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Failure mechanisms of concrete masonry Yao, Chicao 1989

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FAILURE MECHANISMS OF C O N C R E T E MASONRY By CHICHAO Y A O B. ENG. Tong Ji University, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA April 1989 © Chichao Yao, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The behaviour of concrete masonry under in-plane compression combined with out-of-plane bending was examined both experimentally and analytically. Ungrouted and grouted masonry, both fully bedded or face-shell bedded, were included in the study. It was found that the masonry under the above stated loading conditions may suffer loss of capacity either due to splitting or shear type of material failure, or by instability. Different loading conditions yield different failure mechanisms, which in turn correspond to different apparent strengths. Theoretical developments are presented leading to estimates of capacity for each of these cases. An extensive experimental program involving 104 masonry prism specimens, was conducted to assist and to verify these analyses. Theoretical developments include those directed to explain splitting failure phenomena, to investigate the mortar joint effect, the deformation compatibility of grouted masonry, and to examine the slenderness of tall masonry wall. Experimental measurements and observations made on the specimens include capacity, deformation and failure pattern. - iii -T A B L E O F C O N T E N T S P A G E A B S T R A C T -ii-T A B L E O F C O N T E N T S - i i i -LIST O F T A B L E S . . •. -vii-LIST O F F I G U R E S , -ix-N O T A T I O N , .-xvi-A C K N O W L E D G E M E N T -xx-D E D I C A T I O N -xxi-C H A P T E R I I N T R O D U C T I O N 1 1.1 General Remarks 1 1.2 Object and Scope.... 2 II E X P E R I M E N T A L W O R K 3 2.1 Purpose and Scope 3 2.2 Materials 3 2.2.1 Masonry Unit 3 2.2.2 Mortar 7 2.2.3 Grout 10 2.3 Specimens 13 2.4 Testing Device 15 2.5 Instrumentation 15 2.6 Data Acquisition 19 2.7 Summary of Characteristic Results 22 III S O M E B A C K G R O U N D T O C O M P R E S S I O N F A I L U R E O F C O N C R E T E 29 3.1 Purpose 29 - i v -C H A P T E R P A G E 3 .2 B r i t t l e F a i l u r e u n d e r U n i a x i a l C o m p r e s s i o n 30 3 .3 M o d e l s o f I n t e r n a l B r i t t l e F a i l u r e 35 3 .4 P r o p o s e d M o d e l 37 3 .4 .1 C r a c k I n t e r a c t i o n a n d C r i t i c a l S t a t e 37 3 . 4 . 2 S o m e C o n s e q u e n c e s o f t h e M o d e l : P e a k S t r e s s 40 3 . 4 . 3 R e l a t i o n t o T e n s i l e S t r e n g t h 4 1 3 .5 T h e S t r e s s - S t r a i n C u r v e s f o r B r i t t l e M a t e r i a l s u n d e r C o m p r e s s i o n 4 4 3 .5 .1 T h e P r e - P e a k B r a n c h 4 5 3 . 5 . 2 T h e P o s t P e a k B r a n c h 50 3 . 5 . 3 T h e P r e d i c t e d S t r e s s - S t r a i n C u r v e 51 3 .6 S t a t i s t i c a l C o n s i d e r a t i o n 54 3 .7 S u m m a r y a n d C o r o l l a r y 57 I V P L A I N M A S O N R Y W I T H F U L L B E D D I N G 5 8 4 .1 T w o B a s i c a l l y D i f f e r e n t F a i l u r e M o d e s 58 4 . 2 J o i n t E f f e c t A R e v i s i o n o f H i l s d o r P s M o d e l 59 4 . 2 . 1 E x p e r i m e n t a l R e s u l t s 61 4 . 2 . 2 T h e o r e t i c a l A n a l y s i s 64 4 . 2 . 3 C o n c l u s i o n o n H i l s d o r f s M o d e l 7 4 4 . 3 S o m e C o m m e n t s o n S p l i t t i n g F a i l u r e a n d M o d e T r a n s i t i o n P h e n o m e n a 75 4 . 4 J o i n t E f f e c t o n A x i a l C a p a c i t y 76 4 . 5 S t r e s s i n J o i n t V i c i n i t y 78 4 . 6 C a p a c i t y E a t i m a t i o n 84 4 . 7 S u m m a r y 88 V P L A I N M A S O N R Y W I T H F A C E - S H E L L B E D D I N G 8 9 5 .1 I n t r o d u c t i o n • 89 - V -C H A P T E R P A G E 5 .2 E x p e r i m e n t a l W o r k 89 5 . 3 S t r e s s A n a l y s i s 94 5 .4 S o m e C o m m e n t s o n J o i n t E f f e c t 9 7 5 .5 S u m m a r y 98 V I P L A I N M A S O N R Y U N D E R E C C E N T R I C C O M P R E S S I O N : 100 6.1 F a i l u r e M o d e T r a n s i t i o n 100 6 . 2 E f f e c t o f J o i n t C o n d i t i o n s • 105 6 .3 S u m m a r y 106 V I I R E C O M M E N D E D D E S I G N A P P R O A C H F O R P L A I N M A S O N R Y 107 7.1 R e c o m m e n d a t i o n s o n t h e B a s i s f o r D e s i g n 107 . 7 .2 D i s c u s s i o n o f t h e C u r r e n t D e s i g n C o d e 119 V I I I G R O U T E D M A S O N R Y W I T H F U L L B E D D I N G 125 8.1 I n t r o d u c t i o n 125 8 .2 E x p e r i m e n t a l O b s e r v a t i o n s 126 8 .3 A n a l y s i s 132 8 .4 S u m m a r y 152 I X G R O U T E D M A S O N R Y W I T H F A C E - S H E L L B E D D I N G 153 X G R O U T E D A N D R E I N F O R C E D M A S O N R Y U N D E R E C C E N T R I C C O M P R E S S I O N S 10 .1 G e n e r a l R e m a r k s 157 10 .2 E x p e r i m e n t a l O b s e r v a t i o n s 157 1 0 . 3 T h e o r e t i c a l C o n s i d e r a t i o n s 161 1 0 . 4 C o m p a r i s o n o f T h e o r y w i t h E x p e r i m e n t s . . . 1 6 7 X I S L E N D E R N E S S O F C O N R E T E M A S O N R Y 174 11 .1 I n t r o d u c t i o n 174 1 1 . 2 B a c k g r o u n d I n f o r m a t i o n R e v i e w '. . . . 1 7 5 - v i -C H A P T E R P A G E 11.3 Mason ry Character is t ics and Some Assumpt ions .. . . .178 11.4 Di f ferent ia l Equat ions Govern ing Concrete Mason ry w i t h C r a c k e d Sect ion 179 11.5 Resul ts and A p p l i c a t i o n s 191 11.6 Usefulness and L i m i t a t i o n s 199 11.7 Some S imp l i f i ca t ions 202 X I I C O N C L U S I O N S 207 R E F E R E N C E S 209 A P P E N D I X A Expressions for dU and dR i n Chapte r III 216 B So lu t ion of equat ion 3.10 219 C So lu t ion of equat ion 4.1 . 222 D Coeff icients A m , Bm i n stress funct ion <3> specified by equat ion 4.11 223 E D e r i v a t i o n of equat ion 11.5 226 F Integrat ion of equations 11.14 and 11.16 227 G Con f igu ra t ion of a c o l u m n loaded w i t h double curvature bending 229 H E lect ron ic C i r c u i t Used i n Detect ing Macroscop ic S p l i t t i n g (Par t ) 232 G C o m p u t e r P r o g r a m C a l c u l a t i n g B u c k l i n g L o a d and M o m e n t Magn i f ie r of . Concrete Mason ry . . . .233 - v i i -L I S T O F T A B L E S T A B L E P A G E 2.1 F a i l u r e Loads of B l o c k U n i t s 5 2.2 M i x P r o p o r t i o n of M o r t a r 8 2.3 28 D a y M o r t a r C u b e Strength 8 2.4 M i x P r o p o r t i o n of G r o u t 11 2.5 G r o u t St rength by S tandard P r i s m Tests 11 2.6 G r o u t Strength by Tests on Cores T a k e n f rom F a i l e d P r i s m s 11 2.7 P r i s m Specimen 14 2.8 A S u m m a r y of F a i l u r e and C a p a c i t y Character is t ics 23 4.1 Fa i l u re Loads of P l a i n P r i s m s w i t h F u l l Bedd ing 64 5.1 F a i l u r e Loads of P l a i n P r i s m s w i t h Face -She l l Bedd ing 91 6.1 F a i l u r e Loads of P l a i n P r i s m s under Eccent r ic L o a d 100 7.1 C o m p a r i s o n w i t h the Recommended A p p r o a c h : Tests by A u t h o r 115 7.2 C o m p a r i s o n w i t h the Recommended A p p r o a c h : Tests by F a t t a l et a l 116 7.3 C o m p a r i s o n w i t h the Recommended A p p r o a c h : Tests by Hatz in iko las et a l 116 7.4 C o m p a r i s o n w i t h the Recommended A p p r o a c h : Tests by Drysdale et a l 117 7.5 F l e x u r a l to U n i a x i a l Strength: Tests by A u t h o r 122 7.6 F l e x u r a l to U n i a x i a l St rength : Tests by F a t t a l et a l 123 7.7 F l e x u r a l to U n i a x i a l St rength : Tests by Hatz in iko las et a l 123 7.8 F l e x u r a l to U n i a x i a l Strength: Tests by Drysdale et a l 124 8.1 Fa i l u re Loads of G r o u t e d P r i s m s (kips) , w i t h V a r i a t i o n in J o n i t C o n d i t i o n 128 8.2 F a i l u r e Loads of G rou ted P r i s m s (kips), w i t h V a r i a t i o n i n G r o u t 128 8.3 G r o u t e d P r i s m s , Tests by the A u t h o r 144 8.4 G r o u t e d P r i s m s , Tests by H a m i d and Drysdale 145 8.5 G r o u t e d P r i s m s , Tests by Drysdale H a m i d 146 - v i i i -T A B L E P A G E 8.6 G r o u t e d P r i s m s , Tests by W o n g and Drysdale 146 8.7 G r o u t e d P r i s m s , Tests by Pr iest ley and E lder 147 8.8 G r o u t e d P r i s m s , Tests by B o u l t 147 8.9 G r o u t e d P r i s m s , Tests by T h u r s t o n 147 8.10 M o d e l P red ic t i on versus C r a c k i n g Loads , Tests by the A u t h o r 152 9.1 G r o u t e d Mason ry w i t h Face -She l l Bedd ing 154 10.1 Fa i l u re Loads of G rou ted P r i s m s under Eccentr ic L o a d (kips) 158 - i x -L I S T O F F I G U R E S F I G U R E P A G E 2 .1 M a s o n r y U n i t 5 2 .2 C o n i c a l F a i l u r e o f M a s o n r y U n i t 6 2 . 3 S t r e s s - S t r a i n R e l a t i o n o f M a s o n r y U n i t u n d e r C o m p r e s s i o n 7 2 . 4 S t r e s s - S t r a i n R e l a t i o n o f M o r t a r 9 2 . 5 M e a s u r e d V e r t i c a l C o m p r e s s i v e S t r a i n s a l o n g B l o c k U n i t s a n d a c r o s s M o r t a r J o i n t o f P l a i n P r i s m s u n d e r U n i a x i a l C o m p r e s s i o n 10 2 .6 S t r e s s - S t r a i n R e l a t i o n o f G r o u t 12 2 .7 T e s t i n g D e v i c e 16 2 .8 L o a d i n g P l a t e n s 17 2 .9 I n s t r u m e n t a t i o n : L V D T s a n d G l u e d W i r e s 17 2 .10 E l e c t r o n i c D e v i c e D e t e c t i n g W i r e B r e a k O r d e r 20 2 .11 D a t a A c q u i s i t i o n S e t u p 21 2 . 1 2 S p l i t t i n g F a i l u r e o f P l a i n C o n c r e t e M a s o n r y w i t h F u l l B e d d i n g u n d e r U n i a x i a l C o m p r e s s i o n ; 24 2 . 1 3 F a i l u r e o f P l a i n M a s o n r y w i t h F a c e - S h e l l B e d d i n g u n d e r U n i a x i a l C o m p r e s s i o n 25 2 . 1 4 F a i l u r e o f F a c e - S h e l l B e d d e d , F u l l y C a p p e d M a s o n r y u n d e r U n i a x i a l C o m p r e s s i o n 2 6 2 . 1 5 F a i l u r e o f P l a i n M a s o n r y u n d e r E c c e n t r i c C o m p r e s s i o n 2 7 2 . 1 6 F a i l u r e o f G r o u t e d M a s o n r y u n d e r E c c e n t r i c C o m p r e s s i o n 28 3 .1 A S l i d i n g F r i c t i o n a l C r a c k i n a C o m p r e s s i v e S t r e s s F i e l d , S h o w i n g t h e O r i g i n a l D e f e c t a n d i t s E x t e n s i o n 32 3 .2 D e p i c t i o n o f t h e E f f e c t o f a C r a c k 32 3 .3 M o d e l s o f M a t e r i a l D e f e c t s . ( T h e M i s s i n g F o r c e A c t s o n E a c h S i d e i n a D i r e c t i o n O p p o s i t e t o t h a t S h o w n . T h e E f f e c t o f t h e D e f e c t is T h e r e f o r e t o A p p l y F o r c e s i n t h e D i r e c t i o n S h o w n . ) 34 3 .4 A S e r i e s o f C r a c k s i n a C o m p r e s s i v e S t r e s s F i e l d : T w o L e v e l s o f I d e a l i z a t i o n 39 3 .5 A S e r i e s o f C r a c k s i n a T e n s i l e S t r e s s F i e l d 4 3 - X -3 .6 P r e d i c t e d R e l a t i o n b e t w e e n T e n s i l e s t r e n g t h a n d C o m p r e s s i v e S t r e n g t h v e r s u s S i z e / S p a c i n g R a t i o f o r B r i t t l e M a t e r i a l s 4 3 3 .7 E x p e r i m e n t a l S t r e s s - S t r a i n R e l a t i o n s o f C o n c r e t e , u n d e r N o r m a l T e s t C o n d i t i o n s ( W a n g 1 9 7 8 ) "... 44 3 .8 E x p e r i m e n t a l S t r e s s - S t r a i n R e l a t i o n s o f C o n c r e t e , S p e c i m e n s w i t h " A n t i - F r i c t i o n " C a p p i n g ( K o t s o v o s 1 9 8 3 ) : (a ) S t r e s s v e r s u s S t r a i n M e a s u r e d o n t h e S p e c i m e n s ; ( b ) L o a d v e r s u s D i s p l a c e m e n t 4 6 3 .9 E x p e r i m e n t a l S t r e s s - S t r a i n R e l a t i o n s o f s o m e N a t u r a l R o c k s ( W a w e r s i k a n d F a i r h u r s t 1 9 7 0 ) 4 7 3 . 1 0 P r e d i c t e d S t r e s s - S t r a i n R e l a t i o n s o f B r i t t l e M a t e r i a l s 52 3 .11 D e p i c t i o n o f I r r e g u l a r C r a c k i n g P a t t e r n 54 3 . 1 2 S e n s i t i v i t y o f C o m p r e s s i v e S t r e n g t h t o C r a c k C o n f i g u r a t i o n F a c t o r : t h e N o r m a l i z e d S t r e n g t h P r e d i c t e d b y t h e M o d e l i s P l o t t e d a g a i n s t t h e C o n f i g u r a t i o n F a c t o r , W h i c h D e p e n d s o n C r a c k C o n f i g u r a t i o n a n d I n t e r n a l F r i c t i o n 5 6 4 .1 A p p a r e n t S t r e n g t h I n c r e a s e P h e n o m e n o n u n d e r E c c e n t r i c C o m p r e s s i o n 5 9 4 .2 M e a s u r e d L a t e r a l S t r a i n s i n W e b s o f M i d d l e C o u r s e o f P l a i n M a s o n r y P r i s m s u n d e r U n i a x i a l C o m p r e s s i o n 6 3 4 . 3 D e t e c t e d O r d e r s o f M a c r o s c o p i c S p l i t t i n g , i n T e r m s o f 4 S e c t i o n s a l o n g P r i s m s 6 3 4 . 4 A M o r t a r J o i n t S a n d w i c h e d b y B l o c k U n i t s : a ) u n d e r A x i a l C o m p r e s s i o n ; b ) u n d e r B i a x i a l C o m p r e s s i o n c ) u n d e r L a t e r a l T r a c t i o n 66 4 . 5 A M o r t a r J o i n t u n d e r L a t e r a l T r a c t i o n 67 4 . 6 L a t e r a l I n t e r f a c e S h e a r D i s t r i b u t i o n b e t w e e n M o r t a r J o i n t a n d B l o c k U n i t s . 69 4 . 7 D e p i c t i o n o f B o u n d a r y C o n d i t i o n s o f a W e b ( o r F a c e - S h e l l ) u n d e r A c t i o n o f I n t e r f a c e S h e a r s 70 4 . 8 L a t e r a l T e n s i l e S t r e s s a l o n g T o p o f B l o c k I n t r o d u c e d b y t h e L a t e r a l S h e a r s 71 4 . 9 L a t e r a l T e n s i l e S t r e s s a l o n g T o p o f B l o c k , w i t h V a r i a t i o n i n P o i s s o n ' s R a t i o o f J o i n t 71 4 . 1 0 L a t e r a l T e n s i l e S t r e s s a l o n g T o p o f B l o c k , w i t h V a r i a t i o n i n J o i n t T h i c h n e s s 72 4 .11 L a t e r a l T e n s i l e S t r e s s a l o n g T o p o f B l o c k , w i t h V a r i a t i o n i n D o m a i n A s p e c t R a t i o 72 - xi -FIGURE PAGE 4.12 Lateral Strains Measured along webs and Face-shells of Plain Prisms under Uniaxial Compression 73 4.13 A Cross-Sectional View (along the Depth of Block Shells) of Mortar Joint 79 4.14 Compressive Stress, Lateral Confining Stress and Confined Strength in Mortar Joint 79 4.15 Failure Curve of Concrete under Shear and Compression 82 4.16 Prism Strength versus Mortar Cube Strength 87 4.17 Prism Strength versus Mortar Cube Strength, with Joint Thickness Doubled 87 5.1 Depiction of Deep Beam Mechanism 90 5.2 Detected Orders of Macroscopic Splitting, in Terms of 4 Sections along Prisms. (Face-Shell Bedded Prisms) 90 5.3 Measured Deformations at Certain Locations of Face-Shell Bedded Prisms: a) S16-1; b) S16-2; c) M27-1 92 5.4 Measured Deformations at Certain Locations of Face-Shell Bedded Prisms: a) N15-1; b) S15-3; c) N15-4 93 5.5 Lateral Stress Distribution in a Web of Face-Shell Bedded Masonry under Uniaxial Compression: Variation across Top of Block, as well as Vertical Distribution on Centre Line and at Vertical Line when the Tension at the Top is a Maximum 95 5.6 Forces Acting on a Block with Full Capping and Face-Shell Bedding 96 5.7 Lateral Stress Distribution in a Web: Full Capping versus Face-Shell Capping; Variation across Bottom of Block, as well as Vertical Distributionon on Centre Line and Quater Line 96 5.8 Prism Strength versus Unit Strength for Face-Shell Bedded Masonry 99 6.1 Measured Deformations at Certain Locations of Plain Prisms under Eccentric Load: a) N18-1, e=t/6; b) N18-4, e=t/6; c) N19-4, e=t/3; d) M20-2, e=t/3 101 6.2 Measured Deformations at Certain Locations of Plain Prisms under Eccentric Load: a) S21-4, e=t/3; b) S21-3, e=t/3; c) N22-2, e=t/3; d) N22-4, e=t/3 102 6.3 Stress Distributions in a Cracked Section 104 6.4 Lateral Stress along Top of a Web with Face-Shell Bedding under Eccentric Load 104 - Xl l -F I G U R E P A G E 6.5 S t r a i n D i s t r i b u t i o n i n a Sect ion of Masonry under Eccent r ic L o a d 106 7.1 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by the A u t h o r : N18 , N19 , M 2 0 and S21 110 7.2 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by the A u t h o r : N 2 2 (Face-Shel l Bedd ing) 110 7.3 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by F a t t a l and C a t t a n e o . . . 111 7.4 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Hatz in iko las et a l I l l 7.5 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysda le and H a m i d : N o r m a l B l o c k 112 7.6 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysda le and H a m i d : W e a k B l o c k 112 7.7 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysda le and H a m i d : S t rong B l o c k 113 7.8 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysdale and H a m i d : L i g h t W e i g h t B l o c k 113 7.9 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysda le and H a m i d : 7 5 % So l i d B l o c k 114 7.10 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysdale and H a m i d : 6 inch B l o c k 114 7.11 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysda le and H a m i d : 10 i nch B l o c k 115 7.12 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments : S u m m a r y 118 7.13 Dep ic t i on of Recommended A p p r o a c h 118 7.14 Cur rent Design Base: U n i a x i a l St rength versus F l e x u r a l Strength 122 8.1 G r o u t e d P r i s m Strength Versus M o r t a r Strength and G r o u t Strength 128 8.2 Measured Deformat ions at C e r t a i n Locat ions of G rou ted P r i s m s under Concent r ic L o a d : a) M 9 - 1 ; b) M 9 - 2 ; c) S8 -1 ; d) S8-2 129 - x m -F I G U R E P A G E 8.3 Measured Deformat ions at C e r t a i n Locat ions of G r o u t e d P r i s m s under Concent r ic L o a d : a) N10 -3 ; b) N10 -4 ; c) N12 -2 ; d) N12 -4 130 8.4 Measured Deformat ions at C e r t a i n Locat ions of G r o u t e d P r i s m s under Concent r ic L o a d : a) N13 -3 ; b) N13-4 ; c) N14 -3 ; d) N14 -4 130 8.5 A G r o u t e d Mason ry P r i s m w i t h Squre Cross -Sect ion 135 8.6 M o d e l P red ic t i on versus Exper iments , Based on Fa i l u re C o n d i t i o n a) 148 8.7 M o d e l P r e d i c t i o n versus Exper iments , Based on Fa i l u re C o n d i t i o n b) 148 8.8 M o d e l P red ic t i on versus Exper iments , Based on M o d i f i e d E q u a t i o n 151 8.9 M o d e l P r e d i c t i o n versus C r a c k i n g Loads, Based on Fa i lu re C o n d i t i o n a) 151 9.1 Measured Deformat ions at C e r t a i n Locat ions of G rou ted , Face -She l l Bedded P r i s m s under Concent r ic Compress ion : a) N17 -3 ; b) N17 -4 156 10.1 Measured Deformat ions at C e r t a i n Locat ions of G rou ted P r i s m s under Eccent r ic Compress ion : a) N 2 6 - 1 , e = t / 6 ; b) N26 -2 , e = t / 6 ; c) M 2 6 - 2 , e = t / 3 159 10.2 Measured Deformat ions at C e r t a i n Locat ions of G rou ted P r i s m s under Eccent r ic Compress ion : a) M 2 6 - 3 , e = t / 3 ; b) S25-1 , e = t / 3 ; c) S25-1, e = t / 3 160 10.3 A s s u m e d Stress D i s t r i b u t i o n of an Uncracked Sect ion and a C r a c k e d Section 163 10.4 A T y p i c a l Sect ion of a G rou ted W a l l 164 10.5 Stress D i s t r i b u t i o n along a Sect ion and Its C o m p o s i t i o n 164 10.6 C o m p a r i s o n of P red ic ted Interact ion C u r v e w i t h Exper iments by the A u t h o r : N26 , M 2 6 , S25 169 10.7 C o m p a r i s o n of P red ic ted Interact ion Curve w i t h Exper iments by Drysdale and H a m i d : N o r m a l B l o c k , T y p e N G r o u t 169 10.8 C o m p a r i s o n of P red ic ted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : N o r m a l B l o c k , T y p e W G r o u t 170 10.9 C o m p a r i s o n of P red ic ted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : N o r m a l B l o c k , T y p e S G r o u t 170 - x i v -F I G U R E P A G E 10.10 C o m p a r i s o n of Pred icted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : W e a k B l o c k , T y p e N G r o u t 171 10.11 C o m p a r i s o n of Pred icted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : S t rong B l o c k , T y p e N G r o u t 171 10.12 C o m p a r i s o n of P red ic ted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : 7 5 % So l i d B l o c k , T y p e N G r o u t 172 10.13 C o m p a r i s o n of P red ic ted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : F u l l B l o c k , 172 10.14 C o m p a r i s o n of Pred icted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : 6 i nch B l o c k , T y p e N G r o u t 173 10.15 C o m p a r i s o n of Pred icted Interact ion C u r v e w i t h Exper iments by Drysda le and H a m i d : 10 inch B l o c k , T y p e N G r o u t 173 11.1 A L o a d — M o m e n t Interact ion C u r v e and L o a d i n g P a t h s of a Compress ion M e m b e r 176 11.2 A Cross -Sect ion V i e w of A Reinforced Concrete W a l l under Eccent r ic Compress ion 180 11.3 C o l u m n Def lect ion C u r v e 189 11.4 C r i t i c a l L o a d versus C r a c k D e p t h at M i d d l e Sect ion of a P l a i n , So l id M e m b e r Loaded at e = t / 6 192 11.5 C r i t i c a l L o a d versus L o a d i n g Eccent r ic i t y for a So l i d Sect ion 192 11.6 C r i t i c a l L o a d versus L o a d i n g Eccent r ic i ty : X = a/b=0, np Var ies 194 11.7 C r i t i c a l L o a d versus L o a d i n g Eccent r ic i ty : A = 0.5, rap = 0.05, a/b Var ies 194 11.8 C r i t i c a l L o a d versus L o a d i n g Eccent r ic i ty : a / f r=0.65, np = 0, A Var ies 195 11.9 Theoret ica l P — M Interact ion C u r v e and L o a d i n g Pa ths C o m p a r e d w i t h Exper iments by Hatz in iko las et a l : 137 inch H i g h W a l l w i t h Reinforcement 3#3 197 11.10 Theore t ica l L o a d — E c c e n t r i c i t y C u r v e C o m p a r e d w i t h Exper iments by Ha tz in iko las et a l : 137 inch H i g h W a l l w i t h Reinforcement 3#3 . T h e P o i n t s Show the E x p e r i m e n t a l Resul ts whi le the Cont inuous L ines Show the P red ic t i on 197 11.11 Theoret ica l L o a d — E c c e n t r i c i t y Curve C o m p a r e d w i t h Exper iments by Hatz in iko las et a l : 137 i nch H i g h W a l l w i t h Reinforcement 3#6 198 - XV -F I G U R E P A G E 11.12 Theoret ica l L o a d — E c c e n t r i c i t y Curve C o m p a r e d w i t h Exper iments by Ha tz in iko las et a l : 105 i nch H i g h P l a i n W a l l 198 11.13 M o m e n t Magn i f ie r versus L o a d for a P l a i n Sect ion . 203 11.14 M o m e n t Magn i f ie r versus L o a d for a Reinforced Section 203 11.15 M o m e n t Magn i f ie r : E x a c t versus A p p r o x i m a t i o n 204 11.16 C r i t i c a l L o a d and C r i t i c a l M o m e n t Magn i f ie r Versus Eccent r ic i t y : for Purpose of Design A n a l y s i s 204 A l A n E l a s t i c B o d y C o n t a i n i n g a Single C r a c k 216 A 2 Geomet r ic R e l a t i o n between C r a c k Opening and S l i d ing D isp lacement 218 A 3 A C r a c k Ex tended by a P a i r of S p l i t t i n g Forces 218 A 4 A C o l u m n Loaded w i t h Doub le C u r v a t u r e Bend ing 230 - x v i -N O T A T I O N A, B, C, D = c o n s t a n t s i n v a r i o u s c o n t e x t s ; Ag,An = g r o s s a r e a a n d n e t a r e a o f b l o c k u n i t , r e s p e c t i v e l y ; a, b = c o n s t a n t s u s e d f o r d i m e n s i o n s ; a, b = p r e - e x i s t i n g h a l f c r a c k l e n g t h a n d h a l f s p a c i n g , r e s p e c t i v e l y , i n ( C h a p t e r I I I ) ; a, b = h a l f w i d t h o f h o l l o w c o r e a n d b l o c k u n i t , r e s p e c t i v e l y ; c , c m , c ^ , c c = b e d d i n g j o i n t c r a c k d e p t h s o f a w a l l c r o s s - s e c t i o n , d e f i n e d i n F i g . 11.2; c 2 = c o n s t a n t s ; E, E1 = m o d u l u s o f e l a s t i c i t y ; Eu, Eg, Ej, = m o d u l u s o f e l a s t i c i t y o f b l o c k u n i t , g r o u t , m o r t a r j o i n t , r e s p e c t i v e l y ; e, e0 = l o a d i n g e c c e n t r i c i t i e s ; £c, ej, em = v i r t u a l e c c e n t r i c i t i e s c o r r e s p o n d i n g t o d i f f e r e n t b e d d i n g j o i n t c r a c k d e p t h s ; F = c o m p r e s s i v e f o r c e ; F1, F2 = f u n c t i o n s o f b e d d i n g j o i n t c r a c k i n g d e f i n e d i n A p p e n d i x F ; / = f r i c t i o n b e t w e e n p r e - e x i s t i n g c r a c k s u r f a c e s ; fc i ft — c o m p r e s s i v e a n d t e n s i l e s t r e n g t h , r e s p e c t i v e l y ; fmp, fmg — c o m p r e s s i v e s t r e n g t h o f p l a i n m a s o n r y a n d g r o u t e d m a s o n r y , r e s p e c t i v e l y ; At /gi fj — c o m p r e s s i v e s t r e n g t h o f b l o c k u n i t , g r o u t ( p r i s m s t r e n g t h ) , m o r t a r ( c u b e s t r e n g t h ) , r e s p e c t i v e l y ; / • u = u n c o n f i n e d s t r e n g t h o f m o r t a r j o i n t ; fjci fje — c o n f i n e d s t r e n g t h s o f m o r t a r j o i n t ; fut = t e n s i l e s t r e n g t h o f b l o c k ; GLT G2 — f u n c t i o n s o f b e d d i n g j o i n t c r a c k i n g d e f i n e d i n A p p e n d i x F ; Gj, GJQ = e n e r g y r e l e a s e r a t e o f c r a c k e x t e n s i o n a n d i t s c r i t i c a l v a l u e ; H = p a r a m e t r i c f u n c t i o n d e f i n e d i n A p p e n d i x B ; - x v i i -h = h e i g h t o f w a l l o r s p e c i m e n ; hc, hj = w a l l h e i g h t s c o r r e s p o n d i n g t o d i f f e r e n t b e d d i n g j o i n t c r a c k d e p t h s ; h0 = h e i g h t o f b l o c k u n i t ; /, Ig = m o m e n t o f i n e r t i a o f w a l l c o r r e s p o n d i n g t o n e t c r o s s - s e c t i o n a n d g r o s s - s e c t i o n , r e s p e c t i v e l y ; Kj, KIC = s t r e s s i n t e n s i t y f a c t o r a t c r a c k t i p s a n d i t s c r i t i c a l v a l u e ; k = c r a c k c o n f i g u r a t i o n f a c t o r ( i n C h a p t e r III); k, klt k2 = c o n s t a n t s ; /, l0 = e x t e n d i n g c r a c k ( h a l f ) l e n g t h a n d i t s i n i t i a l v a l u e ( i n C h a p t e r III); / = l e n g t h o f w a l l o r s p e c i m e n ; M = n u m b e r o f c r a c k s . i n s p e c i m e n , ( i n C h a p t e r III); M = b e n d i n g m o m e n t ; m = m o d u l u s o f W e i b u l l d i s t r i b u t i o n ; mi, m2 = m o d u l a r r a t i o o f Eu/Eg a n d Eu/Ej, r e s p e c t i v e l y ; n = m o d u l a r r a t i o o f r e i n f o r c i n g s t e e l t o b l o c k s h e l l ; P, Px = t e n s i l e s p l i t t i n g f o r c e ( i n C h a p t e r III); P = a p p l i e d c o m p r e s s i v e l o a d ; Per, P). = E u l e r l o a d ( c o r r e s p o n d i n g t o g r o s s s e c t i o n ) , a n d b u c k l i n g l o a d o f w a l l , r e s p e c t i v e l y ; p = c o n t a c t p r e s s u r e b e t w e e n g r o u t a n d b l o c k s h e l l ; Qii Qni Qti "Tf = t r a c t i o n c o m p o n e n t s o n i n t e r n a l b o u n d a r y a n d e x t e r n a l b o u n d a r y o f a n e l a s t i c b o d y c o n t a i n i n g c r a c k s , r e s p e c t i v e l y ; R = e n e r g y d i s s i p a t e d b y f r i c t i o n ; S = s h e a r f o r c e ; s = c o n t o u r l e n g t h ; T = e f f e c t i v e c r a c k i n d u c e d s h e a r s l i d i n g f o r c e ; - x v i i i -t = t h i c k n e s s o f w a l l ; t0 = t h i c k n e s s o f m o r t a r j o i n t ; U = s t r a i n e n e r g y ; u, v = d i s p a c e m e n t v a r i a b l e s ; u%i v i t "HI vt — d i s p l a c e m e n t c o m p o n e n t s o n i n t e r n a l b o u n d a r y a n d e x t e r n a l b o u n d a r y o f a n e l a s t i c b o d y c o n t a i n i n g c r a c k s , r e s p e c t i v e l y ; V — v o l u m e ; V = w o r k d o n e b y e x t e r n a l l o a d ; W = e n e r g y d i s s i p a t e d t o f o r m n e w c r a c k w = s p e c i m e n w i d t h ( i n C h a p t e r I I I ) ; w, wg = s u m o f t h e m o r t a r e d w e b d i m e n s i o n a n d g r o u t d i m e n s i o n a l o n g w a l l l e n g t h , r e s p e c t i v e l y ; x , y = v a r i a b l e s u n d e r d i f f e r e n t c o n t e x t ; Z = c u m u l a t i v e f u n c t i o n o f a W e i b u l l d i s t r i b u t i o n ; a = i n c l i n i n g a n g l e o f p r e - e x i s t i n g c r a c k s ( i n C h a p t e r I I I ) ; a = r e l e a s e a n g l e o f b l o c k i n n e r c o r e ; r\ , T 2 = e x t e r n a l a n d i n t e r n a l b o u n d a r i e s ; A, S = d i s p l a c e m e n t s ; 6, cij = c r a c k o p e n i n g a n d i t s v a l u e a t t h e s t a r t i n g p o n i t o f t r a n s i t i o n a l i n t e r v a l ( i n C h a p t e r I I I ) ; e = c o m p r e s s i v e s t r a i n ; c x = e x t r e m e fiber s t r a i n o n c o m p r e s s i o n s i d e o f w a l l ; e«, £g, £j = c o m p r e s s i v e s t r a i n i n b l o c k u n i t , g r o u t a n d m o r t a r j o i n t , r e s p e c t i v e l y ; €tp crij = s t r a i n a n d s t r e s s c o m p o n e n t s ; rj = n e t a r e a t o g r o s s a r e a r a t i o o f b l o c k u n i t An/Ag; - xix -0, 0o, = cracking phase and its values at the starting points of crack extension and transitional interval; 0 = Weibull distribution parameter; K = constant related to Poisson's ratio, defined in Chapter IV; A = parameter defining grout and bedding extent, given by Eq. 10.3; u = coefficient of friction (in Chapter III); p = effective Poisson's ratio: i / / ( l - i / ) ; v = Poisson's ratio; vu, v3, vj = Poisson's ratio of block unit, grout and mortar, respectively; £ = average defect size-spacing ratio (a/b, in Chapter III); £ = ratio of moment of inertia I/Ig] p = steel ratio with respect to gross section of wall; cr, at = compressive and tensile stress, respectively; <T(J., Ttj = normal and shear stress components, respectively; a0, CTJ, cr2 = threshold stress for crack extension, stresses at the starting and finishing