(1984)Model for Predicting the Acrosswind Response of Buildings

8/22/2019 (1984)Model for Predicting the Acrosswind Response of Buildings

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M o d e l fo r p r e d i c ti n g th ea c r o s s w i n d r e s p o n s e o fb u i l d i n g sA . K a r e e mDepartment of Civil Engineering, University of Houston, Houston, Texas, USA{Received December 1982;. revised July 1983)

A model is presented for predict ing the acrossw ind respons e of isolatedsquare c ross-sec t ion bu i ld ings to typ ica l a tmo spher ic b oun dary layers ,over d i f fe ren t te r ra ins . C losed- form express ions for the auto- and co-spectra of the a c ro s s w in d fo r c e f lu c tu a t io n s a r e fo r m u la te d , b a s e d onwind tunne l measurements . A s ta tis tica l in tegra t ion scheme is used todevelop a mode-generalized acrosswind spectrum fo r any des i red approachf lo w c ond it ion, i .e. open country, suburban and urban. A s impl i f iede x p re s s io n b a s e d o n r a n d o m v ib r a t io n a n a ly s is i s u s e d to c o m p u te th emodal response. The model prov ides f lex ib i l i ty in the selection ofa p p r o p r ia te in p u t p a r am e te rs , broadening the scope o f i ts app l ica t ion an dserv ing as a use fu l too l fo r ta i lo r ing th e pre l im inary des ign o f ta l l bu i ld ings .K e y w o r d s : buildings, wind Ioadings, acrosswind response

I t is i m p o r t a n t t o r e c o g n i ze t h e u n s t e a d y n a t u r e o f w i n dlo a d s in th e d e s ig n o f b u i ld in g s to e n s u re s t ru c tu ra l s a fe tya n d s e rv ic e a b i l i t y r e q u i re me n t s . T h e a lo n g w in d re s p o n s e o fb u i ld in g s c a n b e c o mp u te d u s in g th e g u s t f a c to r a p p ro a c h . !H o w e v e r , l a c k o f a s u i t a b le t r a n s fe r fu n c t io n b e tw e e n th ev e lo c i ty f lu c tu a t io n s in th e a p p ro a c h f lo w a n d th e p re s s u ref lu c tu a t io n s o n th e f a c e s o f a b u i ld in g w i th s e p a ra t e d f lo wh a s p r o h i b i te d a n y a c c e p t a b l e f o r m u l a t i o n , t o d a t e , o f t h ea c ro s s w in d re s p o n s e b a s e d o n a g u st f a c to r a p p ro a c h .P h y s ic a l mo d e l l in g o f w in d -s t ru c tu re in t e ra c t io n in ab o u n d a ry l a y e r w in d tu n n e l , t h e re fo re , c o n t in u e s to s e rv ea s th e p ra c t i c a l a p p ro a c h re l a t in g s t ru c tu ra l r e s p o n s e a n da e ro d y n a m ic lo a d in g to p ro p e r t i e s o f l o c a l w in d c l ima te s . 2Q u a n t i t a t iv e d e s c r ip t io n o f w in d lo a d s w o u ld p e rmi t t h en u m e r ic a l e s t ima t io n o f a b u i ld in g re s p o n s e a t t h e p re -l im in a ry d e s ign s t ag e s , a llo w in g e a r ly a s s e s s me n t o f t h es t ru c tu ra l r e q u i re m e n t s to r e s i s t o s c i l l a to ry r e s p o n s e toe n s u re o c c u p a n c y c o mf o r t , a n d a s se ss in g th e n e e d fo rd e ta i l e d a e ro e la s t i c w in d tu n n e l t e s t s . T h e e x p e r ime n ta lm e a s u r e m e n t o f a e r o d y n a m i c f o r c e s o n a s c al e m o d e l c a nb e in t ro d u c e d in li e u o f t h e in t r a c t a b le s o lu t io n o f th ee q u a t io n s o f f lu id mo t io n a r o u n d th e m o d e l . 3 -8R e fe re n c e 4 p re s e n t s a d e t a i l e d a c c o u n t o f t h e me a s u re -me n t o f t o t a l f l u c tu a t in g lo a d s o n a b u i ld in g u s in g s u r fa c ep re s s u re s . A c o v a r i a n c e ( s t a t i s t ic a l ) i n t e g ra t io n p ro c e d u rew a s u s e d to d e r iv e th e s p e c t ra o f t h e a c ro s s w in d lo a d sf r o m t h e n u m e r i c a l e s t i m a t e s o f t h e a u t o - a n d c o - s p e c t r ao f p re s s u re f lu c tu a t io n s o n th e b u i ld in g mo d e l s u r fa c e .T h e m o d e l d e v e lo p e d w a s l im i t e d to q u a s i - s t a ti c l o a d in ga n d d id n o t i n c lu d e mo t io n - in d u c e d a e ro d y n a mic lo a d s .

