89
SOME A S P E C T S OF SHEAR WALL D E S I G N
T. P A U L A Y *
I. I N T R O D U C T I O N
T h e u s e f u l n e s s of c e r t a i n w a l l s in the s t r u c t u r a l p l a n n i n g of m u l t i s t o r e y b u i l d i n g s in p a r t i c u l a r h a s l o n g b e e n r e c o g n i s e d . W h e n e x t e r n a l or i n t e r n a l w a l l s a r e s i t u a t e d in a d v a n t a g e o u s p o s i t i o n s they c a n be very e f f i c i e n t in re si st ing l a t e r a l l o a d s o r i g i n a t - ing f r o m w i n d or e a r t h q u a k e s . In a d d i t i o n to the p o t e n t ial s t r e n g t h , w h i c h shear w a l l s p o s s e s s , t h e y o f f e r c o n s i d e r a b l e l a t e r a l s t i f f - n e s s a n d t h u s c a n p r o t e c t a b u i l d i n g a g a i n s t n o n - s t r u c t u r a l d a m a g e that a r i s e s w h e n l a t e r a l d i s p l a c e m e n t or sway b e c o m e s exce ssive d u r i n g a m o d e r a t e s e i s m i c d i s t u r b a n c e .
Only in t h e l a s t f e w y e a r s d i d the
s u b j e c t r e c e i v e m o r e a t t e n t i o n a n d m a n y a s p e c t s of t h e b e h a v i o u r of s h e a r w a l l s are still b e i n g st u d i e d a n d e v a l u a t e d . B e c a u se of n e c e s s a r y s p a c e l i m i t a t i o n s r e l e v a n t f e a t u r e s, w h i c h h a v e be en t r e a t e d in w e l l k n o w n p u b - l i c a t i o n s ^ i f ', w i l l be m e n t i o n e d very
b r i e f l y h e r e , b u t r e f e r e n c e to the a p p r o p r i a t e s o u r c e w i l1 b e m a d e . O t h e r a s p e c t s , w h i c h a r e n o w u n d e r s t o o d a l i t t l e b e t t e r b e c a u se of m o r e r e c e n t r e s e a r c h e f f o r t s , are p r e s e n t e d in m o r e d e t a i l . E m p h a si s is p l a c e d on f e a t u r e s of b e h a v i o u r r e l e v a n t to l o a d i n g of s e i s m i c o r i g i n r a t h e r t h a n on d e s i g n r e c i p e s .
2 . C A N T I L E V E R S H E A R W A L L S
2 • 1 T a l l W a l l s w i t h R e c t a n g u l a r C r o s s S e c t i o n s A s i n g l e c a n t i l e v e r shear w a l l , w i t h a h e i g h t to d e p t h r a t i o of over 2, can be e x p e c t e d to b e h a v e a s an o r d i n a r y r e i n f o r c e d c o n c r e t e f l e x u r a l m e m b e r . T h e s m a l1 w i d t h of s u c h a s e c t i o n s u g g e st s that the p r o b l e m of i n s t a b i l i t y m a y a r i s e . C o n s e q u e n t l y the t h i c k - n e s s at t h e c r i t i c a l r e g i o n n e a r the b a s e of the w a l l m u s t b e c h o s e n a c c o r d i n g l y w h e n c o n - s i d e r i n g the f l o o r h e i g h t as the b u c k l i n g l e n g t h .
T h e s t r e n g t h of s u c h w a l l s can be simply e v a l u a t e d f r o m c o n v e n t i o n a l a x i a l l o a d - b e n d i n g m o m e n t i n t e r a c t ion r e l a t i o n s h i p s . T h e axial c o m p r e s s i o n i s o f t e n r a t h e r s m a l1 a n d t h e r e - f o r e it d o e s n o t a p p r e c i a b l y r e d u c e the d u e t - i l i t y of a p r o p e r l y r e i n f o r c e d r e c t a n g u l a r s h e a r w a l l • S o m e t i m e s the c o d e s p e c i f i e d l o a d d o e s not r e q u i r e m u c h f l e x u r a l r e i n f o r c e m e n t a n d t h e d e s i g n e r is t e m p t e d to p r o v i d e steel u n i f o r m l y d i s t r i b u t e d o v e r the w h o l e d e p t h of t h e sect ion. It m a y be s h o w n f r o m f i r s t p r i n c i p l e s, h o w e v e r , that s u c h a r r a n g e m e n t of r e i n f o r c e m e n t r e d u c e s the d u c t i l i t y a s the t o t a l s t e e l c o n t e n t i n c r e a s e s .
* R e a d e r in C i v i l E n g i n e e r i n g , U n i v e r s i ty of C a n t e r b u r y , C h r i s t c h u r c h .
W h e n the f1e x u r a l s t e e l d e m a n d is l a r g e r it wi 1 1 be m u c h m o r e e f f i c i e n t to f o l l o w the a r r a n g e m e n t u s e d in o r d i n a r y r e i n f o r c e d c o n c r e t e b e a m s , i.e. to a c c o m m o d a t e m u c h of the f l e x u r a l s t e e l n e a r the e x t r e m e f i b r e s. A m i n i m u m of Pv = 0 . 2 5 % v e r t i c a l steel s h o u l d a l w a y s be p r o v i d e d in the Mw e b " of the s h e a r w a l l secti on.
H o w e v e r , other c o n s i d e r a t i o n s , to be d i s c u s s e d l a t e r , m a y r e q u i r e m o r e r e i n f o r c e m e nt in the
" w e b " . T h e c o n c e n t r a t i o n of the f l e x u r a l s t e e l n e a r the e x t r e m i t i e s of a r e c t a n g u l a r s h e a r wall s e c t i o n will also i n c r e a s e c o n s i d e r a b l y the r o t a t i o n a l d u c t i l i t y ' ^ ',
T h e shear s t r e n g t h of tall r e c t a n g u l a r she ar w a l l s can be a s s e s s e d the say way a s in o r d i n a r y b e a m s . It s h o u l d be r e m e m b e r e d , h o w e v e r , that w h e r e y i e l d i n g (near the b a s e of the w a l l ) of the f l e x u r a l r e i n f o r c e m e n t can o c c u r the c o n t r i b u t i o n of the c r a c k e d c o n c r e t e t o w a r d s s h e a r r e s i s t a n c e must be n e g l e c t e d a n d h o r i z o n t a l s t i r r u p r e i n f o r c e m e n t n e e d be p r o v - i d e d for the total shear w h i c h is a s s o c i a t e d w i t h the load c a u s i n g f1e x u r a l y i e l d i n g .
A s s u m i n g a kS d e g r e e p o t e n t i a l d i a g o n a l f a i l u r e c r a c k , this s t i r r u p r e i n f o r c e m e n t w i l l h a v e to be p r o v i d e d over a h e i g h t e q u a l to the d e p t h of the shear wall s e c t i o n . T h i s c a n be m o r e t h a n t h e f l o o r h e i g h t . I n t e r m e d i a t e v e r t i c a l b a r s w i l l not c o n t r i b u t e s i g n i f i c a n t l y t o w a r d s s h e a r s t r e n g t h in tall s h e a r w a l l s .
In the u p p e r p a r t s of the c a n t i l e v e r s h e a r w a l l no f1e x u r a l y i e l d i n g is e x p e c t e d . C o n s e - q u e n t l y a n o m i n a l s h e a r stre ss of a p p r o x i m a t e l y 2 |f£ can a l w a y s be a l1 o c a t e d to the c o n c r e t e . A m i n i m u m h o r i z o n t a l w a l l s t e e l c o n t e n t of 0 . 2 5 $ , w h e n p r o p e r l y a n c h o r e d , w i l1 s u p p l y a b o u t the s a m e a m o u n t of s h e a r s t r e n g t h so that the m i n i m u m shear s t r e n g t h a v a i l a b l e in the e l a s t i c p o r t ion of a r e c t a n g u l a r s h e a r w a l l is a p p r o x - i m a t e l y ^ y f£ , i n t e r m s of n o m i n a l s h e a r s t r e s s e s . 2 . 2 Squat S h e a r W a l l s W i t h R e c t a n g u l a r C r o s s
S e c t i o n s
In c a n t i l e v e r shear w a l l s w h o s e h e i g h t , hw, is c o m p a r a b l e w i t h t h e i r d e p t h , i . e . hw/ 1 -zz 1 , it is no 1 o n g e r p o s s i b l e to s e p a r a t e the c o n s i d e r a t i o n s of f l e x u r e a n d s h e a r from e a c h o t h e r . T h e r e l a t i v e l y l a r g e f l e x u r a l c a p a c i t y of t h e s e w a l l s c a n be a s s o c - i a t e d wit h shear f o r c e s w h i c h are l a r g e e n o u g h to d e s t r o y the s t r u c t u r e in a b r i t t l e m a n n e r . H e n c e the c o n s i d e r a t i o n of s h e a r s t r e n g t h a s s u m e s p r e - e m i n e n c e .
