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Similitude and modelling in wind tunnel testing of bridges
Journal of Wind Engineering and Industrial Aerodynarnics, 33 (1990) 283-300
283
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Invited Paper
S I M I L I T U D E A N D M O D E L L I N G IN W I N D T U N N E L TESTING OF BRIDGES H. TANAKA
University of Ottawa, Ottawa, Ont. (Canada)
i.
Introduction
W h e n the d e s i g n of a b r i d g e r e q u i r e s the c o n s i d e r a t i o n of the likely effects of w i n d u p o n it, it is a common practice to undertake wind tunnel testing. Modelling t e c h n i q u e s e m p l o y e d in these testings v a r y but the s i m i l i t u d e requirements for t e s t i n g are c o n s i d e r e d to h a v e been well established and p r a c t i c e d for q u i t e some time. The o b j e c t i v e of the p r e s e n t p a p e r is to r e v i e w these modelling t e c h n i q u e s and a s s o c i a t e d similitude requirements for the understanding of the s t a t e - o f - t h e - a r t and d i s c u s s i o n for further improvements of our p r a c t i c e . The g e n e r a l requirement for modelling a physical phenomenon is e s s e n t i a l l y the same in any wind engineering problem. Our c o n c e r n is t y p i c a l l y the b e h a v i o u r of w i n d f l o w in a c e r t a i n s p a c e or a r e a and its i n t e r a c t i o n w i t h the g e o m e t r i c a l and/or mechanical characteristics of the b o u n d a r i e s of the f i e l d of c o n c e r n ( F i g . i ) . For c o r r e c t m o d e l l i n g in t h e s e p r o b l e m s , b a s e d on B u c k i n g h a m ' s Pit h e o r e m , it is r e q u i r e d that a set of dimensionless parameters c o n s i s t of suitable combinations of the r e f e r e n c e q u a n t i t i e s are i n v a r l a n t in m o d e l and p r o t o t y p e and with them the governing e q u a t i o n s are also rendered dimensionless. Various boundary c o n d i t i o n s have to be also maintained in d i m e n s i o n l e s s form. We cannot overemphasize the i m p o r t a n c e of this p r i n c i p l e s i n c e m o s t of our p r o b l e m s in the f i e l d of w i n d e n g i n e e r i n g c a n n o t be s o l v e d with theoretical approaches alone. For c o r r e c t modelling, all of t h e s e d i m e n s i o n l e s s parameters in the p r o t o t y p e m u s t be d u p l i c a t e d in the m o d e l . However, almost invariably, complete duplication of these parameters is impractical or impossible. As a m a t t e r of fact, the r e q u i r e m e n t s can be satisfied exactly only when model and prototype are identical. Hence, the decision must be made as to which parameters could be relaxed for each testing based on the understanding of the p h e n o m e n o n and the knowledge of dominant parameters. The less i m p o r t a n t ones s h o u l d be ignored. The p r o c e s s of this d e c i s i o n m a k i n g is e s s e n t i a l l y the c o n c e r n of this paper.
0167-6105/90/$03.50
© 1990 Elsevier Science Publishers B.V.
284
2. 2.1
Development Historical
of
Understanding
in
Bridge
Aerodynamics
Development
The u n d e r s t a n d i n g of the wind induced b e h a v i o u r of b r i d g e s has come a long way t h r o u g h the h i s t o r y [1,2]. One of the difficulties bridge e n g i n e e r s had t h r o u g h the p r o g r e s s of m o d e r n s u s p e n s i o n b r i d g e s was this p r o b l e m . It was a c o n t i n u o u s struggle against wind action (Fig.2). M a n y of the m a j o r s u s p e n s i o n b r i d g e s constructed in 19th c e n t u r y , the M e n a i S t r a i g h t B r i d g e by T h o m a s T e l f o r d to start with, were e i t h e r d e s t r o y e d or s e v e r e l y d a m a g e d by w i n d . In t e r m s of the p r o g r e s s in understanding of wind-bridge interaction, three particular incidents should be perhaps m e n t i o n e d as the t u r n i n g p o i n t s in the h i s t o r y . The f i r s t of them is the c o l l a p s e of the F i r t h of Tay B r i d g e in 1879. Tay B r i d g e was the w o r l d l o n g e s t b r i d g e in those days, c o n s i s t i n g of 84 s p a n s of w r o u g h t iron t r u s s e s . The c e n t r a l p o r t i o n of the b r i d g e was 13 s p a n s of 60 m e t r e truss g i r d e r , s p a n n i n g 27 m e t r e s a b o v e w a t e r , c a l l e d "the H i g h G i r d e r s " . This was the p o r t i o n b l o w n down by the gale and some 80 lives were c l a i m e d w i t h a train. N o t h i n g else gave such a big impact to bridge engineers regarding the consideration of wind loading on bridges. This incident apparently prompted some e a r l y w i n d t u n n e l as w e l l as full s c a l e s t u d i e s of w i n d l o a d i n g on s t r u c t u r e s [3]. The s e c o n d o c c a s i o n to be r e m e m b e r e d is a f a m o u s d i s a s t e r of the T a c o m a N a r r o w s B r i d g e in 1940. What h a p p e n e d then is now very well known and need not be repeated here. It s h o u l d be p o i n t e d out, h o w e v e r , that the b r i d g e was s u p p o s e d to w i t h s t a n d the s t a t i c wind loading up to the d e s i g n w i n d speed, w h i c h was more than twice the s p e e d at w h i c h the b r i d g e a c t u a l l y c o l l a p s e d . The key here was the d y n a m i c a c t i o n i n d u c e d by w i n d . As it is d e s c r i b e d l a t e r , the exact cause of the i n c i d e n t was not i m m e d i a t e l y clear then. But it was at least c l e a r to e n g i n e e r s that the windi n d u c e d d y n a m i c r e s p o n s e had to be in c o n s i d e r a t i o n . Looking back the h i s t o r y , a c t u a l l y there had b e e n some incidents like the Tacoma Narrows b e f o r e , in which bridges v i b r a t e d due to wind. However, apparently the d y n a m i c s of b r i d g e s had n e v e r become a very s e r i o u s topic for b r i d g e e n g i n e e r s . The third t u r n i n g c o r n e r of the h i s t o r y this a u t h o r w a n t s to p o i n t out is the i n t r o d u c t i o n of the n a t u r a l w i n d c h a r a c t e r i s t i c s in c o d i f i e d form for the d e s i g n consideration. This happened around 1960 particularly when Davenport, A. G. successfully reduced enormous a m o u n t of m e t e o r o l o g i c a l information to a set of modelling specifications and s t a r t e d o p e r a t i n g his B o u n d a r y L a y e r Wind Tunnel for a standardized simulation of the atmospheric b o u n d a r y layer [4,5]. Nobody would do any testing of structures without consideration of external loading conditions p r e c i s e l y as w e l l as t h o s e of structures. For the t e s t i n g of b r i d g e s , h o w e v e r , the wind action has b e e n often considered to be simply a uniformly s m o o t h air flow, w i t h or w i t h o u t s m a l l a n g l e of a t t a c k d e v i a t e d f r o m p a r a l l e l to the g r o u n d , and a l w a y s n o r m a l to the l o n g i t u d i n a l bridge axis. This loading condition is something like a consideration of e a r t h q u a k e e x c i t a t i o n by a s i m p l e h a r m o n i c g r o u n d
285 motion, without
which may be exceptions.
a fairly
conservative
assumption
but
not
E a r l y w i n d t u n n e l s t u d i e s of m o d e l b u i l d i n g s in 1930s and 40s indicated significant i n f l u e n c e of w i n d t u r b u l e n c e c h a r a c t e r i s t i c s on wind tunnel test results such as pressure distribution patterns. It was a b o u t the same p e r i o d of time w h e n P r a n d t l ' s b o u n d a r y l a y e r t h e o r y was a p p l i e d by m e t e o r o l o g i s t s to e x p l a i n the s t r u c t u r e of the l o w e r a t m o s p h e r e . H o w e v e r , the m o s t f u n d a m e n t a l p r i n c i p l e for wind tunnel tests in this r e g a r d was first c l e a r l y s t a t e d by J e n s e n , M. [6] as f o l l o w s : "The n a t u r a l w i n d is turbulent, and the (wind engineering) p h e n o m e n a in this b o u n d a r y l a y e r of the w i n d , as s h o u l d be emphasized, are highly dependent on the nature of this boundary layer. - . . . . The c o r r e c t m o d e l test for p h e n o m e n a in the w i n d m u s t be (therefore) carried out in a turbulent boundary layer, and the m o d e l - l a w r e q u i r e s that this b o u n d a r y l a y e r be to s c a l e as r e g a r d s the v e l o c i t y p r o f i l e . " Following the Tacoma Narrows, there was a significant contribution by aeronautical e n g i n e e r s t o w a r d s the d e v e l o p m e n t of aerodynamics with civil engineering applications. At the same time, it became a general practice to do testing of civil engineering structures such as bridges with a conventional aeronautical wind t u n n e l in uniform smooth air f l o w r a t h e r than with simulated natural winds. Davenport's formulation of c o d i f i e d natural wind and s i m u l a t i o n of it in w i n d t u n n e l t e s t i n g was a significant i m p a c t to the e n g i n e e r i n g p r a c t i c e a g a i n s t this trend. 2.2
Recent
Trend
in
Bridge
Aerodynamics
The last half a c e n t u r y has s e e n a n u m b e r of s t u d i e s on w i n d l o a d i n g on bridges, wind-lnduced r e s p o n s e of bridges and wind r e s i s t a n t d e s i g n of b r i d g e s . A l a r g e s c a l e and c o m p r e h e n s i v e wind tunnel testings have been c o n d u c t e d by m a n y r e s e a r c h e r s . Amongst those, the s t u d i e s led by F a r q u h a r s o n , F. B. of University of Washington, USA, and S c r u t o n , C. at the National Physical Laboratory, Teddington, E n g l a n d , w e r e the e a r l i e s t ones. Studies by o t h e r g r o u p s at USA, F r a n c e , Germany, Norway, Japan, Canada, Australia etc. have f o l l o w e d . T h e s e r e s e a r c h e s h a v e b r o u g h t us m u c h b e t t e r u n d e r s t a n d i n g of wind-bridge interaction. At the same time, it is not s u r p r i s i n g that t h e r e are some d i f f e r e n c e of t r e n d in bridge design and construction compared w i t h half a c e n t u r y ago, w h i c h are c r e a t i n g new problems for us. Some of the recent trend in bridge aerodynamics are as f o l l o w s : i) D e v e l o p m e n t in identification s t u d i e s to s u p p o r t them; 2) A c c u m u l a t i o n of both scale comparisons;
2D
and
3)
Development of cable-stayed suspension bridges;
4)
Wind problems of b r i d g e b r i d g e s at t h e i r e r e c t i o n
of
3D
phenomena
test
bridges
members, stage;
results
rather
cables
in
and
and
than
theoretical
some
full
classical
particular,
and
286
5)
A d v a n c e s in c o s t - e f f e c t i v e d e s i g n of more stringent requirement for results.
It w o u l d be a useful exercise to similitude requirements in w i n d tunnel t h e s e a n g l e s in m i n d . 2.3
Identification
of W i n d
Induced
b r i d g e s and, quantitative
review testing
Bridge
the of
as a result, wind study
m o d e l l i n g and bridges with
Response
Immediately after the Tacoma Narrows disaster, the US Government commissioned the B o a r d of D i r e c t o r s to i n v e s t i g a t e the c a u s e of the b r i d g e failure. The Board consisted of three engineers, A m m a n n , O. H., yon K a r m a n , T. and Woodruff, G. B.. Their Official R e p o r t was p r e s e n t e d in the f o l l o w i n g y e a r . It is interesting to note that two e m i n e n t b r i d g e e n g i n e e r s and a w o r l d famous fluid mechanician in this R e p o r t c o n c l u d e d that the c a u s e of the c a t a s t r o p h i c t o r s i o n a l m o t i o n of the b r i d g e was not c l e a r l y identifiable whereas the v e r t i c a l bending motion of the bridge deck observed p r e c e d i n g to the t o r s i o n a l m o t i o n was probably a forced vibration due to wind gust. Obviously it was very d i f f i c u l t to i d e n t i f y the n a t u r e of w i n d i n d u c e d b r i d g e r e s p o n s e at that s t a g e but it was a l r e a d y s u g g e s t e d then that t h e r e w e r e more than one c o n c e i v a b l e c a u s e s of b r i d g e i n s t a b i l i t y . L a t e r , of c o u r s e , yon K a r m a n c o l l a b o r a t e d with Farquharson to do w i n d tunnel investigation of the b r i d g e and t r i e d to a t t r i b u t e the c a u s e of vibration in both modes to vortex excitation. B l e i c h , F., a n o t h e r f a m o u s b r i d g e e n g i n e e r , t r i e d to e x p l a i n it by the a n a l o g y to the aerofoil flutter t h e o r y w i t h some a d d i t i o n a l aerodynamic force terms. Modification of f l u t t e r t h e o r y was also a t t e m p t e d by K l o p p e l , K. of G e r m a n y and S e l b e r g , A. of Norway. H i r a i , A. of University of T o k y o f o u n d the b r i d g e f a i l u r e as the lateral buckling due to lack of torsional rigidity of the stiffening frame. Many wind t u n n e l tests during 1960s and 70s g r e a t l y h e l p e d identifying the v a r i o u s types of w i n d i n d u c e d r e s p o n s e of b r i d g e structures (Fig.3). One w a y of classifying them is as f o l l o w s
[7]:
Static
Dynamic
behaviours
behaviours
Overturning Excessive lateral Divergence Lateral Buckling
deflection
Vortex induced oscillations Self-excited oscillations Vertical bending instability Torsional instability Coupled flutter Buffeting motion
S t a t i c p h e n o m e n a l i s t e d a b o v e are s e l d o m c o n s i d e r e d to be the definitive factor for the wind resistant design of bridges. Besides, it has been well accepted that t h e s e p h e n o m e n a can be p r e d i c t e d by the theoretical calculations with good accuracy p r o v i d e d that the t h r e e major aerodynamic force components, lift force, d r a g f o r c e and p i t c h i n g m o m e n t , are k n o w n .
