1992-h. tanaka a, n. yamamura a, m. tatsumi b

8/6/2019 1992-H. Tanaka a, N. Yamamura a, M. Tatsumi b

http://slidepdf.com/reader/full/1992-h-tanaka-a-n-yamamura-a-m-tatsumi-b 1/12

Journal o f W ind Engineering and Industrial Aerodynamics,41-44 (1992) 1279-1290 ! 27 9Elsevier

C o u p l ed M o d e F l u t t e r A n a l y s i s U s i n g F l u t t e r D e r i v a t iv e sH . T a n a k a a, N . Y a m a m u r a a, M . T a t s u m i b

a b r i d g e D e s i g n D e p t . , H i t a c h i Z o s e n C o r p o r a t i o n , 1 -3 -4 0 , S a k u r a j i m a ,K o n o h a n a - K u , O s a k a 5 5 4 , J A P A N

b T a r u m i C o n s t r u c t i o n O f f i c e , H o n s h u - S h i k o k u B r i d g e A u t h o r i t y , 1 - 1 - 6 6H i r a i s o ,T a r u m i - K u , K o b e 6 5 5, J A P A N

A b s t r a c t

T h e " c l a ss i ca l " c o u p l e d b e n d i n g - t o r s i o n f l u t t e r o f l o n g - s p a n b r i d g e s ,p o s s e s s i n g 3 - d i m e n s i o n a l v i b r a t i o n m o d e s , i s i n v e s t i g a t e d b o t h a n a l y t i c a l l ya n d e x p e r i m e n t a l l y . I n t h e a n a l y s i s , s e l f- e x c it e d f o r ce s a r e d e f i n e d b y u s i n gt h e s o - c a l l e d f l u t t e r d e r i v a t i v e s . T h e v e r t i c a l w i n d p r o f i l e a s w e l l a s t h es p a n w i s e n o n - h o m o g e n e i t y o f w i n d v e l o c it y a r e a l s o i n c o r p o r a t e d . T h ea n a l y s i s p e r m i t s ( i) s t r a i g h t f o r w a r d p r e d i c t i o n o f t h e c o u p l e d f l u t t e r v e l o c it yb y u s i n g t h e f l u t t e r d e r i v a t iv e s a n d ( ii ) r e a s o n a b l e i n t e r p r e t a t i o n o f2 - d i m e n s i o n a l m o d e l t e s t s t o p r e d i c t t h e c o u p le d f l u t t e r b e h a v i o r o f b r i d g e sw i t h 3 - d i m e n s i o n a l f r e e d o m o f m o t i o n .

I . IN T R O D U C T I O N

T h e p r e d i c ti o n o f c o u p le d m o d e ( i . e , b e n d i n g - t o r s io n ) f l u t t e r i s o n e o f t h em o s t i m p o r t a n t a s p e c t s f o r t h e d e s i g n o f l o n g - s p a n b r id g e s . A n a n a l y t i c a la p p r o a c h h a s b e e n a d v a n c e d b y B l e ic h [1 ], S c a n l a n [9 .] a n d r e c e n t l y b yM i y a t a & Y a m a d a [3 ] i n c o r p o r a t i n g s e lf - ex c it ed f o r ce s i n e q u a t i o n o f m o t i o n .B l e i c h a n d S c a n l a n a p p l i e d t h e s e l f - e x c i t e d f o r c e s o n l y t o t h e g i r d e r ,n e g l e c t i n g t h e d a m p i n g e f fe c ts d u e t o t h e m o t i o n o f c a b l e s a n d t o w e r s. M i y a t a& Y a m a d a i n c l u d e d t h e m o t i on o f c a b le s a n d t o w e r s b u t t h e i r a n a l y s i s i ss i m p l y b a s e d o n t h e T h e o d o r s e n ' s a e r o d y n a m i c f o r c e s o n f l a t p l a t e s .

T h e p r e s e n t p a p e r f i r s t p r o v i d e s t h e c o m p l e x e i g e n v a l u e a n d v e c t o rso lu t ions fo r c oup le d mode f l u t t e r w h ic h , u s ing f l u t t e r de r iva t ive s [2 ] , [4 ] ,i nc orpora t e t he se l f -e xc i t e d fo rc e s on a l l s t ruc tu ra l me mbe rs . A n e f fo r t i sa l s o m a d e t o i n c l u d e t h e v e r t i c a l w i n d p r o f i l e a s w e l l a s t h e s p a n w i s en o n - h o m o g e n e i t y o f t h e w i n d v e lo c it y . T h e a n a l y s i s e n a b l e s s t r a i g h t - f o r w a r dp r e d i c t i o n o f c o u p l e d m o d e f l u t t e r b e h a v i o r in c l u d i n g f l u t t e r f r e q u e n c y ,v e l oc i ty , i n t e r - c o u p l i n g a m p l i t u d e s a n d t h e p h a s e - s h i f t o f e a c h m o d e , i f t h ef l u t t e r d e r i v a t iv e s a r e m e a s u r e d . I n n u m e r i c al e x a m p l e, t h e p r e s e n t m e t h o da n d t h e S c a n l a n ' s s o l u t i o n a r e c o m p a r e d f o r a s i m p l e - b e a m m o d e l.