F o r want of o t h e r a n d b e t t e r i n f o r m a t i o n it w a s o f t e n a t t e m p t e d to p r e d i c t the l i k e l y b e h a v i o u r of low ri se s h e a r w a l l s f rom t e s t s c a r r i e d out on d e e p b e a m s . G e o m e t r i c s i m i l - a r i t i e s s u g g e s t e d s u c h p r o c e d u r e s . M o s t t e s t s on d e e p b e a m s h a v e a c o m m o n f e a t u r e , the l o a d Bulletin of N.Z. Society for Earthquake Engineering, Vol. 5 No. 3, September 1972
90
is d i r e c t l y a p p l i e d to the t o p or b o t t o m face s of s i m p l y s u p p o r t e d s p e c i m e n s , a s s h o w n in F i g .
la. A s the d i a g o n a l 1i n e of t h r u s t , or i n t e r n a l a r c h , can d i s p o s e of the shear f o r c e , v e r t i c a l s t i r r u p s w i l l not be e f f e c t i v e . T h e r e f o r e c e r t a i n c o d e s s u g g e s t e d tha t the c o n t r i b u t i o n of w e b r e i n f o r c e m e n t s h o u l d be i g n o r e d a n d the n o m i n a l s h e a r s t r e s s on l o w r i s e s h e a r w a l l s be 1i m i t ed^ '. F i g . lb shows that w i t h a
d i f f e r e n t t y p e of l o a d int r o d u c t i o n no e f f e c t - ive a r c h a c t i o n can d e v e l o p a n d h e n c e no sub- s t a n t i a l s h e a r s t r e n g t h can e x i s t w i t h o u t w e b r e i n f o r c e m e n t c r o s s i n g the p o t e n t i a l d i a g o n a l f a i l u r e c r a c k s . L e o n h a r d t a n d W a l t h e r
d e m o n s t r a t e d ^ ^' that w h e n the she a r . i s i n t r o - d u c e d a s a 1i n e l o a d over the d e p t h of a d e e p b e a m , s t i r r u p r e i n f o r c e m e n t i s f u l l y e f f e c t i v e . One of t h e i r s p e c i m e n s t o g e t h e r w i t h t h e a p p l i e d load p a t t e r n i s s h o w n in F i g . 2 .
A s i m i l a r d i f f e r e n c e in the b e h a v i o u r of low r i s e c a n t i l e v e r shear w a l l s e x i s t s . E a r l y r e s u l t s were d e r i v e d from r a c k i n g t e s t s , w i t h a s i n g l e c o n c e n t r a t e d load b e i n g a p p l i e d
a g a i n s t the t e n s i o n f o r c e of a c a n t i l e v e r . In t h e s e s p e c i m e n s a r c h act ion c o u l d d e v e l o p . In a c t u a l shear w a l l s the l o a d is i n t r o d u c e d n o r m a l l y a s a l i n e l o a d , by m e a n s of a f l o o r slab, a n d a s h e a r d i s p o s a l s i m i l a r to that shown in F i g . 2 e n s u e s .
T o i l l u s t r a t e the b e h a v i o u r of s q u a t s h e a r w a l l s , a n d the e f f e c t s of b o t h the f l e x u r a l ( v e r t i c a l ) and shear ( h o r i z o n t a l ) r e i n f o r c e - m e n t on f a i l u r e m o d e a n d d u c t i l i t y , the m o r e i m p o r t a n t p o i n t s of a study, c a r r i e d out at the U n i v e r s i t y of C a n t e r b u r y ' * ' , a r e b r i e f l y p r e - s e n t e d in the f o l l o w i n g .
S q u a r e s h a p e d ( hw/1w = 1 ) 6 i n. t h i c k c a n t i l e v e r s h e a r w a l1s were s u b j e c t e d to a l o a d in s u c h a way that the s h e a r f o r c e w a s d i s t r i b - u t e d a l o n g t h e i r top e d g e . See F i g . 3. F o r c o n v e n i e n c e thi s is e x p r e s s e d in t e r m s of n o m i n a l s h e a r s t r e s s , v. The p r o p e r t i e s of t h r e e s p e c i m e n s a n d the r e s u l t s are s u m m a r i s e d in T a b l e I. T h e f a i l u r e m o d e of e a c h of the test w a l l s is e v i d e n t f r o m F i g . 3 a n d F i g . ^#
(i) W a l l 2 0 3 w a s d e l i b e r a t e l y u n d e r d e s i g n e d f o r s h e a r . S t i r r u p s w e r e e x p e c t e d to r e s i s t o n l y 50% o f the l o a d , w h i c h w o u l d h a v e b e e n d e v e l o p e d if f l e x u r e g o v e r n e d the s t r e n g t h . ( R o w 8 of T a b l e I ) . F i g . 3 s h o w s the d e v e l o p m e n t of d i a g o n a l c r a c k s d u r i n g 1 2 c y c l e s of the l o a d i n g . Some of the s h e a r w a s st ill t r a n s f e r r e d by a r c h a c t i o n b e c a u s e 123% of the t h e o r e t i c a l s h e a r c a p a c i t y a n d 10 8% of the f l e x u a l c a p a c i t y c o u l d be d e v e l o p e d . N o t h i n g s h o w s b e t t e r t h e u s e f u l - n e s s of s t i r r u p s in s u c h squat s h e a r w a l l s t h a n the f a c t t h a t at f a i l u r e one s t i r r u p f r a c t u r e d .
( i i ) B y p r o v i d i n g m o r e v e r t i c a l r e i n f o r c e - m e n t in W a l l 2 0 1 its f l e x u r a l c a p a c i t y w a s a p p r o x i m a t e l y t w i c e t h a t of the p r e v i o u s w a l l . T h i s f l e x u r a l s t r e n g t h r e q u i r e d the d e v e l o p m e n t of l a r g e s h e a r s t r e s s e s , i.e. a p p r o x i m a t e l y
l O y f £ . T h e d e s i g n shear s t r e n g t h of the w a l l w a s in e x c e s s o f it s f l e x u r a l c a p a c i t y , a n d a f l e x u r a l f a i l u r e c o u l d b e e x p e c t e d . A s F i g . ka s h o w s a s l i d i n g shear f a i l u r e o c c u r r e d in the 1 2t h c y c l e of l o a d i n g at 39% of the t h e o r e t - ical c a p a c i t y o f the s p e c i m e n . It s h o u l d be n o t e d t h a t a d d i t i o n a l s t i r r u p r e i n f o r c e m e n t c a n n o t p r e v e n t t h i s type of f a i l u r e , w h i c h is t y p i c a l in d e e p m e m b e r s , w h e r e h i g h i n t e n s i t y of c y c l i c s h e a r o c c u r s . T h e p h e n o m e n o n w i l l be d i s c u s s e d in g r e a t e r d e t a i l w h e n the b e h a v i o u r .of d e e p s p a n d r e l b e a m s is r e v i e w e d .
(iii) W a l l 2 0 2 w a s i d e n t i c a l to W a l l 203 e x c e p t that s h e a r r e i n f o r c e m e n t w a s p r o v i d e d in e x c e s s of the f1e x u r a l c a p a c i t y . T h e t h e o r e t i c a l f l e x u r a l s t r e n g t h w a s a s s o c i a t e d w i t h m o d e r a t e n o m i n a l s h e a r s t r e s s e s , i.e.
5 . 6 JfJ>. T h e s p e c i m e n e x c e e d e d i t s ul timat e de sign c a p a c i t y in e a c h of the " p l a s t i c " c y c l e s a n d c o u l d not be de s t r o y e d as c o n t e m p l a t e d be c a u s e the l o a d i n g f r a m e w a s u n a b l e to mat ch the s p e c i m e n • s d u c t i l i t y . Fig. ^ b s h o w s W a l l 2 0 2 at t h i s s t a g e . It is p r o b a b l e that f u r t h e r l o a d r e v e r s a l s w o u l d h a v e e v e n t u a l l y c a u s e d a s l i d i n g s h e a r f a i l u r e .