287 The t h r e e components can be u s u a l l y m e a s u r e d u s i n g the m o s t conventional force balance system with reasonable accuracy. Only two f a c t o r s p e r h a p s s h o u l d be k e p t in m i n d as a p o s s i b l e s o u r c e of deviations: w i n d f l o w t u r b u l e n c e and w i n d t u n n e l b l o c k a g e . Though it is not a conventional practice to introduce wind flow turbulence for the f o r c e measurement with 2D sectional model, t h e r e is no r e a s o n w h y it c a n n o t be done and in fact, as it is mentioned in l a t e r s e c t i o n s , it has b e e n p r a c t i c e d r e c e n t l y to an extent. Consideration of s p a n w i s e v e l o c i t y c o r r e l a t i o n w o u l d give better accuracy for p r e d i c t i n g the c r i t i c a l mean wind s p e e d or m a g n i t u d e of d e f l e c t i o n . Experimental confirmation of it by u s i n g aeroelastic models has not b e e n done too e x t e n s i v e l y in a u t h o r ' s knowledge. The d i s c u s s i o n sections, therefore,
3. 3.1
Similitude Similarity
on s i m i l i t u d e requirements in is m o s t l y on d y n a m i c p h e n o m e n a .
Requirements
the
following
in G e n e r a l
of W i n d
The similarity requirements of wind characteristics in physical wind tunnel model and/or mathemetical simulation model have been extensively discussed by b o t h meteorologists and w i n d engineers [8,9]. The s i m u l a t i o n of w i n d can be c o n s i d e r e d in two categories: one is the s i m u l a t i o n of the a v e r a g e c h a r a c t e r i s t i c s of the turbulent boundary~ayer w i n d w h i c h is a p p r o a c h i n g towards the f i e l d of c o n c e r n for e a c h p a r t i c u l a r p r o j e c t and a n o t h e r is the s i m u l a t i o n of the wind structure at the i m m e d i a t e p r o x i m i t y field which of c o u r s e is l a r g e l y i n f l u e n c e d by its particular topographical conditions. T h e s e are s o m e t i m e s r e f e r r e d to as the "'far f i e l d " and "near field" simulation respectively. The "far field" flow characteristics are d e f i n e d g e n e r a l l y by the f o l l o w i n g parameters: Mean
wind velocity distribution vertical directions
in h o r i z o n t a l
and
Spectral quantities of v e l o c i t y c o m p o n e n t s : Intensities of t u r b u l e n c e S c a l e s of t u r b u l e n c e Distribution of s p e c t r a in a r e l e v a n t f r e q u e n c y range Space correlations of t u r b u l e n c e c o m p o n e n t s Temperature and concern Other
moisture
distribution
m i n o r f a c t o r s such as p r e s s u r e g r a d i e n t etc.
Coriolis
in
the
force,
field
of
atmospheric
For the s i m u l a t i o n of n a t u r a l w i n d at a p a r t i c u l a r location, the e f f e c t s of the surrounding t o p o g r a p h y n e e d to be c o n s i d e r e d . T h e s e are u s u a l l y p r o v i d e d by i n s t a l l i n g the g e o m e t r i c a l l y scaled m o d e l of "near f i e l d " p r o x i m i t y at the w i n d t u n n e l test s e c t i o n . To w h a t e x t e n t the s u r r o u n d i n g t o p o g r a p h y n e e d s to be s i m u l a t e d is still a good question but s h o u l d be d e c i d e d in r e l a t i o n to the development of the internal boundary layer flow due to the p r o x i m i t y m o d e l and its i n f l u e n c e on the test r e s u l t s .
288
A good modelling technique for the "far f i e l d " s i m u l a t i o n is a boundary layer wind t u n n e l -the use of a q u i c k l y d e v e l o p e d turbulent boundary layer by various roughness and obstacles installed on the w i n d t u n n e l f l o o r w i t h or w i t h o u t temperature and moisture control devices [5,10]. This technique has b e e n w i d e l y accepted and practiced but is not w i t h o u t any s h o r t c o m i n g unless it is e m p l o y e d with caution. F i r s t of a l l , the m e t e o r o l o g i c a l characteristics of the s i t e of c o n c e r n are not necessarily always the same. Excessive presumption of the w i n d c h a r a c t e r i s t i c s b a s e d on l i m i t e d d a t a m a y lead us to a wrong, sometimes non-conservative conclusion. Turbulence intensity is often a controversial factor. The suppression of the t u r b u l e n c e intensity by t h e r m a l s t r a t i f i c a t i o n is r e p o r t e d to h a v e an e s s e n t i a l effect sometimes on w i n d - i n d u c e d bridge motion [ii]. Another thing to be remembered is t h a t the simulation of rather special meteorological phenomena such as hurricanes, tornados and downbursts has somewhat different aspects. It has been studied recently to s o m e e x t e n t , but t h e r e are s t i l l m u c h to be e x p e c t e d . O n e of the a p p r o a c h e s t r i e d for h u r r i c a n e w i n d s is a Monte Carlo simulation of tropical low pressure models to d e f i n e the s t a t i s t i c s of e x t r e m e w i n d s o v e r a l o n g p e r i o d of t i m e [12]. 3.2
Aeroelastic
Similarity
The wind tunnel modelling requires aeroelastic similarity in addition to the s i m i l a r i t y of w i n d f l o w c h a r a c t e r i s t i c s as w e l l as the c o n s i s t e n t matching of l e n g t h s c a l e and g e o m e t r i c a l s h a p e of the s t r u c t u r e . Aeroelastic similarity, in general, is b a s e d on the c o n s i d e r a t i o n of the length, density, elastic property and internal friction of the s t r u c t u r e , density and viscosity of a i r , wind velocity and the a c c e l e r a t i o n due to g r a v i t y . These physical properties can be conveniently summarized into the following dimensionless quantities: i) 2) 3) 4) 5)
Reynolds number Froude number Density ratio Cauchy number Critical damping
review A brief testing influence on of s o m e w o r t h . Reynolds
ratio
of these quantities in of w i n d - l n d u c e d structural
terms of their response w o u l d be
Number
Reynolds number c a n be d e f i n e d as the r a t i o of the fluid inertia force to the f l u i d v i s c o u s force. In m o s t of w i n d t u n n e l testings it is i m p r a c t i c a l to s a t i s f y R e y n o l d s number similitude. Indeed, the viscous forces are usually at l e a s t the order of magnitude smaller and relatively unimportant compared to the f l u i d inertia forces. However, the c o n s e q u e n c e of t h i s d i s t o r s i o n of the s i m i l i t u d e requirement should be e x a m i n e d carefully for the corret interpretation of the test results. The following three p o i n t s s h o u l d be p a r t i c u l a r l y noted: i) It is cylinder
well known t h a t the flow is very sensitive to the
pattern change
around a circular of the Reynolds
289 n u m b e r b e c a u s e of the s h i f t of f l o w s e p a r a t i o n p o i n t s w i t h it. T h e r e is a c o r r e s p o n d i n g c h a n g e of the w a k e w i d t h and the drag f o r c e as w e l l as the f r e q u e n c y of w a k e v o r t e x f o r m a t i o n . It is i m p o r t a n t that, in m o d e l l i n g a s t r u c t u r e w i t h s m o o t h c u r v e d surface geometry, these effects are properly taken into consideration, though this is not u s u a l l y the case in b r i d g e structures. It s h o u l d be n o t e d that the critical Reynolds n u m b e r is also d e p e n d e n t on the s u r f a c e r o u g h n e s s of the s o l i d b o u n d a r y and the t u r b u l e n c e l e v e l in the a p p r o a c h i n g air flow. 2) In case of the f l o w over the s e c t i o n s w i t h s h a r p c o r n e r s , the f l o w separation points do not s h i f t and the f l o w p a t t e r n is less s e n s i t i v e to the c h a n g e of the R e y n o l d s n u m b e r . This is u s u a l l y the case for b r i d g e deck cross-sections. Broad wake after separation from the u p s t r e a m c o r n e r s may r e a t t a c h to the b o d y s u r f a c e , d e p e n d i n g on the a s p e c t r a t i o of the b o d y cross-section. The f l o w reattachment, of course, results a r e d u c t i o n of drag f o r c e and i n c r e a s e of the S t r o u h a l n u m b e r in general. The c r i t i c a l a s p e c t r a t i o of the b o d y at w h i c h this change occurs d e p e n d s on the R e y n o l d s n u m b e r as w e l l as the corner radius and the airstream turbulence l e v e l [13]. It s h o u l d be also noted that this f a c t o r is i n f l u e n c e d by the w i n d t u u n e l b l o c k a g e r a t i o as w e l l . 3) In the p r o b l e m s i n v o l v i n g e f f e c t s of w i n d t u r b u l e n c e , it is essential to simulate the velocity spectra correctly. Townsend [14] has p o i n t e d out that "--- w h i l e geometrically similar flows are e x p e c t e d to be dynamically and structurally similar if their Reynolds n u m b e r s are the same, t h e i r s t r u c t u r e s are a l s o v e r y n e a r l y s i m i l a r for all R e y n o l d s n u m b e r s w h i c h are l a r g e e n o u g h to allow turbulent flow." This is a significant g o s p e l for the wind tunnelers who e m p l o y an a r t i f i c i a l l y developed turbulent boundary layer flow as a simulated natural wind, since achieving the Reynolds number similitude is i m p r a c t i c a l a n y w a y s . H o w e v e r , it s h o u l d be r e m e m b e r e d that the R e y n o l d s n u m b e r does play a part in the e x i s t a n c e of the i n e r t i a s u b r a n g e of the e n e r g y s p e c t r u m . As the R e y n o l d s number increases, the h i g h f r e q u e n c y end of the distribution w i l l be e x t e n d e d so that the total d i s s i p a t i o n of turbulence energy remains unchanged. On the o t h e r h a n d , w h e n the R e y n o l d s n u m b e r is s m a l l , the r a t i o of the size of the dissipating eddies to the representative size of the predominant e d d i e s b e c o m e s h i g h l y d e p e n d e n t on v i s c o s i t y . The results of this is inaccurate simulation of turbulence s t r u c t u r e due to n a r r o w e r than r e q u i r e d i n e r t i a s u b r a n g e [15]. Froude
Number
F r o u d e n u m b e r is the r a t i o of f l u i d i n e r t i a f o r c e to v e r t i c a l f o r c e due to g r a v i t y and/or buoyancy. Consequently the F r o u d e similitude becomes i m p o r t a n t for the c a s e s such as d i s s i p a t i o n of airborne particles or w i n d - i n d u c e d response of cable-supported structures where g r a v i t y is a dominant factor. Although the F r o u d e n u m b e r s i m i l i t u d e has b e e n w i d e l y a c c e p t e d and e m p l o y e d for many aeroelastic s t u d i e s in the past, it is not an essential requirement unless the g r a v i t y a n d / o r b u o y a n c y play an i m p o r t a n t role. For e x a m p l e , if the r e s t o r i n g f o r c e of a m o d e l s t r u c t u r e is provided only by its linear elastic properties, its a e r o e l a s t i c
290
response does not require Froude number consideration. Simplifications of the r e q u i r e m e n t due to i n t e r a c t i o n of g r a v i t y and elastic f o r c e for a i r - s u p p o r t e d structures have been discussed elsewhere [16]. Density
Ratio
The ratio of the structural material density to a i r d e n s i t y is an important parameter w h i c h has to be considered in any aeroelastic testing. However, the r e q u i r e m e n t c a n be e x p r e s s e d in slightly different way since the a e r o e l a s t i c m o d e l m a y not be an exact replica of the prototype structure. In fact, many aeroelastic models a r e the so-called equivalent models, which simply maintain the g e o m e t r i c a l shape and dynamic characteristics; i . e . , the natural frequencies, corresponding mode shapes and damping ratios. If t h i s is the c a s e , t h e r e is no p o i n t k e e p i n g the d e n s i t y r a t i o as a p a r a m e t e r a n d the m a s s r a t i o s h o u l d r e p l a c e it. T h e m a s s r a t i o ~ is d e f i n e d by the f o l l o w i n g equation a n d it is i d e n t i c a l to the density ratio if the s t r u c t u r e is s o l i d a n d homogeneous:
= m/p~ where
p p L m
~ = = = =
(p
L3 / L ) / ( p L
~ )
=
material density air density linear dimension m a s s per u n i t l e n g t h
?~1?
of
(11
structure.
