Aerodynamic characteristics of continuous box girder bridges relevant to their vibrations in wind

Journal o[ Wind Engineering and Industrial Aerodynamics, 29 (1988) 399-408

399

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

AERODYNAMIC CHARACTERISTICSOF CONTINUOUS BOX GIRDER BRIDGES RELEVANT TO THEIR VIBRATIONS IN WIND

N. Narita, K. Yokoyama, H. Sato and Y. Nakagami Public Works Research I n s t i t u t e , M i n i s t r y of Construction, JAPAN

ABSTRACT The wind tunnel experiments concerning the wind-induced v i b r a t i o n s of continuous box girder bridges are presented in t h i s paper. The v i b r a t i o n s were measured both in smooth and t u r b u l e n t flow. Their c h a r a c t e r i s t i c s and the effects of turbulence were investigated and explained in view-point of the aerodynamic damping and the external l i f t force. NOTATION B: width of the bridge deck

CL: l i f t

coefficient

D: depth of the bridge deck

f : frequency

fi:

fr:

natural frequency of i th mode

reduced frequency (=fB/U)

h: displacement of v e r t i c a l bending v i b r a t i o n HLh: frequency response f u n c t i o n between l i f t

force and heaving motion

Im: imaginary part J i : j o i n t acceptance f u n c t i o n for i th mode l : bridge length

m: mass per u n i t length

Mi: generalized mass of i th mode ( = ~ m ( z ) @ i 2 ( z ) d z ) qi: i th generalized coordinate f o r h S: Power Spectral Density Function

U: mean wind speed

Ur: redeced wind speed (=U/(fB)=l/fr) u, w: f l u c t u a t i n g wind speed of l o n g i t u d i n a l and v e r t i c a l component x: axis along the main flow

y: horizontal axis perpendicular to x

z: axis along the bridge axis 6, 6i: logarithmic s t r u c t u r a l damping (of i th mode) 6a,

6ai: logarithmic aerodynamic damping (of i th mode)

p: a i r density, o: reduced mass (=m/(pB2)), ~ i ; i th mode shape

0167-6105/88/$03.50

© 1988 Elsevier Science Publishers B.V.

400 i.

INTRODUCTION Since the collapse of the Tacoma Narrows Bridge, a great deal of at t ent ion

has been paid to the wind effects on bridges, r e s u l t i n g in a great number of studies on the subject.

These studies have enabled the construction of

long-span bridges to take place.

However, there is much that remains uncertain

concerning the mechanism of the aerodynamic forces and wind-induced vibrations of bridges.

This is due to the complexity of the nature of flow around a

structure, as well as the complicated configuration of the structure. In general, structures are apt to vibrate when the product of t h e i r natural frequency and t h e i r representative length of the section is small compared to the wind speed at the s i t e , and when t h e i r mass and damping are small as well. Therefore, research on wind effects is indispensable for construction of long-span bridges, such as suspension bridges and cable-stayed bridges, whose natural frequencies are very small. Since to increase mass is not economical as well as unfavourable due to the reduction of natural frequency, and to increase damping is not p r a c t i c a l , the most e f f e c t i v e way to reduce wind-induced vibrations is to select the bridge deck section so that i t has favourable aerodynamic c h a r a c t e r i s t i c s as well as s u f f i c i e n t r i g i d i t y .

Such

sections are a truss deck with small s o l i d i t y r a t i o and a streamlined box deck which has a large r a t i o of deck width (B) to depth (D) (say, B/D>5).

These

bridge deck sections have been applied to most of the long-span bridges. However, the recent trend of increasing the span length of continuous box-girder bridges presents a new problem.

The increased span length decreases

the natural frequency of v e r t i c a l bending mode to values as low as those of cable-stayed bridges.

To bear the s t a t i c load by i t s deck alone, the depth of

the bridge section must be larger than that of a suspension or cable-stayed bridge, r e s u l t i n g in a small B/D, which is aerodynamically unfavourble. lift

The

slope, dCF/d~, can be considered as a representative value for the

aerodynamic admittance and the aerodynamic d e r i v a t i v e in the low frequency range (or high wind speed range).

The l i f t

slopes of rectangular prisms

( r e f . 1 ) suggest that the aerodynamic damping would be negative f or a bridge deck with small B/D r a t i o .

