Wind Analysis for Vortex Shedding

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  • 7/25/2019 Wind Analysis for Vortex Shedding

    1/14

    Journal of Wind Engineering and Industrial Aerodynamics

    14 (1983) 153--16 6 153

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

    S I M P L I F I E D A P P R O A C H E S T O T H E E V A L U A T I O N O F T H E A C R O S S - W I N D R E S P O N S E O F C H I M N E Y S

    B . J . V i c k e r y

    U n i v e r s i t y o f W e s t e r n O n t a r i o , L o n d o n , C a n a d a

    and

    R B a s u

    H . G . E n g i n e e r i n g Lt d . , T o r o n t o , C a n a d a

    S U M M A R Y

    A m o d e l f o r t h e p r e d i c t i o n o f th e r e s p o n s e o f c h i m n e y s t o v o r t e x s h e a d i n g

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

    d e s c r i b e d . S i m p l i f i e d e q u a t i o n s s u i t a b l e f o r r o u t i n e o f f i c e u s e a r e d e r i v e d .

    F o r m o d e s o t h e r t h a n t h e f u n d a m e n t a l t h e s i m p l i f i e d f o r m s r e q u i r e a k n o w l e d g e o f

    t h e m o d e s h a p e s a n d f r e q u e n c i e s b u t, f o r t he f u n d a m e n t a l m o d e , i t i s s h o w n t h a t

    a n e q u i v a l e n t s t a t i c l o a d c a n b e d e f i n e d w i t h a k n o w l e d g e o f t he f r e q u e n c y o n l y .

    T h e a p p l i c a t i o n o f t h e s i m p l i f i e d f o r m s i s d e m o n s t r a t e d w i t h s a m p l e c a l -

    c u l a t i o n s p r e s e n t e d f o r t w o c h i m n e y s . T h e r e s u l t s o f t h e s i m p l i f i e d f o r m s a r e

    s h o w n t o be s l i g h t l y c o n s e r v a t i v e i n r e l a t i o n t o e s t i m a t e s o b t a i n e d u s i n g t h e

    d e t a i l e d a p p r o a c h .

    N O T A T I O N

    B b a n d w i d t h o f s p e c t r u m

    C r m s l i f t f o r c e c o e f f i c i e n t

    d d i a m e t e r

    m e a n d i a m e t e r o f t op t h i r d

    o f c h i m n e y

    f f r e q u e n c y

    f s s h e d d i n g f r e q u e n c y

    f i f r e q u e n c y o f i t h m o d e

    g a peak factor

    h h e i g h t o f c h i m n e y

    c o r r e l a t i o n l e n g t h i n di a m e t e r s

    m m a s s p e r u n i t h e i g h t

    m e e q u i v a l e n t m a s s p e r u n i t l e n g t h

    S S t r o u h a l N u m b e r

    S f ) s p e c t r a l d e n s i t y f u n c t i o n

    t taper

    d d z ) / d z )

    V w i n d s p e e d a t m a x i m u m r e s p o n s e

    w l o a d p e r u n i t h e i g h t

    z h e i g h t a b o v e g r o u n d

    B d a m p i n g a s a f r a c t i o n o f c r i t i c a l

    ~ a a e r o d y n a m i c d a m p i n g

    ~ s s t r u c t u r a l d a m p i n g

    K s ; K a m B s / P d 2 ; - m B a / P d 2

    a s p e c t r a t i o ,

    h / d

    a a . r m s m o d a l a m p l i t u d e o f i t h m o d e

    e i ratio of tip to base diam eter

    (other symbols are defined as they arise in the text)

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

    I t is w e l l k n o w n t h a t t a l l s l e n d e r s t r u c t u r e s o f c i r c u l a r c r o s s - s e c t i o n ,

    s u c h as c h i m n ey s , t o w e r s , e t c . , u n d e r w i n d l o a d i n g r e s p o n d d y n a m i c a l l y i n t h e

    a c r o s s - w i n d d i r e c t i o n , a s w e l l a s t h e a l o n g - w i n d d i r e c t i o n . F o r s u c h s t r u c t u r e s

    t h e d y n a m i c r e s p o n s e i n t h e a c r o s s - w i n d d i r e c t i o n i s o f t e n g r e a t e r t h a n a l o n g -

    wind.

    T h e m e c h a n i c s o f a c r o s s - w i n d r e s p o n s e r e s u l t i n g f r o m v o r t e x - s h e d d i n g f o r c e s

    a r e l e s s w e l l - u n d e r s t o o d t h a n a l o n g - w i n d r e s p o n s e t o a t m o s p h e r i c t u r b u l e n c e .

    0167 -610 5/83 / 03.0 0 1983 Elsevier Science Publishers B.V.

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    N e v e r t h e l e s s , r a n d o m v i b r a t i o n t h e o r y h a s b e e n a p p l i e d t o t h e p r o b l e m [I I T h e

    t h e o r y i s f o u n d t o b e a d e q u a t e f o r s m a l l a m p l i t u d e s . F o r l a r g e r a m p l i t u d e s ,

    g r e a t e r t h a n a b o u t I % o f t he d i a m e te r , t h e y t h e o r y y i e l d s u n c o n s e r v a t i v e

    r e s u l ts . T h e c h a r a c t e r i s t i c o f th e l o a d - r e s p o n s e r e l a t i o n s h i p i n t h e a c r o s s -

    w i n d d i r e c t i o n w h i c h g i v e s r i s e t o t h is u n d e r e s t i m a t e i s t h a t t h e l o a d i s

    a m p l i t u d e - d e p e n d e n t . T h u s a m o d i f i c a t i o n b e c o m e s n e c es s a r y to a c c o m m o d a t e t h i s

    f e a t u r e .

