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BuiM. Sci. Vol. 8, pp. 11-22. Perg amo n Press 1973. Printed in Grea t Britain I I( 2 1 ) 1 i ( K )IT he Structural A na lys is o f Ta l l Bui ld ingsHaving Irregular ly Pos i t ioned Shear Wal l s1.. G . J A E G E R *A . A . M U F T I iJ . C . M A M E T

A n ana h ' t i c a l t h e or y f i ) r t he ana l y s i s o f t a l l th r e e - ~ f im e ns i ona l m u l t i p l e s he a rw a l l bu i l d i ngs i s de v e l ope d . T he ba s i s o [" t he t he or l ' i s t he c on t i nu um appr o ac hh i w h i c h t h e f lo m ' s o f t h e b u i l d in g a r e r e p l a c e d b l' a n e q u i v a l e n t c o n t i n u o u sm e d i u m . T h e r e. ~u lt s a r e c o m p a r e d w i t h d a t a o b t a i n e d b y t h e f i n i t e e h ,m e n tm e t h o d a n d e x p e r i m e n t s c o m t u c t e d o n a s e v e n s to r e y m u l t i p h , s h e a r w a l l m o d e l A g o o d c o r r e la t i o n i s a c h ie v e d .

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N O M E N C L A T U R Ever t ical co -ord in a tehor izon ta l co -ord ina teclear span between shear wallsdistance between n eutral axes of shear wallsvertical shear force per unit de pth in floor systemcum ulative vertical shear forc e in floor systemaxia l fo rce in shear wal ls and co lum nshorizo ntal shear force in a shear wallY o u n g ' s M o d u l u scross -sec t iona l a ream om ent o f ine r t iasum of bend ing m om ents in shear wal lsex te rna l ly app l ied over tu rn ing m om ent a t leve l xflexural r igidity per unit de pth o f floor systemheight of the buildin ga param ete r hav ing d im ens ions ( leng th) - z re la t ingfloor stiffness to she ar wall stiffnessano ther param ete r hav ing d im ens ions ( leng th) -a non -d im ens iona l param ete r govern ing def lec t ionp a n e r n sf rac t ion o f to ta l shear wal l E l a t the r th s ta t iond isp lacem ents in x d i rec t ion

I N T R O D U C T I O NI N R E C E N T Y E A R S t h e p r o b l e m o f t h e i n te r -a c t i o n b e t w e e n s h e a r w a l l s a n d f l o o r s y s t e m s in t a l lb u i l d i n g s h a s a t t r a c t e d m u c h a t t e n t i o n . F r e q u e n t l yt h e b a s i s o f a t t a c k o n t h e a n a l y t i c a l p r o b l e m h a sb e e n t o r e p l a c e t h e m a n y f l o o r s o f t h e b u i l d i n g b y ac o n t i n u o u s " m e d i u m " , i .e . b y a n i n fi n it y o f " f lo o r s "h a v i n g t h e s a m e t o t a l E I as t h e a c t u a l s y s t e m .

T h i s c o n c e p t , o f r e t a i n in g d i s c r e te m e m b e r s i no n e d i r e c t i o n o f a n e l a s t i c s t r u c t u r e w h i l s t u t i l i z i n ga c o n c e p t u a l c o n t i n u u m i n a n o t h e r d i r e c t i o n h a sb e e n u s e d i n g r il l a g e a n a l y s i s b y H e t e n y i [ l ] , b y

* Dean o f the Facu l ty o f Eng ineer ing , Univers i ty o f NewBrunswick , F reder ic ton , Canada .I Ass is tan t P rofessor , Depar tm ent o f C iv i l Eng ineer ingand Appl ied Mechan ics , McGil l Univers i ty , Montrea l ,C a n a d a .,* Research Ass is tan t , Depar tm ent o f C iv i l Eng ineer ing

and Appl ied Mechan ics , McGil l Univers i ty , Montrea l ,C a n a d a .

H e n d r y a n d J a e g e r [ 2 ] a n d b y m a n y o t h e r s , i n i tsa p p l i c a t i o n t o s h e a r w a l l s it a p p e a r s t o h a v e b e e nf i rs t u s e d b y C h i t t y [ 3 ] , s u b s e q u e n t l y b e i n g u s e d a st h e b a s i s f o r m o d i f i c a t i o n s a n d e x t e n s i o n s t o t h et h e o r y b y a n u m b e r o f r e se a r c h e rs , f o r e x a m p l eC o u l l [ 4 ] . O f t h e t h e o r e t i c a l t r e a t m e n t s w h i c h u s et h i s c o n c e p t t h e r e a r e t w o m a i n t y p e s , w h i c h o n em i g h t c a t e g o r i z e a s i n e x t e n s i o n a l a n d e x t e n s i o n a lr e s p ec t i ve l y , d e p e n d i n g u p o n w h e t h e r a x i a l c h a n g e si n l e n g th o f th e s h e a r w a l l s a r e i g n o r e d o r t a k e n i n t oa c c o u n t . I n t h e l a t te r c a t e g o r y t h e w o r k o fR o s m a n [ 5 ] m a y b e n o te d .

