SIMULATION OF COUNTER FLOW PEDESTRIAN DYNAMICS IN HALLWAYS USING SPHEROPOLYGONS INTRODUCTION

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1 SIMULATION OF COUNTER FLOW PEDESTRIAN DYNAMICS IN HALLWAYS USING SPHEROPOLYGONS Ferado Aloso-Marroqu 1, Cela Lozao 2, Álvaro Ramírez-Gómez 3, ad Joatha Busch 1 1 School of Cvl Egeerg, The Uversty of Sydey, Sydey, NSW, Australa 2 School of Appled Mathematcs ad Physcs, The Uversty of Navarra, Spa 3 Uversdad Poltécca de Madrd, Spa * Correspodg author Ferado Aloso-Marroqu: ferado.aloso@sydey.edu.au We developed a method for smulatg pedestra dyamcs a large, dese crowd. Our umercal model calculates pedestra moto from Newto s secod laws, takg to accout vsco-elastc cotact forces, cotact frcto, ad groud reacto forces. I our computer smulato, o-sphercal shapes (spheropolygos) modelled the postos of the chest ad arms the packg arragemet of pedestra bodes based o a cross-sectoal profle usg data from the US Natoal Lbrary of Medce. Motve torque was take to arse solely from the pedestras oretato toward ther preferred destato. The objectve was to ga sght to a tragc cdet at the Madrd Area Pavlo Spa, where fve grls were crushed to death. The cdet took place at a Hallowee Celebrato 2012, a log, desely crowded hallway used as etrace ad ext at the same tme. Our smulatos recostruct the mechacs of cloggg the hallway. The hypothetcal case of a total evacuato order was also vestgated. The result hghlghts the mportace of the pedestras desty ad the effect of couter flow the oset of avalaches ad cloggg, ad provdes qualtatve measures of the umber of jures based o a calculato of the cotact force etwork betwee the pedestras. INTRODUCTION Pedestra dyamcs modellg has bee a subject of mathematcal vestgato sce the 1950s [1; 2], ad was orgally used by Hak ad Wrght to study the movemets of pedestras the Lodo Subway to allow for more effcet desg of stars ad corers. The study of pedestra pac, however, bega later the 1970s [3] respose to several of hstory s most horrfc crowd stampedes. The most otable of these occurred Chogqg, Cha, durg World War II. O 6 Jue 1941 a ar rad shelter was evacuated followg a Japaese rad, ad approxmately 4,000 people were klled a resultg pac-drve stampede. The majorty of fataltes were due to asphyxato [4]. The stes where the Muslm plgrmage, or Hajj, coverge o Mecca have a log hstory of deaths due to pedestra pacs. Whe plgrm luggage splled over Mecca s arrow Jamarat Brdge 2006, the crowd was set to pac; at least 345 people were klled, ad 289 were jured [5]. Every year hudreds of people are hurt or lost crowd dsasters. Sce Helbg et al. [6] publshed o termttet flow ad cloggg evacuato processes 2000, there have bee may publcatos modellg pedestra behavour: cellular automata models [7 9], addressg couter flows [7, 8, 10], or force models [11]. Our am s to detfy varables that are helpful for advace warg of crowd dsasters: the mportace of the shape, desty, ad couter flow.

