FREQUENCY-DOMAIN NUMERICAL BUFFETING RESPONSE …uet.vnu.edu.vn/~thle/Frequency-domain numerical buffeting response... · (1) A cable-stayed bridge example for numerical buffeting

1 | L e T h a i H o a – F r e q u e n c y d o m a i n b u f f e t i n g r e s p o n s e p r e d i c t i o n : I I . E x a m p l e

FREQUENCY-DOMAIN NUMERICAL BUFFETING RESPONSE PREDICTION

II. EXAMPLE

Prepared by Le Thai Hoa

2004


FREQUENCY-DOMIAN NUMERICAL BUFFETING RESPONSE PREDICTION: II. EXAMPLE

1. ANALYTICAL PROCEDURES

Step-wise analytical procedure (1) Field wind parameter:

a) Mean wind velocity (U10 & Uz); wind directions

b) Turbulence intensities (Iu, Iw)

c) Correlation and correlation coefficient ()

d) Scales of Turbulence (Lux, L)

e) PSD (Su(n), Sw(n))

(2) Force coefficient parameters:

a) Static force coefficients (CL, CD, CM at zero angle attack 0 )

b) Slope of static force coefficient (d

dCC LL ' ;

ddCC D

D ' ; d

dCC MM ' )

C’L, C’D, C’M

(3) Correction functions and transfer function:

a) Aerodynamic admittance (|L(n)|2 , |D(n)|2 , |M(n)|2)

b) Coherence (|CohL(n,s)|2, |CohD(n,s)|2 , |CohM(n,s)|2

c) Joint acceptance function (|JL(n)|2, |JD(n)|2, |JM(n)|2

d) Mechanical admittance (|Hi(n)|2)

(4) Full-bridge analysis and free vibration characteristics:

a) Free vibration analysis: Modal value and frequencies

b) Modal integral sums


(5) Buffeting response analysis:

a) Response of static wind load (displacements and sectional forces)

b) Response of dynamic wind load

ith mode response

Multimode response combination

Background and resonance response components

c) Total response of full-scale bridges:

Combination of static and dynamic responses

Gust response (loading) factor

(6) Some investigations

2. STRUCTURAL PARAMETERS

2.1. Structural information

Spectrum of Wind Fluctuations

Spectrum of Point- Buffeting Forces

Spectrum of Line- Buffeting Forces

Spectrum of ith Mode Response

Response Estimate of ith Mode

Aerodynamic Admittance Joint Acceptance Function

Mechanical Admittance

Spectral Density Functions

MultimodeResponse Combination

Inverse Fourier Transform


(1) A cable-stayed bridge example for numerical buffeting prediction (as model

of Diapoulsau Bridge, Swistherland) for demonstration of analytical

procedures and investigation

(2) Span arrangement: L=40.5m + 97m + 40.5m

2.2. Geometrical and material characteristics Gider Tower Stayed cables

Material parameters

E =3600000 T/m2

G =1384600 T/m2

=0.3 Poison ratio

Geometrical parameter

A =6.525 m2

I33 =0.11 m4

I22 =114.32 m4

J =0.44m4

Material parameters

E =3600000 T/m2

G =1384600 T/m2

=0.3 Poison ratio


A =1.14 m2; I33=0.257 m4

I22 =0.118 m4;J=0.223m4

A =1.14 m2; I33=0.257 m4

I22 =0.118 m4;J=0.223m4

Material parameters

E = 19500000 T/m2


A =26.355 cm2 Type 19K15

A =16.69 cm2 Type 12K15


3. FREE VIBRATION ANALYSIS

AND MODAL INTEGRAL SUMS

3.1. Structural modelling and free vibration analysis (1) FEM’s 3D space frame model with single grillage of bridge deck has been

used for static and free vibration analyses. Nonlinearity due to cable sagging

has been taken into consideration as Ernst’s secant equivalent elastic

modulus in which innitial cable tenstion has been estimated thanks to deck-

elevated preshaping procedure. Structural analyses have been carried out by

computer-aided structural analysis package SAP2000 Nonlinear

(Computers&Structures 1998)

(2) Free vibration analysis based on the Eigenvector analysis (Modal analysis)

has determined undamped free-vibration mode shape and frequencies of the

system itself. 10 fundamental free modes and their charctericstics were

computed.

