A NEW LOAD MODEL OF THE PEDESTRIANS LATERAL ACTION Fiammetta VENUTI PhD Politecnico di Torino Torino, IT Luca Bruno Aociate Profeor Politecnico di Torino Torino, IT 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 further dicued. Keyword: crowd-tructure interaction; footbridge; ynchronou lateral excitation; pedetrian load; lock-in. 1. Introduction The problem of determining the lateral action exerted by pedetrian walking on vibrating footbridge ha become of great importance in recent year in order to tudy the phenomenon of ynchronou lateral excitation. The problem ha been generally tackled uing an empirical aroach. Laboratory tet involving a ingle pedetrian walking on a moving platform (e.g. [1]) 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 [3] 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, that can be decribed by two variable, the pedetrian denity and the footbridge lateral vibration. In thi paper the deck acceleration i retained a the vibration variable which motly affect the pedetrian force; - 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 [6] and [7], 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 elf-limited 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 model aume that crowd denity and deck acceleration are given, for intance by mean of in itu meaurement, computational imulation with a crowd-tructure interaction model [7][8] or a priori choen within a wort-cae cenario aroach. On one hand, the model i conceived to 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. 2. Formulation of the model The model i 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. The propoed force model can be acribed to the category of time-domain 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) p where F p 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. F p ha the ame frequency f a the excited lateral tructural mode, while the other two term have the ame frequency f pl a the lateral pedetrian foottep. f pl, which i half the walking frequency f p, i aumed to vary a a function of the walking velocity v (Fig. 1), which i in turn dependent on the crowd denity u [9]. The f p -v relation i derived through a fitting of the experimental data in [10], that i: 3 2 f pl = (0.35v 1.59v + 2.93v) / 2 (2) Fig. 1 Relation between tep frequency and walking velocity Each term of the overall force i weighted on the bai of phenomenological conideration, by mean of three weight, n p, 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 p = ns p = ns ( 1 S p ) (3) = n n n p where S p 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 with the tructure, the preence of a high crowd denity reult in a lateral force that trigger the lateral vibration of the bridge. S p repreent the degree of coupling between the crowd and the tructure. Thi i a function of two variable: the tructure lateral acceleration ζ (to be intended a the envelope of the acceleration time hitory & z& ( t) ) and the ratio f r = f pl / f, where f r i defined in the [0 2] domain: the lower bound depend on the minimum value of f pl (f pl =0), when the
walking velocity i null; the uer bound wa obtained from the laboratory tet reported in [1], 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 p veru ζ (Fig. 2a) i given by a fitting of the experimental data in [2], by mean of the interpolating function: S ζ ) = 1 exp[ b( ζ & z& )] (4) p ( c where & z& c =0.2 m/ 2 i the threhold of motion perception defined in [8]. 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 =2.1 m/ 2 [11]. (a) (b) Fig. 2 S p veru ζ and f r S p (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 (Fig. 2b). S p (f r ) i defined a: S p ( f r ) = exp[ η ( f r 1) ] η( ζ ) = 50 exp( 20ζ / π ) 2 The ynchronization coefficient S p (ζ,f r ) i given by the product of equation (4) and (5). (5) The coefficient S (Fig. 3) 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 u ync + uc S ( u) = 1 + erf a u (6) 2 2 where a=3.14, u c =0.3 ped/m 2 i the uer limit for uncontrained free walking [7] and u ync i the denity that correpond to the total ynchronization of pedetrian. It value i etimated to be 1.8 ped/m 2, according to the maximum denitie recorded on crowd event (e.g. [3]). Fig. 3 S veru u The firt component of the total force, F p, can be written, analogouly to [1], a the um of a component in-phae and another 90 out-phae with repect to the tructure lateral diplacement. The in-phae component can be een a 180 out-phae of the acceleration, while the other can be een a in-phae with the lateral velocity:
F p = n [ F ( ζ )in(2πf t + π ) + F ( ν ) co(2πf t)] (7) p where ν i the envelope of the deck lateral velocity time hitory. The amplitude of the two component (Fig. 4) are defined by mean of piecewie function. 