Analysis of Compressed Force in Crowds

JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 7, Issue 2, April 2007 Online English edition of the Chinese language journal Cite this article as: J Transpn Sys Eng & IT, 2007, 7(2), 98−103.

RESEARCH PAPER

Analysis of Compressed Force in Crowds LU Chunxia* School of Naval Architecture and Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China

Abstract: The risk of dying or injuring caused by force accumulated among crowds is great in the crowd accidents. This kind of force could even bend steel barriers or tear down brick walls. Compressed force between pedestrians may result in the deformation and compression of pedestrians’ body, or serious psychological problems like shortness of breath. The movement and force of crowds are analyzed from the view of microcosm at high density. The compression and deformation are emphasized. The social model has been modified to meet the requirements of very high density which is the non-ideal condition. The model allows moving bodies to be touched more closely. Crowds can even be overlapped. The force will accumulate and propagate especially at dense crowds because of the overlap. It is elastical deformation that results in the force spreading and accumulating through the crowds, which is the important cause of the disaster. The elastical deformation and the movement are correlated closely. The failure of crowd simulation software at high density lies in neglecting the compressed phenomena and the force propagating. The objective of this article is to develop a micro-simulation model to reveal crowd movement at the non-ideal condition by modeling and simulating the force and movement of pedestrians. Key Words: crowds; compressed force; micro-simulation; nonideal condition

1

Introduction

China has been on its way to fast urbanization. Urban population increases quickly. A large number of high buildings, business streets, and great stadiums have been built over these years. A great many people may need to be transferred. Therefore, the research on safety of crowds becomes very important. A lot of reasons account for accidents, such as unreasonable design of architecture, wrong message delivery, psychological panic, sudden falling, and so on. From the analysis of crowd accidents, the risk of dying or injuring on account of the force accumulated among crowds is great. This force could even bend steel barriers or tear down brick walls. Compressed force between pedestrians may result in the deformation and compression of a pedestrian’s body, or serious psychological problems like shortness of breath. These accidents are not completely unavoidable, but some of them have happened again and again. A lot of crowd simulation softwares cannot reveal the compressed phenomena. Most of them fail when the density is great. One of the reasons is the overlook of force

exerted on pedestrians. The force will accumulate and propagate, especially in dense crowds, because of the overlapping of pedestrians. It is the compressed force that makes this happen. That is why the micro-simulation models should take the close contact among pedestrians into account and reveal the deformation of bodies caused by the compressed force. The elastic deformation and flow are closely related. The objective of this article is to develop a micro-simulation model to reveal crowd flow in a nonideal condition by modeling and simulating the force and movement of pedestrians.

2

Related researches

Micro-simulation research must focus on each pedestrian including the velocity and interaction. Macro-simulation overlooks the difference of each pedestrian and focuses on integral indexes, such as, correlation between velocity, density, and flow. It emphasizes on the space allocation, such as, the width of the pavement[1,2]. The development of computer technology has caused a

Received date: 2006-10-31 *E-mail: [email protected] Foundation item: the National Natural Science Foundation of China (70473016). Copyright © 2007, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved.

LU Chunxia / J Transpn Sys Eng & IT, 2007, 7(2), 98−103

boom in micro-simulation research because it involves huge calculation. Cellular Automaton model of vehicle traffic is － introduced into pedestrian traffic[3 6]. Cellular Automaton is a dynamic modeling tool to describe complex behavior. It is appropriate to simulate and research a complex system dynamically. A lot has been achieved as it has been introduced into the traffic fields in the 1990s. Its flexibility and adaptation offers additional superiority, to describe more complex pedestrian traffic problems. Superiority, based on rules, provides a tool to describe different behavior of different pedestrians. The Cellular Automaton Model can quickly simulate a large-scale road network with simple rules. It can reveal the characteristics of a macroscopic pedestrian flow, brought by all kinds of microcosmic behaviors. It also can display the correlation between velocity, flow, and density in a statistical way. － Grid-gas model[7 10] is similar to the Cellular Automaton one. It considers each pedestrian as an active particle on the grid. It researches the characteristic of the pedestrian system in a statistical way. The particle is driven by rules and probability. Some Japanese scholars like to use such research methods. The limitation of these two models lies in considering each pedestrian as the same one, without any thinking or interacting, because fluid particles have no feeling of scare or ache, no decision-making, no falling, and no stumbling. But in fact, each pedestrian’s behavior will affect the integrated behaviors greatly. Once one pedestrian stumbles, he will become another person’s barrier, resulting in a lot of injury or death. The most famous micro-model is Helbing’s social force model, in which he considers pedestrians as interacted particles. He modeled the interaction between pedestrians as repellence and attraction, which is quite different from other micro-models. This new research method realizes that each pedestrian can think and react according to the surroundings. The most famous phenomenon is panic[11]. His research makes us rethink on how to design the width of the corridor, number and position of door, and so on. The microcosmic-particle research has analyzed each particle as a pedestrian, and taken into account the force on everyone. It is more accurate and veracious than any other research method. It gets especially more attention in academic research. But it has been found that pedestrians will be deformed and compressed easily by compressed force at high density. During the detailed analysis of the social force model, it was found that Helbing did not definitely consider the compressed force between pedestrians, because the force only existed when the density was over-great. But the chief interaction between pedestrians should be the compression at that time. Movement of pedestrians was constrained by compression. The pedestrians did not have enough space to hold as they should. They overlapped together. The force

