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Investigation of continuity in joints between precast and "cast in place" reinforced concrete members Kratz, Rolf D. 1970

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INVESTIGATION OF CONTINUITY IN JOINTS BETWEEN PRECAST AND "CAST IN P L A C E " REINFORCED CONCRETE MEMBERS by ROLF D. KRATZ B. Sc. (Civil Engineering) University of Capetown Rep. of South Africa, 1965 SUBMITTED IN PARTIAL FULFILMENT OF REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of CIVIL ENGINEERING We accept this thesis as conforming to the required standard A THESIS THE The University of British Columbia In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . R . D . K R A T Z Depar tment o f C IV IL E N G I N E E R I N G The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date SEPTEMBER 1970 i . ABSTRACT The i n v e s t i g a t i o n d e a l t m a i n l y w i t h the s h e a r t r a n s f e r c a p a c i t y of a j o i n t between a p r e c a s t c o n c r e t e column and a c a s t - i n - p l a c e c o n c r e t e beam. Four r e i n f o r c e d c o n c r e t e f r a m e s , each c o n s i s t i n g o f two columns and two beams, were c a s t , a s s e m b l e d and t e s t e d i n a s p e c i a l l o a d i n g f r a m e . To o b t a i n a g e n e r a l p a t t e r n o f f a i l u r e m e c h a n i s m s , a s e r i e s of l o a d s c o n s i s t i n g o f d i f f e r e n t r a t i o s o f moments, s h e a r s and a x i a l f o r c e s were imposed on these f r a m e s . A l l r e c o r d i n g o f t e s t da ta was done e l e c t r o n i c a l l y i n the form o f punched tape to f a c i l i t a t e computer a n a l y s i s . The i n v e s t i g a t i o n showed c l e a r l y t h a t h i g h v a l u e s of s h e a r t r a n s f e r can be r e a c h e d even under the most adverse l o a d c o n d i t i o n s . T A B L E OF CONTENTS A B S T R A C T T A B L E OF CONTENTS L I S T OF T A B L E S L I S T OF F I G U R E S L I S T OF P L A T E S L I S T OF SYMBOLS 1.0 I N T R O D U C T I O N 1.1 PURPOSE 1.2 S Y N O P S I S OF P R E C E D I N G I N V E S T I G A T I O N S 1.3 S C O P E OF T E S T S 1.4 D E S C R I P T I O N OF T E S T FRAMES 1.5 S P E C I M E N P R O P E R T I E S 2.0 A P P A R A T U S 2 . 1 LOADING FRAME 2.2 LOAD AND DEFORMATION MEASUREMENT 3.0 A P P L I E D THEORY 3 . 1 M A T H E M A T I C A L R E P R E S E N T A T I O N OF CONCRETE S T R E S S - S T R A I N CURVE 3.2 R E I N F O R C E D CONCRETE THEORY 4.0 T E S T PROCEDURE AND E V A L U A T I O N 4 . 1 D E S C R I P T I O N OF T E S T S 4.2 C O R R E L A T I O N AND D I S C U S S I O N OF T E S T R E S U L T S 4.3 CONCLUSION L I S T OF R E F E R E N C E S A P P E N D I X : S T R E S S - S T R A I N CURVES OF S T E E L T E S T S P E C I M E N S P L O T S OF T H E O R E T I C A L AND MEASURED CONCRETE S T R E S S - S T R A I N CURVES LIST OF TABLES Table I. Tabulation of shear strength theories Table II. Shear capacity of joint Table III.Joint forces and displacements at fai1ure Table IV. Joint forces and displacements at c r i t i c a l shear LIST OF FIGURES Fi g . 1 . Shear c a p a c i t y o f the j o i n t 11 F i g . 2 . R e i n f o r c e d c o n c r e t e frame 13 F i g . 3 . L o a d i n g frame 16 F i g . 4 . R o t a t i o n m e a s u r i n g d e v i c e 20 F i g . 5 . S t r e s s and s t r a i n d i s t r i b u t i o n s t r e s s f u n c t i o n by 24 F i g . 6 . C rack p a t t e r n o f Frame 3 35 F i g . 7 a . L e f t c r a c k p a t t e r n o f Frame 4 37 7b . R i g h t c r a c k p a t t e r n o f Frame 4 37 F i g . 8 . T h e o r e t i c a l b e n d i n g moment and r o t a t i o n c u r v e s beam 41 F i g . 9 . L o a d i n g c u r v e s f o r Frame 1 43 F i g . 1 0 . SL IP -SHEAR p l o t f o r Frame 1 44 F i g . 1 1 . SLIP-MOMENT p l o t f o r Frame 1 45 F i g . 1 2 . ROTATION-MOMENT p l o t f o r Frame 1 46 F i g . 1 3 . L o a d i n g c u r v e s f o r Frame 2 48 F i g . 14 . SLIP -SHEAR p l o t f o r Frame 2 49 F i g . 1 5 . SLIP-MOMENT p l o t f o r Frame 2 50 F i g . 1 6 . ROTATION-MOMENT p l o t f o r Frame 2 51 F i g . 1 7 . L o a d i n g c u r v e s f o r Frame 3 52 F i g . 1 8 . SLIP -SHEAR p l o t f o r Frame 3 53 F i g . 1 9 . SLIP-MOMENT p l o t f o r Frame 3 54 F i g . 2 0 . ROTATION-MOMENT p l o t f o r Frame 3 55 v i . F i g . 2 1 . L o a d i n g c u r v e s f o r Frame 4 57 F i g . 2 2 . SLIP -SHEAR p l o t f o r Frame 4 58 F i g . 2 3 . SLIP-MOMENT p l o t f o r Frame 4 59 F i g . 2 4 . ROTATION-MOMENT p l o t f o r Frame 4 60 F i g . 2 5 . MOMENT-SHEAR f a i l u r e i n t e r a c t i o n p l o t 64 VI 1 L I S T OF P L A T E S P l a t e 1. F r a m e s e t u p 17 P l a t e 2. F r a m e s e t u p ( c l o s e u p ) 17 P l a t e 3. D e t a i l o f h o r i z o n t a l l o a d i n g s e t u p 19 P l a t e 4„ R o t a t i o n m e a s u r i n g d e v i c e 19 P l a t e 5. T y p i c a l c r a c k p a t t e r n o f b e a m - c o l u m n j o i n t 35 L I S T OF SYMBOLS shear arm, i.e. distance of vertical load from column face (inches) area of steel in tension zone (square inches) area of steel in compression zone (square inches) total area of web reinforcement within a distance "s" measured in a direction parallel to the longitudinal reinforcement (square inches) width of the beam (inches) column width compression force in concrete (kips) compression force in steel (kips) effective depth of flexural sections measured from the compression face to the centroid of the steel (inches) distance from the centroid of the compression reinforcement to the compression face (inches) distance from centroid of tension steel to the tension face of a flexural member (inches) modulus of elasticity of concrete (Ksi) modulus of elasticity of steel (29,000Ksi) concrete stress (psi) hoop stress due to transverse reinforcement compressive strength of concrete (psi) stress in tension steel (psi) stress in compression steel (psi) i x . f = u l t i m a t e s t r e s s r e a c h e d by c o n c r e t e d u r i n g c y l i n d e r s t r e n g t h t e s t f = y i e l d s t r e n g t h of r e i n f o r c i n g s t e e l ( p s i ) f ' t = modulus of r u p t u r e o f c o n c r e t e ( p s i ) h = h e i g h t o f column from bot tom p i n to c e n t e r o f beam ( f t . ) H = a x i a l t e n s i o n i n beam ( k i p s ) HL = e q u a l h o r i z o n t a l l o a d a p p l i e d to both s i d e s of the r e i n f o r c i n g c o n c r e t e frame k = r a t i o o f d i s t a n c e between ext reme c o m p r e s s i v e f i b r e and n e u t r a l a x i s to the depth " d " o f a f l e x u r a l s e c t i o n ky = r a t i o o f d i s t a n c e between ext reme c o m p r e s s i v e f i b r e and n e u t r a l a x i s t o the depth " d " o f a f l e x u r a l s e c t i o n a t the p o i n t when t e n s i o n s t e e l y i e l d s k u = r a t i o of d i s t a n c e between ext reme c o m p r e s s i v e f i b r e and n e u t r a l a x i s t o the depth " d " o f a f l e x u r a l s e c t i o n when the c o n c r e t e r e a c h e s i t s u l t i m a t e s t r a i n i n the ext reme f i b r e k i = r a t i o o f d i s t a n c e s between r e s u l t a n t c o m p r e s s i v e c o n c r e t e s t r e s s and the n e u t r a l a x i s t o kd L = Span l e n g t h between column f a c e s ( i n c h e s ) 1 = d i s t a n c e ove r which beam r o t a t i o n i s measured ( i n c h e s ) M = bend ing moment at the column f a c e ( k i p - i n c h e s ) M u = u l t i m a t e b e n d i n g moment at the column f a c e ( k i p - i n c h e s ) ^ = bend ing moment at the column f a c e when t e n s i o n s t e e l y i e l d s ( k i p - i n c h e s ) N = a x i a l c o m p r e s s i o n i n column ( k i p s ) n = modu la r r a t i o f o r s t e e l and c o n c r e t e E„/E s c c o n c e n t r a t e d p o i n t l o a d on beam ( k i p s ) s t e e l r a t i o A s / b d s t e e l r a t i o A ' s / b d web r e i n f o r c i n g s t e e l r a t i o (A /bd) X d/s v h o r i z o n t a l sway f o r c e ( k i p s ) s p a c i n g of web r e i n f o r c e m e n t ( i n c h e s ) o v e r a l l depth of r e i n f o r c e d c o n c r e t e beam ( i n c h e s ) s h e a r s t r e s s V/bd ( p s i ) c o n c r e t e s h e a r s t r e s s ( p s i ) u l t i m a t e s h e a r s t r e s s c a r r i e d by c o n c r e t e ( p s i ) u l t i m a t e t o t a l s h e a r s t r e s s of a r e i n f o r c e d c o n c r e t e s e c t i o n V u / D d ( P S 1 ) s h e a r f o r c e at column f a c e ( k i p s ) s h e a r f o r c e c a r r i e d by the c o n c r e t e ( k i p s ) c r i t i c a l s h e a r , i . e . s h e a r a t f i r s t c o n c r e t e c r a c k i n g s h e a r f o r c e c a r r i e d by the c o n c r e t e a t u l t i m a t e ( k i p s ) t o t a l s h e a r f o r c e c a r r i e d by a r e i n f o r c e d c o n c r e t e s e c t i o n ( k i p s ) c o e f f i c i e n t used i n s h e a r e x p r e s s i o n c o n c r e t e c o m p r e s s i o n f o r c e of e l e m e n t dx u n i t s t r a i n i n c o n c r e t e c o n c r e t e s t r a i n i n o u t e r c o m p r e s s i o n f i b r e u n i t s t r a i n i n t e n s i o n s t e e l u n i t s t r a i n i n c o m p r e s s i o n s t e e l u n i t s t r a i n i n t e n s i o n s t e e l a t i t s y i e l d p o i n t x i . e = unit strain in concrete at yield point cy of tension steel E ., = ultimate strain for concrete c u e 0 = unit concrete strain corresponding to maximum concrete stress e s u = strain in tension steel at ultimate strength of the concrete 6p = plastic rotation of a concrete section 6 U = ultimate rotation of a concrete section <|> = capacity reduction factor 4>p = plastic curvature of a concrete section <j)u = ultimate curvature of a concrete section <j)v = yield curvature of a concrete section ACKNOWLEDGEMENTS The author expresses his grateful appreciation to Professor S. Lipson of the Civil Engineering Department, University of British Columbia, for his constructive criticism during the investigation and writing of this thesis. The thesis is based on tests carried out in the structural laboratory of the Civil Engineering Department and was made possible by grants from the National Research Council. The co-operation and advice of the technical staff of the Civ i l Engineer-ing Department is greatly appreciated. September 1970 Vancouver, B.C. 1. 