T h e n , t h e a n a l y s i s i s e x t e n d e d t o a m e n d t h e V - 6 c u r v e o b t a i n e d t h r o u g ha s e c t i o n m o d e l t e s t b y p r o v i d i n g a f o r m u l a f o r t h e a d d i t i o n a l d a m p i n ge f f ec t s d u e to t h e m o t i o n s o f c a b l e s a n d t o w e r s. T h e f o r m u l a , e x p r e s s e d a sa w e i g h te d a v e r a g e o f a d d i ti o n a l d a m p i n g t e r m s f o r t h e s i m p l e b e n d i n g o r

t o r s i o n a l f l u t t e r [5 ], i s e x p e c t e d t o e l i m i n a t e t h e i n c o n s i s t e n c y b e t w e e n3 - d i m e n s i o n a l a n d s e c t i o n m o d e l t e s t s .

T h e a n a l y s i s i s a p p l i e d to a s u s p e n s i o n b r i d g e w i t h a c e n t e r - s p a n o f 7 7 0 mw h i c h h a s d i f f e r e n t a e r o d y n a m i c c h a r a c t e r i s t i c s a t th e l e f t - a n d r i g h t - h a n d

g i r d e r s e c t io n s u n d e r e r e c t io n . T h e V -~ c u r v e f r o m t h e 3 - d i m e n s i o n a l t e s ta n d t h a t f r o m t h e a n a l y s i s a n d t h e a m e n d e d V -~ c u r v e o f s e c t i o n m o d e l t e s ta r e s h o w n t o be r e m a r k a b l y c o n s i s t e n t .

2 . D Y N A M I C E Q U A T I O N S

B y e x t e n d i n g S c a n l a n ' s f o r m u l a t i o n [ 2] , [ 4 ] , d y n a m i c e q u a t i o n s c a n b e

d e r i v e d i n m a t r i x f o r m b y t h e d i s p l a c e m e n t m e t h o d [ 5 ]. F i r s t , t h e e q u a t i o n

o f m o t i o n i s e x p r e s s e d a s ( S e e F i g . 1 f o r n o t a t i o n ) ,

(~O( t )

.O ~ ( t ) " ' ~

Li(t)o '.

/ Rt ~ \

Pi(t) Li Ai

F i g . 1 M e m b e r ~ i n g l o b a l c o o r d i n a t e

[M ]. O~ (t) + [C ]. I)~(t) + [K ]. U ~(t) ffi {F~(O~, U~, t)}= {F~(t)} ( 1 )

w h e r e [ M ] i s t h e m a s s m a t r i x , [ C ] i s t h e s t r u c t u r a l d a m p i n g m a t r i x a n d [ K ]

i s t h e s t i f f n es s m a t r i x i n c l u d i n g g e o m e t r i c a l s t i ff n es s . F i (t ) s t h e s e l f - ex c i t e d

f or ce s. T h e d i s p l a c e m e n t U i ( t ) c a n b e e x p r e s s e d a s

U i( t) = ~ ¢ i m ' m ( t ) ( i = 1 ,2 , . . , : M i s t h e a d o p t e d n u m b e r o f m o d e s ) ]

~ , 1= ~ , ( 2 )

{~im = (~}km+ ( ~ l m ) / 2 ( M o d e s h a p e a t t h e c e n t e r o f i - t h m e m b e r )

P r e - m u l t i p l y i n g E q . ( 1 ) b y { ¢ i m }T , i t b e c o m e s

X m ( t ) + 2 h m m • O } m ' X m ( t ) ÷ 0 }m2 " X m ( t ) f fi { ~ im } " {Fi( t )}/Mm 1

T [ M ] I ( 3 )

w h e r e hmma n d O)m a r e , r e s p e c t i v e l y , t h e s t r u c t u r a l d a m p i n g r a t i o i n s t i l la i r a n d c i r c u l a r f r eq u e n c y [ r ad / s] o f t h e m - t h m o d e . U n d e r t h e a s s u m p t i o nt h a t t h e g i rd e r c a n b e r e g a r d e d a s h o r iz o n t a l , t h e c o m p o n e n t s o f t h e w i n dl o a d v e c t o r { F i ( t ) } i n E q , ( 3 ) c a n b e e x p r e s s e d a s

1 2 8 1

Fi(t) = {0, L~(t),Pi(t) ,M~(t), 0, 0}

P i ( t ) = ( p . V i 2 / 2 ) • A i . K i . [~ i (Ki ) , H i (K i ) , V3i (Ki ) ]

• { ~ .~ ( t) /v i, B i - ~ i ( t ) N i . K i . a ~ (t )} • L ~

L i ( t ) = ( p V i 2 / 2 ) B i . K i [ , . l i (Ki), H~.i(Ki), I-I~i(Ki)]

• { : ~ i( t) /V i , B i . & ~ ( t) /V l , ~ . a i ( t ) } . L i

M i ( t) = ( p . V 2 / 2 ) • B i . K i . [ A li (K i ), A 2 i ( K i ) , A ~ i ( K i ) ]

• { y i ( t ) / V i , B i " ~ i ( t ) / V i , K i . ( x i(t)} • L i

K i - B i " o ) /V i : t h e r e d u c e d f l u t t e r f r e q u e n c y

P ~ i = - 2 C D i / K i , P 2 i - 0 [ 2 ] , P 3 i = ( d C D i / d ( x ) / K i ~ = C D i ' / K 2 t( 6 )

w h e r e P i (t ), L i( t) , M i ( t) a r e r e s p e c t i v e l y , t h e d r a g f o rc e , li f t f o rc e a n d m o m e n t .