In t o r s i o n a l problems, the ratio based on m a s s moment of inertia per u n i t l e n g t h of s t r u c t u r e (J) is u s e d to r e p l a c e E q . ( 1 ) = J/pL 4 . by Cauch~
Number
Cauchy number is d e f i n e d fluid inertia force; i.e., Ca frequency of the s t r u c t u r e f ~.~ this
ratio
can Ca
where
[ ( E l / m ) ~a be
~v
the r a t i o of elastic f o r c e to E/pu z . Considering the n a t u r a l
]/L ~
rewritten
(EIILi)I(~uZL
E = Young's f = natural
as =
as
follows:
2 ) -v
modulus of frequency
(fLlu)Z(m/yL
a)
(2)
material
Therefore, if we t a k e the r e d u c e d v e l o c i t y u ~ = u / f L as a n e w dimensionless parameter, the C a u c h y n u m b e r is e q u i v a l e n t to ~ / u ~ ~. In m o s t w i n d t u n n e l t e s t i n g s , rather than considering the o r i g i n a l definition of Cauchy number, it is e a s i e r to t a k e the m a s s r a t i o into consideration for the model design and analyse the dimensionless structural response as a function of reduced velocity. Usually it is n o t d i f f i c u l t to do t e s t i n g o v e r a w i d e r a n g e of r e d u c e d v e l o c i t y to c o v e r its p o s s i b l e r a n g e in r e a l i t y . Critical
Damping
Ratio
The magnitude of s t r u c t u r a l damping is o b v i o u s l y an i m p o r t a n t parameter for the p r e d i c t i o n of s t r u c t u r a l response. The problem, however, is t h a t its magnitude is not k n o w n exactly until the
291 s t r u c t u r e c o m e s into e x i s t a n c e . As a m a t t e r of fact, e v e n for the existing structures, the m a g n i t u d e of s t r u c t u r a l d a m p i n g is o f t e n controversial because of the difficulty in m e a s u r i n g its value particularly with relatively large amplitude motion. Naturally the p r e d i c t i o n of s t r u c t u r a l r e s p o n s e p r i o r to its c o n s t r u c t i o n is done by s p e c u l a t i o n b a s e d on a t y p i c a l d a m p i n g v a l u e f r o m the past experience.
4.
Effects
of M a s s
and
Damping
The b r i e f r e v i e w of the p r e v i o u s s e c t i o n u n d e r l i n e s meaning of c h o o s i n g appropriate mass r a t i o and damping ratio in wind tunnel studies. Mass and d a m p i n g are two i m p o r t a n t f a c t o r s in the d e s i g n and construction of wind tunnel models and these two requirements have b e e n s o m e t i m e s c o m b i n e d t o g e t h e r as the S c r u t o n n u m b e r (or the m a s s - d a m p i n g parameter) requirement in the past [17]. Scruton, through his e x p e r i e n c e , s h o w e d that the joint mass-damping parameter, when used together with the reduced v e l o c i t y ur, was a c o n v e n i e n t d i m e n s i o n l e s s p a r a m e t e r to i n d i c a t e the a e r o d y n a m i c s t a b i l i t y of structures. A simple mathematical m o d e l can d e m o n s t r a t e that the m a g n i t u d e of m a x i m u m v o r t e x - i n d u c e d r e s p o n s e can be d e c i d e d by the product of mass and damping parameters and this has been experimentally verified ( F i g s . 4 and 5). In case of some p a r t i c u l a r b l u f f s e c t i o n s , h o w e v e r , the mass and d a m p i n g not o n l y a f f e c t on the r e s p o n s e m a g n i t u d e a n d / o r the wind speed to have the r e s p o n s e o c c u r , but also can c h a n g e its nature from the instability with diverging amplitude to the " a e o l i a n " type r e s p o n s e of r e s t r i c t e d a m p l i t u d e [18]. Figs. 6 and 7 show two of these peculiar examples observed as 2D s e c t i o n a l m o d e l test r e s u l t s in s m o o t h air flow. In both cases, mass and damping parameters were changed over a range though the geometrical shape of the m o d e l was maintained e x a c t l y the same. Very similar response characteristics are o b s e r v e d in b o t h cases, t h o u g h one is in h e a v i n g and a n o t h e r is in t o r s i o n a l mode of vibration. W h e n mass and/or damping p a r a m e t e r s are small, there are typical vortex-induced type v i b r a t i o n s of restricted amplitude observed at low wind speed range and also there appear instabilities of d i v e r g i n g a m p l i t u d e at h i g h e r w i n d speed. It is interesting to n o t e that these i n s t a b i l i t i e s start approximately at the r e d u c e d w i n d s p e e d of i/St, w h e r e St is the S t r o u h a l n u m b e r d e f i n e d in the w a k e . It m e a n s that the f i r s t v o r t e x - i n d u c e d like peak, w h i c h c a n n o t be o b s e r v e d w i t h h e a v y m o d e l s , a c t u a l l y o c c u r s at a l m o s t a half of the S t r o u h a l s p e e d . H o w e v e r , w h e n the m o d e l mass is increased and/or the higher structural damping is provided, the response started at the Strouhal wind speed is suppressed at h i g h e r s p e e d range, m a k i n g a n o t h e r p e a k r a t h e r than diverging instability. The s i m i l a r p h e n o m e n a h a v e been r e p o r t e d e a r l i e r by N o v a k [19] and M i y a t a et al. [20]. Instability for this case is s o m e t i m e s o b s e r v e d at e v e n h i g h e r s p e e d r a n g e a f t e r the s e c o n d p e a k is p r a c t i c a l l y suppressed. These seem to be the c a s e s w h i c h have been classified by Y a m a d a [21] as the "bluff sections with imperfectly locked-in shedding". The w i n d - i n d u c e d fluctuating pressure on a l o n g - w i n d s i d e s of the m o d e l cross-section for these cases, a c c o r d i n g to Y a m a d a , has two p r e d o m i n a n t frequency components: the Strouhal
292
frequency and the f r e q u e n c y of the body motion. They do not necessarily coincide. When the S c r u t o n number is small, the response develops as the instability whereas it c a n be s e p a r a t e d from instability and become ordinary vortex-induced motion when the S c r u t o n n u m b e r is larger. It m a y be also worthwhile to mention that the e f f e c t of m a s s parameter and d a m p i n g parameter for t h e s e c a s e s is not r e a l l y the s a m e and h e n c e the Scruton number does not w o r k as a good single parameter to d e s c r i b e the response characteristics. These two parameters have to be examined separately [18]. T h i s fact indicates the importance of g o o d m a s s and d a m p i n g simulation not o n l y in dynamic response measurement but a l s o in measurement of aerodynamic derivatives by f r e e vibration method where generally m a s s and damping simulation is not c o n s i d e r e d to be e s s e n t i a l [22].
3.
Structural
3.1
2D-3D
Modelling
and
Response
Prediction
Comparison
Comparison of sectional versus full bridge model wind tunnel testing (Fig.8) has b e e n a classical query [23,24]. The obvious advantages of the f u l l model technique are somewhat retarded because of difficulty with complexity of model design and construction including configuration changes and sometimes unavailability of experimental facilities large enough to accomodate full bridge models. For relatively short span bridges, it m i g h t be e v e n more difficult to j u s t i f y s u c h e n d e a v o u r simply b e c a u s e of financial constraints. The alternative approach to t h i s is the u s e of a c o n v e n t i o n a l sectional m o d e l of the b r i d g e , as has been widely practiced for years. This method can well simulate the s t r u c t u r a l geometry of the b r i d g e but has b e e n a l w a y s facing a debate whether or not it c a n r e a l l y p r o v i d e a l l of the necessary and c o r r e c t information for the b r i d g e d e s i g n e r s . The e f f e c t s of w i n d turbulence, to start with, have been traditionally neglected in 2D testings. Consequently sectional models were considered to be not m u c h of u s e for the p r e d i c t i o n of bridge buffeting. However, an a t t e m p t has b e e n m a d e to e x t e n d this method for the s y s t e m a t i c measurement of a e r o d y n a m i c forces, which can t h e n be applied for response prediction taking into account the w i n d t u r b u l e n c e [25-28]. Sectional testing has b e e n b e l i e v e d to be r e l i a b l e in d e s i g n to a v o i d v o r t e x s h e d d i n g excitation and aerodynamic instabilities. Its larger geometrical scale is well-suited for examining corrective configuration changes and u s e of a e r o d y n a m i c devices. Question remains h o w e v e r as to w h e t h e r or n o t it is e x c e s s i v e l y conservative. T h i s is an i n c r e a s i n g l y important point because of the r e c e n t tendency towards competitive cost-effective design and a strong impetus to d e t e r m i n e the e x t e n t of the a e r o d y n a m i c margin provided by the s e c t i o n a l model studies. Some of the recent comparative studies particularly on turbulence and mass effects h a v e b e e n c a r r i e d out in t h i s c o n t e x t [29,30]. 5.2
as
Taut
Strip
Model
Method
This technique (Fig.9) a m e a n s to t a k e a s e c o n d
was first introduced by l o o k at the e x p e r i m e n t s
Davenport [31] c a r r i e d out in
293 the e a r l y s t a g e of s u s p e n s i o n b r i d g e a e r o d y n a m i c s , i n c l u d i n g this time t h e i r 3D r e s p o n s e characteristics to simulated turbulent wind. T h o u g h the observed model r e s p o n s e did not s e e m to be " e x a c t " , the m e t h o d w a s f o u n d to be q u i t e p r o m i s i n g in r e p r o d u c i n g full scale bridge hehaviour b e c a u s e the possibility of a d a p t i n g the m e a s u r e d data using the s i m p l e half-wave response to more complicated mode s h a p e s was then c o n c e i v e d . This c a l c u l a t i o n was later formulated by D a v e n p o r t [32] including the "cross-wave" contribution in statistical manner for the classical suspension bridges. The chief o b j e c t i v e of the taut strip model method is a simulation of the d y n a m i c c h a r a c t e r i s t i c s of the b r i d g e road d e c k -- its frequency, d a m p i n g and mode s h a p e -- w i t h the same s c a l i n g f a c t o r s as the s i m u l a t i o n of n a t u r a l w i n d . W i t h this c o n c e p t , the linear scale of the m o d e l is c o n s t r a i n e d not so m u c h by the size of the p r o t o t y p e s t r u c t u r e but by h o w l a r g e a s c a l e of t u r b u l e n c e can be in the wind tunnel. It s h o u l d be n o t e d that, in most cases, the basic scaling f a c t o r s for l e n g t h and time are c h o s e n almost independently of each other. It m e a n s that b o t h R e y n o l d s and F r o u d e s i m i l i t u d e s are v i o l a t e d . The f u n d a m e n t a l response characteristics of taut s t r i p m o d e l s have b e e n examined [33] to c o n f i r m that the r e s p o n s e in g e n e r a l are in g o o d a g r e e m e n t w i t h b u f f e t i n g and f l u t t e r t h e o r i e s w h e n the aerodynamic derivatives and a e r o d y n a m i c admittance f u n c t i o n s are m o r e or less k n o w n . As a matter of fact, it is one of the advantages offered by this m e t h o d that w h e n the taut s t r i p m o d e l test r e s u l t s are used as input to the buffeting theory, the complication of d e f i n i n g the aerodynamic admittance f u n c t i o n is avoided. A l s o the possible complication due to n o n - l i n e a r i t y of aerodynamic derivatives need not be c o n s i d e r e d e i t h e r . The m e t h o d also shares m a n y a d v a n t a g e s of full b r i d g e m o d e l method such as the intrinsic inclusion of 3D structural characteristics, availability of t e s t i n g a g a i n s t an o b l i q u e w i n d etc., and yet is still much s i m p l e r in c o n c e p t than the full bridge model and to a c e r t a i n e x t e n t s h a r e s the a d v a n t a g e s of the sectional model of low cost and short lead time. The disadvantages of this method, on the other hand, are some technical difficulties associated with the c e n t r e of rotation, frequency ratio, lateral sway characteristics, adjustment of generalized mass and structural damping and maintaining the a c c u r a c y in m o d e l c o n f i g u r a t i o n [34]. Discussing the similitude problems of taut strip models, special attention s h o u l d be g i v e n to the c o n s i d e r a t i o n of the vibration mode shapes. It is worthwhile to n o t e a g a i n that the c o n c e p t of taut s t r i p m o d e l has b e e n d e v e l o p e d in c o n n e c t i o n w i t h the a e r o d y n a m i c s of c l a s s i c a l suspension bridges. The v i b r a t i o n mode shapes of classical suspension bridges can often be approximately constructed by the combination of half sine-wave shapes, because the m e c h a n i c a l characteristics of the b r i d g e d e c k b e h a v i o u r are m o r e or less e q u i v a l e n t to a s i m p l y s u p p o r t e d b e a m w i t h the axial tensile f o r c e in longitudinal direction. Cablestayed bridges, on the o t h e r h a n d , tend to e x h i b i t v e r y d i f f e r e n t vibration modes [22] s i n c e the b r i d g e behaves basically as a continuous beam on a l a r g e n u m b e r of e l a s t i c s u p p o r t s . H e n c e the simulation of cable-stayed bridges u s i n g taut s t r i p m o d e l s n e e d s special consideration on this point. This tendency is particularly pronounced in N o r t h America, where the two-plane,
294
muiti-cab[,~ type i ~, wi(!ely a,~cepted. A l s o for ti~e c a s e ot c a b l e stayed bridges, m a n y n u m b e r of e i g e n I r e q u e n c i e s are often crowded in v e r y narrow frequency r a n g e a n d t h e r e is a g o o d p o s s i b i l i t y of multi-mode coupling of vibration in reality [22,35] . The s i l a u l a t i o n of t h i s s i t u a t i o n s o m e h o w n e e d s to be c o n s i d e r e d . 5.3
Response
Prediction
Wind tunnel t e s t s g i v e us the s t r u c t u r a l response i n d u c e d by a simulated wind. This simulated w i n d is, at the b e s t , a m o d e l v e r s i o n of the d e s i g n w i n d or the o b s e r v e d w i n d at o n e p a r t i c u l a r occasion. However, very o f t e n w h a t we w o u l d l i k e to k n o w is the maximum possible wind-induced response in the structure's anticipated life s p a n or the p r o b a b l e response for any given future period. The only way to m a k e this prediction, at t h i s moment, is to c o m b i n e the ~ e a s u r e d wind tunnel test results and a statistical model of w i n d o c c u r a n c e at the p a r t i c u l a r s i t e of the structure and calculate the probability of exceeding a certain response level per unit p e r i o d of time. Davenport [36] has outlined such a procedure, in w h i c h m e t e o r o l o g i c a l information on wind speed and direction is u s e d to c o n s i s t the j o i n t p r o b a b i l i t y distribution p ( u , a ) a n d Js ~ n t e ~ r a t ~ d w i t h the s t r u c t u r a l response g i v e n as a function of the s a m e two p a r a m e t e r s y ( u , a ) to p r o d u c e y ( R ) , the response prediction as a f u n c t i o n of the r e t u r n p e r i o d R. Particularly interesting m a t t e r for the c a s e of w i n d - i n d u c e d bridge response compared to the b u i l d i n g response, for example, is in the fact that bridge response t e n d s to be e x t r e m e l y sensitive to the c h a n g e of wind direction. Fig.9 gives an example to explain this. An analysis of w i n d d a t a g e n e r a l l y shows somewhat non-homogeneous directionality of s t r o n g w i n d . The combination of these two facts could yield a drastic difference in p r e d i c t i o n results. F i g . l O g i v e s an i n t e r e s t i n g e x a m p l e in w h i c h the b r i d g e is s e n s i t i v e to w i n d directions within small fractions of yaw angles near e a s t or w e s t w h e r e a s the p r e v a i l i n g wind direction at the s i t e is the south, which is a favourable one [26]. The modifying effect of bridge orientation and the directionally varying statistical properties of the wind climate is also illustrated in Fig.lO. The curves presented are predictions of annual extreme response against return periods when I) 2)
3)
the actual bridge response is predicted, taking directional variation of w i n d c l i m a t e into account; the b r i d g e hypothetically rotated to a p p r o x i m a t e l y align the sensitive orientation with the prevailing wind direction; and the b r i d g e as is, w i t h o u t taking directional variation of wind climate into account.
