Application of Mathematical Model of Evacuation for Large Stadium Building

The statistics of sports arena accidents show that the main reasons which leading to crowd stampede are the exports blockage and the poor surrounding transportations. In the process of evacuation, the most common problem is that there are a large number of people are stranded and also they are the main carrier which leading to crowded stampede. With large amounts of data and reasonable evaluations on staffs and transportation instruments. We propose inflow model in the crowding state, principle of maximum flow on channel design, optimal model of vehicle parking, evacuation model of subways and buses, according to sections of evacuation in stadiums. We analyze their usage area, marginal conditions and real data. Finally, we get some valuable results, which are curves of density and flow, evacuation time, formula for channel design, optimal parking design and formulas for evacuation time of subways and buses. Such data suits the real data from varied references. With the help of models and results, we get the total time of evacuation, simulation of progress and give parts of real situations of evacuation. According to such results, 100000 people’s evacuation can be finished in about 45 min. On such basis, we propose some optimal plans for stadium and its surroundings building.


INTRODUCTION
The public places where have intensive persons are very dangerous, it could cause significant accidents with mass casualties easily. For example, the Atlanta Olympic games in 1996, when a lot of persons gathered in the Olympic Park, there was a sudden explosion which causing hundreds of people injured and dead. Fire, people over excessive. For example, on October 20, 1982, there were 340 people's dead by crowded. Riots, for example, when the game took place between Syria Super League of a Kameshli and Deiral-zour, there were riots in the fans and this leaded to over 100 persons injured and dead. On December 25, 2000, there was a fire accident in East building of Luoyang City, which leads to 309 persons dead. In 2001, the incident "911", which occurred in the high-rise building where people crowed, leading to 25,000 persons were evacuated emergency, 2000 persons dead and 6347 persons missing. In 2003, the subway arson which occurred in the crowd accumulation subway, leading 134 persons dead and 136 persons injured at least. In 2004, during the festival performances in Beijing, there occurred crowded accident leading to 17 persons dead. Therefore, study the safety evacuation of large public places has a great practical significance and social security value and in this process, the speed of evacuation is the most critical factor. At present, the typical models of evacuation velocity are as follows: the former Soviet Union Predtechenski and Milinskii and Togawa Ando in Japan. The space grid evacuation model SGM which developed by Wuhan University and City University of Hong Kong in China and the mathematical model of evacuation and escape speed. Compared the above models of crowd evacuation speed; it could take countermeasures to improve the design of Fire, which has a guiding significance for the study of crowd evacuation.
With the rapid development of sports competition in the world, the scale and the audience are all increasing; the safety management system of sport stadium has become the major constructing objects. The sport stadium is a place where always has crowed people, a large sport stadium can accommodate tens of thousands or even over 10% million persons. The key to hold a tournament is the transportation safety, especially the opening ceremony, closing ceremony and some wonderful competitions. The large numbers of people and centering time make it difficult for transportation and vehicles. The evacuation in stadiums should be considered seriously.
This study takes Chinese National Stadium as an example. It locates in suburbs of Beijing, contains 100 thousand people and has enough high-speed roads. This paper analyzes and designs the problems of exits, channels, places of parking, reasonable car styles, constitution of staff, arrangement of vehicles and acceptable waiting time, etc. This paper builds proper mathematical model of evacuation, simulates real situations and calculates the time for whole evacuation. The innovation of this study is that, according to the characteristics of the development of Beijing, analyze

L OF CROW
The area dep n all directions g. 2) and the ate convenient or square area. an indicated as: (1) And (2) w

WDING STATE
pends on max ; usually the b e Depth of bo tly, we subtra will be E ximum breadth ody dp act the (1) (2)  The data of physiological sizes in different regions are shown in the following, just as the following Table 1: According to the results in Table 1, considering future development of population's quality, adding the convenient calculations, this paper set b p = 0.5 m, d p = 0.25 m, S = 0.125 m 2 .
Group density: Group density is related to hysiological sizes and distances. The spatial occupation data in typical situation is shown as Table 2 (Stephen, 2001).
In view of large density while evacuating, the traverse distance between two people is 100 mm, the vertical distances change with the density.
In general, in view of safety, the maximum density must be less than 40 person/m 2 (Department of National Heritage, 1997). Combining the calculation above, the density can be accepted is (0.4) person/m 2 (the speed is ignored and the ideal value can be shown as follows).