points, of transitional interval (in Chapter III); respectively; a u <72 = outer fibre stresses of wall; a, = lateral confining stress in joint; <rm, a, = compressive stress in masonry (average) and in masonry shell, respectively; cru, ag, a j = compressive stress in block unit, grout and mortar joint, respectively; crut = lateral tensile stress in block unit; <p = rotation (slope) of wall section; tpc, ipj = rotations of wall section corresponding to different bedding joint crack depths; $ = Airy stress function; <f> = density function of Weibull distribution; fij, £22 = functions of bedding joint crack depth defined by Eqs. 11.19 and 11.21. - XX -A C K N O W L E D G E M E N T The author wishes to express his gratitude to Dr. N. D. Nathan, who supervised the research for this program, for his continued interest, unfailing help and much valuable advice during preparation of this thesis. The author also wishes to thank Drs. D. L. Anderson and R. F. Foschi for their many constructive comments and suggestions. The scholarships, i.e. I. W. Killam Memorial Fellowship and the University Graduate Fellowship, received by the author are also gratefully appreciated. The research project is partially supported by the National Science and Engineering Research Council. The masonry units for the experimental program were kindly donated by Ocean Construction Supplies Limited. - xxi -D E D I C A T I O N T O T H E M E M O R Y O F M Y F A T H E R 1 C H A P T E R I I N T R O D U C T I O N 1.1 G e n e r a l R e m a r k s M a s o n r y c o n s t r u c t i o n i s b a s i c a l l y a n a s s e m b l y o f b l o c k s . T h e b l o c k s c a n b e n a t u r a l s t o n e , c l a y b r i c k s o r p r e c a s t c o n c r e t e u n i t s . T h e y a r e j o i n t e d w i t h c e m e n t i t i o u s m a t e r i a l c a l l e d m o r t a r . T h e h i s t o r y ! o f m a s o n r y b u i l d i n g m a y b e as o l d as h u m a n c i v i l i z a t i o n , b u t t h e i n t e r e s t i n m a s o n r y i s s t i l l i n c r e a s i n g t o d a y b e c a u s e o f t h e e c o n o m y o f c o n s t r u c t i o n a n d t h e p l e a s i n g a p p e a r e n c e o f m a s o n r y s t r u c t u r e s . S e r i o u s s t u d i e s o f s t r u c t u r a l m a s o n r y h a v e b e e n c a r r i e d o u t f o r t h e l a s t t w o d e c a d e s . W h i l e k n o w l e d g e o f t h e s t r u c t u r a l b e h a v i o u r o f m a s o n r y h a s b e e n g r e a t l y i m p r o v e d , m a n y q u e s t i o n s s t i l l r e m a i n u n a n s w e r e d i n t h i s a r e a , a n d t h e d e s i g n r e s t s l a r g e l y o n a n e m p i r i c a l b a s e . T h i s s t u d y w i l l f o c u s o n t h e f a i l u r e a n d c a p a c i t y o f t h e c o n c r e t e m a s o n r y u n d e r i n - p l a n e c o m p r e s s i o n c o m b i n e d w i t h o u t - o f - p l a n e b e n d i n g . T h e s t u d y e x t e n d s f r o m a b a c k g r o u n d i n v e s t i g a t i o n o f m a t e r i a l f a i l u r e t o a r a t i o n a l a n a l y s i s o f m a s o n r y s t a b i l i t y . T h e b e h a v i o u r o f m a s o n r y p r i s m s w i t h v a r i o u s b e d d i n g a n d g r o u t i n g c o m b i n a t i o n s u n d e r v a r i o u s l o a d i n g c o n d i t i o n s i s c a r e f u l l y o b s e r v e d t h r o u g h e x p e r i m e n t s . T h i s t h e s i s i s o r g a n i z e d i n t h e f o l l o w i n g m a n n e r . T h e e x p e r i m e n t a l p r o g r a m is f i r s t r e p o r t e d i n C h a p t e r I I ; t h e r e s u l t s w i l l b e q u o t e d a n d s t u d i e d i n d e t a i l i n t h e s u b s e q u e n t c h a p t e r s . T h e b a c k g r o u n d s t u d y o n m a t e r i a l f a i l u r e u n d e r a x i a l c o m p r e s s i o n , w h i c h w i l l b e u s e d t o e x p l a i n s o m e b e h a v i o u r o f c o n c r e t e : m a s o n r y i n l a t e r c h a p t e r s , f o l l o w s i n C h a p t e r I I I . C h a p t e r s I V t o V I I f o c u s o n t h e b e h a v i o u r o f p l a i n c o n c r e t e m a s o n r y ; C h a p t e r s V I I I t o X o n g r o u t e d m a s o n r y . T h e s t u d y o n t h e s l e n d e r n e s s a n d t h e s t a b i l i t y o f c o n c r e t e m a s o n r y i s p r e s e n t e d i n C h a p t e r X I . F i n a l l y , C h a p t e r X I I c o n c l u d e s t h i s s t u d y . I t i s h o p e d t h a t t h e t h e o r e t i c a l f i n d i n g s a n d t h e e x p e r i m e n t a l o b s e r v a t i o n s p r e s e n t e d i n t h i s s t u d y w i l l e n h a n c e e x i s t i n g k n o w l e d g e o f t h e f a i l u r e o f c o n c r e t e m a s o n r y , a n d a s s i s t i n t h e 2 formulation of design rules for concrete masonry structures. 1.2 Object and Scope The object of this thesis will be: a) To review and develop the background for material failure theory. b) To observe the behaviour including deformation, fracture pattern, failure mode and ultimate capacity of concrete masonry prisms with different loading conditions, joint conditions and grouting conditions. c) To examine and develop the existing theories for failure of concrete masonry under various conditions. d) To investigate the slenderness and stability of concrete masonry. 3 C H A P T E R II E X P E R I M E N T A L P R O G R A M 2.1 P u r p o s e a n d S c o p e E x t e n s i v e e x p e r i m e n t a l w o r k o n c o n c r e t e m a s o n r y h a s b e e n c o n d u c t e d p r e v i o u s l y . I n t h i s p r o g r a m , h o w e v e r , e f f o r t s w e r e m a d e t o o b s e r v e m o r e c l o s e l y t h e d e f o r m a t i o n a n d f r a c t u r e p a t t e r n o f m a s o n r y p r i s m s u n d e r c o n c e n t r i c a n d e c c e n t r i c c o m p r e s s i o n . P r i s m s p e c i m e n s w e r e d e s i g n e d t o c o v e r v a r i o u s c o m b i n a t i o n s o f b e d d i n g a n d g r o u t i n g c o n d i t i o n s . I n o r d e r t o r e - e x a m i n e t h e H i l s d o r f m o d e l o f m o r t a r e x p a n s i o n , f o r p l a i n p r i s m s u n d e r c o n c e n t r i c l o a d i n g , p a r t i c u l a r e m p h a s i s w i l l l i e o n o b s e r v a t i o n o f s p l i t t i n g f a i l u r e , a n d t h e e f fect o f j o i n t s o n d e f o r m a t i o n a n d c a p a c i t y o f m a s o n r y . F o r g r o u t e d p r i s m s u n d e r c o n c e n t r i c c o m p r e s s i o n , a t t e n t i o n w i l l b e p a i d t o t h e c r a c k s i n d u c e d b y t h e d i f f e r e n t d e f o r m a t i o n p r o p e r t i e s o f t h e m a s o n r y u n i t a n d t h e g r o u t , a n d t h e f o r c e s s h a r e d b y t h e s e t w o m a t e r i a l s b e f o r e a n d a f t e r c r a c k i n g . F o r m a s o n r y p r i s m s u n d e r e c c e n t r i c l o a d i n g , f a i l u r e m o d e s w i l l b e o b s e r v e d a n d t h e j o i n t b o n d o n t h e u n l o a d e d s i d e w i l l b e m o n i t o r e d t h r o u g h d e f o r m a t i o n g a u g e s . T h e e x p e r i m e n t a l r e s u l t s c o n c e r n i n g t h e p r o p e r t i e s o f t h e m a s o n r y c o n s t i t u e n t s a r e c o n c i s e l y r e p o r t e d i n t h i s c h a p t e r . T h e c h a r a c t e r i s t i c r e s u l t s f o r m a s o n r y s p e c i m e n a r e s u m m a r i z e d . T h e d e t a i l e d r e s u l t s w i l l be r e p o r t e d a n d s t u d i e d i n t h e c o n t e x t o f r e l a t e d a n a l y s i s i n l a t e r c h a p t e r s . 2.2 M a t e r i a l s A l l m a t e r i a l s u s e d i n m a k i n g t h e t e s t s p e c i m e n s a r e c o m m e r c i a l l y a v a i l a b l e a n d t y p i c a l o f t h o s e c o m m o n l y u s e d i n l o c a l c o n s t r u c t i o n . 2.2.1 M a s o n r y U n i t s A l l t h e m a s o n r y p r i s m s t e s t e d w e r e b u i l t b y u s i n g 8 i n c h s t a n d a r d c o n c r e t e b l o c k u n i t s 4 with double end (in accordance with CSA-A165-M85, C-20). The units were kindly donated by Ocean Construction Supplies Ltd, Vancouver, B. C. The dimensions are shown schematically in Fig. 2.1. To determine the compressive strength of the units, 16 blocks were tested with a Baldwin Tate-Emery testing machine. In accordance with ASTM C140, 8 blocks were capped with hydrostone (a gymsum cement). In order to observe the effect of the capping condition, another 8 blocks were tested with fibreboard capping. Table 2.1 gives the results of failure loads. As can be seen, although there is a statistical difference, it is not sufficient to suggest a different failure mechanism. This is consistent with the fact that the two test conditions exhibited similar shear failure patterns, as typically shown in Fig. 2.2. 16 block units with two different capping conditions all exhibited conical type failure, owing to the low height to width ratio. The average failure load is 200.5 kips, which corresponds to an average strength of about 3250 psi based on the net area of the unit (the ratio of net area to gross area of the unit rj is 0.51). Attempts were also made to obtain the deformation properties by measuring the relative displacement of the loading head. However, due to the compliance of the testing machine (the dial gauge was not mounted directly against the loading platens) and the variation in the cappings, the results were not accurate compared with those measured by LVDTs directly mounted on the blocks in prism tests. The latter are given by Fig. 2.3. The initial modulus of the units is about 3.42xlO6 psi. The concrete units were very brittle in the sense that they often failed totally in an explosive manner as soon as the peak load was reached. It was not possible to measure the deformation after the peak strain with the standard test procedure. 7.5" 190mm 130mm 141mmi I 150mm F I G . 2.1 M a s o n r y U n i t S P E C I M E N 1 2 3 4 5 6 7 8 A V G C O V G R O U P 1 169 188 194 196 2 0 7 186 2 1 3 189 193 .0 6 . 5 % G R O U P 2 219 186 232 205 208 187 2 2 0 2 0 3 2 0 8 . 0 7 . 2 % T a b l e 2.1 F a i l u r e L o a d s o f B l o c k U n i t ( k i p s ) G R O U P 1: H y d r o - s t o n e c a p ; G R O U P 2: F i b e r b o a r d c a p 6 FIG. 2.2 Conical Failure of Masonry Unit 0 1 2 3 STRAIN ( 1/1000 IN/IN ) 4 5 F I G . 2 .3 S t r e s s - S t r a i n R e l a t i o n o f M a s o n r y U n i t u n d e r C o m p r e s s i o n 2 .2 .2 M o r t a r T h r e e t y p e s o f m o r t a r w e r e u s e d , i .e . t y p e M , S , a n d N i n a c c o r d a n c e w i t h C S A - 1 7 9 M -1976 . T h e m o r t a r s w e r e m i x e d b y a n e x p e r i e n c e d m a s o n , w i t h a s m a l l e l e c t r i c a l l y d r i v e n m i x e r , d u r i n g t h e c o n s t r u c t i o n o f t h e s p e c i m e n s . T h e m i x p r o p o r t i o n s a r e g i v e n i n T a b l e 2 .2 . M o r t a r c u b e s w e r e s a m p l e d f o r e v e r y b a t c h . T h e 2 8 - d a y c u b e s t r e n g t h s a r e g i v e n i n T a b l e 2 .3 . A t t h e s a m e t i m e , t h e s t r e s s - s t r a i n r e l a t i o n s h i p s w e r e m e a s u r e d fo r t y p e N a n d s o m e o f t h e t y p e S m o r t a r , as s h o w n i n F i g . 2 . 4 . T h e r e s u l t s i n d i c a t e t h a t t h e m o r t a r s a r e m u c h s o f t e r t h a n n o r m a l c o n c r e t e , a n d t h a t t h e y h a v e v e r y l a r g e peak s t r a i n s . T h e i n i t i a l m o d u l i a re a b o u t 0 . 4 x l 0 6 p s i f o r t y p e N m o r t a r , a n d 0 . 5 x l 0 6 p s i f o r t y p e S m o r t a r . T h e p e a k s t r a i n s a r e a b o u t 0 . 0 0 6 fo r t h e f o r m e r a n d 0 .009 fo r t h e l a t t e r . T h e h i g h c o m p l i a n c e o f t h e m o r t a r w a s a l s o i n d i c a t e d b y 8 d e f o r m a t i o n m e a s u r e m e n t s a c r o s s t h e j o i n t s o f p r i s m s p e c i m e n s . A s t y p i c a l l y s h o w n i n F i g . 2 . 5 , t h e r a t i o o f t h e i n i t i a l m o d u l u s o f t h r e e t y p e s o f m o r t a r t o t h a t o f t h e c o n c r e t e u n i t s i s a b o u t 1 t o 6 - 8 . T h e d e f o r m a t i o n p r o p e r t i e s m e a s u r e d d i r e c t l y f r o m m o r t a r c u b e t e s t s a n d t h e u n i t t e s t s a r e v e r y c l o s e t o t h e s e r e s u l t s . M o r t a r P r o p o r t i o n b y V o l u m e T y p e C e m e n t ( T y p e II I M a s o n r y C e m e n t F i n e S a n d W a t e r M 1 1 2 .5 1 S 1 / 2 1 3 1 N — 1 3 0 .68 T a b l e 2 .2 M i x P r o p o r t i o n s o f M o r t a r S P E C I M E N 1 2 3 4 5 6 A V G C O V M T Y P E 4 1 8 8 4 8 3 5 4 6 2 5 5 1 0 0 4 6 9 0 7 . 1 % S T Y P E 3 9 8 5 4 3 2 0 3 8 7 5 4 0 7 5 3 7 5 0 4 0 0 0 4 . 8 % N T Y P E 1450 1 7 1 0 1560 1650 1 3 2 5 1730 1570 9 . 2 % T a b l e 2 .3 28 D a y M o r t a r C u b e (2 i n ) S t r e n g t h ( p s i ) FIG .2 .4 Stress-Strain Re la t ion of Mor ta r : a) T y p e N , b) T y p e S. 10 F I G . 2.5 Measured Ver t ica l Compressive Strains a long B lock U n i t s and across M o r t a r J o i n t of P l a i n P r isms under U n i a x a i l Compress ion . 2.2.3 G r o u t Three types of grout w i th different strengths were designed for the specimens. They are denoted by G S , G N and G W . The m i x proportions are l isted in T a b l e 2.4. T h e water content was adjusted s l ight ly to achieve 3-5 inch s lump. F o r every m i x batch , two or three standard prisms were cast and cured in accordance to C S A - 1 7 9 M - 1 9 7 6 . T h e compressive strengths are given in T a b l e 2.5. T o examine the correlat ion between the strength obtained by the standard test and that actual ly grouted in the masonry, 20 grout pr isms taken f rom the cores of fai led masonry specimens were tested. T h e cores were cut by a d iamond saw and capped with sulfur before testing. T h e results are shown in Tab le 2.6. A s is seen, the strength of the grout, pr isms taken 11 f r o m f a i l e d m a s o n r y s p e c i m e n s i s s u b s t a n t i a l l y h i g h e r t h a n t h a t o f t h e s t a n d a r d p r i s m s . T h i s m a y b e p a r t l y d u e t o t h e d i f f e r e n t h e i g h t t o w i d t h r a t i o s o f t h e s p e c i m e n s ( 1 .4 :1 f o r t h e f o r m e r , 2 :1 f o r t h e l a t t e r , a p p r o x i m a t e l y ) , p a r t l y t o t h e d i f f e r e n c e i n c u r i n g t i m e ( t h e f o r m e r w e r e t e s t e d a b o u t a y e a r l a t e r ) . T h i s s u g g e s t s t h a t t h e s t r e n g t h o b t a i n e d b y t h e s t a n d a r d t e s t i s o n l y m e a n i n g f u l as a r e f e r e n c e p a r a m e t e r . T h e d e f o r m a t i o n p r o p e r t i e s w e r e m e a s u r e d o n t h e c o r e s t a k e n f r o m t e s t e d g r o u t e d m a s o n r y p r i s m s . T h e d e f o r m a t i o n c u r v e s a r e g i v e n b y F i g . 2 . 6 . T h e i n i t i a l m o d u l u s i s 2 .8 x l O 6 p s i f o r t y p e S g r o u t , 2 . 6 x l 0 6 p s i f o r t y p e N g r o u t a n d 1 . 9 x l 0 6 p s i f o r t y p e W g r o u t . G r o u t P r o p o r t i o n b y V o l u m e T y p e C e m e n t ( T y p e I I I C o a r s e S a n d P e a G r a v e l W a t e r G S 1 2 .5 2 .5 0 .6 G N 1 2 , 2 0 .8 G W 1 5 — 1.0 T a b l e 2 .4 M i x P r o p o r t i o n s o f G r o u t S P E C I M E N 1 2 3 4 5 6 7 8 9 A V G C O V S - T Y P E 4 7 2 0 5 4 4 5 4 8 9 0 5 3 2 0 4 7 0 5 5 0 1 5 6 . 1 % N — T Y P E 3 6 8 5 3 8 8 5 3 7 2 0 4 0 1 5 3 3 9 0 3 4 2 5 3 7 4 5 3 6 0 0 3 8 1 5 3 7 0 0 5 . 1 % W - T Y P E 3 1 6 5 3 5 0 0 3 3 0 5 3 3 7 5 3 2 8 5 3 3 2 5 3 . 2 % T a b l e 2 . 5 G r o u t S t r e n g t h , b y S t a n d a r d P r i s m T e s t s ( p s i ) S P E C I M E N 1 2 3 4 5 6 7 8 9 A V G C O V S - T Y P E 6 3 0 5 4 1 7 0 5 5 3 0 6 6 9 0 5 2 4 5 5 5 9 0 1 5 . 7 % N - T Y P E 6 1 8 0 5 6 2 5 5 9 7 0 6 3 5 0 5 8 1 0 6 3 1 0 5 8 7 0 6 6 6 0 6 2 1 5 6 1 1 0 4 . 9 % W - T Y P E 4 5 3 0 4 7 2 5 4 3 3 0 4 4 5 0 4 0 5 0 4 2 1 0 4 3 8 5 5 . 0 % T a b l e 2 .6 G r o u t S t r e n g t h , b y T e s t s o n C o r e s T a k e n f r o m F a i l e d P r i s m s ( p s i ) 12 13 2 . 3 P r i s m S p e c i m e n s 104 3 - h i g h p r i s m s w e r e b u i l t w i t h d i f f e r e n t b e d d i n g , a n d g r o u t i n g c o n d i t i o n s , d e s i g n e d t o b e t e s t e d u n d e r d i f f e r e n t e c c e n t r i c i t i e s . T h r e e h i g h s p e c i m e n s w e r e c h o s e n b e c a u s e i t i s b e l i e v e d t h a t t h e e n d e f fect o f t h e l o a d i n g d e v i c e c a n b e e l i m i n a t e d i n t h e m i d d l e c o u r s e w h e r e a l l t h e m e a s u r e m e n t s w e r e m a d e . T h e s p e c i f i c a t i o n s o f t h e s p e c i m e n s a r e l i s t e d i n T a b l e 2 .7 . A l l t h e p l a i n p r i s m s w e r e b u i l t b y a n e x p e r i e n c e d m a s o n . A l l t h e m o r t a r j o i n t s w e r e c u t f l u s h o n t h e p r i s m f a c e s . T h e p r i s m s w e r e t h e n g r o u t e d 4-5 d a y s l a t e r ( fo r g r o u t e d p r i s m s ) . T h e s p e c i m e n s w e r e s t o r e d i n t h e s t r u c t u r e s l a b o r a t o r y 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 f o r a b o u t a y e a r u n t i l t h e y w e r e t e s t e d . A f e w s p e c i m e n s w e r e d i s c a r d e d b e c a u s e o f t h e d e b o n d i n g o f t h e m o r t a r j o i n t ( t h e d e b o n d i n g h a p p e n e d b e c a u s e s p e c i m e n s w e r e m o v e d o n c e , d u e t o o t h e r e x p e r i m e n t a l a c t i v i t i e s , d u r i n g t h e s t o r i n g p e r i o d , a n d b e c a u s e o f s e t u p h a n d l i n g ) . F o r t h e g r o u t e d p r i s m s , h y d r o s t o n e w a s u s e d t o f i n i s h t h e t o p e n d s p r i o r t o t e s t i n g . A l l t h e p r i s m s w e r e t r a n s p o r t e d b y u s i n g a s m a l l t r o l l e y t o t h e t e s t i n g d e v i c e a n d t h e n ' c a p p e d ( t o p a n d b o t t o m ) w i t h f i b r e b o a r d b e f o r e t h e y w e r e p o s i t i o n e d b e t w e e n t h e l o a d i n g p l a t e n s . S p e c i m e n N o . o i J o i n t G r o u t i n g L o a d A d d i t i o n a l P r i s m s C o n d i t i o n s C o n d i t i o n s E c c e n t . D e s c r i p t i o n S I 4 S M o r t a r — 0 J o i n t t h i c k n e s s i s 3 / 8 N 2 4 N M o r t a r — 0 i n c h e x c e p t o t h e r w i s e M 3 4 M M o r t a r — 0 s p e c i f i e d . N 4 4 N J , t 0 = 6 / 8 " . . . 0 N J = N M o r t a r . P 5 4 t o = 0 . . . 0 C o n t a c t f a c e s w e r e g r i n d e d G 7 4 4 m m g l a s s p l a t e 0 j o i n t e d b y c e m e n t p a s t e . S 8 4 S M o r t a r N G r o u t 0 M 9 4 M M o r t a r N G r o u t 0 N 1 0 4 N J , t 0 = 6 / 8 " N G r o u t 0 N J = N M o r t a r . P l l 4 t o = 0 . N G r o u t 0 N 1 2 4 N M o r t a r S G r o u t 0 N 1 3 4 N M o r t a r N G r o u t 0 N 1 4 4 N M o r t a r W G r o u t 0 N 1 5 4 N J , f a c e - s h e l l — 0 N J = N M o r t a r . S 1 6 4 S J , f a c e - s h e l l . . . 0 S J = N M o r t a r . N 1 7 4 N J , f a c e - s h e l l N G r o u t 0 N J = N M o r t a r . N 1 8 4 N M o r t a r . . . t / 6 N 1 9 4 N M o r t a r . . . t / 3 M 2 0 4 M M o r t a r . . . t / 3 S 2 1 4 S M o r t a r — t / 3 N 2 2 4 N J , f a c e - s h e l l — t / 3 N 2 3 4 N M o r t a r N G r o u t 0 H a l f b l o c k S 2 5 4 S M o r t a r N G r o u t t / 3 N 2 6 4 N M o r t a r N G r o u t t / 6 M 2 6 4 M M o r t a r N G r o u t t / 3 M 2 7 4 M J , f a c e - s h e l l . . . 0 M J = M M o r t a r . T a b l e 2 .7 P r i s m S p e c i m e n s 15 2.4 T e s t i n g Device Since it was expected that for the grouted pr isms the fa i lure loads wou ld be higher than the capac i ty of the exist ing testing faci l i t ies i n the structures laboratory (up to 400 k ips) , a load ing device was bu i l t as shown i n F i g . 2.7. It was formed basical ly w i t h a girder serving as a lever, w i t h appropr iate suppor t ing members; i t had a mechan ica l advantage of 2. T h e device was connected to a hyd rau l i c j ack w i t h 400 k ips capaci ty , and so that i t cou ld app ly a load up to 800 k ips . It was" ca l ib rated up to 600 k ips , but , i n the event, the fai lure loads of the specimens never exceeded 400 k ips . T h e specimens were designed to be compressed w i t h p in -ended condi t ions . T h e top and b o t t o m load ing platens, therefore, were designed w i t h cy l i nd r ica l bearings, as shown i n F igs . 2.7 and 2.8. T h e platens were designed for three load ing eccentricit ies, i.e. e=0, e=t/6 and e = i / 3 . T h e suppor t ing devices were bu i l t w i t h h igh strength steel. T h e hyd rau l i c j ack was contro l led by an M T S cont ro l console ( M o d e l 483.02), w i t h force cont ro l mode (displacement contro l not avai lab le) . T h e load was set to increase a u t o m a t i c a l l y so that a specimen wou ld fa i l i n about 3 minutes , for p la in specimens, and 5 minutes for grouted ones. T h e load was read through an electronic load cel l mounted in the jack . 2.5 Ins t rumentat ion T o measure the deformations of the pr isms, about ha l f of the specimens were inst rumented w i r h s ix quar ter - inch l inear var iable di f ferential transformers ( L V D T , T r a n s - T e k Series 240). T h e arrangement and locat ions of the L V D T s were different for concentr ic and for eccentric compression condi t ions , and are denoted by 1 to 6 in the figures showing the measured curves (cf. F i g . 4.3, for example) . T h e L V D T across the mor ta r j o i n t had a gauge length of 1.8 inches (45 m m ) , whi le a l l rest were 5 inch (125 m m ) . T h e L V D T s were c lamped to a l u m i n u m supports wh ich were then mounted on s m a l l disc screw nuts glued in advance by fast sett ing epoxy, t yp ica l l y as shown in F i g . 2.9. 16 F I G . 2.7 T e s t i n g D e v i c e F i g . 2.9 Instrumentaion: L V D T s and G l u e d Wi res 18 B e c a u s e o f t h e d e s t r u c t i v e n a t u r e o f t h e e x p e r i m e n t s , i t w a s e x p e c t e d t h a t t h e s p e c i m e n w o u l d f a i l i n a s u d d e n , e x p l o s i v e p a t t e r n , e s p e c i a l l y w i t h t h e l o a d c o n t r o l l e d t e s t i n g m a c h i n e . T o r e d u c e t h e i m p a c t o f t h e f a i l i n g s p e c i m e n , t h e L V D T s w e r e s u r r o u n d e d w i t h p l e x i g l a s s s l e e v e s a n d s p o n g e s as c a n b e s e e n i n F i g . 2 . 9 . T o m o n i t o r t h e e f fect o f t h e i m p a c t , t h e y w e r e c h e c k e d f o r n o r m a l f u n c t i o n i n g a f t e r e v e r y t e s t a n d c a l i b r a t e d a g a i n s t g a u g e t h i c k n e s s f o r e v e r y t w o t e s t s , o r w h e n e v e r t h e c e n t r a l c o r e o f a t r a n s d u c e r w a s b e n t ( t h i s h a p p e n e d s e v e r a l t i m e s d u r i n g t h e t e s t s , s t r a i g h t e n i n g w a s o f t e n n e c e s s a r y ) . F o r t u n a t e l y , t h e o u t e r c o i l s s u r v i v e d f o r t h e w h o l e t e s t i n g p r o g r a m , a l t h o u g h t h e l i n k i n g w i r e s b r o k e s e v e r a l t i m e s . A c c o r d i n g t o t h e m a n u f a c t u r e r , t h e L V D T s h a v e a n i n f i n i t e r e s o l u t i o n . H o w e v e r , w h e n t h e d i s p l a c e m e n t m e a s u r e d i s t o o s m a l l , t h e r e a d i n g s m a y be b u r i e d i n t h e n o i s e . It t u r n e d o u t t h a t w h e n t h e d i s p l a c e m e n t w a s l a r g e r t h a n 0 . 0 0 2 5 i n c h ( c o r r e s p o n d i n g t o 0 .5 m i l l i - s t r a i n o f t h e g i v e n g a u g e l e n g t h ) , t h i s w a s n o t a b i g p r o b l e m , a n d t h e r e a d i n g s w e r e s a t i s f a c t o r y f o r m o s t c a s e s . F o r p l a i n p r i s m s u n d e r a x i a l c o m p r e s s i o n , a n e l e c t r o n i c c i r c u i t w a s d e s i g n e d t o s t u d y t h e m a c r o s c o p i c s p l i t t i n g o f p l a i n p r i s m s u n d e r u n i a x i a l c o m p r e s s i o n . F o u r v e r y t h i n c o p p e r w i r e s ( g a u g e 4 2 , d> 0 . 08 m m ) w e r e g l u e d w i t h e p o x y t o d i f f e r e n t l o c a t i o n s o n t h e p r i s m s . T h e s e w i r e s s e r v e d as e l e c t r i c a l c o n d u c t o r s w h i c h g i v e e l e c t r i c a l p u l s e s w h e n t h e y b r e a k . S i n c e t h e w i r e s w e r e f u l l y s u r r o u n d e d b y t h e h a r d e n e d g l u e a n d a d h e r i n g t o t h e s u r f a c e o f t h e s p e c i m e n , t h e y w e r e s u p p o s e d t o b r e a k w h e n t h e s p e c i m e n s p l i t . B y d e t e c t i n g t h e o r d e r o f t h e w i r e s b r e a k i n g , w e o b t a i n t h e r u n n i n g d i r e c t i o n o f a c r a c k w h i c h r u n s a c r o s s t h e w i r e s . T h e c r a c k p r o p a g a t i o n s p e e d i n c o n c r e t e i s a b o u t 1 8 0 m / s e c ( B h a r g a v a a n d R e h n s t o r m 1 9 7 5 ) so t h a t a s p l i t w o u l d r u n t h r o u g h t h e b l o c k h e i g h t i n a b o u t 0 .001 s e c o n d . T h e e l e c t r o n i c c i r c u i t (see a p p e n d i x ) w a s d e s i g n e d b y a n e x p e r i e n c e d e l e c t r i c i a n i n t h e c i v i l e n g i n e e r i n g d e p a r t m e n t , w h i c h w a s c a p a b l e o f d e t e c t i n g t h e b r e a k o r d e r f o r i n t e r v a l s less t h a n 5 x l 0 - 6 s e c o n d . B a s i c a l l y , i t r e c o r d e d t h e e l e c t r i c a l p u l s e s g i v e n b y o p e n c i r c u i t s d u e t o b r e a k a g e s o f t h e w i r e s i n a n o r d e r e d w a y . T h e c i r c u i t w a s b u i l t a n d t h e n t e s t e d i n t h e e l e c t r i c a l e n g i n e e r i n g d e p a r t m e n t a t T J B C . T o g a i n m o r e 19 confidence w i t h the method , i t was first used i n face-shel l bedded pr isms a n d gave consistent results. It was then t r ied w i t h fu l ly bedded masonry w i t h l ines glued to the face-shells as wel l as the webs, and aga in gave consistent results (webs spl i t i n contrast to face-shells) . T h e wires were then a l l g lued to the webs, where sp l i t t i ng a lways occurred. T h e device is shown in F i g . 2.10 (also see F i g . 2.9 for g lued wires), and it is seen that the breaking order is ind icated by four rows of l ight emission diodes ( L E D ) . T o give better insight in to the fai lure processes, a V H S standard v ideo camera was used to record most of the tests. T h i s was found very useful for later observat ion since, as the test ing machine was load contro l led , m a n y specimens were to ta l l y destroyed (often in an explosive manner) as soon as u l t i m a t e load was reached. T h e camera was insta l led to face one of the webs, because fractures were more often observed to occur in webs than in face-shells (compare the deformat ions measured at locations 3, 4 w i t h those at locations 1, 2 given in fo l lowing chapters) . T h e camera was able to record v is ib le cracks on the web faces, usual ly i m m e d i a t e l y before final fa i lure. However , L V D T s were more sensitive to smal ler cracks occur r ing at earlier stages (as inferred by a sudden increase in measured d isplacement) . Fo r the ma jo r i t y of the specimens, the overa l l fa i lure pattern could be examined by slow p layback of the recorder. F o r a few specimens that fa i led in a h igh ly explosive manner , however, the recording was not very sat isfactory. U s u a l l y there was no warn ing , such as c rack ing or spa l l ing , that fa i lure was approach ing in these specimens. 2.6 D a t a A c q u i s i t i o n T h e da ta , i.e. s ix d isplacements and one load , were read by an O p t i l o g system, an electronic d a t a acqu ist ion uni t , wh ich is bas ica l ly a microprocessor d ig i t i z i ng and recording the ana log signals. It was contro l led by an I B M personal computer w i t h O p t i l o g software. T h e whole setup is shown in F i g . 2.11. FIG. 2.10 Electronic Device Detecting Wire Break Order 21 The load cell and all the LVDTs were calibrated through the unit. The system was set so that the load and the displacements were read every two seconds for plain prisms and four seconds for grouted ones. The recorded data were often reviewed during the tests to prevent any abnormal readings. They were then converted to standard format for later processing. FIG. 2.11 Data Acquisition Setup 22 2.7 S u m m a r y o f C h a r a c t e r i s t i c R e s u l t s S i n c e t h e p r i s m s p e c i m e n s c o v e r a w i d e r a n g e a n d e a c h g r o u p h a s i t s o w n e m p h a s i s , i t m a y b e i n a p p r o p r i a t e t o g i v e a l l t h e d e t a i l e d r e s u l t s a t t h i s s t a g e . T h e r e f o r e , t h e r e s u l t s w i l l be r e p o r t e d a n d s t u d i e d i n t h e c o n t e x t o f a n a l y s i s i n t h e r e l a t e d c h a p t e r s . I n o r d e r t o h a v e a n o v e r a l l v i e w o f t h e t e s t r e s u l t s , w e g i v e a s h o r t , d e s c r i p t i v e s u m m a r y o f s o m e o f t h e i m p o r t a n t e x p e r i m e n t a l c h a r a c t e r i s t i c s . T h e y a r e o u t l i n e d i n t e r m s o f t h e f a i l u r e m o d e a n d c a p a c i t y , w h i c h a r e o f c o m m o n i n t e r e s t b u t w h i c h a r e y e t g e n e r a l l y d i s t i n c t b e t w e e n d i f f e r e n t s p e c i m e n s . T h e s p e c i m e n s m a y be r o u g h l y c h a r a c t e r i z e d i n t o 6 m a j o r g r o u p s a c c o r d i n g t o t h e i r b e d d i n g , g r o u t i n g , l o a d i n g c o n d i t i o n s , a s w e l l a s t h e i r f a i l u r e c h a r a c t e r i s t i c s . T h e y a r e : p l a i n m a s o n r y w i t h f u l l b e d d i n g u n d e r c o n c e n t r i c c o m p r e s s i o n ; p l a i n m a s o n r y w i t h f a c e - s h e l l b e d d i n g u n d e r c o n c e n t r i c c o m p r e s s i o n ; p l a i n m a s o n r y ( w i t h b o t h b e d d i n g c o n d i t i o n s ) u n d e r e c c e n t r i c c o m p r e s s i o n ; g r o u t e d m a s o n r y w i t h f u l l b e d d i n g u n d e r c o n c e n t r i c c o m p r e s s i o n ; g r o u t e d m a s o n r y w i t h f a c e - s h e l l b e d d i n g u n d e r c o n c e n t r i c c o m p r e s s i o n ; g r o u t e d m a s o n r y ( w i t h b o t h b e d d i n g c o n d i t i o n s ) u n d e r e c c e n t r i c c o m p r e s s i o n . T h e s u m m a r y i s o r g a n i z e d i n T a b l e 2 . 