ateral vibration of footbridge under crowd-loading: Continuou crowd modeling approach Joanna Bodgi, a, Silvano Erlicher,b and Pierre Argoul,c Intitut NAVIER, ENPC, 6 et 8 av. B. Pacal, Cité Decarte, Champ ur Marne, 77455 Marne-la-Vallée, France a bodgi@lami.enpc.fr, b erlicher@lami.enpc.fr, c argoul@lami.enpc.fr Keyword: Footbridge, Crowd loading, Pedetrian-tructure ynchronization. Abtract. In thi paper, a imple D crowd model i propoed, which aim i to properly decribe the crowd-flow phenomena occurring when pedetrian walk on a flexible footbridge. The crowd i aumed to behave like a continuou compreible fluid and the pedetrian flow i modeled in a -D framework uing the (total) ma (of pedetrian) conervation equation. Thi crowd model i then coupled with a imple model for the dynamical behavior of the footbridge and an optimized modeling of ynchronization effect i performed. Numerical imulation are preented to how ome preliminary reult. Introduction Recent example of footbridge have hown to be enitive to the human induced vibration (Millenium Bridge, ondon; Solférino Bridge, Pari). Several experimental meaurement allowed thi phenomenon to be better undertood [, ]. The crowd walking on a footbridge impoe to the tructure a dynamic lateral excitation at a frequency cloe to Hz. When the firt mode of lateral vibration of a footbridge fall in the ame frequency interval, then a reonance phenomenon i activated, the ocillation amplitude increae and pedetrian are forced to change their way of walking, up to the o-called tructure-pedetrian ynchronization occurring if the ocillation amplitude i large enough. Thi phenomenon ha been often experimentally detected and alo analyzed in everal tudie [, 3, 4]. A erie of implified deign rule for footbridge accounting for thee effect wa recently propoed []. Moreover, the behaviour of a ingle pedetrian i affected by the preence of the crowd around him. In more detail, when the pedetrian denity i very low, the walk i «free» and characterized by the peed, the walk frequency, the tep length, etc. lightly varying from a walker to the other. Nonethele, when the crowd denity become higher, a ingle walker i forced to ynchronize hi peed with that of the other. Thi kind of pedetrian-pedetrian ynchronization occur even when the walk i on a rigid floor and ome crowd model were already developed [5] for thi cae. However, very few exiting tudie concern the modeling of both kind of ynchronization [3]. In thi contribution, an approach that we call Eulerian, i propoed in order to take into account traffic effect and pedetrian-tructure ynchronization. Some experimental reult Single pedetrian walking. The human walking i characterized by time interval where both feet are in contact with the floor and interval where only one foot touche the floor. One can define the beginning of a tep a the beginning of a imultaneou contact period. The end of a tep coincide with the beginning of the following imultaneou contact period. For a given pedetrian walking at contant peed, the time-length T of a tep i approximately contant. The walking frequency i defined by: f T. For a tandard walking, one get f Hz [4]. The force induced by a ingle pedetrian on the floor ha a lateral component related to the mall lateral
ocillation of the centre of gravity of the pedetrian during walk. It act in the direction perpendicular to the walk peed, and with oppoite ign for each foot. Hence, the frequency of the lateral force i: f lat f Hz. The proceing of data provided by Decathlon [6] concerning lateral walking force how a quai-periodic behavior, with a typical right-left cycle like the one indicated in Fig.. The periodicity of the lateral force f l ( t ) ugget a Fourier harmonic decompoition: f ( t) in( i f t ) in( f t ) () l i lat i lat i where i i the i-th Fourier coefficient, i i the i-th phae difference. f l ( t ) i alway bounded and le than 50 N. For an increaing walking peed, the frequency f lat alo increae. Figure - ateral walking force on a rigid floor, for everal walking peed. It can be deduced that the firt harmonic amplitude i of order 30 35N, confirming the reult of SETRA [] and can be related to the walking velocity v by: 0. 