propagated and accumulated among crowds by compression. All of these factors were not taken account into well in the social force model. It is found that the social force model is very close to reality and can reveal many characteristics of pedestrian flow where the density is low, but only in the case when the density is low and pedestrians are not crushed. The purpose of this article is to research on deformation and compression of pedestrians, by compressed force, at high density.

3

The analysis of social force model

The research on pedestrian flow must realize that pedestrians can think and react according to surroundings, which is different from vehicle traffic. Each pedestrian should be modeled as a different and sensible particle. It is also quite different from the Cellular Automaton Model. It is found that each pedestrian’s behavior will greatly affect the whole crowd’s performance. Wrong behavior or trouble on the part of one pedestrian can be seen as a disturbance. It will then produce a wave, which propagates forward or backward through the flow. It will finally result in a shock wave and lead to commotion and chaos. One pedestrian’s stumble is anthers’ barrier and this will cause a lot of injury or death. German professor Helbing’s social force model is the most famous model that takes account of each particle’s difference, moving direction, and velocity, decided by the force exerted on it. He simulates interaction between particles as repellence and attraction, which is the typical particle-oriented microcosmic research. The problem of many micro-models, including the social force model, is that they implicate a hypothesis. The particles are rigid and the space they occupy is the same whether they stop or move or whether they go forward, backward or rotate, but all those are just an ideal case. Fig. 1 shows some prevalent simulation models. They are Cellular Automaton model, social force model, and 3D simulation respectively, from left to right. The common characteristic is that the space of every particle occupied will not change at all.

Fig. 1 Typical simulation models

After analyzing the social force model, it is found that it is in good accordance with reality, at low density. It can demonstrate bi-directional flow forking, stripes of flow crossing, panic phenomena, and so on, but its demonstration is

LU Chunxia / J Transpn Sys Eng & IT, 2007, 7(2), 98−103

different with reality when the density is high, as Figs. 1 and 2 by Helbing[13] show. When the density is high, each particle does not have enough space. They will overlap tightly together. The average width (the width of shoulders) and thickness (the thickness of thorax) are 45.58 cm×28.20 cm. The area of vertical projection is about 0.2 m2. One square meter can load five pedestrians at most. The pedestrians will be deformed and compressed if one square meter has loaded more than five pedestrians. In fact, the density is over eight pedestrians per square meter sometimes, shown as the cases of Figs. 1 and 2 in [13]. Many micro-models fail in simulation at high density because they overlook this point. Huge crowd gathering at public places is very common, such as, the Chinese Spring Festival. There are nearly 700,000 people getting on or off the train per rush hour at Beijing Railway Station. It can often been found that the crowds are deformed and compressed at doors or counters. Most of them cannot occupy enough space (0.2 m2). Their bodies are overlapped. The proceeding directions are out of control. They just go with the preceding crowd called “follow the tide”. Someone is even overhead. Psychological problems like expanse and shrinkage of lung, oppression of breath, anoxia of heart and brain, even injury or death will occur when pedestrians stay at high density (over 6−8 people per square meter) for too long. It is necessary to do a research on dense crowds and then forecast the potential accident site. This phenomenon is very common in daily life, especially in China. Walking system at high density is an important part of the walking system. Eqs. 1 and 2 have expressed social force model (the details in literatures [11,12]):

mi

dvi (t ) mi 0 = (vi e(t ) − vi (t )) + τi dt

∑F

j (≠i)

ww ij

(t )

+ Fiib (t ) + ∑ Fikart (t ) + ξ i (t )