8 . F i g s . 2 . 1 2 t o 2 . 1 6 g i v e s o m e t y p i c a l f a i l u r e m o d e s . 23 S P E C I M E N ' F A I L U R E M O D E C A P A C I T Y C H A R A C T E R I S T I C S 1) P l a i n masonry w i t h f u l l bedding under concentr ic compression. Of ten one major spl i t ran through specimen w i t h i n midd le t h i r d of webs, immed ia te l y before final fa i lure. Spl i ts d id not consistent ly in i t ia te f r o m the mor ta r jo in t . Sp l i ts were cont inuous. J o i n t condi t ions had a s igni f icant influence on the capac i ty . 2) P l a i n masonry w i t h face-shell bedding under concentr ic compression. One or two spl i ts in webs occured at or immed ia te ly before final fai lure. Sp l i ts consistent ly in i t ia ted f r o m jo ints , at locations near two jo in t ends and wandered afterwards. Sp l i ts were discont inuous at jo ints . J o i n t strength had a re lat ive ly s igni f icant inf luence on the capac i ty . C a p p i n g condi t ions had a substant ia l inf luence on the capaci ty . 3) P l a i n masonry under eccentric compression. Fa i l u re was characterized by shear, i.e. by spa l l ing and crushing on the loaded side; and was often local ized in part of the specimen. Jo in ts on unloaded side d i d not effectively transfer tension. J o i n t strength and bedding pattern had a re lat ive ly m ino r effect on the capac i ty . 4) G r o u t e d masonry w i t h fu l l bedding under concentr ic compression. Sp l i ts both in webs and face-shells were observed wel l before final fai lure, some at as low as 4 0 % of fa i lure loads. B l o c k shells s t i l l carr ied substant ia l load after c rack ing . F i n a l fai lure brought by spa l l ing of the shells, fo l lowed by crushing of grout at themidhe ight . J o i n t strength and grout strength had a re lat ive ly m ino r effect on the capaci ty . 5) G r o u t e d p r i s m w i t h face-shel l bedding under concentr ic compression. Sp l i ts in webs occured wel l before f ina l faiure. B l o c k shells carr ied l i t t l e load after cracking . C a p a c i t y was not m u c h higher than that of grout alone. 6) G r o u t e d masonry under eccentric compr . A s described i n 3). B o t h grout and j o i n t have a m ino r effect on the capac i ty . T a b l e 2.8 A S u m m a r y of Fa i l u re and C a p a c i t y Character is t ics 24 F I G . 2.12 S p l i t t i n g F a i l u r e o f P l a i n C o n c r e t e M a s o n r y with F u l l B e d d i n g u n d e r U n i a x i a l C o m p r e s s i o n FIG. 2.13 Failure of Plain Masonry with Face-Shell Bedding under Uniaxial Compression 26 F I G . 2 . 1 4 F a i l u r e o f F a c e - S h e l l B e d d e d , F u l l y C a p p e d M a s o n r y u n d e r U n i a x i a l C o m p r e s s i o n F I G . 2 . 1 5 F a i l u r e o f P l a i n M a s o n r y u n d e r E c c e n t r i c C o m p r e s s i o n 28 F I G . 2 .16 F a i l u r e o f G r o u t e d M a s o n r y u n d e r E c c e n t r i c C o m p r e s s i o n 29 C H A P T E R I I I S O M E B A C K G R O U N D T O C O M P R E S S I O N F A I L U R E O F C O N C R E T E 3.1 P u r p o s e C o n c r e t e m a s o n r y i s b a s i c a l l y a c o n c r e t e m e m b e r w i t h d i s c o n t i n u i t y i n m a t e r i a l p r o p e r t i e s . I n s t r u c t u r a l d e s i g n , i t i s u s u a l l y u s e d t o s u s t a i n c o m p r e s s i v e f o r c e . I n t r a d i t i o n a l a n a l y s i s f o r c o n c r e t e s t r u c t u r e s , a p h e n o m e n o l o g i c a l a p p r o a c h h a s b e e n u s e d : e x p e r i m e n t a l l y o b s e r v e d s t r e s s - s t r a i n r e l a t i o n s h i p s , u s u a l l y f r o m u n i a x i a l t e s t s , h a v e b e e n a p p l i e d . F a i l u r e h a s b e e n d e f i n e d as t h e s t r e s s o r s t r a i n i n t h e m e m b e r w h i c h r e a c h e s s o m e c r i t i c a l v a l u e ( s t r e n g t h o r u l t i m a t e s t r a i n ) , w h i c h i s o b t a i n e d f r o m u n i a x i a l t e s t s a n d a s s u m e d t o b e c o n s t a n t i n a g e n e r a l s t r e s s s t a t e . H o w e v e r , t h i s a p p r o a c h i s s u b j e c t t o c e r t a i n l i m i t a t i o n s . T h e c o m p r e s s i v e s t r e n g t h o f a m a t e r i a l s u c h as c o n c r e t e , w h o s e f a i l u r e i s c h a r a c t e r i z e d b y b r i t t l e c l e a v a g e f r a c t u r e , i s n o t a v e r y m e a n i n g f u l p a r a m e t e r . I t v a r i e s w i t h t h e s t r e s s s t a t e d u e t o t h e s o - c a l l e d s t r a i n g r a d i e n t e f f ec t , a p h e n o m e n o n w h i c h i s m o r e o b v i o u s f o r c o n c r e t e m a s o n r y . T h e a p p r o a c h a l s o f a i l s t o g i v e a n e x p l a n a t i o n f o r t h e s p l i t t i n g f a i l u r e m e c h a n i s m o f t e n o b s e r v e d i n c o n c r e t e m a s o n r y as w e l l a s c o n c r e t e u n d e r u n i a x i a l c o m p r e s s i o n . T h e s e p r o b l e m s h a v e b e e n p a r t i a l l y r e c o g n i z e d b u t n e v e r b e e n f u l l y e x p l a i n e d . T h e s t u d y p r e s e n t e d i n t h i s c h a p t e r w i l l a t t e m p t t o r a i s e t h e q u e s t i o n a n d t o s h e d s o m e l i g h t o n t h e p r o b l e m s b y e x a m i n i n g b r i t t l e m a t e r i a l s u n d e r u n i a x i a l c o m p r e s s i o n . T h e i n t e n t i o n i s t o p r e s e n t a n e x p l a n a t i o n o f t h e b e h a v i o u r o f t h e s e m a t e r i a l s b a s e d o n a f a i l u r e m e c h a n i s m a t t h e f u n d a m e n t a l l e v e l . T h e p r i n c i p l e s o f t h i s s t u d y w i l l b e u s e d t o e x p l a i n s o m e b e h a v i o u r o f c o n c r e t e m a s o n r y a n d t o s u p p o r t a n a l t e r n a t i v e a p p r o a c h i n l a t e r c h a p t e r s . I t is h o p e d t h a t t h i s s t u d y w i l l a l s o h e l p t o d e v e l o p a b e t t e r u n d e r s t a n d i n g o f t h e n a t u r e o f t h e s t r a i n g r a d i e n t e f fect a n d t h e s p l i t t i n g f a i l u r e p h e n o m e n o n c o m m o n l y e x h i b i t e d i n b r i t t l e m a t e r i a l t e s t i n g , a n d l e a d t o m o r e 30 general and consistent fa i lure c r i te r ia for these mater ia ls . 3.2 B r i t t l e F a i l u r e under U n i a x i a l Compress ion A l t h o u g h concrete m a y exhib i t h igh non l inear i ty at work ing compressive stress, i t is essential ly a b r i t t le mate r ia l . T h i s is so m a i n l y because the fai lure of concrete is character ized by b r i t t le cleavage f racture and the p last ic deformat ion due to viscous behaviour of the hardened cement is rather l i m i t e d (Hsu et a l 1963, Ziegeldorf 1983). Extens ive research work at both s t ructura l and phenomenological levels has ind icated that under compression, concrete experiences three d ist inct stages before its f i na l fa i lure: i n i t i a t i o n of the cracks; slow stable crack growth accompanied by crack arrest; a c r i t i ca l cond i t ion character ized by unstable crack propagat ion and an extensive crack network fo rmat ion (for example , see Mindess 1983). T h e crack ing process is reflected in the g loba l non l inear i ty of the mate r i a l , w h i c h appears i n spite of the fact that both aggregate and hardened cement paste are, i n d i v i d u a l l y , essential ly l inear up to the fai lure stress of the concrete. Anothe r i nd ica t ion is the apparent vo lume increase of concrete under compression. A n i m p o r t a n t feature is that g lobal ly , the fractures in the mate r i a l coincide w i t h the d i rect ion of the m a x i m u m pr inc ipa l compressive stress (Kotsovos 1979). F o r the case of u n i a x i a l compression, this corresponds to the wel l k n o w n sp l i t t i ng fa i lure occurr ing overwhelming ly in careful exper iments. T h e con ica l fai lure mode frequently observed in concrete compression tests is due to the latera l conf in ing effect of the load ing p laten ; it w i l l change to a sp l i t t i ng mode i f the f r ic t ion between the specimen and the load ing platen is reduced. A l t h o u g h shear stress does develop on inc l ined planes in un iax ia l compression tests, the classical theories based on shear fracture proposed by C o u l o m b , Nav ier , and M o h r are s i m p l y not born out by exper iment . F racture mechanics based on Gr i f f i th ' s theory provides a powerful methodo logy i n b r i t t le fa i lure analysis . Unfor tunate ly , i ts success in app l ica t ion to concrete, compared w i t h metals , is rather moderate. T h i s is largely because, as a cement-based composite , 31 c o n c r e t e is e s s e n t i a l l y a d i s c o n t i n u o u s , a n i s o t r o p i c , h e t e r o g e n e o u s , m u l t i p h a s e s y s t e m . T h e r e i s n o c l e a r l y d e f i n e d f r o n t f o r a m a j o r c r a c k a n d t h e e n e r g y d i s s i p a t i n g m e c h a n i s m i s n o t m e r e l y c o n f i n e d t o t h e s u r f a c e e n e r g y . D i r e c t a p p l i c a t i o n o f t h e s i n g l e c r a c k m o d e l i n L i n e a r E l a s t i c F r a c t u r e M e c h a n i c s d o e s n o t l e a d t o s a t i s f a c t o r y q u a n t i t a t i v e r e s u l t s . H o w e v e r , r e c e n t d e v e l o p m e n t s i n t h e a p p l i c a t i o n o f f r a c t u r e m e c h a n i c s t o c o n c r e t e h a v e b e e n m o r e e n c o u r a g i n g . T h i s i n v o l v e s t h e use o f a p r o p e r , n o n l i n e a r f o r m o f f r a c t u r e m e c h a n i c s i n w h i c h a f i n i t e n o n l i n e a r z o n e a t t h e f r a c t u r e f r o n t is t a k e n i n t o a c c o u n t , f o r e x a m p l e see B a z a n t ( 1 9 8 5 ) . T h i s finite z o n e c a n m o d e l s t r a i n - l o c a l i z a t i o n d u e t o s t r a i n s o f t e n i n g o f c o n c r e t e ( i n a n a v e r a g e sense o v e r a s m e a r e d c r a c k b a n d ) a t t h e c r a c k f r o n t a n d p r o v i d e a n e n e r g y c r i t e r i o n f o r c r a c k e x t e n s i o n . C o n c r e t e o u t s i d e o f t h i s finite z o n e c a n b e c o n s i d e r e d t o b e h a v e e s s e n t i a l l y e l a s t i c a l l y . I t i s f o u n d t h a t t h e d e t a i l e d d i s t r i b u t i o n s o f s t r e s s a n d s t r a i n a t t h e f r a c t u r e f r o n t h a v e l i t t l e e f fect g l o b a l l y , s i n c e f r a c t u r e p r o p a g a t i o n d e p e n d s e s s e n t i a l l y o n t h e f l u x o f e n e r g y i n t o t h e f r a c t u r e p r o c e s s z o n e , w h i c h i s a g l o b a l c h a r a c t e r i s t i c o f t h e e n t i r e s t r u c t u r e . A l t h o u g h t h e s e findings w e r e o b t a i n e d i n t h e s t u d y o f c o n c r e t e u n d e r t e n s i o n , s o m e o f t h e b a s i c p r i n c i p l e s s h o u l d a l s o b e a p p l i c a b l e t o t h e c a s e o f u n i a x i a l c o m p r e s s i o n . I n t h e c a s e o f u n i a x i a l c o m p r e s s i o n , e x p e r i m e n t s o n d i f f e r e n t b r i t t l e m a t e r i a l s s u c h as c e r a m i c s , g l a s s a n d e s p e c i a l l y o n n a t u r a l r o c k s , h a v e a g a i n r e v e a l e d t h e s a m e s p l i t t i n g f a i l u r e m o d e a n d a s i m i l a r s t a b l e - u n s t a b l e f a i l u r e p r o c e s s . F o r r e l a t i v e l y h o m o g e n e o u s m a t e r i a l s , o f t e n o n l y o n e o r a f e w s p l i t s a r e o b s e r v e d , w h i l e f o r less h o m o g e n e o u s m a t e r i a l s , m o r e v i s i b l e v e r t i c a l c r a c k s a r e f o u n d t o a c c o m p a n y t h e m a i n s p l i t t i n g . ( S e l d e n r a t h et a l 1 9 5 8 , F a i r h u r s t a n d C o o k 1 9 6 6 , B r a c e a n d B y e r l e e 1 9 6 6 , P a t e r s o n 1 9 7 8 ) . T h i s h a s l e d t o r e l a t i v e l y e x t e n s i v e m o d e l s t u d i e s , a t a f u n d a m e n t a l l e v e l , i n t h e s e a r e a s . T h e m o s t f r e q u e n t l y s t u d i e d m o d e l s a r e g r o u n d e d i n t h e i d e a t h a t f r i c t i o n a l s l i d i n g o f a p r e - e x i s t i n g c r a c k p r o d u c e s , a t t h e c r a c k t i p s , t e n s i o n c r a c k s t h a t g r o w i n t h e d i r e c t i o n o f c o m p r e s s i o n , as s h o w n i n F i g . 3 . 1 . I t m a y be w o r t h g i v i n g a b r i e f d e s c r i p t i o n o f t h e e f fect o f t h e p r e s e n c e o f a s l i d i n g c r a c k . C o n s i d e r a b l o c k i n w h i c h a c r a c k a p p e a r s as i n F i g . 3 . 2 ( a ) . B e f o r e t h e c r a c k a p p e a r e d , t h e s t r e s s cr m n 32 F I G . 3.1 A S l i d ing F r i c t i o n a l C rack in a Compress ive Stress F ie ld , Showing the O r i g i n a l Defect and its Extens ion t I I t t a 1 l l l l w i n \ % \ \ t ft (a) I M M (b) F I G . 3.2 Depict ion of the Effect of a C rack 33 field was such that a pa i r of n o r m a l forces N and shear forces T were transferred across the space now occupied by the crack. W h e n the crack forms, N is s t i l l transferred, but T can no longer be carr ied . T h u s the effect of the crack on the or ig ina l stress field is the same as the app l ica t ion of two opposite shear 1 forces T o n the crack zone, as depicted in F i g . 3.2(b) and F i g . 3.1 (i.e. the remova l of 7). S i m i l a r argument m a y also app ly for mater ia l defects w i t h other conf igurat ions. A l t h o u g h this mode l is a rad ica l idea l i zat ion of rea l i ty , i t does capture some of the basic features of the observations made at the microscopic level on rocks and concrete. F r i c t i o n a l s l id ing does occur a long the pre -exist ing interface cavit ies or cleavage cracks, and for concrete this often takes place at the matr ix -aggregate interface. T h e s l id ing - induced tension cracks tend to grow in the d i rect ion of the compression, in spite of local inhomogeniety , in an i n i t i a l l y stable manner . A l t h o u g h mate r ia l defects are also found in the f o r m of cavit ies w i thout contact faces, i t has been observed that the induced tension crack ing has a m u c h lower tendency to grow than does the s l i d ing case (Ziegeldorf 1983). T h i s can also be inferred f rom the ana l y t i ca l work of Panas juk (1976), Zaitsev(1983) , or S a m m i s and A s h b y (1986), wh ich indicates that under compressive stress cr, the energy release rate for a crack w i t h extended length / is in the order of <r 2/ / i f the defect is an inc l ined pre -exist ing crack, and c r 2 / I5 i f the defect is a vo id (see F i g . 3.3). T h u s defects in the approx imate f o r m of s l id ing cracks w i l l dominate the crack extension unless the d i s t r i bu t ion of defects in other forms is overwhelming. A d d i t i o n a l l y , the idea l i zat ion of mate r ia l defects appears to be necessary i f we are to reach an ana l y t i ca l l y manageable approach. P r o b a b l y for a l l these reasons, since it was first proposed by M c C l i n t o c k and W a l s h (1963), the mode l of a s l id ing crack w i t h k inks has received considerable a t tent ion . It has been studied both ana ly t ica l l y and by mode l exper iment ; the latter is often achieved by car ry ing out tests on some synthet ic b r i t t le mate r i a l w i t h m a n - m a d e s l id ing crack(s) . T h e most notable work includes Brace and B o m b o l a k s i (1963), Hoek a n d B i e n i a w s k i (1965), Sant iago and H i lsdor f (1973), K a c h a n o v (1982), Nemat -Nasser and H o r i i (1982), Zaitsev 34 o a n m n m f t t t t . f t t t • a a F I G . 3.3 Models of M a t e r i a l Defects. (The M iss ing Force Ac ts on E a c h Side in a D i rect ion Opposite to that Shown. T h e Effect of the Defect is Therefore to A p p l y Forces in the D i rect ion Shown.) (1983), Steiff (1984), Ho r i i and Nemat-Nasser (1985), A s h b y and Cooksley (1986), and H o r i i and Nemat -Nasser (1986). Because of the complex i ty of the problem, a lmost a l l the ana ly t ica l studies have been based on p la in elast ic i ty . T h i s is, of course, closer to the behaviour of homogeneous br i t t le materials such as ceramics and glass than to rocks and concrete. Nevetherless, they appear to give reasonable explanat ions of some of the characterist ics of these inhomogeneous mater ia ls . In this chapter, some aspects of previous model studies w i l l be briefly reviewed, and a 35 s impl i f ied mode l based on interact ion of the s l id ing cracks w i l l be presented. T h e focus w i l l be on the t rans i t ion f r o m stable to unstable c rack ing under u n i a x i a l compression; and the latter w i l l be shown to manifest the wel l k n o w n sp l i t t i ng fai lure. T h e model w i l l be shown to reveal the character ist ics of the compressive strength, and of the stress-strain re lat ion of b r i t t le mater ia ls under u n i a x i a l compression. T h e mode l w i l l be based on plane e last ic i ty and an ideal ized crack pat tern . W h e n i t is app l ied to the behaviour of concrete and rock, i t m a y be subject to the same l i m i t a t i o n s as the previous ana l y t i ca l work, but , i n v iew of the l i m i t e d p last ic deformat ion of these mater ia ls under u n i a x i a l compression, and of the successful app l icat ion of f racture mechanics to concrete under tension, th is approach should reveal some of the basic features of compression. However , as ind icated above, loca l nonl inear behaviour must be inc luded to give correct quant i ta t i ve predict ions for concrete. T h u s , a l though some quant i ta t ive conclusions d r a w n f rom the proposed mode l w i l l be presented, the basic objective is to i l lust rate rather than quant i fy . It is hoped, that th is theoret ical t reatment , based on a hypothesis for the fai lure mechan ism, w i l l shed some l ight on the actua l fa i lure process, and lead to a better understanding of the p rob lem. 3.3 Mode ls of Internal B r i t t l e Fa i lu re B r i t t l e fa i lure under un iax ia l compression is d ist inct f rom that in tension in that there exists a stable c rack ing process before final unstable fracture. T h i s has been observed exper imenta l ly (for instance, as we reviewed for concrete i n the in t roduct ion ) , and has been ident i f ied f r o m the fracture mechanics point of v iew (for example , see Kostovos and N e w m a n 1981). W i t h crack g rowth , the strain-energy concentrat ion at a crack front tends to increase i n the case of tension, but decrease in the case of compression. T h u s in compression, the fracture occurs i n i t i a l l y in a discrete, stable manner w i t h increasing load ; fai lure occurs when the stable c rack ing reaches a certa in extent, but not at the i n i t i a t i o n of these cracks. Over look ing the subcr i t i ca l crack g rowth w i l l lead to erroneous results, such as Gr i f f i t h ' s p red ict ion of 1:8 for the 36 rat io of tensile to compressive strength - a substant ia l underest imate for m a n y br i t t le mater ia ls (Obert 1972). (In the case of concrete under tension, fracture m a y appear to be temporar i l y s tab i l i zed ; but this is due to arrest by the aggregate rather than the release of the strain-energy concentrat ion. ) T h e model of a single crack w i t h k inks certa in ly exhib i ts this stable feature. Referr ing to F i g . 3.1, the s l id ing shear force, wh ich represents the effect of an inc l ined crack w i t h length 2a in an otherwise compressive stress f ie ld , is the resultant of the d r i v i n g shear stress a long the crack (for example , see Zaitsev 1983) T = 2a a ( s i n a c o s a — u s i n 2 a ) 3.1 where u is the coefficient of f r ic t ion of the mate r ia l . W h e n the extended crack length 21 is long compared w i t h 2 a, the hor i zonta l components of these shear forces may , as far as sp l i t t i ng is concerned, be considered as a pai r of tension forces of magn i tude P= T s i n a . A s the crack extends i n the d i rect ion of the appl ied stress, these forces rema in app rox imate l y constant , and the w e l l - k n o w n fracture mechanics so lut ion (Broek 1978) for this case shows that the stress in tens i ty at the crack t i p attenuates w i th extension ( Kj = P/-fJTl ). It is for this reason that the crack is i n i t i a l l y stable. A n exact fo rmu la t ion of the p rob lem has been g iven by H o r i i and Nemat -Nasser (1985), wh ich gives results very close to th is a p p r o x i m a t i o n . Since the stable extension does not lead to immed ia te fai lure, the detai led tu rn ing pa th of the w ing cracks appears to be un impor tan t . M o d e l exper iments on br i t t le mater ia ls w i t h a man -made s l id ing crack have indeed ind icated this stable, tensile crack extension, tu rn ing in to the d i rect ion of the load ing (Brace and B o m b o l a k i s 1963, Hoek and B ien iawsk i 1965, Sant iago and H i lsdor f 1973, Nemat -Nasser and H o r i i 1982, H o r i i and Nemat -Nasser 1985). T h i s d i rect ion is favored because this is the o r ientat ion i n wh ich the least work is required to open the crack. A l t h o u g h M o d e II stress 37 intens i ty appears i n the crack t ips, shear f racture i n the plane of the prepared s l id ing crack was never observed unless the w i d t h of the specimen was close to the crack length. However , since f ina l fa i lure is brought about by unstable f racture, there must be a t rans i t ion f r o m s tab i l i t y to i ns tab i l i t y i n the crack ing . Recogn iz ing this po int , A s h b y and Cooks ley (1985) developed a model based on the w ing crack interact ion . T h e y hypothesize that when stable cracks are re lat ive ly long , the branches between cracks tend to bend, wh ich intensif ies the stress concentrat ions at the crack t ips and leads to i ns tab i l i t y . However , this bending in teract ion mechan ism appears to be insuff icient to exp la in an unstable sp l i t i n a re lat ive ly short specimen. It m a y be wor th ment ion ing that K e n d a l l (1978) has also developed a s im i la r beam bending mode l to exp la in a x i a l sp l i t t i ng ; but this one requires an indented (i.e. a load w h i c h does not cover the outer edges of the loaded face) compressive load ac t ing on a ver t ica l crack, forc ing the two struts separated by the crack to bend outwards . Obv ious l y , th is mode l fai ls to give an exp lanat ion when the g lobal compressive stress is un i fo rm. Based on their ana l y t i ca l work, H o r i i and Nemat -Nasser (1982, 1985) concluded that the s l id ing - induced tensile crack is very sensitive to latera l stress. T h e crack extension soon becomes unstable i f a s m a l l la tera l tension exists. T h e i r model exper iments on a barrel -shaped specimen gave an excellent i l l us t ra t ion of this point . However, an exp lanat ion is s t i l l needed for the case of u n i a x i a l compression corresponding to zero latera l stress. 3.4 Proposed M o d e l It is clear that the mode l of a single crack w i t h k inks on ly provides the source of the sp l i t t i ng . O ther effects must be inc luded to exp la in the unstable t rans i t ion . W e now present a re lat ive ly s imple model to show that this t rans i t ion can be a consequence of the extension of a group of stable cracks. Some interest ing results w i l l fo l low immed ia te ly . 3.4.1 C r a c k Interactions and C r i t i c a l State 38 Since, as ind icated above, crack extension in compression is i n i t i a l l y stable, there is a h igh p robab i l i t y that , w i t h increasing stress, cracks w i l l extend f r o m a l l defects w i t h s im i la r conf igurat ions. A s a result, compressive fai lure is usual ly not governed by any i n d i v i d u a l defect; this contrasts w i t h tension fai lure, wh ich is governed by the defect w i t h c r i t i ca l conf igurat ion , and in wh ich fracture is h ighly local ized. T h u s it appears necessary to consider a l l the defects w h i c h govern the behaviour . B y the same argument , i t m a y also be reasonable, as w i l l be. discussed later , to treat the crack ing process in an average sense. B y using the mode l of the s l id ing crack w i t h k inks and the described a p p r o x i m a t i o n , every defect in a mate r ia l corresponds to a pai r of sp l i t t i ng forces P{ = a{ cr 3.2 where kt = 2 ( sinaj-cosa,- — / i s i n 2 a 8 - ) s ina, - 3.3 Note that P{ depends on the i n i t i a l , inc l ined , length of the crack and not on the extended length. k{ a,- takes account of the conf igurat ion of the crack. F o r concrete, at m a y be i n the order of the aggregate part ic le size; the coefficient of f r ic t ion /z is about 0.36 (T roxe l l et a l . 1968), so that k for the worst angle is about 0.45 (i.e. the angle corresponding to the biggest force). Let us examine an ideal ized case where a series of defects lies in a l ine as shown i n F i g . 3.4(a). B y the stated approx imat ions , the s i tuat ion i n F i g . 3.4(a) is equivalent to F i g . 3.4(b): a series of cracks w i t h an average length of 21 and an average spacing 26 acted on by pairs of po int forces. F o r an in f in i te m e d i u m , the p rob lem has been studied by I rw in (1957). T h e stress intens i ty factor at the crack t ips for this case is ava i lab le f r o m the Westergaard stress funct ion g iven by h i m : 39 21 P — (a) 2b 2b (b) 21 21 F I G . 3.4 A Series of Cracks in a Compressive Stress F i e l d : T w o Levels of Ideal izat ion 1 'bsin(irl/b) 3.4 or, in terms of energy release rate for plane stra in condit ions: . _ P \ l - v 2 ) 'I Eb sin (W /6 ) 3.5 where E = Young 's modulus; v = Poisson's rat io . C r a c k s extend when Eqs. 3.4 or 3.5 reach some c r i t i ca l value, which is a mater ia l constant. T h e solut ion indicates that, when l/b < 1 /2 , dP/dl > 0, cracks propagate stably ; the propagat ion becomes unstable when l/b > 1 /2, dP/dl < 0. Once l/b reaches 1 /2, cracks w i l l 40 propagate extensively , and one or more w i l l run through the mater ia l immed ia te ly . T h i s point m a y therefore be defined as the c r i t i ca l state. T h i s re lat ively s imple model c lear ly reveals the character ist ics of the stable-unstable fracture process. It shows that c r i t i ca l i ns tab i l i t y can be the result of the stable crack growth itself. In real i ty , pre -ex ist ing defects m a y rarely exist exact ly col inear ly . However , mode l exper iments by H o r i i and Nemat -Nasser (1985), i n wh ich plates of C o l u m b i a resin C R 3 9 conta in ing a number of pre -exist ing s l id ing cracks were tested under un iax ia l compression, have ind icated that the ver t ica l ly d is t r ibuted cracks do indeed tend to j o i n each other to f o r m the f i na l f racture, even though they are not i n a vert ica l l ine. F o r a re lat ive ly homogeneous specimen (without m a n - m a d e cracks) , surface cracks at the top a n d b o t t o m w i l l be l ike ly to govern the behaviour, i.e. ver t ica l cracks w i l l i n i t i a te f r o m top and b o t t o m instead of f r o m inside of mater ia l (this can easily be verif ied by test ing, say, a plexiglass st rut ) . T h e mode l s t i l l appl ies i f we consider the specimen height as 2b, referring to F i g . 3.4(b), and recognize the fact that the so lut ion is s y m m e t r i c w i t h respect to every l ine of sp l i t t i ng forces. P h y s i c a l l y , i t means that the equivalent sp l i t t i ng forces are app l ied at the top and the b o t t o m of the specimen, and the fractures become unstable when the ver t ica l cracks i n i t i a t i n g f r o m the top a n d the b o t t o m both reach approx imate ly one quarter of the specimen height. A s i m i l a r argument appl ies i n the case where the height of a specimen is s m a l l compared to the size of a pre -exist ing crack inside the mate r ia l , so that the final f racture is governed by a sp l i t f r o m this defect. In this case, the specimen height can be s t i l l considered as 2b, but the pai r of sp l i t t i ng forces is appl ied inside. T h e fracture becomes unstable when the spl i t reaches app rox imate l y ha l f of the specimen height. T h u s it appears that the mode l is useful i n m a n y cases. 3.4.2 Some Consequences of the M o d e l : Peak Stress 41 P u t t i n g E q . 3.2 in to E q . 3.4, w i t h l/b = 1 /2 , and so lv ing for a, we can est imate the fa i lure stress (or the so-cal led compressive strength) of concrete, or any other br i t t le mate r i a l , as fc = 3.6 where KJQ= c r i t i c a l stress intens i ty factor; k = average conf igurat ion factor ; 6 = average hal f spacing of defects; £ = average value of a/b. These quant i t ies are a l l considered to be fundamenta l mate r ia l constants. Note that fc is used here to denote the fa i lure stress of a br i t t le mate r i a l , not necessarily concrete. Note that a l l the terms on the r ight hand side of E q . 