Flow model in the crowding state:
Several assumptions: • The flow comes in a unlimited channel with definite width in a single direction, comparative saturate, which is its speed is less than an extreme speed V max = 3 m/sec. • Any person has to conform to common principle: not trying to beyond the front body and without too large distance. • Density ρ (person/m 2 ) equals everywhere and increases with the v decreases. Its values are between (ρ min , ρ max ). • Definite density flow (person/m·sec) is the number of people come across the unit area in a unit time, q = ρ v .
With the help of velocity and foot magnitude from reference (Jiang, 1993 and http://www.crowd dynamics.com/), the relation between density ρ and walking frequency f can be definite: (4) and a deeper proves: K = 1.36, n ≈ 0.5. Indicate the group velocity in density: Definite the flow flux: With the mathematical models above, relative parameters and marginal conditions, draw v-ρcurve and q-ρcurve, which are shown in Fig. 5 and 6: It can be definite, when ρ 0 = 2.22 person/m 2 and v 0 = 1.01 m/sec, q gets the extreme, q* = 2.25 person /m·sec. Conclusion and analysis is as follows: The theoretic prediction conforms to daily experiences and its values are similar to current data. By calculating the trends of flux, the maximum velocity and density can be kept.
The channels in the stadium are all narrow ones, with the enough density. So this model can be used for analyzing. To get the minimal evacuation time, the flow fluxes in all the channels should next to q*.

OPTIMAL DESIGN AND EVACUATION TIME
Maximum flux principle for channel: The analysis above says: to get the minimal evacuation time, the fluxes in all channels should be next to q* and be wide as possible as it could. The reference (Cai, 1997) summaries such principles: According to Chinese shapes, the width of seat is 0.6 m. There are 50 groups of seats, which can hold 1600 people, in a circle. The distance between two groups is 1.0 m. To get the maximum flux, because the density of seats is next to the initial value of flow density, the seat density should be 2 person/m 2 , which is a person occupies 0.5 m 2 , the distance between the neighboring rows is 0.5/0.6 = 0.83 m. The perimeter is The channel between two rows (0 channels) can hold only a flow, whose width can make a person to walk through. The flow in 0 channels cannot reach ideal flux q*. So the length should be as short as possible (suggesting 15 times as the length of seat). The total length of 0 channels is related to the numbers of seats.
Others design depends on varied internal channels. The width should be controlled properly. Promise the inflowing of flow in the last level, to keep the mean flux as high as possible in the stable state, which is shown in Fig. 8. Due to such principle, we can get: K : The total number of joints between n channel and n-1 channel D i : width of i channel Design of exit, the relation formula: The more exits, the smaller total distance between exit and out and it is better for shorting the evacuation time Ts. But the number should not too large, or the flows are too many and scattering, which is bad for controlling. It also increases the loads and makes it easy to forming bottleneck, so as to be dangerous.
Considering foreign large stadiums, n b is 4, with symmetric distribution. The total number of exits reaches 8 or more, to evacuate in any accident.

The number of groups in a channel B:
It is the key that can be controlled. With reference to design criteria and its design scale, estimate the evacuation time T o = 15 min. The number of audiences is 95% of all. N = 100000: Evacuation velocity V (m/min): Flow model in the crowding state quantitatively shows the relation between density and velocity. To get minimal evacuation time, the fluxes in all channels should be next to q*.
The velocities should also be next to v 0 . Evacuation velocity V = v 0 = 60 m/min Evacuation distance S (m): According to the real distance between the entrance and exit, calculate the total distance, which is the weighted distance. The formula is shown as follows: The evacuation distance S should be as small as possible. In view of current references, the audience seats can be classified into two layers. The evacuation form is shown as Fig. 9. The stadium has symmetric structure. To calculate conveniently, only consider the sector. Just as shown in Fig. 10. Due to formula (13), here:     tors of ty on 8. or 7 to ocking, nection g lots, e same 20 occupations, T p is so small to be covered by T b . But it can be predicted, that, the number of private vehicles will go up seriously, then more problems will have to be tackled.