Transcript Dynamic analysis of cantilever grandstand

Modelling of joint crowd-structure
system using equivalent reducedDOF system
Jackie Sim, Dr. Anthony Blakeborough, Dr. Martin Williams
Department of Engineering Science
Oxford University
University of Oxford
Cantilever grandstands
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Dynamic analysis of cantilever
grandstand
Human-structure interaction
Active crowd
Passive crowd
Load model
Crowd model
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Full model
Crowd as 2DOF system
Total mass of crowd = g ms
ms
ms
F
F
x
x
Structure as SDOF system
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Equivalent reduced DOF systems
Equivalent
SDOF
system
ms
F
Equivalent
2DOF
system
x
Full model
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Contents




Crowd model
Response of full model
Equivalent SDOF model
Equivalent 2DOF model
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Individual models – Griffin et al.
y2
y2
m2
k2
y1
DOF 2
m2
DOF 2
c2
k2
c2
mapp i  
y1
m1
m1
DOF 1
DOF 1
k1
k1
c1
m0
F
Seated model
c1
xg
xg
F
Standing model
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F i 
xg i 
Crowd response
Normalized apparent mass
1.6
20
Seated
Standing
0
Phase (degree)
1.4
1.2
-20
1
-40
0.8
-60
0.6
-80
0.4
0.2
0
5
10
15
Frequency (Hz)
20
-100
0
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5
10
15
Frequency (Hz)
20
Crowd model
Transfer functions:
Fourth order polynomial
i.e. 2DOF system
Seated:
 0.13s 4  32.26s 3  3181.27s 2  134673.74s  3794239.28
1.00s 4  74.33s 3  6025.48s 2  132939.73s  381096.01
Standing:
 0.00050s 4  42.10s 3  5040.09s 2  258378.67s  8933006.62
1.00s 4  96.65s 3  9675.75s 2  262198.07s  8899327.20
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Dynamic analysis (1)
2% structural damping,
Crowd mass
= g ms
Natural frequency of 1 to 10 Hz.
50% seated and 50% standing
crowds
ms
F
g = 0%, 5%, 10%, 20%, 30% and
40%
x
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Dynamic analysis (2)
Excitation
force
+
SDOF structure
Displacement
Acceleration
Interaction
force
Seated /
standing crowd
DMF = Peak displacement / Static displacement
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Results – DMF vs Frequency
2 Hz structure
0% 0%
5%
10%5%
20%10%
30%
40%20%
DMF
20
30%
40%
10
0
0
30
20
DMF
30
4 Hz structure
10
1
2
Frequency (Hz)
0
2
3
20
20
DMF
30
DMF
30
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3
4
Frequency (Hz)
5
Summary of results (1):
Resonant frequency reduction factor
Frequency reduction factor
1
0.95
0.9
F.R.F. = Change in frequency /
Frequency of bare structure
0.85
0.8
0.75
0.7
0
2
4
6
8
Natural frequency of bare structure (Hz)
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5%
10%
20%
30%
40%
10
Summary of results (2):
DMF reduction factor
5%
10%
20%
30%
40%
1.4
DMF reduction factor
1.2
1
0.8
DMF R.F. = Change in DMFmax /
DMFmax of bare structure
0.6
0.4
0.2
0
0
2
4
6
8
Natural frequency of bare structure (Hz)
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10
Why reduced-DOF system?

Full crowd-model: 2DOF crowd + SDOF structure

A simplified model for
 Easier analysis
 Insight into the dynamics
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20
30%
40% 1
Equivalent SDOF system
10