F i g . 5 g i v e s the l o a d ro t a t i o n r e l a t i o n - s h i p for all t h r e e b e a m s . ( R o t a t i o n = l a t e r a l d e f1e c t i o n of w a l l d i v i d e d by it s h e i g h t ) . C y c l e s 1 to ^ s h o w the e l a s t i c r e s p o n s e a n d c y c l e s 7 to 8 i n d i c a t e the e l a s t i c re s p o n s e a f t e r one m o d e r a t e e x c u r s i o n b e y o n d the y i e l d r a n g e in e a c h d i r e c t i o n w a s m a d e . T h e p l a s t i c re s p o n s e of e a c h w a l1 c a n be c o n v e n i e n t l y e xpre s sed by the c u m u l a t ive d u c t i1i t y , r o w 10 in T a b l e I. .Not o n l y w a s W a l l 2 0 1 the l e a s t d u e tile o n e , but it a l s o s h o w e d a d i s t i n c t l o s s of the s t r e n g t h . T h e s u p e r i o r p e r f o r m a n c e of W a l1 2 0 2 s u g g e s t s the two i m p o r t a n t c o n c l u s - i o n s for d e s i g n ;
( 1 ) If a d u c t i l e i . e . , f l e x u r a l f a i l u r e me c h a n i sm is de s i r e d in a low r i s e s h e a r w a l1 t h a n the n o m i n a l s h e a r s t r e s s e s , a s s o c i a t e d w i t h the f l e x u r a l c a p a c i t y of the w a l l , m u s t be m o d e r a t e , say vu = 6 yf£. T h i s is n o r m a l l y not d i f f i c u l t to a c h i e v e . The r e q u i r e m e n t w i l l 1i m i t the f l e x u r a l s t e e l c o n t e n t in the w a l l .
( 2 ) B e c a u se the f l e x u r a l f a i l u r e m e c h a n i sm is a s s o c i a t e d w i t h l a r g e c r a c k s no r e l i a n c e s h o u l d be p l a c e d on the c o n c r e t e in c o n t r i b u t i n g t o w a r d s s h e a r s t r e n g t h , C o n s e q u e n t l y , the w h o l e of shear f o r c e s h o u l d be r e s i s t e d by w e b r e i n f o r c e m e n t •
T h e l i k e l y c r a c k p a t t e r on a 1o w r i s e s h e a r w a l1 is s k e t c h e d in F i g . 8. F r o m c o n s i d e r a t i o n s of e q u i l i b r i u m of the t r i a n g u l a r f r e e b o d y
m a r k e d 1, it is e v i d e n t that h o r i z o n t a l s t e e l is r e q u i r e d to r e s i s t the s h e a r i n g s t r e s s a p p l i e d to it s top e d g e . In the free b o d y b o u n d by two d i a g o n a l c r a c k s a n d m a r k e d 2 , on the o t h e r h a n d , v e r t i c a l f o r c e s of e q u a l i n t e n s i t y n e e d be g e n e r a t e d in o r d e r to d e v e l o p the n e c e s s a r y d i a g o n a l compre s s i o n . H e n c e , t h e v e r t i c a l w e b s t e e l n e e d be the same a s the h o r i z o n t a l s t e e l . T h i s is r e f e r r e d to a s s h e a r r e i n f o r c e m e n t e v e n t h o u g h its p r i n c i p a l r o l e is to r e s i s t the m o m e n t w h i c h t e n d s to o v e r t u r n f r e e b o d y 2 . A p p r o p r i a t e r e c o m m e n d a t i o n s for t h i s h a v e n o w b e e n i n c o r p o r a t e d in the A C I C o d e '2 0' .
2 . 3 F l a n g e d C a n t i l e v e r S h e a r W a l l s
T h e r e is no r e a s o n to e x p e c t that f l a n g e d s h e a r w a l1s w o u l d b e h a v e d i f f e r e n t l y f r o m t h o s e wi th r e c t a n g u l a r s e c t i o n s . S t e e l p r o v i d e d in e a c h of the f l a n g e s s h o u l d en sure a d e q u a t e f l e x u r a l d u c t i l i t y . A d d i t ional s a f e g u a r d s , in f o r m s of l o w e r u n d e r - c a p a c i t y f a c t o r s , m a y be a p p r o p r i a t e w h e n , t h e w h o l e t h i c k n e s s of a w i d e f l a n g e is in c o m p r e s s i o n !0) .
T h e f l a n g e s w i l l c o n s i d e r a b l y b o o s t t h e m o m e n t c a p a c i t y of t a l l c a n t i l e v e r s h e a r w a l1s , h e n c e the s h e a r re si s t a n c e of the ir w e b s m a y b e c o m e a c r i t i c a l de s i g n item. T h e l a r g e d e m a n d f o r w e b r e i n f o r c e m e n t can be c o n v e n i e n t l y m e t by u s i n g s t e e l w i t h h i g h e r y i e l d s t r e n g t h (H Y 6 0 ) .
91
In a w e l l d e s i g n e d s h e a r w a l l the s t i r r u p r e i n - f o r c e m e n t is not e x p e c t e d to y i e l d at any stage o f a s e i s m i c d i s t u r b a n c e .
2 , k F l e x u r e _ _: L_ A x i a l L o a d I n t e r a c t i o n in S h e a r W a l l s
F l a n g e d , a n g l e or c h a n n e l s h a p e d c r o s s s e c t i o n s o f t e n a p p e a r in s h e a r w a l l s f o r m i n g the c o r e s of m u l t i s t o r e y b u i l d i n g s. S u c h s e c t i o n s w i l l b e s u b j e c t e d to a x i a l f o r c e s , i n c l u d i n g t e n s i o n , a s w e l l a s b e n d i n g m o m e n t s a b o u t e i t h e r of the p r i n c i p a l a x e s . B e c a u s e the o v e r a l l d i m e n s i o n s of c o r e s r e m a i n s e n s i b l y c o n s t a n t o v e r the full h e i g h t of the s t r u c t u r e a n d b e c a u se m o r e t h a n one l o a d c o m b i n a t i o n m a y h a v e to be c o n s i d e r e d it c o u l d be a d v a n t a g e o u s to c o n s t r u c t the a p p r o p r i a t e i n t e r a c t i o n c u r v e s f o r m o m e n t and a x i a l f o r c e . W i t h the a id of a s m a l l c o m p u t e r thi s is a r e l a t i v e l y e a s y t a s k ^ ? ) .
F i g * ? s h o w s the a x i a l f o r c e - m o m e n t ' i n t e r - a c t i o n for two c h a n n e l s h a p e d s h e a r w a l l s e c t - i o n s . T h e r e i n f o r c e m e n t is a s s u m e d to be u n i - f o r m l y d i s t r i b u t e d w i t h i n the t h i c k n e ss of the w a l l sect ion. C l e a r l y the sect ion is l e s s s e n s i t i v e to c o m p r e s s i o n a n d b e n d i n g m o m e n t .causing c o m p r e s s i o n in the " w e b " of the c h a n n e l
s e c t i o n t h a n to a m o m e n t w h i ch i n d u c e s com- p r e s s i o n at the t i p of the " f l a n g e s " . T h e p o s i t i o n of the n e u t r a l a x i s , at u l t i m a t e , C, r e l a t i v e to t h e c o m p r e s s i o n e d g e is e x p r e s s e d b y the C/D r a t i o, w h e r e D is t h e " f l a n g e "
w i d t h . T h e b a l a n c e d f a i l u r e m e c h a n i s m , c o r r e s p o n d i n g wi th a n a p p r o x i m a t e d u c t i l i t y f a c t o r of u n i t y , is a s s o c i a t e d w i t h a C/D r a t i O o f a p p r o x i m a t e l y 0 . 6 5 in the s e c o n d e x a m p l e . T h e i n t e r a c t i o n c u r v e s a n d the r a d i a l l i n e s i n d i c a t e t h u s to w h a t e x t e n t d o e s the g i v e n l o a d c o m b i n a t i o n l e a d to a d u c t i l e f a i l u r e , s u c h a s w o u l d b e a s s o c i a t e d f o r e x a m p l e w i t h c/D « 0 . 1 . It i s a p p a r e n t f rom the se c u r v e s t h a t o u r e x p e r i e nc e s wi th r e c t a n g u l a r s e c t i o n s c a n n o t be s i m p l y t r a n s p o s e d o v e r s h e a r w a l l s e c t i o n s . T h e a v a i l a b l e r o t a t i o n a l d u c t i l i t y o f the c r i t i c a l s h e a r w a l1 s e c t i o n of g i v e n g e o m e t r y is e v i d e n t l y s t r o n g l y a f f e c t e d b y t h e s e n s e s of t h e m o m e n t s a n d the a x i a l f o r c e s i . e . net c o m p r e s s i o n or t e n s i o n .