O n t h e b a s i s o f p u b l i s h e d w o r k s u c h a s t h a t c i t e da b o v e i t m a y b e s a i d t h a t a s u f f i c ie n t l y a c c u r a t ea n a l y s i s c a n b e m a d e o f a n e l a s t i c s y s t em c o m p r i s -i n g tw o o r t h r e e s h e a r w a l l s c o n n e c t e d b y a l a rg en u m b e r o f h o r i z o n t a l b e a m s a n d s u b j e c t e d t ot r a n s v e r s e ( h o r i z o n t a l ) l o a d i n g i n a v e r t i c a l p l a n ew h i c h p a s s e s t h r o u g h t h e s h e a r w a l l s ( f i g u re I ) .E s s e n t i a l l y t h i s i s a t w o d i m e n s i o n a l t r e a t m e n t .

R e c e n t p r o g r e s s i n t h e a n a l y s i s o f b u i l d i n gs t r u c t u r e s b y t h e f i n i te e l e m e n t m e t h o d h a s p r o v i d e da c o m p l e t e l y d i ff e r en t a p p r o a c h t o t h e p r o b l e m .T h e s e d e v e l o p m e n t s a r e d o c u m e n t e d i n ( 7) a n d a r eu s e d i n t h i s p a p e r f o r a c o m p a r i s o n o f t h e re s u l t s .

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12 L. G, Jaeger, A. ,4,T h e o b j e c t o f t h e p r e s e n t p a p e r i s t o d e v e l o p a

s u f f ic i e n tl y a c c u r a t e t h r e e d i m e n s i o n a l t r e a t m e n t o fm u l t i p l e s h e a r w a l l bu i l d in g s . S u c h a s t u d y g i v e sa s o n e o f it s e n d p r o d u c t s a s ti f f n es s ma t r i x wh i c h i ti s e s s e n t i a l t o h a v e , f o r e x a mp l e , i n f i n d i n g t h en a t u r a l f r e q u e n c i e s o f v i b r a t io n o f th e b u i l d in g . A sa p r e l i m i n a r y t o t h e t h r e e d i m e n s i o n a l a p p r o a c h , ag e n e r a l i z a t i o n o f t h e t w o d i m e n s i o n a l p r o b l e m i sp r e s e n t e d . S y m b o l s a r e d e f i n e d i n t h e te x t a s t h e yo c c u r a n d a r e l is t ed i n th e a p p e n d i x a l o n g w i t h t h em o r e i m p o r t a n t f o r m u l a e .

A N U M B E R O F S H E A R W A L L S I N S E Q U E N C EF i g u r e 2 s h o w s a n u m b e r o f sh e a r w a l l s i n te r -

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

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' n F . d .vl l n ~ - x EA,.

w h e r e E = m o d u l u s o f el a s ti c i tya n d A , = c r o s s s e c ti o n a l a r e a o f n t h s h e a r w a l l.

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L e t a ve r t ic a l c o - o r d i n a t e x b e m e a s u r e d f r o m t h et o p d o w n w a r d s . F o r n s h e a r w a ll s th e r e a r e n - 1" g a p s " b e t w e e n t h e m . L e t t h e v e r t ic a l s h e a rs , p e ru n i t d e p t h , i n t h e g a p s b e q~ , q 2 . . . . . q,-1. T h e nt h e s h e a r f o r c e s a t t h e l e v e l x a r e

T I = i q l d xO

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L e t t h e f l e x ur a t r i g i d it y p e r u n i t d e p t h o f t h eh o r i z o n t a l m e d i u m i n t h e f i r s t g a p b e B j , a n ds i m i l a r l y f o r t h e o t h e r g a p s .