1.0 INTRODUCTION 1.1 Purpose The o b j e c t o f t h i s i n v e s t i g a t i o n was the s t u d y of s h e a r t r a n s f e r c a p a c i t y o f a j o i n t between a p r e c a s t column and a c a s t - i n - p l a c e c o n c r e t e beam s u b j e c t e d to v a r i o u s c o m b i n a t i o n s o f s h e a r , b e n d i n g moments, and a x i a l t e n s i o n . 1.2 S y n o p s i s o f p r e c e d i n g i n v e s t i g a t i o n s Shear t r a n s f e r has been one o f the most w i d e l y i n v e s t i -g a t e d and d i s c u s s e d t o p i c s i n the f i e l d o f r e i n f o r c e d c o n -c r e t e t e c h n o l o g y . V a r i o u s e x p r e s s i o n s f o r c a l c u l a t i n g s h e a r c a p a c i t y have been p r o p o s e d and t h e s e i n v o l v e d i f f e r e n t p a r a m e t e r s . Some o f t h e s e e x p r e s s i o n s are d i s c u s s e d below and t a b u l a t e d on page 8 . Washa (1) and Badoux and H u l s b o s (2) d e a l w i t h h o r i z o n -t a l s h e a r between p r e c a s t beams and c a s t - i n - p l a c e s l a b s . T h i s i s o f a s i m i l a r n a t u r e to v e r t i c a l s h e a r a t a beam-column i n t e r f a c e ; t h e r e f o r e , the e x p r e s s i o n s d e r i v e d from t h e s e e x p e r i m e n t a l r e s u l t s were of i n t e r e s t . Badoux and H u l s b o s p r o p o s e d f u r t h e r t h a t a d i f f e r e n t c o e f f i c i e n t be used f o r r o u g h , i n t e r m e d i a t e , and smooth i n t e r f a c e o f j o i n t . K r i z and Raths (3) d e s c r i b e a p r o j e c t d i r e c t e d towards deve lopment of d e s i g n c r i t e r i a f o r r e i n f o r c e d c o n c r e t e c o r b e l s . As these a re e s s e n t i a l l y s h o r t c a n t i l e v e r beams o f a/d r a t i o s up to 1 . 5 , they f a l l w i t h i n the range o f the 2. t e s t s made i n t h i s i n v e s t i g a t i o n . O f s p e c i a l i n t e r e s t i s t h e e f f e c t o f d i r e c t t e n s i l e l o a d "H" on t h e s h e a r t r a n s f e r s t r e n g t h as was c l e a r l y s h o w n by t h e t e s t r e s u l t s , a n d t h e r e s u l t i n g i n c l u s i o n o f "H" i n t o t h e u l t i m a t e s h e a r e x p r e s s i o n . P.W. a n d H.W. B i r k e l a n d ( 4 ) a n d R . F . M a s t ( 5 ) i n v e s t i g a t e d t h e s h e a r f r i c t i o n h y p o t h e s i s ; t h a t i s , t h e i n c r e a s e o f s h e a r s t r e n g t h w i t h c o m p r e s s i o n a c r o s s t h e s e c t i o n . T h e i n f l u e n c e o f j o i n t f i n i s h ( r o u g h o r s m o o t h ) was s h o w n by t h e f o r m e r , w h i l e t h e l a t t e r f u r t h e r d e v e l o p e d t h e s h e a r e x p r e s s i o n s t o i n c l u d e j o i n t t e n s i o n . T h e s e e x p r e s s i o n s a r e s t r a i g h t l i n e s f i t t e d t o e x p e r i m e n t a l d a t a . K r e f e l d a n d T h u r s t o n ( 6 , 2 ) i n v e s t i g a t e d t h e c o n t r i b u t i o n o f l o n g i t u d i n a l s t e e l t o t h e s h e a r r e s i s t a n c e o f a s i m p l y s u p p o r t e d beam. T h i s s h e a r r e s i s t a n c e i s i n t h e f o r m o f d o w e l a c t i o n o f t h e s t e e l . D i s t i n c t i o n was made b e t w e e n c a s e s w i t h a n d w i t h o u t s t i r r u p s . I t was f o u n d t h a t i n c a s e s w i t h o u t s t i r r u p s t h e d o w e l a c t i o n r e s u l t e d i n t e n s i l e s p l i t t i n g o f t h e c o n c r e t e a l o n g t h e l o n g i t u d i n a l s t e e l . T h e a d d i t i o n o f s t i r r u p s p r o v i d e s f o r c e s w h i c h r e s t r i c t t h e r e l a t i v e d i s -p l a c e m e n t o f t h e s e g m e n t s i n a d i a g o n a l l y c r a c k e d z o n e a f t e r t h e i n c l i n e d c r a c k h a s f o r m e d a n d t h u s r e t a r d s t h e c r a c k p r o p a g a t i o n . T h e d e r i v e d s h e a r e x p r e s s i o n s w e r e b a s e d on t h e a b o v e c o n c l u s i o n s a n d f i t t e d t o t h e i r r e s u l t s . S i n c e t h e e x p r e s s i o n s w e r e d e r i v e d f o r s i m p l y s u p p o r t e d b e a m s , p a r t s o f t h e m w i l l n o t a p p l y t o a b e a m - c o l u m n j o i n t w h i c h was u s e d i n t h i s i n v e s t i g a t i o n . 3 . H a n s o n a n d C o n n o r ( 8 ) p e r f o r m e d t e s t s on a m o n o l i t h i c a l l y c a s t f r a m e w h i c h i n c l u d e d a j o i n t s i m i l a r t o t h a t i n t h i s i n v e s t i g a t i o n . T h e t y p e o f l o a d i n g i s o f a n o t h e r n a t u r e , b e i n g r e p e a t e d l o a d i n g t o d e t e r m i n e s e i s m i c r e s i s t a n c e . S p e c i a l a t t e n t i o n was g i v e n t o c o n f i n e m e n t o f c o n c r e t e b y h o o p s o r s p i r a l s a n d t o s p l i c e s a t t h e c r i t i c a l s e c t i o n s . T h e s h e a r a n d d e s i g n e x p r e s s i o n s w e r e t a k e n o u t o f t h e ACI c o d e a n d E a r t h q u a k e M a n u a l . A l t h o u g h an a t t e m p t was n o t made t o d e t e r m i n e t h e q u a n t i t a t i v e e f f e c t o f j o i n t c o n f i n e m e n t , i t was e s t a b l i s h e d t h a t t h e t e s t s w i t h c o n f i n e d j o i n t s p r o v e d s t r o n g e r a n d m o r e d u c t i l e . H o f b e c k , I b r a h i m a n d M a t t o c k (9) p r e s e n t a t e s t s e r i e s t o s t u d y s h e a r t r a n s f e r i n r e i n f o r c e d c o n c r e t e . T h e s t r e s s s t a t e i n a l l t e s t s was c o n s t a n t ; t h a t i s , c o n s t a n t a x i a l s t r e s s a n d n e a r c o n s t a n t s h e a r s t r e s s a c r o s s t h e j o i n t . A l t h o u g h t h e i r t e s t r e s u l t s show c l o s e a g r e e m e n t w i t h t h e s h e a r f r i c t i o n h y p o t h e s i s a n d v e r y g o o d a g r e e m e n t w i t h t h e Z i a E n v e l o p e t o M o h r C i r c l e s , no a t t e m p t was made a t a m o r e p r a c t i c a l c a s e w i t h v a r y i n g s t r e s s e s a c r o s s t h e j o i n t . T h e f o l l o w i n g c o n c l u s i o n s w e r e p r e s e n t e d : 1) a p r e - e x i s t i n g c r a c k a l o n g t h e s h e a r p l a n e w i l l b o t h r e d u c e u l t i m a t e s h e a r t r a n s f e r a n d i n c r e a s e s l i p a t a l l l e v e l s o f l o a d i n g . 2) S h e a r t r a n s f e r s t r e n g t h i s a f u n c t i o n o f " p f ". 3) D o w e l a c t i o n o f r e i n f o r c i n g b a r s a c r o s s t h e s h e a r p l a n e i s i n s i g n i f i c a n t i n i n i t i a l l y u n c r a c k e d c o n c r e t e , b u t i s s u b s t a n t i a l i n c o n c r e t e w i t h a p r e - e x i s t i n g c r a c k a l o n g 4. t h e s h e a r p l a n e . 4) S h e a r - f r i c t i o n t h e o r y g i v e s a r e a s o n a b l y c o n s e r v a t i v e e s t i m a t e o f s h e a r t r a n s f e r i f t a n <f> = 1.4, p r o v i d e d p f < 1 5 f ' o r < 6 0 0 p s i . y c 5) T h e Z i a F a i l u r e E n v e l o p e c l o s e l y a g r e e s w i t h o b s e r v e d f a i 1 u r e s . S m i t h ( 1 0 ) p r e s e n t s a s u m m a r y o f t e s t s made b y known r e s e a r c h c e n t r e s . He d i v i d e s s h e a r f a i l u r e i n t o t h r e e g r o u p s d e p e n d i n g on a / d r a t i o a s f o l l o w s : D i a g o n a l t e n s i o n f a i l u r e f o r a / d > 2 . 4 S h e a r c o m p r e s s i o n f a i l u r e f o r 1.0<a/d<2.4 S h e a r p r o p e r f a i l u r e f o r a/d<1.0 F o r a / d > 2 . 4 , t h e P a d u a r t e q u a t i o n i s a p p l i c a b l e ; w h i l e f o r a / d < 2 . 4 , t h e L a u p a e q u a t i o n i s a p p l i c a b l e . T h e L a u p a a n d P a d u a r t c u r v e s i n t e r s e c t i n a s e r i e s o f p o i n t s d e p e n d i n g on " p " , w h i c h c o r r e s p o n d s t o t h e t r a n s i t i o n f r o m D i a g o n a l S h e a r t o S h e a r C o m p r e s s i o n f a i l u r e . T h e m a t t e r o f c o n f i n e m e n t o f c o n c r e t e i n t h e j o i n t z o n e c a n i n f l u e n c e t h e f a i l u r e m e c h a n i s m c o n s i d e r a b l y as d e m o n s t r a t e d by Newmark ( 1 1 ) . He s t a t e s : " I t i s s e e n t h a t b o t h t h e s t r e n g t h a n d d u c t i l i t y o f t h e c o n c r e t e i n c r e a s e s a s t h e l a t e r a l p r e s s u r e i s i n c r e a s e d . F o r a l a t e r a l p r e s s u r e o f 4 , 0 9 0 p s i , t h e c o n c r e t e ( u n c o n f i n e d c y l i n d e r s t r e n g t h 3 , 6 6 0 p s i ) a t t a i n s a maximum s t r e s s o f 1 9 , 0 0 0 p s i a t a s t r a i n o f 0 . 0 5 , t h e l a t t e r v a l u e b e i n g a b o u t 25 t i m e s w h a t w o u l d be e x p e c t e d f o r u n c o n f i n e d c o n c r e t e a t maximum s t r e s s . " 5. F o r t h e c o n f i n e d c o n c r e t e c o r e o f a r e c t a n g u l a r c o l u m n t h e f o l l o w i n g e x p r e s s i o n was d e r i v e d : . 4 * 1 A v f y a = 0 . 