2 . 5 H o r i z o n t a l C o n s t r u c t i o n J o i n t s A e r o ss S h e a r W a l l s
E a r t h q u a k e d a m a g e in s h e a r w a l1s w a s o f ten o b s e r v e d at c o n s t r u e t ion j oint s, a l o n g w h i c h s i i d i n g m o v e m e n t s o c c u r r e d . T h e se are m o r e c o m m o n in l o w s h e a r w a l l s w h i c h c a r r y r e l a t i v e l y s m a l l g r a v i t y l o a d s . H o w e v e r , t h e y h a v e a l s o b e e n o b s e r v e d in m u l t i - s t o r e y s t r u c t u r e s (see F i g . 1 ^ ) .
A w e l l p r e p a r e d c o n s t r u c t i o n j o i n t w i l l p r o d u c e a r o u g h s u r f a c e a n d h e n c e it w i l l be a b l e to t r a n s f e r s h e a r i n g f o r c e s by m e a n s of a g g r e g a t e i n t e r l o c k . T h i s m e c h a n i s m is a l s o r e f e r r e d t o a s " s h e a r f r i c t i o n " . A s s o o n a s a s i i d i n g m o v e m e n t c o m m e n c e s , a c r a c k , f o r m e d a l o n g a c o n s t r u c t i o n j o i n t , w i l1 t e n d to w i d e n . A n y m e a n s w h i c h w i l1 p r e v e n t or d e l a y the o p e n - i n g of s u c h a c r a c k w i l l n e c e s s a r i l y b o o s t the s h e a r t r a n s f er c a p a c i t y of a c o n s t r u c t i o n j o i n t . A n y c o m p r e s s i o n f o r c e a c t i n g a c r o s s the j o i n t , e i t h e r a p p l i e d e x t e r n a l l y a s a l o a d or g e n e r a t e d i n t e r n a l l y by m e a n s of v e r t i c a l re i n f o r c e m e n t , c a n b e u t i l i s e d f or t h i s p u r p o s e . It s c l a m p i n g e f f e c t is m a d e u s e of in t h e t r a d i o n a l s e n s e of t h e f r i c t i o n c o n c e p t . W h e n t h e e x t e r n a l com-
pre s s i o n is i n s u f f i c i e n t or a b s e n t the c l a m p i n g f o r c e g e n e r a t e d by the v e r t i c a l r e i n f o r c e m e n t , w h i c h is s t r a i n e d a c r o s s the c r a c k w h e n it w i d e n s , can e n s u r e that s u b s t a n t i a l s h e a r i n g f o r c e s c a n be t r a n s f e r r e d .
A r e c e n t re s e a r c h p r o j e c t ! ^ , s t i m u l a t e d by the N e w Z e a l a n d Soc ie ty of E a r t h q u a k e E n g i n e e r i n g , r e v e a l e d c e r t a i n a s p e c t s w h i c h are b r i e f l y s u m m a r i s e d . Fir stly, h o w e v e r , the b a s i c c r i t e r i a of the d e s i r e d b e h a v i o u r of h o r i z o n t a l c o n s t r u c t i o n j o i n t s a c r o s s s h e a r w a l1s , a s s e e n by the a u t h o r , may be s t a t e d .
(a) If po s s i b l e , a con st rue t i o n j o i n t s h o u l d not form the w e a k e s t 1i n k i n t h e c h a i n of s t r u c t u r a l r e s i s t a n c e . S h e a r i s t h e g o v e r n - ing m o d e of l o a d t r a n s f e r . It is t h e r e fore n e c e s s a r y that the shear s t r e n g t h of s u c h a j o i n t be e q u a l b u t p r e f e r a b l y l a r g e r t h a n the s h e a r ( d i a g o n a l t e n s i o n ) s t r e n g t h of the s h e a r w a l1 i t s e l f .
(b) U n d e r m o d e r a t e 1o a d i n g s h e a r d i s p l a c e - m e n t s a n d c r a c k w i d e n i n g s h o u l d b e s m a1 1 e n o u g h not to i n t e r f e r e w i t h f u n c t i o n a l r e q u i r e m e n t s or r e q u i r e r e p a i r .
(c) T h e r e q u i r e d s u r f a c e p r e p a r a t ion s h o u l d be af f o r d e d by s i m p l e st p o s s i b l e m e a n s .
T h e c u r r e n t r e c o m m e n d e d u p p e r l i m i t s t r e s s of s h e a r t r a n s f e r by a g g r e g a t e int e r l o c k is 0 . 2 fc or 8 0 0 p s i2 0. T h i s is in e x c e s s of the l i m i t i n g shear ( d i a g o n a l ) stre ss o f 1 0/ f £ ( p s i ) a c c e p t e d for s t r u c t u r a l m e m b e r s f o r a l l c o n c r e t e w i t h a c r u s h i n g s t r e n g t h in e x c e s s of 2 5 0 0 p s i . H e n c e f or the u s u a l s h e a r w a l l s t r u e t u r e t h e n o m i n a l shear s t r e s s c a p a c i t y of t h e j o i n t n e e d o n l y be in e x c e s s of lO^fJ ( p s i ) .
T h e c l a m p i n g f o r c e at y i e l d o f t h e v e r t i c a l s t e e l is Tv = Avfy - pvb d fv, H e n c e t h e f r i c t i o n - al s h e a r s t r e s s to be r e l i e d on be c o m e s vu j = /*TVIbd =f &Pvf y > w h e r e the v a l u e of the f r i c t i o n
c o e f f i c i e n t , ^ , can be t a k e n c o n s e r v a t i v e l y a s u n i t y . W h e n w e l l p r e p a r e d c l e a n , r o u g h s u r f a c e s w i t h b o n d are a s s u r e d Jbi = 1.^, r e c o m m e n d e d f o r m o n o l i t h i c a l l y cast c o n c r e t e , c o u l d be u s e d . T h i s w a s e v i d e n t f rom the e x p e r i m e n t s.
De s i g n e r s o f t e n as s u m e d that the p r i n c i p a l m o d e of shear t r a n s f e r is d o w e l a c t i o n . T h e se a n d o t h e r t e s t s h a v e c o n f i r m e d , h o w e v e r , t h a t r e l a t i v e l y l a r g e d i s p l a c e m e n t s are r e q u i r e d to f u l l y m o b i l i s e d o w e l a c t i o n . T h i s a p p l i e s p a r t i c u l a r l y to s m a l l size bar s. In the u s e f u l r a n g e of shear di s p l a c e m e n t s, say o n e t h i r d of a s m a l l c r a c k w i d t h , ( 0 . 0 l / 3 = 0 . 0 0 3 3 i n . ) the c o n t r i b u t i o n of the v e r t i c a l r e i n f o r c e m e n t a s d o w e l s is n e g l i g i b l e . T h i s is e v i d e n t f r o m F i g . 8. O n l y a f t e r t h e b r e a k d o w n of a d h e sive b o n d b e t w e e n the two f a c e s of a co n s t r u c t i o n
j o i n t or the y i e l d i n g of the v e r t i c a l r e i n f o r c e - m e n t d o d o w e l s s i g n i f i c a n t l y r e s i s t s h e a r f o r c e s.