T h i s e x a m p l e s h o w s c l e a r l y the s i g n i f i c a n c e of the p a t t e r n in wind climate statistics f o r the w i n d - i n d u c e d response prediction. T h i s is not a m a t t e r of the " B l u f f Body Aerodynamics" but is mentioned here since, talking a b o u t the s i m u l a t i o n and modelling of w i n d induced bridge response, the m o d e l l i n g of w i n d s t a t i s t i c s is an extremely important m a t t e r in terms of making accurate predictions. Another factor which might be of somebody's interest in similar context is a question of h o w to logically decide the r e t u r n p e r i o d for s t r o n g w i n d p a r t i c u l a r l y f o r the c a s e of b r i d g e s
295
under construction. However the point perhaps going too far away from the title
of this discussion of the Colloquium.
is
Acknowledgement The author expresses his gratitude to those who have, for many years, discussed with him on this subject and helped his understanding. The names of Alan Davenport, Nick Isyumov, Toshio Miyata, Bob Scanlan, Bob Wardlaw and Hitoshi Yamada should be p a r t i c u l a r l y mentioned with thanks.
References i.
Buckland, P.G. and Wardlaw, R.L., "Some Aerodynamic C o n s i d e r a t i o n s in Bridge Design", J. Eng'ng Inst. of Canada, April 1972 (EIC-72-BR & STR 5).
2.
Study Group for Wind Effects, "Analysis of Design and Wind Tunnel Testing Methods for Wind Effects on Bridge Structures", J. Wind Eng'ng (JAWE), No.15, Feb. 1983, pp.5160 (in Japanese).
3.
Aynsley, R.M., Melbourne, W. and Vickery, Aerodynamics, Applied Science Publishers,
4.
Davenport, A.G., "The R e l a t i o n s h i p of Wind Structure to Wind Loading", Proc. Symp. on Wind Effects on Buildings and Structures, Teddington, England, June 1963, Vol.l, PP.53-I02.
5.
Davenport, A.G. and Isyumov, N., "The Application of the Boundary Layer Wind Tunnel to the Prediction of Wind Loading", Proc. Int. Res. Seminar on Wind Effects on Buildings and Structures, Ottawa, Sept. 1967, Vol.l, pp.201230.
6.
Jensen, M., Ingenloren,
7.
- ...... , Bridge Aerodynamics, Proc. London, March 1981, Thomas Telford Ltd.
8.
Plate, E.J. (ed.), Scientific Publishing
9.
Special Issues on "Numerical Simulation for Turbulent Flow" of "Seisan-Kenkyu" -Monthly Journal of Institute of Industrial Science, University of Tokyo, (36) 12, Dec. 1984 & (38) i, Jan. 1986, etc.
i0.
Cermak, J.E., "Laboratory Simulation of Boundary Layer", J. AIAA, (9) 9, Sept. 1971,
B.J., 1977.
Architectural
"The M o d e l - l a w for Phenomena in Natural International Edition, (2) 4, 1958.
Engineering Co., 1982.
Conf.
held
Meteorology,
at
Wind",
ICE,
Elsevier
the Atmospheric pp.1746-1754.
ii. Wardlaw, R.L. et al., "Comparative Wind %unnel Testing of Bridge Road Decks", Proc. 3rd US-Japan Bridge Workshop, Tsukuba, Japan, May 1987, pp.278-288. 12.
Tryggvason, B.V., Surry, D. and Davenport, A.G., "Predicting Wind Induced Response in Hurricane Zones", Proc. ASCE, J. Struc. Div., (102) STI2, Dec. 1976, pp.2333-2349.
296
13. Laneville, A. and Yong, L.Z., "Mean Flow Patterns around Twodimensional Rectangular Cylinders and their Interpretation", J. Wind Eng'ng and Industrial Aerodynamics, (14) Dec. 1983, pp.387-398. 14.
Townsend, Cambridge
A.A., Univ.
The Press,
Structure of Turbulent 1976 (2nd ed.), pp.53-56.
Shear
Flow,
15.
Isyumov, N. and Tanaka, H., "Wind Tunnel Modelling of Stack Gas Dispersion: Difficulties and A p p r o x i m a t i o n s " , Proc. 5th Int. Conf. on Wind Eng'ng, Fort Collins, Colorado, July 1979, Vol.2, pp.987-i001.
16.
Kind, R.J., "Aeroelastic Modelling of Membrane Structures", Proc. Int. Workshop on Wind Tunnel Modeling Criteria and Techniques in Civil E n g i n e e r i n g Applications, Gaithersburg, April 1982, pp.429-439.
17.
Zdravkovich, M.M., "Scruton Number; A Proposal", J. Wind Eng'ng and Indutrial Aerodynamics, (I0) 3, Dec. 1982, pp.263265.
18.
Tanaka, H. and Yamada, H., "Mass and Damping Simulation for the Modelling of A e r o e l a s t i c Responses", Proc. Int. Conf. on Flow Induced Vibrations, B o w n e s s - o n - W i n d e r m e r e , England, May 1987, pp.103-110.
19.
Novak, M., "Galloping and Vortex Induced Oscillations Structures", Proc. 3rd Int. Conf. on Wind Effects Buildings and Structures, Tokyo, Sept. 1971, pp.799-809.
of on
20. Miyata, T., Miyazaki, M. and Yamada, H., "Pressure Distribution Measurements for Wind Induced Vibrations of Box Girder Bridges", J. Wind Eng'ng and Industrial Aerodynamics, (14) Dec. 1983, pp.223-234. 21.
Yamada, H., " I d e n t i f i c a t i o n Response of Shallow Bluff Univ. of Tokyo, Dec. 1983.
and Estimation Bodies", Ph.D.
22.
Scanlan, R.H., "Interpreting stayed Bridges", Proc. ASCE, 1987, pp.555-575.
of Vortex Induced Thesis present.
Aeroelastic Models J. Eng. Mech., (113)
23. Wardlaw, R.L., "Sectional Versus Full Testing of Bridge Road Decks", Public 1980, pp.25-47.
Model Roads,
of CableEM4, April
Wind (44)
Tunnel i, June
24. Wardlaw, R.L., "The Use of Scale Models for Aerodynamic I n v e s t i g a t i o n s into the Effect of Wind on Structures", Proc. Int. Symp. on Scale Modeling, Tokyo, July 1988, pp.155-164. 25.
Scanlan, R.H., "Theory of the Wind Analysis of Long-span Bridges Based on Data Obtainable from Section Model Tests", Proc. 4th Int. Conf. Wind Effects on Buildings and Structures, Heathrow, England, Sept. 1975, pp.259-269.
297
26.
Davenport, A.G. et al., "Wind Induced Response of Suspension Bridges -Wind Tunnel Model and Full Scale O b s e r v a t i o n s " , Proc. 5th Int. Conf. on Wind Eng'ng, Fort Collins, Colorado, July 1979, Vol.2, pp.807-824.
27.
Davenport, A.G. and King, J.P.C., "Dynamic Long Span Bridges", Proc. 12th Congr. IABSE, 1984, pp.705-712.
28.
Tanaka, H. and Bridges", Proc. California, May
29.
Zan, S.J., Yamada, H. and Tanaka, H., "The Influence of Turbulence and Deck Section Geometry on the Aeroelastic Behaviour of a C a b l e - s t a y e d Bridge Model", A e r o n a u t i c a l Note, NAE-AN-40, NRC No.26190, National Research Council Canada, Aug. 1986.
30.
Tanaka, H. and Yamada, H., "On Predicting the Performance under Wind of Full Bridges from Section Model Wind Tunnel Results", J. Wind Eng'ng and Industrial Aerodynamics, (26) 3, Dec. 1987, pp.289-306.
31.
Davenport, A.G., Isyumov, N. and Miyata, T., "The Experimental Determination of the Response of Suspension Bridges to Turbulent Wind", Proc. 3rd Int. Conf. on Wind Effects on Buildings and Structures, Tokyo, Sept. 1971, pp.1207-1219.
32.
Davenport, A.G., "The Use of Taut Strip Models in the P r e d i c t i o n of the Response of Long Span Bridges to Turbulent Wind", Proc. Symp. on Flow-lnduced Structural Vibrations, IUTAM-IAHR, Karlsruhe, Aug. 1972, pp.373-381.
33.
Tanaka, H. and Davenport, Models to Turbulent Wind", EMI, Feb. 1982, pp.33-49.
34.
Tanaka, H., "On Wind Tunnel Models", Proc. 3rd US-Japan May 1987, pp.318-323.
35.
Xie, J., "Theory of Flutter for Bridge Structures and Study of Flutter Behaviours of C a b l e - s t a y e d Bridges", DESc. thesis present. Tongjl University, 1985.
36.
Davenport, A.G., "On the Statistical Prediction of Structural P e r f o r m a n c e in the Wind Environment", ASCE Nat. Struct. Eng'ng Meeting, Baltimore, Maryland, April 1971, Preprint 1420.
37.
Davenport, A.G., "The Prediction of Risk under Wind Loading", Proc. 2nd Int. Conf. on Structural Safety and Reliability, Munich, Sept. 1977, pp.511-538.
Xie, J., "Some 4th US-Japan 1988.
Wind Forces on Vancouver, Sept.
Discussions on Buffeting of Bridge Workshop, San Diego,
A.G., Proc.