8 5 f ' + a c s ( d - d ' ) Newmark a l s o d e v e l o p e d an e x p r e s s i o n f o r s h e a r s t r e n g t h o f a r e i n f o r c e d c o n c r e t e s e c t i o n s u b j e c t e d t o a t e n s i l e f o r c e "H". T h e A C I ( 3 1 8 - 6 3 ) s t a n d a r d s show e x p r e s s i o n s t o be u s e d i n s h e a r t r a n s f e r c a l c u l a t i o n s ( 1 2 ) . T . C . Z s u t t y ( 1 3 ) a p p l i e d t h e t e c h n i q u e s o f d i m e n s i o n a l a n a l y s i s a n d s t a t i s t i c a l r e g r e s s i o n a n a l y s i s t o d a t a o f b e a m - s h e a r t e s t s p e r f o r m e d a t m a j o r r e s e a r c h c e n t r e s . By t h e m e t h o d o f l e a s t s q u a r e s he s h o w s t h a t t h e v a l u e o f a / d = 2 . 5 i s t h e t r a n s i t i o n b e t w e e n beam a c t i o n a n d a r c h a c t i o n . T h e s m a l l p e r c e n t a g e e r r o r s h o w s t h e g o o d f i t o f t h e r e s u l t i n g e x p r e s s i o n s w h i c h w e r e d e r i v e d b o t h f o r c r a c k i n g a n d u l t i m a t e l o a d . B u r t o n , C o r l e y a n d H o g n e s t a d ( 1 4 ) d e v e l o p e d e x p r e s s i o n s t o e v a l u a t e t h e s h r i n k a g e f o r c e s d e v e l o p e d i n beam members o f a m u l t i - s t o r y r e i n f o r c e d c o n c r e t e f r a m e . T h e t e s t s s h o w e d t h e i n c r e a s e o f s h r i n k a g e f o r c e s d e v e l o p e d b y p a r t i a l l y a n d f u l l y - r e s t r a i n e d beams o v e r a t i m e i n t e r v a l . C o m b i n e d w i t h M a t t o c k ' s r e p o r t ( 1 5 ) r e g a r d i n g t h e i n f l u e n c e o f s h a p e f a c t o r on t h e s h r i n k a g e a n d c r e e p a s r e c o r d e d f r o m r e i n f o r c e d c o n c r e t e t e s t b e a m s , t h e a b o v e 6. i n f o r m a t i o n p r o v i d e d a means t o d e t e r m i n e t h e t e n s i l e f o r c e w h i c h was a p p l i e d t o t h e t e s t beams i n t h i s i n v e s t i g a t i o n . C o h e n ( 1 6 ) d e v e l o p e d a s e r i e s o f r o t a t i o n e x p r e s s i o n s f o r r e i n f o r c e d c o n c r e t e m e m b e r s i n t h e p l a s t i c r a n g e . Some o f t h e s e e x p r e s s i o n s w e r e a p p l i e d i n t h e e v a l u a t i o n o f l i m i t i n g c o n d i t i o n s . T h e s e a r e g i v e n b e l o w : 6 = p l a s t i c r o t a t i o n = P <j> dx p 0 o P *p = * u • *y W h e r e : d> = e /k d Y u c u u d> = e /k d * y c y u F o r c o n s t a n t s t r e s s c o n d i t i o n s o v e r t h e p l a s t i c z o n e : e„ = $ 1 = e _ „ l / k d ... 1.2-1 u T u c u u e y = V = e c y 1 / k y d . . . 1 . 2 - 2 W h e r e : " 1 " i s t h e d i s t a n c e o v e r w h i c h r o t a t i o n i s m e a s u r e d . T h e " U l t i m a t e F l e x u r a l A n a l y s i s B a s e d On S t r e s s - S t r a i n C u r v e s o f C y l i n d e r s " by Y o u n g a n d S m i t h ( 1 7 ) p r o v e d u s e f u l i n t h e t h e o r e t i c a l c o r r e l a t i o n a n d e v a l u a t i o n o f c o n c r e t e s t r e n g t h t e s t s p e r f o r m e d on a n u m b e r o f c o n c r e t e c y l i n d e r s o f t h e same b a t c h as t h e t e s t b e a m s . T a b l e I s h o w s a s u m m a r y o f t h e s h e a r t r a n s f e r e x p r e s s i o n s as d e r i v e d by t h e a b o v e m e n t i o n e d a u t h o r s . T a b l e I I s h o w s t h e s e e x p r e s s i o n s as a p p l i e d t o t h e j o i n t t e s t e d i n t h i s i n v e s t i g a t i o n . F i g u r e 1 s h o w s a p l o t o f t h e s e e x p r e s s i o n s . T h e w i d e s t v a r i a t i o n was f o u n d f o r l o w a / d r a t i o s , 7. while towards the generally accepted transition point of a/d=2.5 the agreement was better. 8 . INVESTIGATOR' ULTIMATE SHEAR EXPRESSION SAEMANN and WASH A v c u = 2700+30,000p a/d+5 BADOUX and HULSBOS 3 3 - a / d (a/d) +6a/d+5 3500 + 2 0 , 0 0 0 p y c u = 2 0 0 0 + 2 0 , 0 0 0 p v c u = . 8 5 [ 6 . 5 f F j l - 0 . 5 d / a ) * ( 1 0 0 0 p ) ^ 1 / 3 + Q ' 4 H / V ) ! i  1 0 . 8 H / V v c u = ( A s f y - H u ) t a n 4> bd vc =[ pTd.a+d/d'j + is.ooop] d' /d fiTd v c u=1 .8fF; + 2 6 0 Q P ( M / V d ) x vu=vcu + iilrf y t r f o r f ' =3000psi rough f i n i s h i n t e r m e d i a t e f i n i s h r e s t r i c t i o n s on r e i n f o r c e m e n t : ^ s < . 0 1 3 , A V ^0.5A< bd t a n $=0.7 to 1 .4 a c c o r d i n g to j o i n t roughness l i m i t e d b y : V v c u f o r r f y < 3 ° vu= vcu + l - 5 r f y - 4 4 f o r 3 0 < r f y < 9 0 v = v + r f u cu y f o r 9 0 < r f „ A C I ( 3 1 8 - 1 9 6 3 ) HANSON and CONNOR v c u = <j ) [1.9ff T ^ + 2500pVd  vu= vcu + <*>rfy Q ** V V f w w 1 M - N ( 4 t - d ) / 8 M/Vd<l , <j)=l T a b l e I TABULATION OF SHEAR STRENGTH THEORIES I N V E S T I G A T O R U L T I M A T E SHEAR E X P R E S S I O N REMARKS S M I T H v c u = 1 . 9 f F ^ ( l + 7 5 0 p ) x [ 0 . 1 8 6 + . 0 0 1 5 7 ( 9 - a / d ) 2 ] v c u=1.9 Vf1 c ( Y ' l O O p - i y p J d / a f o r a / d > 2 . 4 l < a / d < 2 . 4 NEWMARK v c u = ( l - H / 8 A c \ / f T ~ c ) 1 . 9 V f , c  v u = v c u + r f y J Z S U T T Y v c r = 5 9 ( f c p d / a ) 1 / 3 v c u = 6 3 . 4 ( f c p d / a ) 1 / 3 T a b l e I ( c o n t i n u e d ) 10, I N V E S T I G A T O R SAEMANN a n d WAS HA BADOUX a n d HULSBOS K R I Z a n d RATHS B I R K E L A N D B I R K E L A N D a n d MAST K R E F E L D a n d THURSTON A C I ( 3 1 8 - 1 9 6 3 ) HANSON a n d CONNOR SMITH NEWMARK Z S U T T Y M=112.5K i n a / d = 0 . 5 M=225K i n a / d = l M=450 a/d=2 17+27=44 14+12=26 8+12=20 13+12=25 1 2 + 1 2 = 2 4 7 . 5 + 1 2 = 1 9 . 5 7+12=19 4.4+3.1=7.5 4.5+3.2=7.7 7 . 7 + 3 3 = 4 0 . 7 3.1+2.2=5.3 4.5+1.6 = 6. 1 6.1+33=39.1 2 . 2 + 1 . 6 = 3 . 8 4.5+0.8=5.3 5 . 3 + 3 3 = 3 8 . 3 4 . 8+2.9=7.7 7 . 7 + 3 3 = 4 0 . 7 9.0 4.8+1.4=6.2 6.2+33=39.2 4.5 2.9 2 . 9 + 3 3 = 3 5 . 9 11.5 1 2 . 4 2.9 2 . 9 + 3 3 = 3 5 . 9 4.8+0.7=5.5 5 . 5 + 3 3 = 3 8 . 5 2.25 9.1 9.8 2.9 2 . 9 + 3 3 = 3 5 . 9 7.2 7.8 c u c u c u c u c r c u T a b l e I I SHEAR C A P A C I T Y OF J O I N T 0.5 1.0 1.5 2.0 a/d rat io . 1 S h e a r c a p a c i t y o f t h e j o i n t 12. 1 .3 Scope o f t e s t s The i n v e s t i g a t i o n i n v o l v e d the t e s t i n g o f two groups o f t h r e e - p i n n e d r e i n f o r c e d c o n c r e t e f r a m e s . Each group c o n s i s t e d of two i d e n t i c a l f rames which were l o a d e d t o f a i l u r e by a c o m b i n a t i o n of p o i n t l o a d s , sway l o a d s and a x i a l l o a d s . The o n l y d i f f e r e n c e between the two groups was the ar rangement of l o n g i t u d i n a l r e i n f o r c e m e n t - the f i r s t h a v i n g c o n t i n u o u s l o n g i t u d i n a l r e i n f o r c e m e n t , and the second h a v i n g l a p p e d l o n g i t u d i n a l r e i n f o r c e m e n t at the beam-column j o i n t . In both c a s e s l a p and anchorage l e n g t h s were adequate to p r e v e n t bond f a i l u r e . The t e s t s were p e r f o r m e d w i t h i n the f o l l o w i n g l i m i t a -t i o n s a p p l i e d to l o a d i n g : 1) the maximum moments were to be s m a l l e r than the u l t i m a t e moment i n o r d e r to r e s t r i c t the f a i l u r e to s h e a r f a i l u r e s a t the beam-column j o i n t . 2) the j o i n t t e n s i o n i n d u c e d was t o s i m u l a t e s h r i n k a g e f o r c e s d e v e l o p e d i n r e s t r a i n e d members. 3) sway was a p p l i e d to produce d i f f e r e n t f a i l u r e c o n d i t i o n s i n the o t h e r w i s e i d e n t i c a l two j o i n t s of the s y m m e t r i c a l t e s t f r a m e s . 1 .4 D e s c r i p t i o n o f t e s t f rames F i g u r e 2 shows one h a l f s e c t i o n of the r e i n f o r c e d c o n c r e t e f r a m e . The columns were c a s t f i r s t w i t h the s i d e o f j o i n t -i n t e r f a c e a g a i n s t s t e e l f o r m s . The weaker beam c o n c r e t e was c a s t s i x weeks l a t e r a g a i n s t the smooth column s u r f a c e 1 3 . 2 - 6 f 4# 10 threaded bar 2#5 2#| 0 threaded bar Pi 2*3 #3 ties at 2^ % b 6 T lap bars in frames 3^4 only #3 ties at 4 ' % f -4#8 ~ 4#l" threded bar "T C D F i g . 2 R e i n f o r c e d c o n c r e t e f r a m e t o r e p r e s e n t a c t u a l c o n s t r u c t i o n c o n d i t i o n s between p r e c a s t columns and c a s t - i n - p l a c e beams. In the l a s t two f rames the l o n g i t u d i n a l s t e e l was l a p p e d as shown i n d o t t e d l i n e s i n F i g u r e 2 . The l a p l e n g t h s a c c o r d i n g t o ACI code w e r e : 18 i n . f o r #5 bars and 12 i n . f o r #3 bars 1 .5 Measurement o f Specimen P r o p e r t i e s For each t e s t frame s i x f o u r - i n c h d i a m e t e r c o n c r e t e t e s t c y l i n d e r s were l o a d e d to f a i l u r e i n d i r e c t c o m p r e s s i o n . Three o f the c y l i n d e r s were o f the same b a t c h as the column c o n c r e t e , and the o t h e r t h r e e were of the same b a t c h as the beam c o n c r e t e . The c y l i n d e r s were c r u s h e d on the same day as the r e s p e c t i v e frame t e s t was p e r f o r m e d . S t r e s s and s t r a i n measurements were taken up to f a i l u r e to a i d w i t h the t h e o r y d e v e l o p e d i n s e c t i o n s 3 . 1 and 3 . 2 . These are r e p r e s e n t e d as s t r e s s - s t r a i n c u r v e s i n the A p p e n d i x A. S e v e r a l r e i n f o r c i n g s t e e l spec imens of both the #5 and #3 s t e e l used i n the beams were l o a d e d to f a i l u r e i n t e n s i o n . The s t r e s s - s t r a i n r e a d i n g s t a k e n are shown i n the A p p e n d i x . 2.0 APPARATUS 2 . 