    I n th e p a r a g r a p h s b e l o w a n o u t l i n e o f t h e t h e o r e t i c a l m o d e l d e v e l o p e d t o

    p r e d i c t r e s p o n s e t o v o r t e x s h e d d i n g f o r c e s i s g i v e n . S i n c e t h e f o r m o f m o d e l i s

    n o t s u i t a b l e f o r d e s i g n o f f i c e u s e a n u m b e r o f s i m p l i f i e d m e t h o d s h a v e b e e n

    f o r m u l a t e d ; t h e s e a r e d e s c r i b e d . A c o m p a r i s o n b e t w e e n t h e d e t a i l e d a n d

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

    d i m e n s i o n s a s e x a m p l e s . A c o m p r e h e s i v e t r e a t m e n t o f t h e d e t a i l e d a p p r o a c h h a s

    b e e n g i v e n b y V i c k e r y & B a s u [ 2, 3] .

    2 . O U T L I N E O F A M O D E L FO R P R E D I C T I N G T H E R E S PO N S E O F T A L L S L E N D E R S T R U C T U R E S

    T O V O R T E X - S H E D D I N G F O R C E S

    T h e f o l l o w i n g p a r a g r a p h s o u t l i n e t h e d e v e l o p m e n t o f a m a t h e m a t i c a l m o d e l

    f o r p r e d i c t i n g t h e l a t e r a l r e s p o n s e o f ta l l s l e n d e r s t r u c t u r e s , s u c h a s

    c h i m n e y s , t o v o r t e x s h e d d i n g f o r c e s . I t i s a s s u m e d t h a t t h e d y n a m i c r e s p o n s e i n

    e a c h m o d e c a n b e t r e a t e d i n d e p e n d e n t l y o f r e s p o n s e i n o t h e r m o d e s . T h e t o t a l

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

    follows:

    y ( z , t ) = ~ a i ( t ) ~ i ( z ) ( i )

    i=i

    w h e r e a i : m o d a l c o e f f i c i e n t f o r m o d e i; ~ i ( z ) : m o d e s h a p e f o r m o d e i

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

    ( i) t h e v o r t e x s h e d d i n g f o r c e s c a n b e m o d e l l e d a s a n a r r o w - b a n d r a n d o m f o r c e

    w i t h a n o r m a l d i s t r i b u t i o n a n d w i t h t h e f o l l o w i n g c h a r a c t e r i s t i c s ;

    (a) the spe ctr um of w ( z , t ) , t h e f o r c e p e r u n i t l e n g t h a t s o m e p o i n t ,

    z, is of the form:

    f S w ( f ) / U w 2 : I / B ~ e x p { - ( ( l - f / f s ) / B ) 2 }

    O w 2 = C L 2 { Q u 2 } 2 d 2 = v a r i a n c e o f w ( z , t )

    N o t e : g w 2 a b o v e r e f e r s t o t h a t p a r t o f t h e t o t a l v a r i a n c e i n t h e v i c n i t y ol

    f = fs, t h e c o m p l e t e s p e c t r u m a l s o c o n t a i n s e n e r g y a t l o w f r e q u e n c i e s

    b u t t h i s i s a s s o c i a t e d p r i m a r i l y w i t h t u r b u l e n c e i n t h e f l o w a n d w h i l e

    i n f l u e n c i n g t h e r e s p on s e a t h i g h v e l o c i t i e s d o e s n o t c o n t r i b u t e

    s i g n i f i c a n t l y w h e n f s i s i n t h e v i c i n i t y o f t h e n a t u r a l f r e q u e n c y , fo,

    i . e .; a t o r n e a r t h e s o - c a l l e d " c r i t i c a l " v e l o c i t y .

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    1 5 5

    w h e r e

    b ) T h e c o - s p e c t r u m d e s c r i b i n g t h e v o r t e x s h e d d i n g f o r c e s a t p o s i t i o n s

    zl, and z3 can be expr esse d in the form;

    C o C W , z l , z 2 = / S w ( z l , f ) R C Z l , z 2 ) ( 3 a )

    R ( z l , z 2 ) = c o s ( 2 r / 3 ) { e x p ( - ( r / 3 i ) 2 } ( 3 b )

    r = 21 z 1 z 2 1 / d z 1 ) + d z 2 ) )

    ( ii ) T h e m o t i o n d e p e n d e n t f o r c e s c a n b e r e p r e s e n t e d b y a n o n - l i n e a r

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

    W d C Z ) = + 4 7 p d 2 f o K a o ( l - ( U y / U L ) 2 )

    w h r

    u

    4 )

    a n d i : i n t e n s i t y o f t u r b u l e n c e

    R e : R e y n o l d s N u m b e r

    ~ y : r . m . s , d i s p l a c e m e n t

    a : a l i m i t i n g r . m . s , d i s p l a c e m e n t e q u a l t o ~ d .

