Multiscale probabilistic evaluation of the footbridge crowding. Part 2: Crossing pedestrian position

Transcription

1 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (ed.) ISSN: ; ISBN: Multicale probabilitic evaluation of the footbridge crowding. Part 2: Croing pedetrian poition Luca Bruno 1, Aleandro Corbetta 2,3 1 Department of Architecture and Deign, Politecnico di Torino, Viale Mattioli 39, Torino, Italia 2 Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Coro Duca degli Abruzzi 24, Torino, Italia 3 Centre for Analyi, Scientic computing and Application Department of Mathematic and Computer Science, Eindhoven Univerity of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherland luca.bruno@polito.it, aleandro.corbetta@polito.it ABSTRACT: In the previou Part of thi tudy, an attempt to etimate the c.o.v. of the pedetrian denity approaching a footbridge ha been made. In analogy to wind engineering, where tatitic on the incoming wind are complemented with the decription of the flow around the obtacle to provide the aerodynamic force on the tructure, the obtained reult i not concluive. In pedetrian dynamic, the denity along the footbridge cannot be confued with the incoming one if the pedetrian dynamic i affected by the walkway feature (ide barrier, walkway geometry, obtacle). On the other hand, differently from fluid dynamic, pedetrian motion i a multicale phenomenon, which can be decribed both at the macrocopic cale (continuou medium) and at the microcopic cale (granular medium). The pedetrian denity decribed in Part I can be acribed to the former. Converely, the uncertain location (in pace and time) of each pedetrian at the entrance of the footbridge mut be ampled and treated at the microcopic cale. Such uncertainty on poition further propagate panwie along the walkway, being tranported by pedetrian themelve. Evidence of thi are available in literature baed either on in-itu obervation or on lab experiment. In thi part of the tudy, the pan-wie propagation of the uncertainty generated by the incoming pedetrian denity i tudied in the framework of the Monte Carlo method. In particular, a tatitical analyi of repeated microcopic imulation of a firt order crowd model accounting for body ize i performed. Inlet condition are precribed on the bai of the tatitic of the crowd denity (decribed in part I), by aigning probability law to the pedetrian chord-wie inlet poition and to the inter-arrival time. The final goal of the model i to etimate the propagation of the incoming variability along the footbridge. The application of the model to an ideal footbridge i provided. KEY WORDS: pedetrian traffic; inter-ubject variability; microcopic model; probabilitic evaluation. 1 INTRODUCTION In a companion paper [1], an attempt to etimate the tatitic of the pedetrian denity approaching a footbridge ha been made. Having in mind a comprehenive uantification of the uncertaintie affecting the footbridge pedetrian traffic, the obtained reult i not concluive. In fact, the pedetrian traffic incoming the footbridge cannot be confued with the one along it. Such demarcation i reuired becaue the latter i expected to be enitive to two main feature of the problem. The firt one include the parameter that define the walkway geometry (e.g. walkway width or longitudinal alignment, ide barrier, obtacle along the walkway). Walkway departing from ideal one (obtacle free, traight, with contant width) are expected to modify the pedetrian flow analogouly to fluid dynamic phenomena. Some effect of the walking geometry on the pedetrian traffic along footbridge have been recently decribed in determinitic term in [2] [3] [4] and in a probabilitic perpective in [14]. The econd cla of feature affecting the pedetrian traffic along the footbridge refer to the inter-ubject and intraubject variability of the pedetrian at inlet. In general, it i worth pointing out that the incoming variability effect hold whichever i the walkway geometry; lie at the pedetrian cale (microcopic) but can involve collective effect at the footbridge cale; cannot be decribed, in a determinitic nor probabilitic ene, by macrocopic uantitie uch a the incoming global denity decribed in the companion paper; have genuinely tochatic origin. In particular, inter-ubject variability include, beide body feature and walking parameter - already widely tudied in literature - the uncertain chord-wie poition and the arrival time (i.e. the relative ditance of two euential walker) of each pedetrian at the entrance of the footbridge. The pioneering work of Matumoto et al [6] ugget the arrival time follow the Poion ditribution and further in itu meaurement [7][8] confirm thi conjecture, while tatitic about the chord-wie poition are not available to the Author knowledge. The uncertainty on the poition in pace and time at inlet further propagate along the walkway, being tranported by the pedetrian themelve along uncontrained trajectorie (differently from e.g. trolleybu on railway or car in lane). Some laboratory [8][9][10] or in-itu [10] obervation of uch phenomenon area available in literature, thank to the recent advance in computer viion techniue applied to the analyi of pedetrian trajectorie. Some of the trajectorie 937