2 I ths paper, we preset a study of pedestra dyamcs uder pac to shed lght o a tragc cdet Spa o 31 October 2012, at the Madrd Area pavlo. The veue was already hghly overcrowded whe more people etered, ad securty persoel faled to cotrol the dstrbuto of the crowd, leadg to the crushg death of fve teeagers whe the passageway to the cetral court was blocked. A few people attempted to leave the court; but a large umber excted by the musc, a desre to be a part of the acto, ad alcohol charged the opposte drecto. I the semdarkess, people were uable to see what lay frot or behd. Securty cameras recorded as may as fve avalaches of people at the three-meter-wde passageway, the last bottleeck lastg for may mutes. The cameras revealed that oe of several freworks explosos set off more pac, wth smoke ad sparks t was tally thought that trggered the dsastrous fal bottleeck. The cdet was characterzed by shovg, chaos, ablty to move ay drecto, rsg temperatures, claustrophoba, screams, pac attacks, ad wag oxyge. Some foud themselves lfted off the groud, supported by elbows agast ther rbs. Those of smaller buld, partcularly grls, were draw dow uder the crush of bodes. Autopses o the fve deceased foud suffocato ad cosequet bra damage as causes of death. Icdets lke these hghlght the mportace of accurate pedestra dyamcs to form safe desg of hallways, ad more specfcally the desg of corrdors ad recommedatos o lmtatos for the movemet of people through them. Ths paper s orgased as follows. Secto 2 descrbes the spheropolygo-based model used to perform the smulatos; Secto 3 descrbes the Madrd Area veue ad the partcular crcumstaces precedg the tragc cdet; Secto 4 cludes the results of smulatos the hallways were the fataltes happeed. We also calculate the cumulatve dstrbuto of cotact forces the hallway, ad compare t wth the dstrbuto the case of evacuato of the whole veue uder pac. Coclusos ad dscusso of our results are preseted Secto 5. FORMULATION OF THE PEDESTRIAN DYNAMICS MODEL The umercal study coducted ths paper s based o a cotact force model for pedestras. The force equatos are mplemeted a spheropolygo-based model prevously used to smulate graular materals [12 14] Force model The velocty v of the pedestra ca be determed accordg to Newto s secod law, dv F =m (1) dt where t s the tme, m s the mass of the pedestra, ad F s the force actg o the pedestra, derved as show equato (2):

3 F =F +F + F (2) 0 b c j c The frst two terms are groud reacto forces exerted o the pedestras by the floor, ad c F c j s the sum of cotact forces produced by other pedestra ad obstacles. The frst term, 0 F, deotg the self-drve compoet of a pedestra s moto [6]: 0 m F = (v0e v ) (3) τ Here, τ s the tme requred by the pedestra to reach the maxmal velocty ad v 0 the desred speed. The ut vector e s the desred drecto of moto of the pedestra. The thrd term equato (2) s the sum of all teractos wth other pedestras, walls, colums or obstacles. Each teracto force of the cotact c betwee partcles ad j s calculated as where the elastc forces are gve by c e, e,t v, v,t F j=f j +F j +F j +F j (7) F =-k δ e, j j j F =-k δt e,t t j t j j (8) (9) ad, j ad t j are the ormal ad tagetal ut vectors. The scalar δ j s the overlappg legth ad deotes the vertex-to-edge dstace betwee the two partcles. The scalar deoted by t δ j accouts for the tagetal elastc dsplacemet gve by the frctoal force, ad satsfes the sldg codto by F e,t j μf (10) where μ s the coeffcet of frcto. Here, e, j k ad k t deote the ormal ad tagetal coeffcets of stffess. The last two terms o the rght sde of equato (7) accout for the coeffcet of resttuto betwee pedestras [14], whch are show equatos (11) ad (12). F =-m γ v v, j j j j F =-m γ v t v,t t j j t j j (11) (12) where the mass, m=ρa (13), the mass of the partcle, ad mj mm j/(m +m j) (14) s the effectve mass of the two partcles cotact. The desty s deoted by ρ; ad A deotes the area of the 2D partcle. The ormal ad tagetal coeffcets of dampg are respectvely

4 gve by γ ad γ t ; velocty. v j ad t v j deote the ormal ad tagetal compoets of the cotact b Addtoally, F s the back force actg o each pedestra. Ths force captures the avodace behavour betwee pedestras, whe pedestras alter ther drecto to avod collso wth ocomg pedestras. It s a groud reacto force, wth two compoets: Oe s related to the teto of the pedestras to step to ther rght to avod collso, ad ts drecto s the cross product e k where k s the ut vector ormal to the floor. The secod compoet s a top-up force the desred drecto e. Ths force s related to the resstace of the pedestras to beg pushed backwards by the crowd. I stuatos of pac, whe alcohol ad crowded hallways may be volved, ths back force wll be larger tha would be observed ormal socal codtos. The back force s mportat, because t s resposble for stream formato, whch s a attrbute of crowd dyamcs both socal ad pac stuatos. Equato (15) descrbes the back force: b mv 0 F= Θ( e.v ) {e +e k} (15) τ where the Heavsde step fucto Θ( x) returs 1 f x>0, ad zero otherwse. Ths fucto actvates the back force oly whe the pedestras are pushed back from ther desred drecto. Shape represetato (a) (b) FIGURE 1: (a) Boudary of the cross-secto chest derved from US Natoal Lbrary Image of Thorax, ad (b) represetato of the thorax shape usg a spheropolygo wth 18 vertces ad a sphero-radus of 0.05 m. No-sphercal shapes have bee cosdered the pursut of a ever-more accurate descrpto of pedestra mechacs [15 18]. I catastrophes volvg hgh destes, the packg arragemet of pedestras more closely reflects the cross-sectoal arragemet of the chest ad arms rather tha the head. The head has a degree of flexblty, that t ca rotate to avod collsos, ad ca ft, from a overhead vew, wth the area of the thorax