T h i s w a s o b s e r v e d at a s h e a r slip in e x c e ss of
one t e n t h of a n i n c h . ! F i g . 9 s h o w s the r e s p o n s e of const r u c t i o n
j o i n t s f o r v a r i o u s s u r f a c e p r e p a r a t i o n s a n d a s t e e l c o n t e n t of 0 . 6 9 % , T h e l o w e s t c u r v e s h o w s the response of ^ in. d o w e l s . It is s e e n t h a t j o i n t s w i t h d i f f er e n t l y p r e p a red r o u g h s u r f a c e s d e v e l o p e d an u l t i m a t e s h e a r s t r e s s m u c h in e x c e s s of t h e s u g g e s t e d v a l u e of 1 . 0 pvfy. T h e e f f e c t of l o s s of b o n d , s u c h a s c o u l d be c a u s e d by a t h i n l a y e r of l a i t a n c e , is s h o w n b y the t h i r d c u r v e f r o m the b o t t o m . T h e s i g n i f i c a n t f e a t u r e
is the l a r g e r s l i p at l o w l o a d s . S u b s e q u e n t tests i n d i c a t e d that t h e r e c o m m e n d e d shear s t r e n g t h c a n b e s u s t a i n e d s e v e r a l t i m e s d u r i n g a l t e r n a t i n g c y c l i c l o a d i n g . T h e apex of e a c h curve in F i g . 9 o c c u r r e d at t h e onset of y i eId- ing in t h e r e i n f o r c e m e n t . It w a s a ls o o b s e r v e d that wi th a v e r t ical r e i n f o r c i n g c o n t e n t equal or m o r e t h a n 0 . 6 9 % s l i d i n g s h e a r f a i l u r e d i d not o c c u r a l o n g the r o u g h b o n d e d c o n s t r u e t i o n joint, but r a t h e r aero ss the o l d c o n c r e te b e l o w the j o i n t . It i s k n o w n that the q u a l i t y of this c o n c r e t e , at the t o p of a p o u r , i s inf e r i o r b e c a u s e of s e d i m e n t a t i o n , w a t e r a n d a i r g a i n . T h i s c o n c r e t e 1a y e r of p o o r q u a l i ty w i l l n o r m a l - ly limit t h e s t r e n g t h of t h e r e g i o n a n d n o t the i n t e r f ace a l o n g the c o n s t r u c t ion j o i n t . It i s sugge sted that the m i n i m u m steel a c r o s s a h o r i z o n t a l const r u c t i o n joint in a shear w a l l be d e t e r m i n e d a s f o l l o w s :
(a) T h e n o m i n a l u l t i m a t e s h e a r s t r e s s
s u s t a i n e d a l o n g a joint is ' v . = (N + A f )/A
J U v y " g
. . . ( l a ) w h e r e N - t h e a x i a l c o m p r e s s i o n r e s u l t ing
f r o m l o a d i n g .
A ^ f = the c l a m p i n g f or ce r e p r e s e n t e d b y the v e r t i c a l steel cros sing the j o i n t .
A ~ h i the g r o s s a r e a of t h e w a l l s e c t i o n .
(b) T h e b e n e f i c i a l e f f e c t of t h e a x i a l c o m p r e s s i o n m u s t not b e o v e r e s t i m a t e d . A r e d u c t i o n of at least 20% s h o u l d b e c o n s i d e r e d to a l l o w f o r n e g a t i v e v e r t i c a l a c c e l e r a t i o n .
(c) B y a s s u m i n g the e f f e c t i v e d e p t h to b e e i g h t t e n t h of the o v e r a l1 d e p t h E q . 1 c a n be r e w r i t t e n t h u s
ju
0.8N + A f
1-1=
0.8 (» + p f > >-1 . 2 5
h d • v y v =... ( l b ) w h e r e v = i s t h e s h e a r ( d i a g o n a l t e n s i o n )
c a p a c i t y of t h e r e m a i n d e r of the w a l l . H e n c e t h e r e i n f o r c i n g c o n t e n t r e q u i r e d is
p >
( 1 . 2 5
v £-)/fV
yo. 0025 . . . ( 2 )
W i t h a s m a l l a m o u n t of a x i a l c o m p r e s s i o n a n d m o d e r a t e s h e a r s t r e s s e s , vu, t h e m i n i m u m steel c o n t e n t o f 0.25$> w i l l s u f f i c e to e n s u r e a d e q u a t e p r o t e c t i o n of t h e c o n s t r u c t i o n j o i n t . F o r h i g h shear s t r e s s e s s t e e l c o n s i d e r a b l y i n e x c e s s of t h e n o m i n a l q u a n t i t y w i l l b e r e q u i r e d . F i g . 10 s h o w s t h e r e q u i r e m e n t s w h e n fc = U00 0 p s i a n d fv
^0.000 p s i .
(d) It i s i m p o r t a n t that at l e a s t t h i s a m o u n t o f s t e e l b e p r o v i d e d o v e r e a c h foot l e n g t h of w a l1 sect ion. H e a v y re i n f o r c e m e n t , s i t u a t e d n e a r t h e e d g e s of t h e s e c t i o n , w i l l p r o v i d e t h e d e s i r e d c l a m p i n g f o r c e in i t s i m m e d i a t e v i c i n i t y o n l y .
It i s e m p h a s i z e d t h a t it d o e s n o t seem to m a t t e r w h a t m e t h o d of p r e p a r a t i o n i s u s e d f o r a h o r i z o n t a l c o n s t r u c t i o n j o i n t a s l o n g a s a r o u g h s u r f a c e w i t h a c o a r s e t e x t u r e i s p r o d u c e d .
to whi ch the f r e s h l y p l a c e d c o n c r e t e c a n b o n d . W i t h a d e q u a t e vert ical re i n f o r c e m e n t, a s o u t - l i n e d a b o v e , a shear s t r e n g t h in e x c e s s of the d i a g o n a l t e n s i o n s t r e n g t h of the she ar w a l l s t r u e t u r e c a n be a s s u r e d .
3 . T H E I N T E R A C T I O N OF S H E A R W A L L S W I T H E A C H O T H E R A N D W I T H F R A M E S
Only s e l d o m will a s i n g l e c a n t i l e v e r w a l1 be c a l l e d u p o n to r e s i s t t h e whole of the l a t e r a l l o a d a c t i n g u p o n a m u l t i s t o r e y s t r u e t - ur e. It i s m o r e 1i k e l y that a n u m b e r of s u c h w a l l s w i1 1 share in t h e t o t a l load r e s i s t a n c e . In t h e m a j o r i t y of m u l t i s t o r e y b u i l d i n g s s h e a r w a l1s w i1 1 o c c u r a r o u n d t h e s e r v i c e core a n d r i g i d j o i n t e d f r a m e s a r e l i k e l y to c a r r y t h e g r a v i t y l o a d over the r e m a i n d e r o f t h e f l o o r .
1 1 i s b e y o n d t h e s c o p e of t h i s r e v i e w to di s c u s s the t e c h n i q u e s of a n a l y s i s whi ch a re a v a i l a b l e ( ^ »* 5 »t o d e a l w i t h t h e p r o b l e m of i n t e r a c t i o n b e t w e e n d i f f e r e n t lat eral l o a d re s i s t i n g s u b a s s e m b l i e s . Howe v e r , o n o n e a spe ct, whi ch m o s t de s i g n e r s a r e a w a r e o f , a w a r n i n g m a y b e w a r r a n t e d ,
T h e r e s p o n s e of r i g i d j o i n t e d f r a m e s a n d c a n t i l e v e r shear w a l l s t o l a t e r a l l o a d s c a n be so m a r k e d l y d i f f e r e n t , p a r t i c u l a r l y i n t h e u p p e r s t o r e y s, that u n d e s i r a b l e i n t e r a c t i o n m a y e n s u e . T h e t w o type s of s t r u c t u r e s m a y w o r k a g a i n s t e a c h o t h e r , a n d an u n u s u a l l y l a r g e d u c t i l i t y d e m a n d m a y p o s s i b l y r e s u l t in the p r o c e s s of d e v e l o p i n g t h e u l t i m a t e st r e n g t h of t h e w h o l e s t r u c t u r e * 9 ) .
k. S H E A R W A L L S W I T H O P E N I N G S
O p e n i n g s f o r w i n d o w s a n d d o o r s i n s h e a r w a l1s c a n n o t b e a v o i d e d . H o w e v e r , t h e de s i g n e r s h o u l d be a w a r e at t h e e a r l y s t a g e s o f t h e p l a n n i n g that t h e a r r a n g e m e n t of t h e s e m u s t e n s u r e a rat i onal s t r u c t u r e . T h e b e h a v i o u r of t h e s e c a n b e a s s e s sed b y b a r e i n s p e c t i o n . I r r a t i o n a l s h e a r w a l l s o n the o t h e r h a n d d e f y an a t t a c k b y t h e c o n v e n t i o n a l t e c h n i q u e s of s t r u e t u r a l a n a l y s i s • M o r e o v e r such s t r u c t u r e s o f t e n i n v i t e d i s a s t e r b y c o n c e n t r a t i n g e n e r g y a b s o r b t ion at a f e w l o c a l i t i e s w h i c h a r e u n a b l e to s u p p l y t h e u n u s u a l l y h i g h d e m a n d f o r d u c t i l i t y . A f e w e x a m p l e s of such s t r u c t u r e s a n d the ir r e s p o n s e to ear t h q u a k e s w e r e g i v e n e l s e w h e r e ( ° s 9 / e
5- C O U P L E D S H E A R W A L L S
T h e c o m m o n shear w a l1 w i l l c o n t a i n o n e or m o r e c o l u m n s or r o w s of r e g u l a r o p e n i n g s . T h e se s t r u c t u r e s l e n d t h e m s e l v e s to rat i o n a l a n a l y s i s . T h e p r o t o t y p e s t r u c t u r e is t h e s h e a r c o r e of a m u l t i s t o r e y b u i l d i n g in w h i c h t w o or m o r e f l a n g e d c a n t i l e v e r w a l l s a r e i n t e r -
c o n n e c t e d by short a n d r e l a t ively d e e p " c o u p l i n g b e a m s . (See F i g . 1 1 ) . T h e a n a l y s i s of the se r e q u i r e s a l s o the c o n s i d e r a t i o n o f a x i a l d e f o r m - a t i o n s in the w a l l s a n d s h e a r d i s t o r t i o n s of t h e c o u p l i n g b e a m s . F o r t h i s r e a s o n c o n v e n t - i o n a l h a n d c o m p u t a t i o n s a r e i n a d e q u a t e .