1 L o a d i n g frame F i g u r e 3 , p l a t e 1 and p l a t e 2 show the l o a d i n g f r a m e . Three h y d r a u l i c l o a d c y l i n d e r s p r o v i d e d the r e q u i r e d l o a d c o n n e c t i o n s between the l o a d i n g frame and the t e s t f r a m e . The two h o r i z o n t a l j a c k s were b o l t e d t o the l o a d i n g frame i n l i n e w i t h the c e n t r e - l i n e of the beams and c o n -n e c t e d to the ends of the t e s t frame by c o n n e c t i n g rods and b a l l j o i n t s , to i n s u r e the h o r i z o n t a l p o s i t i o n o f the l o a d , as shown on p l a t e 3 . The a c c u m u l a t o r c o n n e c t e d i n f r o n t o f the j a c k on the s o u t h s i d e f a c i l i t a t e d a h o r i z o n t a l movement of the f r a m e , a f t e r s h u t t i n g o f f the south s u p p l y v a l v e , w i t h a n e g -l i g i b l e change i n j a c k p r e s s u r e . Thus a c o n s t a n t f o r c e can be m a i n t a i n e d a t one s i d e o f the frame w h i l e the f o r c e on the o t h e r s i d e can be f u r t h e r i n c r e a s e d f o r s i d e - s w a y . The v e r t i c a l l o a d c y l i n d e r i s s i t u a t e d i n s i d e the " l o a d i n g t r i a n g l e " which keeps i t i n a v e r t i c a l p o s i t i o n d u r i n g a l l s t a g e s of frame s i d e - s w a y . The v e r t i c a l l o a d c y l i n d e r i s c o n n e c t e d to the " l o a d i n g beam" by two p i n -j o i n t e d c o n n e c t i o n p l a t e s . The " l o a d i n g beam" i s r e s t i n g on top o f the t e s t frame on two a d j u s t a b l e l o a d p o i n t s . 2.2 Load and d e f o r m a t i o n measurement The two h o r i z o n t a l l o a d s were measured by " S t r a i n s e r t " b o l t s c o n n e c t i n g the h o r i z o n t a l l o a d j a c k s t h r o u g h a b a l l j o i n t to the l o a d i n g f r a m e . J""— horizontal load jack F i g.3 L o a d i n g f r a m e Plate 1 . Frame setup Plate 2 . Frame setup (closeup) 18. T h e c e n t r a l l o a d was m e a s u r e d b y a l o a d c e l l p l a c e d i n s e r i e s w i t h t h e c e n t r a l l o a d c y l i n d e r . A l l d e f o r m a t i o n s w e r e m e a s u r e d by d i r e c t c u r r e n t d i s -p l a c e m e n t t r a n s d u c e r s ( D C D T ' s ) . T h e v e r t i c a l b e am s l i p r e l a t i v e t o t h e c o l u m n was m e a s u r e d b y a 7 D C D T - 5 0 0 w i t h a d i s p l a c e m e n t r a n g e o f + . 5 i n . T h e t r a n s d u c e r s w e r e a t t a c h e d t o t h e c o l u m n , o n e a b o v e a n d o n e b e l o w t h e b e a m . T h e c o r e e x t e n s i o n s r e s t e d on a h o r i z o n t a l p ; l a t e g l u e d on t h e b e a m , t h e b o t t o m o n e b e i n g h e l d i n p l a c e by a r u b b e r s l i n g . Beam a n d c o l u m n r o t a t i o n s w e r e m e a s u r e d b y a s e t u p a s s h o w n i n F i g u r e 4 a n d P l a t e 4. T h e t r a n s d u c e r s u s e d w e r e 7 D C D T - 1 0 0 w i t h a d i s p l a c e m e n t o f * . l i n . T h e r o t a t i o n s w e r e m e a s u r e d a s f o l l o w s : S i n c e t h e p e n d u l u m a l w a y s a s s u m e s t h e v e r t i c a l p o s i t i o n d u e t o g r a v i t y , a n y r o t a t i o n o f t h e f r a m e w i l l move t h e t r a n s d u c e r c o r e , w h i c h i s c o n n e c t e d t o t h e p e n -d u l u m t h r o u g h a b e a r i n g ( F i g . 4 ) . F o r s m a l l r o t a t i o n s t h e h o r i z o n t a l c o r e m o v e m e n t i s p r o p o r t i o n a l t o t h e p e n d u l u m r o t a t i o n . I n o r d e r t o o v e r c o m e t h e f r i c t i o n a l r e s i s t a n c e o f t h e c o r e s l i d i n g i n t h e t r a n s d u c e r a n d i n t h e b e a r i n g s , a b u z z e r o s c i l l a t i n g a t a h i g h f r e q u e n c y , was a t t a c h e d t o t h e t r a n s d u c e r b r a c k e t . An a d j u s t m e n t d e v i c e was a t t a c h e d b e t w e e n t h e m o u n t i n g p l a t e a n d t h e c o n c r e t e ( F i g . 4 i n d o t t e d l i n e s ) . T h u s t h e t r a n s d u c e r r e a d i n g s c o u l d be c a l i b r a t e d a g a i n s t a known a n g l e c h a n g e . F i g u r e 3 s h o w s t h e p o s i t i o n s o f t h e l o a d a n d d e f o r -m a t i o n m e a s u r e m e n t d e v i c e s . A l l m e a s u r e m e n t s w e r e r e c o r d e d P l a t e 4. R o t a t i o n m e a s u r i n g d e v i c e bearing (center of rotation of pendulum ) •mounting plate tension spri ng (between arm and plate ) •connecting pin rr-><=s, (arm to plate ) screw stop (on plate ) rotation adjustment arm (fixed to the concrete) adjustment screw - m a i n fastening screw (joining arm to Concrete) buzzer • transducer(dcdt) •transducer bracket -transducer core attached to pendulum by bearing pi njoi nt (transducer core extension to pendulum) pendulum F i g . 4 R o t a t i o n m e a s u r i n g d e v i c e i n the form o f v o l t a g e s by a V i d a r d i g i t a l v o l t m e t e r and punched out on paper t a p e . T h i s s e r v e d as i n p u t f o r com-p u t e r programs wh ich p e r f o r m e d the n u m e r i c a l a n a l y s i s o f the t e s t d a t a . 3.0 A P P L I E D THEORY  3.1 M a t h e m a t i c a l r e p r e s e n t a t i o n o f c o n c r e t e s t r e s s - s t r a i n c u r v e I n o r d e r t o d e t e r m i n e t h e moment o f r e s i s t a n c e o f a r e i n f o r c e d c o n c r e t e beam s e c t i o n a t a s t r e s s s t a t e b e l o w u l t i m a t e w h e r e n e i t h e r t h e l i n e a r c o n c r e t e s t r e s s v a r i a t i o n ( m o d u l a r r a t i o ) m e t h o d n o r t h e r e c t a n g u l a r ( W h i t n e y s ) s t r e s s b l o c k a p p l i e s , i t was a t t e m p t e d t o r e p r e s e n t t h e a c t u a l s t r e s s d i s t r i b u t i o n b y a m a t h e m a t i c a l e x p r e s s i o n . By e v a l u a t i n g r e s i s t i n g m o m e n t s c o r r e s p o n d i n g t o s p e c i f i c c o n c r e t e o r s t e e l s t r a i n s a n d t h e r e s p e c t i v e p o s i t i o n o f t h e n e u t r a l a x i s , a n d a p p l y i n g t h e s e v a l u e s t o t h e e x p r e s s i o n s as g i v e n by C o h e n ( 1 6 ) i n C h a p t e r 1.2, a c o r r e l a t i o n i s p o s s i b l e b e t w e e n t h e r e c o r d e d d i s p l a c e m e n t a n d l o a d d a t a . F o u r e x p r e s s i o n s w e r e u s e d t o f i n d t h e b e s t f i t t i n g t h e o r e t i c a l r e p r e s e n t a t i o n o f t h e s t r e s s - s t r a i n c u r v e s o b t a i n e d e x p e r i m e n t a l l y . T h e f i r s t e x p r e s s i o n was d e r i v e d by Y o u n g a n d S m i t h ( 1 7 ) : f = f ' G e ( 1 " G ) . . . 3 . 1 - 1 c c W h e r e G = e c / e 0 T h e same e x p r e s s i o n was u s e d w i t h t h e f o l l o w i n g c o r r e c t i v e t e r m a s d e r i v e d by t h e a u t h o r : f c = f c ^ 1 " 1 " 0 * 1 C o s ( t f ^ / 8 0 0 0 ) ( l - G ) G ] e ^ 1 _ G^ . . . 3 . 1 - 2 T h e n e x t e x p r e s s i o n was t a k e n f r o m S a e n z ( 1 8 ) : f c = E i e c / n + ( R + R E " 2 ) G - ( 2 R - 1 ) G 2 + R G 3 ] W h e r e R = R £ ( R f - 1 ) / ( R g - 1 ) 2 - 1/R e R = E /E e c u o R. = f ' / f „ . , Ec F = S e c a n t M o d u l u s = f ' / e C o E..- = I n i t i a l T a n g e n t M o d u l u s = E SE .e T h e f o u r t h e x p r e s s i o n b a s e d on p a r a b o l i c s t r e s s - s t r a i n r e l a t i o n s h i p was d e r i v e d by t h e a u t h o r : I n t h e A p p e n d i x t h e s e e x p r e s s i o n s w e r e s u p e r i m p o s e d on e x p e r i m e n t a l s t r e s s - s t r a i n d a t a f o u n d by t h e t e s t s . Two p l o t s w e r e d o n e , e a c h c o n t a i n i n g t h e i n f o r m a t i o n o b t a i n e d f r o m t h e s i x t e s t c y l i n d e r s b e l o n g i n g t o one b a t c h . F r o m t h e c o m p a r i s o n o f a l l t h e s e p l o t s t h e f o l l o w i n g c a n be s a i d : i ) e x p r e s s i o n 3.1-1 s h o w s c l o s e a g r e e m e n t w i t h t h e e x p e r i m e n t a l v a l u e s , b u t i s s o m e t i m e s s l i g h t l y on t h e h i g h s i d e . i i ) e x p r e s s i o n 3.1-4 a l s o s h o w s c l o s e a g r e e m e n t w i t h t h e e x p e r i m e n t a l v a l u e s a n d i s m o r e c o n s e r v a t i v e i n m o s t c a s e s . B o t h o f t h e s e e x p r e s s i o n s w e r e a p p l i e d t o t h e r e i n f o r c e d c o n c r e t e t h e o r y i n s e c t i o n 3.2. f . = 2 f ' G ( l - G / 2 ) c c . . . 3 . 1 - 4 24. BEAM STRAIN DIAGRAM STRESS DIAGRAM F i g . 5 S t r e s s a n d s t r a i n d i s t r i b u t i o n by s t r e s s f u n c t i o n 3.2 R e i n f o r c e d C o n c r e t e T h e o r y T h e f o l l o w i n g a n a l y s i s i s s i m i l a r t o t h e o n e p r e s e n t e d by Y o u n g a n d S m i t h ( 1 7 ) . F r o m t h e g e o m e t r y o f t h e s t r a i n d i a g r a m we h a v e : k c i s 1 _ k k - d ' / d e 1 = e . . . 3 .2-2 s s 1-k ( a ) e x p o n e n t i a l s t r e s s - s t r a i n r e l a t i o n s h i p g i v e n b y 3 . 1 - 1 : f . fee"" 1 3' C C H = f b dx c c W h e r e G = c c / e 0 Pkd T h e r e f o r e C = f b dx c J o e a n d e = e . x / k d c c t H e n c e by e v a l u a t i n g t h e i n t e g r a l s p k d c Jo ' c 1 Pkd a n d k ! k d = — ( f b d x ) x c c J 0 t h e e x p r e s s i o n s f o r t h e c o n c r e t e c o m p r e s s i o n f o r c e a n d i t s l e v e r a r m c o e f f i c i e n t a r e o b t a i n e d : C r = k b d f ' [ l - ( J + l ) e " J ] e / J . . . 3 . 2 - 3 c 2 - [ J + 2 ( 1 + J ) ] e " J a n d k. = — . . . 3 . 2 - 4 J [ l - ( J + l ) e - J ] W h e r e J = e c t / £ 0 2 6 . T h i s c a n a l s o be w r i t t e n C = k b d f c a Whe r e : f = a v e r a g e c o n c r e t e s t r e s s a = f [ l - ( J + 1 ) e ~ J ] e / J . . . 3 . 