PEDESTRIAN LATERAL ACTION ON LIVELY FOOTBRIDGES: A NEW LOAD MODEL

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1 PEDESTRIAN LATERAL ACTION ON LIVELY FOOTBRIDGES: A NEW LOAD MODEL F. Venuti, PhD tudent, L. Bruno, Aociate Prof., P. Napoli, Full Prof., Department of Structural Engineering and Geotechnic, Politecnico di Torino, Torino, Italy. Summary Thi paper propoe a new load model to predict the lateral force exerted by pedetrian walking on lively footbridge. The aim of the model i to take into account ome important feature of the ynchronou lateral excitation phenomenon, which o far ha not been fully undertood or modelled, e.g. the ditinction between ynchronization among pedetrian, due to crowd denity, and between the pedetrian and the tructure, caued by deck ocillation; the triggering of the lock-in phenomenon and it elf-limited nature. The propoed load model ha been teted with reference to two crowd event that were recorded on the T-bridge in Japan (1993) and on the Millennium Bridge in London (2001). The reult obtained with the preent model are compared to reult predicted by other load model found in literature and are then dicued. Introduction The phenomenon of ynchronou lateral excitation caued by pedetrian walking on footbridge ha attracted increaing public attention in the lat few decade from the earliet cae in the ninetie [1] to the well-known cae of the London Millennium Bridge [2]. Up to date, thi kind of dynamic lateral load ha never led to tructural failure, but ha often caued dicomfort for the uer and the temporary cloure of the footbridge in order to provide proper countermeaure [2]-[3]. Thi reduced erviceability repreent a evere problem for it economic and ocial outcome. Thee reaon have recently motivated intene reearch activity, reviewed in [4], which repreent the cientific background of deign guideline (e.g. [5]-[6]). The problem of determining the force exerted by ynchronized pedetrian i generally tackled uing an empirical aroach. Laboratory tet involving a ingle pedetrian walking on a moving platform (e.g. [7]) allow important data to be obtained on the lateral force exerted by one pedetrian and to etimate the mean probability that individual will ynchronie their tep to the platform motion [2]. Moreover, the obervation of video recorded during crowd event on actual bridge [1] permit a qualitative etimate to be made of the ynchronization phenomena that occur between people walking in a crowd. Several model have been propoed uing the aforementioned data to predict the lateral force exerted by pedetrian. Mot of the model found in literature (e.g. the one reviewed in [4]) are determinitic time-domain model, baed on the aumption that both feet produce exactly the ame periodic force. The load model propoed in the deign code alo belong to thi category. For example, three different load model are propoed in [5] to etimate the force exerted by a ingle pedetrian, by a group of pedetrian or by a crowd uniformly ditributed along the footbridge deck, repectively. To the author' knowledge, many of the force model propoed have o far experienced ome difficultie in taking into account ome not negligible apect of the problem, that i: - the dependence of the pedetrian force on the two-way interaction between two ytem, the crowd and the tructure, which can be decribed by two variable, the pedetrian denity and the footbridge lateral vibration. In thi paper the deck velocity i retained a the vibration variable which motly affect the pedetrian force, a indicated by ome author (e.g. [2]); - the poibility of a inhomogeneou ditribution of the crowd along the deck due to bottleneck, congeted traffic or other non-linear traffic phenomena; - the exitence of two kind of ynchronization, a recently pointed out in [8] and [9], one between the pedetrian and the tructure and the other among the pedetrian. The latter take place when the relative movement of the pedetrian i contrained becaue of high crowd denity; - the preence of different frequency component in the overall force; - triggering of the lock-in phenomena and the reulting elflimited ocillation. The aim of the preent work i to propoe a force model that give due weight to the aforementioned feature of the phenomenon. The propoed mathematical model aume that crowd denity and deck velocity are given, for intance by mean of in itu meaurement, computational imulation with a crowd-tructure interaction model [10] or a priori choen within a wort-cae cenario aroach. On one hand, the model i conceived to