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-phae component [1]. The econd branch i defined qualitatively, baed on the following aumption: i) the DLF reach their maximum when the velocity and the acceleration exceed their erviceability limit (that i, z& =0.25 m/ and & z& =1.35 m/ 2 [11]); ii) the DLF decreae to zero when the velocity and the acceleration reach maximum value, above which pedetrian top walking, i.e. z& M =0.44 m/ and & z& M =2.1 m/2 [11]. Thi trend alo guarantee that the amplitude of F p i elf-limited a i the overall tructural repone. It hould be pointed out that the threhold value uggeted in [11] are baed on a very limited number of oberved cae. In particular, the maximum velocity and acceleration refer to thoe recorded on the Millennium Bridge, while the erviceability value are baed on the field tet performed on the M-bridge in Japan. Recalling that the ratio between the acceleration and the velocity amplitude depend on the natural frequency of vibration f (i.e. ζ/ν =2π f ), it i clear that the threhold value on the velocity and on the acceleration are imultaneouly reached only for a particular frequency. Fig. 4 DLF of the component in phae with ζ (a) and ν(b) The econd force component, F, i defined a: (a) (b) F = n F in( 2πf t) (8) pl 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 [12]. Finally, the component F S i determined according to the model propoed by Matumoto et al. [13], who found that the force due to n uncorrelated pedetrian i n higher than the force due to a ingle pedetrian. Therefore, F S become: F = n F in( 2πf t) (9) pl 3. Alication The propertie of the force model have been evaluated by mean of a enitivity tudy on the two main variable, ζ in the [0 & z& M ] m/2 interval and u in the [0 u M ] ped/m 2 range. The imulation are performed auming a deck acceleration time hitory with contant amplitude ζ: & z ( t) = ζ in(2πf t) (10) It i worthwhile pointing out that 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. The enitivity tudy ha been performed by referring to two cae-tudie, the T-bridge and the outh pan of the London Millennium Bridge, ince they are fully documented. Therefore, the v-u relation, decribed in the accompanying paper [9], ha been characterized for the two cae a for the geographic area and the travel purpoe, that i: Aia and ruh hour traffic for the T-bridge (TB); Europe and leiure traffic for the Millennium Bridge (MB). A ummary of the input data i reported in Table 1.
Table 1 Input data Footbridge v M [m/] u M [ped/m 2 ] γ f [Hz] T-bridge 1.48 7.7 0.273 u M 0.9 Millennium Bridge 1.18 6 0.245 u M 0.8 Fig. 5 a-c-e and Fig. 6 a-c-e 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 for both the cae-tudie: - n p i much more enitive to the deck acceleration 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 (Fig. 5a and Fig. 6a); - n ha the ooite trend to n p : it grow a u increae and a ζ decreae. For u > u ync (S =1), n i complementary to n p (Fig. 5c and Fig. 6c); - 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, i.e. when all the pedetrian are ynchronized (Fig. 5e and Fig. 6e). (a) (b) (c) (d) (e) (f) Fig. 5 Reult for the Millennium Bridge (g) (h)
(a) (b) (c) (d) (e) (f) Fig. 6 Reult for the T-bridge (g) A colour map of the three weight in the u-ζ plane (Fig. 7) 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. (h) Fig. 7 Colour map of the weight of the force component Fig. 5 b-d-f-h and Fig. 6 b-d-f-h 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. Some common feature can be recognized in both cae: - the evolution of F p veru ζ i influenced by both n p and by the ditribution of the DLF (ee Fig. 4), which determine it non monotonic trend and it elf-limited nature with repect to ζ. The dependence of F p on u i almot linear, i.e. it amplitude grow linearly a the number of pedetrian increae (Fig. 5b and Fig. 6b); - 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 pl =0, that i, for u=u M : thi mean that the F component i elf-limited with repect to u (Fig. 5d and Fig. 6d). A imilar conideration can be drawn for F, which alo ha a non monotonic evolution veru u (Fig. 5f and Fig. 6f); - 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 (Fig. 5h and Fig. 6h). Table 2 provide the reult obtained with the model for the actual value of crowd denity and deck acceleration recorded on the cae tudie. The correpondence between the imulated reult and the actual data can clearly be een by looking at the force frequency content. In the cae of the T-bridge, the etimated tep lateral frequency f pl (0.86 Hz) i very cloe to the deck lateral frequency (0.9 Hz), therefore the two component F and F alo excite the firt lateral mode. A for the Millennium bridge, ince f pl i very far from f, it can be tated that the overall force i mainly due to the ynchronization between pedetrian and the tructure. Table 2 Reult obtained with the propoed force model (force in [N/ped]) Input data Output Footbridge u [ped/m 2 ] ζ [m/ 2 ] f [Hz] a(f p ) a(f ) a(f ) f pl [Hz] F rm n p /n n /n T-bridge 1.4 0.34 0.9 3.44 19.0 5.10 0.86 17.46 0.33 0.63 Millennium Bridge 1.4 1.50 0.8 94.3 0.87 1.09 0.66 67.0 0.97 0.03 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. The ingle force per pedetrian [N/ped] obtained with the model i, finally, compared to that predicted with the force model propoed in [3], [14] and [2] (Table 3). 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 p. Table 3 Comparion between the propoed force model and thoe found in literature (force in [N/ped]) Model T-bridge Millennium Bridge Fujino et al. [3] 7 n.a. Nakamura & Kawaaki [14] 4.7 n.a. Dallard et al. [2] n.a. 89.5 Propoed 3.44 94.3 Firt, it i hould be noticed that each model in literature i baed on the data recorded on a particular footbridge and it can not therefore be directly extended to all other tructure. For intance, Nakamura and 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 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 p predicted by the propoed model are very imilar to thoe etimated by Nakamura and Kawaaki for the T-bridge and by Dallard et al. for the Millennium bridge, repectively: thi fact highlight the wide alicability of the model.