    L

    Acc ept ing the descri ptio ns in (i) and (ii) above it can be show n {2,3}

    t h a t t h e r . m . s , m o d a l a m p l i t u d e , a a i , c a n b e c l o s e l y a p p r o x i m a t e d a s;

    U a i / d 0 =

    C 3 / { B s -

    ( P d o 2 / m e ) ( C i - C 2 ( ~ a i / d o ) 2 ) }

    where; d o : t i p d i a m e t e r

    m e

    =

    f h m ( z ) ~ i Z C z ) d z / f h ~ i 2 z ) d z

    C I = f h K a o ( Z ) ( d ( z ) / d o ) 2 ~ i 2 ( z ) d z / f h ~ i 2 ( z ) d z

    c2 1/~2 fh 4 ~ z / f h Cz

    K a o ( Z ) ~ i ( z ) ~ i 2 d z

    o o

    5 )

    6 )

    7 )

    8 )

    / ~ f o c ~ ~a 2

    C 2 m e h 2 ~ f o ) 2 d o

    { f ~ f ~

    { 2 C / - ~ S f s

    l - f / f s 2

    e x p - l - - - ~ - )

    } z : z I

    c ~ p d u ~ u ~ 1 f / ~ J }

    e x p ( - % { z=z2

    { z / ~ B f s B

    2 r

    C O S - - e x p ( - ( r / 3 ) 2 ) ~ i ( z l ) ~ i ( z 3 ) d z l d z 2 } ( 9 )

    3

    T h e c o m p l e x i t y o f E q u a t i o n s ( 5) t o ( 9) i s s u c h t h a t t h e y a r e n o t r e a d i l y

    a d a p a t a b l e t o a d e s i g n s i t u a t i o n ; f u r t h e r t o t hi s , t h e d o u b t s t h a t p r e s e n t l y

    s u r r o u n d t h e d e f i n i t i o n s o f v a l u e s f o r t he v a r i o u s a e r o d y n a m i c c o e f f i c i e n t s

    a r e c o n s i d e r a b l e a n d s om e l o s s o f a c c u r a c y c a n be a c c e p t e d i n d e v e l o p i n g s i m p -

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    56

    l i f i c a t i o n s . B e f o r e p r o c e e d i n g t o t h e s e s i m p l i f i c a t i o n s i t i s o f i n t e r e s t t o

    e x a m i n e t h e g e ne r a l c h a r a c t e r i s t i c s o f s o l u t i o n s t o t h e e q u a t i o ns .

    3 . N A T U R E O F T H E R E S P O N S E P R E D I C T E D B Y T H E P R O P O S E D M O D E L

    For the purposes of exami ning the nature of the solutio n of equations (5)

    to (8) it is sufficie nt to consid er the form taken for uniform motio n of a

    l o n g c y l i n d e r w i t h u ( z ) = u , d ( z ) = d o and ~(z) = i and at the critical

    v e l o c i t y w h e n f s = f o In this case the equatio n is of the form

    ~ a / d o = C / { K s - K a o ( 1 - ( ~ a / ~ d o ) 2 ) } ( i 0 )

    w h e r e K s = m S s / D d 2 and C is dependen t upon the mass and aspect ratio of

    the struct ure and the aerody namic parameters, CL, , B and S. This genera l

    form will hold for more comple x systems and in Fig. I an equation of this form

    is plotted t ogether with the experi menta l results of Wooton [4] obtaine d f rom

    m e a s u r e m e n t s o f a c a n t i l e v e r s t r u c t u r e.

    T h e r es p o n s e r e l a t i o n s h i p d e f i n e d b y E q u a t i o n 1 0 d i v i d e s n a t u r a l l y i n t o

    three regi ons as follows;

    (i) A large amplitu de or "lock-in" region corres pondi ng to low values of

    mass and/or dampi ng and in which the response is independe nt of C

    and hence ind ependent of the forces acti ng on a stationa ry cylinder.

    In this region the respo nse is determi ned only by the nature of the

    n o n - l i n e a r a e r o d y n a m i c d a m p i n g a n d i s g i v e n by ;

    U a / d 0 : { i - K s / K a o } x ~ ( i i )

    W i t h i n t h i s r e g i o n t he r e sp o n s e e x h i b i t s c o m p a r a t i v e l y m i n o r v a r i a -

    t i o n s i n a m p l i t u d e a s d e m o n s t r a t ed b y t h e c o mp u t e d r e s p o n s e t r a c e i n

    Fig. 2a and the peak factor (which approa ches / 7 for a steady

    sinusoi dal motion) shown in Fig. 3.

    (ii) A small ampl itude region in which the respons e can be regarded as

    random for cing with linear p ositive damping at a value below that

    provide d struct urally and in which the response is given by;

    U a / d o = C / { K s - K a o }

    In this region the response is nearly Gau ssian as shown by the trace

    in Fig. 2c and by the pe ak fa ctor s in Fig. 3.

    (iii) A transit ion region in the vicin ity of K s = K a o in which the

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

    e x h i b i t a n e x e e p t i o n a l l y s t r o n g d e p e n d e n c e o n K s .

    T h e s i m p l i f i e d m e t h o d s w h i c h a r e d e v e l o p e d i n t h e f o l l o w i n g s e c t i o n

    a s s u m e t h a t " s m a l l " a m p l i t u d e s r e s u l t a n d t h e e q u a t i o n s d e v e l o p e d

    are of the form;

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    1 5 7

    0.10

    D m a x

    0 0 1

    Lock-ln , ~ I

    Regime t

    Transition

    Regim

    Experimental

    Re No =- 600000

    Height/Diamete~= 11.5

    0 .0 0 4 3 - -

    [ K s - 0 .5 4 ( ] - ( Y ~ ) ) 1

    i , o 2 3 ~ ,

    0 3 0 1 _ _ I t ~ i t _ _ t _ _ - J

    0.1 0.2 0.4 0.6 0 8 1.0 2.0, 40

    K

    F I G . 1

    V a r i a t i o n o f R M S

    A m p l i t u d e w i t h K s

    W o o t o n )

    ( a ) C / C 0 . 2

    0.5 d

    ~ ~

    l~ l~ n ~ l l l l l l l l l l l

    L o . 5 d

    b ) c I c = o ,5 ~

    d

    .