5 ad arms. Ths s cosstet wth a revew coducted from publcly avalable footage of pedestra dsasters [17 20]. To obta a more accurate pedestra shape, a chest cross-secto of a huma thorax cludg the arms was obtaed from the US Natoal Lbrary of Medce, whch was recoloured Matlab to solate the surface. The teral compoets of the cross secto where removed from the recoloured mage, ad the mage was scaled to reflect a typcal sterum wdth of 200 mm. Fgure 1 showsthe processed mage. Ths outle s represeted usg a spheropolygo, whch s the fgure resultg from sweepg a dsk to a polygo, see Fgure 1b. The advatage of the spheropolygos s that the cotact forces amog them ca be effcetly calculated from ther vertex-edge dstaces, as s explaed [12] Torque he methods proposed by Korohoe et al. [15 16] ad Lagsto et al. [18] wll be used as a foudato for refg the rotatoal equatos of moto. The groud reacto torque wll be cosdered as arsg solely from the pedestra s desre to face toward ther preferred destato, as from [15], appled by the legs at a typcal dstace of 20 cm. Wth the torque whch acts o the pedestra, there s a torque correspodg to the drecto of moto, called motve torque, m T, ad a dampg compoet T v. The force compoets the motve torque s appled by the legs, ad they allow the pedestra to rotate the drecto of desred moto. As such, the torque arsg from ths movemet s gve by (16), whch s a dsambguato of the model costat of the approaches used by Lagsto et al. ad Korhoe et al. [15 16,18]. The λ term equato (16) s a dmesoless costat that s derved expermetally, θ D= (v ) represets the agle to the walkg drecto, ad θ s the pedestra s oretato. z m λi v T = 1 exp( ) (θ θ 2 D) (16) τ v0 The expoetal fucto s used to accout for lower motve torque at lower speeds. Ths s a reflecto of the fact that pedestras ca freely rotate f ther velocty s low. The dampg compoet of the torque reflects the pedestras desre ot to rotate. Usg a smlar framework to Helbg s self-drve force model [6], whch a desred walkg velocty determes the fal speed of the pedestra, a desred agular velocty was created. As the pedestra prefers ot to rotate, the desred agular velocty was set at zero. As a cosequece, the pedestra comes to a comfortable stop after a collso rather tha cotug to sp. Equato (17) descrbes ths dampg force, where s the relaxato tme of rotato, ad z I s the momet of erta: z v I T= ω (17) τ

6 The et torque actg at tme t o pedestra, s therefore s descrbed equato (18): T =T +T + F (18) m v c j j j> The last term ths equato s the torque produced by the cotact forces. The term j s the brach vector coectg the ceter of mass of the pedestra wth the pot of applcato of the cotact force. Cotact parameters The parameters of the cotact force model were chose as follows: the thorax resposes to 5 forces k =10 N/m are take from medcal data o thorax deformato [21]; 4 k 6=10 N/m leads to a bulk Posso rato of 0.3; μ=0.4 s close to the coeffcet of frcto betwee peces of cloth fabrc. The dampg parameter -1 γ =10s was chose usg the formula of the coeffcet of resttuto [14] betwee colldg partcles, show equato (19). ε = exp - πγ /2 2 k /m j - γ /4 (19) Sce ot much formato s avalable regardg the coeffcet of resttuto betwee pedestras, a pedulum test was coducted to derve the parameter expermetally. The test was coducted usg observatos of collsos betwee pedestras suspeded by ropes by haresses a rock clmbg veue. By measurg the veloctes before ad after mpact, ad eglectg the partcle rotato after collso, we obtaed coeffcets of resttuto wth the rage Detals of methods to establsh the coeffcet of resttuto from veloctes ca be foud the lterature o graular meda [14]. I the model, a coeffcet of resttuto of ε 0.3 was used, whch allowed the calculato of the dampg parameter from equato (11). The reacto tme τ=2/3 s was used, ad the termal velocty was radomly chose betwee 0.8 ad 1.2 m/s. The surface desty of the pedestras s 3 2 ρ=10 Kg/m ; mass of the pedestras were uformly dstrbuted the rage kg. A summary of the key model parameters s provded below Table 1. Numercal soluto of the equato of moto The -house object-oreted computer program SPOLY was used to coduct the smulatos. SPOLY smulates the pedestra dyamcs based o spheropolygos ad a fve-order predctor-corrector umercal tegrato [12]. The use of a eghbour table ad Verlet dstaces allows real-tme smulato of log hallways wth up to 400 pedestras. For largescale smulatos wth aroud te thousads pedestras the algorthm executes aroud oe mute of smulatos per hour. Detals relatg to the SPOLY code are foud [13 14].