S u c h m o t i o n i n d u c e d l o a ds r e s ul t f r o m a c o m b i n a t i o no f n e g a t iv e a e ro d y n a mic d a mp in g a n d a n in c re a se in th ec o r re l a t io n o f th e f lu c tu a t in g p re s s u re o n th e b u i ld in gs u r fa c e . I t i s a g e n e ra l c o n s e n s u s th a t i n m o s t o f t h e t a l lb u i ld ing s th e in f lu e n c e o f mo t io n - in d u c e d lo a d in g i sin s ig n i fi c a n t fo r t y p ic a l d e s ig n w in d s p e e d s . H o w e v e r , fo re x c e p t io n a l ly s l e n d e r , f le x ib le a n d l ig h t ly -d a mp e d s t ru c -tu re s a t h ig h re d u c e d v e lo c i t i es , t h e q u a s i - s ta t i c fo rc in gf u n c t i o n d e v e l o p ed i n r e f e r en c e 4 m a y u n d e r e s t i m a t e t h es t ru c tu ra l r e s p o n s e . F o r th e s e e x c e p t io n a l s t ru c tu re s , t h emo t io n - in d u c e d a e ro d y n a mic lo a d s c a n e a s i ly b e in c o r -p o r a t e d t h r o u g h t h e m o d i f i c a t i o n o f t h e b u i l d in g m e c h a n -i c a l a d mi t t a n c e fu n c t io n u s in g a p p ro p r i a t e v a lu e s o fa e ro d y n a m ic d a mp in g . 9 O n th e o th e r h a n d , i n v i e w o f th es ig n i fi c a n c e o f th e m o t io n in d u c e d lo a d s , s u ch s t ru c tu re s ,w h ic h a re e x c e p t io n a l ly s e n s i ti v e to w in d e x c i t a t io n , a reg e n e ra l ly s u b je c t ed to e x te n s iv e w in d tu n n e l a e ro e la s ti cmo d e l t e s t in g . H e re , c lo s e d fo rm e x p re s s io n s a re d e r iv e dfo r th e a u to - a n d c o - s p e c t ra o f a c ro s s w in d fo rc e s a t v a r io u sl e v e ls t o d e v e lo p o v e ra l l mo d e -g e n e ra l i z e d s p e c t ra . T h i sp ro c e d u re w o u ld p ro v id e th e f l e x ib i l i t y o f c h a n g in g th ep o w e r l a w e x p o n e n t , t u rb u le n c e in t e n s i ty , a c ro s s w in dfo rc e c o e f f i c i e n t s , me a n w in d s p e e d a n d s t ru c tu ra l p ro p -e r t i e s o f a b u i ld in g , i . e . ma s s , s t i ffn e s s , d a mp in g , f r e q u e n c ya n d m o d e s h a p e s , i n f lu e n c e c o e f f i c i e n t s fo r s h e a r a n dmo me n t a s th e in p u t p a ra me te r s . In th i s p a p e r th e v a l id i tyo f s o m e c o m m o n l y u s e d s i m p l i f ic a t io n s in t h e d e v e l o p m e n to f fo rc in g fu n c t io n s a re e x a m in e d . ~ '1 1 R e s p o n s e e s t ima te sa r e c o m p a r e d w i t h t h e v a l u es o b t a i n e d f r o m t h e s p e c t r ad e r iv e d in r e fe re n c e 9 . T h i s p ro c e d u re c a n b e a d o p te d

