Flutter analysis of an Flutter analysis of an extradosed bridge in Hungaryextradosed bridge in Hungary

MMáátytyáás HUNYADIs HUNYADI

Budapest University of Technology and EconomicsBudapest University of Technology and Economics (BME)(BME)Department of Structural EngineeringDepartment of Structural Engineering

hunyadi@hunyadi@vbt.bme.huvbt.bme.hu

M43 motorway TiszaM43 motorway Tisza--bridgebridge

M43 motorway TiszaM43 motorway Tisza--bridgebridge

Spans: 96,30 + 180,00 + 96,30 mSpans: 96,30 + 180,00 + 96,30 m

Courtesy of PontCourtesy of Pont--TERVTERVFlutter instability investigation:Flutter instability investigation:�� construction phase (cantilever stage)construction phase (cantilever stage)�� final phasefinal phase

Mode shapesMode shapes

nn11=0,55 Hz=0,55 Hz

nn22=0,78 Hz=0,78 Hz nn77=2,47 Hz=2,47 Hz

nn66=2,33 Hz=2,33 Hz nn88=2,80 Hz=2,80 Hz

SzSzéélhatlhatáásoksok

�� KvKváázizi--statikus statikus áállapotllapot�� ererőőttéényeznyezőő

�� Dinamikus vizsgDinamikus vizsgáálatlat�� áátviteli ftviteli füüggvggvéényny

�� aeroelasztikusaeroelasztikus hathatáásoksok((ööngerjesztett erngerjesztett erőők)k)

�� derivatderivatíívumokvumok

Flutter analysisFlutter analysis

�� MiddleMiddle--span cross sectionspan cross section�� 2 DOF2 DOF

Instability criteriaInstability criteria�� ωωcritcrit ,, UUcritcrit

( ) ( ) ( )

( ) ( ) ( )

+++=++

+++=++

B

hAKKAK

U

BKKA

U

hKKABUkcS

B

hHKKHK

U

BKKH

U

hKKHBUhkhchm hh

*

4

2*

3

2*

2

*

1

22

*

4

2*

3

2*

2

*

1

2

2

1

2

1

αα

ρααα

αα

ρ

αα

&&

&&&

&&&&&

Wind tunnel investigationWind tunnel investigation

�� scale 1:100scale 1:100�� forced 2 DOF modelforced 2 DOF model�� ssemiemi--finite and infinite modelsfinite and infinite models

�� angles of attack: angles of attack: --66°°, 0, 0°°, +6, +6°°

�� small turbulent intensity (Ismall turbulent intensity (Iuu≤≤0,5%)0,5%)�� separation of inertial forces, separation of inertial forces, „„nono--windwind”” measuresmeasures

Force coefficientsForce coefficientscc DD

[[ --]]

cc MM[[ --]]

cc LL[[ --]]

infinite model

infinite model

0 5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

Wind tunnel investigationWind tunnel investigation

�� mmotionotion frequencyfrequency f= 1,0; 1,5; 2,0; 2,5; 3,0 Hzf= 1,0; 1,5; 2,0; 2,5; 3,0 Hz

�� wind speedwind speed U=5; 10; 15; 20 U=5; 10; 15; 20 m/sm/s

�� model widthmodel width B=295 mm (scale 1:100)B=295 mm (scale 1:100)

�� measuring by 6 dynamometersmeasuring by 6 dynamometers

�� oscillationoscillation αα=0=0°° ±± 1010°° +6,+6,--6 6 ±± 44°°h= h= ±± 30mm30mm

�� sampling ratesampling rate 100 Hz100 Hz

Bf

UU red =

quasi-steadyhighly non-

stationary

αα[[ °°]]

αα=0 f=2 Hz U=10 =0 f=2 Hz U=10 m/sm/s

t [ms]t [ms] t [ms]t [ms]

FFff--FFf0f0[N

][N

]

FFff

MMzz

MMzz--MM

z0z0[N

m]

[Nm]

αα[[ °°]]

h [mm]

h [mm]

h [mm]

h [mm]

measured force

fitted curve

displacement

Flutter derivativesFlutter derivatives

LLhh

MMα α

--66°° 00°° +6+6°°

0 2 4 6 8 10 12 14-1

0

1

2

3

4

5M43: A* derivatívumok i=0°

Ured

A*1

A*2

A*3

A*4

0 2 4 6 8 10 12 14-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3M43: A* derivatívumok i=-6°

Ured

A*1

A*2

A*3

A*4

0 2 4 6 8 10 12 14-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5M43: A* derivatívumok i=+6°

