Response of a joint passive crowd-SDOF system subjected to crowd jumping load
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Response of a joint passive crowd-SDOF system subjected to crowd
jumping load
Jackie Sim, Dr. Anthony Blakeborough, Dr. Martin WilliamsDepartment of Engineering Science
University of Oxford
University of Oxford
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Vibration problem on cantilever grandstand
Flexible structure with large span and lightweight
Synchronised crowd loadings 1.5 ~ 2.8 Hz
+
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Dynamic analysis of cantilever grandstand
Human-structure interaction
Passive crowdCrowd model
Active crowd
Load model
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Contents
Outline
1. Passive crowd model
How to model the seated and standing crowds?
2. Active crowd model
How to model the jumping crowd?
3. Analysis of active + passive crowds on SDOF structure
What is the structural response?
4. Results
5. Case study
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Passive crowd model (1)
• Griffin et al. – experimental tests and model development
• Measure the apparent mass of 24 seated and 12 standing men:
ix
iFim
gapp
DOF 1
m2
k2
c1
k1
m1
c2
F
y2
y1
DOF 2
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Passive crowd model (2)
• Curve-fitting the crowd apparent mass response
• Crowd model represented as transfer function
• Seated:
• Standing
0 5 10 15 200.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency (Hz)
No
rmal
ized
ap
par
ent
mas
s SeatedStanding
0 5 10 15 20-100
-80
-60
-40
-20
0
20
Frequency (Hz)P
ha
se
(d
eg
ree
)
01.38109673.13293948.602533.7400.1
28.379423974.13467327.318126.3213.0234
234
ssss
ssss
20.889932707.26219875.967565.9600.1
62.893300667.25837809.504010.4200050.0234
234
ssss
ssss
Fourth order polynomial i.e. 2DOF system
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Active crowd model (1)
Experimental tests
- University of Surrey
- 100 test subjects
- Each individual jumping on rigid force plates
- Metronome at 4 beat frequencies (1.5, 2, 2.67 and 3.5 Hz)
- Synchronised test results were analysed
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Active crowd model (2)
0 1 2 3 4 5-500
0
500
1000
1500
2000
Load
(N
)
Time (s)0 0.1 0.2 0.3 0.4 0.5
-500
0
500
1000
1500
2000
2500
Lo
ad
(N
)
Time (s)
Load-time history at 2 HzAverage impulse of
each individual
Average impulse of all individuals => Crowd jumping load
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Active crowd model (3)
Crowd jumping load
0 1 20
1
2
3 1.5 Hz
Time (s)0 1 2
0
1
2
3 2 Hz
Time (s)
0 1 20
1
2
3 2.67 Hz
Time (s)0 1 2
0
1
2
3 3.5 Hz
Time (s)
F'
F'
F'
F'
Beat Frequency
(Hz)1st 2nd 3rd
1.5 0.911 0.150 0.034
2 1.193 0.337 0.040
2.67 1.228 0.311 0.032
3.5 1.020 0.157 0.008
Fourier coefficients
High FC => Better synchronisation
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ms
F
x
Analysis (1)
Passive crowd-SDOF system subjected to crowd jumping
load
SDOF structure
Seated / Standing
crowd
Crowd jumping load
Interaction force
_
+
Feedback system representation
Displacement
Acceleration
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Analysis (2)
Joint passive crowd-SDOF system
H()
Structuralresponse
R()
Crowdjumping load
F()
Frequency domain analysis
Parameters
Natural frequency of empty structure: 1 ~ 8Hz
Structural damping ratio: 2%
Passive crowd mass ratio, : 0 ~ 0.3 (increment of 0.05)
Subjected to crowd jumping load at 1.5, 2, 2.67 and 3.5Hz
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Results - Maximum displacement
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Results – RMS Acceleration
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Case study – Cardiff Millennium Stadium
• First mode at 2.9 Hz• Crowd mass = 16800 kg per
bay• Assume = 0.3• Structure mass = 56000 kg for
one bay• Structure stiffness;
MN/m6.185600029.2 22 sns mk
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Results
• Rugby match between Australia and France in Nov 1999
• Displacement of approximately 50mm reported after the match
• Half full capacity
• Mass ratio, = 0 ~ 0.15
Maximum displacement (mm)
RMS Acceleration (times g = 9.81m/s2)
Crowd jumping frequency (Hz)
1.5 2 2.67 3.5
0 14.2 12.8 38.2 14.0
0.05 7.3 9.0 34.9 8.2
0.1 3.3 4.8 27.1 3.7
Crowd jumping frequency (Hz)
1.5 2 2.67 3.5
0 0.20 0.14 0.70 0.34
0.05 0.08 0.09 0.65 0.2
0.1 0.03 0.05 0.52 0.08
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Concluding remarks
• Passive crowd adds significant damping to the system and alters the resonance frequency
• Preliminary analysis on the Cardiff Millennium Stadium gave good results
• Current work – statistical model of the crowd jumping load – taking into account the timing of each individual