3.6 should be understood in an effective sense when they are not clearly defined by microscopic observat ion. F o r concrete and rocks, the te rm KJC or GIC should be understood as the energy d iss ipated by a l l the mechanisms when a crack propagates, not merely the surface energy. It can be easily shown that , based on this model , the stress intensi ty at the crack t ips w i l l be d rast ica l l y reduced even i f a s m a l l latera l compressive stress is present. T h u s such a stress w i l l lead to a different fa i lure mode corresponding a higher fa i lure stress. T h i s m a y exp la in the shear fai lure mode, wh ich is accompanied by a s igni f icant increase in strength, that is exh ib i ted in a compression test on a conf ined specimen. In practice, the latera l stress is often in t roduced by the load ing p laten in un iax ia l compression tests. E q u a t i o n 3.6 m a y need mod i f ica t ion for specimens of f in i te size and for the inter lock a n d crack arrest mechanisms that are present i n concrete and rocks; and for concrete, inc lus ion of the nonl inear behaviour at the crack front appears to be necessary for precise analysis . Nevertheless, the equat ion should give a reasonable est imate of the compressive strength. 3.4.3 R e l a t i o n to Tens i le Strength B r i t t l e tension fai lure is re lat ively wel l understood. It is governed d i rect ly by the pre-ex is t ing cracks, because extension is unstable under tensile loading. T h e tensile strength can be 42 est imated, based on the pre -exist ing crack conf igurat ion and d is t r ibut ion (see F i g . 3.5) us ing the so lut ion (Broek 1978) Kj = [<rtJWa] [(2b/w a) tan ( 7 r a / 2 6 ) ] 1 / 2 3.7 where the te rm in the f irst bracket is the we l l - known so lut ion for an isolated crack in a background tensile stress f ie ld ; the te rm in the second bracket is inc luded to prov ide an est imate of the effect of adjacent cracks. A l t h o u g h cracks wou ld rarely exist i n the conf igurat ion of F i g . 3.5, tension fa i lure is governed by a single crack w i t h c r i t i ca l conf igurat ion , so that E q . 3.7 need on ly ho ld for a very s m a l l region. E q u a t i o n 3.7 m a y be rearranged to give the tensile strength as ft - • K j C 3.8 \2b tan ( 7 r£/2) so that , in v iew of E q . 3.6 A = 3 9 fe ft tan(7r£/2) T h i s impl ies that the rat io of tensile to compressive strength of a br i t t le mate r ia l is solely dependent on the conf igurat ion and d is t r ibu t ion of the pre -exist ing defects, and the interna l f r i c t ion of the mate r i a l . T h i s ra t io for concrete is p lo t ted against £ in F i g . 3.6. T h e rat io for the extreme case, u = 0 (e.g. for some ceramics) is shown as wel l . E x p e r i m e n t shows that ft/fc ranges f r o m 0.06 to 0.13 for concrete, suggesting a range of £ f r o m 0.05 to 0.25. T h e mode l also suggests that the lower strength rat io is associated w i t h a smal ler £, wh ich tends to ind icate a higher strength mate r ia l . T h i s agrees w i t h the we l l - known non -p ropor t iona l re lat ionship between tensile strength and compressive strength of concrete F I G . 3.6 P r e d i c t e d R e l a t i o n b e t w e e n T e n s i l e S t r e n g t h a n d C o m p r e s s i v e S t r e n g t h v e r s u s S i z e / S p a c i n g R a t i o f o r B r i t t l e M a t e r i a l s 44 (Park and Pauley 1975),. T h e model also predicts an upper bound of about 0.16 for this rat io . T o the author 's knowledge, this extreme case has never been surpassed. A n explanat ion is provided for the wide range in strength rat io observed in other br i t t le materials. Rocks, for example, exhib i t values f rom 0.02 to 0.10 (Obert 1972), which are covered by the model when the pre-exist ing cracks range f rom short to long relative to their average spacing. The model also predicts that no br i t t le mater ia l can have a tensile strength exceeding 28% of its compressive strength. 3.5 T h e Stress-Strain Curves for B r i t t l e Mater ia ls under U n i a x i a l Compression W e first review exist ing knowledge of the force-deformation relationship, which m a y be d iv ided into two parts: the pre- and post-peak branches. A l though the word br i t t le impl ies l im i ted deformat ion before failure, it appears that , even for very br i tt le mater ia ls , there is 120 10.0 8.0 6.0 £ 4.0 in 2.0 NORMAL WEIGHT 40 E 0.001 0.002 0.003 0.0O4 0.005 0.006 STRAIN F I G . 3.7 Exper imenta l Stress-Strain Relat ions of Concrete, under N o r m a l Test Cond i t ions ( W a n g 1978) some nonl inear i ty immediate ly before the peak stress. Fo r less br i t t le mater ia ls such as concrete, there is an essentially l inear response up to a certain stress level , then ^ nonl inear i ty becomes apparent, and increasingly so as the mater ia l approaches fai lure (Wawers ik and Fa i rhurst 1970; Obert 1972; Mindess 1983; B rady 1985). T h e post-peak behaviour is s t i l l more compl icated . Less br i t t le mater ia ls , such as concrete, exhib i t a pronounced long ta i l ( W a n g 1978) in the stress-strain 45 curve under n o r m a l test ing condi t ions , as shown i n F i g . 3.7. However , Kotsovos (1983) indicates that this wide ly held v iew m a y be mis lead ing . H i s exper iments show that end condit ions s ign i f icant ly affect the post-peak behaviour , especial ly for h igh strength concrete. He placed var ious " a n t i - f r i c t i o n " m e d i a between the specimen and load p laten , and found very different behaviour ( F i g . 3.8). He concludes that , i f the f r ic t iona l restraint is e l im inated , the mate r i a l w i l l suffer a complete a n d immed ia te loss of l oad -car ry ing capac i ty . H i s results show an apparent recovery of compressive st ra in after the peak load , but he does not comment at any length on this surpr is ing phenomenon. It is se ldom observed, even in tests of more br i t t le mater ia ls , since recording in this range is very d i f f icu l t w i thout special arrangements. W a w e r s i k and Fa i rhurs t (1970), using very careful test procedures, were able to fo l low, i n part , the descending branch for some f ine-grained rocks. Some st ra in was c lear ly recovered as the load was reduced beyond the peak stress, and the stress-strain curve turned towards the or ig in ( F i g . 3.9). T h e long t a i l , w i t h decreasing stress accompanied by increasing st ra in (assuming that i t is not merely a result of imprecise test procedures), is k n o w n as class I response. T h e a l ternat ive observat ion, when the s t ra in is recovered, is k n o w n as class II response; i t has the def in ing character ist ics that " t h e fracture process is unstable or sel f -sustaining; to cont ro l f racture, energy must be extracted f r o m the mate r i a l " (B rady 1985). T h e c lassi f icat ion has been based entirely on exper imenta l observat ion, and it is hoped that the fo l lowing analysis , based on the proposed mode l , w i l l shed some l ight on the observed phenomena. 3.5.1 T h e P r e - P e a k B r a n c h T h e i n i t i a l de format ion , before the cracks begin to extend, can be ca lcu lated f rom Y o u n g ' s modu lus . A second phase, wh ich w i l l now be studied, is entered when crack propagat ion begins. 46 F I G . 3.8 Exper imenta l Stress-Strain Relat ions of Concrete, Specimens w i th " A n t i - F r i c t i o n " C a p p i n g (Kotsovos 1983): (a) Stress versus St ra in Measured on the Specimens; (b) L o a d versus Displacement 47 ISOTROPIC MATERIAL - UNIAXIAL COMPRESSION 0 0.1 0.2 0.3 0.4 0.5 A x i a l s t r a i n , e , (%) F I G . 3.9 Exper imenta l Stress-Strain Relat ions of some N a t u r a l Rocks (Wawersik and Fa i rhurs t 1970) Consider a rectangular region of height h and w id th w under un iax ia l compression; assume the cracking process is quasi -stat ic . W h e n the cracks extend dl we have, for energy conservation, dV - dU = dW + dR 3.10 where dV = work done by external load; dU = dissipated to form new crack extensions; dR = < : increase of the st ra in energy; dW = energy energy dissipated by the f r ict ion between the 48 contact surfaces of pre -ex ist ing cracks. C l e a r l y dV = FdA 3.11 where F = external load ; A = associated displacement. Since in a br i t t le mate r ia l the p last ic deformat ion is l i m i t e d , the mate r i a l w i l l r ema in essential ly l inear ly elastic regardless of the c rack ing . It can be shown (see appendix ) that as long as overa l l f racture does not occur, so that the region is s t i l l connected, the s t ra in energy can be expressed i n terms of the external load and the associated d isplacement , as If the f r ic t ion between the pre -exist ing crack surfaces is inc luded, this expression becomes where M is the number of pre -ex ist ing cracks i n the region, l0 m a y be cal led the effective i n i t i a l extending crack length, wh ich is a funct ion of the crack conf igurat ion , and A is a constant expressed as U = 1 /2 FA 3.12 3.13 8(l-v2) nka2sma irEw2 3.14 T h u s dU is readi ly ava i lab le by di f ferent iat ion of this expression: dU = 4- FA'+ FJA - 2MAFF1 log ( tan ( 7 r / / 26 ) t a n ( 7 r / o / 2 6 ) ) - MAF2-^ c o s e c ^ dl 3.15 49 T h e new crack surface energy can be expressed as dW = 2M GTn dl 3.16 F i n a l l y , the energy dissipated by the f r ic t ion can be app rox imated as (see appendix) dR « M 8uGrrsma k 1 + -k- cos| tan ( 7 r / / 26 ) t a n ( 7 r / 0 / 2 6 ) 3.17 These equations are v a l i d when cracks are extending, i.e., when Kj or Gj defined by Eqs . 3.4 or 3.5 have reached the c r i t i ca l values. T h e y give the re lat ionship between load F and the d isplacement A i n terms of the independent parameter /, the crack length. F o r a given load , the equations can be solved for A by subst i tu t ing Eqs. 3.11, 3.15, 3.16 and 3.17 in to E q . 3.10, and us ing the i n i t i a l cond i t ion where a0 m a y be cal led the threshold stress for the crack extension; the re lat ion to l0 is obta ined i n v iew of Eqs . 3.4 and 3.6: A = (Toh/E when / = l0 3.18 3.19 N o w , the load F = aw is related to / by Eqs. 3.2, and 3.4; and A = eh 3.20 where e = l ong i tud ina l s t ra in . 50 T h u s we are able to extract an expression for s t ra in in terms of stress. A s s u m i n g the average spacing of the cracks is the same hor i zonta l ly and ver t ica l ly , M _ 1 3.21 wh (2 b)2 we get e = 2/^sina k log t a n ( 8 / 2 ) , fc\ a tan(9o/2) + E ] fc 3.22 where 0 O = arc sin(<x0/y£)2; 0 = arc sin(a/fc)2. Phase II of the pre-peak branch covers the range a0 < a < fc, w i t h Q0 < 0 < 7r/2. 3.5.2 T h e P o s t - P e a k B r a n c h T h e model discussed above is found to represent behaviour of class II mater ia ls in to the post-peak branch . W h e n \x = 0, E q . 3.22 is v a l i d for the fu l l range 0 O < 0 < 7r. W h e n f r ic t ion is i nc luded , however, the equat ion appl ies on ly u n t i l the cracks stop opening somewhere in the descending branch . There is then a compl icated s i tuat ion as the cracks begin to close and the f r ic t ion to change d i rect ion ; a more elaborate t reatment is given in the appendix . A f te r an in terva l , the crack widths decrease and the f r ic t ion force is react ivated in the opposite d i rect ion ; E q . 3.22 is again appl icable , but w i th opposite sign on the terms conta in ing u in the bracket . There is also a different constant of integrat ion i n this range. T h i s app l i ca t ion to class II behaviour is predicated on the assumpt ion that cracks extend ver t ica l l y i n iso lat ion f r o m each other, so that the region is s t i l l connected. Fu r ther , i t is assumed that the cracks are regular, so that hor i zonta l f racture does not occur, and that the crack surfaces are re lat ive ly smooth , so that they close dur ing the descending b ranch wi thout in te r lock ing . These assumpt ions, necessary for cont inued app l icat ion of E q . 3.22, are good for a 51 re lat ive ly homogeneous br i t t le mater ia l . F o r less homogeneous mater ia ls such as concrete and coarse grained rocks these assumpt ions m a y be expected to be approx imate ly fu l f i l led dur ing the load ing stage, when the cracks are less extensive, and s t i l l opening. O n the descending branch , however, the cracks tend to propagate through weak gra in boundaries or aggregate-cement m a t r i x bonds (Ziegeldorf 1983; B r a d y 1985), and the z ig -zag crack paths have a tendency to interconnect para l le l fractures. T h i s leads to type I response as w i l l be discussed below. B u t , even i n this case, the experiments of Kotsovos (1983) suggest that the type I response m a y merely be due to the end f r ic t iona l constraints i n h i b i t i n g ver t ica l crack extension and leading to th is type of fai lure. W i t h " a n t i - f r i c t i o n " capp ing , he observed that ver t ica l c rack ing of the higher strength specimens a lways extended in both direct ions, whi le , for the lower strength ones, i t extended in one d i rect ion only , i nd icat ing that the restra in ing act ion of at least one end zone was s t i l l present. T h u s i t appears that the v a l i d i t y of the assumpt ions of E q . 3.22 depends upon the mate r i a l properties and the load ing condit ions. 3.5.3 T h e P red ic ted Stress -Stra in Curve E q . 3.22 is p lo t ted for a0//c = 0.3 , w i t h u = 0 and a = 0.36, in F i g . 3.10. T h e shape of the T y p e II curve is character ized by 4 points as ind icated on the figure. F r o m O to A , below the threshold stress, I = l0, the cracks do not extend, and d isp lacement is due solely to the l inear elastic response. In real i ty , of course, the onset of stable crack extension is d i f f icul t to ident i fy ; i t is a g radua l process rather than a sudden one, because of the var iety of defect conf igurat ions. T h u s there is a t rans i t ion rather than a wel l -def ined point A . F r o m A to B , add i t iona l deformat ion occurs due to crack extension, and the curve becomes more non- l inear as fc is approached. Inclusion of f r ic t ion increases both peak stress and 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 STRAIN F I G . 3 .10 P r e d i c t e d S t r e s s - S t r a i n R e l a t i o n s o f B r i t t l e M a t e r i a l s 53 st ra in . Greater non - l inear i ty appears due to energy d iss ipat ion through f r ic t ion . A f te r the peak, the cracks cont inue to propagate as the app l ied stress decreases, i n an unstable extension process. A t point C the st ra in reaches i ts m a x i m u m value, and then begins to reduce; the work required for the cracking process beyond C is p rov ided by part of the st ra in energy released f r o m the mate r ia l , but surplus energy must be extracted by the load ing device. T h e area enclosed by the complete curve is, of course, equal to the energy dissipated in creat ing new crack surfaces. A " less b r i t t l e " mate r ia l m a y have a lower threshold stress for crack extension, and a higher coefficient of f r i c t ion . A " m o r e b r i t t le" mate r ia l , on the contrary , m a y have a very h igh threshold a n d low f r ic t ion coefficient, so that i t gives the appearance of a l inear stress-strain curve. B u t the mode l impl ies that , since fai lure is caused by crack ing , there must a lways be some non - l inear i ty before it occurs. Note that fa i lure is f ina l l y brought about by the unstable crack extension sp l i t t i ng the specimen into pieces wh ich are i nd i v i dua l l y unstable, thus reducing the load capaci ty . W i t h convent iona l test arrangements, a load contro l led test ing mach ine w i l l cause mate r i a l fa i lure at po int B ; d isplacement contro l w i l l lead to fai lure at C i f the mach ine is st i f f enough. F a i l u r e w i l l be explosive because of the sudden release of s t ra in energy. M o s t reported results for more br i t t le mater ia ls are incomplete i n this sense, but Wawers ik and Fa i rhurs t showed complete curves that agreed, qua l i ta t i ve ly , w i t h the model pred ict ion , as d id the post-peak curves for h igh strength concrete obtained by Kotsovos . Note that , dur ing unstable crack extension, one or a few cracks w i l l propagate preferential ly , so that the mode l m a y lose some v a l i d i t y . However , for less homogeneous mater ia ls , such as concrete and coarse-grained rocks, the assumpt ion of regular ver t ica l crack extension m a y not be va l id in the post -peak range, as discussed earlier. T h i s wou ld exp la in the frequently observed class I response. Wawers ik and Fa i rhu rs t (1970) found that , in class I behaviour of rocks, ver t ica l f racture is, indeed, 54 accompanied by gradual development of shear fracture in the post-peak range; a s im i la r descr ipt ion is given by Kotsovos for his lower strength specimens. D u r i n g this range, the ratchet - l ike mechan ism of F i g . 3.11 forms, vert ica l deformat ion involves the wedges being dr iven into each other, and the energy of the load is converted to st ra in energy in the wedges, and f r ict ion losses between them, as wel l as surface energy in new cracks. T h e first two are clearly nonl inear, requir ing that the vert ica l load decrease less rap id ly w i th increasing deformat ion , and causing the load-deformat ion curves to be concave upward . Hence the inf lect ion point observed by W a n g et a l . (1978) and the long ta i l thereafter. F r o m the above analysis one may conclude that the fracture pattern determines the post-peak stress-strain re lat ion. V e r t i c a l fracture through the mater ia l w i l l lead to " m o r e b r i t t le" fai lure, whi le the development of shear faults w i l l give the appearance of more duct i le behaviour . T h e fracture pattern, in turn , may often be governed by the loading condit ions; f r ict ion i n the loading platen, for example, may cause shear cracks and , more duct i le behaviour. F I G . 3.11 Dep ict ion of Irregular C r a c k i n g Pa t te rn 3.6 Stat is t ica l Considerat ions W e now consider briefly the sensit iv i ty of the model predictions to stat is t ica l var iat ions in the parameters, which have hitherto been treated in an average sense. Assume that the defects are un i formly d ist r ibuted spat ia l ly but that the conf igurat ions 55 have a r a n d o m character described by some p robab i l i t y d i s t r i bu t ion , w i t h density funct ion $ = $(ka) 3.23 a n d c u m u l a t i v e funct ion Z = Z(ka) 3.24 w i t h ka i n some range (ka)min < ka < (ka)max 3.25 E q u a t i o n 3.6 can be modi f ied by these assumptions to give KIC(b Fa ( l-Z(ka) ka[l-Z(ka)}1/4 ~ ka ka I l-Z(ka) 1 / 4 3.26 where b0 is the average ha l f spacing of to ta l defects under considerat ion, and the barred quant i t ies denote the mean values. A s s u m i n g a n o r m a l d i s t r ibu t ion for defect conf igurat ion , and a p p r o x i m a t i n g by a W e i b u l l d i s t r i bu t ion , we wr i te Z(ka) = 1 - exp [ -(ka/6)m] 3.27 T h e W e i b u l l modu lus m = 3.6 best represents a n o r m a l d i s t r i bu t ion ; 8 is taken corresponding to a coefficient of va r ia t i on of ka of about 0.32. the second te rm of E q . 3.26, wh ich accounts for the s tat is t ica l considerat ions, is p lotted in F i g . 3.12, and shows the var ia t ion of ft as predicted 56 by the model , against the var iat ion in crack conf igurat ion. - If only a few defects w i th larger conf igurat ion factor ka (i.e. w i t h wider crack spacing) have extended to govern specimen behaviour, the fai lure stress w i l l be higher. T h i s is because these defects are sparsely d ist r ibuted, the stress required to br ing them to interact upon each other to reach the cr i t ica l state w i l l be h igh , as indicated at the r ight end of the graph. A l t h o u g h defects w i th smal ler conf igurat ion are more densely d is t r ibuted , the stress needed to cause them to extend w i l l s t i l l be higher, as shown at the left end of the graph. T h e m i n i m u m value is reached when ka is close to the mean value, a round wh ich the va r ia t ion is sma l l for a wide range of ka. T h u s the average parameters do y ie ld a reasonable approx imat ion . T h i s conclusion should be va l id as long as the d i s t r ibu t ion is not extremely d istorted. i -0 -I 1 I 1 I — 1 1 T 1 I I 0 0.2 0.4 0.8 0.8 1 1.2 1-4 1.8 1.8 2 ka / ka F I G . 3.12 Sensi t iv i ty of Compressive Strength to C rack Conf igurat ion Factor : the Norma l i zed Strength Predicted by the Mode l is P lo t ted against the Conf igura t ion Factor , W h i c h Depends on C r a c k Conf igurat ion and Internal F r i c t i o n 57 3.7 S u m m a r y and Co ro l l a ry T h e fa i lure mechan ism of b r i t t le mater ia ls under u n i a x i a l compression has been examined at the fundamenta l level . A fa i lure mode l based on the interna l mechan ism has been proposed to reveal the character ist ics of the compressive strength, and stress-strain re lat ion of these mater ia ls . T h e observed sp l i t t i ng fai lure has been shown to be the result of the cumu la t i ve , subcr i t i ca l , stable crack growth . T h e under ly ing concepts of the mode l have been just i f ied by reported observations. It m a y be further inferred f r o m the study that the fa i lure stress, or the so-cal led compressive strength of a b r i t t le mate r ia l is closely dependent on the interna l fa i lure mechan ism. T h e interna l mechan ism, however, depends not on ly on the mate r ia l texture, but is also affected by the testing or load ing condit ions. S p l i t t i n g fai lure corresponds to the lowest fai lure stress. A n y condi t ions wh ich prevent this fai lure mode f r o m being realized w i l l lead to an apparent ly higher fa i lure stress. These condit ions m a y be la tera l conf inement such as that in t roduced by the end f r i c t ion , or a s t ra in gradient wh ich causes unequal compression in the mate r i a l . A l t h o u g h the compressive strength as a funct ion of these condi t ions is d i f f icu l t to determine, one m a y expect the fai lure stresses to be better correlated i f the in terna l fa i lure mechanisms are s im i la r . T h i s , is of p ract ica l signif icance. In the later chapters, separate t reatments for concrete masonry under different load ing condit ions w i l l be proposed and i t w i l l be seen that this leads to better correlat ions i n terms of the fai lure stresses. 58 C H A P T E R I V P L A I N M A S O N R Y W I T H F U L L B E D D I N G 4.1 T w o B a s i c a l l y Dif ferent Fa i l u re Modes In concrete masonry compression tests, the specimens fa i l basical ly in two modes. One is sp l i t t i ng i n the d i rect ion of the load ; the other shows conica l fai lure planes (see Chapte r II). T h e signif icance of these two different mechanisms arises f r o m the fact that the different fa i lure modes y ie ld different apparent strengths, as ind icated i n the preceding chapter . It has been found repeatedly i n previous exper imenta l studies (for example , F a t t a l and Cat taneo 1976; T u r k s t r a and T h o m a s 1978) that when the eccentr ic i ty of the load on masonry specimens is increased, there is a s igni f icant apparent increase in compressive strength. T h i s phenomenon is also revealed in the tests conducted by the author , as depicted in F i g . 4.1 by compar ing a theoret ical l oad -moment interact ion curve for a masonry p r i s m w i t h exper imenta l results. A l t h o u g h this strength increase was a t t r ibu ted in some earlier studies ( T u r k s t r a and T h o m a s 1978) to the stress gradient effect, i t is now generally accepted that i t is essential ly due to a difference i n the fa i lure mechan ism. W h e n masonry is under u n i a x i a l compression, sp l i t t i ng fa i lure dominates , whether the masonry is fu l l y or face-shell bedded. T h e fai lure mode changes to the shear type when the masonry is under eccentric load ing . T w o obvious questions arise: 1) what is the cause of these two different fai lure mechanisms? 2) what is the i m p l i c a t i o n of these fai lure mechan isms for the compressive strength, the parameter of most p ract ica l concern. In th is s tudy , the fa i lure mechan ism is careful ly re -examined, and some of the exist ing theory is revised, i n the l ight of both exper imenta l and ana l y t i ca l work. W e start w i t h the case of f u l l y bedded p la in concrete masonry under u n i a x i a l compression. 59 0 . 0 40 80 120 160 200 M (KJP-N) F I G . 4.1 A p p a r e n t S t r e n g t h I nc rease P h e n o m e n o n u n d e r E c c e n t r i c C o m p r e s s i o n S p l i t t i n g f a i l u r e u n d e r u n i a x i a l c o m p r e s s i o n h a s b e e n i n d i c a t e d b y n u m e r o u s p r e v i o u s e x p e r i m e n t s , a n d t h e e x p e r i m e n t s c o n d u c t e d b y t h e a u t h o r h a v e a l s o r e v e a l e d t h i s p h e n o m e n o n (see F i g . 2.12). U n d e r u n i a x i a l c o m p r e s s i o n , the o n l y a p p a r e n t d i s t u r b a n c e i n t h e u n i a x i a l c o m p r e s s i o n f i e l d is t h e j o i n t . W e s h a l l d i s c u s s t h e ef fect o f t h e j o i n t o n t h e s t r e n g t h o f m a s o n r y , a n d c o n s i d e r w h e t h e r t h e p r e s e n c e o f t h e j o i n t is t h e c a u s e o f t h e s p l i t t i n g f a i l u r e . i 4.2 J o i n t E f f e c t — A R e v i s i o n o f H i l s d o r f s M o d e l T h e m a i n f u n c t i o n o f m o r t a r j o i n t s is t o p r o v i d e s t r u c t u r a l c o n t i n u i t y , w i n d a n d w a t e r t i g h t n e s s , as w e l l as a r c h i t e c t u r a l e f fect . T h e j o i n t s c a n be i n v a r i o u s p a t t e r n s , s u c h as r u n n i n g b o n d o r s t a c k b o n d , a n d t h e y c a n be r a k e d o r f l u s h . H o w e v e r , f o r r e a s o n s o f s i m p l i c i t y , t h e 60 study i n this chapter is confined to stack bonded, fu l l y bedded masonry w i th unraked jo in ts . It is bel ieved that the bond pattern does not have a s igni f icant effect on masonry strength (Maurenbrecher 1980; Shr ive 1982), a n d that the analysis to be presented is generally app l icab le . T o achieve cohesiveness and workab i l i t y , mor ta r contains certa in proport ions of cement, sand a n d l ime. Mechan ica l l y , i t is usual ly weaker and less st i f f than the sur round ing concrete un i ts (see C h a p t e r II). It is wide ly accepted that the mor ta r jo in ts affect the masonry strength, stronger mor ta r m a k i n g stronger masonry . T h e most in f luent ia l theory for the j o i n t effect was proposed by H i l sdor f (1969). H is theory postulates that when masonry pr isms are under u n i a x i a l compression, the less st i f f mor ta r has a tendency to expand latera l ly ; th is latera l expansion of the mor ta r is conf ined by the masonry units , g i v ing rise to latera l compressive stress in the mor ta r and to latera l tensile stresses in the units , thereby causing tensile sp l i t t i ng fai lure of the blocks. U s i n g the C o u l o m b - N a v i e r fa i lure cr i ter ion and some rather coarse assumpt ions about e q u i l i b r i u m and c o m p a t i b i l i t y , H i lsdor f der ived an equat ion re lat ing the compressive strength of masonry to the strengths of un i t and mortar . T h i s , of course, is very p rac t ica l , since the strengths of the un i t and the mor ta r are comparat ive ly easy to measure. However , there has been a lot of controversy about the correctness of H i l s d o r f s mode l i n the subsequent l i terature. O n the one hand , Hatz in iko las et a l (1978) made a numer ica l analys is based on H i l s d o r f s mode l and concluded that the magn i tude of the tensile stress in the block un i ts due to the lateral expansion of the mor ta r was sufficient to exceed the tensile strength of concrete b locks and thus was responsible for the sp l i t t i ng fa i lure of concrete masonry . Pr iest ley et a l (1983) extended H i l s d o r f s equat ion to grouted concrete masonry and c la imed good agreement ( in terms of masonry strength) w i t h the exist ing exper imenta l da ta . M o s t recently, B i o l z i (1988) app l ied the fai lure model i n an approx imate p last ic analysis for br ick masonry . O n the other hand , Shr ive (1980, 1983) strongly opposed the not ion that the latera l expansion of • mor ta r was the m a i n cause of sp l i t t i ng fai lure. He noted that 1) sp l i t t i ng fa i lure of compression 61 specimens is not unique to masonry . 