2 obtained by [9] and [11] in lab tet and in-itu obervation, repectively, are included in Figure 1 and briefly commented in the following to provide a phenomenological bai of the tudy. In the laboratory experimental etup [9], each pedetrian tart to walk from the inlet to the outlet of a corridor where he/he i intructed to turn back and then move again in the oppoite direction. After about four trip, the pedetrian leave from where he/he enter the corridor. Trajectorie of a ingle pedetrian and the complete et of meaurement are hown in Figure 1(a)-(b), repectively. Figure 1(c) how ome trajectorie recorded in the central egment of an indoor U-haped walkway at the TU/e campu during everyday traffic [11]. [m] [m] (c) Figure 1 - Walking trajectorie of a ingle pedetrian (a) and complete et of the recoded trajectorie (b) (after [1]).Walking trajectorie in indoor U-haped walkway at the TU/e campu (c) (after [11]). Even if accurate tatitic on the trajectorie are till not available in literature, ome ualitative and comparative remark can be drawn from the phenomenological obervation: both the effect of intra-ubject variability (Figure 1 a) and inter-ubject variability (Figure 1b-c) can be eaily recognized; the inter-ubject variability i much higher than the intra-ubject one, a expected; inter-ubject variability look higher in real world condition (Figure 1c) than in laboratory tet (Figure 1b). Among the conjectured caue, the arrival time variability i upected to affect the one of the trajectory ditribution in pace. The preent tudy aim at complementing the companion one by the probabilitic evaluation of the pedetrian traffic along footbridge. Becaue of the tate of the art critically dicued above, the tudy focue on incoming pedetrian variability, and in particular on the inter-ubject one of the chord-wie poition and arrival time. On one hand, having in mind the pedetrian cale at which variability take place, the pedetrian traffic along the footbridge i modelled by a genuine microcopic model. On the other hand, in order to meet practical need of a final coare-grained and probabilitic decription of the crowd dynamic, the model reult are ubeuently proceed by defining macrocopic uantitie (e.g. denity, flow) and their tatitic in the framework of a Monte Carlo like method. In order to point out 938 and iolate the effect of the incoming inter-ubject variability, the adopted etup refer to an ideal walkway having no obtacle, traight alignment and contant width, while common walking feature for every pedetrian are retained. 2 PROBABILISTIC ANALYSIS FRAMEWORK In the firt Part of thi tudy, we pointed out the preence of an intrinic uncertainty aociated to the uantification of the global crowd denity incoming a footbridge. Such uncertainty cumulate with other and i naturally propagated, i.e. advected and poibly modified, by the complex pedetrian motion. The aim of thi ection i to introduce in a general etting an analyi framework addreed to the probabilitic characterization of the crowd traffic along the footbridge face to inter-ubject variability at inlet. The developed framework allow one to obtain the probabilitic/macrocopic decription of crowd traffic on the bai of given macrocopic inflow data via a microcopic crowd model. Hence, the final probabilitic analyi i poible after two conecutive cale tranition (ee Figure 1). Given macrocopic data about the incoming pedetrian traffic (e.g. the global denity tatitic provided in [1]), the Macroto-micro (M2m) tranition allow to enrich them by including inter-ubject variability at the proper pedetrian cale and to convert them into a microcopic form. Such form i uitable to be handled by the direct microcopic modeling and imulation of the pedetrian traffic: the microcopic model i the pivoting core of the approach, allowing one to handle and decribe the propagation of uncertain condition form inlet along the pan. The micro-to-macro (m2m) tranition i finally accomplihed to obtain the deired macrocopic probabilitic evaluation of the crowd dynamic. Figure 2 - Scheme of the multicale probabilitic analyi framework Remarkably, the introduced framework omewhat reflect at the procedural level the intrinic multicalarity of the crowd ytem, in which collective macrocopic phenomena are generated by the individual (i.e. microcopic) motion of ingle [20]. The following two Section detail the component of the framework introduced above: Section 3 i focued on the microcopic model, while Section 4 decribe the cale tranition from the macrocopic viewpoint to the microcopic one and vice vera.

