University of Oxford Modelling of joint crowd-structure system using equivalent reduced- DOF system Jackie Sim, Dr. Anthony Blakeborough, Dr. Martin Williams.

University of Oxford

Modelling of joint crowd-structure system using equivalent reduced-DOF system

Jackie Sim, Dr. Anthony Blakeborough, Dr. Martin Williams

Department of Engineering ScienceOxford University

University of Oxford

Cantilever grandstands

University of Oxford

Dynamic analysis of cantilever grandstand

Human-structure interaction

Passive crowdCrowd model

Active crowd

Load model

University of Oxford

Full model

ms

F x

Total mass of crowd = ms

ms

F x

Crowd as 2DOF system

Structure as SDOF system

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Equivalent reduced DOF systems

Equivalent SDOF system

Equivalent 2DOF system

ms

F x

Full model

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Contents

Crowd model Response of full model Equivalent SDOF model Equivalent 2DOF model

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DOF 2m2

k2

c1

k1

m1

c2

m0

F

y2

y1

DOF 1

gx

DOF 1

m2

k2

c1

k1

m1

c2

F

y2

y1

DOF 2

gx

Seated model Standing model

Individual models – Griffin et al.

ix

iFim

gapp

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Crowd response

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

)

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Crowd model

Transfer functions:

Seated:

Standing:

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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Dynamic analysis (1)2% structural damping,

Natural frequency of 1 to 10 Hz.

50% seated and 50% standing crowds

= 0%, 5%, 10%, 20%, 30% and 40%

ms

F x

Crowd mass = ms

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Dynamic analysis (2)

DMF = Peak displacement / Static displacement

SDOF structure

Seated / standing crowd

DisplacementExcitati

on force

Interaction force

Acceleration+

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0 1 2 30

10

20

30

Frequency (Hz)

DM

F

2 3 4 50

10

20

30

Frequency (Hz)D

MF

4 5 6 70

10

20

30

Frequency (Hz)

DM

F

6 7 8 90

10

20

30

Frequency (Hz)

DM

F

0%5%10%20%30%40%

0%5%10%20%30%40%

Results – DMF vs Frequency

2 Hz structure 4 Hz structure

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Summary of results (1): Resonant frequency reduction factor

0 2 4 6 8 100.7

0.75

0.8

0.85

0.9

0.95

1

Natural frequency of bare structure (Hz)

Fre

qu

en

cy r

ed

uct

ion

fa

cto

r

5%10%20%30%40%

F.R.F. = Change in frequency / Frequency of bare structure

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Summary of results (2): DMF reduction factor

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Natural frequency of bare structure (Hz)

DM

F r

ed

uct

ion

fa

cto

r

5%10%20%30%40%

DMF R.F. = Change in DMFmax / DMFmax of bare structure

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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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Equivalent SDOF system

DMF2

1

SDOF system transfer function:

Curve-fit DMF frequency response curve over bandwidth

0 1 2 30

10

20

30

Frequency (Hz)

DM

F

2 3 4 50

10

20

30

Frequency (Hz)

DM

F

Frequency (Hz)

DM

F

6 7 8 90

10

20

30

Frequency (Hz)

DM

F

0%5%10%20%30%40%

0%5%10%20%30%40%

Frequency (Hz)

peak2

1DMF

peakDMF

0 1 2 30

10

20

30

Frequency (Hz)

DM

F

2 3 4 50

10

20

30

Frequency (Hz)

DM

F

Frequency (Hz)

DM

F

6 7 8 90

10

20

30

Frequency (Hz)

DM

F

0%5%10%20%30%40%

0%5%10%20%30%40%

kcsmsk

2

11

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Dynamic properties

1 2 3 4 5 6 7 8 9 100.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Natural frequency of bare structure (Hz)

Mas

s ra

tio

(%

)

=5%=10%=20%=30%=40%

1 2 3 4 5 6 7 8 9 100.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Natural frequency of bare structure (Hz)

Sti

ffn

ess

rati

o (

%)

=5%=10%=20%=30%=40%

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Natural frequency of bare structure (Hz)

Dam

pin

g r

atio

(%

)

=5%=10%=20%=30%=40%

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Error analysis (1)

Peak DMF relative error

DM

F

Frequency (Hz)

Peak DMF

F*

Full model

EquivalentSDOF model

%100modelFullofDMFPeak

DMFPeak

Resonant frequency relative error

%100modelFullofFrequencyResonant

F*

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Error analysis (2)

1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3

4

5

Natural frequency of bare structure (Hz)

Pe

ak

DM

F r

ela

tive

err

or

(%)

=5%=10%=20%=30%=40%

1 2 3 4 5 6 7 8 9 10-2

0

2

4

6

8

10

Natural frequency of bare structure (Hz)

Res

on

ant

freq

uen

cy r

elat

ive

erro

r (%

)

=5%=10%=20%=30%=40%

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Equivalent 2DOF system

ms

F

x

Crowd modelled as SDOF system

Structure remains the same SDOF system

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SDOF crowd model

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

Seated men

Mean responseFitted SDOFFitted 2DOF

01

22

012

2*

bsbsb

asasasmapp

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

Standing men

Mean responseFitted SDOFFitted 2DOF

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Dynamic analysis

SDOF structure

SDOFSeated / standing crowd

DisplacementExcitati

on force

Interaction force

Acceleration+

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Error analysis

1 2 3 4 5 6 7 8-5

-4

-3

-2

-1

0

1

2

3

4

5

Natural frequency of bare structure (Hz)

Pe

ak

DM

F r

ela

tiv

e e

rro

r(%

)

=5%=10%=20%=30%=40%

1 2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

Natural frequency of bare structure (Hz)

Re

so

na

nt

fre

qu

en

cy

re

lati

ve

err

or

(%)

=5%=10%=20%=30%=40%

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Bode diagrams

1.2 1.4 1.6 1.8 2 2.20

10

20

30

Frequency (Hz)

DM

F

2 Hz structure

2 2.5 3 3.5 4 4.50

5

10

15

Frequency (Hz)

DM

F2 4 6 8

0

1

2

3

4

5

Frequency (Hz)

DM

F

6 Hz structure

2 4 6 8 100

1

2

3

4

Frequency (Hz)D

MF

7 Hz structure

Full model2DOFSDOF

Full model2DOFSDOF

1.2 1.4 1.6 1.8 2 2.20

10

20

30

Frequency (Hz)

DM

F

2 Hz structure

2 2.5 3 3.5 4 4.50

5

10

15

Frequency (Hz)D

MF

2 4 6 80

1

2

3

4

5

Frequency (Hz)

DM

F

6 Hz structure

2 4 6 8 100

1

2

3

4

Frequency (Hz)

DM

F

7 Hz structure

Full model2DOFSDOF

Full model2DOFSDOF

1.2 1.4 1.6 1.8 2 2.20

10

20

30

Frequency (Hz)

DM

F

2 Hz structure

2 2.5 3 3.5 4 4.50

5

10

15

Frequency (Hz)

DM

F

2 4 6 80

1

2

3

4

5

Frequency (Hz)

DM

F

6 Hz structure

2 4 6 8 100

1

2

3

4

Frequency (Hz)

DM

F

7 Hz structure

Full model2DOFSDOF

Full model2DOFSDOF

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Conclusions

Passive crowd adds significant damping 1 to 4 Hz – behaviour of a SDOF system > 4 Hz – behaviour of a 2DOF system