7 TABLE 1 SUMMARY OF MODEL PARAMETERS Parameter Uts Parameter ame Value Commet k N/m Normal stffess 1x10 5 Expermetally calculated usg forcedsplacemet relato k t N/m Tagetal stffess 1x10 4 Numercally calculated usg elastc aalyss of two partcles μ dmesoless Coeffcet of frcto 0.4 Obtaed from tables of frcto betwee clothes ϵ dmesoless Coeffcet of resttuto 0.4 Expermetally calculated usg observatos of pedulum test λ dmesoless Torso stffess 27 Ftted from comparsso of smulato wth vdeo footage. Γ dmesoless Back force parameter 1.2 Ftted from comparsso of smulato wth vdeo footage. EMPIRICAL OBSERVATIONS The sghts of ths study to crowd pac behavour are mportat for the orgazato of safer mass evets as the Hallowee party the Madrd Area. Ths s a door area located Madrd, the captal cty of Spa. It s dstrbuted o three floors (access, termedate, ad low). It was plaed to house basketball compettos ad later featured electroc musc artsts ad a costume cotest. The groud floor s the ma dace floor, ad ts dmesos are 55 x 35 m 2 ; see Fgure 2. We recreate the Madrd Area veue order to set the scee. The ma problem was that the party s orgazers were allowed to sell 9,000 tckets but may more people etered, the judge fdg that 19,000 tckets had bee ssued. Polce vestgato estmated that durg the Hallowee party 7,700 of the attedees were o the groud floor. Ths correspods to a desty of four people per square meter. (a) (b) (c) FIGURE 2: (a) Detals of the groud floor of the Madrd Area (ELPAIS, 5/11/2012). (b) The red crcle shows the etrace of the hallway were the cdet happeed. (c) shows the groud floor, the ma ext, ad the posto of the stage.

8 Polce vestgato reports a stampede the ma hallway (dmeso 12 m x 3 m) that coected the ma dace floor to aother area (Fgure 2a). Itally t was thought that the stampede was caused by someoe who set off a flare or a frework earby, however later t was reported that some frecrackers ad some flares were set off dfferet places oe the corrdor but 20 mutes after the last stampede. Whe the performace of the vted artst started, the groud floor was packed; however, a crowd of people that were aroud the veue tred to get (Fgure 3). At ths tme, some people were tryg to get out through the aforemetoed corrdor. The fact that people were movg out through the hallway, whereas other people were usg the same hallway to eter the veue, trggered avalaches alog the hallway. Fve avalaches were detected before the fatal cdet, the last of whch eded wth people fallg dow cloggg the hallway, ad causg the death of fve grls by crushg ad asphyxato. Oe wtess sad the stampede left partygoers pled atop oe aother as hgh as ther shoulders. Where ths fatal cdet was happeg the hallway, thousads of partygoers were uaware of the stampede utl the polce arrved ad bega to move people out slowly. Based o the formato about crowd desty ad couter flow take from vdeo footage, ad the avalable plas of the Madrd Area veue, we smulated the lae formato, avalaches, ad cloggg formato the hallway. We also compared the hallway case wth smulatos of the hypothetcal case of total evacuato of the veue. Evaluato of the cotact forces amog pedestras gves us a sght to how the dagerous pressures are buldg uder the crowd codtos these two cases. FIGURE 3: Two sapshots from the vdeo footage showg fve dagerous avalaches Madrd Area precedg the fatal oe. These sapshots are of the ffth avalache, at ed of the corrdor. The terval betwee these sapshots s 0.2 secods, suggestg pedestra veloctes up to 3 m/s. Vdeo footage shows creasg desty wth tme the hallways. The frst avalaches happeed at 2:18 am, 2:19 am, 2:41 am, 2:59 am, ad 3:29 am. I the fatal cdet, regstered at 3:33 am, people fell dow, cloggg the hallway. Smulato of the hallways RESULTS I the frst seres of smulatos pedestra flow the 3 m x 12 m hallway was vestgated, wth replcato of Madrd Area s corrdor. A radom dstrbuto of pedestras was placed the smulato area. Followg ths, a percetage of the pedestras were assged a desred

