Loads generated by jumping crowds: Numerical modelling

Abstract

This paper is concerned with modelling the loads generated by groups of people jumping rhythmically. The principal objective is to replicate the results that were obtained in an earlier experimental study in which measurements were made with groups of up to 64 people. The experiments showed how the Fourier components of the loads attenuate with increasing group size and this defines a load model which can be used to calculate structural response. The measurements also showed the variations that can occur for similar sized groups. A model for the loads produced by an individual jumping is used as the basis of this study, with variations to three main parameters being examined. The first parameter being the jump height which the individual selects subconsciously; the second is the jumping frequency which may not align perfectly with the requested frequency; and finally the phase differences between individuals in a crowd. It is assumed that the variations in jump height and frequency will follow normal distributions and that the standard deviations of the distributions can be determined from the available measurements. A load-time history can then be generated for an individual jumping using the basic load model but including the chosen variables selected at random from the distributions. Groups of people are represented by the combination of the appropriate number of individual loadtime histories and here the phase difference between individuals can be introduced. The variation in phase difference can be determined from the experimental data. The modelling is based upon the measurements and attempts to reproduce the experimental data. Although this provides a method for determining a load model, it is not suggested that this should be used for calculating structural response because the model derived directly from the experiments is far easier to use. However, this serves to explain some of the characteristic variations that were observed in the experiments and provides a better understanding of this important load case. It also enables the loads produced by larger groups to be calculated.

Full text

7 September 2004 – The Structural Engineer|35
Synopsis
This paper is concerned with modelling the loads
generated by groups of people jumping rhythmically. The
principal objective is to replicate the results that were
obtained in an earlier experimental study in which
measurements were made with groups of up to 64 people.
The experiments showed how the Fourier components of
the loads attenuate with increasing group size and this
defines a load model which can be used to calculate
structural response. The measurements also showed the
variations that can occur for similar sized groups.
A model for the loads produced by an individual
jumping is used as the basis of this study, with variations to
three main parameters being examined. The first
parameter being the jump height which the individual
selects subconsciously; the second is the jumping
frequency which may not align perfectly with the
requested frequency; and finally the phase differences
between individuals in a crowd. It is assumed that the
variations in jump height and frequency will follow normal
distributions and that the standard deviations of the
distributions can be determined from the available
measurements. A load-time history can then be generated
for an individual jumping using the basic load model but
including the chosen variables selected at random from
the distributions. Groups of people are represented by the
combination of the appropriate number of individual load-
time histories and here the phase difference between
individuals can be introduced. The variation in phase
difference can be determined from the experimental data.
The modelling is based upon the measurements and
attempts to reproduce the experimental data. Although
this provides a method for determining a load model, it is
not suggested that this should be used for calculating
structural response because the model derived directly
from the experiments is far easier to use. However, this
serves to explain some of the characteristic variations that
were observed in the experiments and provides a better
understanding of this important load case. It also enables
the loads produced by larger groups to be calculated.
Introduction
It is recognised that the dynamic loads generated by crowds
need to be considered in the design of some structures, but
the available information on this topic is limited. This has
been highlighted by SCOSS and identified as an area of
concern for engineers1.The largest dynamic loads are gener-
ated by rhythmic jumping, which may be encountered with
some types of dancing, and this can be quite a severe load
case especially as it may generate a resonant response in
some structures. However, it is important to recognise the
significant difference between the situation where everyone
is jumping and the situation which is often encountered at
real events, like pop concerts,where only some of the crowd
are jumping. These two situations will lead to large differ-
ences in structural response. This paper considers the
extreme condition when everyone in a crowd is jumping.
A reasonable load model for an individual jumping is avail-
able2but the determination of the loads generated by groups
jumping is not a simple extrapolation based upon this model.
A previous study3obtained experimental measurements of
the response of two floors for groups of up to 64 people
jumping and derived a load model based on this experimen-
tal data. Although this model provides a reasonably simple
method for calculating the average structural response for
various sized groups it fails to replicate the variations
observed in the experiments. These variations are seen as
varying peak amplitudes of response for any one event (i.e.
not steady-state) and different averaged responses for similar
sized groups, this being accentuated for the smaller groups.