(3) Eigenvectors (or modal amplitudes) of each mode shapes have been

normalized (so-called the mass-matrix-based normalization or

transformation into the normalized coordinates) by the structural mass

matrix M as follows:

1iTi M

(4) The modal participataion factors are defined as participaed contributions of

masses and inertia moments of mass associated with the three global

coordinates X, Y, Z have been computed for each mode as:

dxxmf Xi

L

iXi )(0

x: spanwise coordinate


dxxmf Yi

L

iYi )(0

dxxmf Zi

L

iZi )(0

Mode Eigenvalue Frequency Period Modal Feature

shape 2 (Hz) (s)

1 1.47E+01 0.609913 1.639579 S-V-1

2 2.54E+01 0.801663 1.247406 A-V-2

3 2.87E+01 0.852593 1.172893 S-T-1

4 5.64E+01 1.194920 0.836876 A-T-2

5 6.60E+01 1.293130 0.773318 S-V-3

6 8.30E+01 1.449593 0.689849 A-V-4

7 9.88E+01 1.581915 0.632145 S-T-P-3

8 1.05E+02 1.630459 0.613324 S-V-5

9 1.12E+02 1.683362 0.594049 A-V-6

10 1.36E+02 1.857597 0.53830 S-V-7

Note : S: Symmetric mode T: Torsional mode shape

A: Asymmetric mode P: Horizontal mode shape

V: Heaving mode shape


3.2. Mode shapes and normalized modal amplitudes

Mode 1 f=0.6099Hz

Mode 2 f=0.8016Hz

Mode 3 f=0.8522Hz

Mode 4 f=1.1949Hz

Mode 5 f=1.2931Hz

Mode 6 f=1.4495Hz

Mode 7 f=1.5819Hz

Mode 8 f=1.6304Hz


Second mode shape

(Second heaving mode)

-0.15

-0.1

-0.05

0

0.05

0.1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Mod

e am

plitu

de

Third mode shape

(First torsional mode)

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Mod

e am

plitu

de

Fourth mode shape(Second torsional mode)

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Mod

e am

plitu

de

Fiveth mode shape (Third heaving mode)

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mod

e am

plitu

de

Sixth mode shape(Fourth heaving mode)

-0.15

-0. 1

-0.05

0

0.05

0. 1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Mod

e am

plitu

de

Seventh mode shape(Third torsional mode)

-2.00E-02

-1.50E-02

-1.00E-02

-5.00E-03

0.00E+00

5.00E-03

1.00E-02

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Mod

e am

plitu

de

Eighth mode shape(Fiveth heaving mode)

-0.08-0.06-0.04-0.02

00.020.040.060.08

0.10.12

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29Gi ̧

trÞ

d¹n

g

First mode shape (First heaving mode)

-0.15

-0.1

-0.05

0

0.05

0.1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Mod

e am

plitu

de


3.3. Modal integral sums Modal integral sums is considered as parameters to account to mode shapes and their

modal amplitude, modal spanwise distribution associated with any individual free

mode as well as coupling between modes. Modal intergral sums can be computed as

following formula:

L

kskrrmsn dxxxLG0

,, )().()( (x: spanwise direction)

N

knksmkrkrmsn LG

1,, )()(

(Unit: length unit as meter)

N: Number of deck nodes

Lk: Length interval between 2 nodes

k: Nodal indicator

r, s: Modal indicator

m, n: Combination indicator

r, s=h, p or : Heaving, lateral or rotational

m, n=i or j

mkr )( , : rth modal value at node k

rmrmG : Same modes and same coordinate (r=s, m=n); (Auto-modal sums)

rmsmG : Same modes and different coordinate (r#s, m=n); (Cross-modal sums)

rmrnG : Different modes and same coordinate (r=s, m#n)

rmsnG : Different modes and different coordinate (r#s, m#n)