2 - 5 c T h i s i s u s e d i n t h e h o r i z o n t a l e q u i l i b r i u m e q u a t i o n : H = T - C c - C s I n t h e f o l l o w i n g a n a l y s i s e i t h e r "e " i s k n o w n ; t h a t i s , c t e =e a t u l t i m a t e f l e x u r a l s t r e n g t h , o r " c " i s k n o w n ; i . e . c t c u s e =e a t o n s e t o f y i e l d i n g o f t e n s i o n s t e e l . T h u s e i t h e r s s y J 3 t h e s t e e l s t r a i n s a r e g i v e n i n t e r m s o f t h e c o n c r e t e s t r a i n , o r t h e c o n c r e t e s t r a i n a n d c o m p r e s s i o n s t e e l s t r a i n a r e g i v e n i n t e r m s o f t h e t e n s i o n s t e e l s t r a i n t o e v a l u a t e t h e m o m e n t o f r e s i s t a n c e o f t h e s e c t i o n . F o r e c t = e c u : H = A f - k b d f - A ' E C ( l - k , , d ' / d ) e s y u a s s v u ' ' c u T h i s p r o d u c e s a q u a d r a t i c e x p r e s s i o n i n " k u " w h i c h i s s o l v e d t o p r o d u c e : k = [\/(A f + A ' E c e + H ) 2 + 4 b d f A' E e d ' / d -u L v v s y s s c u a s s c u (-A f +A'E e + H ) ] / 2 b d f . . . 3 . 2 - 6 s y s s c u a Hen c e : M = A f ( t / 2 - d 1 ) + k b d % [ . 5 t / d - k ( 1 - k . ) ] + u s y u a u 1 A ' f , ( t / 2 - d ' ) . . . 3 . 2 - 7 s s W h e r e f 1 = E ( 1 - k d ' / d ) e < f s s u c u y F o r e = e s s y ( k v - d ' / d ) H = A f - k b d f - A ' f — 1  s y y a s y 1_ k y b u t f a = C [ l - ( F + l ) e ' r ] e / F w h e r e F = ^ s y y e 0 ( l - k y ) T h i s c a n be s o l v e d f o r " k ^ " by a p p r o x i m a t e m e t h o d s , b u t i t i s t o o c u m b e r s o m e f o r p r a c t i c a l c a l c u l a t i o n s . T h e r e f o r e , "k " i s f o u n d b y t h e p a r a b o l i c s t r e s s - s t r a i n d i s t r i b u t i o n y as s h o w n i n s e c t i o n ( b ) . ( b ) P a r a b o l i c s t r e s s - s t r a i n r e l a t i o n s h i p g i v e n b y 3 . 1 - 4 : f = 2 f ' ( l - G / 2 ) G c c W h e r e G = E„/e c o U s i n g t h e same a p p r o a c h a s a b o v e we g e t : C = k b d f ' ( l - J / 3 ) J c c o r C = k b d f c a W h e r e A l s o f = f ( l - J / 3 ) J . . . 3 . 2 - 8 a c 2 - 3 J / 4 k = .. . 3 . 2 - 9 3 - J i W h e r e J = £ / e „ c t 0 E q u a t i o n s 3.2-6 a n d 3 . 2 - 7 , a s d e r i v e d b e f o r e i n t h e c a s e w h e r e e f = e , s t i l l a p p l y i f t h e a b o v e e x p r e s s i o n s f o r " f " a n d " k , " a r e u s e d . F o r e = e p„: s s y S u b s t i t u t i n g i n t o t h e h o r i z o n t a l e q u i l i b r i u m e q u a t i o n we g e t k - d ' / d H = A f - k b d f - A' f s y y a s 1 - k y W h e r e : f a = f'F[l-F/3] a c S u b s t i t u t i n g a n d r e a r r a n g i n g t e r m s p r o d u c e s t h e f o l l o w i n g e q u a t i on i n " k ^ " : k 3 b d f e ( 1 + e /3) + k * ( A + A ' - b d f e - H / f ) -y r r r y v s s r r y k [ 2 A + A ' ( l + d ' / d ) - 2 H / f ] + A + A ' d ' / d - H / f = 0 y s s y s s y . . . 3 . 2 - 1 0 W h e r e : f = f ' / f r c y e = e /e r s y o T h i s c a n be s o l v e d b y t h e a p p r o x i m a t i o n t e c h n i q u e : (k ) = ( k ) - f ( k ) / f ' ( k ) y n + l y ' n y n v y n T h e moment e q u a t i o n i s : t 2 t M = A f ( — - d ' ) + k bd f [ k ( 1 - k , ) ] + y s y 2 y a 2 d y t A ' f ( d') . . . 3 . 2 - 1 1 s s £ Whe r e : k - d ' / d In o r d e r to v e r i f y the v a l i d i t y of the t h e o r y d e v e l o p e d i n t h i s c h a p t e r , a c o m p a r i s o n was done w i t h the c o r r e s p o n d i n g e x p r e s s i o n s as d e r i v e d f rom the r e c t a n g u l a r s t r e s s b l o c k a t u l t i m a t e s t r e s s s t a t e , a p p l i e d to the beam used i n t h i s i n v e s t i g a t i o n . ( i ) U l t i m a t e moment from r e c t a n g u l a r s t r e s s b l o c k : Assuming a x i a l t e n s i o n i n beam as 5 K i p s , i t was found t h a t : k u = . 2 2 0 M u = 165.OK i n c h e s ( i i ) U l t i m a t e moment f rom e x p o n e n t i a l s t r e s s d i s t r i b u t i o n a x i a l t e n s i o n = 5 K i p s k u = . 2 1 2 M u = 166.2K i n c h e s ( i i i ) U l t i m a t e moment f rom p a r a b o l i c s t r e s s d i s t r i b u t i o n : a x i a l t e n s i o n = 5 K i p s k u = . 2 1 8 M u = 166.8K i n c h e s ( i v ) For the same a x i a l t e n s i o n , u s i n g the p a r a b o l i c s t r e s s d i s t r i b u t i o n , i t was found t h a t a t y i e l d i n g of the t e n s i on s t e e l : k y = . 3 7 8 M y = 138.OK i n c h e s (v) To show the e f f e c t o f a x i a l t e n s i o n , the u l t i m a t e moment as found f rom the r e c t a n g u l a r s t r e s s b l o c k f o r z e r o a x i a l t e n s i o n i s : k = . 2 4 8 u M u = 180.5K i n c h e s T h e u l t i m a t e m o m e n t s f o u n d i n s e c t i o n s ( i ) , ( i i ) a n d ( i i i ) s how c l o s e a g r e e m e n t . T h e r e f o r e , t h e e x p r e s s i o n s 3 . 2 - 6 , 3 . 2 - 7 , 3.2-10 a n d 3.2-11 w e r e u s e d i n t h e f o l l o w i n g c h a p t e r t o d e t e r m i n e t h e b e n d i n g m o m e n t s c o r r e s p o n d i n g t o a n y s p e c i f i c s t r e s s l e v e l i n t h e c o n c r e t e o r r e i n f o r c i n g s t e e l . 4.0 T E S T PROCEDURE AND E V A L U A T I O N 4.1 D e s c r i p t i o n o f t e s t s F o r a l l t e s t f r a m e s , e x c e p t t h e l a s t , t h r e e l o a d i n g c y c l e s w e r e p e r f o r m e d . T h e y c o n s i s t e d o f t h e f o l l o w i n g s e q u e n c e : T h e f i r s t c y c l e was p e r f o r m e d w i t h t h e v e r t i c a l l o a d p o s i t i o n a t 5 i n c h e s f r o m t h e b e a m - c o l u m n j o i n t , f r o m z e r o l o a d up t o w o r k i n g s t r e s s r a n g e a n d b a c k t o z e r o l o a d . T h e s e c o n d c y c l e was p e r f o r m e d w i t h t h e v e r t i c a l l o a d p o s i t i o n a t 22 i n c h e s f r o m t h e b e a m - c o l u m n j o i n t . W o r k i n g s t r e s s h e r e r e f e r s t o a s h e a r s t r e s s l e v e l c o r r e s p o n d i n g t o c r a c k i n g s h e a r . T h e t h i r d c y c l e was p e r f o r m e d w i t h t h e v e r t i c a l l o a d p o s i t i o n a t 5 i n c h e s f r o m t h e b e a m - c o l u m n j o i n t up t o f a i l u r e . T h e l a s t f r a m e was l o a d e d t o f a i l u r e d u r i n g t h e f i r s t c y c l e w i t h t h e v e r t i c a l l o a d p o s i t i o n a t 5 i n c h e s f r o m t h e b e a m - c o l u m n j o i n t . F a i l u r e was a s s u m e d t o h a v e o c c u r r e d when t h e l o a d i n c r e a s e b e c a m e e r r a t i c , o r d r o p p e d o f f , o r when t h e d i s p l a c e m e n t s a n d c r a c k i n g b e c a m e s e v e r e t o t h e p o i n t w h e r e c o n t i n u e d l o a d i n g c o u l d c a u s e damage t o t h e e q u i p m e n t . T h e a x i a l t e n s i o n was a p p l i e d t o t h e beams f i r s t , by l o a d i n g t h e t w o h o r i z o n t a l j a c k s u n t i l t h e y r e g i s t e r e d a b o u t 5 K i p s . T h e n t h e v a l v e was c l o s e d on t h e s o u t h j a c k t o p r o d u c e an a p p r o x i m a t e l y c o n s t a n t f o r c e on t h i s s i d e d u r i n g t h e s u b s e q u e n t a l t e r n a t e v e r t i c a l a n d s w a y l o a d i n g . T h e s w a y a n d v e r t i c a l l o a d i n g w e r e i n c r e a s e d a l t e r n a t e l y i n s u c h a way t h a t t h e moment a t t h e s o u t h j o i n t r e m a i n e d l o w w h i l e t h e moment a t t h e n o r t h j o i n t k e p t i n c r e a s i n g . T h u s t w o d i f f e r e n t s t r e s s s t a t e s w e r e c r e a t e d i n t h e two j o i n t s . T h e s o u t h j o i n t r e a c h e d a s h e a r s t r e s s s l i g h t l y l o w e r t h a n t h e n o r t h j o i n t , w h i l e t h e moment on t h e s o u t h j o i n t was k e p t c o n s i d e r a b l y l o w e r t h a n a t t h e n o r t h j o i n t , a t t h e same a x i a l t e n s i o n f o r c e . T h e s l i p was r e c o r d e d o n l y b y t h e b o t t o m DCDT's s i n c e t h e w e d g e o f c o n c r e t e u n d e r n e a t h t h e t o p DCDT's d i d n o t move r e l a t i v e t o t h e c o l u m n . A l l r o t a t i o n s r e c o r d e d b e l o w a r e t h e r o t a t i o n o f a s e c t i o n on t h e beam 2.5 i n c h e s away f r o m t h e c o l u m n , w i t h r e s p e c t t o t h e c o l u m n . I n t h e f o l l o w i n g p a r a g r a p h s t h e m a g n i t u d e s o f v e r t i c a l l o a d s a r e a s r e c o r d e d on e a c h s i d e o f t h e h o r i z o n t a l f r a m e m ember, o r h a l f t h e f o r c e i n t h e j a c k . B e h a v i o r o f t h e f r a m e s d u r i n g t h e l o a d i n g t o f a i l u r e was as f o l l o w s : F r a m e N u m b e r 1 : T h e f r a m e was l o a d e d h o r i z o n t a l l y a t b o t h e n d s u n t i l a l o a d o f 5.5K was r e a c h e d . No v i s u a l o b s e r v a t i o n s w e r e made. T h e s w a y - l o a d was t h e n a p p l i e d up t o 2 . 7 5 K . A t t h e e n d o f t h i s l o a d i n g s t a g e , t e n s i l e c r a c k s b e c a m e v i s i b l e a t t h e b o t t o m o f t h e beam u n d e r t h e s o u t h v e r t i c a l l o a d p o i n t . T h i s s i g n i f i e s t h e l o w moment c a p a c i t y o f t h e beams t o p o s i t i v e m oment due t o t h e w e a k e r l o n g i t u d i n a l r e i n f o r c e m e n t on t h e b o t t o m o f t h e beam. V e r t i c a l l o a d was t h e n a p p l i e d up t o 5.5K on e a c h b e am. T h e c r a c k i n g d i d n o t i n c r e a s e a s m o m e n t s i n b o t h j o i n t s i n c r e a s e d i n t h e n e g a t i v e s e n s e a n d no o t h e r o b s e r v a t i o n s w e r e made. As the v e r t i c a l l o a d was i n c r e a s e d t o 6K a c r a c k c o u l d be seen on the bot tom south i n t e r f a c e between column and beam. The c r a c k ran v e r t i c a l l y up to about 2/3 o f the beam h e i g h t , then 2 i n c h e s toward the v e r t i c a l l o a d p o i n t . At i n c r e a s e d v e r t i c a l l o a d i n g t h i s d i a g o n a l c r a c k l e n g t h e n e d and more d i a g o n a l c r a c k s appeared p a r a l l e l t o i t . No c r a c k s had y e t appeared on the n o r t h j o i n t , a l t h o u g h i t r e c o r d e d a much h i g h e r moment. With f u r t h e r i n c r e a s e o f v e r t i c a l l o a d i n g , the n o r t h j o i n t a l s o s t a r t e d showing the same p a t t e r n o f s h e a r c r a c k -i n g . These c r a c k s d e v e l o p e d r a p i d l y and both s i d e s showed about 1/2" s l i p . At t h i s s t a g e the r a t e o f i n c r e a s e o f s l i p was c o n s i d e r e d e x t e n s i v e and f u r t h e r l o a d i n g was t e r m i n a t e d . F a i l u r e was assumed to have taken p l a c e . Frame Number 2 ; Both h o r i z o n t a l j a c k s were l o a d e d to 4K. V e r t i c a l l o a d was a p p l i e d up to 5K. No v i s u a l o b s e r v a t i o n s were made. S w a y - l o a d was a p p l i e d up to 2 . 