as app op ia e. In Figu e 10, he nume ical es ima ion o he change o he i s e ical ib a ion equency ( 1, e ) o he s uc u e unde he c ossing o en pedes ians is shown. In he igu e h ee cu es a e ep esen ed. The “Expe imen al Resul s” shows he change o he expe imen al i s na u al equency ac- co ding o he esul s o he modal analysis (7), he “Passi e Pedes ians” shows he change o he p oposed in e ac ion model wi h he pedes ian pa ame e s es ima ed om he medium passi e pedes ian beha iou (ma=95.5 %, ζh=51.50 % and h=6.86 Hz ob ained om Table 5), and inally, he “In- e ac ion Model” co esponds o he i ing om he solu ion o he in e se dynamic p oblem. The analysis o he Figu e 10 shows ha he es ima ion o i s na u al equency o he s uc u e by he p oposed mod- el, conside ing he medium alues o he passi e pedes ian pa ame e s es ima ion, p oduces a ela i e e o o only 5.00 %. Besides he a ia ion o he dynamic p ope ies o he oo - b idge p esen s a simila shape o he es ima ed ones expe i- men ally. Good ag eemen is obse ed be ween he expe i- men al and nume ical esul s. A compa ison o he calcula ed alues wi h he es ima ed om he s udies on he dynamic e ec o passi e pedes ians in s adiums, shows mainly a loss o s i ness o each pedes- ian when hey a e in mo ion. F om he s udy o he p e ious esul s, h ee main conclu- sions can be ex ac ed: • Fi s , he s abili y ha he es ima ed model pa ame e s p esen o he di e en load scena ios, allowing ha his model iden i ica ion may be used as a eliable s a ing poin o he cha ac e iza ion o he e ec o he pedes ians on he dynamic beha iou o he s uc u es. • Then, he app op ia ely i ing be ween he expe imen- al and nume ical cu es ha show he change o he i s na u al equency o he oo b idge, a e he adjus men o he pedes ian pa ame e s. The goodness o he co ela ion e i ies he abili y o he p oposed model o cha ac e ize he human-s uc u e in e ac ion p oblem. In Table 7, he esul s ob ained o he nine load scena ios conside ed a e shown. The las wo ows o his able show he medium alue and he s anda d de ia ion ob ained o he di e en pa ame e s. The alues o he pa ame e s ha cha ac e ize he human-s uc u e in e ac ion model emain Figu e 9. Flowcha o he human-s uc u e pedes ian model i ing p ocedu e. Table 7. Es ima ed pa ame e s o he human-s uc u e in e ac ion model. LS ma [%] ζh [%] h [Hz] 1.1 83.046 45.007 2.852 1.2 84.996 45.000 2.600 1.3 84.990 45.583 2.502 2.1 84.861 43.293 2.870 2.2 84.539 49.212 2.759 2.3 84.974 49.975 2.600 3.1 82.139 48.001 2.997 3.2 83.171 51.014 2.908 3.3 83.007 47.486 2.743 Medium 83.969 47.175 2.759 De ia ion 1.054 2.611 0.165 Figu e 10. Nume ical es ima ion i s na u al damped equency (10 pedes ians). 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 9 in his wo k. To cha ac e ize he accu acy o he p esen ed model, he model pa ame e s a e adjus ed om he esul s p esen ed in a scien i ic s udy (7) in which he change o he dynamic p ope ies o a labo a o y oo b idge, unde di - e en pedes ian lows, is ca ied ou . The adequa e i ing be ween he expe imen al and nume ical cu es ha ma k he e olu ion o he i s na u al equency o he oo b idge unde he s ep o he di e en g oup o pedes ians and he s abili y o he pedes ian pa ame e s o he nine adjus - men s e i y he co ec pe o mance o he p oposed model o s udy he p oblem o human-s uc u e in e ac ion. A me- dium alue, in he e ical di ec ion, o he pedes ian sys- em pa ame e s, is p oposed, sp ung mass a io 84.00 %, hu- man damping a io 47.00 % and human equency 2.75 Hz. Howe e , u he s udies a e needed in o de o cha ac e ize he dynamic pa ame e s in he h ee spa ial di ec ions unde mo ing si ua ions in eal oo b idges and u he mo e, o he s ac o s mus be conside ed in he model in o de o include he synch oniza ion phenomena be ween he pedes ians and he s uc u e. • Finally, an impo an esul o he p esen esea ch is ha he main pa ame e ha con ols, in he e ical di ec ion, he a ia ion o he na u al equencies o a oo b idge unde a pedes ian low is he inc ease o he damping a io o he sys em due o he con ibu ion o he human damping. 7. CONCLUSIONS The models adop ed by he cu en codes o he s udy o he dynamic beha iou o oo b idges do no allow a e y accu- a e es ima ion o he eal esponse ha occu s a he di e - en poin s o he deck. The mos ad anced me hodology (4) (5), al hough i is e y e ec i e o es ima e he sensi i i y o he s uc u es o pedes ian low, i is qui e conse a i e o es ima e hei e ec s on he s uc u e. As he accele a ion is he main ac o ha de e mines he in-se ice condi ions, a pedes ian-s uc u e in e ac ion model has been p oposed in o de o ob ain a mo e accu a e es ima ion o he esponse. The model has been p esen ed and i s applicabili y p o en REFERENCES (1) Wolmu h, B., Su ees, J. (2003). C owd- ela ed ailu e o b idges. Ci il Enginee ing, 156(3): 116-123. (2) Dalla d, P.,Fi zpa ick, A.J., Le Bou a, S., Low, A., Ridsill Smi h, R., Will o d, M., Flin , A. (2001). The London Mille- nium Foo b idge. The S uc u al Enginee , 79(22): 17-33. (3) 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, doi: h p://dx.doi.o g/10.1016/j.js .2004.01.019. (4) SYNPEX Guidelines. (2007). Eu opean P ojec on Ad anced Load Models o synch onous Pedes ian Exci a ion and Op imised Design Guidelines o S eel Foo b idges. (5) Se a. (2006). Guide mé hodologique passe elles pié onnes (Technical guide oo b idges: Assemen o ib a ional be- ha iou o oo b idges unde pedes ian loading). Se a. (6) Jones, C.A., Reynolds, R. , A. Pa ic. A. (2011). 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, 330(8): 1531-1566, doi: h p://dx.doi.o g/10.1016/j. js .2010.10.032. (7) Geo gakis, C., Jo gesen, N.G. (2013). Topics in Dynamics o B idges, Volume 3. En P oceedings o he 31s IMAC. A Con e ence on S uc u al Dynamics, 2013. Chap e 4. Change in Mass and Damping on Ve ically Vib a ing Foo b idges Due o Pedes ians. The Socie y o Expe imen al Mechanics, Inc. (8) Gonzalez, M.N., Cobo, A., Fuen e, J.V. (2013). Ob ención de modelos de cálculo de sis emas p o isionales de p o ección de bo de median e la écnica de Análisis Modal Ope acional. In o mes de la Cons ucción, 65(529): 99-106, doi: h p:// dx.doi.o g/10.3989/ic.11.133. (9) Ghee, C., Pe y, M.J. (2010). S uc u al Iden i ica ion and Damage De ec ion using Gene ic Algo i hms: S uc u es and In as uc u es Book Se ies, Vol. 6. London: CRC P ess 2010. * * *