2) the tensile stress caused by mor ta r expansion iri the b lock is too s m a l l to exceed the tensile strength of the block. T h e latter conclus ion was based on the numer ica l analyses of S m i t h et a l (1971), T u r k s t r a et a l (1978), H a m i d (1978) a n d of Shr ive h imsel f w i t h Jessop (1980), wh ich indicate that the tensile stress is m u c h less than that required to break the tensile bonds in the block. Drysdale and H a m i d (1979) suggested that the mechan ism of the la tera l expansion of mor ta r needed reconsiderat ion because i n their exper iments the mor ta r jo in ts had a re lat ively m ino r influence on the capac i ty of concrete masonry . T h e emergence of these controversies is not surpr is ing, since some points were not c lar i f ied in the previous studies. W h e n postu lat ing a tensile stress wh ich w i l l i n i t i a te a crack, i t is i m p o r t a n t to ind icate the locat ion where it w i l l occur. T h i s provides a log ica l way to check the correctness of the mode l by examin ing whether the locat ion is correct ly predicted in exper imenta l studies. T h i s was somehow overlooked i n the previous work. T h e arbi t rar iness invo lved i n the assumpt ion of the mate r ia l constants used in numer ica l analyses m a y also have cont r ibuted to the controvers ia l nature of some previous f indings. A n d so far, there has been no direct exper imenta l evidence wh ich cou ld lead to a conclusive assessment of the model . Because of that , some exper imenta l and ana ly t i ca l work is d irected here to eva luat ion of this theory. It should be ind icated that a l l the previous work i m p l i c i t l y takes one not ion for granted: that fa i lure is a local ized effect. Whether masonry fai ls depends on whether tensile stress at some point exceeds the tensile bonds of the mater ia l . In the l ight of the study in C h a p t e r III, we know that in the case of compression, loca l tensile c rack ing is on ly a necessary cond i t ion for g loba l fa i lure ; i t m a y not be suff icient. In our approach, we w i l l consider both the causes of tensile c rack ing , and whether this is tan tamount to fai lure. 4.2.1 E x p e r i m e n t a l Resul ts • T h e exper imenta l "results indicate: 62 1) for a l l fu l l y bedded p l a i n concrete masonry pr isms tested under un iax ia l compression, ver t ica l sp l i t t i ng (paral le l to the d i rect ion of loading) was the predominant fa i lure mode. T h e sp l i t t i ng occurs i n the m i d d l e t h i r d of the web, cont inuously runs through the specimen, as t yp ica l l y i l l us t ra ted in F i g . 2.12. S i m i l a r observations were reported i n previous exper imenta l work (for example , Ha t z in iko las et a l 1978). 2) There is a latera l expansion effect due to the mortar , as evidenced by the st ra in measurements on the block units . F i g 4.2 shows some t y p i c a l results recorded in exper iments. T h e average la tera l s t ra in at locat ion # 3 , w h i c h is closer to the mor ta r j o in t , is appreciably larger than that measured at locat ion #4, wh ich is at the mid -he ight of the web. 3) However , there is some randomness i n where the macrocrack is i n i t i a ted . B y detect ing the order of b reak ing of the wires crossing spl i ts (A detai led descr ipt ion of this procedure was given i n C h a p t e r II), cracks were found to in i t ia te at a locat ion close to the mor ta r j o i n t on ly in about two th i rds of the fu l l y bedded specimens, as depicted in F i g . 4.3. T h i s does not support H i l s d o r f s model , since the mode l suggests that crack should in i t ia te consistent ly f rom the j o i n t . If we assume that a crack wh ich in i t ia ted f rom this locat ion is a r a n d o m event and relax H i l s d o r f s hypothesis such that there is on ly a 90% chance of this event occurr ing , then this hypothesis is rejected at a 0.1 level of signif icance. W e also note the test result is in sharp contrast to that observed in face-shell bedded p la in concrete masonry, where cracks in i t ia ted consistent ly at a locat ion close to the j o in t , (cf. F i g . 5.2) T h i s strengthens the assertion that the test result does not support H i l s d o r f s postu lat ion that the sp l i t t i ng is due to the latera l expansion of the mor ta r jo in t . 4) T h e j o in t condi t ions have a b ig influence, on the capaci ty of a masonry p r i s m (see T a b l e 4.1). It is noted that stronger mor ta r does not necessarily make a stronger p r i s m , as ind icated by c o m p a r i n g the fa i lure loads of the pr isms w i t h type S mor ta r w i th those w i t h type M mor tar . T h e lower fai lure, loads of the pr isms w i t h type M mor ta r are believed to be due to the poorer adhesion of that type of mortar , wh ich appeared dur ing the exper iments. T h e effect of the 63 LATERALSTRAN (1/1000 N/N) LATERAL STRAN (1/1000 N/N) F I G . 4 .2 M e a s u r e d L a t e r a l S t r a i n s i n W e b s o f M i d d l e C o u r s e s o f P l a i n M a s o n r y P r i s m u n d e r U n i a x i a l C o m p r e s s i o n 1 III I III - II -Ul -1 II 111 - iv -II - iv -1 V 1 V M3-1 S1-3 N4-3 - II It II II -1 III - Ill 1 III - iv -1 - iv -1 V N2-1 N2-2 N2-3 F I G . 4 . 3 D e t e c t e d O r d e r s o f M a c r o s c o p i c S p l i t t i n g , i n T e r m s o f 4 S e c t i o n s a l o n g P r i s m s . 64 S P E C I M E N 1 2 3 4 A V G C O V M 3 ( M - M O R T A R ) 187.0 147.0 123.0 140.0 149.0 15.7% S I ( S - M O R T A R ) 204.0 194.0 178.0 168.0 186.0 7 .5% N 2 ( N - M O R T A R ) 125.0 140.5 143.0 164.0 143.0 9 .7% N 4 ( t 0 = 3 / 4 in ) 120.0 105.0 133.0 119.0 9 .6% P 5 ( t o = 0 in ) 103.0 123.0 112.0 103.0 110.0 7 .5% T a b l e 4.1 F a i l u r e Loads of P l a i n P r i s m s w i t h F u l l Bedd ing (kips) adhesion on masonry strength w i l l be discussed later. 5) V e r t i c a l s t ra in measurements ind icate that the mor ta r j o in ts are m u c h softer than the concrete uni ts . T h e rat io of the i n i t i a l modu lus of concrete to that of three mor ta r types is about 6-8 to 1. (See F igs . 2.3, 2.4 a n d 2.5 in Chapter II.) T o study j o i n t effects, some of the pr isms were bu i l t w i t h zero j o i n t thickness, and one group w i t h glass p late . T h e glass was chosen because of i ts h igh modu lus of e last ic i ty and re lat ive ly low rupture strength (£=8xl0 6 psi, frup = 5000 ps i , obta ined by exper iment ) . It was expected that the glass p late f i l led jo in ts wou ld m i n i m i z e the Poisson's effect and at the same t i m e prov ide l i t t l e la tera l conf inement. However, these specimens wi thout mor ta r bedded jo in ts fa i led at re lat ive ly low loads (see Tab les 4.1, and 8.1 for grouted pr isms) ; the experiments were not conclusive. T h e low fai lure loads are believed to have been caused by stress concentrat ions in the v i c i n i t y of the jo in ts w i thout a mor ta r cushion. T h e ind icat ions are that ver t ica l cracks occurred d u r i n g the load ing stage of these pr isms; and for the specimens w i t h glass plates, the crack ing noise of the glass was also heard. 4.2.2 Theoret ica l A n a l y s i s W e proceed now to revise H i l s d o r f s model i n the l ight of a stress analysis . A l t h o u g h numerous stress analyses (ment ioned above) i nc lud ing some based on 3 d imens iona l mode l ing ( H a m i d a n d Chukwunenye , 1986) of this p rob lem have been conducted, they were a l l based on 65 numerical approaches. To gain some direct insight into the problem, we derive some analytical solutions based on plane elasticity for the configuration of a mortar joint being sandwiched between concrete block units. Consider the case shown in Fig. 4.4(a), which depicts a view of either web face or face-shell face (joint length a may be either equal to web width or face-shell width). The mortar joint, being much softer than the concrete block, as indicated by experimental observation, may be considered as squeezed by two rigid platens, and by the principle of superposition, the loading situation may be decomposed as shown in Fig. 4.4(b) and (c). It is case (c) which will cause interface shear between the mortar and the masonry unit and hence cause tensile stress in the unit. By symmetry, we only need to consider half of the joint, as shown in Fig. 4 . 5 Since the joint is bounded by two rigid platens, the lateral strains due to the traction must be localized at its ends. Thus the vertical displacements v in the middle region of the joint, which are mainly caused by to Poisson's effect, will be small (recall we are considering case c) alone here). Further, since a is much larger than t0, the variation of the vertical displacements with x must also be small. Therefore we assume Assuming that the mortar joint in the plane of the cross-web or the face-shell is in a state of plane stress, Lame's equations (solving the problem in terms of displacements, X u 1979) reduce to throughout the region u — u(x,y) + 0 < x < a/2 0 < y < to 4.1 with boundary conditions 0 a -V A V -X--x-q=y<T — V 0" -V (a) (b) (c) F I G . 4.4 A M o r t a r J o i n t S a n d w i c h e d b y B l o c k U n i t s : a ) u n d e r A x i a l C o m p r e s s i o n ; b ) u n d e r B i a x i a l C o m p r e s s i o n c ) u n d e r L a t e r a l T r a c t i o n . T i x,u q = va •a/2-y-.v F I G 4.5 A M o r t a r J o i n t u n d e r L a t e r a l T r a c t i o n u(0,y) = 0 u(x,0) = 0 u(x,l0) = 0 ^ - g ux (a/2,y) = q l-v 0 < 2/ < <o (by symmet ry ) 0 < x < a/2 0 < x < a/2 0 < y < U 67 4.2 4.3 4.4 4.5 where E is Y o u n g ' s modu lus a n d v is Poisson's rat io . A series so lut ion for this boundary value p rob lem can be found (see appendix ) 4 ( l - t / 2 ) g t 0 £2, sinh[(2n-l)K7T2:/ i 0 ] s in[ (2n- l )7ry /< 0 K2KE ^ (2n-l)2cosh[(2n-l)«:7ra/2<<,] where K= ^(1-U)/2 F r o m w h i c h we deduce the shear stress a long the j o in t 4.6 xy — 2$k) w " ( * ' 0 ) _ 4nq sr^ s inh [(2 n-1) K TT X/ t0] ^ ( 2 n - l ) c o s h [ ( 2 n - l ) K 7 r a / 2 / 0 ] 4.7 w i t h the resultant force a/2 S = T xy dx — *2 h ( 2 -D 2 L c o s h [ ( 2 n - l ) « ; 7 r a / 2 f 0 ] 4.8 Since a ^ > t0, cosh[(2n- l ) « 7 r a /2 i 0 ] 3> 1, the second te rm in the bracket of E q . 4.8, wh ich represents the van ish ing ly sma l l force transferred by the midd le of the jo in t , can be neglected. Fu r ther , by not ing that q = va and 68 oo ( 2n - l ) 1 2 ' 8 2 n = l we o b t a i n S = veto 2 4.9 T h e point of act ion of this resultant is a/2 J" X Txy dx 0 2 ( 2 n - l ) 2 a K 7 r ( 2 n - 1 ) 3 to a _ to_ 2 K 7 r 4.10 w h i c h lies near the end of the j o in t . B y inspect ing Eqs . 4.7 and 4.10, i t is concluded that the interface shear must be h igh ly concentrated near two ends of the jo in t . It is also clear i n v iew of E q . 4.9, that this shear is d i rect ly p ropor t iona l to the appl ied compressive stress, the thickness and the Poisson 's ra t io of the mor ta r j o in t . It is these shear forces act ing l ike la tera l po int loads wh ich introduce the tensile stresses in the web and face-shel l . E q . 4.7 is compared w i t h a numer ica l so lut ion using the boundary element method . T h e computer p rog ram ( T W O F S ) is given by C r o u c h et a l (1983), and 67 elements were used for th is p rob lem. F o r i / = 0.3 and t 0 / o = 3 / 6 4 , the solut ions are p lo t ted i n F i g . 4.6, together w i t h a dep ict ion of how these shears act on a web. T h e ana ly t i ca l so lut ion is i n good agreement w i t h the numer ica l one, wh ich supports the assumpt ion that the ver t ica l d isplacement can be neglected. W e proceed to perform stress analysis for a web or a face-shell under the act ion of these shear forces. A s shown in F i g . 4.7, this is a plane p rob lem in a rectangular d o m a i n w i t h stress specified boundary condi t ions . T h e shear d is t r ibut ions on the boundaries are given by E q . 4.7. 0.4 0 .35 -x / 0.5 a FIG 4.6 Lateral Interface Shear Distribution between Mortar Joint and Block Units. y 1 <Pxx = CJ d> = - T Y xy xy a h =0 V 4 ^ =0 b <txy=0 <t)xx = o (t>xy= T x y given by Eq. 4.7 FIG 4.7 Depiction of Boundary Conditions of a Web (or Face-Shell) under Action of Interface Shears. 70 T h e c o m m o n approach for this k i n d of prob lem is to f ind a stress funct ion ( A i r y stress funct ion) . F o r this par t icu la r case we can wr i te the stress funct ion in a series fo rm as * = £ A m ( s i n h - g m t a n h a m c o s h - y / 2 > ) s i n ^ m=l V / frmr(x-a/2) . mir(x-a/2) mir(x-a/2)\ . miry . + Bm I b S l n n b — / ? m t a n h / 3 m c o s h v f e ' ' s i n — 4 . 1 1 where a m = z ^ » / ? m = = ! ^ a < a n d i r a , - S m are determined by hav ing E q . 4.11 satisfy the boundary condi t ions depicted i n F i g . 4.7 (for detai led der ivat ion see appendix ) . B y inspect ion, the m a x i m u m tensile stress w i l l occur at the top and b o t t o m boundaries of the d o m a i n . So f ina l l y , the tensile stress d i s t r ibu t ion we are interested in is <rx — ®yy = 2 £ Am (Vf) c o s h a m s i n ^ 4.12 m = l F o r a square d o m a i n , as i n the geometry of the web, the series so lut ion is p lot ted in F i g . 4.8, together w i t h a numer ica l so lut ion . A numer ica l so lut ion for the more real ist ic case of (yj/Ej)/(vu/Exl) = §, where subscripts j and u denote mor ta r j o i n t and block un i t respectively, is also inc luded i n the graph . T h e changes of this d i s t r ibu t ion due to var iat ions in Poisson 's rat io and the thickness of the j o in t , the aspect rat io of the rectangular d o m a i n (corresponding to a web and a face-shell) are p lot ted in F i g . 4.9, F i g . 4.10 and F i g . 4.11 respectively. T h e above stress analysis c lear ly indicates that the tensile stress reaches its m a x i m u m at a locat ion close to the two ends of the top or b o t t o m edge of the doma in and a m i n i m u m in the m i d d l e of the edge; changing the parameters i n the stress analysis does not alter the basic features of th is stress d i s t r i bu t ion . T h i s is not surpr is ing in v iew of the point load l ike shear o.s i i i i i i i 1 r~ 0 0.2 0.4 0.6 0.8 x/0.5a FIG 4.8 Lateral Tensile Stress along Top of Block Introduced by the lateral Shears x/0.5a FIG 4.9 Lateral Tensile Stress along Top of Block, with Variation in Poisson's Ratio of Joint 0.6 ° 0.2 0.4 0.6 0.8 x / 0 . 5 a F I G 4 .10 L a t e r a l T e n s i l e S t r e s s a l o n g T o p o f B l o c k , w i t h V a r i a t i o n i n J o i n t T h i c h n e s s 0.5 -i . x / 0 . 5 a F I G 4 .11 L a t e r a l T e n s i l e S t r e s s a l o n g T o p o f B l o c k , w i t h V a r i a t i o n i n D o m a i n A s p e c t R a t i o 73 LATERAL STRAW ( 1 / 1000 N/N) LATERAL STRAN (1 /1000 N/N) F I G 4.12 La te ra l Strains Measured a long webs a n d Face-shel ls of P l a i n P r i sms under U n i a x i a l Compress ion d is t r ibut ion specified by E q . 4.7. A n d it is consistent w i t h the exper imenta l observation that the strains paral le l to the jo in t were higher at the ends than in the midd le of the jo in t , as indicated by gauges in these locations. F i g . 4.12 gives the typ ica l results of this measurement. No appreciable st ra in was measured at locat ion #1 or at locat ion #2 , wh ich are in the midd le of a face-shell and does not cover two ends of the jo int . T h i s is in contrast to those measured in locations # 3 and #4, which cross the whole length of the web. In other words, the tensile stra in is h ighly concentrated near two ends of the jo in t where tensile stress reaches m a x i m u m . 74 4.2.3 C o n c l u s i o n on H i l sdorPs M o d e l Based on above study, the conclusion is obvious, that sp l i t t i ng in masonry can not be  s i m p l y a t t r i bu ted to the lateral expansion of the mortar . F i r s t , if lateral expansion of ' the mor ta r were the m a i n cause of the sp l i t t i ng , i t wou ld occur somewhere near two edges of a web or a face-shel l where the tensile stress reaches its m a x i m u m , (see F i g . 4.8 — F i g . 4.11) If the thickness changes of the face-shell and the web i n the f i l lets near the corner of a un i t are taken in to account , one m a y conclude that sp l i t t i ng wou ld occur somewhere near the web — face-shell j o in t , a conclusion wh ich is not supported by exper imenta l observat ion. Second, the interface shear, wh ich is responsible for the tensile stresses in the web and the face-shel l , is a monoton ica l l y increasing funct ion of the mor ta r j o i n t length a, as clearly ind icated by E q . 4.8. T h u s the shear forces a long the face-shel l are not less than those a long the web. If mor ta r expansion were the m a i n cause for the sp l i t t i ng of a masonry p r i s m , the s p l i t t i n g wou ld be more l ike ly to occur or at least have an equal p robab i l i t y of occurr ing , in the face-shel l , a conclus ion w h i c h contrad icts the exper imenta l observations. T h i r d , the exper imenta l m o n i t o r i n g of crack i n i t i a t i o n , as we have shown earlier, indicates that the latera l expansion of the mor ta r being the cause of the sp l i t t i ng mechan ism is not acceptable. These points alone are sufficient to rule out the correctness of H i l s d o r f s model , since even the necessary condit ions for fa i lure can not be just i f ied by his mode l . Moreover , r igorously , speaking, the under ly ing concepts of H i l s d o r f s mode l m a y be mis lead ing . A s ind icated at the beginning, i t is not suff icient to focus on a local tensile event in the case of compression. E v e n i f the vert ica l sp l i t t i ng were in i t ia ted by j o i n t expansion, for this to lead to direct catast rophic fai lure of the masonry needs further j us t i f i ca t ion . F r o m a fracture mechanics po int of v iew, the energy required to open this crack wou ld come f r o m the s t ra in energy released in the mor ta r j o in t , as a result of pa r t i a l re laxat ion of the la tera l conf in ing stress. It can be shown that the amount of this energy is l i m i t e d , so that the crack wou ld J stabi l ize . E v e n i f we assume this crack cou ld run through a masonry block, the latter wou ld s t i l l • 7 5 not lose vertical load transfer ability; it would have failed only in the sense of the serviceability condition. Certainly, the above stress analysis may be subject to some limitations because it is based on two-dimensional elasticity, which does not take account of the material nonlinearity or of the complete specific geometry of the concrete block. Nevertheless, this does not detract from the useful conclusions deduced from the above study. Nor would nonlinear behaviour in the joint itself change the essential feature of the stress distribution; it would only cause limited shifting in the locations where maximum tensile stress occurs. 4.3 Some Comments on Splitting Failure and Mode Transition Phenomena It is clear that the splitting failure of masonry cannot be attributed to the lateral expansion of the mortar joint alone; rather, it is inherent in the failure of the material as we explained in Chapter III. Although it is difficult at this stage to explain fully the splitting failure for the specific geometry of masonry, or the transition to the shear mode with an increase of loading eccentricity, certain hypotheses may be made in the light of the concepts illustrated in Chapter III. a) The main splitting probably develops in the web rather than the face-shell because this leads to the weakest structure. b) Under eccentric loading, vertical crack surfaces tend to be forced into contact by the transfer of shear from the loaded to the unloaded side. This contacting may in turn increase the friction across the crack, which may prevent splitting failure from occurring. c) Because two different failure mechanisms are involved, the apparent strengths will be different, and a one parameter failure criterion will not be satisfactory. 76 4.4 J o i n t Effect on A x i a l C a p a c i t y It has been demonstrated that a p la in mor ta r j o i n t is not the governing factor for the fa i lure pat tern of concrete masonry . H i l s d o r f s mode l is not appropr iate for assessing the effect of the j o in ts on masonry strength. A v a i l a b l e exper imenta l results on the j o i n t effect, i nc lud ing the tests done by the author , appear to be scattered. A possible way to assess this effect w o u l d be to compare tests on pr isms w i t h mor ta r j o i n ts a n d w i t h dry j o in ts not a very p ract ica l approach. T h e masonry un i t strength is not a good reference, since under standard test ing cond i t ions , i t w i l l exh ib i t a con ica l fa i lure mechan ism as a result of the end f r ic t ion , w i t h a substant ia l l y higher apparent strength. Because of these di f f icult ies, i n most exper imenta l work, the j o i n t effect has been examined by va ry ing the jo in t condit ions. Some exper imenta l observations may be wor th reviewing, a) U s u a l l y the compressive strength of un i t is higher than that of p r i sm, wh ich is i n tu rn higher than that of mor ta r . However, a l though the mor ta r strength is lower than the p r i s m strength (calculated on the mor ta red area), j o i n t fai lure has never been observed. It should be noted that , when t a l k i n g about mor ta r strength, we i m p l i c i t l y assume the unconf ined compressive strength. T h e strength obta ined by the s tandard cube test is actua l ly par t l y conf ined since its height to w i d t h ra t io is s m a l l . Expe r iments by Hatz in iko las et a l (1978) have shown that the unconf ined strength can be as low as 6 3 % of the cube strength. T h e mor ta r in the j o i n t cou ld have even lower strength due to poorer cur ing condit ions. T h i s observat ion is also revealed by the author 's tests. T h e average uni t strength fu is 3250 ps i . F o r most pr isms N type mor ta r was used, wh ich has an average cube strength of 1570 ps i . T h e average fa i lure load of masonry pr isms w i t h this mor ta r is 143 k ips , corresponding to an fm of about 2320 ps i . (cf. T a b l e 4.1) A l t h o u g h the exact corre lat ion between the strength of the m o r t a r cube and that of the mor ta r in the j o i n t is unknown, and type M and S mortars appear to have very h igh cube strengths, i t seems reasonable, to accept that fu > fm > fju, 77 where / - u denotes the unconf ined mor ta r strength. F o r a l l the pr isms tested, no j o i n t fa i lure was observed. T h i s is also evidenced by the deformat ion measured across the j o i n t (see F i g . 2.5 in C h a p t e r II). b) B o t h mor ta r type and j o in t thickness have an influence on the masonry strength, a l though there is s t i l l an uncerta inty about the degree and nature of this influence. T h i s is reflected in that the tests done by Drysdale and H a m i d (1979) have shown the influence to be re lat ively minor , whi le in the author 's tests the influence is s igni f icant (see T a b l e 4.1 and F i g . 4.16); and a l though the ava i lab le test d a t a tend to indicate that stronger mor ta r makes stronger masonry , bo th exper iments have ind icated that this is not a lways true. c) T h i s inf luence becomes re lat ively minor w i t h increase of load ing eccentr ic i ty , (see T a b l e 6.1 in Chapte r V I ) d) Reinforcement by meta l plates enhances both the capaci ty and d u c t i l i t y of masonry (Pr iest ley and E lde r 1982), whi le reinforcement by steel bars i n the j o i n t reduces the strength (Hatz in iko las 1978). It can be conjectured that the mor ta r j o i n t affects the strength of masonry bas ica l ly in that the j o i n t introduces d iscont inui t ies in the mater ia l properties, such as strength and stiffness. These d iscont inu i t ies w i l l compl icate the stress d i s t r ibu t ion in the v i c i n i t y of the j o i n t and thus affect the ver t ica l load transfer ab i l i t y . It m a y be reasonable to assume that as long as the j o i n t condi t ions do not provide la tera l conf inement to prevent sp l i t t i ng fai lure, as i n the case of plate reinforcement (observation d) , the j o i n t w i l l general ly have a negative effect on the masonry strength in the presence of u n i a x i a l compression. T h i s is because the jo in t w i l l generally alter the otherwise un i fo rm compressive stress i n its v ic in i t y , and thus the force is effectively transferred by a smal ler area. T h i s can be i l lust rated by fo l lowing analysis . 78 4.5 Stress i n J o i n t V i c i n i t y A s shown by the stress analysis in section 4.2.2, when masonry is under u n i a x i a l compression, the less st i f f mor ta r j o i n t is subject to ver t ica l compressive force as wel l as latera l interface shear. A l t h o u g h the p rob lem was solved i n a plane co inc id ing w i t h webs or face-shells, the pr inc ip le can be extended to the perpendicular plane representing the cross-section of webs or face-shells. T h u s the mor ta r is ac tua l l y conf ined b i la tera l ly ; and because of that , the apparent strength (the conf ined strength) is increased. T h i s explains why j o in t fa i lure is not observed i n tests a l though fm > /JU. A n ind ica t ion of this conf inement is found in the ver t ica l de fo rmat ion curves of mor ta r j o i n ts recorded i n the tests, wh ich reveal that the jo in ts became stiffer w i t h increase of load (cf. F i g . 2.5). A s a consequence of this conf inement, the otherwise un i fo rm compressive stress d i s t r i bu t ion in the v i c i n i t y of the j o i n t is changed. F i g . 4.13 depicts a cross-sectional view of a mor tar j o i n t and a free body d i a g r a m of the j o i n t . It is obvious that latera l conf in ing stress results f r o m the interface shear and that i t reaches a m a x i m u m i n the midd le of the jo in t . A s an es t imat ion of the conf in ing stress d i s t r i bu t ion , we use the s impl i f ied approach as presented in section 4.2.2. T h e p rob lem approx imates plane st ra in condi t ions , since the j o i n t is a lmost fu l ly confined in the d i rect ion a long its length. Reca l l i ng the fo rm of so lut ion for the latera l d isplacement u as g iven in E q . 4.6, we m a y wr i te the conf in ing stress as (referring to the cross-sectional plane shown in F i g . 4.13): 4.13 T a k i n g the average of this stress over the jo in t thickness leads to 79 F I G 4 .14 C o m p r e s s i v e S t r e s s , L a t e r a l C o n f i n i n g S t r e s s a n d C o n f i n e d S t r e n g t h i n M o r t a r J o i n t 80 to to = ^ f l - J - V c o s h [ ( 2 n - l ) « 7 r r / < 0 ] \ V n- 2 ^ ( 2 n - l ) 2 c o s h [ ( 2 n - l ) K 7 r a 0 / 2 V | / where t0 and a0 are the thickness and the w i d t h (the transverse d imension of the block face-shell or web) of a j o i n t . T o correspond to plane st ra in condit ions, q and K become 4.15 l - i / l ~ 2 v 4.16 > 2 ( 1 - 1 / ) F o r z / a 0 « 1 /4 , t y p i c a l of the geometry of concrete masonry condi t ions , a n d v = 0.3, this conf in ing stress d i s t r i bu t ion is p lot ted in F i g . 4.14. It is clear that the j o in t is not un i fo rmly conf ined. Under this non -un i fo rm conf inement, the j o in t w i l l develop a v a r y i n g conf ined strength. Since the j o in t is more conf ined in the d i rect ion a long its length, the increase in m o r t a r compressive strength w i l l m a i n l y depend on the conf in ing stress in the j o in t w id th d i rect ion (the x d i rect ion i n F i g . ( 4.13) . T h u s , we m a y use the k n o w n emp i r i ca l re lat ion (Park and Pau ley 1975) fjc = / j o + 4.1<T, 4.17 w i t h cr, = C |(x) here. T h e conf ined compressive strength / - c of the jo in t , based on the conf in ing stress g iven by E q . 4.14, is also p lot ted in F i g . 4.14. T h i s m a y underest imate the strength somewhat i n v iew of the fu l l conf inement a long the length of the j o in t . W h e n the appl ied compressive stress is s m a l l compared to / . u , the unconf ined compressive strength of mor tar , a good a p p r o x i m a t i o n for the compressive stress d i s t r i bu t ion in 81 the j o i n t w i l l be g iven by the elastic case : cr • ~ cr — = -3— l - u 2 dx = ( T ( l _ ^ V cosh[(2n-l)K7rx/to] \ 4 V TV2 ^ ( 2 n - l ) 2 c o s h [ ( 2 n - l ) K 7 r a 0 / 2 i o ] / wh ich departs s l ight ly f r o m un i fo rm d is t r ibu t ion , as p lot ted in F i g . 4.14. W h e n cr exceeds / - u , the end part of the j o i n t w i l l " y i e l d " , because the conf ined strength of the end part is less than cr. T h e stress d is t r ibut ion i n the j o i n t is then d r a m a t i c a l l y comp l ica ted . T h e latera l conf in ing stress given by E q . 4.14 is no longer v a l i d since the end parts of the j o i n t have developed substant ia l nonl inear i ty . A precise stress analysis for the j o i n t is di f f icu l t , but we shal l give an approx imate approach to this p rob lem. Because the inner part of the j o i n t is more conf ined, i t develops higher strength and therefore transfers more stress. T h u s we m a y d iv ide the jo in t in to two parts w i t h the d i v i d i n g point x0, w i t h i n w h i c h the mate r ia l remains elastic i n the sense that i t does not fa i l or develop substant ia l non l inear i ty , as shown in F i g . 4.13. W e assume, as is suggested by F i g . 4.14, that the compressive stress is approx imate ly un i fo rmly d is t r ibuted w i t h i n this range. Since the end part is subjected to compressive stress as wel l as h ighly concentrated shear force and latera l conf in ing force as ind icated in the preceding stress analys is , i t w i l l f a i l under the c o m b i n a t i o n of these forces. T h e k n o w n empi r ica l fa i lure curve of concrete under shear and compression, as shown in F i g . 4.15 (Park and pauley, 1975), m a y serve as a good a p p r o x i m a t i o n on ly for the very end of the j o in t , where the latera l conf in ing stress is negl igible. In the presence of large latera l conf in ing stress, we m a y mod i fy the fa i lure curve by assuming that i t is character ized by the confined strength instead of the u n i a x i a l compressive F o r e q u i l i b r i u m , the peak value wou ld be s l ight ly higher than the average stress cr given by E q . 4.18. 