I t may be c o r r e c t l y anticipated that the

wind-induced v i b r a t i o n s of such a bridge would be v i o l e n t . In the Public Works Research I n s t i t u t e , Ministry of Construction, Japan, several wind tunnel experiments have been conducted concerning the wind induced v i b r a t i o n of continuous box girder bridges.

Among these studies, the

followings are described in this paper. 1) the f u l l model test f o r the proposed bridge design, 2) the sectional model test for the measurement of aerodynamic damping of box girder bridges, and 3) the single span e l a s t i c model test f o r the measurement of external l i f t

401

force 2. FULL MODEL TEST FOR THE PROPOSEDBRIDGE DESIGN The proposed bridge was a four-spanned continuous box girder bridge (150+190+190+150m).

B/D changes from 2.2 to 4.1 along the bridge axis.

The

general view of the bridge is shown in F i g . l .

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Side View

7

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8000

. 750

Cross S e c t i o n ~

I. Fig. 1

The 1/120 f u l l

Bcoo

,I

General view of the proposed continuous box girder bridge

e l a s t i c model was constructed in the boundary layer wind

tunnel at PWRI, whose test section is 6m wide, 3m high and 27m long.

The

s t i f f n e s s of the model was simulated by two steel spines, on which c o r r e c t l y scaled deck segments were mounted. mode (Ml/(f:@12dz)/(pB2))-- was 32. 0.02 for the f i r s t

The reduced generalized mass for the f i r s t The logarithmic damping in s t i l l

a i r was

mode, and somewhat smaller f o r the higher modes. The

natural frequencies of the model and the prototype are shown in Table i . Table 1. Natural frequencies 1st mode model prototype

5.6 (Hz) 0.40

2nd mode 3rd mode 5th mode 7.3 0.57

9.9 0.81

11.2 0.99

The 4th mode was horizontal bending The v e r t i c a l v i b r a t i o n a l displacement at the midpoint of the center span was measured both in smooth and turbulent flow. by spires.

The turbulence was generated

The turbulence i n t e n s i s t i e s lu and lw were about 12% and 8%, and

the i n t e g r a l scales Lxu and Lxw were about O.8m and O.3m, r e s p e c t i v e l y . measured v i b r a t i o n a l displacements are shown in Fig. 2.

The

402

h/8

h/B OO6

OO6

in smoothflow z~ : maximum

005

El : r . m

in turbulent flow

S

0,04

0.04 003

|st mode

A2nd

5th mode I

o

o.o

51hmode

0.01

0.01

2

Ii

4

i

&

i

o8

Experimentol wind speed (m/s) Fig. 2

'~ /

~j//~

o.o3

!,

0.0~

o{

: moximum E]:r m s

0.05

2 4 6 Experimental wind speed (m/s)

The v e r t i c a l displacement at mid-point of the center span

From the test r e s u l t s , i t was found that the proposed bridge had strong p o s s i b l i t y of limited amplitude vibrations of i s t mode, 2nd mode, 3rd mode and 5th mode in the low wind speeds, and divergent amplitude vibrations in higher wind speed in smooth flow.

On the other hand, the magnitude of limited

amplitude v i b r a t i o n s decreased remarkably in the turbulent flow, and the divergent amplitude v i b r a t i o n observed in smooth flow turned to be the less divergent but more random v i b r a t i o n in the turbulent flow. 3. MEASUREMENT OF AERODYNAMIC DAMPING In general, wind-induced vibrations are caused e i t h e r by s e l f - e x c i t e d forces (negative aerodynamic damping) or external forces (approaching turbulence, vortex e x c i t a t i o n ) .

When the summation of structural damping and aerodynamic

damping is 0 or negative, the aerodynamic i n s t a b i l i t y w i l l take place. The aerodynamic damping of the box girder bridges was measured in the Low-Speed Wind Tunnel-B of PWRI, whose test section is lm wide, 2m high and 3m long.

The turbulence was generated by the coarse grid whose mesh size is 0.25m

and whose bar size is O.05m. The i n t e n s i t i e s of the turbulence lu and lw are 6.2% and 5.0%, r e s p e c t i v e l y , and the i n t e g r a l scales Lxu, Lxw, Lyu and Lyw are O.09m, O.04m, O.04m and O.04m, respectively.

The measurement was made by the

dynamic balance developed at PWRI ( r e f . 2). The sectional models were shaken in heaving mode with an amplitude of B/IO0. bridge deck models are shown in Fig.3.