2 5 K . T e n s i l e c r a c k s opened as i n the f i r s t f r a m e . V e r t i c a l l o a d was i n c r e a s e d a g a i n and the s o u t h j o i n t s t a r t e d c r a c k i n g i n the same way as the p r e v i o u s f r a m e . U n f o r t u n a t e l y the l a s t f i v e o b s e r v a t i o n c y c l e s were l o s t due to f a u l t y r e c o r d i n g . At a v e r t i c a l l o a d o f 23K (2K a f t e r r e c o r d i n g t e r m i n a t e d ) the b o t t o m - s l i p t r a n s f o r m e r s were r e -moved as damage from the s l i p p i n g beam was e x p e c t e d . At t h i s s t a g e the n o r t h j o i n t had a l s o c r a c k e d e x t e n s i v e l y and s l i p o c c u r r e d . At a v e r t i c a l l o a d o f 2 4 . 5 K l o a d i n g was t e r m i n a t e d as the l o a d i n c r e a s e became e r r a t i c and the s l i p s e v e r e . T h i s t e s t run was r e m a r k a b l y s i m i l a r to c r a c k i n g and s l i p b e h a v i o u r i n the f i r s t f r a m e . Frame Number 3 : T h i s f r a m e } a n d the n e x t , d i f f e r e d f rom the f i r s t two by h a v i n g l a p p e d i n s t e a d o f c o n t i n u o u s l o n g i t u d i n a l r e i n f o r c e -ment th rough the j o i n t . Both h o r i z o n t a l j a c k s were l o a d e d to 5 . 2 K . V e r t i c a l l o a d was a p p l i e d up to 5K. No v i s u a l o b s e r v a t i o n s were made. S w a y - l o a d was a p p l i e d up to 2 . 7 5 K . A s l i g h t o p e n i n g o f the t e n s i l e c r a c k s was o b s e r v e d under the s o u t h v e r t i c a l l o a d p o i n t . The v e r t i c a l l o a d was a g a i n i n c r e a s e d , and s l i g h t s h e a r c r a c k i n g was o b s e r v e d at 10K on the s o u t h j o i n t . F i g u r e 6 shows the e x t e n t o f t h e s e c r a c k s at the f o l l o w i n g l o a d s t a g e s : (1) At a v e r t i c a l l o a d o f 1 1 . 3K (2) A t a v e r t i c a l l o a d o f 17K (3) At a v e r t i c a l l o a d o f 21K At the v e r t i c a l l o a d o f 21K the f i r s t c r a c k s had opened up c o n s i d e r a b l y and v i s u a l s l i p of the sou th beam r e l a t i v e to i t s column was n o t i c e d . The n o r t h j o i n t a l s o s t a r t e d to show s l i g h t s h e a r c r a c k i n g s i m i l a r to #1 a b o v e . The s o u t h j o i n t then showed the f o l l o w i n g i n c r e a s e d c r a c k s 35. Fig.6 South crack pattern of Frame 3 Scale: 1/4 inch = 1 foot ( 4 ) A t a v e r t i c a l l o a d o f 22K ( 5 ) A t a v e r t i c a l l o a d o f 23K T h e n o r t h s i d e a l s o s h o w e d some v i s u a l s l i p . T h e s w a y l o a d h a d s t a r t e d t o d r o p a t t h i s s t a g e a n d was down t o 2K. T h e v e r t i c a l l o a d was i n c r e a s e d f u r t h e r u n t i l 2 9 . 5 K a t w h i c h s t a g e an a p p a r e n t maximum h a d b e e n r e a c h e d a n d f a i l u r e was a s s u m e d t o h a v e t a k e n p l a c e . F r a m e N u m b e r 4: B o t h h o r i z o n t a l j a c k s w e r e l o a d e d t o 5.5K. S w a y - l o a d was i n c r e a s e d up t o 1.5K. No o b s e r v a t i o n s w e r e m a d e. V e r t i c a l l o a d was i n c r e a s e d up t o 15K. A t 8K t h e f i r s t s h e a r c r a c k s a p p e a r e d on t h e s o u t h s i d e a l o n g t h e b o t t o m o f t h e j o i n t . T h i s i n c r e a s e d i n a s i m i l a r way as f o r F r a m e 3. S w a y - l o a d was i n c r e a s e d t o 4.75K. T h i s p r o d u c e d t h e t e n s i l e c r a c k i n g a t t h e b o t t o m o f t h e s o u t h b eam u n d e r t h e v e r t i c a l l o a d p o i n t , t h a t was o b s e r v e d i n p r e v i o u s f r a m e s . T h e v e r t i c a l l o a d was i n c r e a s e d a g a i n . T h e s w a y d i s p l a c e m e n t i n c r e a s e d t o an e x t e n t w h e r e t h e o i l p r e s s u r e i n t h e r i g h t j a c k ( p r o d u c i n g s w a y - l o a d ) d r o p p e d o f f a n d c o u l d n o t be r e c o v e r e d a s i t d r a i n e d t h e h y d r a u l i c h a n d pump. A t a v e r t i c a l l o a d o f 16.5K t h e s h e a r c r a c k s on t h e s o u t h j o i n t b e c a m e c l e a r l y d e f i n e d . As t h e v e r t i c a l l o a d was i n c r e a s e d t o 21K a n d t h e s w a y -l o a d d r o p p e d t o z e r o , t h e s o u t h s i d e c r a c k e d e x t e n s i v e l y a n d 3 7 . F i g . 7 a S o u t h C r a c k P a t t e r n S c a l e : 1/4 i n c h = 1 f o o t F i g . 7 b N o r t h C r a c k P a t t e r n S c a l e : 1/4 i n c h = 1 f o o t C r a c k P a t t e r n s F o r F r a m e 4 v i s u a l s l i p took p l a c e w h i l e the n o r t h s i d e a l s o s t a r t e d c r a c k i n g . At 2 2 . 5 K the r o t a t i o n a l d e f o r m a t i o n s had become s e v e r e and the top s l i p t r a n s f o r m e r s were removed to p r e v e n t damage. At 24K the c r a c k i n g of the n o r t h s i d e (under n e g a t i v e sway f o r c e ) had d e v e l o p e d to a s t a g e where the beam pendulum p l a t e s t a r t e d t o come o f f due to c r a c k s e x t e n d i n g under i t . O ther c r a c k s appeared at the bot tom of the n o r t h beam a t the l o a d p o i n t . A p p a r e n t maximum l o a d s were r e a c h e d at 2 9 . 7 K and f u r t h e r l o a d i n g was t e r m i n a t e d . At t h i s s t a g e the c r a c k i n g had r e a c h e d the s t a g e shown i n F i g u r e 7a & b f o r the s o u t h and n o r t h . G e n e r a l l y the s o u t h j o i n t c r a c k e d a t a lower l o a d and showed l a r g e r s l i p s than the n o r t h s i d e , a l t h o u g h the s h e a r was s l i g h t l y l e s s and the b e n d i n g moment was c o n s i d e r a b l y l e s s than the n o r t h s i d e . The f i r s t two f rames w i t h the c o n t i n u o u s l o n g i t u d i n a l r e i n f o r c e m e n t showed c r a c k i n g and s l i p at a lower l o a d than the l a s t two frames w i t h l a p p e d l o n g i t u d i n a l r e i n f o r c e m e n t , and the f i n a l s l i p v a l u e s were b i g g e r , and were r e c o r d e d a t s m a l l e r u l t i m a t e l o a d s . T a b l e I I I shows the l o a d s and d i s p l a c e m e n t s as r e c o r d e d f o r each frame a t f a i l u r e . Table III JOINT FORCES AND DISPLACEMENTS AT FAILURE 4 . 2 C o r r e l a t i o n and d i s c u s s i o n o f t e s t r e s u l t s The p l o t t e d t e s t da ta f o r the l o a d i n g c y c l e to f a i l u r e o f the f o u r f rames are compared i n t h i s s e c t i o n . In o r d e r to e x p l a i n the i r r e g u l a r i t i e s o b s e r v e d i n the d i s p l a c e m e n t p l o t s and to c o r r e l a t e t h e o r e t i c a l and o b s e r v e d d a t a , the c o n c r e t e t h e o r y as d e v e l o p e d i n c h a p t e r 3 . 0 was a p p l i e d . The r e l e v a n t t h e o r e t i c a l e x p r e s s i o n s were p l o t t e d a g a i n s t j o i n t t e n s i o n i n F i g . 8 . To i n v e s t i g a t e the e f f e c t w&ich the y i e l d i n g of the t e n s i o n s t e e l has on the d i s p l a c e m e n t o b s e r v a t i o n s , the y i e l d moment and c o r r e s p o n d i n g beam r o t a t i o n between column f a c e and p o i n t a t wh ich r o t a t i o n was m e a s u r e d , were e v a l u a t e d f o r each f r a m e . The t h e o r e t i c a l u l t i m a t e moment and the c o r r e s p o n d i n g beam r o t a t i o n were a l s o e v a l u a t e d i n o r d e r to comple te the t h e o r e t i c a l moment r o t a t i o n c u r v e . The r o t a t i o n c a l c u l a t i o n s are based on c o n s t a n t c o n -d i t i o n s ove r the 2 . 5 i n c h e s o v e r wh ich the r o t a t i o n s were m e a s u r e d . T h i s i s c l o s e enough f o r c o m p a r i s o n f o r the n o r t h s i d e , but can not be a p p l i e d to the s o u t h s i d e s i n c e the moment v a r i a t i o n i s too l a r g e , sometimes r e s u l t -i n g i n c u r v a t u r e r e v e r s a l i n s i d e the 2 . 5 i n c h t e s t l e n g t h . T h e r e f o r e o n l y the p l o t o f the n o r t h s i d e can s e r v e as a c o m p a r i s o n w i t h the t h e o r e t i c a l r e s u l t s . The l o a d s and c o r r e s p o n d i n g j o i n t t e n s i o n , j o i n t s h e a r and j o i n t b e n d i n g moment are as f o l l o w s : P = v e r t i c a l l o a d on each beam 41 . 0 20 40 GO 80 100 120 UO 160 180 200 BENDIN6 MOMENT (KIP IN.) o V RATIO F 0 R 2 D E P T H OF N E U T R A L AXIS ROTATION 2OF B E A M ( D E G R E E S ) 5 F i g . 8 T h e o r e t i c a l b e n d i n g moment a n d beam r o t a t i o n c u r v e s 4 2 . HL = h o r i z o n t a l l o a d a p p l i e d a t e a c h e n d o f t h e f r a m e S = h o r i z o n t a l s w a y - l o a d a p p l i e d a t t h e n o r t h s i d e o f t h e f r a m e o n l y H = j o i n t t e n s i o n = HL + S/2 - P x ( a + c / 2 ) / h V = j o i n t s h e a r = P - S x h / l f o r s o u t h s i d e = j o i n t s h e a r = P + S x h/1 f o r n o r t h s i d e M = j o i n t b e n d i n g moment P x a - S ( 1 - c ) h / ( 2 x 1 ) f o r s o u t h s i d e = j o i n t b e n d i n g moment P x a + S ( 1 - c ) h / ( 2 x l ) f o r n o r t h s i d e F r a m e N u m b e r 1: F i g . 