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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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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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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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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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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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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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= 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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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
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