H o w e v e r , b y s u i t a b l e m o d e l l i n g of t h e s t r u c t u r e the a n a l y s i s c a n b e c o n s i d e r a b l y s i m p l i f i e d f o r hand or c o m p u t e r p r o c e s s i n g . T h e e s s e n t i a l i n g r e d i e n t of thi s t e c h n i q u e is t h e r e p l a c e m e n t of t h e d i s c r e t e b e a m s b y a c o n t i n u o u s set of e l a s t i c 11 l a m i n a e " , he nee t h e r e f e r e n c e " l a m i n a r a n a l y s i s " or " c o n t i n u u m a p p r o a c h " . T h e t e c h n i q u h a s b e e n w e l l c o v e r e d in t h e l i t e r a t u r e ( 1 0» 1 1»
1 2 , 1 3 ) a n (i d e t a i 1 s of it a r e b e y o n d t h e s c o p e of t h i s r e v i e w . T h e m a t h e m a t i c a l m o d e l a n d t h e
93 e q u i v a l e n t p r o p e r ties are shown in F i g . 1 1 *
5• 1 T h e E l a s t ic Re sponse of C o u p l e d S h e a r W a l l s In s i m i l a r i t y to a c a s t e l l a t e d be am the c o u p l e d s h e a r w a1 1 s t r u c t u r e r e s i s t s the e x t e r - nal o v e r t u r n i n g m o m e n t , M , (with r e f e r e n c e to F i g .
1 1 )
by( a ) i n t e r n a l m o m e n t s M l and M2• g e n e r at ed in W a l l 1 and W a l1 2 r e s p e c t i v e l y and by
{ b ) a x i a l f o r c e s, T, g e n e r a t e d in e a c h w a l l a n d o p e r a t i n g on a le ver arm, 1. T h e a x i a l f o r c e , w h i c h at any l e v e l is the sum of all the s h e a r - i n g f o r c e s a c r o s s the c o u p l i n g be a m s or l a m i n a e s i t u a t e d a b o v e that l e v e l , g e n e r a t e s t e n s i o n in one wall a n d compre ssion in the o t h e r . The a i m of a l a m i n a r a n a l y s i s is to d e t e r m i n e the r e l a t i v e m a g n i tude s of the se c o m p o n e n t s whi ch a r e r e l a t e d to e a c h o t h e r by the s i m p l e e q u i l i b r i u m s t a t e m e n t ;
Mq a M + Mg + IT ...
(3)
T h e i n t e r p l a y b e t w e e n the m o d e s of i n t e r n a l m o m e n t r e s i s t a n c e , Eq. 3• d e p e n d s on t h e
s t r e n g t h a n d s t i f f n e s s of the c o u p l i n g s y s t e m . C l e a r l y it i s m o r e e f f i c i e n t to r e s i s t the e x t e r n a l m o m e nt p r e d o m i n a n t l y by i n t e r n a l f o r c e s , T, w h i c h o p e r a t e w i t h a l a r g e l e v e r a r m , 1, t h a n by c o m p o n e n t i n t e r n a l m o m e n t s , M1 a n d M2• T h e r e l a t i v e p r o p o r t i o n s of the con- t r i b u t i o n of the i n t e r n a l c o u p l e , IT, in r e s i s t i n g the e x t e r n a l m o m e n t s , MQ, at variou s l e v e l s of an e l a s t i c 20 s t o r e y s h e a r core are s h o w n in F i g . 12, It is seen that t h e c o u p l i n g i s e f f i c i e n t for the top h a l f of the s t r u c t u r e for all but the s h a l l o w e s t b e a m s . At the b a s e l i t t l e d i f f e r e n c e in b e h a v i o u r is i n d i c a t e d for 2k±n. d e e p or i n f i n i t e l y s t i f f b e a m s . The l a t t e r repre sent the case whe n no di s t o r t i o n s o c c u r in the p r o c e s s of s h e a r t r a n s f e r f r o m one w a l l to a n o t h e r , i.e., a c o n t i n u o u s l i n e a r s t r a i n dis t r i b u t ion o c c u r s a c r o s s the e n t i r e s h e a r w a l l s t r u c t u r e . T h e l o w e f f i c i e n c y of 6 i n . d e e p c o u p l i n g b e a m s s h o w s the a p p r o a c h to t h e othe r 1i m i t ing s i t u a t i o n , w h e n IT = 0, i.e.
w h e n t h e e nt ire e x t e r n a l m o m e n t is r e s i sted by f1e x u r e in the c o m p o n e n t w a l l s , M | a n d M2•
T h e r o l e o f c r a c k i n g in the e l a s t i c b e h a v i o u r of s h e a r w a l l s h a s b e e n e x a m i n e d t h e o r e t i c a l l y * 9) an a e x p e r i m e n t a l l y ( *3» 1 ° ) . B e c a u s e of t h e very l a r g e d i f f e r e n c e s b e t w e e n t h e s t i f f n e s ses of the c o m p o n e n t s a n d the d r a s t i c l o s s of s t i f f n e s s in t h e c o u p l i n g s y s t e m a f t e r d i a g o n a l c r a c k i n g , d e f l e c t i o n a n d w a l l m o m e n t (Mi or M2) i n c r e a s e s of 75% to 100%
h a v e b e e n o b t a i n e d in case s t u d i e s, w h e n a l l o w a n c e w a s m a d e for c r a c k i n g .
5-2 T h e E l a s t o - P l a s t i c R e s p o n s e of C o u p l e d S h e a r Walls""" ' ~ ~ "
T h e u l t i m a t e s t r e n g t h of c o u p l e d she ar w a l l s , s u b j e c t e d to s e i s m i c type of l a t e r a l 1o a d i n g , is a t t a i n e d w h e n a s t a t i c a l l y a d m i s s - i b l e m e c h a n i s m is f o r m e d in w h i c h e a c h of the r e q u i r e d p l a s t i c h i n g e s p o s s e s s e s a d e q u a t e r o t a t i o n a l c a p a c i t y . T w o h i n g e s in e a c h c o u p - 1 i n g b e a m are r e q u i r e d to t e r m i n a t e its a b i l i t y t o a c c e p t f u r t h e r s h e a r . In a d d i t i o n , one p l a s t i c h i n g e n e e d be d e v e l o p e d in e a c h of the c a n t i l e v e r w a l l s , n o r m a l l y at t h e i r b a s e , to c o m p l e t e t h e c o l l a p s e m e c h a n i s m . T h e s e q u e n c e o f h i n g e f o r m a t i o n w i l1 d e p e n d u p o n the r e l a t i v e s t r e n g t h and s t i f fne ss of the c o m p o n e n t s.