9 s h o w s t h e a p p l i e d l o a d i n g c u r v e a s p l o t t e d a g a i n s t t h e o b s e r v a t i o n n u m b e r s . F i g . 1 0 , t h e S L I P - S H E A R p l o t , s h o w s a r a p i d i n c r e a s e i n s l i p a f t e r r e a c h i n g a s h e a r o f 13 K i p s f o r t h e s o u t h s i d e a n d a s i m i l a r i n c r e a s e i n s l i p a f t e r r e a c h i n g a s h e a r o f 18 K i p s f o r t h e n o r t h s i d e . T h e c u r v e s a r e p a r a l l e l o v e r t h e r e g i o n w h e r e m o s t o f t h e s l i p t a k e s p l a c e , e n d i n g w i t h a l m o s t t h e same f a i l u r e s h e a r . F i g . 1 1 , t h e SLIP-MOMENT p l o t , s h o w s t h e r a p i d i n c r e a s e o f s l i p f o r t h e s o u t h s i d e o c c u r s a t a p o i n t w h e r e t h e b e n d i n g m o ment i s q u i t e s m a l l ( 3 5 t o 50 K i p i n c h e s ) . On t h e n o r t h s i d e t h e r a p i d i n c r e a s e o f s l i p c o i n c i d e s w i t h t h e m o m e n t a t w h i c h t h e t e n s i o n s t e e l s t a r t s t o y i e l d (My = 138K i n c h e s a t a j o i n t t e n s i o n o f 5.5K) a n d p r o g r e s s e s r a p i d l y e v e n a f t e r t h e moment h a s f a l l e n o f f s l i g h t l y . F i g . 1 2 s h o w s t h e ROTATION-MOMENT p l o t . T h e s o u t h s i d e s h o w s an e r r a t i c r e s p o n s e due t o > t h e moment r e v e r s a l a n d 43. F i g . 9 L o a d i n g c u r v e s f o r Frame 1 4 4 . •s l ip 1 • s l i p sign convention 30 T 201 VERTICAL SL IP ( INCHES) Fig. 1 0 S L I P - S H E A R p l o t f o r F r a m e 1 300 •SLIP +SLIP sign convention 250 A CL *-? 2 0 0 LU 150 CD LJ CD north 0.1 0.2 0.3 0.4 VERT ICAL SLIP ( INCHES) F i g . 1 1 SLIP-MOMENT p l o t f o r F r a m e 1 +M ^ ROTATION +M +ROTATION F i g . 1 2 ROTATION-MOMENT p l o t f o r F r a m e 1 t h e s u m m i n g o f p o s i t i v e a n d n e g a t i v e c u r v a t u r e m e n t i o n e d a b o v e . T h e n o r t h s i d e s h o w s a s o m e w h a t f l a t t e r c u r v e f o r t h e r e c o r d e d d a t a t h a n e x p e c t e d f r o m t h e t h e o r e t i c a l p l o t . F r a m e N u m b e r 2: F i g . 1 3 s h o w s t h e a p p l i e d l o a d i n g c u r v e a s p l o t t e d a g a i n s t t h e o b s e r v a t i o n n u m b e r s . F i g . 1 4 , t h e S L I P - S H E A R p l o t , a n d F i g . 1 5 , t h e S L I P -MOMENT p l o t , show a r a p i d i n c r e a s e i n s l i p f o r t h e s o u t h s i d e a f t e r r e a c h i n g a s h e a r o f 18 K i p s a n d a moment o f 50 K i p i n c h e s . T h e n o r t h s i d e s h o w s a r a p i d i n c r e a s e i n s l i p o n l y a f t e r r e a c h i n g a s h e a r =25 K i p s a n d a moment o f o v e r 180 K i p i n c h e s . T h e l a s t p a r t o f t h e r e c o r d s w e r e l o s t d u e t o f a u l t y e q u i p m e n t s o t h a t t h e d a t a c o r r e s p o n d i n g t o t h e o b s e r v e d v i s u a l c r a c k i n g o f t h e n o r t h s i d e c o u l d n o t be p l o t t e d . T h e MOMENT-ROTATION c u r v e s , F i g . 1 6 , a r e v e r y s i m i l a r t o t h e c u r v e s o b s e r v e d f r o m F r a m e 1. F r a m e N u m b e r 3: F i g . 1 7 s h o w s t h e a p p l i e d l o a d i n g c u r v e as p l o t t e d a g a i n s t t h e o b s e r v a t i o n n u m b e r s . F i g . 1 8 , t h e S L I P - S H E A R p l o t , a n d F i g . 1 9 , t h e S L I P -MOMENT p l o t , show a r a p i d i n c r e a s e i n s l i p f o r t h e s o u t h s i d e a f t e r r e a c h i n g a s h e a r o f 13 K i p s a n d a moment o f 2 5 K i p i n c h e s . T h e n o r t h s i d e s h o w s a r a p i d i n c r e a s e i n s l i p o n l y a f t e r r e a c h i n g a s h e a r o f 24 K i p s a n d a moment o f 1 6 3 K i p i n c h e s . T h e t e n s i o n s t e e l s t a r t s t o y i e l d a t a moment o f 48. F i g . 1 3 L o a d i n g c u r v e s f o r F r a m e 2 . i* * + s l i p + s l i p s i g n c o n v e n t i o n "XT n o r t h s o u t h l | — f = _ , t , , =4 . . 0.0 0.1 0.2 0.3 0.4 V E R T I C A L SL IP ( I N C H E S ) F i g . 1 4 S L I P - S H E A R p l o t f o r F r a m e 2 5 0 . + M ••SLIP +SLIP sign convention 300 -cr 250 -F i g . 1 5 SLIP-MOMENT p l o t f o r F r a m e 2 +M +ROTATION +M +ROTATION s i g n c o n v e n t i o n 300 250 + ROTATION (DEGREES) 16 ROTATION-MOMENT p l o t f o r F r a m e 2 52. F i g . 1 7 L o a d i n g _ c u r v e s f o r F r a m e 3 53. + v +V HCZtfZ t t •slip +slip sign convention cn on < LU X CO 0.0 0.1 0.2 0.3 0.4 0.5 VERTICAL S L I P ( INCHES) F i g . 1 8 S L I P - S H E A R p l o t f o r F r a m e 3 F i g . 1 9 SLIP-MOMENT p l o t f o r F r a m e 3 +M +ROTATION +M +ROTATION sign convention 300 + 250 | i—t F i g . 2 0 R O T A T I O N - M O M E N T , p l o t f o r F r a m e 3 143K i n c h e s f o r a j o i n t t e n s i o n o f 2.4K. T h e f i n a l s h e a r c a r r i e d by t h e t w o j o i n t s i s a g a i n a l m o s t t h e s a m e . F i g . 2 0 s h o w s t h a t t h e ROTATION-MOMENT p l o t i s a g a i n a l m o s t i d e n t i c a l w i t h t h e r e s p o n s e o b t a i n e d f r o m t h e f i r s t t w o f r a m e s . F r a m e N u m b e r 4: F i g . 2 1 s h o w s t h e a p p l i e d l o a d i n g c u r v e . D u r i n g t h e l a s t s t a g e o f l o a d i n g t h e s w a y - l o a d was r e v e r s e d a n d t h e j o i n t t e n s i o n i n c r e a s e d up t o 10K. T h i s r e s u l t e d i n a d i f f e r e n t d i s p l a c e m e n t r e s p o n s e i n t h e f i n a l l o a d s t a g e . F i g . 2 2 , t h e S L I P - S H E A R p l o t , a n d F i g . 2 3 , t h e S L I P -MOMENT p l o t , a g a i n s h ow a r a p i d i n c r e a s e i n s l i p f o r t h e s o u t h s i d e a f t e r r e a c h i n g a s h e a r o f 14K a n d a moment o f 40K i n c h e s . T h e n o r t h s i d e s h o w s a r a p i d i n c r e a s e i n s l i p o n l y a f t e r r e a c h i n g a s h e a r o f 20K a n d a moment o f 170K i n c h e s . T h e t e n s i o n s t e e l y i e l d moment i s 136K i n c h e s f o r a j o i n t t e n s i o n o f 6K. F i g . 2 4 , t h e ROTATION-MOMENT p l o t , s h o w s a r e s p o n s e as b e f o r e f o r t h e n o r t h s i d e up t o t h e p o i n t w h e r e t h e s w a y -l o a d was r e v e r s e d . T h e r e v e r s a l o f s w a y - l o a d ( a n d t h u s s w a y -m o m e n t ) r e s u l t e d i n a d e c r e a s e o f r o t a t i o n on t h e n o r t h s i d e a n d an i n c r e a s e o f r o t a t i o n on t h e s o u t h s i d e d u r i n g t h e f i n a l s t a g e s o f l o a d i n g . T h e moment r e v e r s a l d i d n o t a f f e c t t h e s l i p r e s p o n s e , as t h e f i n a l s l i p a n d s h e a r i s a g a i n a l m o s t t h e same f o r b o t h j o i n t s . S i n c e t h e f a i l u r e s h e a r a n d s l i p r e s u l t s o f t h i s f r a m e , u n d e r a much l a r g e r j o i n t t e n s i o n , w e r e o f t h e same m a g n i t u d e a s f o r t h e f i r s t t h r e e f r a m e s , i t m u s t be c o n c l u d e d t h a t t h e j o i n t t e n s i o n h a s n o 30 i 25 A 20 i (Si CL *— 15 4 •z. %—i O 10 < O 5 4 + H L + p +p j L sign convention OBSERVATION N U M B E R S F i g . 2 1 L o a d i n g c u r v e s f o r F r a m e 4 58. 3 0 + 25 20 4 15 4 10 5 4 s o u t h 0.0 0.1 0.2 0.3 0.4 0.5 VERTICAL SL IP ( INCHES) F i g . 2 2 S L I P - S H E A R p l o t f o r F r a m e 4 F i g . 2 3 SLIP-MOMENT p l o t f o r F r a m e 4 +M +ROTATION +M +ROTAT ION ROTATION ( D E G R E E S ) F i g . 2 4 ROTATION-MOMENT p l o t f o r F r a m e 4 m a j o r e f f e c t on the u l t i m a t e s h e a r t r a n s f e r c a p a c i t y . A j o i n t t e n s i o n o f 10K c o r r e s p o n d s t o an average c o n c r e t e s t r e s s of 185ps i o r a s t e e l s t r e s s of 12000ps i i n the t e s t beam. S i n c e the modulus of r u p t u r e i s about 2 0 0 p s i f o r the beam c o n c r e t e , 10 K i p s c o n s t i t u t e s an upper l i m i t o f j o i n t t e n s i o n wh ich s h o u l d not be exceeded under normal a c t i o n of s h r i n k a g e and t e m p e r a t u r e v a r i a t i o n . To r e l a t e the main o b s e r v a t i o n s made w i t h r e s p e c t to c r i t i c a l s h e a r , an i n t e r a c t i o n p l o t was a t t e m p t e d , F i g . 