Crowd dynamics studies the formation and movement of crowds above a certain density threshold. Understanding crowd dynamics is important for crowd control and safety planning to prevent needless loss of life. One way lives are commonly lost in crowds is through stampedes, which are sudden, rapid movements in response to a stimulus that can cause crushing or trampling. The deadliest stampede on record occurred in Chongqing, China over 70 years ago during bombings, killing many people who could have been saved with better crowd dynamics understanding.

1. Crowd
Dynamics

2. • 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.

3. • 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.

4. • 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.

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

6. • 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.

8. 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.

9. 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

10. 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.

11. 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.

12. 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.

13. • Therefore we state
• This is formulated as
Speed and Potential
Relationship

14. • 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.

15. • 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.

16. • 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.

17. 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 – Bρ
where 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.

18. Greenshields 1934 Model
• The speed, flow and density are formulated by the
following relationship
F = V x ρ
where 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.

19. Greenshields 1934 Model
• The following diagram summarises the findings of
Greenshields.

20. 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.

21. 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.

22. Thank you