    I

    :~ ~ ~ A ~ `~ ~ ` ~ ~ `~ ~ `~ ~ ~ ` , ~ L i~ U ~ W ~ ]~ g ~ i~ W ~ [~ [[~ [[[~ [[~ [~ [~ [~ [

    o jlllllllillllllilllJilll[lmmmmlrmmUlllnl.,..,.mmrllltrmmmmlmll]l

    ] ] l l l l l J l l l l l l l l l l l l t l l l l l l I ]

    I I I I I I I 1

    m i l~ f I In l ll ll il il ll n i li H n ~ l ll n l t lm l ll g l ll J l l ll jl ll l l j ri [ l il l ll l ll l lI l il l jl l lm i J l l t m ~

    o . t d

    (c ) C /c o - 2~

    - 0.03 d

    F I G . 2 NUMERICALLYSIMULATED RESPONSE TO SHEDDING

    IN LARGE SCALE (L/D> lO0) TURBULENCE

    Cl I 0.20, i - O.lO, M/od ' lO0

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    158

    Narl~w i n d G e u s s ~ n

    o

    0% 50

    30

    ~ o

    3O

    o

    1 ~ - - - 4 1

    0.2

    i i i i i i i i i i t i

    0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0

    K s / K %

    F I G . 3

    V a r i a t i o n o f P e a k F a c t o r W i t h K s / K a F o r V a r y i n g

    S c a l e s a n d I n t e n s i t ie s o f T u r b u l e n c e

    3.0

    2 .5

    r n

    2 . 0

    - I N

    u.i

    N i o

    m

    . 5

    / q

    \ J

    m

    .5 i.O 1.5 2) 2.5

    K V / V C R I T

    F I G . 4 ~ B , k ) ; I n f l u e n c e o f B a n d w i d t h o n R e s p o n s e

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    5 9

    ~ a / d 0 : C / { K s - K a o } { 1 3 )

    w h i c h i s c o n s e r v a t i v e b u t d o es n o t y i e l d a s o l u t i o n f o r

    K s < K a o .

    S i n c e a m p l i t u d e s i n t h e v i c i n i t y o f

    K s = K a

    a r e e x t r e m e l y

    d i f f i c u l t t o p r e d i c t d u e to th e s t r o n g d e p e n d e n c e o n t wo p o o r l y

    d e f i n e d p a r a m e t e r s ( K s , K a o ) i t w o u l d n o r m a l l y b e w i s e t o e n s u r e t h a t

    B s a n d m a re s u f f i c i e n t t o a v o i d l a r g e m o t i o n s .

    4 . S I M P L I F I E D F O R M S O F T H E R E S P O N S E E Q U A T I O N S

    4 . 1 C h i m n e y s o f C o n s t a n t o r N e a r C o n s t a n t D i a m e t e r

    F o r f r e e - s t a n d i n g c h i m n e y s e x c i t e d i n t he f i r s t or s e c o n d m o d e t h e b u l k

    o f t h e e x c i t a t i o n i s d u e t o f o r c e s o v e r t h e t o p o n e - t h i r d ( t y p i c a l ly , t h e r e s -

    p o n s e c o m p u t e d a s s u m i n g f o r c e s o v e r t h e t o p t h i r d o n l y a m o u n t s t o 9 0 % + o f th a t

    c o m p u t e d a s s u m i n g e x c i t a t i o n o v e r t h e c o m p l e t e h e i g h t ) . I t i s t h e r e f o r e

    r e a s o n a b l e t o n e g l e c t t h e v a r i a t i o n o f w i n d s p e e d w i t h h e i g h t a n d a s s u m e a

    c o n s t a n t s p e e d e q u a l t o t h e a v e r a g e o v e r t h e to p o n e t h i r d . T h e r e s p o n s e

    e q u a t i o n t h e n b e c o m e s ;

    U a i C L Pd 2 (2__.X_) , { B , k ) / { ~ f h * i 2 ( z ) d z } { S s - d K a ( P d ) }

    d = ~ m e n o m e

    w h e r e

    1 k ~ 2 l - k -

    2}

    ~ ( B , k ) = ~ B e x p { - ( ) ( 1 4 )

    B

    1 f i d

    = V / V C R I T , ; V C R I T = - ~

    T h e f u n c t i o n

    ~ ( B ~ k )

    is shown in Fig. 4 and it is appa rent that for

    v a l u e s o f B e n c o u n t e r e d i n t u r b u l e n t f l o w ( a b o u t 0 . 1 0 to 0 . 3 0 ) t h a t t h e p e a k

    v a l u e i s a b o u t 2 . 5 a n d t h a t t h i s o c c u r s i n t h e v i c i n i t y o f k = I . i , i.e. the

    p e a k r e s p o n s e o c c u r s a t a w i n d s p e e d

    ( V M A x )

    w h i c h i s a b o u t 1 0 % g r e a t e r t h a n

    t h e c r i t i c a l s p e e d d e f i n e d b y t h e S t r o u h a l N u m b er . T h e m a x i m u m r e s p o n s e i s

    then;

    O a i C L P d 2 / ~ Z 2

    ~_ _ _

    ( ) 2. 5 z--------2 ( )/( ~s _K a ( L fh ~i (z )d z) (1 5)

    d 8 W S m e 2 1 m e h o

    and occu rs at a wind speed;

    V M A X ~ i . i ( l i d ~ S ) ( 1 6 )

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

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

    e q u i v a l e n t s t a t i c l o a d c a n b e d e f i n e d . T h e l o a d d i s t r i b u t i o n f o r t h i s s t a t i c

    l o a d s h o u l d f o l l o w t h e d i s t r i b u t i o n o f t h e i n e r t i a l l o a d s f o r v i b r a t i o n i n t h e

    f u n d a m e n t a l m o d e . F o r r e i n f o r c e d c o n c r e t e c h i m n e y s G h o [ 7] h a s s h o w n t h at t hi s

    c a n b e a p p r o x i m a t e d b y a s i m p l e l i n e a r d i s t r i b u t i o n o f t he f o rm ; w ( z ) = W o ( Z / h ).