"Response of Taut ASCE, J. Eng. Mech.,
Testing of Taut Strip Bridge Workshop, Tsukuba,
Strip (108)
Bridge Japan,
298
A
T t i
Fig. i
The Flow Field Concerned
Cupletion of Hajor Bridges
1800
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Collapse of Bridaes due to Wind
1801 Jacobs Creek Bridge by Jaees Finley
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AA
1820 Union Bridge by Samuel Brown 1826 nenai Straits Bridge by Thomas Te]ford
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183G Brighton Chain Pier 1838 Hontrose 1850
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1850 Britannia Bridge 1852 Roche-Bernard 1854 Wheeling (185G Bessemer's Converter) 18&4 Niagara-lewiston (18G7 Siemens~ Open-hearth) 18G9 Niagara-Clifton 1874 St, Louis Arch 1879 Firth of Tay 1889 Eiffel Tower 1890 Forth Railway Bridge
J
1900 Fig. 2
Bridges
and Wind Effects
Fig. 3
Various Types of Wind Induced Responses
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Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Invited Paper
S I M I L I T U D E A N D M O D E L L I N G IN W I N D T U N N E L TESTING OF BRIDGES H. TANAKA
University of Ottawa, Ottawa, Ont. (Canada)
i.
Introduction
W h e n the d e s i g n of a b r i d g e r e q u i r e s the c o n s i d e r a t i o n of the likely effects of w i n d u p o n it, it is a common practice to undertake wind tunnel testing. Modelling t e c h n i q u e s e m p l o y e d in these testings v a r y but the s i m i l i t u d e requirements for t e s t i n g are c o n s i d e r e d to h a v e been well established and p r a c t i c e d for q u i t e some time. The o b j e c t i v e of the p r e s e n t p a p e r is to r e v i e w these modelling t e c h n i q u e s and a s s o c i a t e d similitude requirements for the understanding of the s t a t e - o f - t h e - a r t and d i s c u s s i o n for further improvements of our p r a c t i c e . The g e n e r a l requirement for modelling a physical phenomenon is e s s e n t i a l l y the same in any wind engineering problem. Our c o n c e r n is t y p i c a l l y the b e h a v i o u r of w i n d f l o w in a c e r t a i n s p a c e or a r e a and its i n t e r a c t i o n w i t h the g e o m e t r i c a l and/or mechanical characteristics of the b o u n d a r i e s of the f i e l d of c o n c e r n ( F i g . i ) . For c o r r e c t m o d e l l i n g in t h e s e p r o b l e m s , b a s e d on B u c k i n g h a m ' s Pit h e o r e m , it is r e q u i r e d that a set of dimensionless parameters c o n s i s t of suitable combinations of the r e f e r e n c e q u a n t i t i e s are i n v a r l a n t in m o d e l and p r o t o t y p e and with them the governing e q u a t i o n s are also rendered dimensionless. Various boundary c o n d i t i o n s have to be also maintained in d i m e n s i o n l e s s form. We cannot overemphasize the i m p o r t a n c e of this p r i n c i p l e s i n c e m o s t of our p r o b l e m s in the f i e l d of w i n d e n g i n e e r i n g c a n n o t be s o l v e d with theoretical approaches alone. For c o r r e c t modelling, all of t h e s e d i m e n s i o n l e s s parameters in the p r o t o t y p e m u s t be d u p l i c a t e d in the m o d e l . However, almost invariably, complete duplication of these parameters is impractical or impossible. As a m a t t e r of fact, the r e q u i r e m e n t s can be satisfied exactly only when model and prototype are identical. Hence, the decision must be made as to which parameters could be relaxed for each testing based on the understanding of the p h e n o m e n o n and the knowledge of dominant parameters. The less i m p o r t a n t ones s h o u l d be ignored. The p r o c e s s of this d e c i s i o n m a k i n g is e s s e n t i a l l y the c o n c e r n of this paper.
0167-6105/90/$03.50
© 1990 Elsevier Science Publishers B.V.
284
2. 2.1
Development Historical
of
Understanding
in
Bridge
Aerodynamics
Development
The u n d e r s t a n d i n g of the wind induced b e h a v i o u r of b r i d g e s has come a long way t h r o u g h the h i s t o r y [1,2]. One of the difficulties bridge e n g i n e e r s had t h r o u g h the p r o g r e s s of m o d e r n s u s p e n s i o n b r i d g e s was this p r o b l e m . It was a c o n t i n u o u s struggle against wind action (Fig.2). M a n y of the m a j o r s u s p e n s i o n b r i d g e s constructed in 19th c e n t u r y , the M e n a i S t r a i g h t B r i d g e by T h o m a s T e l f o r d to start with, were e i t h e r d e s t r o y e d or s e v e r e l y d a m a g e d by w i n d . In t e r m s of the p r o g r e s s in understanding of wind-bridge interaction, three particular incidents should be perhaps m e n t i o n e d as the t u r n i n g p o i n t s in the h i s t o r y . The f i r s t of them is the c o l l a p s e of the F i r t h of Tay B r i d g e in 1879. Tay B r i d g e was the w o r l d l o n g e s t b r i d g e in those days, c o n s i s t i n g of 84 s p a n s of w r o u g h t iron t r u s s e s . The c e n t r a l p o r t i o n of the b r i d g e was 13 s p a n s of 60 m e t r e truss g i r d e r , s p a n n i n g 27 m e t r e s a b o v e w a t e r , c a l l e d "the H i g h G i r d e r s " . This was the p o r t i o n b l o w n down by the gale and some 80 lives were c l a i m e d w i t h a train. N o t h i n g else gave such a big impact to bridge engineers regarding the consideration of wind loading on bridges. This incident apparently prompted some e a r l y w i n d t u n n e l as w e l l as full s c a l e s t u d i e s of w i n d l o a d i n g on s t r u c t u r e s [3]. The s e c o n d o c c a s i o n to be r e m e m b e r e d is a f a m o u s d i s a s t e r of the T a c o m a N a r r o w s B r i d g e in 1940. What h a p p e n e d then is now very well known and need not be repeated here. It s h o u l d be p o i n t e d out, h o w e v e r , that the b r i d g e was s u p p o s e d to w i t h s t a n d the s t a t i c wind loading up to the d e s i g n w i n d speed, w h i c h was more than twice the s p e e d at w h i c h the b r i d g e a c t u a l l y c o l l a p s e d . The key here was the d y n a m i c a c t i o n i n d u c e d by w i n d . As it is d e s c r i b e d l a t e r , the exact cause of the i n c i d e n t was not i m m e d i a t e l y clear then. But it was at least c l e a r to e n g i n e e r s that the windi n d u c e d d y n a m i c r e s p o n s e had to be in c o n s i d e r a t i o n . Looking back the h i s t o r y , a c t u a l l y there had b e e n some incidents like the Tacoma Narrows b e f o r e , in which bridges v i b r a t e d due to wind. However, apparently the d y n a m i c s of b r i d g e s had n e v e r become a very s e r i o u s topic for b r i d g e e n g i n e e r s . The third t u r n i n g c o r n e r of the h i s t o r y this a u t h o r w a n t s to p o i n t out is the i n t r o d u c t i o n of the n a t u r a l w i n d c h a r a c t e r i s t i c s in c o d i f i e d form for the d e s i g n consideration. This happened around 1960 particularly when Davenport, A. G. successfully reduced enormous a m o u n t of m e t e o r o l o g i c a l information to a set of modelling specifications and s t a r t e d o p e r a t i n g his B o u n d a r y L a y e r Wind Tunnel for a standardized simulation of the atmospheric b o u n d a r y layer [4,5]. Nobody would do any testing of structures without consideration of external loading conditions p r e c i s e l y as w e l l as t h o s e of structures. For the t e s t i n g of b r i d g e s , h o w e v e r , the wind action has b e e n often considered to be simply a uniformly s m o o t h air flow, w i t h or w i t h o u t s m a l l a n g l e of a t t a c k d e v i a t e d f r o m p a r a l l e l to the g r o u n d , and a l w a y s n o r m a l to the l o n g i t u d i n a l bridge axis. This loading condition is something like a consideration of e a r t h q u a k e e x c i t a t i o n by a s i m p l e h a r m o n i c g r o u n d
285 motion, without
which may be exceptions.
a fairly
conservative
assumption
but
not
E a r l y w i n d t u n n e l s t u d i e s of m o d e l b u i l d i n g s in 1930s and 40s indicated significant i n f l u e n c e of w i n d t u r b u l e n c e c h a r a c t e r i s t i c s on wind tunnel test results such as pressure distribution patterns. It was a b o u t the same p e r i o d of time w h e n P r a n d t l ' s b o u n d a r y l a y e r t h e o r y was a p p l i e d by m e t e o r o l o g i s t s to e x p l a i n the s t r u c t u r e of the l o w e r a t m o s p h e r e . H o w e v e r , the m o s t f u n d a m e n t a l p r i n c i p l e for wind tunnel tests in this r e g a r d was first c l e a r l y s t a t e d by J e n s e n , M. [6] as f o l l o w s : "The n a t u r a l w i n d is turbulent, and the (wind engineering) p h e n o m e n a in this b o u n d a r y l a y e r of the w i n d , as s h o u l d be emphasized, are highly dependent on the nature of this boundary layer. - . . . . The c o r r e c t m o d e l test for p h e n o m e n a in the w i n d m u s t be (therefore) carried out in a turbulent boundary layer, and the m o d e l - l a w r e q u i r e s that this b o u n d a r y l a y e r be to s c a l e as r e g a r d s the v e l o c i t y p r o f i l e . " Following the Tacoma Narrows, there was a significant contribution by aeronautical e n g i n e e r s t o w a r d s the d e v e l o p m e n t of aerodynamics with civil engineering applications. At the same time, it became a general practice to do testing of civil engineering structures such as bridges with a conventional aeronautical wind t u n n e l in uniform smooth air f l o w r a t h e r than with simulated natural winds. Davenport's formulation of c o d i f i e d natural wind and s i m u l a t i o n of it in w i n d t u n n e l t e s t i n g was a significant i m p a c t to the e n g i n e e r i n g p r a c t i c e a g a i n s t this trend. 2.2
Recent
Trend
in
Bridge
Aerodynamics
The last half a c e n t u r y has s e e n a n u m b e r of s t u d i e s on w i n d l o a d i n g on bridges, wind-lnduced r e s p o n s e of bridges and wind r e s i s t a n t d e s i g n of b r i d g e s . A l a r g e s c a l e and c o m p r e h e n s i v e wind tunnel testings have been c o n d u c t e d by m a n y r e s e a r c h e r s . Amongst those, the s t u d i e s led by F a r q u h a r s o n , F. B. of University of Washington, USA, and S c r u t o n , C. at the National Physical Laboratory, Teddington, E n g l a n d , w e r e the e a r l i e s t ones. Studies by o t h e r g r o u p s at USA, F r a n c e , Germany, Norway, Japan, Canada, Australia etc. have f o l l o w e d . T h e s e r e s e a r c h e s h a v e b r o u g h t us m u c h b e t t e r u n d e r s t a n d i n g of wind-bridge interaction. At the same time, it is not s u r p r i s i n g that t h e r e are some d i f f e r e n c e of t r e n d in bridge design and construction compared w i t h half a c e n t u r y ago, w h i c h are c r e a t i n g new problems for us. Some of the recent trend in bridge aerodynamics are as f o l l o w s : i) D e v e l o p m e n t in identification s t u d i e s to s u p p o r t them; 2) A c c u m u l a t i o n of both scale comparisons;
2D
and
3)
Development of cable-stayed suspension bridges;
4)
Wind problems of b r i d g e b r i d g e s at t h e i r e r e c t i o n
of
3D
phenomena
test
bridges
members, stage;
results
rather
cables
in
and
and
than
theoretical
some
full
classical
particular,
and
286
5)
A d v a n c e s in c o s t - e f f e c t i v e d e s i g n of more stringent requirement for results.
It w o u l d be a useful exercise to similitude requirements in w i n d tunnel t h e s e a n g l e s in m i n d . 2.3
Identification
of W i n d
Induced
b r i d g e s and, quantitative
review testing
Bridge
the of
as a result, wind study
m o d e l l i n g and bridges with
Response
Immediately after the Tacoma Narrows disaster, the US Government commissioned the B o a r d of D i r e c t o r s to i n v e s t i g a t e the c a u s e of the b r i d g e failure. The Board consisted of three engineers, A m m a n n , O. H., yon K a r m a n , T. and Woodruff, G. B.. Their Official R e p o r t was p r e s e n t e d in the f o l l o w i n g y e a r . It is interesting to note that two e m i n e n t b r i d g e e n g i n e e r s and a w o r l d famous fluid mechanician in this R e p o r t c o n c l u d e d that the c a u s e of the c a t a s t r o p h i c t o r s i o n a l m o t i o n of the b r i d g e was not c l e a r l y identifiable whereas the v e r t i c a l bending motion of the bridge deck observed p r e c e d i n g to the t o r s i o n a l m o t i o n was probably a forced vibration due to wind gust. Obviously it was very d i f f i c u l t to i d e n t i f y the n a t u r e of w i n d i n d u c e d b r i d g e r e s p o n s e at that s t a g e but it was a l r e a d y s u g g e s t e d then that t h e r e w e r e more than one c o n c e i v a b l e c a u s e s of b r i d g e i n s t a b i l i t y . L a t e r , of c o u r s e , yon K a r m a n c o l l a b o r a t e d with Farquharson to do w i n d tunnel investigation of the b r i d g e and t r i e d to a t t r i b u t e the c a u s e of vibration in both modes to vortex excitation. B l e i c h , F., a n o t h e r f a m o u s b r i d g e e n g i n e e r , t r i e d to e x p l a i n it by the a n a l o g y to the aerofoil flutter t h e o r y w i t h some a d d i t i o n a l aerodynamic force terms. Modification of f l u t t e r t h e o r y was also a t t e m p t e d by K l o p p e l , K. of G e r m a n y and S e l b e r g , A. of Norway. H i r a i , A. of University of T o k y o f o u n d the b r i d g e f a i l u r e as the lateral buckling due to lack of torsional rigidity of the stiffening frame. Many wind t u n n e l tests during 1960s and 70s g r e a t l y h e l p e d identifying the v a r i o u s types of w i n d i n d u c e d r e s p o n s e of b r i d g e structures (Fig.3). One w a y of classifying them is as f o l l o w s
[7]:
Static
Dynamic
behaviours
behaviours
Overturning Excessive lateral Divergence Lateral Buckling
deflection
Vortex induced oscillations Self-excited oscillations Vertical bending instability Torsional instability Coupled flutter Buffeting motion
S t a t i c p h e n o m e n a l i s t e d a b o v e are s e l d o m c o n s i d e r e d to be the definitive factor for the wind resistant design of bridges. Besides, it has been well accepted that t h e s e p h e n o m e n a can be p r e d i c t e d by the theoretical calculations with good accuracy p r o v i d e d that the t h r e e major aerodynamic force components, lift force, d r a g f o r c e and p i t c h i n g m o m e n t , are k n o w n .