0141-0296/84/02136-06/$03.001 3 6 E n g n gStru e t . , 1984 , Vol . 6 , Apri l 1984 But te rworth & Co. (Publishers)Ltd


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con ven i en t l y fo r desk- t op com put e r s i n a sma l l des i gnof f i ce .D e v e l o p m e n t o f f o r c i n g f u n c t i o nT he f l uc t ua t i ng ac rosswi nd fo r ce on a bu i l d ing i s g iven by :

F y ( z , t ) = I p U ~ C L ( z , t ) A (1 )where CL ( z , t ) is the r and om ac rosswi nd fo r ce coe f f i c i en t ,p i s a ir dens i t y an d Uh is t he w i nd ve l oc i t y a t t he bu i l d ingh e i g h t. T h e p o w e r s p e c t r a l d e n s i t y f u n c t i o n o f F y ( z , t )f r o m e x p e r i m e n t a l m e a s u r e m e n t s i n d i c a t es e n e r g y c o n c e n -t r a t i o n n e a r t h e v o r t e x s h e d d in g f r e q u e n c y a n d t h e s p e c tr a lb a n d w i d t h d e p e n d s o n t h e g e o m e t r y o f t h e b u i ld i n g a n dt he ap proac h f l ow cha rac t e r i s t i c s f l xa S y ~ , ( z , n ) i s general lydesc r i bed by a G auss i an- t ype expres s i on fo r s l ende r (ve ryh i gh a spec t r a t i o ) s t r uc t u r es , e .g . ch i mneys . / -1~ S uch aGauss i an desc r i p t i on bes t desc r ibes t he na r row band r eg i ona r o u n d t h e S t r o u h a l f r e q u e n c y b u t f a i ls t o a d e q u a t e l yr epresen t t he spec t r a l ene rgy a t f r equenc i es o t he r t hant h e S t r o u h a l f r e q u e n c y .T o p red i c t t he s t r uc t u r a l r e sponse fo r non- r esonan tc o n d i t i o n s , n D / U h 4= S , whi ch i s gene ra l l y t he case fo rbu i l d i ngs , t he func t i on used t o r epresen t S p ~ ( z , n ) m u s tbe ab l e t o desc r i be t he spec t r a l con t e n t s ove r t he en t i r er ange o f f r equenc i es o f i n t e r e s t a l ong t he he i gh t o f t hes t ruc t u r e . A num ber o f expres s ions r epresen t i ng f i l t e r edw h i t e n o i se w e r e u s e d t o m o d e l S F y ( Z , n ) . D u e t o t h ea s y m m e t r y o f t h e s p e c tr a l d e n s i t y f u n c t i o n , S F y ( z , n ) ,a b o u t t h e S t r o u h a l f r e q u e n c y t h e f o l l o w in g t w o e x p r e s -s io n s , w h i c h a r e f u n c t i o n s o f t h e p o w e r l a w e x p o n e n ta n d t u r b u l e n t i n t e n s i t y p r o f i le o f t h e a p p r o a c h f l o w a n dand t he bu i l d ing he i gh t , bes t desc r i be t he m easured va l ues :

n S F y ( Z , n ) = trot [3f3 ( n / n s ) '9 n < ~ n so 2= oa x [3[3x ( n / n s ) a ' n > 1 n s