This paper considers the numerical modelling of the loads
generated by crowds and relates this to the experimental
measurements. It attempts to model the observed experi-
mental variations in order to understand the key variables
involved in the process. Although this provides an improved
understanding of the topic, for calculations for small groups
(say up to 64 people) it may be sensible to use a simpler load
model, like that derived directly from the experiments.
As the objective of the work is to replicate the experi-
mental measurements, some of the experimental data are
analysed to provide information on a number of variables;
e.g. the variation in an individual’s jumping frequency
during one event and the variation in jumping height. The
model for an individual jumping is used as the basis of the
study with variables introduced into this model in order to
calculate a load-time history. Groups of people are repre-
sented by the combination of the appropriate number of
individual load-time histories, but no variation in the weight
of the different jumpers is considered. The modelling
assumes that the sum of the response for a number of indi-
viduals jumping separately is the same as the response for
them all jumping together, and this pragmatic approach
provides a significant simplification of the situation.
In the following sections the basic load model is considered
first as this introduces the terms used throughout the paper.
The experimental work is then described. The numerical
modelling is explained and calculations compared with the
experimental measurements. The combined load model is
used to calculate the floor’s displacement-time history for 32
people jumping and compared with the experimental data.
The load model is then used to determine the loads produced
by larger groups and the significant points arising from the
work are discussed.
The load model for an individual jumping
The basic model for the vertical loads produced by an indi-
vidual moving cyclically is:
. sin
F
tG r T
nt10 2
n
ss n
pn
1
=+ +
rz
3
=
!
J
L
K
K
K
_
d
N
P
O
O
O
i
n
...(1)
where
Fs(t) = the time varying load
Gs= the weight of the individual
n = number of Fourier terms
rn= Fourier coefficient (or dynamic load factor)
Tp= the period of the cyclic load or the inverse of the cyclic
frequency
φ
n= phase lag of the nth term
For jumping, the motion is defined by the ratio of the
period that the person is on the ground to the period of the
jumping cycle, which is termed the contact ratio ‘
α
’, with the
load during the contact period being represented by a half-
sine wave.This can be used to determine the Fourier compo-
nents of equation (1)2, 4
cos
r
n
n
for n
for n
2
12
2
21
21
n
2!
=
-
=
r
a
ra
a
a
__ii
Z
[
\
]
]
]
]
]
_
`
a
b
b
b
b
b
...(2)
paper: ellis/ji
Loads generated by jumping crowds:
numerical modelling
B. R. Ellis
BSc, PhD, CEng,
MIStructE
BRE Centre for
Structural and
Geotechnical
Engineering
T. Ji
BSc, MSc, PhD,
CEng, MIStructE
Manchester Centre for
Civil and Construction
Engineering, UMIST.
Received: 12/03
Accepted: 06/04
Keywords: Crowds,
People, Jumping,
Dynamic loads,
Experiments,
Numerical Modelling
© Building Research
Establishment Ltd 2004

36|The Structural Engineer – 7 September 2004
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..(3)
Hence if
α
is defined, the load model can be evaluated.
From experimental observations
α
= 1/3was suggested as
representing normal jumping5.However, this model is for
perfect repetitive movement, but perfection is impossible to
achieve and there will be variations about this ideal model.
Assume that an individual is jumping in response to an
audible prompt at a given frequency and consider the possi-
ble variables.
•First, the contact ratio
α
and hence the height of jumping
could vary, thus providing a variation in rnand
φ
n.
•Second, although the jumper will try to jump at the
prompt frequency, the jumping may not be perfectly
aligned with the prompt giving a variation in the jumping
frequency.
•Finally there will be a phase or time difference between
the prompt and the individual jumping and this difference
will vary between individuals.
These variations may be too small to have a significant effect
on the loading produced by an individual; however, their
influence may be significant when a group of jumpers is
considered. Within this paper these possible variations will
be examined to see their influence and identify any charac-
teristics that are likely to be observed in experiments.