287 The t h r e e components can be u s u a l l y m e a s u r e d u s i n g the m o s t conventional force balance system with reasonable accuracy. Only two f a c t o r s p e r h a p s s h o u l d be k e p t in m i n d as a p o s s i b l e s o u r c e of deviations: w i n d f l o w t u r b u l e n c e and w i n d t u n n e l b l o c k a g e . Though it is not a conventional practice to introduce wind flow turbulence for the f o r c e measurement with 2D sectional model, t h e r e is no r e a s o n w h y it c a n n o t be done and in fact, as it is mentioned in l a t e r s e c t i o n s , it has b e e n p r a c t i c e d r e c e n t l y to an extent. Consideration of s p a n w i s e v e l o c i t y c o r r e l a t i o n w o u l d give better accuracy for p r e d i c t i n g the c r i t i c a l mean wind s p e e d or m a g n i t u d e of d e f l e c t i o n . Experimental confirmation of it by u s i n g aeroelastic models has not b e e n done too e x t e n s i v e l y in a u t h o r ' s knowledge. The d i s c u s s i o n sections, therefore,
3. 3.1
Similitude Similarity
on s i m i l i t u d e requirements in is m o s t l y on d y n a m i c p h e n o m e n a .
Requirements
the
following
in G e n e r a l
of W i n d
The similarity requirements of wind characteristics in physical wind tunnel model and/or mathemetical simulation model have been extensively discussed by b o t h meteorologists and w i n d engineers [8,9]. The s i m u l a t i o n of w i n d can be c o n s i d e r e d in two categories: one is the s i m u l a t i o n of the a v e r a g e c h a r a c t e r i s t i c s of the turbulent boundary~ayer w i n d w h i c h is a p p r o a c h i n g towards the f i e l d of c o n c e r n for e a c h p a r t i c u l a r p r o j e c t and a n o t h e r is the s i m u l a t i o n of the wind structure at the i m m e d i a t e p r o x i m i t y field which of c o u r s e is l a r g e l y i n f l u e n c e d by its particular topographical conditions. T h e s e are s o m e t i m e s r e f e r r e d to as the "'far f i e l d " and "near field" simulation respectively. The "far field" flow characteristics are d e f i n e d g e n e r a l l y by the f o l l o w i n g parameters: Mean
wind velocity distribution vertical directions
in h o r i z o n t a l
and
Spectral quantities of v e l o c i t y c o m p o n e n t s : Intensities of t u r b u l e n c e S c a l e s of t u r b u l e n c e Distribution of s p e c t r a in a r e l e v a n t f r e q u e n c y range Space correlations of t u r b u l e n c e c o m p o n e n t s Temperature and concern Other
moisture
distribution
m i n o r f a c t o r s such as p r e s s u r e g r a d i e n t etc.
Coriolis
in
the
force,
field
of
atmospheric
For the s i m u l a t i o n of n a t u r a l w i n d at a p a r t i c u l a r location, the e f f e c t s of the surrounding t o p o g r a p h y n e e d to be c o n s i d e r e d . T h e s e are u s u a l l y p r o v i d e d by i n s t a l l i n g the g e o m e t r i c a l l y scaled m o d e l of "near f i e l d " p r o x i m i t y at the w i n d t u n n e l test s e c t i o n . To w h a t e x t e n t the s u r r o u n d i n g t o p o g r a p h y n e e d s to be s i m u l a t e d is still a good question but s h o u l d be d e c i d e d in r e l a t i o n to the development of the internal boundary layer flow due to the p r o x i m i t y m o d e l and its i n f l u e n c e on the test r e s u l t s .
288
A good modelling technique for the "far f i e l d " s i m u l a t i o n is a boundary layer wind t u n n e l -the use of a q u i c k l y d e v e l o p e d turbulent boundary layer by various roughness and obstacles installed on the w i n d t u n n e l f l o o r w i t h or w i t h o u t temperature and moisture control devices [5,10]. This technique has b e e n w i d e l y accepted and practiced but is not w i t h o u t any s h o r t c o m i n g unless it is e m p l o y e d with caution. F i r s t of a l l , the m e t e o r o l o g i c a l characteristics of the s i t e of c o n c e r n are not necessarily always the same. Excessive presumption of the w i n d c h a r a c t e r i s t i c s b a s e d on l i m i t e d d a t a m a y lead us to a wrong, sometimes non-conservative conclusion. Turbulence intensity is often a controversial factor. The suppression of the t u r b u l e n c e intensity by t h e r m a l s t r a t i f i c a t i o n is r e p o r t e d to h a v e an e s s e n t i a l effect sometimes on w i n d - i n d u c e d bridge motion [ii]. Another thing to be remembered is t h a t the simulation of rather special meteorological phenomena such as hurricanes, tornados and downbursts has somewhat different aspects. It has been studied recently to s o m e e x t e n t , but t h e r e are s t i l l m u c h to be e x p e c t e d . O n e of the a p p r o a c h e s t r i e d for h u r r i c a n e w i n d s is a Monte Carlo simulation of tropical low pressure models to d e f i n e the s t a t i s t i c s of e x t r e m e w i n d s o v e r a l o n g p e r i o d of t i m e [12]. 3.2
Aeroelastic
Similarity
The wind tunnel modelling requires aeroelastic similarity in addition to the s i m i l a r i t y of w i n d f l o w c h a r a c t e r i s t i c s as w e l l as the c o n s i s t e n t matching of l e n g t h s c a l e and g e o m e t r i c a l s h a p e of the s t r u c t u r e . Aeroelastic similarity, in general, is b a s e d on the c o n s i d e r a t i o n of the length, density, elastic property and internal friction of the s t r u c t u r e , density and viscosity of a i r , wind velocity and the a c c e l e r a t i o n due to g r a v i t y . These physical properties can be conveniently summarized into the following dimensionless quantities: i) 2) 3) 4) 5)
Reynolds number Froude number Density ratio Cauchy number Critical damping
review A brief testing influence on of s o m e w o r t h . Reynolds
ratio
of these quantities in of w i n d - l n d u c e d structural
terms of their response w o u l d be
Number
Reynolds number c a n be d e f i n e d as the r a t i o of the fluid inertia force to the f l u i d v i s c o u s force. In m o s t of w i n d t u n n e l testings it is i m p r a c t i c a l to s a t i s f y R e y n o l d s number similitude. Indeed, the viscous forces are usually at l e a s t the order of magnitude smaller and relatively unimportant compared to the f l u i d inertia forces. However, the c o n s e q u e n c e of t h i s d i s t o r s i o n of the s i m i l i t u d e requirement should be e x a m i n e d carefully for the corret interpretation of the test results. The following three p o i n t s s h o u l d be p a r t i c u l a r l y noted: i) It is cylinder
well known t h a t the flow is very sensitive to the
pattern change
around a circular of the Reynolds
289 n u m b e r b e c a u s e of the s h i f t of f l o w s e p a r a t i o n p o i n t s w i t h it. T h e r e is a c o r r e s p o n d i n g c h a n g e of the w a k e w i d t h and the drag f o r c e as w e l l as the f r e q u e n c y of w a k e v o r t e x f o r m a t i o n . It is i m p o r t a n t that, in m o d e l l i n g a s t r u c t u r e w i t h s m o o t h c u r v e d surface geometry, these effects are properly taken into consideration, though this is not u s u a l l y the case in b r i d g e structures. It s h o u l d be n o t e d that the critical Reynolds n u m b e r is also d e p e n d e n t on the s u r f a c e r o u g h n e s s of the s o l i d b o u n d a r y and the t u r b u l e n c e l e v e l in the a p p r o a c h i n g air flow. 2) In case of the f l o w over the s e c t i o n s w i t h s h a r p c o r n e r s , the f l o w separation points do not s h i f t and the f l o w p a t t e r n is less s e n s i t i v e to the c h a n g e of the R e y n o l d s n u m b e r . This is u s u a l l y the case for b r i d g e deck cross-sections. Broad wake after separation from the u p s t r e a m c o r n e r s may r e a t t a c h to the b o d y s u r f a c e , d e p e n d i n g on the a s p e c t r a t i o of the b o d y cross-section. The f l o w reattachment, of course, results a r e d u c t i o n of drag f o r c e and i n c r e a s e of the S t r o u h a l n u m b e r in general. The c r i t i c a l a s p e c t r a t i o of the b o d y at w h i c h this change occurs d e p e n d s on the R e y n o l d s n u m b e r as w e l l as the corner radius and the airstream turbulence l e v e l [13]. It s h o u l d be also noted that this f a c t o r is i n f l u e n c e d by the w i n d t u u n e l b l o c k a g e r a t i o as w e l l . 3) In the p r o b l e m s i n v o l v i n g e f f e c t s of w i n d t u r b u l e n c e , it is essential to simulate the velocity spectra correctly. Townsend [14] has p o i n t e d out that "--- w h i l e geometrically similar flows are e x p e c t e d to be dynamically and structurally similar if their Reynolds n u m b e r s are the same, t h e i r s t r u c t u r e s are a l s o v e r y n e a r l y s i m i l a r for all R e y n o l d s n u m b e r s w h i c h are l a r g e e n o u g h to allow turbulent flow." This is a significant g o s p e l for the wind tunnelers who e m p l o y an a r t i f i c i a l l y developed turbulent boundary layer flow as a simulated natural wind, since achieving the Reynolds number similitude is i m p r a c t i c a l a n y w a y s . H o w e v e r , it s h o u l d be r e m e m b e r e d that the R e y n o l d s n u m b e r does play a part in the e x i s t a n c e of the i n e r t i a s u b r a n g e of the e n e r g y s p e c t r u m . As the R e y n o l d s number increases, the h i g h f r e q u e n c y end of the distribution w i l l be e x t e n d e d so that the total d i s s i p a t i o n of turbulence energy remains unchanged. On the o t h e r h a n d , w h e n the R e y n o l d s n u m b e r is s m a l l , the r a t i o of the size of the dissipating eddies to the representative size of the predominant e d d i e s b e c o m e s h i g h l y d e p e n d e n t on v i s c o s i t y . The results of this is inaccurate simulation of turbulence s t r u c t u r e due to n a r r o w e r than r e q u i r e d i n e r t i a s u b r a n g e [15]. Froude
Number
F r o u d e n u m b e r is the r a t i o of f l u i d i n e r t i a f o r c e to v e r t i c a l f o r c e due to g r a v i t y and/or buoyancy. Consequently the F r o u d e similitude becomes i m p o r t a n t for the c a s e s such as d i s s i p a t i o n of airborne particles or w i n d - i n d u c e d response of cable-supported structures where g r a v i t y is a dominant factor. Although the F r o u d e n u m b e r s i m i l i t u d e has b e e n w i d e l y a c c e p t e d and e m p l o y e d for many aeroelastic s t u d i e s in the past, it is not an essential requirement unless the g r a v i t y a n d / o r b u o y a n c y play an i m p o r t a n t role. For e x a m p l e , if the r e s t o r i n g f o r c e of a m o d e l s t r u c t u r e is provided only by its linear elastic properties, its a e r o e l a s t i c
290
response does not require Froude number consideration. Simplifications of the r e q u i r e m e n t due to i n t e r a c t i o n of g r a v i t y and elastic f o r c e for a i r - s u p p o r t e d structures have been discussed elsewhere [16]. Density
Ratio
The ratio of the structural material density to a i r d e n s i t y is an important parameter w h i c h has to be considered in any aeroelastic testing. However, the r e q u i r e m e n t c a n be e x p r e s s e d in slightly different way since the a e r o e l a s t i c m o d e l m a y not be an exact replica of the prototype structure. In fact, many aeroelastic models a r e the so-called equivalent models, which simply maintain the g e o m e t r i c a l shape and dynamic characteristics; i . e . , the natural frequencies, corresponding mode shapes and damping ratios. If t h i s is the c a s e , t h e r e is no p o i n t k e e p i n g the d e n s i t y r a t i o as a p a r a m e t e r a n d the m a s s r a t i o s h o u l d r e p l a c e it. T h e m a s s r a t i o ~ is d e f i n e d by the f o l l o w i n g equation a n d it is i d e n t i c a l to the density ratio if the s t r u c t u r e is s o l i d a n d homogeneous:
= m/p~ where
p p L m
~ = = = =
(p
L3 / L ) / ( p L
~ )
=
material density air density linear dimension m a s s per u n i t l e n g t h
?~1?
of
(11
structure.
