flutter analysis of an extradosed bridge in hungary · flutter analysis of an extradosed bridge in...
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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@[email protected]
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