M od e / fo r p red i c t i ng the ac rossw ind response o f bu i l d ings : A . K a r e emo_ 0 - - M E A S U R E M E N T

o PREDICTIONURBAN B LZ = I / 2 H

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F i g u r e 1 P r e d i c t e d a n d m e a s u re d n o r m a l i z e d r e d u c e d s p e c t r a o fa c r o s s w i n d f o r c e

(2 )

B[1 ( n t ~ ] 2 n 2

+ O . l S 4 ( , _ ]i n wh i ch n s is shedd i ng f r eq uen cy = S U ( z ) [ D ; o 2 i s them e a n s q u a r e v a lu e o f f l u c t u a t i n g a c r o s sw i n d f o r c e ; S i st h e S t r o u h a l n u m b e r ; a i s t h e e x p o n e n t t e r m i n t h e p o w e rl a w ; B i s th e b a n d w i d t h c o e f f i c i e n t = 2 v 2 I ( z ) ; a n d I ( z ) ist h e t u r b u l e n c e i n t e n s i t y a t h e i g h t z .T he abo ve expres s i on is p l o t t ed i n F i g u r e I a l ong wi t ht he me asured spec t rum a t t he mi d-he i gh t o f t he bu i l d ing .T he sp ec t rum of t he f l uc t ua t i ng ove ra l l ac ros swi nd fo r cei n t he Rh m ode i s g i ven by :

H H ~

o o o i ( z l ) i ( z 2 ) d z t d z 2 ( 3 )

T he t e rm i n t he b r ack e t s i s t he co- sp ec t rum o f t he ac ros s-wi nd fo r ce f l uc t ua t i on s a t z t and z2 l oca t i on s ; i ( z = )i s t he i t h m od e shape eva l ua t ed a t z t ; and H i s t he bu i l d ing

he i gh t . E qu a t i on (3 ) i s r ewr i t t en i n t e rms o f t he no rma l i zedc o - s p e c t r u m :

H HS F i ( n ) = f f N / S F y (Z I , n ) S F , ( z 2 , n ) C o ( z I , z 2 , n )

o oX i ( z l ) i ( z 2 ) d z l d z 2 ( 4 )

i n whi ch Co (z t , z2 , n ) is t he n orma l i zed co- sp ec t rum , andS F y ( z t , n ) i s t he l i f t f o r ce spec t rum a t l oca t i on zv F ur t h e rr ea r r ang ing t e rms i n equa t i on (4 ) l eads t o t h e fo l l owi nge x p r e s s io n f o r t h e r e d u c e d n o r m a l i z e d s p e c t r u m o f t h eac rosswi nd l oad i ng i n t he i t h mod e :H Hn S & ( n ) 1 f f / n S F ~ ( z t , n ) / n S F ~ ( z 2 , n )

[ ~ p U ~ D H I ~ - H = J 4 o b (z O 40 ox Co(z1,z2, n ) Cz(zO Cz(z2) i( zO i(z2) dzz dz2

(5 )i n whi ch :

o F ( z , )C z ( z ) = - -~ p D U ~t he ac ros swi nd fo r ce coe f f i c i en t .T he co- sp ec t rum fo r t he ac ros swi nd fo r ce i s gene ra l l ya n e x p o n e n t i a l ly d e c a y i n g f u n c t i o n w i t h a p e a k n e a r t h eS t rouha l f r equency . / 3 '14 T he fo l l owi ng ana l y t i ca l expres -s io n s , as a f u n c t i o n o f f r e q u e n c y a n d t h e s e p a r a ti o nd i s t ance , a re de r i ved t o m a t ch t he m easured va l ues in va r i ousb o u n d a r y l a y e r f l o w s w h i c h a r e c h a r a c t e ri z e d b y t h e i rp o w e r l a w e x p o n e n t s a n d t h e b u i l d in g h e i g h t :