Experimental data
Experimental procedure
In 1997 and 2001 controlled experiments were undertaken
at BRE’s Cardington laboratory to determine the loads
generated by crowds jumping, focusing on the variation in
load with crowd size. Tests were undertaken on two floors
with different sized groups jumping at selected frequencies3.
The group sizes were selected to be powers of 2 with 64
being the largest group that could be accommodated safely.
The jumping was co-ordinated using a musical beat at
selected frequencies and both the vertical acceleration and
displacement of the centre of the floor were recorded.
For the experiments only the lowest three Fourier coeffi-
cients (FCs) were determined directly. The frequency of the
4th FC was near to the principal frequency of the floor and
here the significant resonance effects would affect the accu-
racy of the evaluation method.
Variation in Fourier coefficients with group size
To determine the loads, the displacement measurements
were used and subjected to band-pass filtering around the
load frequency and two and three times that value. This
provides the response to the different Fourier components of
the load. The dynamic displacements for each Fourier compo-
nent were then obtained by evaluating the rms value of the
filtered displacement and multiplying this value by √2, to
obtain an ‘average’ peak value rather than determining the
absolute peak value. The individual FCs were determined by
dividing the peak dynamic displacement for the selected
Fourier term by the static displacement of the group,with a
correction for any dynamic amplification. The FCs were then
plotted against the number of people in the group on a log/log
plot, together with regression lines for each FC based on a
power relationship. The data are shown in fig 1.
The values which define the regression lines provide the
numerical values of the FCs including their variation with
crowd size and these were used in the load model to evalu-
ate both acceleration and displacement for comparison with
experiment. It was apparent that although this model gave
similar amplitudes of acceleration and displacement to the
average values that were measured it did not replicate the
variations seen in the experimental data.
The regression lines are defined by the following equa-
tions which characterise the variation in FCs with the
number of people ‘p’ in the group and these can be used
within the load model:
First Fourier coefficient r1,p = 1.61 ×p–0.082
Second Fourier coefficient r2,p = 0.94 ×p–0.24
Third Fourier coefficient r3,p = 0.44 ×p–0.31
Variations in frequency and contact ratio for individuals
From the measurement or observation of a group of people
jumping in response to a rhythmic beat it would be very
difficult to determine the variation in jumping frequency of
the individuals within that group. However, determining
variations in frequency is possible for an individual jumping
alone. For this work it has been assumed that the load gener-
ated by a group of people jumping is the same as the sum of
the loads generated by those same people jumping individ-
ually. This hypothesis removes the difficulty of examining
the frequency variations between individuals in a group.
To represent a group the jumping frequencies generated
by several individuals should be considered. Therefore, the
frequency variation of one person jumping for a number of
cycles is examined first. Then the same procedure is used for
several individuals to generate a set of data which describes
the frequency variation of jumping loads.
The method for investigating the variation of jumping
frequency can also be applied to study the variation of the
contact ratio.
There were eight records for individuals, four jumping to
a beat of 1.90Hz and four to a beat of 2.15Hz. For analysis
the recorded vertical displacement time histories were
band–pass filtered to leave the response to the first Fourier
component of the load. For, example for the first test, the
person was jumping at 1.90Hz, so the data were band-pass
filtered between 0.95Hz and 2.85Hz. The filtered data were
then divided by the static displacement for the individual
and the appropriate amplification factor for excitation of the
principal mode by a load at 1.90Hz, to yield a normalised
load in which the peaks are equivalent to the first FC of the
load. The resulting time history is shown in fig 2.
From the figure it can be appreciated that the amplitude
of the peaks varies throughout the test. It should be recog-
nised that the digital filtering does affect the amplitude of
the first and last cycles. A computer program was used to
determine the amplitude of each peak and the time inter-
val between the peaks. For this example the peaks vary
about a mean value of 1.61 which equates to a contact ratio
of 0.47.
The examination of the individual time histories suggests
that the jumper selects a particular jump height (or contact
ratio) and retains this throughout the short jumping event,
paper: ellis/ji
Fig 1.