4. Concluding remark 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 acceleration 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 [8]; once the footbridge ha been built, the model i able to define the pedetrian force on the bai of the deck acceleration 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. Acknowledgement Thi reearch ha been carried on with the financial uort of IABSE Foundation and of the Italian Minitry of Education, Univerity and Reearch M.I.U.R. within the project Aeroelatic phenomena and other dynamic interaction on non-conventional bridge and footbridge. Reference [1] PIZZIMENTI A.D., Experimental evaluation of the dynamic lateral loading of footbridge by walking pedetrian, 6th International Conference on Structural Dynamic, Pari, 2005. [2] DALLARD P., FITZPATRICK T., FLINT A., LE BOURVA S., LOW A., RIDSDILL R. M., WILLFORD M., The London Millennium Footbridge, The Structural Engineer, vol. 79, No. 22, 2001,. 17-33. [3] FUJINO Y., PACHECO B.M., NAKAMURA S., WARNITCHAI P., Synchronization of Human Walking Oberved during Lateral Vibration of a Congeted Pedetrian Bridge, Earthquake Engineering and Structural Dynamic, No. 22, 1993,. 741-758. [4] ŽIVANOVIC S., PAVIC A., REYNOLDS P., Vibration erviceability of footbridge under human-induced excitation: a literature review, Journal of Sound and Vibration, vol. 279, 2005,. 1-74. [5] FIB Federation International du Beton. Guideline for the deign of footbridge, FIB Bulletin No. 32, Lauanne, 2006. [6] RICCIARDELLI F. Lateral loading of footbridge by walker. Proceeding Footbridge 2005, Venice, 2005 [7] VENUTI F., BRUNO L., BELLOMO N. Crowd tructure interaction: dynamic modelling and computational imulation. Proceeding Footbridge 2005, Venice, 2005. [8] VENUTI F., BRUNO L., Synchronou lateral excitation on lively footbridge. Modelling and alication to the T- bridge in Japan, Proceeding Footbridge 2008, Porto, 2008. [9] BRUNO L., VENUTI F., The pedetrian peed-denity relation: modelling and alication, Proceeding Footbridge 2008, Porto, 2008. [10] BERTRAM J. E., RUINA A. Multiple walking peed-frequency relation are predicted by contrained optimiation. Journal of theoretical Biology, No. 209,. 445-453, 2001. [11] NAKAMURA S. Field meaurement of lateral vibration on a pedetrian upenion bridge. The Structural Engineer, No. 81(22),. 22-26, 2003. [12] BACHMANN H. and AMMANN W., Vibration in tructure induced by man and machine. Structural Engineering Document, volume 3a. IABSE, Zurich, 1987. [13] MATSUMOTO Y., NISHIOKA T., SHIOJIRI H., MATSUZAKI K., Dynamic deign of footbridge. IABSE Proceeding, P17/78,.1 15, 1978. [14] NAKAMURA S., KAWASAKI T. Lateral vibration of footbridge by ynchronou walking, Journal of Contructional Steel Reearch, No. 62,. 1148-1160, 2006.