9 drecto towards the ext, ad the others were assged a drecto towards the veue to represet the couter flow. Wth the am to replcate observatos from the Madrd Area veue, these smulatos were repeated usg dfferet destes. Perodc boudary codtos were used the edges of the corrdor. Depedg o the desty of pedestras ad the couter flow percetage, three dfferet regmes are observed the smulatos, see Fgure 4: ) lae formato, whch occurs at low desty ad s characterzed by well-defed les of pedestra flow at the statoary stage; ) avalaches, whch are observed at termedate destes ad couter flow values, characterzed by temporal cloggg of the pedestra flow, followed by a sudde release of potetal eergy that leads to a short-tme acceleratg flow; ad ) cloggg, characterzed by a froze stage where the pedestras are clogged deftely the corrdor. (a) (b) Fgure 4: Two dfferet regmes observed teh smulato of the hallway.. A sapshot of th lae formato s show (a) ad a sapshot wth the cloggg s show (b). Fgure 5 shows the phase dagrams of these three regmes. The avalache regme s sadwched betwee the cloggg ad the lad-formato regme. Below 4.4 pedestras/m 2 ad below a couter flow of 20%, the model predcts a formato of laes the statoary stage. The formatos of avalaches wthout cloggg appear up to a pedestra desty of 5.8 pedestras/m 2, ad a couter flow up to 34%. These avalaches are crtcal pedestra safety as they ca produce people fallg dow ad cloggg etre hallways. The most dagerous stuatos are observed above 5.8 pedestras/m 2, ad couter flow betwee 34% ad 66%. I these cases the model predcts complete cloggg, whch leads to log-lastg pressure amog the pedestras ad, f the pressure are too hgh, possble jury by asphyxato. It s terestg to ote that the phase dagram depeds o the pedestra shape used the model. Thoracc shapes leads to more coservatve results tha crcular shapes; the later t suggests cloggg above 6.3 pedestras/m 2. The avalaches the thorax model are less

10 proouced tha the model of crcular partcles. Ths s probably due to the shoulder rotato of the thoracc-shape pedestras, whch reduces the accumulato of stress that produces avalaches ad eddy-lke moto the boudares betwee laes. We should ote that the resstace of the pedestras to rotato or to beg pushed back depeds o several psychologcal codtos ad cultural backgroud. These are accouted by the parameters Γ ad λ of Eqs. 15 ad 16 our model, whch has bee ftted from comparso of smulatos wth vdeo footage observatos. Our smulatos show that creasg the parameter Γ of the back force reduce the zoe of avalaches the phase dagrams, ad creasg the torso stffess parameter λ creases the zoe of cloggg. FIGURE 5: Phase dagram of the smulatos of couter-flow the corrdor wth crcles. The regmes observed are: lae formato (dsks), avalaches (crosses) ad cloggg (squares). The arrow shows the tme evoluto of these two parameters the hallways the Madrd Area. It started form a dese crowd of people leavg the party. The a couter-flow of people jog the evet creased the desty, ad fally t produced the avalaches ad evetually the fatal cloggg. Cumulatve sze dstrbuto of forces ad fracto of jures Rsk of asphyxa by crushg There s a lack of specfc data cocerg ths type of force these stuatos that cause asphyxa by crushg. Some data are foud from major crowd catastrophes [22] such as those Bolto 1946 [23], Glasgow 1971 [24], or Sheffeld 1989 [25]. Smth ad Lm [26] carred out a expermetal vestgato of the loads that people ca stad whe beg pushed agast a barrer. A lower-boud comfort level level that could be wthstood for 30 secods or so wthout major dstress was determed. I prevous studes Ao (udated) at the Uversty of Surrey [27] pulled the subject agast a barrer by meas of cords attached to a pack frame o