69 v 35. 57. For the analyi of the crowd effect on the lateral motion of footbridge, only the firt harmonic (frequency f lat ) i retained in mot tudie and thi will be the cae in thi paper. Thi f lat frequency i often very cloe to the lateral modal frequency of footbridge. Hence footbridge may be ubject to large ocillation becaue their behavior i often lightly damped even if the pedetrian lateral force i relatively mall. In the cae of non-rigid floor, the lateral force amplitude of the human walk i aumed to be imilar to that of the rigid-floor cae. Thi trong aumption i upported by the experimental reult of SETRA []. Converely, the walking velocity and frequency are affected by the tructure vibration, a een in the following paragraph. Influence of the tructural ocillation on pedetrian walking. When the amplitude of the lateral vibration acceleration for the footbridge i larger than a certain threhold: a min 0. m /, the lateral vibration become perceptible for pedetrian, who tend to change their walking frequency to ynchronize their walk with the tructure ocillation [4]. If a certain number of pedetrian i ynchronized with the tructure (and therefore, with each other), the total lateral force they produce further increae the tructure ocillation, inducing other pedetrian to ynchronize their walk. If the lateral velocity of the footbridge floor reache u 0. 5m /, pedetrian top walking, otherwie they would looe equilibrium [3]. Interaction between pedetrian. For low crowd denitie c 0. 3p / m, every ingle pedetrian walk freely, with an average peed v M. 5m [3], lightly varying from one walker to the other. Converely, when the crowd denity i higher than c, the walking velocity decreae
in order to avoid colliion between pedetrian. For very high crowd denitie (, 68, pedetrian m [3]), pedetrian top walking. M Simplified footbridge modeling The lateral motion of a footbridge can be approximately repreented by an Euler-Bernoulli beam equation with vicou damping []: 4 u u u u m ( x) ( xt, ) cx ( ) ( xt, ) k( x) ( xt, ) F(, ) (, ) (, ) 4 l xt mp xt xt t t x t () where x i the coordinate along the beam axi; t the time; u(x,t) the lateral diplacement; m (x) the ma per unit length of the beam [kg/m]; c(x) the vicou damping coefficient [N. /m ]; k(x) the tiffne per unit length [ Nm ] ; F l (x,t) the pedetrian lateral force per unit length [ N / m] and m p ( x,t ) the linear ma of pedetrian [ kg / m]. Under the aumption of doubly hinged beam of length, the firt lateral mode hape can be approximated by a inu having the half-period equal to, which i exact when m, m p, c and k are contant along the beam axi. Since thi mode play a major role in the lateral footbridge dynamic, the olution i aumed of the uxt (, ) Ut ( ) ( x) Ut ( )in x/ and Eq. () become form m ( t )U( t ) c U( t ) k U( t ) F ( t ) (3) p l where m m ( t ) m ( t ) [ m ( x ) m ( x,t ) ] dx,, c c(x)ψ (x)dx, 0 p d k k(x) (x)dx, and F l ( t ) 0 dx F l ( x,t ) ( x ) dx. One can oberve that the ma m ( t ) i made of the claical tructural ma contribution plu the total ma of pedetrian walking on the footbridge deck and having the intantaneou ditribution given by m p ( x,t ). Hence, the intantaneou modal frequency of the ytem footbridge+pedetrian read: m ( x, t) x fp () t k m () t f in dx) m 0 p 0 (4) The ma m p ( x,t ) and the force F l ( x,t ) mut be defined to olve Eq. (3) and are related to the approach ued for modeling the crowd. Two different approache are undertudy: (i) an Eulerian approach coniting in a macrocopic modeling of the crowd which i conidered a a whole and (ii) a agrangian approach coniting in a microcopic modeling of the crowd where each pedetrian i modeled. Under after, only the firt approach i preented. The Eulerian crowd model (ECM) An Eulerian crowd model (ECM) potulate that the crowd behave like a compreible fluid [3]. Thi kind of analyi i intended to repreent the pedetrian behaviour for high crowd denitie. The crowd motion i characterized by it local denity ( x,t ) and it local peed v ( x,t ). In general three equation govern the motion of a fluid, i.e. the ma conervation (.v ) 0 t x (5)