3 3 MICROSCOPIC MODEL OF THE PEDESTRIAN TRAFFIC The aim of thi ection i to introduce the mathematical model for pedetrian dynamic conidered throughout the paper. The model i firtly deducted in a fully general fahion, agnotic of the pecific crowd cenario or geometric etting; then, feature needed to deal with pecific cenario conidered in Section 5 are decribed. According to the conidered mathematical model, the dynamic of pedetrian i treated from a microcopic perpective in a two dimenional etting. Specifically, the crowd i conidered a a collectivity of active particle which act, namely move, to atify peronal deire and need e.g., the neceity to reach a given target location [11]. Moreover, during the walking proce, uch active particle interact one another and with the environment. The mathematical tructure of the microcopic model conidered follow the general modeling framework introduced in [20] and [24] and further developed for elongated geometrie in [14]. According to thi framework, given N individual whoe poition are defined by the vector z1 = ( x1, y1), z2 = ( x2, y2 ),, zn = ( xn, yn ), the velocity of the j-th individual i defined by vj = dz j dt = v d ( z j ) + v ( z j ; z1, z 2,..., z N ), (1) where, repectively, vd, the deired velocity, repreent the velocity that ingle peron would keep in abence of other (i.e., it i a fixed background velocity field) and v, the ocial velocity, account for the interaction that the individual ha with the urrounding collectivity (for further detail on the concept of deired and ocial velocity ee, e.g., [13]). (a) (b) Figure 3 - Normal component of the deired velocity in the proximity of the boundarie. Scale ditance (a). Magnitude of vdw (x-axi) a function of the ditance from the wall (on y axi) (b). In Section 5, pedetrian croing cenario happening in traight and elongated walkway are conidered (ee alo Figure 1); in that contet, we chooe the following expreion for the free velocity vdf = vm (1,0). (4) being vm i the maximum allowable walking velocity. The ocial velocity term v, on the other hand, provide a model for the fact that in normal crowding condition (i.e. when no-panic occur), individual - while reaching their target - interact with peer that are ufficiently cloe to avoid them. Interaction are uually repulive-like, limited in range to the o called enory region and aniotropic [15]. Actual body hape of the interacting pedetrian mut alo be repected; in fact, conidering hapele (e.g. point-wie) pedetrian could lead to unphyically large occupancie or denitie. In order to give a pecific form to the conidered term, we plit them in impler, additive, contribution. Specifically, we write the deired velocity a a um vd ( z j ) = vdf ( z j ) + vdw ( z j ), (2) where vdf i an expreion of the free deired velocity (i.e., in abence of geometrical contraint), and vdw add uitable impermeability contraint to boundarie, wall and obtacle. We choe to model impermeability contraint via a repulive perturbation of the deired velocity around obtacle. In particular, we et α w vd ( z j ) = 0 w wall d w ( z j ) d ( α ) ( β d w0 d0 )β nw, (3) + where the um i carried over every wall w. Repectively, d w i the ditance between the conidered pedetrian and w, d w0 i 0 the interaction range of the wall and d i the cale ize of the human body (ee alo Figure 1). Figure 4 - Scale ize of enory region. In agreement with the framework [20], we hypotheize linearly additive pairwie interaction between individual, i.e. K (z ; z ); v ( z j ) = j (5) j following [16], we expre the pairwie interaction between individual j and individual by the function 1 1 K z j ; z = v m 1 exp γ, (6) r z j, z δ z j, z + ( ) ( ) ( ) where 939

4 4 r ( z j, z ) = z z j MULTISCALE ASPECTS In thi ection the two cale tranition introduced in Section 2 are detailed. Repectively, micro-to-macro tranition i detailed in Section 4.1, while tranition macro-to-micro i detailed in Section 4.2. δ ( z j, z ) = d 0 + δ f (1 + co(ϑ (vd, z z j ))) [ x]+ = x[ x > 0] ϑ (a, b) = angle between a and b (ee Figure 1, and Table 1 for numerical value of parameter). Figure 5 - Level et of intenity of pairwie interaction of pedetrian in poition z = (0,0) normalized to v m. A previouly reuired, according to Euation [6], the aniotropic interaction i repulive and decreaing. Moreover, the interaction i larget in a central region having the characteritic ize of the human body ( d 0 ). Thi contruction i aimed at keeping into account, although in a oft-core fahion [25], the cale of the human body and the naturally limited poibility of body-on-body uperpoition. Furthermore, the interaction region ha global characteritic ize d 0 + δ f, which enure the expected boundedne of the 4.1 From