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    16

    I f W o i s c h o s e n a s W o = C p v 2 / 2 d t h e n t h e v a l u e s o f c a n d V

    c a n b e d e t e r m i n e d b y e q u a t i n g t h e m o d a l a m p l i t u d e , a e , d u e t o t h e l o a d w ( z )

    t o th e p e a k m o d a l a m p l i t u d e , g d a l , d e f i n e d b y E q u a t i o n 1 7. T h i s p r o c e d u r e

    y i e l d s t h e r e s ul t ;

    Ua i

    F ) - ~

    d

    w h e r e x

    a n d h e n c e

    c ~ pd2 -~ p L

    2 . 5 q _ _ ( ) / ( 6 s _ K a P d ) f h ~ i 2 ( z ) d z ) ( 1 7 )

    8 n 2 S 2 m e 2 ~ m e h o

    i 1

    ( C / 8 W 2 ) ( p d 2 / m e ) ( V / f o d ) f ~ ( x ) x d x / f ~ 2 ( x ) x d x

    o o

    z / h ; g = a peak facto r w ith a value of about 4

    f o I 9 2 ( x ) d x )

    2.5gC r, (/____-~) . ( f Q d ) 2 x

    ( 6 s - K a P d 2 / m e 2 1 V S f o l ~ ( x ) x d x

    I f V i s c h o s e n t o b e t h e m e a n s p e e d a t w h i c h t h e r e s p o n s e a t t a i n s a

    m a x i m u m t h e n f o d / V S ~ I / i . I 0 ; a c c e p t i n g a c o r r e l a t i o n l e n g t h o f o n e d i a m e t e r

    a n d n o t i n g t h a t t h e t e r m d e p e n d e n t o n m o d e s h a p e v a r i e s o n l y s l i g h t l y

    w i t h ~ ( x ) { 1 . 7 3 for ~ ( x ) = x and 1 . 7 9 for ~ ( x ) = x 2} it foll ows that;

    C = 3 .4 g C L / F ~ / { ~ s - K a P d 2 / m e }

    T h a t i s , t h e e q u i v a l e n t s t a t i c l o a d

    w ( z )

    is giv en by;

    w ( z ) ~ 3 . 4 g C ( P V 2 ) d ( z / h ) / V ~ / { ~ s - K a P d 2 / m e }

    wher e; v = I . i f o d / S

    4 . 2 T a p e r e d C h i m n e y s

    A s l i g h t l y m o d i f i e d f o r m o f a n a p p r o x i m a t i o n t o t h e rm s m o d a l a m p l i t u d e ,

    d a i , d e r i v e d b y V i c k e r y a n d C l a r k ~1 1 i s

    C L P d 4 ( Z e ) ~ i ( Z e ) ( ~ / 2 t )

    = z s )

    a i 8 ~ 2 S 2 m e h l o I ~ i 2 ( x ) d x ( B s + 6 a )

    w h e r e Z e = h e i g h t a t w h i c h ~ (z ) = I / S l i d ( z )

    d d z ) d z )

    = ) + ~ ) Z = Z e

    t (-( dz z

    ~ ( Z e ) = { 2 A Z e - A ( z ) d z } %

    6 t Z = Z e

    : t h e l e n g t h o f t h e c h i m n e y o v e r w h i c h t h e d i a m t e r c h a n g e s b y o n e s i x t h .

    T h e m a x i m u m r e s p o n s e o c c u r s w h e n z i s s u c h t h a t d ~ f z ) @ ( z ) / / ~ is a

    m a x i m u m .

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    6

    F o r t h e c a s e o f a c o n s t a n t t a p e r i n u n i f o r m f l o w t h e m a x i m u m r e s p o n s e i n

    t h e f u n a d e m e n t a l m o d e c a n b e e v a l u a t e d f o r a m o d e s h a p e o f t h e f o r m ~ ( z ) =

    z . I n t h i s c a s e t h e h e i g h t , z a t w h i c h C a a t t a i n s a m a x i m u m g i v e n b y;

    Z M / h = N / ( ( N + 4 ) ( I - 8 ) ) w h e r e e = d ( h ) / d ( o ) = t i p d i a m . ~ b a s e d i a m .