2 5 , wh ich c l e a r l y shows c r i t i c a l s h e a r d i f f e r e n c e s between low and h i g h moment -shear f a i l u r e s . C r i t i c a l s h e a r i n t h i s case was d e f i n e d as the s h e a r which i n i t i a t e s s l i p and c r a c k i n g o f the j o i n t . T a b l e IV shows the l o a d s and d i s p l a c e m e n t s as o b s e r v e d at c r i t i c a l s h e a r f o r each f r a m e . In o r d e r to e x p l a i n the two a p p a r e n t s h e a r f a i l u r e modes, the p h y s i c a l components wh ich t r a n s f e r the s h e a r have to be i n v e s t i g a t e d i n more d e t a i 1. The s h e a r i s t r a n s f e r r e d a c r o s s the j o i n t by the f o l l o w i n g t h r e e s t r u c t u r a l c o m p o n e n t s : 1) beam t e n s i on s t e e l 2) beam c o m p r e s s i o n s t e e l 3) beam c o n c r e t e W i t h o u t more d e t a i l e d e x p e r i m e n t a l t e s t s no magn i tudes of s h e a r can be a l l o c a t e d to t h e s e components . S i n c e the mechanism o f the dowel a c t i o n of the t e n s i o n ( t o p ) s t e e l i s d i f f e r e n t f rom the dowel a c t i o n o f the c o m p r e s s i o n (bot tom) s t e e l , no g e n e r a l i z a t i o n i s p o s s i b l e . The t e s t r e s u l t s show t h a t the s h e a r t r a n s f e r c a p a c i t y of the j o i n t s i s h i g h e r i n t h e p r e s e n c e o f a l a r g e b e n d i n g moment t h a n f o r a s m a l l b e n d i n g m o m e nt. S i n c e an a x i a l t e n s i o n was i n d u c e d i n t h e b e a m , e n o u g h c o m p r e s s i o n was n o t d e v e l o p e d i n t h e c o n c r e t e t o t r a n s f e r a s i g n i f i c a n t a m o u n t o f s h e a r ( s h e a r f r i c t i o n h y p o t h e s i s ( 4 ) , ( 5 ) ) . B u t u n d e r t h e c o u p l e o f a l a r g e b e n d i n g m o m e n t , t h e c o m p r e s s i o n b l o c k i n t h e beam c o n c r e t e , e n h a n c e d due t o t h e c o n f i n i n g f o r c e s o f t o p a n d b o t t o m s t e e l a n d s t i r r u p s ( a x i a l c o m p r e s s i o n c a p a c i t y i n t h e p r e s e n c e o f l a t e r a l c o n f i n e m e n t (11) ) , c o u l d t r a n s f e r a s i g n i f i c a n t l y h i g h e r a m o u n t o f s h e a r t o t h e c o l u m n . T h i s t e n d e n c y was o b s e r v e d i n a l l f o u r f r a m e s , e s p e c i a l l y d u r i n g t h e i n i t i a l s t a g e s o f j o i n t c r a c k i n g . O n c e t h e j o i n t c o n c r e t e h a s c r a c k e d t o a h i g h e r d e g r e e , a s s h o w n i n t h e p r e v i o u s c h a p t e r , o n l y t h e d o w e l a c t i o n o f t h e s t e e l r e m a i n e d a n d i n a l l c a s e s t h e s h e a r t r a n s f e r r e d a t u l t i m a t e a p p r o a c h e d t h e same v a l u e o f a b o u t 30 K i p s i r r e s p e c t i v e o f moment. FRAME NUMBER SHEAR K I P S MOMENT K I P I N C H E S 1 S L I P I N C H E S R O T A T I O N | D E G R E E S 1 - S o u t h ,3.0 3 5 . 0 .0 30 .06 1 - N o r t h 1 8 . 0 1 4 0 . 0 . 0 2 8 - . 2 0 2 - S o u t h 18.0 5 0 . 0 . 014 -.12 2 - N o r t h ± 2 5 . 0 ± 1 8 0 . 0 .016 - . 2 0 3 - S o u t h 1 3 . 0 2 5 . 0 .0 40 + .20 3 - N o r t h 2 4 . 0 1 6 3 . 0 . 0 2 4 - . 2 8 4 - S o u t h - .14.0 4 0 . 0 .010 + .12 4 - N o r t h 2 0 . 0 1 70.0 .010 - . 2 4 J O I N T FOR T a b l e I V CES AND D I S P L A C E M E N T S AT C R I T I C A L SHEAR 6 4 . theoretical ultimate shear 100 M c r (KIPSlN) 150 2 0 0 F i g . 2 5 M o m e n t - s h e a r f a i l u r e i n t e r a c t i o n p l o t 4.3 C o n c l u s i o n T h e j o i n t i n t e r f a c e was p u r p o s e l y made s m o o t h t o o b t a i n u n i f o r m t e s t c o n d i t i o n s a n d t o p r o d u c e t h e w o r s t s h e a r - t r a n s f e r c o n d i t i o n w i t h r e g a r d t o t h e c o n c r e t e . T h i s j o i n t s t i l l s h o w e d a c o n s i d e r a b l e s h e a r - t r a n s f e r c a p a c i t y , e v e n u n d e r a d v e r s e l o a d i n g c o m b i n a t i o n s . T o c a l c u l a t e t h e u l t i m a t e s h e a r t r a n s f e r c a p a c i t y o f s u c h a j o i n t , t a k i n g i n t o a c c o u n t a l l t h e f a c t o r s c o n t r i b u t i n g t o i t , i s n o t p o s s i b l e f r o m t h e l i m i t e d r e s u l t s o b t a i n e d f r o m t h i s i n v e s t i g a t i o n . I t i s f e l t , h o w e v e r , t h a t t h e m a j o r o b s e r v a t i o n s d e s c r i b e d i n t h e p r e v i o u s c h a p t e r s c l e a r l y i n d i c a t e t h e f o l l o w i n g : 1) T h e t y p e o f j o i n t i n v e s t i g a t e d c a n t r a n s f e r t h e f u l l beam s h e a r t o an a d j a c e n t c o l u m n . 2) S h e a r f a i l u r e i n t h e f o r m o f c o m b i n e d d i a g o n a l t e n s i o n c r a c k i n g a n d v e r t i c a l j o i n t s l i p o c c u r s a t two l e v e l s o f s h e a r - m o m e n t a c t i o n . One d e v e l o p e s u n d e r t h e a c t i o n o f a l o w moment a n d f a i r l y l o w s h e a r f o r c e . T h e s e c o n d d e v e l o p e s u n d e r t h e a c t i o n o f a much h i g h e r m o m e n t , w h i c h a p p a r e n t l y c o i n c i d e s w i t h t h e y i e l d i n g o f t h e t e n s i o n s t e e l , a n d a c o n s i d e r a b l e s h e a r f o r c e . T h e u l t i m a t e s h e a r , h o w e v e r , r e a c h e s a p p r o x i m a t e l y t h e same v a l u e i n b o t h c a s e s . 3) L a p p i n g o f t h e l o n g i t u d i n a l r e i n f o r c e m e n t h a s no d e t r i m e n t a l e f f e c t on t h e j o i n t c a p a c i t y . I t was e v e n o b s e r v e d t h a t t h e f i r s t two f r a m e s s h o w e d s l i g h t l y h i g h e r s l i p d e f l e c t i o n s a t l o w e r l o a d s t h a n t h e l a s t t w o f r a m e s w h i c h w e r e r e i n f o r c e d w i t h l a p p e d l o n g i t u d i n a l r e i n f o r c e m e n t . 4) T h e s h e a r t r a n s f e r c a p a c i t y o f t h e j o i n t i s n o t a p p r e c i a b l y a f f e c t e d by t h e p r e s e n c e o f a j o i n t t e n s i o n . M o r e d e t a i l e d t e s t s a r e r e q u i r e d t o p r o d u c e a f a i l u r e e n v e l o p e u n d e r t h e a c t i o n o f t h e t y p e o f l o a d i n g w h i c h was a p p l i e d i n t h i s i n v e s t i g a t i o n . M o r e t e s t s a r e a l s o r e q u i r e d t o f i n d t h e c o n t r i b u t i o n o f t h e r e i n f o r c i n g s t e e l a n d t h e c o n c r e t e t o t h e s h e a r c a p a c i t y o f t h e j o i n t . 6 7 . R e f e r e n c e s 1. SAEMANN, J . C . a n d WASHA, G.W. " H o r i z o n t a l S h e a r C o n n e c t i o n s B e t w e e n P r e c a s t Beams a n d C a s t - I n - P l a c e S l a b s " , A C I , N o v e m b e r 1 9 6 4 . 2. BADOUX, J . C . a n d H U L S B O S , C . L . " H o r i z o n t a l S h e a r C o n n e c t i o n i n C o m p o s i t e C o n c r e t e Beams U n d e r R e p e a t e d L o a d s " , A C I , D e c e m b e r 1 9 6 7 . 3. K R I Z , L . B . a n d R A T H S , C.H. " C o n n e c t i o n s i n P r e c a s t C o n c r e t e S t r u c t u r e s - S t r e n g t h o f C o r b e l s " , P C I , F e b r u a r y 1 9 6 5 . 4. B I R K E LAN D , P.W. a n d B I R K E L A N D , H.W. " C o n n e c t i o n s i n P r e c a s t C o n c r e t e C o n s t r u c t i o n " , A C I , M a r c h 1 9 6 6 . 5. MAST, R . F . " A u x i l i a r y R e i n f o r c i n g i n C o n c r e t e C o n n e c t i o n s " , A S C E , J u n e 1 9 6 8 . 6. K R E F E L D a n d T H U R S T O N . " C o n t r i b u t i o n o f L o n g i t u d i n a l S t e e l t o S h e a r R e s i s t a n c e o f R e i n f o r c e d C o n c r e t e B e a m s " , A C I , M a r c h 1 9 6 6 . 7. K R E F E L D a n d T H U R S T O N . " S h e a r a n d D i a g o n a l T e n s i o n S t r e n g t h o f S i m p l y S u p p o r t e d R e i n f o r c e d C o n c r e t e B e a m s " , A C I , M a r c h 1 9 6 6 . 8. HANSON, N.W. a n d CONNOR, H.W. " S e i s m i c R e s i s t a n c e o f R e i n f o r c e d C o n c r e t e Beam-C o l u m n J o i n t s " , A S C E , O c t o b e r 1 9 6 7 . 9. H O F B E C K , J . A . , I B R A H I M , 1.0. a n d MATTOCK, A . H . " S h e a r T r a n s f e r i n R e i n f o r c e d C o n c r e t e " , A C I , F e b r u a r y 1 9 6 9 . 1 0 . S M I T H , R.B.C. " I n t e r a c t i o n o f M o m e n t a n d S h e a r on t h e F a i l u r e o f R e i n f o r c e d C o n c r e t e Beams W i t h o u t Web R e i n f o r c e m e n t " , C I V I L E N G I N E E R I N G & P U B L I C WORKS REVIEW ( L O N D O N ) , J u n e , J u l y a n d A u g u s t , 1 9 6 6 . 1 1 . PORTLAND CEMENT A S S O C I A T I O N . D E S I G N OF M U L T I - S T O R Y R E I N F O R C E D CONCRETE B U I L D I N G S FOR EARTHQUAKE MOTIONS, 1 9 6 1 . 6 8 . 1 2 . AMERICAN CONCRETE INSTITUTE. BUILDING CODE REQUIREMENTS FOR REINFORCED CONCRETE, ACI ( 31 8 - 1 9 6 3 ) 1 3 . ZSUTTY, T . C . "Beam Shear S t r e n g t h P r e d i c t i o n s by A n a l y s i s of E x i s t i n g D a t a " , A C I , November 1968. 14 . BURTON, CORLEY and HOGNESTAD. " C o n n e c t i o n s i n P r e c a s t C o n c r e t e S t r u c t u r e s -E f f e c t s o f R e s t r a i n e d Creep and S h r i n k a g e " , P C I , A p r i l 1967 . 1 5 . MATTOCK, A . H . " C r e e p and S h r i n k a g e S t u d i e s " , PCA RESEARCH & DEVELOPEMENT LABORATORY, May 1 9 6 1 . 1 6 . COHEN, M.Z . " R o t a t i o n C o m p a t i b i l i t y i n the L i m i t Des ign of R e i n f o r c e d C o n c r e t e C o n t i n u o u s B e a m s " , REINFORCED CONCRETE SYMPOSIUM. 1 7 . YOUNG, L . E . and SMITH, G . M . " U l t i m a t e F l e x u r a l A n a l y s i s Based on S t r e s s - S t r a i n Curves o f C y l i n d e r s " , A C I , December 1956 . 1 8 . SAENZ, L . P . D i s c u s s i o n o f " E q u a t i o n f o r the S t r e s s - S t r a i n Curve of C o n c r e t e by P . D e s a y i and K r i s h n a n " , A C I , September 1964. A P P E N D I X : S T R E S S - S T R A I N CURVES OF T E S T S P E C I M E N S 100 utt. 73.6ksi (turned down to 0=.29in) 0 +• 0 2 4 6 8 10 12 S T R A I N (%IN/IN ) S t r e s s - s t r a i n d i a g r a m f o r 3/8 i n c h d i a m e t e r r e i n f o r c i n g s t e e l 100 ult. 87.2ksi (0 = 5/8in.nominal) CO 80 60 cn cn LU CC 40 h -cn ult.78.5ksi (turned down to 0=.423in.) ult.72.4ksi(turned down to 0=.42in) ult.73.gksi(tured down to 0=.447in) ult.72. 8 ksi (turned down to 0=.42in.) 20 0 1 10 1 2 2 4 6 8 STRAIN (% IN/IN ) S t r e s s - s t r a i n d i a g r a m f o r 5/8 i n c h d i a m e t e r r e i n f o r c i n g s t e e l 14 1 6 5000 t 4 0 0 0 4 0.0 0.1 0.2 0.3 0.4 STRAIN <% IN/IN) S t r e s s - s t r a i n p l o t o f 6 t e s t c y l i n d e r s t a k e n f r o m b e am-c o n c r e t e a n d s u p e r i m p o s e d t h e o r e t i c a l c u r v e s 6000 S T R A I N (%IN/IN) S t r e s s - s t r a i n p l o t o f 6 t e s t c y l i n d e r s t a k e n f r o m c o l u m n -c o n c r e t e a n d s u p e r i m p o s e d t h e o r e t i c a l c u r v e s 

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