2 provide an accurate decription of the phenomena and, on the other, to become a ueful tool for footbridge deigner and engineer, in order to predict footbridge behaviour under pedetrian load. The model wa conceived on the bai of a macrocopic decription of crowd dynamic, which mean that the pedetrian are not viewed a ingle individual but a cluter characterized by a mean walking velocity and a mean tep frequency. Neverthele, an adaptation of the model to a microcopic or tatitical decription of the crowd i poible. Firt, a detailed decription of the model i given; then, it propertie are evaluated and, finally, two alication are preented and dicued. Formulation of the model The propoed force model can be acribed to the category of timedomain model. It i baed on the aumption that the force exerted by a number n of pedetrian walking along a portion of the bridge pan i given by the um of three component: F = F + F + F (1) where F i the term due to the ynchronization between the pedetrian and the tructure, F i due to the ynchronization among pedetrian and F i the part due to uncorrelated pedetrian. Figure 1: Relationhip between tep frequency and pedetrian velocity F ha the ame frequency f a the excited lateral tructural mode, while the other two term have the ame frequency f p a the lateral pedetrian foottep. f p i aumed to vary a a function of the walking velocity v (Figure 1), a reported by Bertram & Ruina [11] and Wheeler [12]. In turn, the walking velocity v can be expreed a a function of the crowd denity u, and of the deck velocity ζ ( intended a the envelope of the velocity time hitory), a propoed in [10]. Figure 2: Relation between v and u (a) and v and ζ (b) Several relation between v and u, the o-called fundamental diagram, have been propoed by different author (e.g. [5], [13]) on the bai of experimental tet and the obervation of real event (Figure 2a). It i intereting to notice that the only diagram reported in a deign guideline [5] i clearly different from the other one: it i baed on only a few meaurement referring to fat pedetrian tranit in train tation, and can therefore be conidered meaningle for the alication dicued in thi paper. The light difference that can be oberved among the other diagram are due to different meaurement ituation (location, travel purpoe, pedetrian attribute). The linear fundamental diagram propoed by the author ha been conceived to decribe the behaviour of pedetrian croing the footbridge on the occaion of particular event and in the cae of unidirectional flow. A for the relation between v and ζ, a qualitative law i propoed (Figure 2b): pedetrian walk i uncontrained when ζ i le than a critical value z& c = 0,02 m/, inferred from the experimental data reported in [2]; the walking velocity linearly decreae a ζ grow and the pedetrian are no longer able to walk a ζ reache a maximum admiible value z& M = 0,44 m/ (uggeted by Nakamura in [14]). A a conequence of the aforementioned relation, the lateral tep frequency f p i a function of both the crowd denity u and the lateral footbridge velocity ζ (Figure 3). Thi relation aure that, in the preence of exceive lateral deck motion or crowd denity, pedetrian top walking, o that the amplitude of the term F and F are elflimited. Figure 3: f p veru u and ζ Each term of the overall force i weighted on the bai of phenomenological conideration, by mean of three weight, n, n and n, that can be conidered, repectively, a the number of pedetrian in the cluter that are ynchronized with the tructure, ynchronized to each other and uncorrelated: n n n = ns = ns = n n ( 1 S ) (2) n where S and S are the ynchronization coefficient, which both vary in the [0 1] range. Thank to the ditinction of pedetrian in three categorie, the model i able to repreent the triggering of lock-in: even though no one i ynchronized

3 with the tructure, the preence of a high crowd denity reult in a lateral force that trigger the lateral vibration of the bridge. S repreent the degree of coupling between the crowd and the tructure. Thi i a function of two variable: the tructure lateral velocity ζ and the ratio f r = f p / f, where f r i defined in the [0 2] domain: the lower bound depend on the minimum value of f p (f p =0), when the walking velocity i null; the uer bound wa obtained from the laboratory tet reported in [7], according to which pedetrian are not influenced by tructural ocillation under 0,6 Hz and the maximum recorded lateral walking frequency i 1,1 Hz. The variation of S veru ζ (Figure 4a) i given by a fitting of the Dallard et al. experimental data [2], by mean of the interpolating function: b( ζ z& c ) S ( ζ ) = 1 e (3) Pedetrian tart to ynchronize with the tructure for value of ζ higher than z& c, and they are completely ynchronized when ζ reache the maximum value z& M. Figure 4: S veru ζ (a) and f r (b) S (f r ) i uoed to have a normal ditribution, with a variance that grow when ζ increae. Thi mean that, for increaing value of ζ, the pedetrian who walk with a tep frequency that