The cross sections of the tested Model A corresponds to the cross

section at 1/6 point of the center span of the bridge mentioned in chapter 2. Model B is a rectangular prism whose B/D=2. Model C is the t y p ic al streamlined box girder f o r cable stayed or suspension bridges. The s e l f - e x c i t e d l i f t

force was transformed into mass-damping parameter as

.

.

.

.

MODELA

~

o

:)oo5~4'

fB/U

.,,~,~ " ( ,~! ;r[,~' ~r

II , i| 1 111 J

J/rX/

0.010~.-.

0015

1.

200.0

-I

~14 86.7:1

[~

-30

-20

0.01 -10]

0

k

fBIU

,,, ,TO

.J

0.03_%

%

~n

0.04,,,

~t,.8

f467.6

0

,

.

.

t

.

.

.

'''"1

0.t

I

x

,

,

r'''~

i

1

fB/U

k

'L

Aerodynamic damping of Model C

,

.

flow

o-~o in turbulent

,

.

B/D =8.7

in smooth flo

0.01

2

3

4

~o

MODEL C

Aerodynamic damping of ModeIB Fig. 4.3

C)I

, , , , ,,,

\

I"

cr~o 10- in smooth flow

20'

30.

40' in turbulenf~ow ~

5o- ~ o

fvB/U fv: frequencyof shedding 60-fSLL(f)/(.SpU2BI~ vortex in smoothflow(3) in l u r b u l e n ~

MODELB B/D=2.0 Fig. 3 Cross section of the model

BID=2.9

o

%,0

Aerodynamic damping of Mode] A Fig. 4.2

101 .....

"

iI

Iii

I

g"/

"!,t I" , 'I

ino-~a smoothfIow~, // o-ao I[ / in turbulent--~ : I fSLL(f)I(.5 in turbulent ~l JJ

Fig. 4.1

-5

5"

15

20]

~o

.

.i"

f304.2

o

404

follows. ~6a= -Im(HLh)/(8~Tfr2)(O.5oU21))


In t h i s transformation, the e f f e c t of aerodynamic s t i f f n e s s was neglected, and the uniform mode shape was assumed. ~¢a of Model A in smooth flow is shown in Fig. 4.1.

Positive aerodynamic

damping appears in high reduced frequencies (fr>O.6), and then negative damping appears in the lower frequencies (O.4
The negative damping

corresponds to the limited amplitude v i b r a t i o n observed in the proposed bridge design t e s t .

Then the sharp peak of p o s i t i v e damping appears at fr=O.23.

negative damping follows t h i s p o s i t i v e peak.

The c r i t i c a l

The

reduced frequency at

which the damping turns to negative is about 0.2, which corresponds to the onset of the divergent v i b r a t i o n observed in the proposed bridge design test. o6a measured in the turbulent flow is also shown in the f i g u r e . The pattern is s i m i l a r to that for smooth flow, however, the p o s i t i v e damping peak s h i f t s to the lower reduced frequency and the peak becomes broader in the turbulent flow.

I t results the decrease of the c r i t i c a l

increase of the c r i t i c a l to negative.

reduced frequency or the

reduced wind speed where the aerodynamic damping turns

The aerodynamic damping for the thinner box girder models, whose

B/D r a t i o s (3.4 and 4.0) correspond to the cross section at 1/3 and i / 2 point of the center span of the continuous box girder bridge mentioned in chapter 2, was also measured.

Their c h a r a c t e r i s t i c s were s i m i l a r to those of Model A.

These r e s u l t s suggest that the less divergant but more random v i b r a t i o n observed in the turbulent flow was caused by the external l i f t

force but not by

the negative aerodynamic damping. In Fig. 4.2. shown is

o~a of the rectangular prism (Model B).

The

c h r a c t e r i s t i c s of aerodynamic damping of the prism seem s i m i l a r to those of the box girder bridge (Model A). These c h a r a c t e r i s t i c s of ~6a of the box girder bridge are quite d i f f e r e n t from those of the streamlined bridge deck (Model C).

As is shown in Fig. 4.3,

the aerodynamic damping of the streamlined bridge deck is p o s i t i v e both in smooth and turbulent flow. The e f f e c t of turbulence on the aerodynamic damping seems n e g l i g i b l e for t h i s kind of bridge deck. 4. MEASUREMENTOF THE EXTERNAL LIFT FORCES Since the aerodynamic damping of the box girder bridge becomes p o s i t i v e in the t u r b u l e n t flow, the random v i b r a t i o n can be explained by the external l i f t forces induced by the approaching turbulence or by the v o r t i c i e s behind the deck. The external l i f t forces were estimated from the Power Spectral Density Function (PSDF) of the wind-induced v i b r a t i o n of the single span e l a s t i c model.