P ~i(K i) • H ~ i(K i) • A ~ ( K i) ( jf fi l, 2 ,3 ) a r e d i m e n s i o n l e s s f l u t t e r d e r i v a t i v e s o f i - t hm e m b e r ( cf . T a b l e I N .B . ). B y e x p r e s s i n g t h e c o m p o n e n t s o f e i g e n - m o d e

f u n c t i o n {(~im} a s {(~Xm,~)Ym, ~Zm (~i~m,~ )~ m, (~ m } a n d a f t e r s o m e m o d i f i c a ti o n s o f E q .( 3) , o n e o b t a i n s t h e f o l lo w i n g c o u pl e~ l m o d e f l u t t e r e q u a t i o n s .

X m (t) + 2h m m • (COrn/CO)" CO. X m (t) + corn " Xm (t) = Z E,m " CO" ~[n(t) + ZF m n" CO " X n(t)n n

Em n --(p /2M m ) ' YBi" {(~Ym,~ m , ~ ) i m } ' [ H I " {~Y~,~ .~ , ~ , } . L ii

I Hli(K i) ' Bi 0 H~.i(Ki) . B~

[ H I = 0 P ] i ( K i ) . A i P ~ .i (K i )' A i ' B i

A1i (Ki ). B 2 0 A~ i (Ki) . B i

F m = ( p/ 2M m ) ' ~ -B i2 " { ¢ ~ , ¢ ~m , ¢ ~ }Ti

• {H;i(Ki) • B i , P; i (K i) • Ai , A~i(Ki) • Bi2} • * ~ . Li

3 . C O M P L E X E I G E N V A L U E E Q U A T I O N S

T h e c o m p l e x g e n e r a l i z e d c o o r d i n a t e s X m ( t), a s s o c i a t e d w i t h t h e c o m p l e xf l u t t e r f r e q u e n c y co, a r e i n t r o d u c e d a s f o ll o w s :

X m ( t ) = X m o ' e i° ~ , X m o = x R o + i . XImo

co = COR+ i . ¢ ~ = ( I + i . h ) . ¢ ~ t (9)

I Xm o I = [(XRmo) + (X~mo)2]~ : t h e a m p l i t u d e o f t h e m - t h m o d e 1

O m = ta n- l(X I m o / XRmo) : t h e p h a s e - s h i f t o f t h e m - t h m o d e ( t a d )(10)

w h e r e COR i s t h e f l u t t e r c i r c u l a r f r e q u e n c y [ r a d / s ] a n d h =c oi/co R ( ~G / 2 ~ ) i s t h es u m o f s t r u c t u r a l a n d a e r o d y n a m i c d a m p i n g . T h e c o m p l e x e i g e n v a l u ee q u a t i o n s , d e r i v e d f r o m i n s e r t i n g X m (t) o f E q . ( 9 ) i n t o E q . ( 7 ) , a r e a s f o ll o w s :.nJ[]

. . G .

(rG=1-E l) • { X . o } = . . . . - o

Gm l Gin2 Gmn - Xmo

( 1 1 )

Gram = [From + 1 + i . {Em m- 2hm m • (~/c o)} l/COrn

Gmn = (Finn + i . Em n) / (0m2 ( m ~ n )

[~,] = D i a g [1 /{ o ] ( D i a g o n a l m a t r i x o f e i g e n v a l u e ~ ) t ( 1 2 )

T h e n , f o r a s e t o f e i g e n v e c t o r s {X mo} t o e x i s t f o r a s s u m e d K oj,

d e t ( [ G m n ] - [ ~, ]) d e t ( [ G m n ] - D i a g [ 1 /o ) 2 ] ) = 0 ( 1 3 )

T o s o ! r e E q . ( 1 3) , a r b i t r a r y i n i t i a l v a l u e s ( e .g . , co rn /co =l) m a y b e g i v e n a n dw s t h l t e r a t i v e c a l c u l a t i o n o f E q s . ( 1 2 ), ( 1 3 ), co a n d {Xm o} c a n b e d e t e r m i n e db y th e f o l l o w i n g c o n v e r g e n c e c r i t e r i o n w i t h s u g g e s t e d e - v a l u e o f 10 -8 ~ 10-4 :

l e l < - c o l ,- 1 I I I e k l < e (14)

W h e n t h e r a n k o f t h e m a t r i x i n E q . ( 1 3 ) i s M , a n M - s e t o f f l u t t e r f r e q u e n c i e scon a n d a m a t r i x o f e i g e n v e c t o r s [ X ~ m ] ( n = 1 , 2 , . ., ,M ) w i l l b e o b t a i n e d . I ts h o u l d b e n o t e d , h o w e v e r , t h a t t h e c a l c u l a t io n o f E q s . ( 12 ) a n d ( 1 3) s h o u l dn o t n e c e s s a r i l y b e c a r r ie d o u t f o r a ll m o d e s . O n e c a n e a s i l y f in d t h e f l u t t e rf r e q u e n c y e0n a n d a m p l i t u d e I X °m I max i n w h i c h t h e m - t h ( e .g . , t o r s i o n a l )m o d e i s d o m i n a n t w i t h t h e f o l l o w i n g E q . ( 1 5 ) :

t x ~ o t ~ a , = ~ A X [ t x ° ~ s t , I x % I , . . . , t x ~ t ] ( 1 5 )