macrocopic incoming denity to microcopic inflow condition The common notion of pedetrian inflow or pedetrian denity at domain boundarie [18] genuinely own to the macrocopic cale; a uch, they cannot be directly impoed to microcopic ytem. Microcopic ytem, in fact, reuire finer-grain time and pace information pecified on a pedetrian bai which mut be a realization of the original macrocopic data. Finer microcopic information realizing the macrocopic notion of inflow can be obtained by characterizing probabilitically the pedetrian arrival proce. Specifically, given a macrocopic pedetrian inflow per unit length (or an inlet denity ρ 0 ), and uppoing the arrival proce through an inlet chord (of ize B) to be Poion, we have that B 1, < Δt > (7) where the variable Δt Exponential (λ ) (8) repreent the inter-arrival time between two conecutively entering pedetrian, i.e. λ = (B) 1. (9) interaction. It i worth pointing out that the parameter introduced are not fixed once and for all, rather they depend on anthropologic and ethnical apect uch a average body ize and average allowed courtey ditance. Finally, the free parameter γ which determine the rate at which the interaction decay, allow one to model different kind of traffic (ee alo [19]). In Table 1, the choice of parameter for the microcopic model ued in Section 5 i reported. Euation 8 define probabilitically an entrance chedule for the conidered pedetrian; hence, time-like information ha been matched through the cale. Probabilitic enrichment of the chord-wie entering ite can be introduced by defining a probability law for the poition along the chord. For the ake of implicity, in the following, a chord-wie homogeneou entrance behavior i adopted. Specifically, let y 0 be the chord-wie entrance poition, we aume that Table 1. Value of parameter ued throughout the imulation y0 Uniform( B / 2, B / 2). Parameter vm Value 1.34m/ Parameter δf γ 6m α δ f β d0 d w0 Value 0.5m 10 m1+β / m It i worth pointing out that the conidered velocity i continuou (in mathematical ene) with repect to pedetrian poition; thi enure well-poedne of the aociated initial value problem. Moreover, in order to integrate reliably the euation of motion and thu being conitent with the complex interaction dynamic choen, multitep AdamBahforth-Moulton implicit predictor-corrector cheme ha been ued. 940 (10) In cae an inlet crowd denity ρ 0 i given intead of the influx B, a uitable fundamental diagram can be ued to convert the denity information into the correpondent flux. 4.2 From microcopic reult to macrocopic probabilitic evaluation of the crowd dynamic Once the entrance proce ha been pecified, the model in Euation (1) can be ued to imulate the conidered crowd event. The data generated by the imulation can be ued to repreent a ueful bae to proceed toward the macrocopic and probabilitic characterization. In particular, given a partition of the domain in Q region C1, C2,..., C..., CQ, and given a regular time ampling t1, t 2,..., t...with tep Δt, we conider the random variable

5 ρ = { # pedetrianin C at time t } Area (C ) = # P (11) Area (C ) which uantify the local crowd denity in region at time. Generally peaking, other kinematic variable φ can be conidered, e.g., net pedetrian fluxe (i.e., Q = { } { # ped. entering C at time t # ped. exiting C at time t Δt } = # P +1 # P Δt ) average velocitie, load and o on. In order to obtain empiric etimate of the law of the variable φ, extenive microcopic crowd imulation in a Monte Carlo-like etting i needed. The obtained collection of empiric law indexed by both region () and time () give thu a probabilitic picture of the crowd event along the footbridge. In Section 5, we conider tationary crowd croing event, i.e. we expect the behavior of pedetrian (once a poible tranient regime i exhauted) to be independent on time. In 5.1 Application etup We call elongated thoe domain featuring a mall apect ratio B/L among the chord B and the pan L (in the conidered cae L=200m and B=4m). Thee domain, for intance, provide geometrical model for lender walkway and pedetrian footbridge. In thi geometric etting, imple tationary croing event are conidered, namely involving pedetrian entering from one ide and leaving from the other ide after a regular walk (i.e. no paue or change of ultimate direction are expected, ee Figure 1). The intauration of a uai-tationary crowd flow with a random component dependent on the arrival proce i thu expected. In order to perform the analyi detailed in Section 4.2, we conider a regular pan-wie partition of the domain in 40 eual egment ( C0, C1,..., C..., C39 ) of length 5m. Several pedetrian inflow (B = {.6, 1.2, 1.8, 2.4, 3.0, 3.6} ped/) are conidered; they are obtained by etting