405

The experiment were made in the Low Speed Wind Tunnel-A in PWRI, whose test section is 2.5m wide, 4m high and 10m long.

The three kinds of turbulence were

generated by spires and floor roughness. The characteristics of the turbulent flow are shown in Table 2. Table 2 Characteristics of the turbulent flow Turbulent flow i Power Exponent lu lw Lxu Lxw

Turbulent flow 2 Turbulent flow 3

0.12 7.7% 5.3% 0.34m 0.15m

0.16 11.4% 7.4% 0.35m 0.22m

0.30 20.3% 13.0% 0.36m 0.13m

The span length, deck width and depth were 1.65m, 0.1m and O.05m, respectively.

The cross section and mass were uniform along the span. The

reduced mass and logarithmic structural damping were 33 and 0.02, respectively. The e l a s t i c i t y of the model was provided by the simply supported alminum spine, and the fundamental mode shape of the model was half-sine. The cross sectional shape was similar to the box girder bridge in chapter 2, except that B/D (=2.0) is a l i t t l e smaller. The wind-induced vibrations were measured both in smooth and turbulent flow. The characteristics of the vibrations (Fig.5) were similar to those described in chapter 2, however, the effect of turbulence intensity on the rms values of the response was not simple.

While the turbulence intensity increases in the

order of turbulent flow 1, 2, 3, the rms value around the reduced wind speed 6 increases in the order of 2, 3, i .

The aerodynamic damping estimated from the

auto-correlation function of the vibrational displacement suggested that the r e l a t i v e l y large responses in the turbulent flow 1 were caused by the small aerodynamic damping. r.m.s,

of

h/B

0.05 0.04

o smooth flow v turbulent flow 1

0,03

n turbulent flow 2 z~ turbulent flow 3

o o

0.02

0

V

~AZ~

A

~

D

0.01 0

t

2

3

4

5

6

7

8 Ur

Fig. 5 r.m.s, of wind-induced vibration

406

The PSDF of the reduced generalized force for the f i r s t

mode was estimated

as f o l l o w s . f S C L C L ( f ) I J i ( f ) 1 2 / l 2= f S q i q i ( f ) ( M i ( 2 ; r f i ) 2 ) 2 / ( O . 5 p U 2 B l ) 2 / i H s i ( f ) 1 2

(2)

I H s i ( f ) l 2= ( ( 1 - ( f / f i ) 2 ) 2 + ( ( S i + ~ a i ) ( f / f i ) / T r ) 2 ) -1

(3)

fSCLCL(f) IJ1(f)12/12 The r e s u l t s are shown in Fig. 6.

Since the

PSDF of the external force has sharp peak in smooth flow, i t seems that the primary

smooth flow \

1

0.06

cause for t h i s force is v o r t e x - e x c i t a t i o n . The peak reduced frequency becomes smaller in t u r b u l e n t flow than in smooth flow.

It

0.02

seems that the turbulence broadens and

turbulent flow I,

lowers the peaks of the PSDF when the i n t e n s i t y is not so high (smooth flow, t u r b u l e n t flow 1 and 2), and that the turbulence increases the power of the

0.01

"'

L|

external force in case of high i n t e n s i t y ( t u r b u l e n t flow 3).

/ ,,

5. DISCUSSION The magnitude of the random v i b r a t i o n of the box girder bridge in turbulent flow is affected by both the external lift

fB/U force

Fig. 6 Generalized external l i f t

force and the aerodynamic damping.

In the experiment described in chapter 3, the l i f t

force on the models at rest

was measured in the turbulent flow as well as the aerodynamic damping. PSDF of the l i f t damping.