i different from f gradually become involved in the ynchronization phenomenon. For ζ = z& M, everyone i ynchronized with the tructure, whatever the value of f r (Figure 4b). S (f r ) i defined a: γ ( ζ ) = 50e 2 [ γ ( f r 1) ] S ( f r ) = e (4) ( 40ζ ) The ynchronization coefficient S (ζ, f r ) i given by the product of equation (3) and (4). Figure 5: S veru u The coefficient S (Figure 5) repreent the degree of ynchronization among pedetrian and, becaue of the lack of experimental data, it ha been defined in a qualitative way a a function of the crowd denity u [ped/m 2 ]: 1 S ( u) = 2 ( uync uc ) u* = 2 { 1+ erf [ au ( u u *)]}, M (5) where: - a=3,14; - u c i a critical denity that repreent the uer limit for uncontrained free walking. According to [5], it value i et to 0,3 ped/m 2 ; - u M i the maximum admiible denity, correponding to a pedetrian velocity equal to zero. Different value of u M are propoed in literature (e.g. [13]): in the following u M = 5,4 ped/ m 2 ; - u ync i the value that correpond to the total ynchronization of pedetrian. It value i etimated to be 1,8 ped/ m 2, according to the maximum denity recorded on the T- bridge [3]. It i worthwhile pointing out that the propoed law for the ynchronization coefficient S and S can be interpreted a cumulative denity function of an exponential and a Gauian probability denity function, repectively. From thi point of view, the model can be adapted to a tatitical decription of the phenomenon a oon a a greater amount of data i available. The firt component of the total force, F, can be written, according to [7], a the um of a component in phae with the tructure acceleration & z& and another one in phae with the velocity z& : F = n [ F in(2π f t) F in(2πf t)] (6) & z + The amplitude of the two component (Figure 6) are defined by mean of piecewie function. Figure 6: DLF of the component in phae with velocity (a) and acceleration (b) The firt branch come from a quadratic fitting of the experimental data concerning the medium Dynamic Load Factor (DLF) of the in-phae and out-of-phae component [7]. The data correponding to DLF cloe to zero for non null z& or & z&, and to z& or & z& above their erviceability limit (that i, z& =0,25 m/ and & z& =1,35 m/ 2 [14]) have been dicarded. The econd branch i defined z&

4 qualitatively, baed on the following aumption: - the DLF reach their maximum when the velocity and the acceleration exceed their erviceability limit; - the DLF decreae to zero when the velocity and the acceleration reache maximum value, above which pedetrian top walking, i.e. z& M =0,44 m/ and & z& M =2,1 m/2 [14]. Thi trend alo guarantee that the amplitude of F i elf-limited a i the overall tructural repone. The econd force component, F, i defined a: component and the overall force in the cae of u=1,3 ped/m 2 and ζ=0,06 m/ (condition recorded on the T- bridge [16]). Figure 9: Colour map of the pedetrian ynchronization F = n F in( 2πf t) (7) p where F i the medium amplitude of the force exerted by a ingle pedetrian in the cae of a motionle deck, whoe DLF=0,04 [4]. Finally, the component F i determined according to the model propoed by Matumoto et al. [15], who found that the force due to n pedetrian walking with Gauian ditributed frequencie and random phae i n higher than the force due to a ingle pedetrian. Therefore, F become: Figure 7: Time hitorie of the force component F = n F in( 2πf t) (8) p Propertie of the model The propertie of the propoed force model have been evaluated by mean of a enitivity tudy on the two main variable, ζ in the [0 0,5] m/ interval and u in the [0 u M ] ped/m2 range. The imulation are performed auming f =0,93 Hz (the firt lateral natural frequency of the T-bridge in Japan [16]) and a platform velocity time hitory with contant amplitude ζ: z& = ζ co( 2πf ) (9) Thi type of imulation can be compared to an experimental tet on a moving treadmill, except that the ingle pedetrian i ubtituted by a cluter of pedetrian. A an example, Figure 7 graph the time hitorie of the three force Figure 8: Weight of the force component caled with repect to n Figure 10: Amplitude of the force component and rm value of the overall force Figure 8 how the weight of the force component, caled with repect to the number of pedetrian n, veru ζ and u. The following conideration can be made:

5 - n i much more enitive to the deck velocity ζ, than to the crowd denity u and grow monotonically a ζ increae and a u decreae; it reache the highet value for ζ > z&, whatever the value of u (Figure 8a); - n ha the ooite trend to n : it grow a u increae and a ζ decreae. For u > u ync (S =1), n i complementary to n (Figure 8b); - n i obtained from ubtraction of the other two term, therefore it reache the maximum value when ζ and u are under their critical value and i null for ζ > z& and u > u ync, when all the pedetrian are ynchronized (Figure 8c). A colour map of the three weight in the u-ζ plane (Figure 9) can be obtained for each tructure, in order to quickly etimate, in the preliminary deign phae, which kind of ynchronization will play the main role. Figure 10 repreent the amplitude a of the three force term (in N/m 2 ) and of the overall force F, which i expreed in term of it root mean quare (rm) value (F rm ), veru ζ and u. It can be oberved that: - the evolution of F veru ζ i influenced by both n and by the ditribution of the DLF (ee Figure 6), which determine it non monotonic trend. The dependence of F on u i almot linear, i.e. it amplitude grow linearly a the number of pedetrian increae (Figure 10a); - the amplitude of F ha the ame evolution a n, ince it come from the product of n and the contant F. It i worthwhile pointing out that F goe abruptly to zero when f p =0, that i, for u=u M or ζ > z& M : thi mean that the pedetrian who have toed do not exert lateral excitation and the elf-limited nature of the deck repone i guaranteed (Figure 10b). A imilar conideration can be drawn for F, which alo ha a non monotonic evolution veru u (Figure 10c); - the evolution of F rm veru u and ζ i not trivial, ince it come from the um of periodic ignal with different frequencie and it therefore cannot be determined by imply umming the three component amplitude (Figure 10d). The amplitude of the force component can alo be analyed with repect to their frequency content (Figure 11). The term F evolve in time with a contant frequency equal to f ; the component F, intead, ha a frequency f p which depend on both u and ζ (ee Figure 3). When the two variable are under their critical value, f p i at it maximum value, while, a u and ζ grow, the walking velocity decreae and f p decreae in turn. Figure 11: Amplitude of F and F veru f p, for ζ=0,06 m/ (a) and u=1,3 ped/m 2 (b) Alication - The force model ha been teted adopting the condition recorded in two crowd event on the T-bridge in Japan (Figure 12) and on the Millennium Bridge in London (Figure 13). Figure 12: The T-bridge (photo wa taken by S. Nakamura [3]) The data ued for the imulation are the following: - T-bridge: u=0,8 1,8 ped/m 2 ; f =0,93 Hz; ζ=0,06 m/ [1]-[16]; - Millennium bridge: u=1,3 1,5 ped/m 2 ; f =0,8 Hz (outh pan); ζ=0,25 m/ [2]. Figure 13: The Millennium Bridge (from [2]) The ingle force per pedetrian [ N/ped] obtained with the model i compared to that predicted with the force model propoed in [1]-[16]- [2] (Table 1) : the range of the reported value refer, repectively, to the lower and higher crowd denitie ued for the imulation. It hould be pointed out that, ince the cited model do not ditinguih between the two type of ynchronization, the predicted force i conidered to be only due to the pedetrian-tructure ynchronization and it i therefore compared to the term F. Table 1: Comparion between the propoed force model and thoe found in literature (force in N/ped) Model a(f ) T-br. Fujino et al [1] 7 Nakamura [16] 4,7 Dallard et al [2] 18 Propoed 5,7 4,7 M. br. Dallard et al [2] 75 Propoed 81 Firt, it i hould be noticed that the reult provided by the model in literature are rather different for the T-bridge: thi i probably due to the fact that each model i baed on the data recorded on a particular footbridge and it can not therefore be extended to all other tructure. For intance, Nakamura & Kawaaki oberved that the model they propoed for the T-bridge only agreed with that of Dallard et al. for deck lateral velocitie under 0,015 m/. The propoed model, intead, ha a more general validity, ince it

6 i not baed on one ingle event but on the phenomenological decription of the component of the coupled ytem in their fundamental contitutive law. The amplitude of the term F predicted by the propoed model are very imilar to thoe etimated by Nakamura & Kawaaki for the T- bridge and by Dallard et al. for the Millennium bridge, repectively: thi fact highlight the wide alicability of the model. Table 2 provide a deeper inight into the other reult obtained with the model. The correpondence between the imulated reult and the actual data can clearly be een looking at the force frequency content. In the cae of the T-bridge, the etimated tep lateral frequency f p (0,89 0,8 Hz) i very cloe to the deck lateral frequency (0,93 Hz), therefore the two component F and F alo excite the firt lateral mode. A for the Millennium bridge, ince f p i cloe to 0,5 Hz, it can be tated that the model i alo able to predict what occurred on the central pan, which vibrated at both the firt (0,5 Hz) and the econd (1 Hz) lateral mode. Hence, it can be argued that the central pan econd mode (1 Hz) wa excited by the