λ = 1 / B in Euation 8. thi etting, we can introduce the new variable φ obtained from φ via a ummation (marginalization) over time: ( ) Pr ob(φ ) Pr ob φ = (12) Hence, according to thi framework, the variable uantifie probabilitically the denity in region independently on the pecific intant of time. ρ, Once the law of the variable φ are obtained via imulation, they can be approached in a more practical way by uing their ynthetic tatitic. In particular, the following region-wie tatitic are conidered a metric of the crowding tatu: mean m (φ ) =< φ >, tandard deviation td (φ ) = td (φ ), cov (φ ) = cov(φ ) coefficient = td (φ ) m (φ ) of variation th and region-wie 95 percentile. Repectively, aide from the mean behavior expreed φ95 Figure 6 - Conidered rectangular domain and relative longitudinal ubdiviion in 40 egment having length 5m 5.2 Reult dicuion To check convergence for the probabilitic crowd data conidered, a pecific tudy i carried out. In particular, given the partition of the domain in Figure 1, we conider the indicator Δm which i the maximum - computed upon all the egment - of the relative difference between average denitie in two ucceive intant of time. In formula it read m ( ρ ) m ( ρ 1 ) Δm = max m ( ρ ). (13) by the firt term, the econd and the third term are ued a index of uncertainty diperion (repectively abolute and relative to the mean), while the fourth one i ued a indicator of extreme cae. 5 APPLICATIONS AND RESULTS In thi ection, the framework introduced in Section 2 and developed in Section 3 and 4 i applied to crowd croing event happening in a imple elongated geometry. A imple geometry i choen in order to iolate and evaluate contribution to the uncertaintie genuinely due to the crowd traffic. In other word, we want to neglect any uncertainty coming from the interaction among pedetrian and the pecific geometry of the walkable built environment (bottleneck, corner, croway, etc.). Figure 7 - Maximum relative difference between average denitie acro time. 941

6 In the conidered cenario, the threhold Δm <10 3 ha been targeted. In other word, the microcopic imulation are carried out for an interval of time long enough uch that the relative variation on the crowd average denity i maller than 3 10 (ee alo Figure 1). Remarkably, with the model parameter lited in Table 1, the model predict a crowd tationary behavior comparable with etablihed literature data. In particular, if we conider the crowd flux, or average velocity, againt the average denity, a behavior cloe to the one decribed by the fundamental diagram introduced by Kladek ([23] and [19]) in leiure traffic regime i recovered (ee Figure 1). It i worth noticing that the explored combination of fluxe and denitie refer to the firt part of the fundamental diagram, i.e. the conidered denitie are below the capacity threhold. Figure 8 - Comparion between model behavior (average and abolute diperion) and Kladek fundamendal diagram in term of flux-denity (a), velocity-denity (b). When the macro to micro converion decribed in Section 4.1 i operated, an uncertainty i naturally introduced in the flow. Such uncertainty get propagated by pedetrian to different extent depending on the influx. We chooe to uantify the uncertainty propagation in term of the regionwie coefficient of variation of the crowd denity. Specifically, we conider both the abolute coefficient of variation ( cov ( ρ) ) and the one normalized to the inlet value (a) ( cov ( ρ)/cov0 ( ρ) ). Thi i aimed at uantifying repectively the abolute uncertainty a well a the one genuinely due to the dynamic which follow the pedetrian entering act. In Figure 1, we report the probability law of the variable ρ z for the conidered value of pedetrian inflow. The law (on the y axi) are reported a io-contour of the probability denity function depending on pan-wie egment (x axi). A further clarified by ynthetic tatitic in Figure 1, the mean crowd denitie m (ρ) grow monotonically when the influx increae. On the other hand, the egment-wie tandard deviation td (ρ), meauring the abolute diperion of the denity, remain approximately contant independently on the inflow. In Figure 9, 95 th percentile are conidered a well; they are an indicator of extreme denity cae. (b) Figure 9 - Probability denity function of crowd denity along the facility pan for the conidered pedetrian influx value. Bearing in mind the firt Part of thi work, in which coefficient of variation for tructural parameter have been etimated, everal obervation can be made. Coefficient of variation obtained range among 0.1 and 0.5: remarkably, thee value are in the ame range or even larger with repect to the coefficient of variation actually recognized in probabilitic force