4 . E Q U I V A L E N C Y F O R P R O T O T Y P E B R I D G E A N D S E C T I O N M O D E L

T h e r e q u i r e m e n t s f o r a se c t io n m o d e l te s t to d u p l i c a t e p r o p e r l y t h e f l u t t e rb e h a v i o r o f a 3 - d i m e n s i o n a l p r o t o t y p e b r i d g e c a n b e s u m m a r i z e d a s (i)r e d u c e d f r e q u e n c y c o'. b / v ( m o d e l ) ffi c o. B / V ( p r o t o t y p e ) , ( ii ) m a s s a n d m o m e n to f i n e r t i a e q u i v a l e n cy [ 5 ] a n d ( i i i ) s t r u c t u r a l a n d a e r o d y n a m i c d a m p i n ge q u a l i t y . T h e s e c o n d c r i t e r i o n c a n b e w r i t t e n a s f o ll ow s :

1 2 8 3

m ~lq = ( b / B ) 2 - M ~ ) = ( b / B ) 2 . M ~ / [ Z (~,Yn) . I~ ]

m(2)q = (b / B ) 4" M ~ ) ff i (b / B ) 4" M n / [ T '. (~ )~ )2 . L i ](16)

w he r e m (k) a nd M (k) ( k = l , 2 ) de no te the e qu iva le n t m a ss [ t /m ] f o r k = l a n dq eqt h e e q u i v a l e n t m o m e n t o f i n e r t i a It- m 2 /m ] f o r k = 2 . C a p i t a l a n d s m a l l l e t t e r sr e p r e s e n t 3 - d i m e n s i o n a l p r o t o t y p e b r i d g e a n d s e c t i o n m o d e l , r e s p e c t i v e l y .

T h e t h i r d c r i t e r io n i s a l so d e v e l o p e d [5 ] f o r s i n g l e - d e g r e e f l u t t e r c a s e s t os u p p l e m e n t ( w i th a d d i t i o n a l d a m p i n g A hm ) t h e V - 5 c u r v e o f t h e s e c t io n a lm o d e l t e s t , w h i c h c a n n o t s i m u l a t e t h e d a m p i n g e f fe c ts a d d e d b y th e m o t i o no f c a b l e s, t o w e r s a n d t h e l a t e r a l m o v e m e n t o f t h e g i r d e r . H e r e , t h ec o r r e s p o n d i n g c r i t e r io n i s d i sc u s s e d a n d s u p p l e m e n t a r y d a m p i n g A hj i sf o r m u l a t e d f o r t h e m o r e c o m p l e x c a s e o f m u l t i - m o d e ( e .g . , b e n d i n g - t o r s i o n a l )c o u p l e d f l u t t e r .

F i r s t , t h e su pp le m e n ta r y d a m pin g Ahj f o r the c o up le d - f lu t t e r m ode {~bFj}a t K f K o j i s e x p r e s s e d b y t h e f o l l o w i n g E q . (1 7 ), th e d e d u c t i o n o f w h i c h i sm a d e i n t h e s a m e w a y a s f o r s i n g l e - d e g r e e f l u t t e r [ 5] .

Ahj =-(p/4M ~) • YBi. {(~, ~ ) ~ i , ~)~i}T" [HF]" {~)~i, } ~ i , ~)~i}" Lii

M ~ = {+ ~ ,t . E r a 1 . { , ~ , }( 1 7 )

w h e r e t h e c o u p l e d - f l u t t e r m o d e { ~ ) F i } i s g ive n a s f o l lows :

{~)Fi} -- -Z ~ ) i m " X m o ( re = l, 2 ,. .. , t he nu m be r o f c oup l ing m ode s ) ( 18 )m

S u b s t i t u t i n g E q . ( 1 8 ) i n t o E q . ( 17 ) , u s i n g o r t h o g o n a l i t y o f m o d e s , M ~ i s w r i t t e na s

M ~ = z X m o 2 . , ~ } " . [ M ] . { ,,m } z X ~ o 2 • ~m m

( 1 9 )

U n d e r t h e a s s u m p t i o n t h a t t h e g i r d e r i s s t r a i g h t i n p l a n a n d h a s s y m m e t r i c

s e c t i o n s , t h e m a t r i c e s [ H F ] a r e s i m p l y w r i t t e n a s :

[H F C ] I H ~ i (K i) .B 0 0 10 P ~ i ( K i ) " i 0

( 2 0 a )

[ H F ] = [ H ~ ] = P ~ i ( K i ) ' A i 0

w h e r e [ H C ], [H F ] a n d [ H ~ ] d e n o t e [H F ] f o r c a b l e s , g i r d e r s a n d t o w e r s ,

r e s p e c t i v e l y .T h e c o m p o t e n t s o f { ~ F i } i n E q . ( 1 8 ) a r e e x p r e s s e d a s f o l l o w s :

{ ~ i = ~ ' { ~ Y " X m o , { ~ } i= X ~)iZ m X m o , ¢ ~ i = X ( ~ z in . X m o ( 2 1 )m m m

W i t h t h e a s s u m p t i o n o f o r t h o g o n a l i t y f o r t h e e i g e n - m o d e f u n c t i o n s { ~ F i }

f o r t h e m a t r i x o f a e r o d y n a m i c c o e f f i c i e n t s [ H F ] , t h e R . H . S . o f E q . ( 1 7 ) b e c o m e s

{ (~ } i, ~ } i , ~ i } T " [ H F ] " { ~ i , ~ } i , ~ } i }

T- " ~ X r n o 2 " { { ~ Y m ,O i Z m ,~ } i m } " [ H F ] " { ~)Y m l, ~ iZ m , }'~ m } ( 2 2 )

T h e s u p p l e m e n t a r y d a m p i n g A h j c a n b e o b t a i n e d b y i n s e r t i n g E q s . ( 1 9 ) a n d