( A z , = e x p - - A z l ' n ( n D ~ l

+ ,A z ~

E ngng S t ruc t. , 1 9 8 4 , V o l . 6 , Apr i l 1 3 7


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M o d e / f o r p r e d i c t i n g t h e a c r o s s w i n d e s p o n s e o f b u i ld i n g s : A . K a r e emr I s i I 0 " ] " \ \ - - , - - - - , - - 1 ; - - ~ < F *UaAx z, F ~ " "A z "1"2-nO "1/2 n DC o , ) - - ~ > F *e x p ( - - 2 0 ( - - f f ) ( - U h - h ) ) ; U h

(6 )i n w h i c h A z = I z~ - z : l a n d F * = 1 . 2 5 ( 1 - - a ) / 1 0 .T h e p l o t s f o r C o ( x z/ n , n ) ar e g ive n in F i g u r e s 2 a n d 3f o r a = 0 . 1 2 a n d a = 0 . 3 4 , r e s p e c t i v e l y .

E q u a t i o n s ( 2 ) a n d ( 6 ) a r e su b s t i t u t e d i n e q u a t i o n ( 5 )t o o b t a i n t h e s p e c t r a o f t h e a c r o s s w i n d l o a d i n g f o r t h el i n e a r m o d e s h a p e . T h e p r e d i c t e d a s w e l l a s t h e e x p e r i -m e n t a l s p e c t r a a r e g i v e n i n F i g u r e s 4 a n d 5 f o r o p e nc o u n t r y a n d u r b a n e n v i r o n m e n t s . T h e m e a s u r e d a n d t h ep r e d i c t e d v a l u e s a r e i n g o o d a g r e e m e n t , e s p e c i a l l y i n t h ef r e q u e n c y r a n g e o f t a l l b u i l d i n g s .A n i m p o r t a n t f e a t u r e in t h e m o d e l p r e s e n te d h e r e i st h e c h o i c e o f a p p r o a c h f l o w c h a r a c t er i st i c s a n d t h e b u i l d in gh e i g h t . T h i s b r o a d e n s t h e s c o p e o f a p p l i c a t io n s a n d m a k e st h i s m o d e l h i g h l y u s e f u l f o r t a i lo r i n g t h e p r e l i m i n a r yd e s i g n o f a n i s o l a t e d t a l l b u i l d i n g i n a n y d e s i r e d e n v i r o n -m e n t . I n F i g u r e 6 a s e t o f t y p i c a l s p e c t r a , b a s e d o n e q u a t i o n( 5 ) f o r t h e f u n d a m e n t a l m o d e , a r e p r e s en t e d f o r a b u i ld i n gw i t h a n a s p e c t r a t io o f 1 : 6 i n o p e n c o u n t r y , s u b u r b a n , an du r b an f l o w c o n d i t i o n s .

I n th e f o l l o w i n g s e c t io n s o m e c o m m o n l y u s e d s i m p li -f i c a t io n s i n t h e f o r m u l a t i o n o f m o d e - g e n e r a l i z e d s p ec t r a

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a re e x a m i n e d . T h e f i rs t s u c h s i m p l i f i c a ti o n i n t r o d u c e s ap r e s su r e c o r r e l a t i o n l e n g t h s c a l e i n e q u a t i o n ( 5 ) t o f a c i l i t a t ee v a l u a t i o n o f a t w o - f o l d i n t e g r a l b y r e d u c i n g i t t o s i n g l e -f o l d . T h e s e c o n d s i m p l i f i c a t i o n i s a p r o c e d u r e f o r a d j u s t i n ga m o d e - g e n e r a l i z e d s p e c t r u m f r o m o n e m o d e s h a p e t oa n o t h e r w i t h o u t r e -e v a l u a ti n g e q u a t i o n ( 5 ) .