Ured

A*1

A*2

A*3

A*4

0 2 4 6 8 10 12 14-25

-20

-15

-10

-5

0

5

10

15M43: H* derivatívumok i=0°

Ured

H*1

H*2

H*3

H*4

0 2 4 6 8 10 12 14-20

-15

-10

-5

0

5

10M43: H* derivatívumok i=-6°

Ured

H*1

H*2

H*3

H*4

0 2 4 6 8 10 12 14-25

-20

-15

-10

-5

0

5

10M43: H* derivatívumok i=+6°

Ured

H*1

H*2

H*3

H*4

Flutter derivativesFlutter derivativesαααα =0°

-50

-40

-30

-20

-10

0

10

0 10 20 30 40 50 60 70 80 H1*

H4*

A1*

A4*

H1*s

H4*s

A1*s

A4*s

αααα =0°

-600

-500

-400

-300

-200

-100

0

100

200

0 10 20 30 40 50 60 70 80

H2*

H3*

A2*

A3*

H2*s

H3*s

A2*s

A3*s

( ) ( ) ( )

( ) ( ) ( )

+++=

+++=

B

hAKKAK

U

BKKA

U

hKKABUM

B

hHKKHK

U

BKKH

U

hKKHBULh

*

4

2*

3

2*

2

*

1

22

*

4

2*

3

2*

2

*

1

2

2

1

2

1

αα

ρ

αα

ρ

α

&&

&&

Polynomial fitting of 3rd order

Polynomial fitting of 4th order

Ured

Bf

UU red =

Bf

UU red =

redUU

BfK

ππ 22==

VerificationVerificationFitted values

-45,00

-40,00

-35,00

-30,00

-25,00

-20,00

-15,00

-10,00

-5,00

0,00

5,00

0,00 0,50 1,00 1,50

H4*s

-K×H2*

Fitted values

-50,00

-45,00

-40,00

-35,00

-30,00

-25,00

-20,00

-15,00

-10,00

-5,00

0,00

0,00 0,50 1,00 1,50

H1*s

K×H3*-Cd/K

Fitted values

0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

8,00

0,00 0,50 1,00 1,50

A1*s

K×A3*

Fitted values

0,00

0,50

1,00

1,50

2,00

2,50

3,00

3,50

4,00

4,50

0,00 0,50 1,00 1,50

A4*s

-K×A2*

a=0a=0°°

*

2

*

4KHH −=

K

cKHH D−=

*

3

*

1

*

2

*

4KAA −= *

3

*

1KAA =

K

KK

K

redUK

π2=

Force Force coeffscoeffs. . -- derivativesderivatives

a=0a=0°°Limit state values

-5,00

-4,00

-3,00

-2,00

-1,00

0,00

1,00

2,00

0,00 0,20 0,40 0,60 0,80 1,00 1,20 K×H1*

K^2×A3*

dCL/da

dCM/da

αd

dcKH LK

→→ 0*

1

αd

dcAK MK

→→ 0*

3

2

K

Flutter conditionFlutter condition

Affinity in the crossAffinity in the cross--terms:terms:

43514,30 T4,85-6

22615,30 B2,450

3,84*15,28 *0,040,90

2,472,33

+6

56,1214,40 T0,62-6

47,93,46 B2,200

12,87*3,47 *0,59

1,002,800,55

+6

U0crit

[m/s]

ωcrit

[1/s]

U red,crit

[-]

ΦfT

[1/s]

fB

[1/s]

α

[°]

*

3

*

2

*

4

*

1,,, HHAA ΦΦΦΦ

Damping factor: 3,5%Damping factor: 3,5%

Max. wind speedMax. wind speed

�� Lack of wind map for Hungary Lack of wind map for Hungary �� middlemiddle--GermanyGermany

�� 10 min average wind velocity10 min average wind velocity

�� 2020--30 oscillations for instability 30 oscillations for instability �� duration of 5 sec duration of 5 sec

�� p=0,02 annual probability of exceedencep=0,02 annual probability of exceedence

�� maximum wind speed 35,5 maximum wind speed 35,5 m/sm/s

�� safety against flutter instability safety against flutter instability ((UUccritrit //UUmaxmax))22 = (47= (47,,9/359/35,,5)5)22 = = 11,,8282

ConclusionConclusion

�� Enough safety against flutter instabilityEnough safety against flutter instability

�� More measures in the highly nonMore measures in the highly non--stationary domain,stationary domain,at low reduced velocitiesat low reduced velocities

�� Acceptable setAcceptable set--up, but a special test rig could be up, but a special test rig could be designeddesigned

�� More verification during the wind tunnel testMore verification during the wind tunnel test

Thank you for your kind attentionThank you for your kind attention

AcknowledgementsAcknowledgements�� dr. I. dr. I. HegedHegedűűss, BME, BME�� dr. I. dr. I. KovKováácscs, Dynamic, Dynamic--Consulting (BRD)Consulting (BRD)for their comments and help in this workfor their comments and help in this work