( 2 2 ) i n t o E q . ( 1 7 ) .

a TAhj = - t o / ( 4 £ Xmo . M ~) ]. Z Xmo2Z Bi" {¢Ym,¢ ~ , ~ i m } ' [HF]I n I n i

• { ¢ Y m , ~ )iZ m ,~)~m}"Li = X (X=o M m )" A hi n/ X (Xino - Mm) (23)m I l l

Ahm = -( p / 4M m)" ~ B i" {¢Ym,~bzin,~biam} ' [H F ]" {¢Ym,~b~, ¢~m}" Li (2 4)i

w h e r e A hin i s t h e a d d i t i o n a l d a m p i n g f o r s i m p l e - f l u t t e r o f m - t h m o d e [5 ].W h e n t h e e i g e n - m o d e s a r e n o r m a l i z e d a s

E q . ( 2 3 ) i s s i m p l i f i e d t o b e c o m e

A h j = Z X m o 2 . A h m / Z X m o 2 ( 26 )m m

A s t h e c o u p l e d f l u t t e r a m p l i t u d e o f t h e m - t h : m od e X ino i s a c o m p l e x n u m b e ri n E q s . ( 1 7 ) - ( 2 6 ) , A h j g i v e n b y E q . ( 2 3 ) o r ( 2 6 ) i s r i g o r o u s l y a c o m p l e x n u m b e r .H o w e v e r , t h e p h a s e - s h i f t 0m ( m ff il ,2 . .. M ) i s u s u a l l y f o u n d t o b e 6m ~ 0 , o re , , ~ ± ~ , w h e n f l u t t e r i s e v o k e d , t h e r e f o r e ,

X = = I X m o I . e i" G m - [(XRmo)2+ ( X I m o ) 2 ] ~ (cosein + i . s i n era)

'ffi. S ig n( co s e ra). I Xmol (27)

w h e r e t h e o p e r a t o r S i g n d e n o t e s t h e s i g n o f c ose in i n p a r e n t h e s i s .S u b s t i t u t i n g E q . ( 2 7 ) i n t o E q . ( 2 3 ) , o n e o b t a i n s

A h j = X ( t X i no 2 . M ~ ) . A h i n / X ( I X m o I~ . M ~ )I n I n

w h e r e A h j i s e x p r e s s e d a s a w e i g h t e d a v e r a g e o f Ahm b y m o d a l e n e r g y .C o r r e s p o n d i n g t o E q . ( 2 5 ) , E q . ( 2 8 ) b e c o m e s

~ t h j - ~ '. [ X m o [ 2 . ~ l m / • I X m o [ 2 (29)m m

5. NUMERICAL AND EXPERIMENTAL EXAMPLES

The present method and th e Scanlan' s solution are compared in a numeri calexample at first. Next, the present anal ysis is applied to a suspension bridgewith a center-span of 770 m which has different aerodynamic characteristicsat the left- and right-hand girder sections under erection.

5 .1 C o m p a r i s o n w i t h S e a n la n ' s m e t h o dScanl an showed the numeri cal example of coupled mode flutter of a bridge

with f lat box girder and gave critical fl ut te r velocity Vc in Ref. [6] (pp. 39-47).The m echa nical dampi ng 8ram is take n as 0.0628; the flut ter deriva tives arelisted in Table 1. The deck width Bf30.5m, while the span L=1220 m (Fig.2). The pol ar mome nt of ine rti a is tak en as M=3822 (tf. m2/m), while the

mass of the deck is m=34.11 (tf/m). It is a ssu med tha t tea = 20~h and t ha t0~h= 2~. (0.1 H,.). For an as sum ed simple-beam model (Fig. 2) to sa tis fy thesefrequencies, the sectional moment of ine rtia I and torsional resistance J areas sumed to be I=1488 (m 4) and J=11.56 (m4), respect ively.

V .. /

. /A A A A ..

t0xt22 000=1 220 000

I V..___~ Bffi30-500• " 1

Fig.2 Simple -beam model for coupled flutt er analysi s

Table 1 Flu tt er der iva tives H~ • A~(i = 1,2,3)

H] H~. H~ A~ A~

2 . 0 - 1 . 3 4 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0

4 . 0 - 3 . 0 0 0 . 0 0 0 . 1 0 0 . 0 0 - 0 . 0 6 0 . 0 0

6.0 -4,10 -1.40 2,50 -1,50 -0.10 1.00

8.0 -6.50 -4.50 6.70 -1.40 -0.20 2.00

10.0 -8.50 -8.50 8.00 -1.36 -0.28 2.9212.0 -11.00 -17.80 10.00 -1.40 -0.32 3.38

N.B.) All derivatives are double of those given by Scanlan [6] andH~i. H~i. A~i have reversed sign due to upward y-axis here.

The following three cases are solved by the present method.Case-(1): Coupled vertical bending and torsion flutterCase-(2): Vertical bending (single DOF) instability (Galloping)Case-(3): Torsional (single DOF) flutter

As the solutions of-these cases, the change of flu tter fr equency t~ (= ¢ ~ R j / 2 K )

a n d t h a t o f l o g a r i t h m i c d e c r e m e n t ( V - 5 c u r v e ) o f t h e b r i d g e c o r r e s p o n d i n g

t o w i n d v e l o c i t y a r e r e s p e c t i v e l y s h o w n i n F i g s . 3 ( a ) a n d ( b) . A c c o r d i n g t ot h e s o l u t i o n o f C a s e - ( 1 ) :

(i) C r i t i c a l f l u t t e r f r e q u e n c y f c i s 0 . 1 6 2 ( H z ) a n d a g r e e s w i t h f c = 0 . 1 6 2

( H z ) g i v e n b y S c a n l a n [ 6 ] .( i i ) C r i t i c a l f l u t t e r v e l o c i t y V c f r o m V - 5 c u r v e ( F i g . 3 ( b ) ) i s 5 8 . 4 ( m / s ) a n da g r e e s w e l l w i t h V c = 5 8 . 0 ( m / s ) g i v e n b y S c a n l a n w i t h t h e d i f f e r e n c eo f 0 . 7 % .