(1)

k

The formula expresses the movement of every pedestrian dynamically based on Newton law. The left part of the r equation is ma , and the right parts are all types of typical forces. The first is the driving force. The second is influence force among units. The third is psychology and other social factors. The last are other possible factors.

ma = ∑ f r dv a= dt r dx v= dt

during simulation. The force on each pedestrian is a discrete point per updating step if a pedestrian is selected randomly. The force is calculated based on the position related to other pedestrians, walls, or other barriers. The sum of the force vectors makes the velocity and displacement change at every update. Then the force comes down to zero and is calculated again, based on the position related to other affecting factors. Therefore, every pedestrian’s force is finally transferred into the change of position, namely the force is digested by displacement. The force is not propagated to any other pedestrians around. The influence of any two pedestrians is only involved in their relative position, which is the right case only when the density is less than five people per square meter. The compressed force built up by crowds can even bend steel barriers or tear down brick walls as Fig. 2 by Helbing shows[13]. This is why the force of pedestrians can accumulate and propagate forward. Once it reaches the threshold, the front barriers are torn down and the front pedestrians fall down suddenly. Many micro-models, including the social force model, cannot explain this phenomenon. First of all, it is the accumulation of interacting forces that causes these accidents. Therefore, it is inappropriate to overlook the force between pedestrians. Second, although they have taken into account the force between pedestrians, the force is absorbed by them resulting in the change of velocity and displacement. In fact, because there is not enough effective space to move when the density is great, the force exerted on each pedestrian leads to compression and deformation rather than effective displacement. The compression will also result in the accumulation and propagation of force. The force propagates and accumulates from back to front. The front barriers are pushed down if the force is finally not released.

4

The force at high density

Now to analyze the above nonideal movement of crowds at high density:α,β are two random pedestrians in a highly dense crowd. They move in the direction of one dimension positive X axis. The position of β is x β (t ) at moment t.

x α

(2)

The change of displacement is the result of force exerted on each pedestrian. Updating of each displacement is caused by the sum of force vectors during every update of the time step

α

x β (t +1)

β

x ′β ( t +1) x β (t )

β t

t+1

t

Fig. 2 The movement of pedestrians at high density

LU Chunxia / J Transpn Sys Eng & IT, 2007, 7(2), 98−103

4.1 Analysis of movement β will be at position x ′β ( t +1) at most at moment (t+1) and overlap α closely (dashed circle in Fig. 2) at moment (t + 1). If it is supposed that the body is rigid, as other micro-models do, the pedestrians will crush each other at high density as shown in Figs. 1 and 2 in [13]. Therefore, β may be at x β ( t +1) . Displacement, ( x β ( t +1) − x ′β ( t +1) ), is the compressed variable of a body. The force exerted on each pedestrian produces two different displacements,

∑f

(3) → ( x ′β ( t +1) − x β ( t ) ) + ( x β ( t +1) − x ′β ( t +1) ) Displacement, ( x′β ( t +1) − x β (t ) ), is the rigid displacement, which is expressed in a social force model in an ideal case. But it tends to come down to zero at high density. The compressed variable, ( x β (t +1) − x′β (t +1) ), is very obvious and cannot be neglected when the free space is limited. 4.2 Analysis of expressed force The deformation and compression of pedestrians shown in Figs. 1 and 2 in [13] illustrate that their bodies are elastic. The deformation is caused by force. The major force is the compressed force, which puts constrains on a pedestrian’s movement. The social force model does not consider the compressed force definitely as shown in Eq. 1; therefore, the modified formula should be as follows: r r r dv (t ) (4) mi i = ∑ F − ∑ f crushing dt r ∑ F denotes all r types of forces described in the social f crushing denotes the compressed force. force model. r f crushing = ki xi . k i denotes the compressed coefficient, and xi denotes the compressed variable, which is the function of the distance d between pedestrians,

d < ( r1 + r2 ) , d ≥ (r1 + r2 ) ,

f crushing = g (d )

f crushing = 0

(5)

Eq. 5 shows that the compression only comes into being when the distance of the center of pedestrians is less than the sum of radii; otherwise they are free. The compressed force affects velocity and displacement as Fig. 2 shows. Eq. (6) takes the place of (4). Eq. (7) is the movement equation of β, namely, the relationship between velocity and displacement,

mβ

r dvβ (t + 1) dt

r r k β + ( xβ (t +1) − x′β (t +1) ) = ∑ F β

r dx β (t + 1) r dvβ (t + 1) = dt

(6) (7)