11 the subject s back. Part of these tests attempt to measure a subjectve load level of dscomfort estmated by the subject to become tolerable after a few secods. The mea load determed at chest level from 17 subjects was 418 N wth a mmum of 116 N ad a maxmum of 774 N. Hopks et al. [28] also measured pressures a publc evet. The data were collected durg a 85-mute performace by the Eglsh bad Rde, at a heavy rock cocert Brta. These crowd pressures were measured from the load measurg trasducers attached o eght paels arraged frot of the stage, each of whch was of 1.5 m legth. It s dffcult to assess the load ad the tme eeded to produce asphyxa the fve grls at the Madrd Area; however, takg to accout the vsual observato of the cdet a reasoable estmato of the load mght be 400 N, cosderg the umber of stacked pedestras. These data wll be used our smulatos to provde estmates of the umber of people at rsk. Cumulatve sze dstrbuto Two dfferet cumulatve sze dstrbuto scearos were cosdered: the clogged hallway ad the etrace the case of packg evacuatos. Vdeo footage from the Madrd Area dsaster showed that after fve huma avalaches the hallway some pedestras fell dow, whch subsequetly created a blockage ear the ext of the corrdor. To smulate ths blockage some addtoal rules were cluded the dyamcs regme: f pedestras accelerate above 5m/s 2 they were assumed to fall dow. A falle pedestra was assumed to behave lke a obstacle; that s, they caot ether rotate or traslate. A sapshot of the smulato of ths cloggg s show Fgure 6 below. The falle pedestras create bottleeck codtos, whch lead to a stable clog. At ths stage, most of the pedestras eter to physcal cotact, creatg a cotact etwork. We represet ths cotact etwork by drawg a le from the ceter of mass of the pedestra to the cotact pot, ad ecodg the magtude of the force by the thckess of the le. The cotact etwork betwee the partcles, show Fgure 6 as a example, demostrates that the forces o partcles are strogly heterogeeous, as ther dstrbuto s ot uform but orgazed the form of flametary structures that graulologsts call force chas. Whe appled to pedestra moto, these force chas ca determe whch pedestras suffer the largest forces ad are most lkely to choke. At ay gve pot, a pedestra may bear eormous forces, whle a adjacet pedestra may experece almost o force. I Fgure 6 we also see the percetage of pedestras who are save, ucomfortable, ad dager. A dramatc dstrbuto of cotact forces s observed the hypothetcal case of a evacuato a large room flled wth pedestras. A smulato was coducted usg a evacuato of 6,800 pedestras a room wth the same dmesos of the Madrd Area. By calculatg the cotact etwork we were able to vestgate theoretcally the severty of the cotact forces o each pedestra ad demostrate the umber of pedestras at severe rsk. Due to the stress supported by the walls, the forces bult a hallway such as the corrdor of the Madrd Area are ot large eough for may pedestras to accumulate ( the order of ~10), but due to the

12 large devatos the force dstrbuto we would expect few fataltes. Besdes, due to the large durato of ths cloggg the hallways (several mutes) the strogest forces were eough to produce asphyxato a few pedestras, accoutg for the fve fataltes o Hallowee the Madrd Area. (a) (c) (b) (d) FIGURE 6: Cotact force etwork the Madrd Area. (a) shows the cotact forces durg the blockage cdet the hallway of the Madrd Area; the thckess of the le ecodes the magtude of the cotact force. (c) shows the cumulatve dstrbuto of forces (CDF) scaled by F 0 =400N. (b) shows the cotact force durg the packg evacuato of the veue (hypothetcal). The CDF of ths case s show (d). The colours (c d) represet the safe zoe (gree), ucomfortable zoe (yellow), ad the dager zoe (red). The percetage of fataltes the total evacuato case s fve tmes larger tha the blocked corrdor. CONCLUSIONS We preseted a pedestra teracto model based o cotact forces, groud reacto forces ad torso, ad represetato of thorax shapes usg spheropolygos. The model s sutable for calculatos of flow rates ad cotact force etwork uder crowd codtos, where pedestras teract more by chest ad arm arragemet tha by avodg physcal cotact. For less dese scearos where the prvate space of each pedestra s relevat, the model may requre the troducto of log-rage socal force as proposed by Helbg [6].