( i i i ) A m p l i t u d e r a t i o a n d p h a s e s h i f t b e t w e e n v e r t i c a l b e n d i n g ( M o d e - l )a n d t o r s i o n ( M o d e - 2 ) a r e s h o w n i n F i g s . 3 ( c ) a n d ( d ) .

) . 0.2S

~ 0 1 S

" ~ O l O

- - C A S E d ! ) - - - C A S E d 2 ) . . . . .. C A l l| .( 3 ) ,

,~ + , + , " ,~ . ' k ~ , -o , . ,

W I N D V E L O C I T Y ( m / e ) . ,+ - o 1 o

O3O - - C A I I E - ( I ) - - - - C A I I E . ( ; I ) . . . . . C A I E . (: I)0; 5

i0 15 / iIo,oO W ,

0 .10 ~0 Xl 40 W ~ ' ,~ ~

W I N D v E L O C I T Y ( m l s )

- - U O O l i . ( 2 ) - -- M O O E , I ) . ~ ZOO - - M O O I - i : I ) - - - - - - I d O O E . l t )

°. i I

o i o ;¢ Io +o !o Io ~ - ~ "~ oo+ L / WIND VELO CITY (m /s ) " ' ' " -

W I N D V E L O C I TY ( m / e ) ~ : ~(c) (d)

Fig. 3 Flutter characteristics of simple-beam model

6 . 2 C o m p a r i s o n w i t h w i n d t u n n e l t e s tThe secoud model is a suspension bridge with a center span of 770 m

(Fig.4). Consideration is given to wind aerodynamic stability just before theclosing of the girder, on which bogie-girders are located upstream anddownstream at the left- and right-hand sections, respectively (Fig.4).

With the aerodynamic characteristics of the left- and right-hand girdersections being completely different, sectional model test have been carried

out usiug both sections [8], then the lower flutter velocity has beenconservatively taken. In addition, a full model test has been made [7].Drag, lift and moment coefficients of the girder measured on the sectionalmodel [8] are shown in Fig.5. For other members, the drag coefficients aretaken as CD=0.7 for main cables and hangers, and CD=I.8 for towers.

The flutter derivatives H~, A~ (i=1,2,3) for girder have been measured bythe forced vibration method (Fig.6).

The following three cases of analyses are compared with the experimentalresults:

Case-(1): the coupled flutter (l st sym. bending mode and 1st sym. torsionalmode)

! 2 5 0 0 0 0 ( 2 5 0 0 )

BoSm.Rirdet. . 5 5

y [ 7 "- - + ~ I T - / ~ - ' / I J ~ T ~ NOIeDinlenllionlenotep ototvpe

~ A - A ~

B o B ie .g i rd e r - - I 46 9

,~05 6

B - B it

7 7 0 0 0 0 ( 7 7 0 0 ) _ , 2 5 0 0 0 0 ( 2 5 0 0 )

Fig. 4 F ra me mod el of a suspension bridge

C o e L C .l + o o . o . + I 0 0 . 0 " - / 1 1

$01) • + 3 1 G 0 + P

3 t | { l , 0 5 1 O 0 | l

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-----co - ~ .,'-P+"~¢ , ' - .,, +p~-++.]

• {II + @ 0 |

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+ . j +-+°+l , 1 0 . . l L t l . O 0 0 t

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f . M o . p . a . + - ~ o : : M t A C K - A N C L E ( D E O )

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D r a g , l i f t a n d m o m e n t c o e f f i ci e n ts o f s t i f f e n i n g t r u s s

C a s e -( 2 ): t h e s i n g l e - d e g r e e t o r s i o n a l f l u t t e r ( l s t s y m . t o r si o n a l m o d e )C a s e -( 3 ): t h e 4 - m o d e s o f c o u p l e d f l u t t e r ( l s t a n t i - s y m , a n d s y m . b e n d i n g

m o d e s , p l u s 1 s t a n t i - s y m , a n d s y m . t o r s i o n a l m o d e s )T h e s u p p l e m e n t a r y d a m p i n g & Sin ( = 2 ~ . A h m ) b y E q . ( 2 4 ) a n d A S j b y E q . ( 2 8 )

f o r t h e s e c t i o n a l m o d e l a r e l i s t e d i n T a b l e 2 . T h e s e c t i o n m o d e l t e s t s a r el i m i t e d t o t h e 1 s t s y m m e t r i c m o d e s , t h e r e fo r e , A 8m a n d A 8j g i v e n i n T a b l e2 a r e t h e a d d i t i o n a l d a m p i n g f o r t h e m . T h e V - 8 c u r v e f or t h e s e c t i o n m o d e l ,

t h e n , i s o b t a i n e d b y t h e f o l l o w i n g f o r m u l a :

8 = ( 8 1 + + = Ahj ( 3 0 )

1 2 8 8

0 , ° ° I H ;