E q u a t i o n ( 5 ) c a n b e s i m p l i f ie d o n t h e a s s u m p t i o n t h a tt h e s p a n w i s e c o r r e l a t i o n o f t h e a c r o s s w i n d f o r c e i s s m a llc o m p a r e d w i t h t h e b u i l d i n g h e i g h t a n d t h a t i t is f r e q u e n c yi n d e p e n d e n t :

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F i g u r e 4 P r e d i c t e d a n d m e a s u r e d m o d e - g e n e r a l i z e d s p e c t r a f o ro p e n c o u n t r y f l o w

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F i g u r e 5 P r e d i c t e d a n d m e a s u r e d m o d e - g e n e r a l i z e d s p e c t r a f o ru r b a n f l o w

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UHF igure 6 Mode-genera l ized spect ra for open c oun t r y , s ubu rban a n du r b a n f l o w c o n d i t io n s

Model for predicting the acrosswind response of bui ldings: A. Kareemthe desi red shape , and i t i s g iven by:

H1S F ( n ) = - ~ f S F ( n ) ~ 2 ( z ) d z0H

C = ~ @2(z) dz (9)i i /0

in which ~b(z) s the or ig ina l mod e shap e ; ~b(z) i s the m odi-f i e d mode sha pe , a nd the se c ond subsc r ip t i n S F ( n )r e p re se n ts t he m ode sha pe w i th r e spe c t to w h ic h thespe c t rum ha s be e n w e igh te d .How ever , it appears tha t the va l id i ty o f th is s impli fica-t ion is inf luenced by an increase in the nonl inear i ty of themode shape , i .e . depar ture f rom l inear mode shape . A simila rexpression for adjust ing the l inear mod e genera l ized spec t rato any desi red mode shape has been g iven in re fe rence 15.In F i g u r e 7 , t he mod i f i e d values a nd tha t o b t a ine d f romthe so lu t ion o f t he c omp le t e e qua t ion (5 ) a re c ompa re d fo rtw o d i f f e re n t mod e sha pe s . R e su l t s show good a g re e me n t .N e ve r thel e ss , fo r m os t o f t he bu i ld ing s t ruc tu re s t he p ro -cedu re g iven in equa t io n (9) can be used to adjust a g ivenspe c t rum fo r a ny de s i r e d mod e sha pe . A c om mo nlyoc c u r r ing p rob le m in s t ruc tu ra l dyna mic s , i n w h ic h suc h amod i f i c a t ion be c ome s ne c e ssary , ha s l e d to t he ve r i f i c a tiono f t he p roc e du re g ive n in e qua t ion (9 ) fo r w h ic h e qua t ion(5 ) p rov ide s a good re fe re nc e fo r c ompa r i son .

a c rossw ind fo rc e . The D i ra c de l t a func t ion i s u se d fo rma the m a t i c a l c onve n ie nc e he re , s inc e in t e g ra t ion o f t hec o -spe c t rum, a ssume d to be on ly a func t ion o f se pa ra t iondistance , wi th respec t to he ight , reduces to a pressurelength sca le . This subst i tu t ion reduces two-fold in tegra t ionin equa t ion (5) to the fo l lowing single in tegra t ion:

Hn S F i (n ) k f n S F y ( z , n )[ Z 2 p U ~ D H ] 2 -I ~ o h ( z ) C ~ ( z ) @ 2 ( z ) d z (8 )0

P r e d i c t io n o f r e s p o n s eThe a c rossw ind dyna m ic r e sponse ca n be c om pu te d u s ingra ndom v ib ra t ion the o ry . The m e a n squa re r e sponse isg iven by:

o ~ ( r ) (z ) = f S y ( r ) ( Z ,n ) d n (10a )0

S y ( r ) ( Z , n) = E dp~(z)H~ ) (n)2SFi (n) ( ] 0 b )iA c rossw ind spe c t ra ob ta ine d f rom the s imp l i f ie d p roc e du re(e qua t ion (8 ) ) a nd the fu l l - e qua tion a pp roa c h (e qua t ion(5 ) ) a re c om pa re d in F i g u r e s 4 and 5 . The simpl i f iedp roc e du re p rov ide s e x t re me ly c onse rvat ive va lue s . In a smuc h a s the c o - spe c t rum i s a func t ion o f se pa ra t ion d is -t a nc e a nd the f r e que nc y , the a ssum pt ion w h ic h l e d toe qua t ion (8 ) m a y ha ve ove r s imp l if i e d the p roc e du re .Me thods ba se d on suc h a ssumpt ions shou ld the re fo re beviewed careful ly .H ow e ve r , i t is impor t a n t t o no te t ha t t he se mode l s ma ypre d ic t a c c u ra t e ly t he r e sona n t r e sponse o f a s t ruc tu ree xc i t e d by a na r row ba nd fo rc ing func t ion . H a nse n 1 ha sused a s imi la r approach for predic t ing the resonant responseo f a t a l l c h imne y due to v o r t e x she dd ing. In suc h a c a se,a t t he r e son a n t f r e que n c y the c o - spe c t rum i s on ly a func -t i on o f se pa ra t ion d i s ta nc e w h ic h ju s ti f i es t he u se o fe qua t ion (8 ) .The m ode -ge ne ra li z e d spe c tra , i . e. t hose ob ta ine d f rome qua t ion (5 ) , c a n be mod i f i e d to a ny de s i r e d mode sha pew i thou t r e -eva luat ing the tw o- fo ld in t e g ral . Th i s m od i f i c a -t i on a l so be c ome s ne c e ssa ry w he n on ly a mode -ge ne ra l iz e dspec tru m is ava ilable in a mo de shape wh ich d i ffe rs f rom

MODE SHA PE EO .5 E0 .9%0 (Z ) = (Z /H )1 .5 0d / ( Z ) = ( Z / H } 2 A

E3Q.

Z

F i g u r e 7spectra

I I I0 .00 1 0.01 0.1 1 .0N * D / U H

C om p a r is on o f p r e d i c t e d a nd a d j us t e d m ode - ge ne r a l iz e d

E n g n g S t r u c t . , 1 9 8 4 , V o l . 6 , A p r i l 1 3 9


5/6

M od e / fo r p red i c t i ng the ac rossw ind response o f bu i l d ings : A . Ka reemin which I H ~ 0 ( n ) l ~ i s t h e f r e q u e n c y r e s p o n s e f u n c t i o n i nt h e i t h m o d e f o r r t h d e r i v a t iv e o f d i s p l a c e m e n t r e s p o n s e ;r = 0 , 1 , 2 , 3 , r epresent s d i sp lacement , ve loc i ty , acce le ra-t io n a n d j e r k c o m p o n e n t s o f r e s po n s e . S F i ( n ) is given bye q u a t i o n ( 5 ) . 9 I f o n l y t h e f u n d a m e n t a l m o d e o f v i b r a t i o n c ~i s of in te res t then i --- 1 and the sum ma t ion of equ a t ion I---z(10) i s r educed to a s ingle va lue . Equ at ion (10) involves a la.It wo - f o l d i n t e g r a t i o n t o e v a l u a t e SFi ( r t ) and a single integra l t )r rt o c o m p u t e t h e m e a n s q u a r e v a l u e. Th i s i n t e g r a t i o n c a n wb e e x p e d i t i o u s l y c a r ri e d o u t u s i n g a M o n t e Ca r l o p r o - a .cedure . 16 The in tegra t ion in equ a t ion (10a) can be a l so Zp e r f o r m e d u s i n g t h e r e s id u e t h e o r e m " 7

= E , , i s p i ( , i ) 2 , o_i 4(27rni)4~im~ ( l l )


6/6

M o d e / f o r p r e d i c t i n g t h e a c ro s sw i n d r e sp o n se o f b u i l d i n g s : A . K a r e e maspect ratio equal or greater than 1:4 for typical designwind speeds.