13 The flowablty of the pedestras log hallways was modelled as a fucto of the desty of pedestra, ad the percetage of pedestra couter flow. We obta three regmes: lae formato, avalached flow ad cloggg. These regmes allowed us to recostruct the tragedy of the Madrd Area. From our smulatos we foud that the crowd the hallway tured from a lae formato regme to the avalaches regme due to crease of desty ad the couter-flow. These trastos suggest that the Madrd Area cdet may have bee produced by a ever-creasg crowded couter-flow codto rather tha by pac derved by frework, as suggested by early offcal reports Ideed after the vdeos the polce report clamed that the frework was 20 m after the last ad fatal avalache. A partcularly dffcult aspect of the modellg was smulatg the people fallg the floor. We assumed that these people act as obstacles for the other pedestras. Ths s a qute crude smplfcato, as smplfes a three dmesoal problem a two dmesoal model. Yet our two dmesoal model allowed us to calculate the force experece by the pedestras, ad to estmate a small, but mportat, fracto of pedestras who suffered from log-lastg pressure that derved to asphyxato. We oted from the reports o the cdet that the authortes allowed the party to cotue spte of the tragedy. Smulatos showed that f further pac was caused by hasteed evacuato, t mght have produced a bottleeck stuato the etrace of the corrdor. Uder these codtos, our model predcts that the umber of causaltes would be fve tmes larger. The force dstrbuto foud our smulatos was qute heterogeeous, suggestg that t s orgazg force chas that carry more of the load produced by the crowd. These force chas carry large forces ad persst for several mutes, potetally leadg to crushg ad asphyxato. It remas uclear whether a dvdual pedestra could avod bee trapped by these force chas by movg ther arms ad rotatg ther chest (Actually there s a emergecy procedure these cases to help agast asphyxa: to cross the arms, left had rght shoulder, rght had left shoulder). Uder dese crowd codtos, the pedestra would have lttle space ad would therefore fd t dffcult to re-arrage themselves. Eve f a sgle force cha s temporarly dsspated by a pedestra escapg, the large pressure from other packg pedestras would lead to the force cha beg re-establshed. Two ma parameters were detfed the cotact force model: the coeffcet of frcto, whch s the ma cotroller of the flow rate dese graular flow, followed mportace by the coeffcet of resttuto. Gve the mportace of these parameters t s recommeded that further emprcal work be coducted to provde greater model accuracy. Ethcal ssues ca be avoded by usg sad sde sttched clothes or crash-test dummes. The test dummes would have the coveece of already havg bult sesors to measure pressuredeformato relatos. Psychologcal parameters that eed to be vestgated further studes clude the λ factor related wth the stffess of the shoulder rotato, ad the Γ factor related to stregth of the back force used by pedestras to avod beg push back from ther desred drecto. We

14 presume that these parameters depeds o cultural aspects ad specal crcumstaces of the crowd, ad they would eed to be calbrated by observatos of crowd dyamcs uder smlar codtos before makg predctos. Further studes should focus provdg recommedatos of desg for hallways ad solve orgazatoal ssues for ths type of evets lookg for crowd safety. ACKNOWLEDGMENTS C. L. thaks Asocaco de Amgos la Uversdad de Navarra for a scholarshp. F.A.M. ackowledges dscusso wth Iker Zurguel, ad Dael Pars, ad techcal support from Cadace Lu, Shumao Che, Foa Tag Huey Mg, ad Gue Mao. REFERENCES 1. Yu, W., Che, R., Dog, L., & Da, S. (2005). Cetrfugal Force Model for Pedestra Dyamcs. Physcal Revew E, 72(026112). 2. Hak, B. D., & Wrght, R. A. (1958). Passeger Flow Subways. Operatoal Research Socety, 9(2), Hrak, K., & Taru, K. (1975). A Smulato of the Behavor of a Crowd Pac. Paper preseted at the 1975 Iteratoal Coferece o Cyberetcs ad Socety, Sa Fracsco. 4. Colombo, R. M., Goat, P., & Ros, M. (2010). A Macroscopc Model For Pedestra Flows Pac Stuatos. Gakuto Iteratoal Seres Mathematcal Sceces ad Applcatos: Curret Advaces Nolear Aalyss ad Related Topcs(2010), Helbg, D., Johasso, A. ad Al-Abdee, H. Z., (2007). The Dyamcs of Crowd Dsasters: A Emprcal Study. Physcal Revew E 75(046109). - Hudreds Klled Hajj Stampede. (2006). BBC News Retreved 11th March 2011, 2011, from 6. Helbg, D., Farkas, I., ad Vcsek, T., (2000). Smulatg dyamcal features of escape pac, Nature, 407, Muramatsu, M., ad Nagata, T., (2000). Jammg trasto two-dmesoal pedestra traffc, Physca A, 275, Burstedde, C., Klauck, K., Schadscheder, A., (2001). J Smulato of pedestra dyamcs usg a two-dmesoal cellular automato, Physca A, 295, Helbg, D., Isobe, M., Nagata, T., Takmoto, K., (2003). Lattce gas smulato of expermetally studed evacuato dyamcs, Phys. Rev. E, 67, (067101). 10. Weg, W. G., Che, T., Yua, H. Y., Fa, W. C., (2006). Cellular automato smulato of pedestra couter flow wth dfferet walk veloctes. Phys. Rev. E, 74, (036102). 11. Helbg, D., & Molar, P. (1998). Socal force model for pedestra dyamcs. Physcal Revew E, 51, Aloso-Marroqu, F. (2008). Spheropolygos: A ew method to smulate coservatve ad dsspatve teractos betwee 2D complex-shaped rgd bodes. EPL (Europhyscs Letters), 83(1), Aloso-Marroquí, F., & Wag, Y. (2009). A Effcet Algorthm for Graular Dyamcs Smulatos wth Complex-Shaped Objects. Graular Matter, 11, 13.