In t o r s i o n a l problems, the ratio based on m a s s moment of inertia per u n i t l e n g t h of s t r u c t u r e (J) is u s e d to r e p l a c e E q . ( 1 ) = J/pL 4 . by Cauch~
Number
Cauchy number is d e f i n e d fluid inertia force; i.e., Ca frequency of the s t r u c t u r e f ~.~ this
ratio
can Ca
where
[ ( E l / m ) ~a be
~v
the r a t i o of elastic f o r c e to E/pu z . Considering the n a t u r a l
]/L ~
rewritten
(EIILi)I(~uZL
E = Young's f = natural
as =
as
follows:
2 ) -v
modulus of frequency
(fLlu)Z(m/yL
a)
(2)
material
Therefore, if we t a k e the r e d u c e d v e l o c i t y u ~ = u / f L as a n e w dimensionless parameter, the C a u c h y n u m b e r is e q u i v a l e n t to ~ / u ~ ~. In m o s t w i n d t u n n e l t e s t i n g s , rather than considering the o r i g i n a l definition of Cauchy number, it is e a s i e r to t a k e the m a s s r a t i o into consideration for the model design and analyse the dimensionless structural response as a function of reduced velocity. Usually it is n o t d i f f i c u l t to do t e s t i n g o v e r a w i d e r a n g e of r e d u c e d v e l o c i t y to c o v e r its p o s s i b l e r a n g e in r e a l i t y . Critical
Damping
Ratio
The magnitude of s t r u c t u r a l damping is o b v i o u s l y an i m p o r t a n t parameter for the p r e d i c t i o n of s t r u c t u r a l response. The problem, however, is t h a t its magnitude is not k n o w n exactly until the
291 s t r u c t u r e c o m e s into e x i s t a n c e . As a m a t t e r of fact, e v e n for the existing structures, the m a g n i t u d e of s t r u c t u r a l d a m p i n g is o f t e n controversial because of the difficulty in m e a s u r i n g its value particularly with relatively large amplitude motion. Naturally the p r e d i c t i o n of s t r u c t u r a l r e s p o n s e p r i o r to its c o n s t r u c t i o n is done by s p e c u l a t i o n b a s e d on a t y p i c a l d a m p i n g v a l u e f r o m the past experience.
4.
Effects
of M a s s
and
Damping
The b r i e f r e v i e w of the p r e v i o u s s e c t i o n u n d e r l i n e s meaning of c h o o s i n g appropriate mass r a t i o and damping ratio in wind tunnel studies. Mass and d a m p i n g are two i m p o r t a n t f a c t o r s in the d e s i g n and construction of wind tunnel models and these two requirements have b e e n s o m e t i m e s c o m b i n e d t o g e t h e r as the S c r u t o n n u m b e r (or the m a s s - d a m p i n g parameter) requirement in the past [17]. Scruton, through his e x p e r i e n c e , s h o w e d that the joint mass-damping parameter, when used together with the reduced v e l o c i t y ur, was a c o n v e n i e n t d i m e n s i o n l e s s p a r a m e t e r to i n d i c a t e the a e r o d y n a m i c s t a b i l i t y of structures. A simple mathematical m o d e l can d e m o n s t r a t e that the m a g n i t u d e of m a x i m u m v o r t e x - i n d u c e d r e s p o n s e can be d e c i d e d by the product of mass and damping parameters and this has been experimentally verified ( F i g s . 4 and 5). In case of some p a r t i c u l a r b l u f f s e c t i o n s , h o w e v e r , the mass and d a m p i n g not o n l y a f f e c t on the r e s p o n s e m a g n i t u d e a n d / o r the wind speed to have the r e s p o n s e o c c u r , but also can c h a n g e its nature from the instability with diverging amplitude to the " a e o l i a n " type r e s p o n s e of r e s t r i c t e d a m p l i t u d e [18]. Figs. 6 and 7 show two of these peculiar examples observed as 2D s e c t i o n a l m o d e l test r e s u l t s in s m o o t h air flow. In both cases, mass and damping parameters were changed over a range though the geometrical shape of the m o d e l was maintained e x a c t l y the same. Very similar response characteristics are o b s e r v e d in b o t h cases, t h o u g h one is in h e a v i n g and a n o t h e r is in t o r s i o n a l mode of vibration. W h e n mass and/or damping p a r a m e t e r s are small, there are typical vortex-induced type v i b r a t i o n s of restricted amplitude observed at low wind speed range and also there appear instabilities of d i v e r g i n g a m p l i t u d e at h i g h e r w i n d speed. It is interesting to n o t e that these i n s t a b i l i t i e s start approximately at the r e d u c e d w i n d s p e e d of i/St, w h e r e St is the S t r o u h a l n u m b e r d e f i n e d in the w a k e . It m e a n s that the f i r s t v o r t e x - i n d u c e d like peak, w h i c h c a n n o t be o b s e r v e d w i t h h e a v y m o d e l s , a c t u a l l y o c c u r s at a l m o s t a half of the S t r o u h a l s p e e d . H o w e v e r , w h e n the m o d e l mass is increased and/or the higher structural damping is provided, the response started at the Strouhal wind speed is suppressed at h i g h e r s p e e d range, m a k i n g a n o t h e r p e a k r a t h e r than diverging instability. The s i m i l a r p h e n o m e n a h a v e been r e p o r t e d e a r l i e r by N o v a k [19] and M i y a t a et al. [20]. Instability for this case is s o m e t i m e s o b s e r v e d at e v e n h i g h e r s p e e d r a n g e a f t e r the s e c o n d p e a k is p r a c t i c a l l y suppressed. These seem to be the c a s e s w h i c h have been classified by Y a m a d a [21] as the "bluff sections with imperfectly locked-in shedding". The w i n d - i n d u c e d fluctuating pressure on a l o n g - w i n d s i d e s of the m o d e l cross-section for these cases, a c c o r d i n g to Y a m a d a , has two p r e d o m i n a n t frequency components: the Strouhal
292
frequency and the f r e q u e n c y of the body motion. They do not necessarily coincide. When the S c r u t o n number is small, the response develops as the instability whereas it c a n be s e p a r a t e d from instability and become ordinary vortex-induced motion when the S c r u t o n n u m b e r is larger. It m a y be also worthwhile to mention that the e f f e c t of m a s s parameter and d a m p i n g parameter for t h e s e c a s e s is not r e a l l y the s a m e and h e n c e the Scruton number does not w o r k as a good single parameter to d e s c r i b e the response characteristics. These two parameters have to be examined separately [18]. T h i s fact indicates the importance of g o o d m a s s and d a m p i n g simulation not o n l y in dynamic response measurement but a l s o in measurement of aerodynamic derivatives by f r e e vibration method where generally m a s s and damping simulation is not c o n s i d e r e d to be e s s e n t i a l [22].
3.
Structural
3.1
2D-3D
Modelling
and
Response
Prediction
Comparison
Comparison of sectional versus full bridge model wind tunnel testing (Fig.8) has b e e n a classical query [23,24]. The obvious advantages of the f u l l model technique are somewhat retarded because of difficulty with complexity of model design and construction including configuration changes and sometimes unavailability of experimental facilities large enough to accomodate full bridge models. For relatively short span bridges, it m i g h t be e v e n more difficult to j u s t i f y s u c h e n d e a v o u r simply b e c a u s e of financial constraints. The alternative approach to t h i s is the u s e of a c o n v e n t i o n a l sectional m o d e l of the b r i d g e , as has been widely practiced for years. This method can well simulate the s t r u c t u r a l geometry of the b r i d g e but has b e e n a l w a y s facing a debate whether or not it c a n r e a l l y p r o v i d e a l l of the necessary and c o r r e c t information for the b r i d g e d e s i g n e r s . The e f f e c t s of w i n d turbulence, to start with, have been traditionally neglected in 2D testings. Consequently sectional models were considered to be not m u c h of u s e for the p r e d i c t i o n of bridge buffeting. However, an a t t e m p t has b e e n m a d e to e x t e n d this method for the s y s t e m a t i c measurement of a e r o d y n a m i c forces, which can t h e n be applied for response prediction taking into account the w i n d t u r b u l e n c e [25-28]. Sectional testing has b e e n b e l i e v e d to be r e l i a b l e in d e s i g n to a v o i d v o r t e x s h e d d i n g excitation and aerodynamic instabilities. Its larger geometrical scale is well-suited for examining corrective configuration changes and u s e of a e r o d y n a m i c devices. Question remains h o w e v e r as to w h e t h e r or n o t it is e x c e s s i v e l y conservative. T h i s is an i n c r e a s i n g l y important point because of the r e c e n t tendency towards competitive cost-effective design and a strong impetus to d e t e r m i n e the e x t e n t of the a e r o d y n a m i c margin provided by the s e c t i o n a l model studies. Some of the recent comparative studies particularly on turbulence and mass effects h a v e b e e n c a r r i e d out in t h i s c o n t e x t [29,30]. 5.2
as
Taut
Strip
Model
Method
This technique (Fig.9) a m e a n s to t a k e a s e c o n d
was first introduced by l o o k at the e x p e r i m e n t s
Davenport [31] c a r r i e d out in
293 the e a r l y s t a g e of s u s p e n s i o n b r i d g e a e r o d y n a m i c s , i n c l u d i n g this time t h e i r 3D r e s p o n s e characteristics to simulated turbulent wind. T h o u g h the observed model r e s p o n s e did not s e e m to be " e x a c t " , the m e t h o d w a s f o u n d to be q u i t e p r o m i s i n g in r e p r o d u c i n g full scale bridge hehaviour b e c a u s e the possibility of a d a p t i n g the m e a s u r e d data using the s i m p l e half-wave response to more complicated mode s h a p e s was then c o n c e i v e d . This c a l c u l a t i o n was later formulated by D a v e n p o r t [32] including the "cross-wave" contribution in statistical manner for the classical suspension bridges. The chief o b j e c t i v e of the taut strip model method is a simulation of the d y n a m i c c h a r a c t e r i s t i c s of the b r i d g e road d e c k -- its frequency, d a m p i n g and mode s h a p e -- w i t h the same s c a l i n g f a c t o r s as the s i m u l a t i o n of n a t u r a l w i n d . W i t h this c o n c e p t , the linear scale of the m o d e l is c o n s t r a i n e d not so m u c h by the size of the p r o t o t y p e s t r u c t u r e but by h o w l a r g e a s c a l e of t u r b u l e n c e can be in the wind tunnel. It s h o u l d be n o t e d that, in most cases, the basic scaling f a c t o r s for l e n g t h and time are c h o s e n almost independently of each other. It m e a n s that b o t h R e y n o l d s and F r o u d e s i m i l i t u d e s are v i o l a t e d . The f u n d a m e n t a l response characteristics of taut s t r i p m o d e l s have b e e n examined [33] to c o n f i r m that the r e s p o n s e in g e n e r a l are in g o o d a g r e e m e n t w i t h b u f f e t i n g and f l u t t e r t h e o r i e s w h e n the aerodynamic derivatives and a e r o d y n a m i c admittance f u n c t i o n s are m o r e or less k n o w n . As a matter of fact, it is one of the advantages offered by this m e t h o d that w h e n the taut s t r i p m o d e l test r e s u l t s are used as input to the buffeting theory, the complication of d e f i n i n g the aerodynamic admittance f u n c t i o n is avoided. A l s o the possible complication due to n o n - l i n e a r i t y of aerodynamic derivatives need not be c o n s i d e r e d e i t h e r . The m e t h o d also shares m a n y a d v a n t a g e s of full b r i d g e m o d e l method such as the intrinsic inclusion of 3D structural characteristics, availability of t e s t i n g a g a i n s t an o b l i q u e w i n d etc., and yet is still much s i m p l e r in c o n c e p t than the full bridge model and to a c e r t a i n e x t e n t s h a r e s the a d v a n t a g e s of the sectional model of low cost and short lead time. The disadvantages of this method, on the other hand, are some technical difficulties associated with the c e n t r e of rotation, frequency ratio, lateral sway characteristics, adjustment of generalized mass and structural damping and maintaining the a c c u r a c y in m o d e l c o n f i g u r a t i o n [34]. Discussing the similitude problems of taut strip models, special attention s h o u l d be g i v e n to the c o n s i d e r a t i o n of the vibration mode shapes. It is worthwhile to n o t e a g a i n that the c o n c e p t of taut s t r i p m o d e l has b e e n d e v e l o p e d in c o n n e c t i o n w i t h the a e r o d y n a m i c s of c l a s s i c a l suspension bridges. The v i b r a t i o n mode shapes of classical suspension bridges can often be approximately constructed by the combination of half sine-wave shapes, because the m e c h a n i c a l characteristics of the b r i d g e d e c k b e h a v i o u r are m o r e or less e q u i v a l e n t to a s i m p l y s u p p o r t e d b e a m w i t h the axial tensile f o r c e in longitudinal direction. Cablestayed bridges, on the o t h e r h a n d , tend to e x h i b i t v e r y d i f f e r e n t vibration modes [22] s i n c e the b r i d g e behaves basically as a continuous beam on a l a r g e n u m b e r of e l a s t i c s u p p o r t s . H e n c e the simulation of cable-stayed bridges u s i n g taut s t r i p m o d e l s n e e d s special consideration on this point. This tendency is particularly pronounced in N o r t h America, where the two-plane,
294
muiti-cab[,~ type i ~, wi(!ely a,~cepted. A l s o for ti~e c a s e ot c a b l e stayed bridges, m a n y n u m b e r of e i g e n I r e q u e n c i e s are often crowded in v e r y narrow frequency r a n g e a n d t h e r e is a g o o d p o s s i b i l i t y of multi-mode coupling of vibration in reality [22,35] . The s i l a u l a t i o n of t h i s s i t u a t i o n s o m e h o w n e e d s to be c o n s i d e r e d . 5.3
Response
Prediction
Wind tunnel t e s t s g i v e us the s t r u c t u r a l response i n d u c e d by a simulated wind. This simulated w i n d is, at the b e s t , a m o d e l v e r s i o n of the d e s i g n w i n d or the o b s e r v e d w i n d at o n e p a r t i c u l a r occasion. However, very o f t e n w h a t we w o u l d l i k e to k n o w is the maximum possible wind-induced response in the structure's anticipated life s p a n or the p r o b a b l e response for any given future period. The only way to m a k e this prediction, at t h i s moment, is to c o m b i n e the ~ e a s u r e d wind tunnel test results and a statistical model of w i n d o c c u r a n c e at the p a r t i c u l a r s i t e of the structure and calculate the probability of exceeding a certain response level per unit p e r i o d of time. Davenport [36] has outlined such a procedure, in w h i c h m e t e o r o l o g i c a l information on wind speed and direction is u s e d to c o n s i s t the j o i n t p r o b a b i l i t y distribution p ( u , a ) a n d Js ~ n t e ~ r a t ~ d w i t h the s t r u c t u r a l response g i v e n as a function of the s a m e two p a r a m e t e r s y ( u , a ) to p r o d u c e y ( R ) , the response prediction as a f u n c t i o n of the r e t u r n p e r i o d R. Particularly interesting m a t t e r for the c a s e of w i n d - i n d u c e d bridge response compared to the b u i l d i n g response, for example, is in the fact that bridge response t e n d s to be e x t r e m e l y sensitive to the c h a n g e of wind direction. Fig.9 gives an example to explain this. An analysis of w i n d d a t a g e n e r a l l y shows somewhat non-homogeneous directionality of s t r o n g w i n d . The combination of these two facts could yield a drastic difference in p r e d i c t i o n results. F i g . l O g i v e s an i n t e r e s t i n g e x a m p l e in w h i c h the b r i d g e is s e n s i t i v e to w i n d directions within small fractions of yaw angles near e a s t or w e s t w h e r e a s the p r e v a i l i n g wind direction at the s i t e is the south, which is a favourable one [26]. The modifying effect of bridge orientation and the directionally varying statistical properties of the wind climate is also illustrated in Fig.lO. The curves presented are predictions of annual extreme response against return periods when I) 2)
3)
the actual bridge response is predicted, taking directional variation of w i n d c l i m a t e into account; the b r i d g e hypothetically rotated to a p p r o x i m a t e l y align the sensitive orientation with the prevailing wind direction; and the b r i d g e as is, w i t h o u t taking directional variation of wind climate into account.