82 0.3 £ 0.2 O.I 0 a - O I - * — i \ 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FIG. 4.15 Failure Curve of Concrete under Shear and Compression: Solid Line, after Park and Pauley (1975); Dashed Line, Fitted by Eq. 4.19 strength. This basically enlarges the failure curve in an absolute stress space. The left and right ends of this curve can be fitted by segments of two ellipses. For ease of analysis, we further simplify the situation by considering only the average failing compressive and shear stresses in the joint. It may be useful to list here the notations that will be used in the following paragraphs: (see Fig. 4.13) a0 = width of the joint x0 = half width of the middle part of the joint xe = width of the end part of the joint to = thickness of the joint fju = unconfined strength of the mortar joint •fje = confined strength of the mortar joint in the centre part fjt = confined strength of the mortar joint in the end part cr = average compressive stress in entire joint a, = lateral confining stress in joint a je = average compressive failing stress in the end part Tjt = average shear failing stress in the end part Thus the failure criterion for the end part is .22/,-for crje/fje < 0.6. And for crje/fje > 0.6 / < r i e / J j . g - u . D y / rje y _ I 0.4 ) + V 0.22/,, ) ~ 4.196 0-22/ i £ fje can be written /;,= t t L t ^ . 4.20 We assume a loading path by noting the relation given by Eq. 4.9 and 5 ~ xeTje, from which it follows that, at failure Vertical equilibrium requires <xa0 2 = xofjc{x?) + Xe(Tje 4.22 84 where fjc is defined by fjc = fju + 4.1 cr, 4.23 and <rt can be found by lateral equilibrium of the end part a, t0= 2rjexe 4.24 Eq. 4.19 to Eq. 4.24 together with the relation that x0+ xe= a0/2 can be used to find the 7 unknowns; namely, xe, x0, /,c, a,, fje, aje and r j e . As an example, we use this approach to examine the prisms with type N mortar. Recall that the cube strength of this type mortar is 1570 psi; we set it equal to fju. This may underestimate the distortion of the uniform stress, since the actual value for / - u is even lower. At failure of the N type masonry, a= fm = 2320 psi, then fju/<r = 0.677, leading to x 0 / (a o /2) = 0.84, fjc/<r =1.15, Cje/v = 0.27. i.e., about 16% of the joint (outer part) failed with an average failing stress 27% of the average stress a. As a consequence, the outer part of the joint sheds forces to the inner part, leading to an increase in stress to 1.15 times <r in that part. 4.6 Capacity Estimation The above' study indicates that the mortar joints can alter the otherwise uniform compressive stress distribution considerably when masonry is under uniaxial compression. According to the approximation, at failure of the masonry with N type mortar tested by the author, about 97% of the compressive force is transferred through a strip with 84% of the width of the web or face-shell. This means that the compressive force is only transferred by part of the joints and the masonry units are actually not fully loaded, which will certainly have a negative effect on the strength of the masonry. 85 However , if. one wishes to generalize the j o i n t effect on masonry strength by corre lat ing the mor ta r cube strength and the un i t strength w i t h the p r i sm strength, as is often desired in pract ice, some uncertaint ies have to be recognized: 1) T h e corre lat ion between mor ta r cube strength and the strength of the mor ta r placed in the j o i n t . 2) T h e corre lat ion between un i t strength and the strength of " a p r i s m w i t h dry j o i n t s " , a desired reference parameter . 3) Uncerta int ies i n fai lure c r i te r ia of concrete. 4) M a t e r i a l properties such as those governing deformat ion and adhesion, wh ich are often assumed but not measured. T h e y are by no means un impor tan t to masonry strength. Based on above analysis and arguments, we present here a semi -emp i r i ca l approach. Since no better fa i lure cr i ter ion is ava i lab le , we assume that the masonry un i t w i l l f a i l when the average compressive stress in the m idd le part of the mor ta r jo in t , wh ich w i l l be higher t h a n the average stress in the masonry , reaches some c r i t i ca l value; and this c r i t i ca l value m a y be l inear ly correlated to the masonry uni t strength. T h u s we m a y write the fa i lure cond i t ion fjc(x0) = kJ* 0 < *x < 1 4.25 Fur ther , we assume 1 that the unconf ined strength of the mor ta r placed in j o in ts is p ropor t iona l to the cube mor ta r strength fju = hfj 0 < k2 < 1 4.26 where f- denotes the cube mor ta r strength, and kl and k2 are some assumed corre lat ion factors. A t fa i lure, equations f r o m 4.19 to 4.24 are assumed to be satisf ied. T h u s the masonry strength, i.e. the average fai lure stress in masonry, is ac tua l l y expressed in E q . 4.22, f r o m wh ich I 86 we can wr i te where < 7 j e a n d r j e can be exp l ic i t l y expressed in terms of fj and fu i n v iew of Eqs . 4.19 — 4.21, 4.23 a n d 4.24. F o r kx = 0.95, k2 = 0.75, i/a0 - 0.25 and Poisson's rat io v = 0.3, E q . 4.27 is p lo t ted in F i g . 4.16 and compared w i t h the exper imenta l data . T h e value of the corre lat ion factor k2 is very close to the conversion factor between concrete cy l inder strength and cube strength recommended by L ' H e r m i t e (Nevi l le 1965). F o r the cube strength ranging f r o m 2000 psi to 3000 ps i , th is factor is between 0.73 and 0.76. T h e model curve is essential ly ident ica l to a l inear regression curve of the data , i.e. i f = 0.68 + 0.19 4.28 /» fu T h e four points on the r ight have been excluded f r o m this analysis ; they were type M mor tar , and are bel ieved to represent a different phenomenon — fai lure of the adhesion between block mor tar . T h e mode l also gives a reasonable corre lat ion w i t h the l i m i t e d d a t a on masonry capaci ty when the m o r t a r j o i n t is doubled ( t = 3 / 4 inch) , as shown in F i g . 4.17. T h e mode l m a y be used to est imate the masonry strength. However, as ind icated before, some uncertaint ies heed further invest igat ion . One of them, is of course, the corre lat ion between the cube mor ta r strength and the strength of mor ta r placed in the j o in t , since the cur ing condi t ions are so different. T h e other is the effect of the jo in t adhesion. W e m a y conclude f r o m the above analysis that since the load transfer capab i l i t y of mor ta r j o i n ts depends largely on the existence of the latera l conf in ing stress, in t roduced bas ica l ly by the interface shear between the jo in t and block uni t , that the adhesion between j o in t and uni t in 2 1.2 1.1 H 1 0.9 0.8 0.7 -\ 0.6 0.5 -0.4 -0.3 -0.2 0.1 -0 MODEL Lt£AR REGRESSION AUTHOR DRYSDALE&HAME HATZNKOLAS 1~ i r 0.2 — I — 0.8 0.4 0.6 .  1 MORTAR CUBE STRENGTH / UNIT STRENGTH 1.2 1.4 F I G . 4.16 P r i s m Strength versus M o r t a r C u b e Strength i 0.9 .0.8 -0.7 -0.8 -0.5 0.4 0.3 -0.2 -0.1 -0 (to-3/4 in) 0.2 MODEL • AUTHOR • DRYSOALE&HAMD "i r 0.4 0.6 0.8 1 MORTAR CUBE STRENGTH / IMT STRENGTH — I — 1.2 1.4 F I G . 4.17 P r i s m Strength versus M o r t a r Cube Strength , w i t h Jo in t Th ickness Doubled 88 should be important for masonry strength. These uncertainties may have contributed to the experimental observation that stronger mortar does not necessarily make stronger masonry. In the experiments conducted by the author, the stronger mortar, here type M, did not j make a stronger prism, probably because it contains less lime than does type S mortar. This not only causes poorer adhesion to the blocks, (a phenomenon noticed by the author in his experiments), but also may lead to poorer water retaining ability. In other words, type M prisms not only had poorer adhesion between joint and unit, but also may actually have a lower joint strength due to poorer curing conditions. Therefore, it is recommended that, in practice, attention should be paid to the overall quality of the mortar. Proper mix design should be specified and the cohesive requirement should be enforced. 4.7 Summary In this chapter, the failure and capacity of plain concrete masonry under concentric compression has been studied. Hilsdorfs model of splitting failure has been reviewed in the light of both experimental and analytical work. It is concluded that the splitting failure mode cannot simply be attributed to the lower stiffness of the mortar joints; it is a manifestation of compression failure as discussed in Chapter III. The less stiff mortar joint tends to be confined laterally, developing higher compressive strength in the inner part. On the one hand, it prevents joint failure. On the other hand, it tends to alter the uniform compressive stress in the vicinity of the joint, i.e. more compressive force tends to be transferred by the inner part of the joint. A failure criterion based on failure of masonry unit under this intensified compressive stress was proposed, which gives reasonable capacity estimation. 89 C H A P T E R V P L A I N M A S O N R Y W I T H F A C E - S H E L L B E D D I N G 5.1 In t roduct ion In N o r t h A m e r i c a concrete masonry is often mor ta red only on the face-shells. E v e n when a mason a t tempts to app ly mor ta r to the cross-webs as wel l , he m a y not be able to ensure ver t ica l a l ignment so that the webs can t ransmi t force effectively. T h e mechan ica l propert ies of face-shel l bedded masonry , therefore, have been studied extensively. T h e s p l i t t i n g fa i lure of face-shel l bedded masonry is re lat ively wel l understood. Shr ive (1982) concluded that tensile stress is developed at the centre of the webs, by a mechan ism somewhat analogous to deep beam bending, i.e. the top and b o t t o m halves of the web are taken as deep beams, bending up and down respectively (see F i g . 5.1), thus causing the sp l i t t i ng fa i lure i n face-shel l bedded masonry . T h e author is in f u l l agreement w i t h the reasoning in Shr ive 's paper. T h e present s tudy of face-shel l bedded masonry was intended to con f i rm his mode l exper imenta l ly , to s tudy the t rans i t ion to a fa i lure mechan ism for eccentr ical ly loaded specimens, to explore the re lat ionship to fu l l y bedded masonry , and to develop some related quant i ta t i ve results. 5.2 E x p e r i m e n t a l W o r k Sixteen face-shel l bedded pr isms were bu i l t and twelve of them were tested under u n i a x i a l compression i nc lud ing two w i t h f u l l capp ing . T a b l e 5.1 summar izes the fa i lure loads of these specimens. Under u n i a x i a l compression, sp l i t t i ng of the webs was again revealed by the tests. B y observing the wire breaking order as previously described, cracks were found to in i t ia te consistent ly f r o m the top or b o t t o m of the webs i n m idd le course (see F i g . 5.2). T h i s supports Shr ive 's mode l . B o t h the fa i lure process recorded on video and the latera l deformat ion 90 tension compression tlttlHMIHI I H H H H I I I I compression tension F I G . 5.1 Depict ion of Deep B e a m Mechan ism 11 HI II III - 1 III - 1 l l III - IV -II - IV -1 V 1 V IV M27-2 S16-1 S16-2 - IV - - II " III -1 --1 - Ill -- II - - IV IV deep beam mechanism N15-1 N15-3 N15-4 F I G . 5.2 Detected Orders of Macroscopic Sp l i t t i ng , i n T e r m s of 4 Sections along Pr isms. (Face-Shel l Bedded Pr isms) 91 S P E C I M E N 1 2 3 4 A V G c o v M 2 7 ( M - M O R T A R ) 118.0 99.0 86.0 75.0 94.5 19.9% S16 ( S - M O R T A R ) 119.0 127.0 140.0 109.0 123.8 9 .2% N15 ( N - M O R T A R ) 100.0 115.0 107.5 7 .0% N15 ( N - M O R T A R ) 53.0* 48.0* 50.5 5 .0% T a b l e 5.1 Fa i l u re Loads of P l a i n P r i s m s w i t h Face -She l l Bedd ing (kips) * Tested w i t h f u l l capp ing measurements ind icated that sp l i t t i ng occurs at or immed ia te l y before f i na l fa i lure . F igs . 5.3, 5.4 give the deformat ion curves. A deep beam mechan ism is suggested by the substant ia l difference i n the deformat ion measurement at locations # 3 #4; sp l i t t i ng is c lear ly evidenced by the j u m p s in these curves. T h e final fa i lure is character ized by peeling off ( fu l ly or par t ly ) of the face-shells, as shown in F i g . 2.13 i n Chapter II. However , for most of the specimens w i t h face-shell capp ing cracks were found to in i t ia te i n the web somewhere near two ends of the mor ta r jo in ts , and tended to wander afterwards, as t yp ica l l y i l l us t ra ted in F i g . 2.13. T h i s appears somewhat different f r o m what one wou ld expect by the deep beam bending model , which suggests that sp l i t t i ng wou ld occur in the centre of the web. S p l i t t i n g i n the centre of webs was found i n the specimens tested w i t h f u l l capp ing (see F i g . 2.14), usual ly occurr ing in the top course. These specimens fa i led at very low loads (about 5 0 % of that of the face-shel l capp ing , see T a b l e 5.1), immed ia te l y after web sp l i t t i ng ; the two halves of b locks col lapsed by h ing ing about the inside toes of the mor ta r jo in ts . T h e h ing ing mechan ism is i m p l i e d by the ver t ica l d isplacement measured across the outside of the j o i n t of specimen N15 -3 , wh ich contracted first because of compression then tended to open due to the j o i n t ro ta t ion . FIG. 5.3 Measured Deformations at Certain Locations of Face-Shell Bedded Prisms F I G . 5.4 Measured Deformations at Cer ta in Locat ions of Face -She l l Bedded Pr isms 94 5.3 Stress A n a l y s i s Shr ive d i d a 3 -d imens iona l stress analysis by mode l ing a 2 -h igh face-shel l bedded p r i sm us ing the f in i te element method . However , the analysis was on ly for the case of u n i a x i a l compression and the results given in his paper are l i m i t e d to certain locat ions. Therefore, some a d d i t i o n a l numer ica l stress analysis is performed here. W e mode l a web as a plane elastic p rob lem for s imp l i c i t y . T h e author believes the 2 — d i m e n s i o n a l mode l has some value, a l though this is actua l ly a 3 — d i m e n s i o n a l p rob lem requi r ing the exact geometry of the p r i sm. T h e stress f ield was solved by using the boundary element method (C rouch 1983). T h i r t y four elements per edge length were used, and the results g iven on the boundaries i n the fo l lowing figures are the stresses evaluated at the centre of each element. T h e stress d is t r ibut ions determined for certa in locat ions i n the face-shel l loaded web are g iven in F i g . 5.5. It is interest ing to note that la tera l tensile stress i n the top of the web remains app rox imate l y constant w i t h i n the midd le range and reaches its m a x i m u m at about the quarter points instead of i n the centre. T h e h igh latera l tension at the quarter points can be a t t r ibu ted to the local stress concentrat ion ar is ing f r o m the compressive forces in the face-shel l , whi le the centre part is stressed in tension because of the beam bending mechan ism (cf. F i g . 5.1). T h i s is i m p l i e d by the tensile stress d is t r ibut ions along the depth at these two corresponding locat ions; the former has a m u c h sharper stress gradient , as shown in F i g . 5.5. T h e elastic analys is gives the astonishingly h igh value of the m a x i m u m tensile stress, read as 4 9 % of the ver t ica l compressive stress act ing on the face-shel l . T h i s result is comparab le to that g iven by Shr ive (1982). However, we m a y argue that , since nonl inear developments in the concrete a l low some degree of stress red is t r ibut ion , the tensile stress m a y be expected not to reach such a h igh value at the moment of fa i lure. T h e stress analysis suggests that the two sides of the web are not on ly car ry ing higher loca l tensions than the centre part , but are also under a complex stress state, i.e, under tension, 95 > to "5 .5 0.6 - 0 . 6 x / O . S a FIG. 5.5 Lateral Stress Distribution in a Web of Face-Shell Bedded Masonry under Uniaxial Compression: Variation across Top of Block, as well as Vertical Distribution on Centre Line and at Vertical Line where the Tension at the Top is a Maximum compression and shear. This clearly explains why splitting initiates at these locations, and suggests, furthermore, that because of the beam bending mechanism, the crack will run through the web once it is initiated. Since the splitting occurs near the face-shell, after splitting, the force is transferred by the face-shell alone without effective lateral support. Thus vertical stability is unlikely to be maintained even if the face-shell is still not crushed. Therefore, in practice, we may consider that splitting signifies failure. The deep beam bending mechanism is more obvious when face-shell bedded masonry is fully capped. The masonry block is loaded as depicted in Fig. 5.6. Unlike the face-shell capped prism, in this case the internal shear between face-shell and web cannot be neglected, lf we assume that the compressive stress on the capping side is uniformly distributed and that the internal shear resultant introduced thereby acts on the midheight of the web, then the lateral tensile stress distribution is plotted in Fig. 5.7. In this figure, the result is compared with that of FIG. 5.6 Forces Acting on a Block with Ful l Capping and Face-Shell Bedding FIG. 5.7 Lateral Stress Distribution in a Web: Ful l Capping versus Face-Shell Capping; Variation across Bottom of Block, as well as Vertical Distribution on Centre Line and Quarter Line 97 the face-shell only capping conditions under the same total prism load. Note that the entire web acts as a single deep beam in the top capped block. The maximum tensile stress is found at the bottom centre of the web. This tensile stress is not only higher than that of the face-shell capped prisms, but also extends to a larger depth. This explains why splitting is prone to occur at the centre of the web of fully capped prisms, and these prisms fail at a lower loads than their face-shell capped counterparts. One practical implication is that plain concrete masonry should be either built totally fully bedded or totally face-shell bedded. Mixed bedding patterns should be avoided. If a wall is going to be built by face-shell bedding, one must ensure that the whole wall is face-shell mortared, and detail the top and bottom of the wall so that the vertical load will be effectively transferred on the face-shell only. Otherwise one may inadvertantly sacrifice as much as half of the wall's capacity (see Table 5.1). 5.4 Some Comments on Joint Effect The deep beam bending mechanism suggests that the mortar type should have a relatively minor effect on the capacity of face-shell bedded masonry, and thus it appears possible to estimate the capacity of such a system using the modulus of rupture of the masonry units. The known correlation between the compressive strength and the modulus of rupture of concrete suggests that the capacity of masonry should be in a form such as fm = k ^ jd ( in Imperial units ) 5.1 or fm _ k (j 2 where k is a constant. When k = 40, Eq. 5.2 is plotted in Fig. 5.8 with four groups of experimental data, which gives a reasonable correlation considering the scatter of the data. 98 However , exper iments conducted by both Shr ive (1982) and by the author by va ry ing mor ta r strength have ind icated that the effect of mor tar type m a y not be to ta l l y neglected (cf. T a b l e 4 .1 ; the va r ia t i on i n the masonry capac i ty w i t h mor ta r strength is also reflected i n the scatter of the author 's d a t a i n F i g . 5.8, wh ich includes pr isms w i th three different types of mor ta r ) . A g a i n , i t is noted that the stronger mor ta r does not necessarily make stronger masonry . In the tests conducted by the author , the strongest mor ta r made the weakest masonry pr isms. W e m a y argue that a l though the deep beam mechan ism dominates the fa i lure , p a r t i a l fai lure of the mor ta r j o i n t m a y s t i l l occur at the fai lure stress. T h i s is because the fai lure stress based on mortared area is s t i l l h igh compared w i th that of the fu l ly bedded masonry . T h e outside edges of the mor ta r tend to fa i l and spal l out, leaving a narrow st r ip of m o r t a r d o w n the centre of the face-shel l . T h i s pa r t i a l j o i n t fa i lure w i l l not on ly cause a loca l stress concentrat ion in the v i c i n i t y of the j o in t , as studied in deta i l i n the preceding chapter, but m a y also change the j o i n t essential ly f rom a f lat -base to a h inge- l ike support , wh ich provides l i t t l e ro ta t ion constra int . T h e deep beam bending mechan ism m a y be intensif ied by this support change. T h e above argument suggests that the adhesion of mor ta r j o i n t to block uni t is i m p o r t a n t to face-shell masonry capac i ty as wel l . E q . 5.2 gives an est imate of masonry capac i ty based on the un i t strength. Fu r the r invest igat ion is needed to include the j o i n t effect quant i ta t i ve ly . 5.5 S u m m a r y T h e behaviour of p la in concrete masonry w i t h face-shel l bedding under concentr ic compression has been studied. T h e deep beam bending mode l for sp l i t t i ng in webs proposed by Shr ive has been veri f ied by exper iments. T h e effect of capp ing condi t ions on capac i ty and fa i lure mode has been invest igated. J o i n t effect has also been discussed. 99 FIG. 5.8 Prism Strength versus Unit Strength for Face-Shell Bedded Masonry 100 C H A P T E R V I P L A I N M A S O N R Y U N D E R E C C E N T R I C C O M P R E S S I O N 6.1 Fa i l u re M o d e T r a n s i t i o n W h e n p la in masonry (whether fu l l y mor ta red or face-shell mortared) is under eccentric compression, i t fa i ls i n a rather different mode a n d at a higher apparent stress than i t does under u n i a x i a l compression. 5 groups of p la in pr isms were tested under eccentric compression. M o s t of the specimens exh ib i ted shear type fai lure, i.e. fai lure is roughly character ized by an inc l ined fracture plane (or more precisely, a f racture zone in wh ich mate r ia l is h igh ly cracked or crushed) separat ing the mate r i a l . Because of this mode, the fa i lure appeared to be re lat ively sudden. A l l specimens fa i led on the loaded compression side, a n d fa i lure was often local ized in some part of the p r i s m . F i g . 2.15 in Chapter II i l lustrates the t y p i c a l fa i lure pat tern . T a b l e 6.1 summar izes the fa i lure loads. F igs . 6.1, 6.2 give the measured deformat ion curves. T h e apparent increase in strength phenomenon is depicted by compar ing a theoret ical P — M interact ion curve I S P E C I M E N 1 e / t 1 2 3 4 A V G C O V N18 ( N - M O R T A R ) 1 /6 150.5 107.0 120.0 121.0 124.6 12.8% M 2 0 ( M - M O R T A R ) 1 /3 77.5 79.0 86.5 95.0 84.5 8 .2% S21 ( S - M O R T A R ) 1 /3 96.0 90.0 100.0 93.0 94.8 3 .9% N19 ( N - M O R T A R ) 1 /3 83.0 81.0 85.0 69.0 79.5 7.8% N22 ( N - M O R T A R ) * 1 /3 64.0 78.0 69.0 73.0 71.0 7 .3% T a b l e 6.1 Fa i l u re Loads of P l a i n P r i s m s under Eccent r ic L o a d (kips) * Face-shel l Bedded 160 150 140 130 120 -110 100 -90 -eo 70 -80 50 -40 -30 -20 10 0 (\ (\ 1 t -4- I 1 Loaded Side Unloaded Side AVERAGE STRAIN ( 1/1000 IN/IN ) (a) - 2 2 AVERAGE STRAIN ( 1/1000 IN/IN ) (c) 2 o < s 130 120 110 100 90 80 70 60 50 -40 30 20 10 -0 Loaded Side -4 - 2 AVERAGE STRAIN ( 1/1000 IN/IN ) (b) AVERAGE STRAIN ( 1/1000 IN/IN ) (d) FIG. 6.1 Measured Deformations at Certain Locations of Plain Prisms under Eccentric Load: a) N18-1, e=t/6; b) N18-4, e=t/6; c) N19-4, e=t/3; d) M2D-2, e=t/3 AVERAGE STRAIN ( 1/1000 IN/IN ) AVERAGE STRAIN ( 1/1000 IN/IN ) (a) (b) AVERAGE STRAIN ( 1/1000 IN/IN ) AVERAGE STRAIN ( 1/1000 IN/IN ) (C) (d) FIG. 6.2 Measured Deformat ions at Cer ta in Locat ions of P l a i n P r i s m s under Eccent r ic L o a d : o a) S21-4, e = t / 3 ; b) S21-3, e = t / 3 ; c) N22-2 , e = t / 3 ; d) N22-4 , e = t / 3 103 based on the uniaxial compressive strength with the eccentric compression test data, as shown by Fig. 4.1 in Chapter IV. There are some differences in the detailed failure modes among the specimens. Vertical splitting in the web before or at failure of the loaded side was observed in some of the prisms with high eccentricity (e=i/3). A similar phenomenon was observed by Hatzinikolas et al in their experiments (1978), and it worth giving a brief explanation. For those prisms which were under large eccentricity, the joints on the tension side of some specimens debonded before the compression side failed. (This is shown by the deformation measurement across the joint on the tension side, see Figs. 6.1, 6.2) Because of this debonding, the prisms were actually only loaded on the compression side, as depicted in Fig. 6.3. The resultant force acting on the compression side of the web is an axial force with a bending moment. Therefore, it is not surprising that some transverse tensile stress can develop in the web. For an ideal elastic case in which the compressive stress is triangularly distributed, a numerical study shows that the maximum magnitude of this transverse tensile stress can be as high as 25% of the maximum compressive stress, as depicted in Fig. 6.3. However, it can be visualized that the splitting caused by this tensile stress does not directly lead to final failure of a prism, or of a low wall. This view is supported by the experimental observation that splitting can occur before the loaded face-shell fails, and that failure is essentially characterized by a shear mechanism. Nevertheless, it again implies the importance of sound adhesion in the joints. Although plain masonry is not usually designed to sustain load with high eccentricity, sound bond may ensure the wall's integrity in the case of the wall being accidentally loaded in the unfavorable condition (with tensile stress occurring on one side of the wall). For the case of face-shell bedded masonry, with increasing eccentricity, the deep beam mechanism may no longer dominate the failure. A stress analysis, keeping the vertical compression stress on the loaded side constant, indicates that the magnitude of the lateral 104 F I G . 6 .4 L a t e r a l S t r e s s a l o n g T o p o f a W e b w i t h F a c e - S h e l l B e d d i n g u n d e r E c c e n t r i c L o a d 105 tensile stress due to this mechan ism is substant ia l l y reduced w i t h increasing eccentr ic i ty , as shown i n F i g . 6.4. However , to fu l l y exp la in the preference of the shear fa i lure mode when masonry is under eccentric load ing needs a thorough understanding of the fa i lure of a concrete- l ike b r i t t le m a t e r i a l under var ious condi t ions . In Chapte r III we have proposed a fa i lure mode l exp la in ing the sp l i t t i ng fa i lure under u n i a x i a l compression. However, i t appears no easy extension can be made when the mode l mechan ism is under a compressive stress w i t h gradient . It cou ld be that the uneven compression due to the stress gradient intensifies the f r ic t ion a n d inter lock mechan ism between crack surfaces and thus prevents the sp l i t t i ng mode f r o m occurr ing . 6.2 Effect of J o i n t Cond i t i ons A s shown i n T a b l e 6.1, under large eccentr ic i ty , change of mor ta r strength apparent ly has a re lat ive ly m ino r effect on the capac i ty of the p r i s m . T h i s m a y be expla ined as fo l lows. W h e n p la in concrete masonry is under h igh ly eccentric compression, the compressive force is m a i n l y transferred by the face-shell on the loaded side. Since there is a s t ra in gradient across the face-shel l , the stress d i s t r ibu t ion across i t , at a po int remote f r o m the j o in t , must be h u m p shaped because of the non l inear i ty of the mate r ia l . T h i s is quite different f r o m that under u n i a x i a l compression, where the stress wou ld a lways be un i fo rm ly d is t r ibuted in the absence of the j o in t , regardless of the development of mate r i a l nonl inear i ty . T h e h u m p shaped stress d i s t r ibu t ion suggests that the force wou ld be largely transferred by the m idd le part of the face-shel l . W e know by the analysis of the preceding sections that the mor ta r j o i n t can develop re lat ively h igh strength i n its m idd le part (cf. F i g . 4.14). T h e presence of the j o i n t , therefore, m a y not alter the n o r m a l stress d is t r ibut ion as m u c h as the j o i n t under u n i a x i a l compression w i l l do. Moreover , because of the eccentric loading, the outer f iber of the loaded face-shel l w i l l deform more than the inner f ibre w i l l do. F o r load ing w i t h eccentr ic i ty equal to one th i rd of the F I G . 6.5 S t ra in D is t r ibu t ion in a Section of Masonry under Eccentr ic L o a d 106 w id th of the p r i sm, e i = 0.64e o , as depicted in F i g . 6.5. Here we neglect the tensile strength of concrete and assume e,- = 0 in the midd le of the cross-section after the tensile part of the p r ism has debonded. W h e n this stra in is imposed on the jo in t , the j o i n t is actua l ly under a combinat ion of un iax ia l compression and bending. A stress analysis shows that the bending stresses w i l l lend add i t iona l latera l confinement to the more compressed side of the jo in t and thus enhance the j o i n t strength in that part , i.e., under eccentric compression the more compressed part of the jo in t w i l l develop more strength. T h i s ensures that the j o i n t does not fa i l dur ing the loading to the final stress d is t r ibut ion discussed above. T h u s , in practice we m a y neglect the jo in t condit ions in designing walls under eccentric loading. T h i s approach is further studied in the next chapter. 