15 14. Aloso-Marroquí, F., Ramírez-Gómez, Á., Gozález-Motellao, C., Balaam, N., Haaor, D. A., Flores-Johso, E. A., Ga Y., Che S. & She, L. (2013). Expermetal ad umercal determato of mechacal propertes of polygoal wood partcles ad ther flow aalyss slos. Graular Matter, Korhoe, T., Helovaara, S., Ehtamo, H., & Hostkka, S. (2009). Collso Avodace ad Shoulder Rotato Pedestra Modellg. 22. Retreved from Korhoe, T., & Hostkka, S. (2007). FDS+ Evac Techcal Referece+ User s Gude. VTT Techcal Research Cetre of Flad, PO Box, Retreved from t.pdf 17. Lakoba, T. I., Kaup, D. J., & Fkelste, N. M. (2005). Modfcatos of the Helbg- Molár-Farkas-Vcsek Socal Force Model for Pedestra Evoluto. Smulato, 81(5), Lagsto, P., Maslg, R., & Asmar, B. N. (2006). Crowd Dyamcs Dscrete Elemet Mult-Crcle Model. Safety Scece, 44(2006), Wllams, P. (2010). Love Parade Dsaster, 19 Dead 25 July ABC News. ABC News. Retreved from aggrostar69. (2010). Massepak auf der Loveparade 2010 (mass pac at the loveparade 2010). Retreved from kazpmk. (2010). Crowd pac ad stampede at a rave party Los Ageles. Retreved from 金边踩踏事件造成千余人死伤 (Phom Peh, more tha a thousad deaths ad jures caused by the stampede). (2011). Sohu. Retreved from shtml 21. Busch, J. (2011). A Realstc Dscrete Elemet Model of Pedestra Pac Behavour. The Uversty of Sydey Udergraduate Theses Dcke (Eds.), Egeerg for Crowd Safety. Elsever Scece B.V., Amsterdam, pp Hughes, R.M., Equry to the dsaster at Bolto Waderers Football Groud o 9th March, Commad Paper HMSO, Lodo 24. Walker, Sr Ala, Proceedgs of the Fatal Accdet Iqury to the Ibrox Stadum Dsaster. 25. Taylor, Lord Justce, The Hllsborough Stadum dsaster 15 Aprl 1989, Fal Report. Commad Paper 962. HMSO, Lodo. 26. Smth, R.A., Lm, L.B. (1995) Expermets to vestgate the level of comfortable loads for people agast crush barrers. Safety Scece 18, Ao. Prelmary Report coecto wth Crowd cotrol Barrers. Uversty of Surrey 28 Hopks I.H.G., Poutey, S.J., Hayes, P. Sheppard, M.A. (1993) Crowd pressure motorg. I R.A. Smth, J.F. Dcke (Eds.), Egeerg for Crowd Safety, Elsever, Amsterdam (1993), pp

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