I f c h a n g e s o f le n g t h o f t h e f l o o r b e a m s a r e n e g -l e c te d , t h e h o r i z o n t a l d e f l e c t io n ( y ) o f al l s h e a r w a l lsm a y b e t a k e n t o b e t h e s a m e a t a n y f l o o r l e v e l . I fa l s o t h e f l e x u r al r i g i d it y o f e a c h w a l l i s la r g e c o m -p a r e d t o t h a t o f a h o r i z o n t a l b e a m , i t f o ll o w s t h a t ,s u f f ic i e n tl y cl o s e l y , t h e d e f o r m a t i o n o f t h e h o r i z o n -t a l m e d i u m m a y b e t a k e n t o b e s k e w s y m m e t r i c i ne a c h g a p , i .e . w i t h a p o i n t o f c o n t r a f l e x u r e h a l f w a ya c r o s s t h e g a p . T h e s i t u a t i o n i n a t y p i c a l " g a p "is th e n a s s h o w n i n f i g u re 3 a n d a p p l i c a t i o n o f t h e

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The Structural Analys is o f Tall Buildings Having I rregular ly Posi t ioned Shear Walls 13s l o p e d e f l e c t io n e q u a t i o n s g i v e s w h e r e

q ,b 3 d v f ~ F , d x j ' " F , d x M = E ( ~ . I , ) d2 y- - = " / : 1 ~ X 21 2 B ~ - I ~ - ~ x - ~ + ~ E A 2 le t= - / + - + W + + + E A +

': l" ' ' h +" ' - I . d y I A r F . _ t d x ~ U F . d xI ' _a , , _ , - ' a -V .,- -O ~ 0 .~ ] - - 7 + E A , (3 )

D i f f e r e n t i a ti n g e a c h o f e q u a t i o n s ( 3 ) w i t h r e s p e c tt o x and us i ng r e l a t i ons h i ps ( I ) and ( 2 ) g i ves

b 3 d2T t d2y / / I 1 ), 2 e , - - - / , T ,

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h ~ _ d2T . _ d2y 1t t = _ 1 . _ 1 7 . . _ 21 2 B , , _ , d x ' d x " E A . _ l+ + T ._ , (4 )

The var iab le y is readi ly e l iminated from equat ions(4) by using moment /curvature re la t ionships.L e t M , = t h e s t a t i c a l l y d e t e r m i n a t e ( e x t e r n a l l y

a p p l i e d ) o v e r t u r n i n g m o m e n t a t l e ve l x . T h e n ,r e f e r r i n g t o f i g u r e 4 , t h e s u m o f t h e b e n d i n gmoment s i n t he s hea r w a l l s a t t h i s he i gh t i s

M = M . - ~ . T i l ii= l

and / = ~ I iI

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/ 2 I n - I+ . . . + T , _ t ~ ( 6 )E1E q u a t i o n s ( 6 ) a r e s ee n t o b e a b a n k o f ( n - I )s i m u l t a n e o u s o r d i n a r y d i f f e re n t i a l e q u a t i o n s w i t hc o n s t a n t c o e f f i c i e n t s . F o r a n y g i v e n f u n c t i o n M ~ ,t h e se c a n b e s o l v e d f o r t h e u n k n o w n f u n c t io n sT 1 , 7"2 . . . T ,_ 1 . O nc e t he s e f u nc t i on s a r e kno w n ,t h e b e n d i n g m o m e n t s a n d s h e a r s t h r o u g h o u t t h es y s t e m a r e r e a d i l y w r i tt e n d o w n .

F o r s i m p l i c it y , in t h i s p a p e r , s o l u t i o n s o f e q u a -t i o n s (6 ) a r e d e m o n s t r a t e d f o r o n l y t h e f o l lo w i n gcas es :

( i) a p a i r o f s h e a r w a l ls w i t h o n e " g a p "b e t w e e n ;

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1 4 L . G . J a e g e r , A . A . M u f t i a n d J . C . M a m e t( ii ) t h r e e s h e a r w a l ls , w i th s y m m e t r y a b o u t t h e

c e n t re w a l l ;( J i l l a s in g le s h e a r w a l l , f l a n k e d b y m u c h w e a k e r

c o l u m n s , w i t h s y m m e t r y .T h e t h r e e c a s e s a r e s h o w n d i a g r a m m a t i c a l l y i nf ig u re 5 a n d th e th i rd c a s e r e q u i re s a s l i g h t m o d i f i c a -t io n o f t r e a tm e n t , w h ic h i s g iv e n b e lo w .

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R e fe r r in g to f ig u re 5 (b ) , t h e re a re tw o " ' g a p s "a n d th e re a re tw o e q u a t io n s (6 ) . B e c a u s e o fs y m m e t r y , h o w e v e r , t h e s e t w o e q u a t i o n s a r e i d e n -t i c a l , s o th a t a s in g le d i f f e re n t i a l e q u a t io n c h a ra c -t e r i z e s t h e b e h a v i o u r . T h i s s i n g l e e q u a t i o n m a y b ee x p r e s s e d i n t e r m s o f t h e s u m o f th e b e n d i n gm o m e n t s in th e s h e a r w a l l s , M, a n d i s i n f a c t t h es a m e a s e q u a t io n (8 ) b u t w i th d i f f e re n t d e f in i t i o n sof %, ),~.

d 2 M d 2 M ~d .v 2 : ~ M - d x 2 y Z , . M , {8 )

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