It can be concluded from Eq. (6) and Eq. (7) that the compression not only produces compressed force, but also produces compressed variables, related to the displacement of pedestrians at high density. The space for every pedestrian to move freely is quite small, and as a result the change of position is not obvious at high density. Subsequently, the

compressed variable cannot be neglected when the position is updated. The accumulation of micro-change in pedestrians will influence the whole crowd greatly when the number is huge. It is especially significant to urgently evacuate the whole crowd from public places with large pedestrian flows. 4.3 Force propagating Pedestrians cannot move freely at high density because of the limitation of space. The movement is out of control. The effective displacement is very limited just as Fig. 2 shows: x ′β (t +1) − x β (t ) → 0 . r If β gets another force f at this moment, it will produce compressed deformation: f → x β ( t +1) − x ′β ( t +1) . If there is a barrier before α, a wall for example, the effective displacement of α is 0 as Fig. 2 shows, so xα (t +1) − xα (t ) → 0 . The compressed deformation of β affects the compressed deformation of α directly, namely, x β (t +1) − x ′β (t +1) ⇒ xα (t +1) − xα′ (t +1)

Deformation will produce force just as analyzed in this article. α will produce a compressed force, which has resulted from the r deformation of β. The deformation of β is caused by force f :

f → ( x β ( t +1) − x ′β ( t +1) ) ⇒ xα ( t +1) − xα′ ( t +1) → f α

(8)

Eq. (8) illustrates that force can propagate at high density by the way of compressed deformation.

5

The analysis of result

The modified social force model has been used to simulate the environment of [11]. The original number of pedestrians is set as 500. The diameter of the pedestrians is 0.6m ± 0.1m. The size of the room is 15 m×15 m. The width of the door is 1.2 m. The original density of crowds is about two pedestrians per square meter. The result (Fig. 3) shows that evacuating people still encloses the exit in the shape of a semi-circle after a moment, but there are obvious compressed phenomena and overlapping around the exit. The velocity of pedestrians at the overlapping place is nearly the same because there is not enough space for people to move freely. Most of pedestrians’ movement are out of control and just follow the footsteps of the pedestrians in front. This is called “follow the tide”, which is very close to reality. i) Compression only occurs at high density. A pedestrian pushes in the direction which he/she wants to go or he/she pushes to keep his safe space. ii) Force propagates by the way of pushing in the crowd. Only after deformation and compression occur can the force be propagated. Otherwise, force will be transferred into displacement. Concisely, force only propagates by the way of compression. iii) Compression and deformation accelerate the propagation of force and gather pedestrians into a mass.

LU Chunxia / J Transpn Sys Eng & IT, 2007, 7(2), 98−103

iv) Density of a crowd decides the propagation of force directly. The higher the density is, the fewer the interstices are, and the easier it is for the crowd to be compressed. Therefore the compressed force propagates more effectively. Interstice prevents force from being built up and propagated. The propagation of force will be interdicted by the interspace existing in crowds when the density is low. The analysis of the result can help us to understand the reason for accidents on account of overcrowding. They are caused by the compressed force and its propagation. The deformation will not happen and the force will not propagate if the density is low even if one feels crowded. The interspace interdicts the propagation of force. This is important to guide the management when huge crowds gather at public places. The crowd will be evacuated once they are compressed. Some interspaces should be set to release accumulated force, which is significant to avoid accidents caused by overcrowding.

Any mathematical model should reveal reality honestly. It is important to distinguish the different actual conditions when the crowd is modeled. Otherwise, it will lead to serious consequences. It also gives valuable knowledge if one is aware of when and what model should be used when one is faced with an emergency. Maybe perfect safety can not been obtained, but at least the modified model can make people safer in overcrowded conditions.

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6

Conclusions

The social force model describes the less dense system in which pedestrians are not compressed and deformed. It is the model under ideal conditions. This article researches on a highly dense system in which pedestrians are compressed and deformed. It is the model under nonideal conditions. It is the compression and deformation that cause propagation and accumulation of force in crowds, and result in accidents.

[11] Helbing D, Farkas I, Vicsek T. Simulating dynamic features of escape panic. Nature, 2000, 407: 487−490. [12] Helbing D, Farkas I, Vicsek T. Physica Rev. Lett. 2000, 84: 1240. [13] Helbing D, Buzna L, Johansson A. Self-organized pedestrian crowd dynamics: experiments, simulations, and design solutions. Transportation Science, 2005, 39(1): 1−24.