Mode Frequency Modal Modal integral sums Grmsn

shape (Hz) Character Ghihi Gpipi Gii

1 0.609913 S-V-1 5.20E-01 7.50E-11 0.00E+00

2 0.801663 A-V-2 4.95E-01 7.43E-09 1.35E-09

3 0.852593 S-T-1 3.79E-09 5.23E-05 1.14E-02

4 1.194920 A-T-2 1.78E-07 1.82E-05 1.07E-02

5 1.293130 S-V-3 5.07E-01 1.36E-07 23.62E-09

6 1.449593 A-V-4 4.99E-01 2.10E-09 9.42E-09

7 1.581915 S-T-P-3 2.67E-07 1.10E-03 1.10E-02

8 1.630459 S-V-5 5.03E-01 1.43E-07 1.27E-08

9 1.683362 A-V-6 1.64E-06 1.77E-04 1.09E-02

10 1.857597 S-V-7 4.16E-06 2.78E-03 1.11E-02


4. STATIC AERODYNAMIC COEFFICIENTS

AND FIRST-ORDER DEVIATIVES

(1) Static aerodynamic force coefficients have been determined by force

measurements generated on sectional model in wind tunnel test as following

formula (determined at balanced position 00 ):

BU

LCL2

21

; BU

DCD2

21

; 22

21 BU

MCM

(2) First-order derivatives of force coefficients (as slope of curve of force

coefficients vs. attack angle at balance position) determined by following

formulas thanks to least square technique:

ddC

C LL '

; ddC

C DD '

; ddC

C MM '

CD

0

0.02

0.04

0.06

0.08

0.1

-8 -4 0 4 8

Attack angle (degree)

Forc

e co

effic

ient


CL

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

-8 -4 0 4 8


Forc

e co

effic

ient

CM

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-8 -4 0 4 8


Forc

e co

effic

ient

Aerodynamic force coefficients:

CD=0.04102; CL=-0.1576; CM=0.17386

First-order derivatives of aerodynamic force coefficients by the least square

technique:

C’D0; C’L=0; C’M=


5. POWER SPECTRAL DENSITY (PSD) OF FLUCTUATIONS

(1) Deck elevation: Z=20m; static wind velocity U=20m/s

(1) One-sided power spectral density (PSD) functions determined by empirical

formulas in which PSD of horizontal wind fluctuation determined by

Kaimail’s spectrum, whereas PSD of vertical wind fluctuation computed by

Buches and Panofsky’s spectrum

3/5

2*

501200)(

fnfunSu

(Kaimal’s spectrum)

f: Non-dimensional Monin coordinates, Unzf

n: Frequency (Hz)

U,z: Mean velocity (m/s) and structural altitude z (m), respectively

u*: Friction or shear velocity (m/s), )/ln(* zozkUu

k, z0: Scale factor and roughness length (m)

k=0.4 and z0=2.5 : Simui&Scanlan (1976)

3/5

2*

10136.3)(

fnufnSw

(Panofsky’s spectrum)


10-3

10-2

10-1

100

101

0

100

200

300

400

500

600PSD of horizontal wind fluctuation

Freqency n(Hz)

Su(

n) m

2 .s/s

2

Kaimal's spectrumU= 40m/sZ= 20mu*= 2.5m/s

10-3

10-2

10-1

100

101

0

2

4

6

8

10

12PSD of vertical wind fluctuation

Freqency n(Hz)

Sw

(n) m

2 .s/s

2

PSD

Kaimal's spectrumU= 40m/sZ= 20mu*= 2.5m/s


6. AERODYNAMIC ADMITTANCE (1) Some empirical formulas can be applied for aerodynamic admittance such as

Sears’s function(1935), Liepmann’s function(1952), Davenport’s function

(1963) and Irwin’s function (1977).

(2) Liepmann’s function is often used in analytical buffeting response prediction

UBn

ni

i 22

21

1)(

(Liepmann’s function)

10-2 10-1 100 1010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency Log(n)

Aer

odyn

amic

adm

ittan

ce

(Inputs: B=15m, U=20m/s)


7. COHERENCE AND JOINT ACCEPTANCE FUNCTION (1) Coherence function of spanwise distribution of buffeting forces can be used

by the Davenport’s coherence function for wind fluctuations or modified

Davenport’s function that determined as the best-fit solution for measured

datas.

)exp(),(U

ynynCoh iiu

(Davenport’s coherence function)

: Decay factor (taken between 8 and 16, in which the less value, the safer in

analysis)

Davenport (1961) =7

10-2 10-1 100 1010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Log(n)

Coh

eren

ce

y=0.1m

y=0.3

y=0.5

y=1

y=5

y=10

y=30

Note: the bigger span-wise separation, the smaller coherence function


(2) In this numerical example, spanwise separation holds 5m that is distance

between every two deck nodes.