    I n m a n y c a s e s t he f u n d a m e n t a l m o d e s h a p e i s w e l l a p p r o x i m a t e d w i t h N = 2

    a n d t h e n ;

    Z M / h = 1 / 3 ( 1 - 6 ) a n d d ( z s ) / d ( o ) = 2 / 3

    T h e c r i t i c a l s p e e d i s

    I / S 2 / 3 d ( o ) f o

    a n d a n e q u i v a l e n t s t a t i c l o a d c a n

    b e e v a l u a t e d a s b e f o r e . T h e e q u i v a l e n t l o a d i s a g a i n s e t a s ;

    w ( z ) = W o ( Z / h ) w h e r e w o = C ( p V C z ) d ( o ) ( 1 9 a )

    a n d C =

    g (

    4 )2 CL {

    n d ( o ) Z } 1

    9 ( 1 - @ ) 2 ( - O ) h ( B s + B e )

    p u t t i n g l o :

    h / d ( o )

    c

    .~ ~ - - r ~ i ~ ) ( 1 9 b )

    c = ( 4 ) 2 {

    } ~ cL, (/~,~o ~ s + ~ s ~

    9 - 0 ) 2 ( ~ - @ )

    T h e r e s u l t o b t a i n e d f o r a t a p e r e d c h i m n e y i s i n v a l i d f or 8 n e a r I a n d

    t h e r e s u l t s f o r a u n i f o r m s t r u c t u r e m u s t b e e m p l o y e d . T h e t r a n s i t i o n v a l u e o f

    d, c a n e e e v a l u a t e d b y e q u a t i n 6 t h e v a l u e s o f w o c o m p u t e d f r o m E q u a t i o n s 1 7

    a n d 1 9 . I t i s t h e n n e c e s s a r y t o d e f i n e a r e p r e s e n t a t i v e d i a m e t e r , d, w h i c h

    w i l l b e t a k e n a s t h e a v e r a g e d i a m e t e r o v e r t h e t o p t h i r d o f t h e ch i m n e y . T h e

    e q u i v a l e n t s t a t i c l o a d s m a y t h e n b e e x p r e s s e d a s ;

    w { z ) : % 9 ~ v ~ : ~ ( = / h ) 2 e )

    w h e r e f o r O n e a p I;

    -~ (2 a

    y i . I / S f c c

    C = 3 . U T C L ( / l } 4 / ( ~ s + ~a, ( 2 1 Z

    ~ a - K a p c 2, 'S me ( 2 1 c )

    a n d f o r s m a l l 0 ;

    3 . 6 4 g C L c , 1

    C = (--/:~

    ( 2 2 a )

    { D s + ~ :~ ) ~ : X ~ _ - i~ ) 5 / 2 i + f C ) 3 / 2

    ~ 4 f o ~ / S ( i + 5 0 ) ( 3 2 b )

    x = ~ i ~

    T h e r a t i o o f t h e t w o v a l u e s o f w ( z ) i s t h e n ;

    (r?(=) for 0 : . /(Te(z) fo r {~ < < i } : 0 . 0 7 ( - C ) s i Z ( z + 5 0 ) 7 1 2

    = i @ 0 = 0 5

    T h u s , E q u a t i o n 2 1 is a p p l i c a b l e f o r a ti p t o b a s ~ i a m e t e r P a t i o b e t w e e n 0 . 5

    a n d

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    1 . 0 a n d E q u a t i o n 2 2 f o r r a t i o s l e s s t h a n 0 . 5 . F o r v a l u e s o f e l e s s t h a n 0 . 5

    t h e n e g a t i v e a e r o d y n a m i c i s w e a k e n e d f r o m t h a t a c t i ng o n u n i f o r m o r l i g h t l y

    t a p e r e d s t a c k s a n d 8 a c a n b e e v a l u a t e d f r o m t h e e m p i r i c a l r e l a t i o n s h i p

    8 a ( @ < ) : - ( K a P d 2 / m e ) ( 0 . 6 + 0 . 8 @ ) ( 2 3)

    5 C O M P A R I S O N O F D E T A I L E D A N D A P P R O X I M A T E M E T H O D S

    T h e f i r s t m o d e r e s p o n s e o f t he t w o c h i m n e y s s h o w n i n F i g . 5 w e r e c o m p u t e d

    u s i n g t h e d e t a i l e d m e t h o d d e f i n e d b y E q u a t i o n s ( 5) t o (8 ) a n d c o m p a r e d w i t h

    t h e a p p r o x i m a t i o n s d e f i n e d b y E q u a t i o n s 2 0 t o 2 2. T h e d y n a m i c p r o p e r t i e s o f

    t h e t w o c h i m n e y s a r e s h o w n i n F i g s . 6 a n d 7 a n d t h e p r e d i c t e d p e a k b a s e m o m e n t s

    i n F i g s . 8 a n d 9 . T h e r e l e v a n t d a t a e m p l o y e d i n t h e c o m p u t a t i o n s w e r e ;

    S t r o u h a l N u m b e r , S = 0 . 2 2 R M S L i f t C o e f f i c i e n t s , C L = 0 . 2 0

    C o r r e l a t i o n L e n g t h, = 1 S t r u c t u r a l D a m p i n g ~ s = 0 . 0 1

    A e r o d y n a m i c D a m p i n g

    K a = 0 . 6

    T e r r a i n R o u g h n e s s z =

    . O 0 8 m

    g = / 2 ~ o T + 0 . 5 7 7 / / 2 1 n f o T

    T h e c h o s e n p a r a m e t e r s a r e no t u n r e a s o n a b l e b u t s h o u l d n o t be a c c e p t e d f o r

    g e n e r a l a pp l i c a t i o n . T h e c h o i c e o f s u i t a b l e a e r o d y n a m i c p a r a m e t e r s i s

    d i s c u s s e d b y B a s u [ 5] w h o s u g g e s t s th a t t h e v a l u e s o f 0 . 2 f o r C i s

    e x c e s s i v e f o r " s m o o t h " s t r u c t u r e s i n t h e a b s e n c e o f s m a l l s c a l e t u r b u l e n c e .

