Crowd Dynamics

• Crowd dynamics or crowd science is regarded as the study of the formation and movement of crowds and where they move higher than a certain quantity, called the density per square metre.

• The importance of this branch of physical science stems from the fact that many incidences in the past have occurred where many lives have been needlessly lost and could have been saved if there was better crowd control and planning.

• A way in which many lives are lost in a crowd is a stampede.

• A stampede is a sudden rapid movement or reaction of a mass of people in response to a particular circumstance or stimulus.

• Over 70 years ago today, the most disastrous stampede occurred in Chongqing, China, which was caused by mass panic during the bombings by a Japanese air fleet.

• The following table summarises historic stampedes, highlighting the importance of crowd dynamics as the majority of the pedestrians killed could have been saved.

• Human stampedes can be categorised into two types. • The first type is generated by pedestrians rushing towards something

(acquisitive panic). • The second type is generated by panicked pedestrians surging away

from danger with no particular direction.• The following picture shows the victims of the stampede in

Chongqing, China. The photograph was taken by the American photographer, Carl Mydans.

As the bodies are generally intact, this suggests that the crowd density was high.But what is crowd density ?

• Crowd density is a fundamental factor which characterises pedestrian flow.

• Crowd density can be thought of as the number of pedestrians per square meter.

• We can have high crowd density which is unsafe as there is a high risk of asphyxiation (suffocation and crushing of the torso). Thus, asphyxia is the most common cause of death in a high crowd density stampede.

• We can have lower crowd density which poses a lower risk to pedestrians as there is sufficient space to maneuver. Thus, death in a low crowd density stampede is usually caused by trampling.

• The following graph illustrates the relationship between crowd density and the flow rate.

Hughes Model• In the project a study of Hughes model was

undertaken and its application in the analysis of crowd flow over Jamaraat Bridge, near Mecca, was examined.

• Hughes model is based on three fundamental concepts relating to the nature of pedestrian motion.

• These three concepts taken into consideration with the continuity equation yields Hughes model.

Speed and Density Relationship

• This concept states that the speed of a pedestrian is only dependent on the density of the flow of pedestrians.

• Hence the velocity components V=(u,v) are

where

Potential • This concept states that pedestrians have a

common sense of the task called potential. • If pedestrians are such that they perceive no

advantage in exchanging places in order to reach their desired destination, then they are at the same potential.

• Potential is open to our own formulation. • Let us clarify potential with a diagrammatic

example.

Potential• The blue circles represent the pedestrians at

potential 1 (P1).• The green circles represent the pedestrians

at potential 2 (P2). • The yellow circles represent the pedestrians

at potential 3 (P3). • The pink circle represents the pedestrian at

potential 4 (P4). • The red triangle represents the destination

point of all pedestrians (e.g. A fire exit, the entry to a bunker).

• As each curve is a quadrant, it is a loci around the destination point. Therefore pedestrians standing along any curve are equidistant from the destination point as anyone else standing on the same curve.

• Pedestrians on the same curve will thus perceive no advantage in trading positions with any other pedestrian on the same curve.

• Therefore they will be at the same potential.

Speed and Potential Relationship

• This concept states that the path chosen by pedestrians is a function of the estimated travel time and the density of the moving crowd.

• This is because pedestrians choose to take a path which results in the shortest travelling time, however, alter this behaviour in order to avoid high crowd densities.

• From the diagram in the previous slide, we infer that moving pedestrians of which are initially at the same potential will be at the same potential again at some time in the future.

• Thus the distance between potentials must be proportional to the pedestrian speed, regardless of the pedestrians initial position along the same line of constant potential.

• Therefore we state

• This is formulated as

Speed and Potential Relationship

• By substituting the previously stated equations into the continuity equation given by

• We obtain the Governing Equations for Pedestrian Flow, given by

The Governing Equations for

Pedestrian Flow.

• The obtained equation is utilised in observing pedestrian flow in two dimensions.

• For a flow around a curved path for instance, cylindrical polar coordinates may be used.

• Two different flow types are analysed, they are Supercritical and Subcritical flow.

• Supercritical flow occurs when we have fast moving, low density flow.

• Subcritical flow occurs when we have slow moving, high density flow.

• The following diagram, which coincides with Greenshields (1934) model summarises these two types of flows.

• We see that at the critical density value of ρ= A/2B, we have maximum flow.

• The flow to the left of the critical value is deemed Supercritical.

• The flow to the right of the critical value is deemed Subcritical.

Greenshields 1934 Model

• The fundamentals of crowd dynamics are based on Greenshields findings from 1934.

• Bruce D. Greenshield carried out tests to measure traffic flow using photographic measurement methods.

• He measured the vehicular flow, speed and density of moving traffic flows.

• Greenshield assumed the following linear relationship between speed and density

V = A – Bρwhere V is the speed in miles per hour, the constants A and B are determined from experimentation and observations, and the density ρis measured in vehicles per mile.

Greenshields 1934 Model

• The speed, flow and density are formulated by the following relationship

F = V x ρwhere F is the flow measured in vehicles per hour, V is given on the previous slide and ρis the density.• By the use of the equation given in the previous

slide, we hence obtain the quadratic equation F = (A-Bρ) ρ.

Greenshields 1934 Model

• The following diagram summarises the findings of Greenshields.

Conclusion• Human behaviour is usually rational and is straight

forward to mathematically formulate. • Although the equations discussed are non-linear with

time dependence, a useful property is that they are conformally mappable.

• There is however difficulty in choosing the appropriate boundary conditions in accordance to the characteristics (or physiological state) of the pedestrians.

• Factors affecting the psychological state of a group of pedestrians dramatically varies the behaviour of the overall flow of pedestrians, thus making it hard to analyse the flow pattern observed of the pedestrians.

Conclusion • There is a great urgency in the improvement in

safety of pedestrians in major events and buildings.

• As the location of potential pedestrian accidents can be accurately forecasted, major accidents can be avoided.

• This is not only rewarding, but fascinating as a continuum approach to crowd dynamics makes one realise how similar humans behave to fluids.

Thank you