SDOF system transfer function:
2
DMF
DMF
30%
40% 20%
1
1
 2
k ms  cs  k
0
0
1
2
3
Frequency (Hz)1
 Curve-fit DMF frequency response curve over
DMF
2
bandwidth
3
DMFpeak
2
DMF
1
DMF
2
DMFpeak
1
Frequency
(Hz)
Frequency (Hz)
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Dynamic properties
1.8
g=5%
g=10%
g=20%
g=30%
g=40%
1.7
1.05
1
Stiffness ratio (%)
1.5
1.4
1.3
1.2
0.95
0.9
0.85
1.1
0.8
1
0.75
0.9
1
2
3
4
5
6
7
8
Natural frequency of bare structure (Hz)
25
9
10
0.7
1
g=5%
g=10%
g=20%
g=30%
g=40%
2
3
4
5
6
7
8
Natural frequency of bare structure (Hz)
g=5%
g=10%
g=20%
g=30%
g=40%
20
Damping ratio (%)
Mass ratio (%)
1.6
1.1
15
10
5
0
1
2
3
4
5
6
7
8
Natural frequency of bare structure (Hz)
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9
10
9
10
Error analysis (1)
Peak DMF relative
error
 Peak DMF
 Peak DMF
 100%
Peak DMF of Full model
DMF
Full model
Equivalent
SDOF model
Resonant frequency
relative error
 F*
Frequency (Hz)
 F*
 100%
Resonant Frequency of Full model
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Error analysis (2)
5
3
Resonant frequency relative error (%)
4
Peak DMF relative error (%)
10
g=5%
g=10%
g=20%
g=30%
g=40%
2
1
0
-1
-2
-3
1
2
3
4
5
6
7
8
Natural frequency of bare structure (Hz)
9
10
g =5%
g =10%
g =20%
g =30%
g =40%
8
6
4
2
0
-2
1
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2
8
7
6
5
4
3
Natural frequency of bare structure (Hz)
9
10
Equivalent 2DOF system
Crowd modelled as SDOF system
ms
F
x
Structure remains the same
SDOF system
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SDOF crowd model
m
*
app
s  
a 2 s 2  a1s  a 0
b2 s 2  b1s  b0
Seated men
Standing men
1.6
1.6
Mean response
Fitted SDOF
Fitted 2DOF
1.4
Normalized apparent mass
Normalized apparent mass
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Mean response
Fitted SDOF
Fitted 2DOF
1.2
1
0.8
0.6
0.4
5
10
Frequency (Hz)
15
20
0.2
0
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5
10
Frequency (Hz)
15
20
Dynamic analysis
Excitation
force
+
SDOF structure
Interaction
force
SDOF
Seated /
standing crowd
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Displacement
Acceleration
Error analysis
5
3
Resonant frequency relative error (%)
4
Peak DMF relative error(%)
3
g =5%
g =10%
g =20%
g =30%
g =40%
2
1
0
-1
-2
-3
g =5%
g =10%
g =20%
g =30%
g =40%
2
1
0
-1
-2
-4
-5
1
2
3
4
5
6
Natural frequency of bare structure (Hz)
7
8
-3
1
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2
3
4
5
6
Natural frequency of bare structure (Hz)
7
8
DMF
10
DMF
20
Bode
diagrams
10
10
2.2
10
15
4
Hzstructure
structure
6 2Hz
4
DMF
DMF
0
1.2
3
5
1
4
0 01.2
2
3
DMF
210
1.4 1.6 1.8
2
Frequency (Hz)
2.2
6 Hz structure
3
10
0
2
2
54
1
1.424 1.6 1.8
2
6
Frequency(Hz)
(Hz)
Frequency
0022
2
2.2
8
1
3
6 Hz structure
4
6
Frequency (Hz)
4
8
3
DMF
DMF
4
0
2
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2
Full
Fullmodel
model
2DOF
2DOF
2.5
3
3.5SDOF
4
4.5
SDOF
Frequency (Hz)
Full
Fullmodel
model
7 Hz 2DOF
structure
2DOF
SDOF
SDOF
2.5
3
3.5
4
Frequency (Hz)
4.5
7 Hz structure
3
1
5
5
DMF
20
0
2
DMF
30
15
2 Hz structure
1.4 1.6 1.8
2
Frequency
(Hz)
20
DMF
0
1.2
5
DMF
DMF
30
5
SDOF
SDOF
0
2
2.54 3 6 3.5 84
Frequency
Frequency(Hz)
(Hz)
4.5
10
7 Hz structure
10
4
6
8
Frequency (Hz)
Conclusions



Passive crowd adds significant damping
1 to 4 Hz – behaviour of a SDOF system
> 4 Hz – behaviour of a 2DOF system
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