    T h e c a l c u l a t i o n s f o r t he a p p r o x i m a t e e v a l u a t i o n s a r e p r e s e n t e d b e lo w ;

    C h i m n e y No . I

    0 =

    v M =

    d

    g

    k

    m / p d 2 =

    C =

    w f z =

    =

    B a s e M o m e n t =

    =

    C h i m n e y N o . 2

    8 =

    vn =

    d =

    g =

    0 . 9 6 ( > 0 . 5 )

    i . I x 1 / 0 . 2 2 x 0 . 3 6 4 x 1 7 . 6 3 = 3 2 . 1 m / s

    1 7 . 6 3

    3 . 6 5

    1 9 3 . 6 / 1 7 . 6 3 = 1 0 . 9 8

    1 0 3

    3 . 4 x 3 . 6 5 x 0 . 2 0 1

    x ( ) = I i . 6

    ( 0 . 0 1 - 0 . 6 0 / 1 0 3 ) 1 0 . 9 8

    1 1 . 6 x 1 7 . 6 3 x x 1 . 2 0 x 3 2 . 1 2 x z / h

    1 2 6 ( z / h ) k N / m

    1 2 6 x x 1 9 3 . 6 x 2 / 3 x 1 9 3 . 6

    1 5 7 6 x 1 0 6 N m

    0 . 3 3 3 ( < 0 . 5 )

    4 / ( 1 + 5 / 3 ) x 1 / 0 . 2 2 1 6 . 8

    1 6 . 8 m

    3 . 5 5

    252 = 28 9 m/s

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    63

    30 3

    2C~

    10C

    Chimn~

    = 17.63m

    = 17.63rn

    = 18.4m

    No. 1

    Chlmn~

    , 12.6m

    F I G 5

    >= 37.8m

    No. 2

    O v e r a l l D i m e n s i o n s o f C h i m n e y s

    1.0

    / 1

    0.8

    0.6

    0.2

    I 2

    ~(z)

    (~z

    H

    0.4

    H = 193.6m

    f= = 0.364 Hz

    M I = 7.44 x 106 kg

    F I G 6 M o d a l P r o p e rt ie s o f C h i m n e y N o 1

    10 20 3O 411

    F I G 7 M o d a l P r o p e rt ie s o f C h i m n e y N o 2

    -3 -2 -1 0 1 2 3

    H = 365 8m

    ft = 0.252 Hz

    f2 = 0,88 Hz

    J

    MI = 13.3 x 106 kg /

    MZ= 15,5 x 106 kg /

    /

    1

    \

    X

    x BM 2

    -40 -20

    l

    20 ~ ~ 80

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    1500

    0

    ; ,

    o

    r o t

    1000

    500

    L

    10

    ~ /s = 0 . 0 1

    FIG. 8

    15 20 25 30 35

    U 1o m / s )

    Vortex Shedding Response Calculated by Detailed Metho d

    Chimney 1

    164

    2000

    x 1500

    z

    ~E I000

    . E

    m 5 0 0

    175 = 0.01 /

    ~ /~ M ode

    - - I J

    10 20 30 40 50

    U lo lm/s l

    FIG. 9

    Vortex Shedding Response Calculaled by Detailed Method

    Chimney 2

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    65

    = 3 6 5 . 8 / 1 6 . 8 = 2 1 . 8

    m / D d 2 = 1 0 7

    3 . 6 4 3 . 5 5 0 . 2 0 1 1

    C = ) = 4 . 5

    0 . 0 1 - 0 . 6 0 / 1 0 3 ) 2 1 . 8 . 6 7 ) 5 / 2 2 . 6 7 ) 3 / 2

    w = 4 . 5 0 x 1 6 . 8 x 1 . 2 0 x 2 8 . 9 2 z / h )

    = 3 8 z / h ) k N / m

    B a s e M o m e n t = 3 8 x x 3 6 5 . 8 x 2 / 3 x 3 6 5 . 8

    = 1 6 9 0 x 1 0 6 N m

    T h e s i m p l i f i e d f o r m s r e q u i r i n g a k n o w l e d g e o f t h e m o d e s h a p e m a y a l s o b e

    u s e d t o o b t a i n e s t i m a t e s o f t h e m a x i m u m b a s e m o m e n t a n d t h e s e a r e i n c l u d e d i n

    T a b l e I t o g e t h e r w i t h t he r e s u l t s of t h e d e t a i l e d a p p r o a c h . A s w o u l d b e e x p e c t e d

    f r o m t h e n a t u r e o f t h e a p p r o x i m a t i o n s u s e d i n d e r i v i n g t h e s i m p l i f i e d m e t h o d s

    t h e s e y i e l d c o n s e r v a t i v e e s t i m a t e s . I n t h e tw o e x a m p l e s t h e s t a t i c l o a d a p p r o a c h

    y i e l d s e s t i m a t e s w h i c h a r e 8 % a n d 1 6 % h i g h w h i l e t h e s i m p l i f i e d m o d al a p p r o a c h

    y i e l d s I st m o d e e s t i m a t e s w h i c h a r e h i g h b y 7 %, 1 8 % w h i l e t h e e s t i m a t e f o r t h e

    s e c o n d m o d e i s 3 % g r e a t e r t h a n t h a t o f t h e d e t a i l e d a p p r o a e h .