4 . 0 0 _.O O

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1 ,O O 5 .O f I 0 . 0 1 5 . 0 2 0 . 0 2 5 . 0u I I I 1 I I ' , ; , ~ -

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- " 0 t- 4 , o o i - - - L E A S T " " ~ Q t, ,I ~ R E S- ~ + o o ] I " I E T H O D

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" ° o.o ,o.o ,,.o ,0,o , ,.o °" °t +.o . _ p ~ ' ~ ' , ' 0 ' ' " oo o ~ - " ~ ~ • ' + ' . ~ -~ ' ' ~ ' - ' ~ - -

+ , + ~ .. r , ~ . + , , p ,~ , . ~ , ,T ' ~ + " + + " l , . p,+, m ~ . . ,

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U /N O - 0,3D"0 1̀0 '0 , 3 0 ' Q / H O

L E A S T - 5 Q U A n E S . o .4 o | " L ( A S T - S Q U A R E SH E i f O 0 | t IE T ) IO O

( b ) D o w n s t r e a m b o s i e . s i r d e r

F i g . 6 F l u t t e r d e r i v a t i v e s ( E r e c t i o n - s y s t e m ( I V ) . S t e p ( 1 1 ) ; (~ = + 3 0

Z 0.10

i 0.04

< 00 2

~ C A S E - ( I ) - - - C A S E - - ( 2 ) . .. .. . C A S E - - ( 3 )

- - O - - 3 - d i m e n g i o n a l m o d e l

- - 4 - - S e c t i o n m o d e l

- - - - t - - - - - -- -4 I t- I I I I 4 II0 20 3o 40 SO 60 tO gO gO IO0

WIND VELOCITY (MF$)

F i g . 7 L o g a r i t h m i c d e c r e m e n t F i g . 8f o r t o r s i o n a l a n d c o u p l e df l u t t e r

( E r e c t i o n - s y s t e m ( IV ) ,S t e p ( 1 1 ); u = + 3 °)

o Sectionmodel /B o

' V ° ~ o0 0 O0

tO 20 30 40 50 6 0 7 0 8 0

-- Prolo ly i~eV (m/s)

R o t a t i o n a l c e n t e r a t P t . 4 8 4( E r e c t i o n - s y s t e m ( IV ) .S t e p (1 1 ) w i t h d o w n s t r e a mb o g i e - g i r d e r ; (z =+ 3 °

w h e r e 8z a n d 82 a r e t h e l o g a r i t h m i c d e c r e m e n t s o f t h e s e c t i o n a l m o d e l sw i t h u p s t r e a m a n d d o w n s t r e a m b o g i e -g i rd e r s, re s p e c t i v e ly .

T h e V - 8 c u r v e o b t a i n e d b y E q .( 3 0 ) u s i n g s e c t io n m o d e l t e s t s a n d t h a to b t a i n e d b y 8 - d i m e n s i o n a l m o d e l t e s t s a r e c o m p a r e d w i t h t h e a n a l y s i s u s i n gf l u t t e r d e r i v a t i v e s i n F i g . 7 . I t i s o b s e r v e d t h a t ( i ) c a s e - (3 ) g i v e s t h e l o w e s td a m p i n g b y a n a l y s i s a n d i s r e m a r k a b l y c o n s i s te n t w i t h t h e m o d e l te s t s , ( ii )t h e V - 8 cu r v e o f t h e s e c t i o n m o d e l s h o u l d b e a m e n d e d b y E q . (3 0 ) a n d ( ii i)s e c t i o n m o d e l t e s t s w i t h o u t a d d i t i o n a l d a m p i n g ASj m a y b e t oo c o n s e r v a t i v e .

T h e l a t e r a l e x c u r s i o n o f r o t a t i o n a l c e n t e r a t P t . 4 3 4 b y a n a l y s i s i s c o m p a r e dw i t h t h a t o b t a i n e d i n s e c t i o n a l m o d e l t e s t s i n F i g . 8 . T h e g o od a g r e e m e n tb e t w e e n t h e m m a y b e r e g a r d e d a s a v e r if i ca t i o n t o t h e a c c u r ac y o f t h ep r o p o s e d a n a l y s i s .

T a b l e 2 S u p p l e m e n t a r y d a m p i n g

E r e c t i o n s y s t e m ( I V ) . S t e p ( l l ) ( a = + 3 °)

10 0.0031 0.0009 0.0030 549.2 0.003020 0.0126 0.0017 0.0060 549.3 0.006030 0.0305 0.0025 0.0089 549.5 0.008940 0.0543 0.0033 0.0118 550.3 0.011850 0.0824 0.0041 0.0148 551.6 0.014760 0.1140 0.0049 0.0177 553.9 0.017670 0.1483 0.0057 0.0207 557.1 0.020480 0.1851 0.0065 0.0236 561.5 0.023290 0.2243 0.0073 0.0265 567.3 0.0259

100 0.2664 0.0081 0.0295 574.7 0.0285

N . B . ) My = 359 . 6 ( t . m 2 /9 ), M~ = 549 . 2 ( t . m ~ /9 ), I X ~ , ~ I = 1.0