6.3 S u m m a r y In this chapter, the behaviour of plain masonry under eccentric compression has been investigated. The eccentric behaviour differs f rom the concentr ic one not on ly in fai lure mechanism but also in the j o i n t effect on the strength. In the fo l lowing chapter, we w i l l propose a design approach based on these findings, and conclude the study on p la in concrete masonry . 107 C H A P T E R V I I R E C O M M E N D E D D E S I G N A P P R O A C H F O R P L A I N M A S O N R Y 7.1 Recommendat ions on the Bas is for Design It has been demonstrated that under different load condit ions p la in masonry w i l l fa i l by different modes. Under u n i a x i a l compression, masonry w i l l fa i l by vert ica l sp l i t t i ng , but not due to the mechan ism proposed by Hi lsdorf . F o r face—she l l bedded masonry , sp l i t t i ng can be a t t r ibu ted to a mechan ism s i m i l a r to deep beam bending. Under eccentric load ing , masonry tends to fa i l i n a mode a p p r o x i m a t i n g shear fa i lure. These two different fa i lure modes w i l l y ie ld different apparent strengths. T h e j o in t condi t ions w i l l affect the capac i ty of the masonry to a different extent under each of these two basic load patterns. In pract ice, one wishes to est imate the capaci ty of masonry f r o m the block un i t compressive strength and the mor ta r cubic strength, since the latter are re lat ive ly easy to measure exper imenta l ly . T h e corre lat ion given by Eqs . 4.27 or 4.28 and E q . 5.2 m a y serve this purpose. However , when using these relations, one must keep in m i n d that some uncertaint ies are invo lved as was ind icated in the development of the equations. In par t icu la r , we have uncertaint ies in the fa i lure c r i te r ia of the mater ia l itself, i n the mate r i a l properties other than strength, i n the corre lat ion between strengths, and last but not least, i n the workmansh ip . These uncertaint ies are reflected in the scattered d a t a of numerous exper iments. Therefore, i t is recommended that in pract ice either we use the relat ions such as given by Eqs . 4.27 or 4.28 and E q . 5.2 i n a conservative manner or we reta in the masonry p r i sm test to est imate fm, the design base of p la in concrete masonry, under u n i a x i a l compression. However , for the case of eccentr ical ly loaded masonry , an approach wh ich differs f r o m 108 the t rad i t i ona l one w i l l be recommended. In the t rad i t i ona l approach, the eccentric capaci ty est imat ion is also based on the value fm associated w i t h concentr ic load ing . T h e apparent increase i n strength is taken into account by a (strain gradient) factor . T h i s factor as a funct ion of eccentr ic i ty has been frequently studied through exper iments (for example , T u r k s t r a a n d T h o m a s 1978; Drysda le and H a m i d 1983). In the current design code (CAN3-S304-M84 1984) the factor is given as a fixed value (1.3, reflected in the eccentr ic i ty coefficient Ce). T h e usefulness of this approach depends on an assumed close a n d fixed corre lat ion between the concentr ic capac i ty and the eccentric capaci ty . In the l ight of preceding studies, we know that this corre lat ion is questionable since different fai lure mechanisms are invo lved . In view of the fai lure mechanisms, the eccentric capaci ty of concrete masonry m a y be better correlated w i t h the un i t compressive strength fu instead of fm. A s shown in F igs . 2.2 and 2.15 i n C h a p t e r II, the fai lure pattern of the un i t block is very s im i la r to that of p r isms under eccentric compression. T h u s i t is recommended here that the eccentric capaci ty est imat ion be d i rect ly based on the un i t compressive strength fu, whi le the concentr ic capac i ty is based on the p r i s m strength fm. T h e j o i n t effect is neglected since apparent ly i t is re lat ively m ino r for the case of eccentric load ing . A l t h o u g h the apparent compressive strength of masonry m a y vary w i t h the eccentr icty, the va r ia t i on is ignored for p ract ica l reasons. It is believed that this approach w i l l y ie ld better correlat ions since i t is based on recognit ion of the fai lure mechanisms. Fur ther , of course, this recommended approach considers the fact that i t is not p ract ica l to test pr isms under eccentric load ing to assess the capac i ty . T h e t rans i t iona l po int where the fai lure mode changes f r o m sp l i t t i ng to shear fa i lure needs to be indent i f ied . It is suggested by the ava i lab le exper imenta l work that this occurres at a s m a l l eccentr ic i ty (e < t/6). T h i s impl ies that the cross-section capaci ty curve is d iscont inuous somewhere between e = t/6 and e = 0. T h e detai led behaviour of the cross-section capac i ty in 109 this range needs further invest igat ion . A t this point we recommend that this part of the curve be interpo lated between the capacit ies at zero eccentr icity and at </6 eccentr ic i ty , but not to exceed the ver t ica l load capac i ty of the zero eccentr icity case. T h i s is on the conservative side, as w i l l be shown later , since the capac i ty at z /12 is also wel l correlated w i t h the un i t strength. T o examine this p ract ica l a l ternat ive of basing the eccentr ical ly loaded capac i ty on the un i t strength, we compare ava i lab le test d a t a w i t h the recommended capac i ty curve. T h e capac i ty curves are generated by a convent iona l method , i.e. l inear elastic behaviour a n d plane cross-section are assumed and the extreme fibre stress is set equal to the unit strength. T h e general expressions based on this method (for both grouted and ungrouted masonry) are der ived i n Chapte r X . These expressions were checked against a computer p rog ram developed by N a t h a n (1985), w h i c h performs a ra t iona l analysis for beam co lumns based on m a t e r i a l properties and cross-section geometry. Since, under large eccentr ic i ty , masonry fai ls in a s im i la r pattern regardless of the bedding condi t ions , exper imenta l d a t a for both bedding condit ions ( ful l and face-shell) are inc luded. T h e compar ison is i l lust rated in F igs . 7.1 to F i g . 7.11, wh ich include the tests done by the author , by F a t t a l and Cat taneo (1976), by Hatz in iko las et a l (1978) and by Drysda le a n d H a m i d (1983). Tab les 7.1 to 7.4 summar ize the numer ica l results. F o r the 58 cases compared , the average value of the rat io of fa i lure load to predicted load is 1.026 w i t h a coefficient of va r ia t ion of 11.36%, corresponding to an expected rat io of -1.026 w i t h 9 5 % confidence l i m i t s equal to 0.996 and 1.056. T h e agreement is extremely good consider ing the scatter of the d a t a and the errat ic nature of the mater ia l . F i g . 7.12 summar izes the compar ison of the fai lure loads predicted by the recommended method w i t h the exper imenta l da ta . T h e coefficient of correlat ion is 0.956 and the ma jo r i t y of d a t a po ints l ie w i t h i n the 99 percent confidence l i m i t s , wh ich is h ighly s ign i f icant . T h e recommendat ions for design are concisely depicted in F i g . 7.13 by a P— M capac i ty curve. C u r v e O — Blt the masonry capaci ty under eccentric load , should be determined 110 200 280 M(KP-N) F I G . 7.1 C o m p a r i s o n between Recommended A p p r o a c h a n d Exper iments by the A u t h o r : N18, N19, M 2 0 and S21 140 80 120 160 200 M(KP-N) F I G . 7.2 C o m p a r i s o n between Recommended A p p r o a c h a n d Exper iments by the Autho r : N22 (Face-Shel l Bedding) Ill F I G . 7.3 Compar i son between Recommended A p p r o a c h a n d Exper iments by F a t t a l and Cat taneo 2 400 M ( K P - N ) F I G . 7.4 Compar i son between Recommended Approach and Exper iments by Hatz in iko las et a l 112 400 350 300 -250 -200 150 -100 -50 -M(KN-M) F I G . 7.5 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysdale and H a m i d : N o r m a l B l o c k M (KN-M) F I G . 7.6 Compar i son between Recommended A p p r o a c h and Exper iments by Drysdale and H a m i d : W e a k B lock 113 490 2" M(KN-M) F I G . 7.7 Compar i son between Recommended A p p r o a c h a n d Exper iments by Drysdale and H a m i d : St rong B l o c k M (KN-M) F I G . 7.8 Compar i son between Recommended A p p r o a c h a n d Exper iments by Drysdale and H a m i d : L i g h t We igh t B l o c k 0 2 4 6 8 10 12 14 M (KN-M) F I G . 7.9 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysdale and H a m i d : 7 5 % So l i d B l o c k 280 -r 1 v i i i. i ; i 0 2 4 6 M(KN-M) F I G . 7.10 C o m p a r i s o n between Recommended A p p r o a c h and Exper iments by Drysdale and H a m i d : 6 inch B lock 400 a. 0 2 4 6 B 10 12 14 16 M (KN-M) F I G . 7.11 Compar ison between Recommended A p p r o a c h a n d Exper iments by Drysdale and H a m i d : 10 inch B l o c k N18 (e= = l /6 t ) N19 (e= = l /3 t ) M 2 0 (e: = l / 3 t ) S21 (e= : l / 3 t ) N22 (c= = l / 3 t ) P 0 =121.7k ips P 0= 84.4kips P 0= 84.4kips P 0= 84.4kips P 0= 70.1kips (predicted) (predicted) (predicted) (predicted) (predicted) P -k ips P/Po P -k ips P/Po P -k ips P/Po P -k ips P/Po P -k ips P/Po 150.5 1.24 83.0 0.98 77.5 0.92 96.0 1.14 64.0 0.91 107.0 0.88 81.0 0.96 79.0 0.94 90.0 1.07 78.0 1.11 120.0 0.99 85.0 1.01 86.5 1.02 100.0 1.18 69.0 0.98 121.0 0.99 69.0 0.82 95.0 1.13 93.0 1.10 73.0 1.04 A V G 124.6 1.02 79.5 0.94 84.5 1.00 94.8 1.12 71.0 1.01 C O V 12.8% 7.8% 8.2% 3.9% 7.3% Tab le 7.1 Compar ison wi th the Recommended A p p r o a c h : Tests by A u t h o r e = l / 1 2 ' t e = l / 6 t e = l / 4 b e = l / 3 t P 0 = 116.3 k ips P 0 = 97.1 k ips P 0 = 83.3 k ips P 0 = 72.9 k ips P (kips) P/Po P (kips) P/Po P (kips) P/P 0 P (kips) P/Po 120.0 1.03 115.1 1.19 82.5 0.99 62.2 0.85 87.8 0.75 108.9 1.12 84.4 1.01 77.0 1.06 160.0 1.38 117.1 1.21 82.3 0.99 68.0 0.93 A V G 122.6 1.05 113.7 1.17 83.1 1.00 69.1 0.95 C O V 2 4 . 1 % 3 . 1 % 1.1% 8 .8% T a b l e 7.2 C o m p a r i s o n w i t h the Recommended A p p r o a c h : Tests by F a t t a l et a l e = l / 6 t e = l / 3 t P 0 = 185.4 k ips P 0 = 138.7 k ips P (kips) P/Po P (kips) P/Po 180.0 0.97 119.3 0.86 196.0 1.06 158.7 1.14 150.1 0.81 A V G 175.4 0.95 139.0 1.00 C O V 10.8% 14.2% T a b l e 7.3 C o m p a r i s o n w i t h the Recommended A p p r o a c h : Tests by Hatz in iko las et a l S P E C I M E N e / t P (kN) P (kips) Po(kips) P/Po n o r m a l b lock 1 /6 247 55.5 54.4 1.02 ( N B ) 1 /3 206 46.3 40.8 1.13 5 / 1 2 158 35.5 36.1 0.98 weak block 1 /6 171 38.4 37.1 1.04 ( W B ) 1 /3 133 29.9 28.0 1.07 5 / 12 99 22.2 22.5 0.99 strong block 1 /6 301 67.7 60.5 1.12 ( S B ) 1 /3 236 53.1 45.4 1.17 5 / 1 2 194 43.6 40.2 1.09 l ightweight b lock 1 /6 228 51.3 43.8 1.17 ( L B ) 1 /3 169 38.0 32.8 1.16 5 / 1 2 149 33.5 29.1 1.15 7 5 % so l id 1 /6 258 58.0 62.8 0.92 ( Q B ) 1 /3 190 42.7 45.5 0.94 5 / 1 2 100 22.5 22.2 1.01 6 inch block 1 /6 185 41.6 40.7 1.02 ( 6 " B ) 1 /3 137 30.8 30.3 1.02 5 / 1 2 94 21.1 21.4 0.99 10 inch block 1 /6 200 45.0 54.7 0.82 ( 1 0 " B ) 1 /3 172 38.7 41.5 0.93 5 / 12 132 29.7 28.7 1.03 A V G 1.04 C O V 8.7% T a b l e 7.4 C o m p a r i s o n w i t h the Recommended A p p r o a c h : Tests by Drysda le et a l 118 x UJ 200 180 160 140 120 100 80 60 40 20 0 — n AUTHOR A HATZlhJKOLAS v DRYSDALE 0 FATTAL s s ' A . • f 1 s A y '1 y / —Ar-t s 1 / • 'Y I s l • V 99% COhFIDENCE LIMFTS \ y • \ Y' • V LINEAR REGRESSION w s V V s ft y s r / y s S " ' 20 40 60 80 100 120 140 160 180 200 PREDICTIONS (KPS ) F I G . 7.12 Compar i son between Recommended A p p r o a c h and Exper iments : S u m m a r y F I G . 7.13 Dep ict ion of Recommended A p p r o a c h 119 based on the masonry un i t strength fu and the geomerty of the cross-section, where B x denotes the kern eccentr ic i ty capac i ty (e = t/&). A.± or A 2 , wh ich stands for the concentr ic capaci ty , should be determined either by the p r i s m test or by the correlat ion w i t h the masonry uni t strength fu and the mor ta r strength fj . T h e whole capac i ty curve is represented either by A : — Bi~O or by A 2 — B 2 — O depending whether the concentr ic capaci ty is greater than the kern eccentr ic i ty capac i ty . 7.2 Discussion of T h e Cur rent Design Code T h e current design code ( C A N 3 - S 3 0 4 - M 8 4 , 1984) permits two ways of designing wal ls for car ry ing in -p lane ax ia l compression and out of plane bending due to eccentr ic i ty of the ver t ica l load . T h e y are the so-cal led "coefficient method" a n d the " load def lect ion m e t h o d " . T h e former gives an a d d i t i o n a l a l ternat ive to determine the eccentr ic i ty coefficient. Therefore, one actua l ly cou ld develop three different P — M design curves for the same member . W e denote them by method 1, 2 and 3 for ease of discussion. T h e basic i n fo rmat ion needed for design is fm, the u l t imate compressive strength of masonry . T h e code specifies two methods to determine fm. M e t h o d A requires testing of at least f ive pr isms in u n i a x i a l compression. fm is then taken as the average strength minus 1.5 t imes the s tandard dev ia t ion of the sample. (Th is value m a y be reduced by a factor depending on whether the specified uni t strength is consistent w i t h the tested uni t strength.) M e t h o d B a l lows one to test un i t and mor ta r separately (the latter is to ensure that the specified mor ta r type is adequate) and fm is taken f r o m the tabled value based on uni t strength and mor ta r type. T h e most obvious object of a design code is to ensure consistent re l iab i l i t y i n structures. F o r f lexura l design of masonry wal ls , consistent re l iab i l i t y for var ious load combinat ions is required. T h e current code recognizes the apparent strength increase when wal ls are under eccentric load ing , and some modi f icat ions are inc luded in the convent iona l b e a m - c o l u m n approach. T h i s is reflected in method 1 by an increase in Ce, the eccentr ic i ty coefficient, of 3 0 % 120 (this is fa i r l y reasonable when compared w i t h the author 's exper iments) when load ing eccentr ic i ty is greater than r /20. F o r methods 2 and 3, this is reflected i n the different a l lowable stresses that are used (compressive, f lexural ) i n developing the P—M i n teract ion curve. A l t h o u g h the author has d i f f icu l ty i n understanding why the code a l lows quite different results by the three different methods for wal ls loaded w i t h equal eccentricit ies, we w i l l consider here the imp l ica t ions of present research w i t h respect to these provis ions. F i r s t of a l l , i t was concluded that the p r i s m strength does not correlate wel l w i t h the strengths of mor ta r and unit . T h i s suggests that there is considerable uncerta inty in the use of method B to avo id m a k i n g a x i a l p r i s m tests. F o r the fu l l y bedded pr isms tested by the author , method A wou ld give fm equal to 1760, 2625 and 1930 psi for pr isms w i t h M , S, and N type mor ta r respectively. M e t h o d B, wh ich is based on the corre lat ion between p r i s m strength and strengths of un i t and mor tar , wou ld give fm equal to 1855 psi for pr isms w i t h M and S type mor ta r and 1430 psi for pr isms w i t h type N m o r t a r 2 . T h i s is very inconsistent , especial ly in terms of the p robab i l i t y of non-exceedence of the strength value. fm determined by method A corresponds to a non-exceedence p robab i l i t y of about 6 .7% for a l l three type mor ta r pr isms, whi le fm determined by method B gives the non-exceedence p robab i l i t y of about 1 2 % for M type mor ta r p r i s m , 0 .001% for S type and 0 .03% for N type mor ta r pr isms. T h e conclus ion is clear: method B cannot be recommended, or i t should be used very conservat ively . T h e current code m a y already be on the very conservative side in most cases, but i t can not prevent unfavorable results i n some par t icu la r cases, such as the type M p r i s m in the above example . Second, the code requires that the design be based on fm. However , fm is the strength of masonry under u n i a x i a l compression. It was concluded that , because of the different fa i lure mechanisms, this strength does not correlate wel l w i t h the f lexura l compressive strength, wh ich 2 H e r e we on ly have a sample size of 4 a n d we relax the 15% restr ict ion on coefficient of va r ia t i on for M type mor ta r masonry p r i sm. 121 i s m u c h less d e p e n d e n t o n t h e j o i n t c o n d i t i o n a n d b e d d i n g p a t t e r n . T h e c o d e o n l y t a k e s a c c o u n t o f t h e a p p a r e n t i n c r e a s e i n s t r e n g t h b y a l l o w i n g a n i n c r e a s e i n t h e d e s i g n s t r e s s f o r m a s o n r y u n d e r e c c e n t r i c l o a d i n g o f a f i x e d a m o u n t , i .e . 3 0 % i n m e t h o d 1 a n d 2 0 % i n m e t h o d 2 a n d 3 ( s o m e i n t e r p o l a t i o n i s i n v o l v e d i n m e t h o d s 2 a n d 3 ). T h i s i m p l i e s t h a t a r e l a t i o n s h i p i s a s s u m e d b e t w e e n u n i a x i a l s t r e n g t h a n d f l e x u r a l s t r e n g t h o f m a s o n r y w h i c h i s i n d e p e n d e n t o f j o i n t p a t t e r n a n d m o r t a r t y p e . I n o t h e r w o r d s , s i n c e u n i a x i a l c o m p r e s s i o n s t r e n g t h d e p e n d s l a r g e l y o n t h e j o i n t c o n d i t i o n a n d b e d d i n g p a t t e r n , t h e s t r e n g t h u n d e r e c c e n t r i c l o a d i n g a l s o d e p e n d s l a r g e l y o n t h e s e v a r i a b l e s , a c c o r d i n g t o t h e c o d e . T h e s t u d y i n t h e p r e c e d i n g s e c t i o n s i n d i c a t e s t h a t t h i s is n o t t h e c a s e . T h e s t r e n g t h o f m a s o n r y u n d e r e c c e n t r i c c o m p r e s s i o n m a y b e b e t t e r c o r r e l a t e d w i t h t h e u n i t c o m p r e s s i v e s t r e n g t h . T h u s , i n c o n s i s t e n c y i n t h e r e l i a b i l i t y o f w a l l c a p a c i t y as d e s i g n e d b y t h e c u r r e n t c o d e m a y b e e x p e c t e d . F o r e x a m p l e , f o r t h e 5 8 e x p e r i m e n t a l c a s e s s t u d i e d a b o v e , i n c l u d i n g e c c e n t r i c a l l y l o a d e d p r i s m s o f w a l l s e c t i o n s t e s t e d b y t h e a u t h o r a n d o t h e r s , t h e f l e x u r a l c o m p r e s s i v e s t r e s s a t f a i l u r e w a s c a l c u l a t e d ( b a s e d o n t h e a s s u m p t i o n t h a t s t r e s s i s l i n e a r l y d i s t r i b u t e d i n t h e c r o s s - s e c t i o n ) , a n d l i s t e d i n T a b l e 7 .5 t o T a b l e 7.8 a n d s u m m a r i z e d i n F i g . 7 . 1 4 . T h e a v e r a g e r a t i o o f t h e f l e x u r a l s t r e n g t h t o u n i a x i a l c o m p r e s s i v e s t r e n g t h w a s 1.21 w i t h a c o e f f i c i e n t o f v a r i a t i o n e q u a l t o 1 3 . 3 % . A l t h o u g h t h e a v e r a g e 21 p e r c e n t h i g h e r f l e x u r a l c o m p r e s s i v e s t r e n g t h is c l o s e t o t h e e c c e n t r i c c o e f f i c i e n t s p e c i f i e d b y t h e c o d e ( m e t h o d 2 a n d m e t h o d 3 ) , t h e c o e f f i c i e n t o f v a r i a t i o n i s h i g h e r t h a n t h a t o f t h e r e c o m m e n d e d m e t h o d ( 1 1 . 4 % ) . F i g . 7 .14 s h o w s t h e c o m p a r i s o n b e t w e e n a x i a l a n d f l e x u r a l s t r e n g t h s f o r t h e r e p o r t e d t e s t s , w h i c h h a s a c o e f f i c i e n t o f c o r r e l a t i o n o f o n l y 0 . 8 7 5 , c o m p a r i n g u n f a v o r a b l y w i t h t h a t o f 0 . 9 5 6 f o r t h e r e c o m m e n d e d m e t h o d . T h e s u p e r i o r i t y o f t h e r e c o m m e n d e d m e t h o d o v e r t h e c o d e s p e c i f i e d m e t h o d is o b v i o u s i n v i e w o f F i g s . 7 . 1 2 a n d 7 . 1 4 . o -Y-0 1 2 3 4 5 UNIAXAIL STRENGTH ( KSI ) FIG. 7.14 Current Design Base: Uniaxial Strength versus Flexural Strength N18 (e= =l/6t) N19 (e= =l/3t) M20 (e: =l/3t) S21 (e= : l /3t) N22 (e= =l/3t) fm=2.32 ksi fin = 2.32 ksi fm=2.42 ksi =3.02 ksi fm = 2.52 ksi (prism test) (prism test) (prism test) (prism test) (prism test) il(ksi) f'e/f'm fl(ksi) f'e/f'm fl(ksi) f'e/f'm f'e(ksi) f'e/f'm fe(ksi) f'e/f'm 4.01 1.73 3.38 1.37 2.97 1.23 3.68 1.22 2.96 1.17 2.85 1.23 3.11 1.34 3.03 1.25 3.45 1.15 3.60 1.43 3.20 1.38 3.26 1.41 3.32 1.37 3.84 1.27 3.19 1.26 3.22 1.39 2.65 1.14 3.65 1.51 3.57 1.18 3.37 1.34 AVG 3.32 1.43 3.05 1.31 3.24 1.34 3.63 1.21 3.28 1.30 COV 12.8% 7.8% 8.2% 3.9% 7.3% Table 7.5 Flexural to Uniaxial Strength: Tests by Author e = l / 1 2 t 4 = 1.89 ks i e = l / 6 t 4 = 1.89 ks i e = l / 4 t 4 = 1.89 ksi e = l / 3 t 4 = 1.89 ks i it (ksi) ft/4 it (ksi) it/4 it (ksi) it/4 ft (ksi) fe/4 2.28 1.67 3.04 1.20 0.88 1.61 2.61 2.47 2.66 1.38 1.31 1.41 2.18 2.23 2.18 1.15 1.18 1.15 1.88 2.33 2.05 0.99 1.23 1.09 A V G 2.33 1.23 2.58 1.37 2.20 1.16 2.09 1.10 C O V 2 4 . 1 % 3 .1% 1.1% 8.8% T a b l e 7.6 F l e x u r a l to U n i a x i a l Strength: Tests by F a t t a l et a l e = l / 6 t 4 = 1.96 ksi e = l / 3 t 4 = 1.96 ks i it (ksi) fe/4 fe (ksi) f'e/4 2.28 2.49 1.90 1.17 1.28 0.97 2.20 2.68 1.03 1.37 A V G 2.22 1.14 2.35 1.20 C O V 10.8% 14.2% T a b l e 7.7 F l e x u r a l to U n i a x i a l St rength : Tests by Hatz in iko las et a l S P E C I M E N e / t ft (mpa) fe (ksi) fe/4 n o r m a l b lock 1 /6 28.0 4.06 1.12 ( N B ) 1 /3 27.8 4.03 1.12 f m = 24.9 m p a 5 / 1 2 27.0 3.91 1.08 weak block 1 /6 19.4 2.81 1.08 ( W B ) 1 /3 19.9 2.88 1.11 fm = 18.0 m p a 5 / 1 2 16.8 2.43 0.93 strong block 1 /6 34.0 4.93 1.14 ( S B ) 1 /3 35.1 5.09 1.17 fm = 29.9 m p a 5 / 1 2 32.9 4.77 1.10 l ightweight block 1/6 25.8 3.74 1.24 ( L B ) 1 /3 25.1 3.64 1.21 f m = 20.8 m p a 5 / 12 25.3 3.67 1.22 7 5 % sol id 1 /6 19.1 2.77 1.17 ( Q B ) 1 /3 . 17.6 2.55 1.08 fm = 16.3 m p a 5 / 1 2 21.1 3.06 1.29 6 inch block 1 /6 26.5 3.84 1.11 ( 6 " B ) 1 /3 26.1 . 3.78 1.10 fm = 23.8 m p a 5 / 1 2 23.6 3.42 0.99 10 inch block 1 /6 20.8 3.01 0.96 ( 1 0 " B ) 1 /3 23.4 3.39 1.08 f m = 21.6 m p a 5 / 1 2 22.5 3.26 1.04 A V G 1.11 C O V 7.7% T a b l e 7.8 F l e x u r a l to U n i a x i a l Strength: Tests by Drysdale et a l 125 C H A P T E R VI I I G R O U T E D M A S O N R Y W I T H F U L L B E D D I N G 8.1 In t roduct ion In the west coast area of C a n a d a , where earthquake resistance is a m a i n concern in s t ructu ra l design, masonry wal ls and co lumns are required to be grouted and reinforced to improve s t ructu ra l cont inu i ty and duc t i l i t y . T h e a x i a l capac i ty of grouted and reinforced masonry is of interest not on ly because it d i rect ly determines the design thickness of a w a l l p rov id ing a x i a l and la tera l load resistance in a m u l t i — s t o r y bu i l d ing , but also, because it is closely related to the design d u c t i l i t y (Pr iest ley and H o n 1983). T h e methods for determin ing the compressive strength of grouted concrete masonry specified in the current code ( C A N 3 - S 3 0 4 - M 8 4 , 1984) are essential ly the same as those for p la in concrete masonry , as reviewed in the preceding chapter. M e t h o d A , wh ich requires a p r i s m test, is not very p ract ica l . Since the fa i lure load of a s tandard 8 inch grouted p r i s m w i l l usua l ly be wel l above 300 k ips , test ing faci l i t ies w i th adequate capaci ty are extremely l i m i t e d . M e t h o d B, wh ich estimates the compressive strength based on the uni t strength and mor ta r type, tends to be excessively conservative due to the uncertaint ies invo lved . T h e code does not correlate the masonry compressive strength w i t h grout strength, but merely requires that the grout strength be at least equal to that of the block shel l . O n the one hand , this does not a l low one to take advantage of h igh strength grout ing , and on the other hand , i t is a d i f f icu l t speci f icat ion to meet since the un i t strength is often much higher than the specified value (cf. Tab les 8.3 — 8.9). If the fa i lure mechan ism is dependent on the relat ive strengths, i t w i l l be correspondingly uncerta in . T h e a x i a l behaviour of grouted concrete masonry has been studied both exper imenta l ly and ana l y t i ca l l y . Drysda le and H a m i d (1979) f irst addressed the c o m p a t i b i l i t y p rob lem between masonry uni t and grout based on their exper imenta l observations. A k i o B a b a and O s a m u Senbu (1986) proposed the concept of the grout efficiency, and found it var ied considerably w i t h 126 different combinat ions of masonry uni t and grout they tested. A few fai lure models have been suggested to predict the u l t imate strength considering the in teract ion between un i t , grout and m o r t a r ( A h m a d and Drysda le 1979, Pr iest ley and H o n 1983). However , the in terna l forces of these models were ent i rely based on a state that a l l the three mater ia ls reach some assumed fracture cr i te r ia . T h i s is not a lways a real ist ic descr ipt ion. Fur ther , the fracture is not necessarily equivalent to u l t imate state. It is clear that a better understanding of the mechanica l behaviour is needed and a more accurate est imate for masonry strength is desirable. In this study, the exper imenta l behaviour of grouted masonry pr isms is careful ly examined and a better corre lat ion of the masonry strength w i t h the un i t st rength, grout strength and mor ta r strength is sought. 8.2 E x p e r i m e n t a l Observat ions 23 grouted pr isms were tested under u n i a x i a l compression, w i t h var ious j o i n t a n d grout ing condi t ions . T h e fa i lure loads of the pr isms are summar i zed in Tab les 8.1 and 8.2. T h e grout strengths, evaluated by testing in accordance w i t h C S A Standard (A179 —1976) , are l isted in T a b l e 2.5. T h e exper iments ind icate : 1) B o t h mor ta r strength and grout strength affect the p r i sm strength. A p p a r e n t l y stronger mor ta r and grout make stronger masonry (see Tab les 8.1 and 8.2). However , the increase in masonry capac i ty is m inor , even w i t h a substant ia l increase in the const i tuent strengths, as depicted in F i g . 8.1. T h i s is especially true for grout, suggesting that the cont r ibu t ion of grout and b lock shel l ( inc lud ing mor ta r j o in t ) to the capaci ty of masonry is a funct ion of their compat ib le deformat ions, and is not s i m p l y given by superposit ion of their i n d i v i d u a l capacit ies. T h i s observat ion conf i rms that by Drysdale and A h m a d (1979). 2) De fo rmat ion measurements and direct observat ion (recorded by a video camera) ind icate that a lmost a l l the pr isms were cracked before f ina l fa i lure. C r a c k s were found in the webs as wel l as 127 in the face-shel l , occurr ing at loads as low as 4 0 % of the fa i lure value, as evidenced by the recorded deformat ions i n F igs . 8.2, 8.3 and 8.4. S i m i l a r observations have been reported by Sturgeon and L o n g w o r t h (1985). U s i n g acoustic measurement, A k i o B a b a and O s a m u Senbu (1986) have also observed more detai led c rack ing process wel l before u l t i m a t e state for grouted pr isms w i t h bond beam concrete uni t . T h i s premature c rack ing m a y have been caused by the i n c o m p a t i b i l i t y between the grout and the block shel l , as w i l l be further studied below. However , closer inspect ion shows that the block shell s t i l l carr ied substant ia l load after c rack ing , i nd ica t ing that premature c rack ing is not necessarily equivalent to fai lure. W e know this m a i n l y f r o m two facts: a) T h e ver t ica l deformat ion measurement of the block shell shows that the compressive s t ra in remained at a h igh level after c rack ing had occurred (cf. F igs . 8.2-8.4). b) T h e capac i ty of the p r i s m is usual ly substant ia l ly greater than that of g rout ing concrete alone. 