8. STRUCTRURAL AND AERODYNAMIC DAMPINGS

8.1. Strucrural damping ratio (1) Two type of dampings can be used in dynamic analyses

Damping ratio: ii

isi km

c2

Logarithmic decrement (logdec): )()(log1

tynTty

nTsi

.100%

Relationship: 2

sisi

(2) System damping ratio in wind-induced vibrations combines between

structural damping and aerodynamic damping that are associated with any

mode in the mode-based analysis

aisii

(3) Structural damping can be estimated by model-scaled and full-scaled

free/forced/ambiemt vibration tests

(4) It is generally agreed to assume that structural damping ratio is taken 0.03

for all modes, corresponding to logarithmic decrement 0.5% (Actually,

structrural damping increases with the order of high-frequency modes)

03.0si

8.2. Aerodynamic damping ratio


(1) Aerodynamic damping ratio can be determined by physical measurement of

aerodynamic damping force on either 2D sectional model or 3D elastic model

(2) Aerodynamic damping ratio can be estimated for all modes in three

displacement components by either of:

From quasi-steady aerodynamic forces: (The same for all modes)

mB

BfUC

Z

LaZ

2'

8

(Vertical response prediction)

mB

BfUC

X

DaX

2

4

(Horizontal response prediction)

IB

BfUCM

a

4'

8

(Rotational response prediction)

From flutter derivatives: (Different from each modes)

mBfHf jjaZ 2

)()(2

*1

mBfPf jjaX 2

)()(2

*1

IBfAf jja 2

)()(4

*2

(3)System damping ratios of 5 fundamental modes have been computed as follows:

Modes s,i a,i i

Mode 1 0.005 0.00121 0.00621 Mode 2 0.005 0.000912 0.005912


Mode 3 0.005 0.0001 0.0051 Mode 4 0.005 0.0000716 0.005072 Mode 5 0.005 0.0000571 0.005057

9. MECHANICAL ADMITTANCE (1)Mechanical admittance is determined by following formula:

12

222

2

222 )]}4)1[({

ii

iii f

fffknH

ki: Generalized mass of ith mode

10-2 10-1 100 10110-4

10-2

100

102

104

106

Frequency Log(n/ni)

Am

plitu

de L

og(|H

(n/n

i)|2 )

Damping ratio 0.003

Damping ratio 0.01 Damping ratio 0.015 Damping ratio 0.02

is , =0.003, 0.01, 0.015, 0.02 ( 0.02, 0.06, 0.13)


Noting that with different damping ratio, the background response component

stays the constant, whereas the resonance response differs at peak values that

depends on damping value (the smaller damping ratio, the bigger mechanical

admittance)

10. RESULTS AND REMARKS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 20 30 40 50 60

Mean wind velocity

RM

S o

f Ver

tical

dis

p. (m

)

RMS response of vertical displacement at midspan node

Mode 1

Mode 2 Mode 5


00.10.20.30.40.50.60.70.8

10 20 30 40 50 60

Mean wind velocity U(m/s)

Res

pons

e (D

egre

e)

RMS response of rotational displacement at midspan node

RMS of Total response (Degree)

0

0.2

0.4

0.6

0.8

10 20 30 40 50 60


RMS of total response of rotational displacement at midspan node

(SRSS response combination)

Mode 3

Mode 4


RMS of total response (m)

00.10.20.30.40.50.60.7

10 20 30 40 50 60


RMS response of vertical displacement at midspan node

(SRSS response combination)

0

0.1

0.2

0.3

0.4

0.5

0.6

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

RM

S o

f rot

atio

n

RMS of rotational displacement at spanwise deck nodes

Mode 3

Mode 4


00.050.1

0.150.2

0.250.3

0.350.4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

RM

S o

f ver

tical

dis

p. (m

)

RMS response of vertical displacement at spanwise deck nodes

Mode 1

Mode 2

Mode 5

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FREQUENCY-DOMAIN NUMERICAL BUFFETING RESPONSE …uet.vnu.edu.vn/~thle/Frequency-domain numerical buffeting response... · (1) A cable-stayed bridge example for numerical buffeting