model (e.g. the one related to tructural damping, pedetrian body or walking parameter, ee Part 1, Table 1). A a coneuence, uncertaintie originated by pedetrian kinematic are relevant from a deign point of view and hould be taken into account when pedetrian facilitie are concerned. It i worth noticing that the coefficient of variation ha a decreaing monotonic trend when the denity increae (ee Figure 1). Thi trend can be related to the pecific choice of crowd model. In fact, according to the modeled local pairwie interaction between individual, pedetrian attempt to avoid one another; a a coneuence, their poition give rie to organized collective pattern. When the macro to micro converion decribed in Section 4.1 i operated, an uncertainty i naturally introduced in the flow. Such uncertainty get propagated by pedetrian to different extent depending on the influx. We chooe to uantify the uncertainty propagation in term of the regionwie coefficient of variation of the crowd denity. Specifically, we conider both the abolute coefficient of variation ( cov ( ρ) ) and the one normalized to the inlet value ( cov ( ρ)/cov0 ( ρ) ). Thi i aimed at uantifying 942

7 repectively the abolute uncertainty a well a the one genuinely due to the dynamic which follow the pedetrian entering act. In Figure 1, we report the probability law of the variable ρ z for the conidered value of pedetrian inflow. The law (on the y axi) are reported a io-contour of the probability denity function depending on pan-wie egment (x axi). Figure 10 - Statitic of crowd denity ditribution along the facility pan for the conidered pedetrian influxe. A further clarified by ynthetic tatitic in Figure 1, the mean crowd denitie m (ρ) grow monotonically when the influx increae. On the other hand, the egment-wie tandard deviation td (ρ), meauring the abolute diperion of the denity, remain approximately contant independently on the inflow. In Figure 9, 95 th percentile are conidered a well; they are an indicator of extreme denity cae. Figure 11 Span-wie average coefficient of variation of crowd denity: abolute and normalized to inlet value. Data ha been fit with exponential function. Figure 12 - Effect of the tochatic entering chedule on the emergence of poition pattern - influx F=3.6ped/. Diordered condition cloe to the inlet (5m<x<20m, t=100) (a). Pattern at different time in 100m<x<115m - t=540 (b), t=740 (c), t=840(d). It i worth to remark that the coefficient of variation obtained, already in range with the one in probabilitic force model, have been evaluated by avoiding geometric effect and by uing a model which tend to regularize pedetrian poition. A a coneuence, the obtained value are expected to be lower bound to actual value to be obtained e.g. experimentally. Indeed, every perturbation introduced in the flow by pedetrian (e.g. pedetrian top walking or pedetrian walking in oppoite direction) or by the walkway geometry (e.g. narrowing/widening of the walkway, obtacle along it) i conjectured to increae the uncertainty preent in the flow and hence enlarge the coefficient of variation involved. 6 CONCLUSION (b) (c) (d) In thi paper, the uncertain poition of incoming pedetrian, further propagated along the pan, ha been tudied via a probabilitic analyi framework. Such framework i grounded on Monte Carlo like microcopic pedetrian imulation whoe outcome are detailed in term of local-in-pace tatitic of crowd obervable (e.g., denity, flux, velocity, etc.). The ource of propagated uncertainty i inter-ubject variability between individual (for a comprehenive dicuion on uncertainty referring to pedetrian incoming a facility, one can refer to the companion paper [1]). It i modeled by a tochatic inflow condition in term of Poion ditributed arrival time and uniformly ditributed chord wie poition. Specifically, we choe to uantify and characterize uncertainty in term of local-in-pace coefficient of variation of the crowd denity. Such coefficient of variation expre the denity diperion with repect to it mean value in different region of the domain. Even for regular crowd event in imple cenario - where the geometry i not expected to play a role - ignificant value of c.o.v. in the range have been obtained. Hence, they are comparable with the coefficient of variation actually recognized in probabilitic force model in tructural engineering. Such finding confirm the key role played by pedetrian traffic related uncertaintie. ACKNOWLEDGMENTS The Author warmly acknowledge Andrea Toin and Vittorio Nacé for the timulating dicuion along the tudy, and Federico Tochi and Alex Liberzon who collaborated to the in progre in-itu experimental meaurement. (a) 943

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