the dynamic equilibrium and a contitutive law. However, for traffic flow modelling, it i uual to ubtitute thee lat two equation with a impler cloure equation [3, 6], relying on and v. For the pedetrian flow, the following cloure equation i adopted [3]: v(,u ) g( ) h( u ) (6) vm c u u u with g( ) exp( ( c ) ( M c )), h( u ) u vm ( ) c exp( ) 0 u u The crowd velocity v(,u ) depend on the g( ) function, repreenting the interaction between pedetrian decribed above. ( x,t ) i a parameter decribing the traffic condition. The mechanical analogy ugget for the role of a vicoity parameter of the equivalent crowd fluid. A larger value indicate more difficult traffic condition (highly vicou crowd fluid). Fig. preent the crowd peed on a rigid floor ( h( u ) ) for different value. Experimental data lead to the value of being between 0 and 0 [3]. Figure - Walking peed v. crowd denity for different value The econd function h( u ) allow accounting for the influence of the lateral footbridge vibration on the crowd peed, in the ene that it impoe a peed reduction when pedetrian walk on a footbridge undergoing large ocillation. In addition, it i aumed that after a top occurring when u u, the pedetrian peed remain zero during five econd. After thi delay, they begin to walk if u u, otherwie, the top lat five more econd. Oberve that Eq. (6) govern the crowd peed but nothing i aid about the pedetrian-tructure ynchronization phenomenon which alo involve the phae of pedetrian and of the tructure. Once the denity (x,t) i known, the linear ma denity follow mp( x, t) m p ( x, t) ( x), where ( x) i the deck width and m p i the ma of a ingle pedetrian. The linear force denity read:, Fl( x, t) fl( t) ( x, t) l x S( u, ) fl( t) np( x, t) S( u, ) fl( t) np eq( x, t) (7) where S( u, ) [ 0; ] i a coefficient introduced to repreent the ynchronization effect [3]. According to the aumption of Venuti [3], the pecial cae S occur when all the pedetrian walk with the frequency and phae of the tructural velocity Ut ( ). And Eq.7 tate then that the linear force denity i given by the product of the force of a ingle pedetrian with the number n p ( x,t ) of pedetrian per unit length. Thi cae i the mot evere for the tructure, in the ene that the ame number of pedetrian alway produce a le important tructural motion when they are not in thi ituation of full ynchronization.
Converely, if pedetrian are not ynchronized, the total force i defined a the product of the force of a ingle pedetrian and an equivalent ynchronized pedetrian number. Thi equivalent number i defined to give a fictitiou total force Fl ( x, t) acting at the modal frequency and phae of the tructure whoe effect (tructural motion) are the ame a thoe of the true total force, deriving from the true non-ynchronized pedetrian (ee []). The ynchronization coefficient, i.e. the ratio between the number of ynchronied pedetrian per unit length and the number of pedetrian per unit length, i defined by Venuti [3] a follow S( x,t ) Spp( ( x,t )) Sp( u( x,t )) (8) The firt term account for the fact that for high denitie pedetrian are ynchronized each other: for c, the walk i free, every pedetrian ha a different frequency and/or phae and the contribution to the total force i therefore null. The econd term derive from the obervation that for large ocillation, pedetrian have a tronger tendency to be ynchronized with the tructure. The function Sp( u ) ha two