T h e b e h a v i o u r of some c o u p l e d s h e a r w a l l s , w h i c h w e r e expo sed to s e v e r e e a r t h q u a k e s ,
i n d i c a t e d that all or most c o u p l i n g b e a m s f a i l e d b e f o r e the u l t i m a t e s t r e n g t h of t he c o u p l e d w a l l s was a t t a i n e d. A c l a s s i c e x a m p l e of thi s is shown in F i g . 13 a n d F i g . 1^. T h i s i n d i c a t e s that d u c t i l i t y d e m a n d s in e x c e s s of the se b e a m s * c a p a c i t y may h a v e e x i s t e d . In t h e f o l l o w i n g , an e x a m p l e will be u s e d to show ho w a n d w h e r e thes e e x c e s s i v e d u c t i l i t i e s may d e v e l o p . 5.3 A D e t e r m i n i s t i c D e s i g n A p p r o a c h
One m a y a d o p t a d e t e r m i n i s t i c a p p r o a c h and di c t a t e a d e s i r a b l e s e q u e n c e of p l a s t ic h i n g e f o r m a t i o n s f o r a g i v e n lat eral s t a t i c l o a d whi ch is a s s u m e d to i n c r e a s e m o n o t o n i c a l l y to i t s ult imate v a l u e . T h e p l a s t i c h i n g e s in t h e w a l l s , w h i c h a r e a l s o m a j o r g r a v i t y l o a d c a r r y - ing unit s, should be the last one s to f o r m . E a c h of the two w a l I s is s u b j e c t e d to a x i a l f o r c e s , g e n e r a t e d b y the lat eral l o a d , whi c h c o u l d r e s t r i c t t h e i r d u c t i l i t i e s . T h e r e f o r e , the a t t e m p t of d e l a y i n g h i n g e f o r m a t i o n in the w a l l s s h o u l d a l s o re suit in b e n e f i t s in t e r m s of r e d u c e d d u c t ility d e m a n d .
T h e maj or s t e p s of a d e s i g n a p p r o a c h b a s e d on the se p r i n c i p l e s are o u t1i n e d v e r y b r i e f l y h e r e wi th the a i d of an e x a m p l e s t r u c t - u r e . T h e b a s i c d i m e n sions of the core of a 20 s t o r e y b u i l d i n g are tho se shown in F i g . 1 2 . T h e l a t e r a l c o d e l o a d c o n s i s t s of a t r i a n g u l a r b l o c k , W = 6 0 0 K , a n d a point l o a d at r o o f l e v e l A W = 8 0 K , a s s k e t c h e d in F i g . 1 2 . T h e e l a s t o - p l a s t i c a n a l y s i s is b a s e d on the m o d e l of l a m i n a r c o u p l i n g * ) . T h e b e a m s a n d the w a l l s are as s u m e d to c o n f o r m wi th i d e a l i sed e l a s t o - p l a s t i c b i l i n e a r b e h a v i o u r d u r i n g the p r o c e s s of p l a s t i c h i n g e f o r m a t i o n .
(i ) T h e stif fne ss p r o p e r t i e s , b a s e d on t h e g r o s s area of the c o m p o n e n t s n e e d be e s t a b l i s h e d f i r s t . T h e l o s s of s t i f f n e ss cau sed by c r a c k i n g can be e s t i m a t e d to e n s u r e a m o r e r e a l i s t i c e v a l u a t i o n of the e l a s t i c re s p o n s e . T y p i c a l r e d u c t i o n mult iplie r s, a p p l i c a b l e to the p r o p e r t i e s of t h e c o m p o n e n t s in the u n c r a c k e d s t a t e , are as f o l l o w s :
S e c o n d m o m e n t of a r e a of t e n s i o n w a l l 0.50 A r e a of t e n s i o n w a l l 0 . ?0 F l e x u r a l s t i f f n e ss of c o u p l i n g b e a m s ,
w h i c h a l s o i n c l u d e s an a l l o w a n c e f o r 0. 20 s h e a r d i s t o r t i o n
P r o p e r t i e s of c o m p r e s s i o n w a l l 1.00 W i t h t h e s e p r o p e r t i e s the i n t e r n a l act i o n s
c a n be d e t e r m i n e d . T h e t y p i c a l d i s t r i b u t i o n of t h e l a m i n a r s h e a r f o r c e , q, l a m i n a r r o t a t i o n , -8-, in t e r m s of the l a m i n a r y i e l d r o t a t i o n , -By, a n d the d e f l e c t i o n at r o o f l e v e l , y, a r e s h o w n f o r the e x a m p l e s t r u c t u r e in F i g . 15a. T h i s is S T A G E 1. A l l r e s u l t s are m a r k e d in F i g . 15 a c c o r d i n g l y w i t h the c i r c l e d n u m b e r 1.
( i i ) A f t e r an e x a m i n a t i o n of the l a m i n a r s h e a r i n t e n s i t y it m a y be d e c i d e d h o w m u c h o v e r l o a d w o u l d be a c c e p t a b l e b e f o r e the y i e l d ( u l t i m a t e ) s t r e n g t h of the c r i t i c a l l y s i t u a t e d l a m i n a or c o u p l i n g b e a m is a t t a i n e d . In t h e e x a m p l e 30% was c h o s e n , T h e r e f o r e , at up to
1 , 3 t i m e s the c o d e l o a d a l1 c o m p o n e n t s a r e a s s u m e d to r e m a i n in t h e e l a s t i c r a n g e . T h i s is S T A G E 2 and it is m a r k e d a c c o r d i n g l y in F i g . 1 5 a . A s thi s is the e l a s t i c l i m i t of t h e s t r u c t u r e , i.e. the p o i n t of f i r s t y i e l d ,
r
94
the o v e r a l l d u c t i l i t y can be expre ssed in terms of the d e f l e c t i o n at roof l e v e l at this 1oad, Ye-
T h e r e q u i r e d c a p a c i t y o f the c o u p l i n g beam at x ~ 0. 7 H is n o w e st a b l i shed a n d
c o n s e q u e n t l y the a p p r o p r i a t e r e i n f o r c e m e n t can be d e t e r m i n e d . D e t a i l s of t h i s are d i s c u s s e d in a s u b s e q u e n t s e c t i o n . A c c o r d i n g to a d e c i s i o n , m a d e in t h is e x a m p l e , a l1 b e a m s t h r o u g h o u t t h e he ight w i l l r e c e i v e the same r e i n f o r c e m e n t .
( i i i ) A n a d d i t ional l o a d , A W ( 1 +A) , c a n n o w be i m p o s e d so t h a t the u l t i m a t e - s t r e n g t h of at l e a s t the u p p e r 90% of the l a m i n a e , qu, is d e v e l o p e d . T o e n s u r e this the r e l a t i v e s t i f f - n e s s e s of t h e c o m p o n e n t s n e e d be t a k e n into a c c o u n t 'l QK A t this S T A G E 3 of the l o a d i n g the w a l l s a r e s t i l l e l a s t i c . H o w e v e r , the c o u p l i n g b e a m s w e r e f o r c e d t h r o u g h v a r i o u s a m o u n t s of p l a s t ic di s to r t i o n s . T h i s i s m e a s u r e d by the i m p o s e d d u c t i l i t y f a c t o r as s h o w n by the a p p r o p r i a t e c u r v e in F i g . 15 a . It s h o w s that a b e a m s i t u a t e d at x = 0 . 6 H
w i l l h a v e to s u p p l y a d u c t i l i t y f a c t o r a l i t t l e over 3s w h e r e a s the t o p m o s t l a m i n a just y i e l d s .
T h e m a x i m u m a x i a l load g e n e r a t e d in the w a l l s by t h i s e x t e r n a l l o a d of ( 1 . 3 + A ) ( 1 + A ) W i s d e r i v e d f r o m t h e s u m m a t i o n of a l l l a m i n a r s h e a r s . T h i s i s Tu » 0 . 9 5 H qu. In a b s e n c e of s t r a i n h a r d e n i n g t h i s v a l u e c a n n o t b e e x c e e d e d d u r i n g f u t u r e l o a d i n g . In the e x a m p l e s t r u c t u r e T ^ = 2 0 0 0 K .
T h e g r a v i t y l o a d on t h e s t r u c t u r e is k n o w n a n d t h u s t h e u l t i m a t e i n t e n s i t y of a x i a l f o r c e in b o t h w a l l s c a n n o w be d e t e r m i n e d . U s u a l l y Tu is t he m o s t s i g n i f i c a n t c o m p o n e n t of the a x i a l f o r c e . In e v a l u a t i n g the g r a v i t y l o a d e f f e c t d u e a l l o w a n c e n e e d be m a d e f o r p o s i t i v e a n d n e g a t i v e v e r t i c a l a c c e l e r a t i o n so a s to o b t a i n t h e m o s t a d v e r s e c o m b i n a t i o n of a x i a l l o a d i n g s e p a r a t e l y f o r e a c h w a l l .