6 . C O N C L U D I N G R E M A R K S

A c o u p le d f l u t t e r a n a l y s i s a s w e l l a s t h e e q u i v a l e n c y f or s e c ti o n m o d e la n d p r o t o t y p e b r i d g e ( o r 3 - d i m e n s i o n a l m o d e l ) a r e d i s c u s s e d . T h e r e s u l t s

a r e s u m m a r i z e d a s , ( i) c o u p l e d f l u t t e r b e h a v i o r c a n be p r e d i c t e d b yE q s .( 7 )- (1 4 ) w h e n t h e f l u t t e r d e r i v a t i v e s a r e m e a s u r e d a n d ( ii ) t h e V - 5 c u r v eo b t a i n e d b y s e c t i o n m o d e l t e s t s , w h i c h s a t i s f y re d u c e d f r e q u e n c y a s w e l l a sm a s s a n d m o m e n t o f i n e r t i a e q u i v a le n c y , m a y b e a m e n d e d b y a n a d d i t i o n a ld a m p i n g t e r m A h j o f E q s .( 2 0) -( 2 9) w i t h o u t m e a s u r i n g t h e f l u t t e r d e r i v a t i v e s .F o r t h i s p u r p o s e , t h e m o d e l a m p l i t u d e r a t i o s [ Xmo [ c a n b e t a k e n f r o m t h et e s t s a n d H ~i (K i) i n E q . (2 0 ) i s o b t a i n e d b y q u a s i - s t e a d y f o r m u l a .

I t s h o u l d b e a d d e d t h a t E q s . ( 2 0 ) - ( 2 9 ) a r e g i v e n u n d e r t h e a s s u m p t i o n o fa n a l og o u s b e n d i n g a n d t o rs i o n a l m o d e s h a p e s, th e r e f o re , f u r t h e r a m e n d m e n ti s n e c e s s a r y w h e n b e n d i n g a n d t o r s i o n a l m o d e s h a p e s d i f f e r w i d e l y , e . g , i nc a s e o f a m o n o - c a b le s u s p e n s i o n b r i d g e o r a s i n g l e - p l a n e c a b l e - s t a y e d b r i d g e .

A c k n o w l e d g e m e n tT h e a u t h o r s e x t e n d s s i n c e r e a p p r e c i a t i o n t o P ro f . N . S h i r a i s h i o f K y o t o

U n i v e r s i t y f o r h i s v a l u a b l e s u g g e s t i o n s o n t h i s s t u d y .

R E F E R E N C E S

1 . F . B l e i c h , D y n a m i c I n s t a b i l i t y o f T r u s s - S t i f f e n e d S u s p e n s i o n B r i g d e su n d e r W i n d A c t i o n , T r a n s . A S C E , V o l . 1 1 4 , p p . 1 1 7 7 - 1 2 3 2 ( 1 9 4 9 )

2 . R . H . S c a n l a n , T h e A c t i o n o f F l e x i b l e B r i d g e s u n d e r W i n d , P a r t I. ( F l ut t e rT h e o r y ) , J o u r n a l o f S o u n d a n d V i b r a t i o n , 6 0 ( 2 ) , p p . 1 8 7 - 1 9 9 ( 1 9 7 8 )

3 . T . M i y a t a , H . Y a m a d a , a n d H . O t a , F l u t t e r A n a l y s i s o f A T r u s s S t i f f e n e dS u s p e n s i o n B r i g d e b y . 3 D M o d e l M e t h o d , P r o c . o f J S C E , V o l 4 0 4 , p p .267-275 , Ap r i l (1989) (l.n . Ja p an es e ) . _ _ •

4 . R .H . S c a n l a n , I n t e r p r e t i n g A e r o e l a s t l c M o d e ls o f C a b l e - S t a y e d B r i g a e s ,J o u r n a l o f E n g i n e e r i n g M e c h a m c s , A S C E , V o l. 1 1 3, N o . E M 4 , p p. 5 5 5 . 5 7 5 ,A r i l (1987)

5. ~I~.T a n a k a , N . Y am a . m u ra , a n d M . T a t s u m l , I n t e r p r e t i n g s e c ti o n a l m o d e lt e s t s t o p r e d i c t 3 - d lm e n m o n a l f l u t t e r b e h a v i or o f l o ng - sp a n b n g d e s ,C a n a d a - J ~ a p a n W o r k s h o p o n B r i d g e A e r o d y n a m i c s , N R C N o . 3 1 8 7 1 , p p .2 3 9 -2 4 8 , O t t a w a , C a n a d a , S e p t . 2 5 - 2 7 ( 1 9 8 9 ) _ _

6 .R . H . S c a n l a n , S t a t e - o f - t h e - A r t M e t h o d s fo r C a l c u l a t i n g F l u t t e r ,V o r t e x - In d u c e d , a n d B u f f e t in g R e s p o n s e o f B r i d g e S t r u c t u r e s , F i n a l R e p o r tt o F H W A N o . F H W A / R D - 8 0 / 0 5 0 ( 1 9 8 1 ) .

7. M . It o, E x p e r i m e n t a l R e s e a r c h o f W i n d S t a b i l i t y fo r t h e I n n o s h i m a B r i d g ei n G i r d e r E r e c t i o n S t e p s , B E L - R e p o r t N o . 8 1 3 0 1 , t h e U n i v e r s i t y o f T o k y o ,( 1 9 8 1 ) ( i n J a p a n e s e )

8 . T . U e d a a n d A . K u m a g a i , 2 - D i m e n s i o n a l W i n d T u n n e l T e s t s f o r S a f e t y o ft h e I n n o s h i m a B r i d g e i n E r e c t i o n a n d C o m p l e t i o n , T h e H i t a c h i Z o s e nT e c h n i c a l R e v i e w , V o l . 4 2 ( 1 9 8 1 ) ( i n J a p a n e s e )

1992-h. tanaka a, n. yamamura a, m. tatsumi b

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