T h i s e x a m p l e s h o w s c l e a r l y the s i g n i f i c a n c e of the p a t t e r n in wind climate statistics f o r the w i n d - i n d u c e d response prediction. T h i s is not a m a t t e r of the " B l u f f Body Aerodynamics" but is mentioned here since, talking a b o u t the s i m u l a t i o n and modelling of w i n d induced bridge response, the m o d e l l i n g of w i n d s t a t i s t i c s is an extremely important m a t t e r in terms of making accurate predictions. Another factor which might be of somebody's interest in similar context is a question of h o w to logically decide the r e t u r n p e r i o d for s t r o n g w i n d p a r t i c u l a r l y f o r the c a s e of b r i d g e s
295
under construction. However the point perhaps going too far away from the title
of this discussion of the Colloquium.
is
Acknowledgement The author expresses his gratitude to those who have, for many years, discussed with him on this subject and helped his understanding. The names of Alan Davenport, Nick Isyumov, Toshio Miyata, Bob Scanlan, Bob Wardlaw and Hitoshi Yamada should be p a r t i c u l a r l y mentioned with thanks.
References i.
Buckland, P.G. and Wardlaw, R.L., "Some Aerodynamic C o n s i d e r a t i o n s in Bridge Design", J. Eng'ng Inst. of Canada, April 1972 (EIC-72-BR & STR 5).
2.
Study Group for Wind Effects, "Analysis of Design and Wind Tunnel Testing Methods for Wind Effects on Bridge Structures", J. Wind Eng'ng (JAWE), No.15, Feb. 1983, pp.5160 (in Japanese).
3.
Aynsley, R.M., Melbourne, W. and Vickery, Aerodynamics, Applied Science Publishers,
4.
Davenport, A.G., "The R e l a t i o n s h i p of Wind Structure to Wind Loading", Proc. Symp. on Wind Effects on Buildings and Structures, Teddington, England, June 1963, Vol.l, PP.53-I02.
5.
Davenport, A.G. and Isyumov, N., "The Application of the Boundary Layer Wind Tunnel to the Prediction of Wind Loading", Proc. Int. Res. Seminar on Wind Effects on Buildings and Structures, Ottawa, Sept. 1967, Vol.l, pp.201230.
6.
Jensen, M., Ingenloren,
7.
- ...... , Bridge Aerodynamics, Proc. London, March 1981, Thomas Telford Ltd.
8.
Plate, E.J. (ed.), Scientific Publishing
9.
Special Issues on "Numerical Simulation for Turbulent Flow" of "Seisan-Kenkyu" -Monthly Journal of Institute of Industrial Science, University of Tokyo, (36) 12, Dec. 1984 & (38) i, Jan. 1986, etc.
i0.
Cermak, J.E., "Laboratory Simulation of Boundary Layer", J. AIAA, (9) 9, Sept. 1971,
B.J., 1977.
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"The M o d e l - l a w for Phenomena in Natural International Edition, (2) 4, 1958.
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Tryggvason, B.V., Surry, D. and Davenport, A.G., "Predicting Wind Induced Response in Hurricane Zones", Proc. ASCE, J. Struc. Div., (102) STI2, Dec. 1976, pp.2333-2349.
296
13. Laneville, A. and Yong, L.Z., "Mean Flow Patterns around Twodimensional Rectangular Cylinders and their Interpretation", J. Wind Eng'ng and Industrial Aerodynamics, (14) Dec. 1983, pp.387-398. 14.
Townsend, Cambridge
A.A., Univ.
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15.
Isyumov, N. and Tanaka, H., "Wind Tunnel Modelling of Stack Gas Dispersion: Difficulties and A p p r o x i m a t i o n s " , Proc. 5th Int. Conf. on Wind Eng'ng, Fort Collins, Colorado, July 1979, Vol.2, pp.987-i001.
16.
Kind, R.J., "Aeroelastic Modelling of Membrane Structures", Proc. Int. Workshop on Wind Tunnel Modeling Criteria and Techniques in Civil E n g i n e e r i n g Applications, Gaithersburg, April 1982, pp.429-439.
17.
Zdravkovich, M.M., "Scruton Number; A Proposal", J. Wind Eng'ng and Indutrial Aerodynamics, (I0) 3, Dec. 1982, pp.263265.
18.
Tanaka, H. and Yamada, H., "Mass and Damping Simulation for the Modelling of A e r o e l a s t i c Responses", Proc. Int. Conf. on Flow Induced Vibrations, B o w n e s s - o n - W i n d e r m e r e , England, May 1987, pp.103-110.
19.
Novak, M., "Galloping and Vortex Induced Oscillations Structures", Proc. 3rd Int. Conf. on Wind Effects Buildings and Structures, Tokyo, Sept. 1971, pp.799-809.
of on
20. Miyata, T., Miyazaki, M. and Yamada, H., "Pressure Distribution Measurements for Wind Induced Vibrations of Box Girder Bridges", J. Wind Eng'ng and Industrial Aerodynamics, (14) Dec. 1983, pp.223-234. 21.
Yamada, H., " I d e n t i f i c a t i o n Response of Shallow Bluff Univ. of Tokyo, Dec. 1983.
and Estimation Bodies", Ph.D.
22.
Scanlan, R.H., "Interpreting stayed Bridges", Proc. ASCE, 1987, pp.555-575.
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23. Wardlaw, R.L., "Sectional Versus Full Testing of Bridge Road Decks", Public 1980, pp.25-47.
Model Roads,
of CableEM4, April
Wind (44)
Tunnel i, June
24. Wardlaw, R.L., "The Use of Scale Models for Aerodynamic I n v e s t i g a t i o n s into the Effect of Wind on Structures", Proc. Int. Symp. on Scale Modeling, Tokyo, July 1988, pp.155-164. 25.
Scanlan, R.H., "Theory of the Wind Analysis of Long-span Bridges Based on Data Obtainable from Section Model Tests", Proc. 4th Int. Conf. Wind Effects on Buildings and Structures, Heathrow, England, Sept. 1975, pp.259-269.
297
26.
Davenport, A.G. et al., "Wind Induced Response of Suspension Bridges -Wind Tunnel Model and Full Scale O b s e r v a t i o n s " , Proc. 5th Int. Conf. on Wind Eng'ng, Fort Collins, Colorado, July 1979, Vol.2, pp.807-824.
27.
Davenport, A.G. and King, J.P.C., "Dynamic Long Span Bridges", Proc. 12th Congr. IABSE, 1984, pp.705-712.
28.
Tanaka, H. and Bridges", Proc. California, May
29.
Zan, S.J., Yamada, H. and Tanaka, H., "The Influence of Turbulence and Deck Section Geometry on the Aeroelastic Behaviour of a C a b l e - s t a y e d Bridge Model", A e r o n a u t i c a l Note, NAE-AN-40, NRC No.26190, National Research Council Canada, Aug. 1986.
30.
Tanaka, H. and Yamada, H., "On Predicting the Performance under Wind of Full Bridges from Section Model Wind Tunnel Results", J. Wind Eng'ng and Industrial Aerodynamics, (26) 3, Dec. 1987, pp.289-306.
31.
Davenport, A.G., Isyumov, N. and Miyata, T., "The Experimental Determination of the Response of Suspension Bridges to Turbulent Wind", Proc. 3rd Int. Conf. on Wind Effects on Buildings and Structures, Tokyo, Sept. 1971, pp.1207-1219.
32.
Davenport, A.G., "The Use of Taut Strip Models in the P r e d i c t i o n of the Response of Long Span Bridges to Turbulent Wind", Proc. Symp. on Flow-lnduced Structural Vibrations, IUTAM-IAHR, Karlsruhe, Aug. 1972, pp.373-381.
33.
Tanaka, H. and Davenport, Models to Turbulent Wind", EMI, Feb. 1982, pp.33-49.
34.
Tanaka, H., "On Wind Tunnel Models", Proc. 3rd US-Japan May 1987, pp.318-323.
35.
Xie, J., "Theory of Flutter for Bridge Structures and Study of Flutter Behaviours of C a b l e - s t a y e d Bridges", DESc. thesis present. Tongjl University, 1985.
36.
Davenport, A.G., "On the Statistical Prediction of Structural P e r f o r m a n c e in the Wind Environment", ASCE Nat. Struct. Eng'ng Meeting, Baltimore, Maryland, April 1971, Preprint 1420.
37.
Davenport, A.G., "The Prediction of Risk under Wind Loading", Proc. 2nd Int. Conf. on Structural Safety and Reliability, Munich, Sept. 1977, pp.511-538.
Xie, J., "Some 4th US-Japan 1988.
Wind Forces on Vancouver, Sept.
Discussions on Buffeting of Bridge Workshop, San Diego,
A.G., Proc.
"Response of Taut ASCE, J. Eng. Mech.,
Testing of Taut Strip Bridge Workshop, Tsukuba,
Strip (108)
Bridge Japan,
298
A
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Fig. i
The Flow Field Concerned
Cupletion of Hajor Bridges
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Collapse of Bridaes due to Wind
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1820 Union Bridge by Samuel Brown 1826 nenai Straits Bridge by Thomas Te]ford
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183G Brighton Chain Pier 1838 Hontrose 1850
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1900 Fig. 2
Bridges
and Wind Effects
Fig. 3
Various Types of Wind Induced Responses
299
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