Fourier coefficients
determined from the
tests

7 September 2004 – The Structural Engineer|37
albeit with slight variations about the selected value. The
distribution of contact ratios when normalised by the
average contact ratio for the event is shown in fig 3. It can
be appreciated that this can reasonably be represented by a
normal distribution with a standard deviation of 0.082.
Although the jumping is co-ordinated by a musical beat,
it may not be perfectly in time with the prompt giving a vari-
ation in the jumping frequency about the prompt frequency.
The jumper probably tries to align the jumping with the
prompt frequency and hence tries to compensate for any
error in one jump by a slight adjustment to the next jump,
thus leading to a process of successive corrections.This may
be more severe at jumping frequencies which the jumper
finds difficult to achieve, but for this study frequencies
around 2Hz were selected which are reasonable comfortable
jumping frequencies.
By examining the time of each peak response in fig 2 the
variation in jumping frequency throughout the displace-
ment-time history is revealed. The average frequency
throughout this record is 1.91Hz (the requested frequency
being 1.90Hz). The distribution for the jumping frequencies,
when normalised by the requested frequency is shown in fig
4. It can be appreciated that this can be reasonably repre-
sented by a normal distribution with a SD of 0.037.
Numerical modelling
The variables to consider and determination of Fourier
coefficients
In the introduction three possible variations in jumping
were mentioned, namely contact ratio, jumping frequency
and relative phase. In this section these variables will be
examined. This is still a simplification of the real situation
because these variables may themselves change whilst an
individual is jumping and be affected by factors like the
individual’s ability and physical state, and possibly his inter-
action with others within a crowd. However, it is reasonable
to assume that the variables can be specified for a particu-
lar event and hence their influence on the generated loads
can be assessed.
The basic analysis procedure is to determine a load-time
history for an individual jumping based on the load model
given in equation 1, but taking account of variability in the
parameter being considered. To determine a load-time
history for a crowd, an appropriate number of individual
load-time histories are combined, with the final load-time
history being divided by the number of individual records.
The combined load-time history is then analysed using a
similar procedure to that used for the experimental data, i.e.
the data are band-pass filtered and the average peak load for
each Fourier component determined.
Generation of data obeying the normal distribution
The calculations consider variations to several parameters
based upon a normal distribution with a specified standard
deviation. To generate a normal distribution a method
similar to that developed for ref 6 is used. It makes use of a
random number generator that generates numbers between
0 and 1. The process generates 8192 numbers between 0
and 1 in 32 bands of equal width. The number of elements
in each band is calculated, to the nearest integer, based on
the normal distribution, and covers ±3 standard deviations.
The random numbers are generated and stored within each
band. When a band has the required number of elements,
further numbers that would fall into the band are discarded.
The process continues until all bands are full. All the
numbers are then combined in a single array and the
elements re-arranged in a random order to remove the
effects of the selection process.The data are then normalised
to leave the array with an average value of 0.0 and a
Standard Deviation of 1. The choice of ±3 standard devia-
tions includes 99.7% of values within the true distribution.
To determine a random number for a given normal distri-
bution with average value Xand standard deviation Y,a
value from the array is selected, multiplied by Yand then
added to X.
Modelling variations in contact ratio and frequency for an
individual
To replicate the load for an individual jumping two vari-
ables are considered. First the contact ratio is selected and
the target frequency is specified. Then random variables
(based on the selected normal distributions) are introduced
paper: ellis/ji
Fig 2.
Normalised load for
first Fourier
component, test 1
Fig 3.
Normalised contact
ratios for all single
person events
Fig 4.
Normalised
frequencies for all
single person events
Fig 5.
Data generated with
characteristics of test 1

38|The Structural Engineer – 7 September 2004
to modify the contact ratio and the frequency. One load cycle
is then determined. Values for frequency and contact ratio
are then selected for the next load cycle and the load-time
history calculated and appended to that of the previous cycle.
This process is repeated for the duration of the jumping.
The variation in the contact ratio is selected from the exper-
imentally determined normal distribution but that for the
frequency is based on a series of successive corrections upon
which the random variation is applied. The successive
correction procedure determines the time for one cycle based
on the selected frequency which will be slightly different to
the time defined by the musical prompt. The next frequency
is calculated based on the time required to re-align with the
musical prompt, but with another variation imposed, etc.
