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

Vibration problem on cantilever grandstand

Flexible structure with large span and lightweight

Synchronised crowd loadings 1.5 ~ 2.8 Hz

+

Dynamic analysis of cantilever grandstand

Human-structure interaction

Passive crowdCrowd model

Active crowd

Load model

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

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

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

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

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

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

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

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

Results - Maximum displacement

Results – RMS Acceleration

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

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

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