( i v ) F o r a n y s u b s e q u e n t a n d a d d i t i o n a l l o a d i n g the t w o w a l l s c a n be a s s u m e d to act a s e l a s t i c i n t e r c o n n e c t e d c a n t i l e v e r s till the f l e x u r a l c a p a c i t y of o n e i s r e a c h e d . In t h e c a s e of t h e e x a m p l e it is a s s u m e d that a n i n c r e m e n t a l l o a d o f A* ( 1 + A) W , e q u a l to % of the o r i g i n a l l o a d o n the c o u p l e d s h e a r w a l l s t r u c t u r e , c a n be a p p l i e d w h e n the f l e x u r a l c a p a c i t y of W a l l 1, Mx ? u, i s a t t a i n e d in the p r e s e n c e of a n a x i a l f o r c e , d e t e r m i n e d in the p r e v i o u s s t a g e . A l l i n f o r m a t i o n r e l a t e d to this S T A G E k of t h e i d e a l i s e d b e h a v i o u r is a c c o r d i n g l y l a b e l l e d in F i g . 1 5 a .
(v) T h e f i n a l l o a d i n c r e m e n t to be a p p l i e d a g a i n st t h e n e a r l y f u l l y p l a s t i c s t r u c t u r e , A " ( 1 + A) W , i s a s s u m e d to d e v e l o p the u l t i m a t e f l e x u r a l c a p a c i t y of W a l l 2, M2, u > s i m u l t a n e - o u s l y w i t h the m a x i m u m c o m p r e s s i o n e x e r t e d on that w a l l • T h e v a l u e of A m h a s a g a i n been t a k e n 0 . 0 5 in t h e e x a m p l e . In thi s S t a g e 5 the c o l l a p s e m e c h a n i s m is a s s u m e d to h a v e b e e n c o m p l e t e d . A n y a d d i t onal t i l t i n g o f the
s t r u c t u r e w i l l a c c o r d i n g l y c a u s e p l a s t i c r o t a t i o n s in a l l p r e v i o u s l y f o r m e d p l a s t i c h i n g e s .
A l l i n f o r m a t i o n f o r t h e d e s i g n of t h e s h e a r w a l l s e c t i o n s is n o w a v a i l a b l e , i . e .
pl , u , Ml , u , P2,u > M2, u ' w i t h the u se of
a p p r o p r i a t e i n t e r a c t i o n d i a g r a m s or f r o m f i r s t p r i n c i p l e s the r e q u i r e d steel c o n t e n t s , p a n d
p2, a r e d e t e r m i n e d . C l e a r l y , in a s y m m e t r i c a l s t r u c t u r e , such as s h o w n in the e x a m p l e , t h e r e c a n n o t be d i f f e r e n t r e i n f o r c e m e n t r e q u i r e m e n t s for e a c h of the w a l l s . W h i c h e v e r is l a r g e r of p1 or p2 is l i k e l y to b e a d o p t e d f o r bo th w a l l s . T h e c a p a c i t y of the s t r u c t u r e is t h u s i n c r e a s e d a n d t h e small change in b e h a v i o u r c a n be
r e t r a c e d if d e s i r e d .
T h e total l a t e r a l l o a d re s i s t a n c e is n o w k n o w n . Wu = (1 . 3 + A + Ae + A " ) (1+ A) W i n the e x a m p l e . H e n c e the a v a i l a b l e l o a d f a c t o r at the end of the first t r i a l i s f o u n d , Wu/ w . In c a s e of the e x a m p l e s t r u c t u r e t h i s is 1 1 3 0 / 6 0 0 = 1 . 8 8 . A s no a l l o w a n c e h a s be en m a d e f o r u n d e r c a p a c i t y f a c t o r s , </>, so far t h i s l o a d f a c t o r a p p e a r s to be a d e q u a t e . C h a n g e s of d i m e n s i o n s m a y be m a d e n o w if d e s i r e d a n d w i t h t h e s e the p r e v i o u s de s i g n s t e p s c o u l d be r e p e a t - ed if n e c e s s a r y .
( v i ) In o r d e r to e v a l u a t e the d u c t i l i t y d e m a n d f o r th e c o u p l i n g b e a m s , w h i c h i s o n e of t h e p r i n c i p a l a i m s o f thi s p r o c e s s, t h e r e q u i r e d o v e r a l l d u c t i l i t y m u s t be i m p o s e d on the st r u c t u r e . In the e x a m p l e s t r u c t u r e an i n e l a s t i c r o o f d i s p l a c e m e n t of 3 t i m e s that at the e l a s t i c l i m i t , S t a g e 2, h a s b e e n i m p o s e d in a c c o r d a n c e w i t h a c u r r e n t l y a c c e p t e d v a l u e of k f o r a d e s i r a b l e o v e r a l l d u c t i l i t y f a c t o r • T h i s i s S T A G E 6 of the b e h a v i o u r . T h e l o a d - d e f l e c t i o n c u r v e in F i g . 1 5a i n d i c a t e s that the d u c t i l i t y f a c t o r s r e q u i r e d for t h e w a l l h i n g e s are a p p r o x i m a t e l y 2 . 5 . T h e c o r r e s p o n d i n g v a l u e for t h e c r i t i c a l l y s i t u a t e d c o u p l i n g b e a m is h o w e v e r 1 1 . 8 . T h i s is of the o r d e r of the m a x i m u m d u c t i l i t i e s a t t a i n e d in c o n v e n t i o n a l l y r e i n f o r c e d test b e a m s w i t h a span to d e p t h r a t i o of t w o ' , T h e d u c t i l i t y d e m a n d of d e e p e r a n d s t i f f e r b e a m s of c o u p l e d s h e a r w a l l s i n c r e a s e s. Ho we v e r, t h e i r p l a s t i c r o t a t i o n a l c a p a c i t y d e c r e a s e s b e c a u s e of i n c r e a s e d d e t r i - m e n t a l s h e a r e f f e c t s ^1 6' .
In the e x a m p l e c h o s e n , a l l c o u p l i n g b e a m s w e r e p r o v i d e d w i t h the s a m e s t r e n g t h c a p a c i t y . H o w e v e r , it is f e a s i b l e that t h e s t r e n g t h o f e a c h b e a m i s m a d e p r o p o r t i o n a l (or a p p r o x i m a t e l y s o ) to t h e e l a s t i c s t r e n g t h d e m a n d . S u c h a n e x a m p l e w a s a l s o c o n s i d e r e d a n d t h e re sult s of the p r e v i o u s l y o u t l i n e d st e p w i se de s i g n a r e g i v e n in F i g . 1 5 b . In t h i s s t r u c t u r e a l l l a m i n a e are a s s u m e d to y i e l d at t h e s a m e t i m e , S t a g e 2 a n d S t a g e 3• It i s s e e n that s m a l l e r d u c t i l i t y d e m a n d s e n s u e d f o r t h e m o r e h e a v i l y r e i n f o r c e d b e a m s in the l o w e r p o r t ion of t h e s t r u c t u r e . F o r the l i g h t e r b e a m s , s i t u a t e d in t h e t o p t h i r d of t h e h e i g h t , h o w e v e r , c o n - s i d e r a b l y l a r g e r d u c t i l i t y d e m a n d s e x i s t . F o r m o s t real c o u p l e d s h e a r w a l l s t r u c t u r e s i n t e r m e d i a t e d u c t i l i t y d e m a n d s f o r t h e c o u p l i n g s y s t e m c a n be e x p e c t e d .
5 . ^ T h e S t r e n g t h of t h e C o u p l e d W a l l s
T h e b e n d i n g m o m e n t p a t t e r n o n o n e of t h e c o u p l e d shear w a l l s w i l 1 n o r m a l l y be s i m i l a r to t h a t of simple c a n t i l e v e r w a l l s . H o w e v e r , b e c a u s e of the c o u p l i n g s y s t e m c o n s i d e r a b l e a x i a l f o r c e s m a y a l s o be p r e sent• T h e s e m a y p r o d u c e net t e n s i o n in o n e of t h e w a 1 1 s . T h e d e s i g n c o n s i d e r a t i o n a p p l i c a b l e to a n y s e c t i o n of a c o u p l e d s h e a r w a l l s a r e t h e r e f o r e no d i f f e r e n t from those o u t l i n e d in 2 . 3 s w h e r e the i n t e r a c t i o n of f l e x u r e a n d a x i a l l o a d w a s c o n s i d e r e d . See a l s o F i g . 7.
5 • 5 T h e S t r e n g t h a n d D u c t i l i t y of C o u p l i n g B e a m s O b s e r v a t i o n s of e a r t h q u a k e d a m a g e h a v e