    6 . C O N C L U S I O N S

    T h e s i m p l i f i e d f o r m s d e v e l o p e d h a v e b e e n d e m o n s t r a t e d t o p r o v i d e a d e q u a t e

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

    c h i m n e y s t o v o r t e x s h e d d i n g . T h e i r s u c c e s s f u l us e , h o w e v e r, i s d e p e n d e n t u p o n

    t h e s e l e c t i o n o f t h e r e q u i r e d a e r o d y n a m i c c o e f f i c i e n t s . T h i s i s a n a r e a w h i c h

    i s n ot c o v e r e d i n t he p a p e r b u t w h i c h h a s b e e n a d d r e s s e d b y B a s u 1 5J i n c o n s i d e r -

    a b l e d e p t h . T h e w o r k o f B a s u d r a w s a t t e n t i o n to t h e d e p e n d e n c y o f S , K a a n d

    C o n s u r f a c e r o u g h n e s s , t he p r e s e n c e o f s m a l l s c a l e t u r b u l e n c e a n d o n a s p e c t

    r a t i o . U n f o r t u n a t e l y , t h e r e i s a d e a r t h o f d a t a a t f u l l s c a l e v a l u e s o f R e y n o l d s

    N u m b e r a n d , a s a c o n s e q u e n c e , a h i g h l e v e l o f u n c e r t a i n t y m u s t b e a c c e p t e d i n

    t h e p r e d i c t i o n s o f v o r t e x i n d u c e d r e s p o n s e . E v e n w i t h a c a r e f u l a p p r a i s a l o f

    a p a r t i c u l a r s i t u a t i o n w i t h k n o w n r o u g h n e s s a n d t u r b u l e n c e t h e l e v e l o f

    a c c u r a c y ( e s t i m a t e d 8 0 % c o n f i d e n c e l i m i t s ) i s r o u g h n e s s 2 5 % f o r C L , I 0 %

    f o r S , 2 5 % f o r K a a n d 2 0 % f o r Z ; c o u p l e d w i t h a t y p i c a l a c c u r a c y i e v e l

    o f 2 0 % f o r t h e s t r u c t u r a l d a m p i n g t h e r e s u l t a n t l e v e l o f r e l i a b i l i t y o n t he

    p r e d i c t i o n is o f t h e o r d e r o f 4 0 % . T h i s f i g u r e i s n o t i n c o n s i s t e n t w i t h t he

    c o m p a r i s o n o f p r e d i c t e d a n d m e a s u r e d r e s p o n s e s p r e s e n t e d b y V i c k e r y a n d B a s u

    1 6] a l t h o u g h t h e l a t t e r s u g g e s t t h a t t h e f u l l s c a l e o b s e r v a t i o n s a r e m o r e

    l i k e l y t o f a l l b e l o w t h e p r e d i c t e d a n d i n a r a n g e f r o m a b o u t + 2 0 % t o - 50 % .

    W i t h t h e s e a c c u r a c y e s t i m a t e s i n v i e w i t is c l e a r t h a t t he e r r o r s a s s o c i a t e d

    w i t h t h e a p p r o x i m a t i o n i n t r o d u c e d i n d e r i v i n g t h e s i m p l i f i e d r e s p o n s e

    e q u a t i o n s a r e b a r e l y i f a t a l l s i g n i f i c a n t .

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    66

    T A B L E I : C o m p a r i s o n o f R e s p o n s e P r e d i c t i o n s

    Max. Base Speed (@ 10m)

    Mome nt at Max. Resp.

    Meth od Nm x 106 m/s

    Chimn ey I: Mode I

    D e t a i l e d 1 4 3 0 2 2 . 6

    S i m p l i f i e d 1 5 3 3 2 3 . 0

    S t a t i c L o a d A p p r o x . 1 5 7 6 2 3 . 0

    Chim ney 2: Mode I

    D e t a i l e d 1 4 6 0 2 0 . 0

    S i m p l i f i e d 1 7 3 0 2 0 . 6

    S t a t i c L o a d A p p r o x . 1 6 9 0 2 0 . 6

    Chim ney 2 : Mode 2

    Detai led 1790 39

    S i m p l i f i e d 1 8 5 0

    S t a t i c L o a d A p p r o x . n o t a p p l i c a b l e

    38

    R E F E R E N E S

    I. Vickery, BJ. and Clark, A.Q., "Lift or acros s-wind resp onse of tapered

    stac ks," Proc. A.S.C.E ., J. Stru ct. Div.; Vol. 98, 1972, (pp. 1-20).

    2. Vickery, B.J. and Basu, R., "Aros s-win d vibrat ions of struc tures of

    circul ar cross -sectio n. Part I: Deve lopme nt of a model for two-

    dimen siona l cond itions, " to be published, J.W.E. and I.A.

    4. Wooton, L.R., "The osci llati on of large circula r stacks in wind," Proc.

    Inst. Civ. Eng., Vol. 43, 1969 (pp. 573-598 ).

    5 . B a s u, R. , " A c r o s s - w i n d r e s p o n s e o f s l e n d e r s t r u c t u r e s o f c i r c u l a r c r o s s -

    section to atmos pheric turbul ence, " Ph.D. Thesis, Fac. Eng. Sc., Univ.

    Weste rn Ont., London, Canada, 1982.

    6. Vickery, B.J. and Basu, R., "A compa rison of model and full-sc ale

    b e h a v i o u r i n w i n d o f t o w e r s a n d c h i m n e y s , " P r o c . I n t . W o r k s h o p o n W i n d

    T u n n e l M o d e l l i n g C r i t e r i a a n d T e c h n i q u e s , N . B . C . , G a i t h e r s b u r g , A p r i l

    1982.

    7. Gho, B.T., "Alon g-win d respon se of chimneys ," M.E.Sc. Thesis, Fac. of

    Eng. Sc., Univ. West ern Ont., London, Canada, 1983.