The model provides flexibility in the selection of appro-priate input parameters, broade ning the scope of itsapplication, and providing a useful tool for tailoring thepreliminary design of isolated tall buildings in any desiredflow conditi on. The response predictions based on theproposed mod el offer excellent agreement with earlierstudies .S, 9,~s

The study indicates that the mode-generalized spectrafor typical tall buildings can be adjusted for different modeshapes by a simple transformati on.

It is also conclu ded that the simplification which leadsto the formul ation of the acrosswind forcing spectrum,obtained through the in troduction of correlation lengthscale, may lead to significant overestimation of the struc-tural response.

AcknowledgmentsThe author tha nks Dr J. E. Cermak for his encourag ementand invaluable suggestions during the exp erimental part ofthe work, and Mr Chii-Ming Cheng for assistance with thecomputer work. This work was partly supported by theNational Science Foun dat ion under Grant Nos. ENG-76-03135 and CEE-8019392. Their support is gratefullyacknowledged.

References1 Simiu, E. 'Modern developments in wind engineering -Part IV', Eng. Struct., 1983, 5, 272 Kareem, A. and Cermak, J. E. 'Wind tunnel simulation ofwind structure interactions',ISA Trans. 1979, 18 (4),3 Tschanz, T. 'Measurement of total dynamic loads usingelastic models with high natural frequencies', Prepr., Int.

Workshop on Wind Tunnel Modeling Criteria and Techniquesfor Ovil Engineering Applications, NBS, 14-16 April 19824 Kareem,A. 'Measurement of total loads using surfacepressures', Prepr., Int. Workshop on Wind Tunnel ModelingCriteria and Techniques for Civil Engineering Applications,NBS, 14-16 April 1982. (Also published by CambridgeUniversity Press)5 Reinhold, T. 'Measurement of simultaneous fluctuating loadsat multiple levels on a model of a tall building in a simulatedurban boundary layer', PhD Thesis, Virginia PolytechnicInst itute and State University, Blaksburg, Virginia, 19776 Saunders, J. W. and Melbourne, W. H. 'Tall rectangularbuilding response to cross-wind excitation',Prec. FourthInt. Conf. Wind Effects on Buildings and Structures,Heathrow, England, 19757 Vickery,B. J. 'Load fluctuations in turbulent flow', Prec.Am. Soc. Cir. Eng., J. Eng. Mech. Div. 1968, 108 (EM1),948 Kwok, K. C. S. 'Cross-wind response of tall buildings', Eng.Struct. 1982, 4, 2569 Kareem,A. 'Acrosswind response of buildings', J. Struct.Div., ASCE 1982, 108 (ST4)10 Hansen, S. O. 'Cross-wind vibration of a 130-m taperedconcrete chimney, J. Wind Eng. and lndustr. Aerodynamics1981, 1811 Davenport, A. G. 'The influence of turbulence on the acre-elastic response of tall structures to wind', IAHR-IUTAMSymp. Practical Experiences with Flow-Induced Vibrations,Karlsruhe, 1979, Springer-Veriag, 198012 Vickery,B. J. 'A model for prediction of the response ofchimneys to vortex excitation',3rd Int. Symp. ChimneysOct. 1978, Munich, Germany13 Kareem, A. and Cermak, J. E. 'Pressure fluctuations on asquare building model in boundary layer flows', J. Wind Eng.and lndust. Aerodynamics (to be published, 1984)14 Kareem, A. 'Wind excited motion of buildings', Thesis,Colorado State University, Fort Collins, Colorado, 1.97815 Saunders, J. W. and Melbourne, W. H. 'Wind-excitedbuildings - design sway stiffness', Royal Melbourne Inst ituteof Technology, Department of Mechanical and ProductionEngineering, Australia, Nov. 197716 Kareem,A. and Hseih, J. 'Reliability of concrete chimneysunder winds', Department of Civil Engineering, University ofHouSton,Report No. CER83-UH83-AK-JH1, May 1983

Engng Struct ., 19 84 , Vo l . 6 , Apr i l 141

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(1984)Model for Predicting the Acrosswind Response of Buildings