crowd-tructure interaction, while the firt wa excited by the ynchronization of pedetrian and by uncorrelated pedetrian. Finally, let u conider the number of ynchronized pedetrian. Even though both event are characterized by high crowd denity, a the two tructure have different lenderne, the Millennium bridge how higher ocillation than the T-bridge, which mean a greater number of pedetrian ynchronized with the tructure. Since only a few pedetrian are captured in the tructure-induced ynchronization on the T-bridge, the crowd denity play a leading role in determining the overall force (ee alo Figure 9). Concluion The propoed force model atifie the preet objective. The main feature of the pedetrian lateral excitation phenomenon are taken into account, in particular the fact that both the crowd denity and the motion of the footbridge deck influence pedetrian behaviour. The model i alo accurate compared to the data and model found in literature, and can therefore be conidered a a general predictive tool for the deign and analyi of footbridge under pedetrian load. Becaue of it veratility, it can be ued for different purpoe and with different degree of accuracy: during the preliminary deign phae, it allow the wort load cenario to be outlined if the expected value of crowd denity and deck velocity are a priori choen; in the final deign phae, it can be ued to determine the load that hould be alied to the tructural model or to a complete computational imulation of the crowd-tructure interaction [10]; once the footbridge ha been built, the model i able to define the pedetrian force on the bai of the deck velocity and crowd denity meaured in itu during actual event. It hould be pointed out that ome of the law preented in thi paper come from qualitative conideration and are not adequately uorted by experimental data. For thi reaon, the model, which contitute a valid reference framework, can certainly be improved with a proper tuning of the parameter by mean of ad hoc conceived experimental tet. Finally, the model framework i uitable to be extended to pedetrian vertical loading. Acknowledgement The author wih to expre their acknowledgement to profeor F. Ricciardelli, Y. Fujino and S. Nakamura for kindly making their experimental data available. Reference [1] FUJINO Y., PACHECO B. M., NAKAMURA S., WARNITCHAI P. Synchronization of human walking oberved during lateral vibration of a congeted pedetrian bridge. Earth. Eng. Struct. Dyn., No. 22, , [2] DALLARD P., FITZPATRICK T., LE BOURVA S., LOW A., RIDSDILL R. M., WILLFORD M. The London Millennium Footbridge. The Structural Engineer, No. 79(22), , [3] NAKAMURA S., FUJINO Y. Lateral vibration on a pedetrian cable-tayed bridge, Structural Engineering International, No. 12(4), , [4] ŽIVANOVICH S., PAVIC A., REYNOLDS P. Vibration erviceability of footbridge under human-induced excitation: a literature review. J. of ound and vibration, No. 279,. 1-74, [5] Federation International du Beton. Guideline for the deign of footbridge, FIB Bulletin No. 32, Lauanne, [6] SETRA/AFGC. Paerelle piétonne Evaluation du comportement vibratoire ou l action de piéton. Guide méthodologique. Pari, [7] PIZZIMENTI A.D., Experimental analyi of the lateral pedetrian-induced mechanim of excitation of footbridge, PhD Thei, Univerità degli tudi di Catania, 2004 (in italian). [8] RICCIARDELLI F. Lateral loading of footbridge by walker. Proceeding Footbridge 2005, Venice, [9] VENUTI F., BRUNO L., BELLOMO N. Crowd tructure interaction: dynamic modelling and computational imulation. Proceeding Footbridge 2005, Venice, [10] VENUTI F., BRUNO L., BELLOMO N. Crowd dynamic on a moving platform: mathematical modelling and alication to lively footbridge. Mathematical and Computer Modelling, No. 45, , 2007 [11] BERTRAM J. E., RUINA A. Multiple walking peed-frequency relation are predicted by contrained optimiation. J. theor. Biol., No. 209, , [12] WHEELER J. Prediction and control of pedetrian induced vibration in footbridge. ASCE J. of the Structural Diviion, No. 108, , [13] DAAMEN W. Modelling paenger flow in public tranport facilitie. PhD Thei, Delft Univerity of technology, [14] NAKAMURA S. Field meaurement of lateral vibration on a pedetrian upenion bridge. The Structural Engineer, No. 81(22), , [15] MATSUMOTO Y., NISHIOKA T-, SHIOJIRI H., MATSUZAKI K. Dynamic deign of footbridge. IABSE Proceeding, No. P-17(78),. 1-15, [16] NAKAMURA S., KAWASAKI T. Lateral vibration of footbridge by ynchronou walking, J. of Contructional Steel Reearch, No. 62, , 2006.

7 Table 2: Reult obtained with the propoed model (force in N/ped) Benchmark a(f ) a(f ) a(f ) f p F rm n /n n /n T-bridge 5,7 4,7 2,5 19,9 24,9 0 0,89 0,8 19,7 14,5 0,33 0,38 0,08 0,66 M. bridge 81 1,9 2,1 2,6 1 0,56 0,54 57,7 57,6 0,93 0,06 0,07