branche (Figure 3-b): the firt one i a quadratic approximation of the ARUP data []. It define an increae of the ynchronization when the tructural ocillation are larger. However, after reaching a imum for a particular amplitude of tructural ocillation, the ynchronization (and the equivalent pedetrian number) i aumed then to decreae, a repreented by the decreaing branch. a) /M u b) Figure3- a) Coefficient Spp( ), b) Coefficient Sp( u ) The previou definition of the ynchroniation coefficient (Eq. 8) have ome drawback: (i) the um of two contribution i rather artificial, (ii) the Spp coefficient i arbitrary and not baed on experimental data, (iii) it i difficult to ditinguih the effect in the cae of a vibrating floor between the two ynchroniation: the one between pedetrian and the other with the vibration of the tructure ; the ynchroniation between pedetrian being induced by the ynchroniation with the deck vibration. Hence, a new definition of the ynchroniation coefficient i introduced. According to SETRA [], the lateral force induced by a crowd of N pedetrian on a rigid floor can be approximated by the force induced by N eq pedetrian walking in place and having the ame phae and frequency. Thu, denoting the proportional modal damping ratio, the ynchroniation coefficient in the cae of non-vibrating floor i defined a follow Sp( ( x,t )) N eq 8. 6 ( ( x,t ) ( x ) ) ( x,t ) / N. 75 ( ( x,t ) ( x ) ) if ( x,t ) otherwie c Finally, a new definition for the ynchroniation coefficient i given by Sp( ( x,t )) S( x,t ) Sp( u( x,t )) if u( x,t ) a min otherwie 0. m / (9) Numerical example : the Millennium Bridge
Eq. (4) i olved by a Runge-Kutta cheme of order 4-5, and Eq. (5) i dicretized by a finite difference cheme (ax-friedrich), fulfilling the ax Friedrich condition for the ratio between the patial tep dx and the time tep dt. The model i validated uing the Millennium Footbridge main deck parameter. The pedetrian ma i aumed to be 75 kg, hence m p ( x,t ) 75( x,t ) l. Moreover the following parameter are adopted: ( x,t ) 0, U( 0 ) U( 0 ) 0and ( x, 0 ).. Figure 4-a and 4-b repreent U ( t ) and F l ( t ) repectively. The initial high denity of pedetrian induce a ynchroniation between pedetrian which lead to a lateral force of high amplitude. A a conequence, the tructure lateral vibration are more important. When the vibration velocity i high ( u ( x,t ) u ), the pedetrian top, hence the lateral force amplitude decreae and o do the tructural vibration. a) b) Figure 4 -a) Footbridge velocity U ( t ) ; b) total lateral force F l ( t ) induced by the crowd Concluion In thi paper, the crowd flow ha been decribed by an Eulerian approach, in a D framework, viz. pedetrian are uppoed to walk along traight trajectorie parallel to the longitudinal dimenion of the footbridge. The non-tationary behaviour ha been analyzed, a well a the ynchronization phenomenon. Some improvement have been propoed. Work i in progre on a agrangian model including ynchronization and the D generalization of both approache. Reference [] Dallard P., Fitzpatrick A.J., Flint A., e Bourva S., ow A., Riddill Smith R.M., Willford M. (00) The ondon Millenium footbridge. The Structural Engineer, Vol. 79(), p. 7-3. [] SETRA (006) Paerelle piétonne - Evaluation du comportement vibratoire ou l'action de piéton. Document technique Bagneux, 9 page. [3] Venuti F., Bruno., Bellomo N. (005) Crowd-tructure interaction: dynamic modelling and computational imulation. Proc. Footbridge 005: nd Int. Conf.,CDRom. [4] Živanovic S., Pavic A., Flint A., Reynold P. (005) Vibration erviceability of footbridge under human induced excitation: a literature review. Journal of Sound and Vibration, 79, -74. [5] Hoogendoorn S. (004) Pedetrian flow modelling by adaptive control. Proc. TRB 004 Annual Meeting,CDRom, page. [6] Bodgi J., Erlicher S., Argoul P. (006) Vibration et amortiement de paerelle piétonne, JSI 006, CDRom.