This method of determining frequency is due to the fact that
the person is jumping in response to a prompt and hence the
jumper tries to compensate for any error in one jump by a
slight correction to the next jump, thus leading to a process
of successive corrections. This may be considered to be a
feedback process. This contrasts with the variations for the
contact ratio where there is no prompt or target. An example
of the generated output for the first Fourier component for
the first test is shown in fig 5, with the initial selection of
alpha being 0.47 and the frequency being 1.90Hz. It can be
seen that the artificially generated data exhibits similar
characteristics to the measurements (fig 2).
Modelling variations in contact ratio and frequency for
groups
From the measurements it appears that the individual
selects a jumping style and tries to maintain this through-
out any one event. Hence in the previous section it was possi-
ble to select values of contact ratio and frequency for the
event that was considered. For the group tests, the target
frequency is known, but the selection of the contact ratio (
α
),
or jump height, is the choice of the individual jumpers.Based
on the data shown in fig 1 and focusing on the single
jumpers, an average value of
α
of 0.4 is selected to model the
observed variations in the three FCs. From analysis of the
first FCs a standard deviation of 0.08 was derived. Whether
a normal distribution is appropriate here is questionable,
but as the objective is to illustrate how the observed exper-
imental variations can occur, an exact distribution of the
loading is not required and, in fact, this cannot be derived
from the available measurements. Thus for each individual
a value of
α
is selected, the target frequency is known and
the variations based upon the experimental results for the
individuals is known, hence the load-time history can be
calculated.
If various sized groups with these characteristics are
considered, and analysed in a similar manner to the exper-
imental data, then the resulting variation in FCs with group
size can be obtained. The load was evaluated six times for
groups of 1, 2, 4 …64 and the results shown in fig 6. The
figure indicates that small variations in the contact ratio and
frequency do not produce a significant reduction in FCs for
increasing group sizes in comparison with the measure-
ments shown in fig 1. Increases in contact ratio and corre-
sponding decreases in FC will be balanced by the equally
likely decreases in contact ratio and corresponding increases
in FC.
In contrast to the results shown in fig 6, if the frequency
feedback loop is removed, to look at a random selection of
frequencies (within the distribution), a severe reduction in
FCs with number of people is seen (in comparison with the
measurements) especially for the first FC.This would repre-
sent the situation without a musical prompt and must there-
fore be rejected.
Variation in phase within a group
Phase is the parameter that those involved with modelling
have previous considered6, 7.Equation 4 provides the model
for an individual jumping but includes a further phase
difference
ψ
.
. sinFt G r T
ntn10 2
n
ss n
pn
1
=+ ++
rz}
3
=
!
J
L
K
K
K
_
d
N
P
O
O
O
i
n
...(4)
In this model it is assumed that the individual jumps at
the correct frequency and at a constant height (or
α
) but
there is a phase difference
ψ
between the prompt and the
response which is different for individuals. By considering
the individual Fourier terms,it is relatively simple to deter-
mine the FCs for groups of people when the variation in
ψ
is defined. This has been undertaken for
α
= 1/3and evalu-
ated for various sized groups. Following a few trials a SD of
0.22
π
was found to provide a reasonable correlation with the
measurements based on the attenuation in FCs with group
size. The results are given in fig 7.
It can be seen that although this is exhibiting attenuation
of FCs with increasing group size similar to the experi-
mental values, there is no variation for FCs for the indi-
viduals as variations of jumping frequency and contact ratio
are not considered here, which is clearly at variance with
the measurements. Also the group-time histories would
provide a steady-state loading, again in contrast to the
measurements.
paper: ellis/ji
Fig 6.
FCs against group
size – no phase
variation
Fig 7.
FCs against group
size considering only
phase variation
Fig 8.