The

force is shown in Fig. 4.1 together with the aerodynamic

From the f i g u r e ,

i t can be found that the large power of the l i f t

force is associated with the large p o s i t i v e damping. I t may be i n t e r e s t i n g to note the close c o r r e l a t i o n between the PSDF of the external l i f t force and the aerodynamic damping plotted against reduced frequency f o r the box girder bridges. F i r s t of a l l , the effects of turbulence are s i m i l a r , namely, turbulence decreases the peak reduced frequency of both the l i f t force and p o s i t i v e damping, and turbulence broadens t h e i r peaks. Secondly, the peak reduced frequencies are almost i d e n t i c a l as is shown in Fig. 4.1. These c o r r e l a t i o n s were also observed in case of the rectangular prism whose B/D=2. As is shown in Fig. 4.2, the peak reduced frequency of the p o s i t i v e damping in smooth flow coincides with the reduced frequency of vortex shedding ( r e f . 3). The frequency of vortex shedding is i d e n t i c a l with the peak frequency of the PSDF of the external l i f t

force.

In the turbulent flow, both

407

of these peaks s h i f t to the lower reduced frequency, and they are almost identical as is shown in the same figure.

These findings suggest that the

vorticies behind the body are the primary factor for both the external l i f t force and the large positive damping in case of the box girder bridge deck. The effects of turbulence on the wind-induced vibrations of the continuous box girder bridges are shown schematically in Fig. 7.

The Power Spectral

Density Function of the external l i f t force has a sharp peak in case of smooth flow.

The turbulence shifts the peak to lower reduced frequency and broadens

the peak, The aerodynamic damping has a sharp positive peak in case of smooth flow.

The negative damping follows the positive damping at the lower reduced

frequency, which corresponds to the onset of galloping.

The turbulence shifts

the positive damping peak to lower reduced frequency and broadens the peak, which remarkably decreases the reduced frequency for the onset of negative damping. The broad-banded l i f t

force in the turbulent flow is associated with

the broad-banded positive damping. This results the less divergent but more random vibration in the turbulent flow.

EXTERNAL

f,ow

LIFT

FORCE

AERODYNAMIC DAMPING

I~L~--~,

EFFECT OF TURBULENCE 1. decrease the frequency of vortex- shedding 2. broaden the spectral peak

turbu,e°, flow

3. increase the Ur for positive damping peak 4. broaden the positive damping peak 5. increase the Ur for the onset o f negative damping

i~ 5

6. increase the Ur for the onset of galloping 7. less divergent but more random vibration

WIND INDUCED VIBRATION

E

linear scale

Ur

Fig. 7 Schematic explanation of the wind-induced vibration of the countinuous box girder bridges 6. CONCLUDINGREMARKS The wind-induced vertical bending vibrations of continuous box girder bridges have been investigated through the wind tunnel tests.

The main

408

f i n d i n g s are as follows. a. The limited amplitude v i b r a t i o n s in the low wind speed and the divergent amplitude v i b r a t i o n s (galloping) in the high wind speed were observed in case of smooth flow.

They were caused by negative damping.

b. The aerodynamic damping has a sharp p o s i t i v e peak in case of smooth flow. The nagative damping follows the p o s i t i v e damping at the lower reduced frequency, which corresponds to the onset of galloping. c. The turbulence s h i f t s the p o s i t i v e damping peak to lower reduced frequency and broadens the peak, which remarkably decreases the reduced frequency for the onset of negative damping. d. The broad-banded p o s i t i v e damping in the turbulent flow is associated with the broad-banded l i f t

force, which changes the galloping into the less

divergent but more random v i b r a t i o n in the t u r b u l e n t flow. e. The p o s i t i v e aerodynamic damping of the continuous box girder bridges was found to be closely correlated with the external l i f t

force p r i m a r i l y due to

the v o r t e x - e x c i t a t i o n . ACKNOWLEDGEMENTS The authors would l i k e to thank Mr. K. Kanzaki and Mr. M. Fukuda of the Structure Division of PWRI, who prepared for the experiments and processed the experimental data. REFERENCES i

T. Mizota and A. Okajima: Experimental Studies of Time Mean Flows around Rectangular Prisms, Proc. of Japan Society of C i v i l Engineers, Voi.312, 1981 (in Japanese) 2 T. Okubo, N. Narita and K. Yokoyama: Some Approaches for Improving Wind S t a b i l i t i y of Cable-Stayed Girder Bridges, Proc. 4th I n t . Conf. on Wind Effects on Bldgs. and Sts., Heathrow, 1975 3 K Washizu, A. Ohya, Y. Otsuki and K. F u j i i : Aeroelastic I n s t a b i l i t y of Rectangular Cylinders in a Heaving Mode, J. of Sound and Vibration, 59(2), 1978