FCs against group
size – combined
model

7 September 2004 – The Structural Engineer|39
A combination of all of the variations
Although the variables have been examined in isolation, it is
apparent that they actually occur together. Hence all the
variables have been included in one program and a series of
load-time histories calculated. It is assumed that the
frequency is determined by a process for successive correc-
tions upon which a variation is imposed, rather than as a
purely random variable. The values of the variables that
appear to give similar characteristics to those observed in the
experiments are: a frequency variation with normalised SD
of 0.037, a phase variation with a SD of 0.18
π
,and a value of
α
of 0.4 with a normalised SD of 0.08, with a variation
throughout each individual time history with a SD of 0.082.
A plot of the variation in FCs with crowd size is given in
fig 8, which compares reasonably well with the experimen-
tal data. However, with so many variables in the numerical
generation of the data, some differences in the results are to
be expected, in the same way that slight differences would
inevitably result if the experiments were repeated. The SD
of the phase variation is slightly smaller than that seen in
the previous section, primarily because of the weighting due
to the FCs for the tests with single jumpers, which are based
on an average
α
of 2/5here, but on an
α
of 1/3in the previous
section.
Although the results presented are for the first three FCs,
the calculation procedure can evaluate more Fourier terms,
and this will be considered later.
Calculated response
Having developed a method of generating a load-time history
it is of interest to see how this can be used to evaluate
response. The normalised load intensity for 32 people is
determined for the first test floor using the method given in
ref 3.
G32=Gave
γ
(32)=67.6 × 9.81 × 320.79=10249N
The calculated load-time history, which is based on unity
load and includes four Fourier terms, is increased by the
above load intensity to represent the actual load due to 32
people. This is used along with the floor’s characteristics
and the frequency measured for the particular tests to deter-
mine a displacement-time history using the Duhamel inte-
gral method. Measured and calculated displacement time
histories are given in fig 9 and 10.
It can be appreciated that the calculations are broadly in-
line with measurements, although an exact match is not to
be expected. For comparison the displacement-time history
calculated using the load model derived directly from the
experimental results is shown in fig 11.
Evaluating the loads for larger groups
The testing was restricted to 64 people because of safety
concerns related to the test floor, but the development of the
load model allows evaluation for larger groups. This has
been undertaken for groups of up to 8192, using the values
defined in the section entitled a combination of all of the
variations and selecting the group sizes to be powers of 2.
These are shown graphically in fig 12 for the first three FCs.
With the results for up to 64 people based upon six calcu-
lations for each group size, it was seen that the spread in
calculated FCs became less for the larger groups. This was
probably due to the statistical variations being smoothed by
a larger sample size. This becomes even clearer when groups
up to 8192 people are examined. It can be seen that the FCs
are no longer reducing with increasing crowd size but attain
a constant value, suggesting that the power relationships
derived from the experimental data are only appropriate for
the smaller groups.
Although it is possible to calculate loads for such large
groups, it is important to realise that other factors,which are
not included in the study, may become important. For
paper: ellis/ji
Table 1: Evaluation of FCs for different values of phase
variation for groups of 8192
SD Fourier coefficients of the resulting loads
Phase FC1 FC2 FC3 FC4 FC5 FC6
0.18 π1.40 0.469 0.072 0.0070 0.0024 0.0018
0.12 π1.52 0.655 0.157 0.0281 0.0073 0.0019
0.00 π1.62 0.852 0.283 0.0831 0.0434 0.0245
Fig 12.
FCs against group
size – combined
model for up to 8192
people
Fig 9. (left)
Measured
displacement-time
history for 32 people
Fig 10. (right)
Calculated
displacement-time
history for 32 people
Fig 11.
Displacement-time
history for 32 people
– experiment model

40|The Structural Engineer – 7 September 2004
example, for a large group there could be a time delay
between the people nearest the loudspeakers hearing the
musical prompt and those furthest away and this has been
observed on a grandstand8.This can result in significant
phase differences and therefore different loads.
With the finding that the FCs settle to final values for
large crowds it is possible to quantify the effects of chang-
ing the Standard Deviations for the phase variation. For
illustration purposes the calculations will be repeated for
groups of 8192 people, but examining three values for phase
variation, i.e. 0.18
π
,0.12
π
and 0.00. The first of these values
is the best-fit value based on the experimental results; the
second is an hypothetical value to reflect a much better co-
coordinated group of jumpers, and the third value assumes
no phase variation (as considered earlier). The results of the
calculations are given in table 1 and six FCs are presented.
It might reasonably be assumed that a group of profes-
sional dancers would have these improved all round char-
acteristics, hence a lower phase variation (0.12
π
) than that
seen in the experiments (0.18
π
), but it is of interest to see
that this has only a relatively small effect on the first FC, but
a larger effect for the some of the higher FCs. However, no
information is available for the co-ordination of different
groups of people, so these cannot yet be quantified.
Discussion
The work presented in this paper has used variations on the
load model for an individual jumping and combinations of
load-time histories for individuals in an attempt to replicate
some experimental results that were obtained for groups of
people jumping on two floors. It is not suggested that this is
the best method for determining a load model, because the
model derived directly from the experiments is far easier to
use for calculating structural response for small groups and
the values given in table 1 are easier to use with large
groups. However the modelling serves to explain some of
the characteristic variations that were observed in the
experiments. It is clear that there are many variables to
consider with a group of people jumping, and each load-time
history will be different even for the same group of people
jumping at different times.The modelling suggests that the
variations will be smaller for larger groups. However, this
model is based upon one set of measurements and some
variations are to be expected with different groups, hence
there will never be one definitive solution.
The range of variables used in the modelling is based on
limited experimental results. It is apparent that different
groups of people will exhibit different characteristics. For
example a group of professional dancers would be far better
co-ordinated than a group of construction workers and there-
fore generate much higher loads. For the experiments the
groups were students with an interest in music, and this
may be reasonably representative of people at discotheques.
Whether the same group of students would achieve better co-
ordination if the same exercise was repeated several times
remains unanswered; but it does seem probable that co-ordi-
nation would improve with experience.
The data on the variations in frequency and contact ratio
have been determined from a relatively small number of
jumping records for individuals, although it would be rela-
tively simple to obtain a large amount of data on these items
from a few more experiments. More data could be used to
provide information on the differences between individuals
and a better understanding of the statistical distributions;
but this was not undertaken here as the intention was to
limit the study to replicating the data measured at
Cardington. Nevertheless, the data presented do help to
explain the significant variations that are seen in the exper-
iments. The information on phase variation has been deter-
mined from the experimental data with the different sized
groups and it would be difficult to get a reasonable estimate
of this parameter by studying individuals or small groups.
This is perhaps the most important variable for determin-
ing attenuation in loads with increasing group size and it
may be difficult to obtain more data here unless experiments
like that undertaken at Cardington are repeated.
One item that the modelling assumes is that everyone is
jumping. In reality this is an extreme case which generates
very large dynamic loads, but in many situations some
members of a group may not be jumping. Stationary people
would serve to reduce the structural response significantly9
as they not only fail to contribute to the load but also provide
a large increase in the system damping. However, for some
structures, like dance floors,it would seem wise to consider
the situation with everyone jumping because this extreme
load case may be encountered. For other structures, like
inclined seating decks on grandstands, this load case may
not be appropriate even for pop concerts.
Conclusions
The paper investigates the numerical modelling of the loads
produced by crowd jumping and considers variations of
contact ratio, frequency and phase lag in the models based
on measurements obtained on two floors with groups of up
to 64 jumpers. The main findings from this work are:
•A numerical method has been developed for determining
the loads generated by groups of people jumping. The
calculated loads correspond reasonably well with those
measured in an earlier experimental study.
•The standard deviations for contact ratios and jumping
frequencies have been determined from records of indi-
viduals jumping. The normalised standard deviation for
contact ratio is 0.082 and for frequency is 0.037.
•The variations in frequency and contact ratio do not signif-
icantly affect the FCs for different group sizes although
they explain some of the variations observed in the exper-
iment.
•The standard deviation for phase lags determined from
the group tests is 0.18π.
•The phase lags have a significant effect on the variation
of FCs with crowd size. For small groups (say up to 64
people) the reduction in FCs is more marked for higher
FCs, but for large crowds the FCs reach a constant value.
Acknowledgments
The authors would like to thank the UK Department of the
Transport, Local Government and the Regions for sponsor-
ing this work.
paper: ellis/ji
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