3) B l o c k shells are stiffer than concrete grout wh ich in tu rn are stiffer than mor tar . T h e peak s t ra in is between 0.0015 to 0.002 for the block uni t and 0.0025 to 0.003 for three types of grout (see "Figs. 2.3, 2.4 and 2.6 in Chapte r II). A s im i la r phenomenon has been ind icated by the research on concrete masonry in New Zealand (Pr iest ley a n d E lder 1985). T h e Y o u n g ' s modu lus of var ious concrete un i ts (used in Japan ) are also appearent ly higher than that of grout accord ing to A k i o B a b a a n d O s a m u Senbu (1986). T h e mor ta r even exh ib i ted higher peak strains, wh ich exceeded 0.005 (measured dur ing the cube strength testing) . T h e difference in the de format ion properties is p robab ly due to different mate r ia l textures and cur ing condi t ions . T h i s observat ion supports the v iew that c o m p a t i b i l i t y p lays an impor tan t role in concrete masonry capac i ty . T h e ver t ica l deformat ion measurements, indeed, ind icated that the block shells carr ied a larger share of l oad relat ive to the grout before they cracked; after c rack ing , the shell cont inued to carry a substant ia l por t ion of the load , a l though in some cases there was a sl ight decrease. SPECIMEN 1 2 3 4 AVG COV M9 (NG, MJ)* 333.0 333.0 310.5 325.5 3.3% S8 (NG, SJ) 303.0 264.0 321.0 296.0 8.0% N13 (NG, NJ) 237.0 332.0 284.5 16.7% N10 (NG, t 0=6/8") 302.0 300.0 273.5 291.8 4.5% P l l (NG, t0 = 0) 274.0 234.0 312.5 273.5 11.7% N17(NG,Face-Shell) 252.0 240.0 258.0 250.0 3.0% Table 8.1 Failure Loads of Grouted Prisms (kips), with Variation in Joint Condition * NG - Type N Grout; NJ - Type N Mortar Joint, etc. SPECIMEN 1 2 3 4 A V G COV N12 (SG, NJ) 316.0 291.0 254.0 287.0 8.9% N13 (NG, NJ) 237.0 332.0 284.5 16.7% N14 (WG, NJ) 257.0 241.0 289.0 262.3 7.6% Table 8.2 Failure Loads of Grouted Prisms (kips), with Variation in Grout I 0.9 0.8 0.7 O . S 0.4 0.3 0.2 • I • • • • • • • 1 1 • • I • GROUT 1 1— 1 1 1 1— • MORTAR 0.2 0.4 0.6 0.8 1 1.2 1 4 1.8 1.8 QROUT STRENGTH. MORTAR CUBE STRENGTH / UMT STRENGTH FIG. 8.1 Grouted Prism Strength Versus Mortar Strength and Grout Strength 132 8.3 A n a l y s i s A l t h o u g h there is considerable scatter in the strength d a t a obta ined by the author and in numerous previous studies, one conclusion can be d rawn w i t h certa inty : that the capac i ty of the b lock uni t and that of the grout are not s imp ly add i t ive . T h i s obv ious ly results f r o m the difference i n de format ion properties of the mater ia ls , as discussed i n the preceding paragraph. W e consider two aspects of this deformat ion c o m p a t i b i l i t y p rob lem: F i r s t , ve r t ica l c o m p a t i b i l i t y . Since the grout and the block un i t usual ly have different peak strains, as shown by exper iment , they are not able to reach their f u l l capacit ies at the same st ra in . F r o m this v iewpoint i t is obvious that s imple capaci ty add i t ion is not v a l i d . T h e efficiency of the grout ing w i l l depend on how close the deformat ion properties of the two mater ia ls are. Second, hor i zonta l (or cross-sectional) c o m p a t i b i l i t y . T h i s includes two parts . One is due to the different Poisson 's effect of grout and block shel l . T h e other is due to the geometry: for manu fac tu r ing reasons, the ho l low core of a concrete masonry block is actua l ly tapered w i t h a release angle 1° —3° . T h i s m a y introduce an add i t iona l cross-sectional c o m p a t i b i l i t y p rob lem when grout and block shel l undergo different ver t ica l strains. T h e premature c rack ing observed i n exper iments is certa in ly caused by these cross-sect ional i ncompat ib i l i t i es . T h u s , a fa i lure mode l of grouted masonry based on deformat ion c o m p a t i b i l i t y w i l l be closer to rea l i ty than one based on strength superposit ion. T h i s w i l l serve as a guidel ine for the fo l lowing mode l development. It m a y be useful to l ist a l l the notat ion appl ied in the model : Ag, An — 2a, 26 = Eu, Eg, Ej = gross area and net area of b lock uni t , respectively; w i d t h of b lock inner core and block uni t , respectively; modu lus of e last ic i ty of block un i t , grout, mor ta r j o in t , respectively; 133 fmp, fmg = compressive strength of p la in masonry and grouted masonry , respectively; fu, fg, fj = compressive strength of b lock un i t , grout (pr ism strength) , mor ta r (cube strength) , respectively; fut = tensile strength of b lock un i t ; h0 = height of block unit ; m 1 ? m 2 = modu la r rat io of Eu/Eg and Eu/Ej, respectively; p = contact pressure between grout and block shel l ; t0 = thickness of mor ta r j o in t ; a = release angle of b lock inner core; eu, eg, €j = compressive st ra in in block uni t , grout and mor ta r j o in t , respectively; 77 = net area to gross area rat io of b lock uni t An/Ag; Vn-, Vg, vj = Poisson's rat io of block uni t , grout and mor tar , respectively; <rm, CTS = compressive stress in masonry (average) and in masonry shel l , respectively; <7 U , (Tg, (Tj = compressive stress in b lock un i t , grout and mor ta r jo in t , respectively; aut = la tera l tensile stress in b lock un i t ; In general, the capac i ty of grouted masonry depends on m a n y factors, most i m p o r t a n t l y : * the strength of the mater ia ls fu, fut, fg, fj * the deformat ion properties of the mater ia ls E«, Eg, Ej, vu, vg, vj * geometr ic properties such as the shape of the block units , the thickness of the mor ta r j o in t , bond pat tern , etc. * w o r k m a n s h i p , test method T o make the mode l p ract ica l l y useful, we neglect those effects wh ich are not easy to quant i fy , such as w o r k m a n s h i p or test method . W e w i l l also t ry to avo id exp l ic i t l y i nc lud ing the deformat ion properties i n the model , since they are d i f f icul t to measure. Fu r ther , we use TJ, the 134 net to gross cross-sectional area ratio and a, the inner core release angle to characterize the geometry of a block unit. We will assume that the grout core is approximately square, and thus the model may be generally useful for grouted hollow concrete masonry with various dimensions. It may be useful in the following derivation to first find some simple approximate relations between rj and the dimensions of a masonry unit. By the definition and the assumption stated, we can write „ = 1 - ( f ) 2 8.1 or -f = J 1 - r, 8.2 In the derivation, an expression for {b—a)/a is needed. Eq. 8.1 can be rewritten as b - a _ V b 1 + a/b When a/b is expanded as 1— n/2 in view of Eq. 8.2, the above expression becomes 8.3 which gives good approximations even when r\ is as large as 0.6. Based on Eq . 8.3, it is easy to obtain » . 2 \ 8.4 o 4 — 3n Although the determination of the stress state in grouted masonry is actually a three dimensional problem, which is further complicated by the inelastic behaviour of the materials, 135 for the sake of s imp l ic i t y and pract ica l i ty , we adopt a quasi -e last ic approach. T h a t is, we use the theory of elast icity and i m p l i c i t l y assume that the deformat ion properties involved are taken as secant or effective values. Further , we assume that stress and s t ra in in the mater ia ls are un i fo rm, or, i n other words we treat the stress and s t ra in in an average sense in each mater ia l . F o r the p r ism shown in F i g . 8.5, the fo l lowing relations can be wr i t ten . A ) In Ver t ica l D i rect ion E q u i l i b r i u m : <Tm = r]<J, + (l-Tj)crg 8.5 If the shear force between the block shell and grout is neglected, we have 136 Ca = cr u = cr j C o m p a t i b i l i t y : ~ h0 + U ~ C u + U , JC> 8.6 Stress — S t ra in Re la t i on : F o r grout we have eg = ' r y f 8.7 F o r the block shel l , an expression for the ver t ica l s t ra in due to the contact pressure p is needed. T h i s can be obta ined by Be t t i ' s law. Referr ing to F i g . 8.5, we have A(2a)h0p 6(crs) = Ancrsho eu(p) where S(crs) is the latera l d isplacement due to the ver t ica l stress, expressed as CTsvua Eu a n d e u (p ) is the ver t ica l s t ra in i n the uni t due to the latera l pressure p. U p o n subst i tu t ion and rearranging, one f inds _ 2pi/ u(l-I?) T h u s the ver t ica l s t ress—st ra in re lat ion of the block shel l is 137 _ v> + 2p i / u ( l -»?) / t? tu — o o.o The stress—strain relation for the mortar joint is B) In Lateral Direction Equilibrium: 2ap = 2(6—a) crut or ap w (4-3r/)p 6—a ~ 277 in which the relation given by Eq. 8.4 has been used. Compatibility: (ugcrg-p)a _ (<Tui + <Tsl'u)a Eg EU 8.10 where we assume that the lateral deformation of the block unit is the sum of the Poisson's effect and the stretch due to the lateral tensile stress in the block unit. If the lateral deformation due to the tapered core is included, which may be modeled as the grout acting as a wedge being driven into the block core, the compatibility condition can be rewritten as « (4r) ( e f - e . ) + iV'TP) = 8 .11 138 T h e above seven equations can be used to determine the stress a n d s t ra in state i n a grouted masonry p r i s m when a ver t ica l stress or s t ra in is imposed. C ) Fa i l u re C o n d i t i o n s There are several possible ways for a grouted concrete masonry assembly to fa i l . T h e y inc lude chief ly : a) P rematu re sp l i t t i ng of the block shell due to the i ncompat ib le mate r ia l propert ies of the grout and the block, wh ich give rise to tensions. in the shel l , b) If the assembly survives th is cond i t i on , fa i lure m a y occur when the s u m of the shel l and grout resistances reaches a m a x i m u m , at a deformat ion between their respective peak strains, c) T h e assembly m a y fa i l when the grout reaches its fu l l capaci ty ; at this po int the substant ia l vo lume increase due to the in te rna l c rack ing of the grout causes the block shell to f a i l . W h i c h e v e r the case, the lower bound of the masonry strength should be a lways sat isf ied: fmg > ( 1 - r, ) fg 8.12 w h i c h corresponds to the fa i lure load being carr ied by the grout alone. F o r n o r m a l range of n, fa i lure cond i t ion c) requires that the peak s t ra in of the grout be reached f i rst . However this is un l ike ly to be the case, i n view not on ly of the exper iments by the author a n d of those done in New Zealand, wh ich have ind icated that the concrete block units were stiffer than grout, but also of the exper imenta l observations by several other researchers that the grout core was in tact even after masonry specimens had fa i led (Drysdale and H a m i d 1979; Ha t z in iko las et a l 1978). Fa i l u re cond i t ion a) , of course, is governed by the block shel l . However , to determine the fa i lure load for cond i t ion b) , a knowledge of the deformat ion properties of the mater ia ls over entire st ra in range is needed. Since this i n fo rmat ion is d i f f icul t to establ ish, as a p ract ica l a l ternat ive , we m a y inquire wh ich mater ia l , b lock shel l or grout, is closer to its f u l l capaci ty at the point of fai lure of the assemblage. In the l ight of the study i n Chapte r II, one m a y tend to 139 believe that the block shell, which is formed by fine aggregate concrete, would be "less ductile" than the grouting concrete in the post—peak range; thus the failure strain of masonry would be closer to the peak strain of the block shell, if the latter is assumed to be the stiffer component. Or in other words, masonry is likely to fail immediately after the full capacity of the block shell is reached, because the stress in the block shell will decrease drastically once its peak strain is exceeded. Although it is difficult to justify this assumption directly by experimental observation, it may be verified statistically, i.e. by correlating the masonry capacity with the block shell strength and with the grout strength. A multiple linear regression on the experimental data from 7 different sources (including the experiments conducted by the author) indicates that the masonry capacity is much more closely correlated to the block unit strength. Thus the assumption is supported by the statistical implications. Of course, this also strengthens the argument that failure condition c) is unlikely to occur. Therefore, whether grouted masonry fails by condition a) or b), it is reasonable to assume the block shell will govern the failure state. We may only consider the solution for failure condition a), since condition b) may be included as a particular case. If the Coulomb-Navier failure criterion is assumed, one writes mp + = 1 8.13 Eq. 8.5 to Eq. 8.11 with Eq. 8.13 can be used to find the 8 unknowns; namely C u , £ €j, <Ts, ag, <rm, V a n d The capacity of the grouted masonry, in terms of the masonry strength fmg, is equal to <rm, since once Eq. 8.13 is included, the group of equations is actually solved at the critical condition. Upon substituting and neglecting higher order terms, we get fmg — [77 + ( l - 7 7 ) ( l + m 2 ^ ) i ] / m p fmp fut 140 8.14 It is interest ing to examine the phys ica l in terpretat ion of this so lut ion . T h e numerator represents the capac i ty of grouted masonry determined by consider ing on ly the ver t ica l c o m p a t i b i l i t y of the deformat ion properties. It means that w i thout the difference i n lateral properties, w h i c h w i l l lead to fai lure condi t ion b), the fa i lure load of grouted masonry wou ld be composed of two parts . One is the p la in masonry strength t imes the net area of the block uni t . T h e other is the product of the stress in the grout, wh ich , due to the difference in stiffness, reaches 1/TOJ t imes the stress in the block shel l , w i th the grouted area. T h e factor, m2to/h0 takes account of the "softer" mor ta r j o in t , i nd ica t ing that the latter tends to increase the stress i n the grout. T h e denominator accounts for the cross-sectional c o m p a t i b i l i t y . It can be seen that the masonry strength fmg is an increasing funct ion of 77, i nd ica t ing that the cross-sectional i n c o m p a t i b i l i t y becomes less impor tant as the thickness of the block shell increases. T h e b ig square bracket contains some very s m a l l quant i t ies. T h e te rm (vg — vu), w h i c h is i m p l i c i t l y assumed to be greater than or equal to zero in the der ivat ion , represents the i n c o m p a t i b i l i t y due to the difference i n Poisson's effect of the two mater ia ls . T h e te rm m2a(t0/a) accounts for the i n c o m p a t i b i l i t y caused by the taper ing of the core. T h e te rm m2vg(t0/'h0) impl ies the effect on the i n c o m p a t i b i l i t y of the softer mor ta r jo in t , wh ich needs more detai led exp lanat ion . Since the mor ta r is usual ly m u c h softer than the block uni ts , as ind icated by exper iment , the grout is actua l l y strained more than the block uni t in the ver t ica l d i rect ion , due to the presence of the j o in t . T h u s even i f the block uni t and grout have the same value of Poisson's rat io , the grout w i l l expand more, latera l ly , than the block shel l , causing add i t i ona l cross-sectional i n c o m p a t i b i l i t y . It is clear that the masonry strength fmg is a, decreasing funct ion of these 3 terms appear ing in the square bracket of the denominator . 141 Obv ious l y , the denominator wou ld reduce to un i t y i f there were no latera l c o m p a t i b i l i t y effect. In other words, the numerator predicts the u l t imate fai lure load of fa i lure cond i t ion b) when the p r i s m survives fa i lure cond i t ion a). ( T o make the mode l p ract ica l l y useful, some s impl i f icat ions are necessary. Since the term m2t0/h0 is usual ly s m a l l compared w i t h 1, e.g., for s tandard 8 inch block units , i0/h0 is smal ler t h a n 0.05 and m2, the secant modu la r ra t io is around 3 according to the author 's exper iments (see F i g . 2.5 i n C h a p t e r II), we m a y neglect i ts var ia t ion by assuming (l + m2t0/h0) i n the numerator to be a constant s l ight ly bigger than un i ty . B y a s i m i l a r argument , the te rm ?7(3 — 2 m x ) in the denominator can be neglected since i t is s m a l l compared w i t h the te rm 4. T h e va r ia t i on of the te rm at0/a m a y also be ignored since i t is s m a l l compared w i t h isg(t0/h0); and for the geometry of a standard 8 inch block these two terms give a p p r o x i m a t e d 0.015. A l t h o u g h it appears that the cross-sectional i n c o m p a t i b i l i t y wou ld be m a i n l y caused by the lateral expansion due to Poisson 's effect on the grout, as believed by some researchers (Drysdale and H a m i d 1979), we m a y neglect the va r ia t ion of the te rm (yg — uu) by replacing it w i t h a constant t , say, not exceeding 0.1. Due to the difference i n the stiffness between grout and block shel l , the block shel l tends to be more stressed at fai lure. T h i s view is also supported by the s tat is t ica l a rgument stated above that masonry capac i ty is governed more by the block shel l . T h u s the effective vg w i l l not increase as m u c h as v'u a round the c r i t i ca l state due to in terna l c rack ing . T h e last te rm fmplfuu the rat io of the block shel l strength to b lock un i t tensile strength, may be assumed to be a constant £ in the order of 10. T h u s for the geometry of a s tandard 8 inch block, E q . 8.14 can be s imp l i f i ed to _ ( n + (1 -^ )4 )fmp fmg — 4 _ Q n 8.15 1 + C ( * + 0 .015m 2 ) ?—p-E q . 8.15 is based on fracture of the block shel l , wh ich m a y or m a y not lead to collapse of the masonry assemblage, as discussed earlier. T h u s E q . 8.15 predicts the load for fa i lure 142 cond i t ion a) : f racture of b lock shell leads to f i na l fa i lure. However i f the assembly survives this cond i t ion , the u l t i m a t e fa i lure load is given by the numerator of E q . 8.15 by neglecting cross-sect ional i n c o m p a t i b i l i t y . E q . 8.15 then corresponds to merely the crack ing load of the block shel l . F r o m the exper iments conducted by the author and by numerous other researchers, i t appears that either fa i lure cond i t ion can occur. T h i s poses the p rob lem i n pract ice as to wh ich so lut ion should be used i n pred ict ing masonry capaci ty . T h i s quest ion, aga in , m a y only be answered in a s ta t i s t ica l sense. W e m a y examine the ava i lab le exper imenta l d a t a to see whether one of the two fai lure condit ions has a p robab i l i t y of occurrence h igh enough to dominate the fa i lure mode. T o correlate the ava i lab le exper imenta l da ta , expressions for fmp, m1 a n d m 2 are needed. In v iew of the study i n Chapte r III, for n o r m a l j o i n t thickness, the expression for fmp m a y be taken in the f o r m as given by E q . 4.28. i.e. fmp = cju + c2fj 8.16 Fur ther , the modu la r rat ios m a y be related to the strength values as and 8.17 8.18 where c l 5 c2, kx a n d k2 are constants. T h e square root re lat ion between the strength and the modu lus of e last ic i ty is adopted by m a n y bu i ld ing codes. W e proceed to give an est imate for the constants invo lved in these relat ions. W e w i l l do this based on the 77 ava i lab le d a t a points f r o m 7 different sources (Presents tests; H a m i d and 143 Drysda le 1978; B o u l t 1979; Drysda le and H a m i d 1979; T h u r s t o n 1981; Pr iest ley and E lder 1982, 1985; W o n g and Drysda le 1983). These actua l l y include m a n y more than 77 specimens because several of these d a t a points were reported as average values. T h e d a t a are s u m m a r i z e d in Tab les 8.3 — 8.9. T h e New Zealand results include fj values based on pr isms. These have been converted to equivalent cube strength using the THermite equat ion (Nevi l le 1965). T h e k values i n E q . 8.17 and E q . 8.18 should make the equations y ie ld the average values of m1 and m 2 when the fu, fg and fj take their mean values. A c c o r d i n g to the d a t a , the rat ios of the average uni t strength to grout strength and to mor ta r strength are 1.04 and 1.5 respectively. T h e average value of mu accord ing to the exper imenta l results by the author and by the New Zealand researchers, m a y be taken as 1.32. T h e mean value of m 2 m a y be taken to be 3 as ment ioned earl ier. T h i s leads to kx = 1.29 and k2 = 2.54. Fu r the r c1 and c 2 may be awarded the values g iven by E q . 4.28. F o r fa i lure cond i t ion a) E q . 8.15 is used whi le for fa i lure cond i t ion b) on ly the numerator of the equat ion is app l ied . T h e results are also summar i zed in T a b l e 8.3 — 8.9. It appears that predicted fai lure loads based on fai lure cond i t ion a) substant ia l l y underest imate those obta ined by experiments. T h e results based on cond i t ion b), however, correlate wel l w i t h the exper imenta l data , a l though it appears that they overest imate strength in the lower range. T h e corre lat ion coefficient for the former is 0.894, whi le for the latter is 0.918. These results are p lo t ted in F igs . 8.6 and 8.7 as predict ions versus exper iments. It is clear that the difference between the two methods is s igni f icant . T h u s , i t m a y be concluded, based on the above study and on the ava i lab le exper imenta l d a t a f rom var ious sources that , under no rma l construct ion condit ions, the strength of grouted masonry is m a i n l y governed by the vert ica l c o m p a t i b i l i t y of the grout and block shel l . Fur ther , since the block shel l is stiffer i n the pre-peak range of s t ra in , and less duct i le in the post-peak range than grout, the masonry w i l l tend to fa i l when the fu l l capac i ty of the block shell is reached; thus the capac i ty of masonry is more closely correlated w i t h b lock un i t strength than 144 w i t h grout strength. It should be noted that the above conclusion does not e l iminate the poss ib i l i ty that the fa i lure m a y occur i n the f o r m of cond i t ion c), and , especially cond i t ion a). It on ly means that fa i lure cond i t ion b) has a predominant p robab i l i t y of governing. V fu 4 fmg a b c 0.51 3.25 1.57 3.70 1.97 1.57 2.30 2.22 0.51 3.25 1.57 3.70 2.76 1.57 2.30 2.22 0.51 3.25 4.00 3.70 2.52 1.97 2.72 2.52 0.51 3.25 4.00 3.70 2.20 1.97 2.72 2.52 0.51 3.25 4.00 3.70 2.67 1.97 2.72 2.52 0.51 3.25 4.69 3.70 2.77 2.07 2.84 2.61 0.5.1 3.25 4.69 3.70 2.77 2.07 2.84 2.61 0.51 3.25 4.69 3.70 2.58 2.07 2.84 2.61 0.51 3.25 1.57 5.02 2.63 1.68 2.46 2.48 0.51 3.25 1.57 5.02 2.42 1.68 2.46 2.48 0.51 3.25 1.57 5.02 2.11 1.68 2.46 2.48 0.51 3.25 1.57 3.33 2.14 1.53 2.24 2.14 0.51 3.25 1.57 3.33 2.00 1.53 2.24 2.14 0.51 3.25 1.57 3.33 2.40 1.53 2.24 2.14 T a b l e 8.3 G r o u t e d P r i sms , Tests by the A u t h o r fmg — E x p e r i m e n t a l value of p r i s m strength a — Theore t ica l pred ict ion of p r i s m strength by fa i lure cond i t ion a) b — Theoret ica l p red ict ion of p r i sm strength by fai lure cond i t ion b) c — Theoret ica l p red ict ion of p r i s m strength by fai lure cond i t ion c) ( A l l i n ks i , same for the fo l lowing tables) 145 n A fmg a b c 0.62 2.85 2.06 1.80 1.51 1.45 1.99 1.61 0.62 2.85 2.06 1.80 1.55 1.45 1.99 1.61 0.62 2.85 2.06 1.80 2.01 1.45 1.99 1.61 0.62 2.85 2.06 1.80 1.45 1.45 1.99 1.61 0.62 2.85 2.06 1.80 1.67 1.45 1.99 1.61 0.62 2.85 2.63 2.07 1.77 1.57 2.12 1.75 0.62 2.85 2.63 2.07 1.78 1.57 2.12 1.75 0.62 2.85 2.63 2.07 1.67 1.57 2.12 1.75 0.62 2.85 2.63 2.07 1.78 1.57 2.12 1.75 0.62 2.85 0.82 2.07 1.49 1.25 1.82 1.48 0.62 2.85 0.82 2.07 1.59 1.25 1.82 1.48 0.62 2.85 0.82 2.07 1.43 1.25 1.82 1.48 0.62 2.85 0.82 2.07 1.51 1.25 1.82 1.48 0.62 2.85 2.29 2.52 1.83 1.56 2.13 1.79 0.62 2.85 2.29 2.52 1.86 1.56 2.13 1.79 0.62 2.85 2.29 2.52 2.06 1.56 2.13 1.79 0.62 2.85 2.29 2.52 1.75 1.56 2.13 1.79 0.62 2.85 2.29 2.52 1.78 1.56 2.13 1.79 0.62 2.85 1.95 3.65 2.12 1.60 2.20 1.94 0.62 2.85 1.95 3.65 1.96 1.60 2.20 1.94 0.62 2.85 1.95 3.65 1.78 1.60 2.20 1.94 0.62 2.85 1.95 3.65 1.90 1.60 2.20 1.94 0.62 2.85 1.95 2.05 1.77 1.46 2.01 1.65 0.62 2.85 1.95 2.05 1.78 1.46 2.01 1.65 0.62 2.85 1.95 2.05 1.67 1.46 2.01 1.65 0.62 2.85 1.95 2.05 1.78 1.46 2.01 1.65 0.62 2.85 1.97 5.52 1.99 1.74 2.38 2.20 0.62 2.85 1.97 5.52 2.30 1.74 2.38 2.20 0.62 2.85 1.97 5.52 2.28 1.74 2.38 2.20 0.62 2.85 1.97 5.52 2.23 1.74 2.38 2.20 T a b l e 8.4 G r o u t e d P r i sms , Tests by H a m i d a n d Drysda le V /« A fmg a b c 0.62 2.85 2.50 2.21 1.64 1.57 2.12 1.76 0.62 2.85 0.83 2.53 1.51 1.29 1.88 1.58 0.62 2.85 2.06 2.21 1.64 1.50 2.05 1.70 0.62 2.85 2.64 2.53 1.75 1.62 2.19 1.85 0.62 2.85 2.29 3.09 1.86 1.62 2.20 1.90 0.62 2.85 1.96 4.48 1.94 1.67 2.29 2.06 0.62 2.85 1.96 2.52 1.75 1.51 2.07 1.74 0.62 2.85 1.97 6.85 2.20 1.81 2.49 2.37 0.59 4.67 2.06 2.87 2.45 2.10 2.99 2.73 0.59 4.67 2.06 2.87 2.38 2.10 2.99 2.73 0.70 3.19 2.06 3.19 1.91 1.79 2.39 1.99 0.69 3.08 2.06 3.19 2.05 1.74 2.32 1.94 0.63 2.92 2.06 2.87 1.76 1.59 2.17 1.85 0.73 2.90 2.06 3.19 2.13 1.71 2.24 1.81 0.61 2.27 2.06 3.10 1.34 1.37 1.86 1.56 0.62 2.85 1.86 2.39 1.73 1.48 2.04 1.70 0.62 2.85 1.86 2.39 1.93 1.48 2.04 1.70 T a b l e 8.5 G rou ted P r i s m s , Tests by Drysdale H a m i d V fu u /, fmg a b c 0.51 2.78 2.72 4.93 2.16 1.75 2.45 2.35 0.51 2.78 2.72 4.93 2.10 1.75 2.45 2.35 T a b l e 8.6 G r o u t e d P r i s m s , Tests by W o n g and Drysdale 147 V fu h fmg a b c 0.55 5.54 2.24 4.03 3.91 2.43 3.55 3.47 0.55 5.54 2.24 4.03 3.77 2.43 3.55 3.47 0.55 5.54 2.24 4.03 3.93 2.43 3.55 3.47 0.61 5.54 1.70 5.30 3.90 2.54 3.70 3.59 T a b l e 8.7 G rou ted P r i s m s , Tests by Pr iest ley and E lde r 1 fu 1 fmg a b c 0.55 5.80 2.20 2.25 2.61 2.28 3.34 3.07. 0.55 5.80 2.20 2.25 2.77 2.28 3.34 3.07 0.55 5.80 2.20 2.25 2.99 2.28 3.34 3.07 0.48 5.28 2.20 2.25 2.61 1.99 2.98 2.84 0.48 5.28 2.20 2.25 2.58 1.99 2.98 2.84 0.48 5.28 2.20 2.25 2.99 1.99 2.98 .2.84 T a b l e 8.8 G rou ted P r i s m s , Tests by B o u l t V fmg a b c 0.52 2.41 2.79 3.75 2.12 1.54 2.13 1.93 0.52 2.41 2.79 3.75 2.16 1.54 2.13 1.93 0.61 2.83 2.79 3.75 1.68 1.74 2.35 2.07 0.54 4.12 2.79 3.75 2.70 2.08 2.93 2.80 T a b l e 8.9 G r o u t e d P r i s m s , Tests by T h u r s t o n 148 PREDICTED FAILURE STRESS (KSI) - FAILURE CONDITION a) F I G . 8.6 M o d e l Pred ict ion versus Exper iments , Based on Fa i lu re C o n d i t i o n a) 1 1.4 1.8 2.2 2.6 3 3.4 3.8 PREDICTED FAILURE STRESS (KSI) - FAILURE CONDITION b) F I G . 8.7 M o d e l Pred ict ion versus Exper iments , Based on Fa i lu re C o n d i t i o n b) 149 Therefore, i t is not surpr is ing to see that the pred ict ion based on cond i t ion b) appears to overest imate the masonry capac i ty i n the lower strength range, since i f masonry fai ls i n cond i t ion a), i.e. as the result of the premature fai lure of block shel l , i t w i l l lead to a lower fa i lure l oad . It is clear that a l though we cou ld use the numerator of E q . 8.15 d i rect ly to est imate the grouted masonry capac i ty based on un i t strength, grout strength, mor ta r strength and area rat io , some discrepancy should be expected since occasional ly fa i lure condi t ions other than cond i t ion b) m a y occur. Moreover , i t is not desirable i n pract ice to overest imate the masonry capac i ty . Therefore the equat ion m a y need empi r ica l mod i f i ca t ion . One mod i f i ca t ion m a y be to adjust the coefficient i n the equat ion to best fit the ava i lab le exper imenta l da ta . Subs t i tu t ing Eqs . 8.16-8.18 and neglecting s m a l l quant i t ies , the numerator of E q . 8.15 m a y be expanded in the fo rm of fmg = Anfu + B(\-r))\fgfu + Cnfj + D ( in ksi) 8.19 A m u l t i p l e regression analys is of the d a t a gives: A = 0.53 B = 0.94 C = 0.24 D = -0.45 E q . 8.19 together w i t h the lower bound given by E q . 8.12 m a y be used to give an est imate for the u l t imate capac i ty of grouted masonry . T h i s est imate is also l isted i n T a b l e 8.3 — 8.9 and plot ted i n F i g . 8.8 versus the d a t a base. T h e re lat ion has a correlat ion coefficient of 0.934, which is s igni f icant . However , the d a t a used to evaluate parameters certa in ly do not a l l refer to fai lure 150 cond i t ion b) , and the corre lat ion is affected by add i t i ona l uncertaint ies such as w o r k m a n s h i p and test method ; and this is p robab ly why a number of points fa l l outside the 99 percent confidence l i m i t (see F i g . 8.8). T h e mode l equat ion clearly reflects the fact that masonry capac i ty is not very sensit ive to the grout strength, as observed by Drysdale and H a m i d (1979) and by the author (see Tab les 8.3, 8.5). T h e masonry strength is better correlated w i t h the square root of the grout strength, based on the deformat ion c o m p a t i b i l i t y . Indeed, l inear regression on the basis of E q . 8.19 in wh ich \ fgfu is replaced by fg, a f o r m often seen i n l i terature, indicates that i t leads not on ly to a lower corre lat ion coefficient of 0.907 but also to a m u c h higher D value, wh ich is not desirable. T h e above analysis suggests that E q . 8.15 m a y be used to est imate the c rack ing load of concrete grouted masonry . Unfor tunate ly , no exper imenta l d a t a are ava i lab le in terms of c rack ing loads except those recorded by the author . F o r these very l i m i t e d da ta , the compar ison is l isted in T a b l e 8.10 a n d p lotted i n F i g . 8.9, in which E q . 4.16 is scaled d o w n by a factor of 0.92. T h e c rack ing loads for the specimens w i t h vary ing jo in t thickness are also inc luded. Except for two d a t a points (S8) the agreement is reasonable, consider ing the crack ing load is a rather r a n d o m event. T h e corre lat ion coefficient for this case is 0.618, whi le pred ict ion of fa i lure loads it is 0.563, i nd ica t ing that the load est imated by E q . 8.15 is indeed more closely correlated w i th the c rack ing load than w i t h the u l t imate load . One p ract ica l i m p l i c a t i o n of the above study is that one should consider the crack ing load est imated by E q . 8.15 as the lower l i m i t load in design, since block shel l c rack ing is, in any case, not a desired event under n o r m a l service condit ions. T h i s usual ly can be achieved, since, in most s m a l l masonry bui ld ings , the ax ia l load levels are low and therefore the ac tua l value of a l lowable stress is not c r i t i c a l . (It is noted that the crack ing load est imated by E q . 8.15 is a round 7 0 % of the u l t imate load est imated by E q . 8.19 or by the numerator of E q . 8.15.) However , the u l t imate load est imated by the model can be used as the final l i m i t load under severe service condi t ions . Fo r example , under earthquake load ing , the a x i a l capac i ty of masonry 151 1 1.4 1.8 2.2 2.6 3 3.4 3.8 PREDICTED FAILURE STRESS (KSI) - REGRESSION FORMULA F I G . 8.8 Mode l P red ic t ion versus Exper iments , Based on Mod i f i ed Equat ion 1.2 1.4 1.6 1.8 2 2.2 2.4 PREDICTION BASED ON CONDITION a) (KSI) F I G . 8.9 Mode l P red ic t ion versus C r a c k i n g Loads, Based on Fa i lu re C o n d i t i o n a) 152 can become c r i t i ca l not on ly because of the d u c t i l i t y requirement but also because of the ine r t ia force itself. One m a y then take advantage of the higher u l t imate strength by a l l owing a higher a l lowable stress based on E q . 8.19. T h i s is economical and certa in ly agrees w i t h the risk ph i losophy c o m m o n l y adopted in the earthquake engineering design, that some damage, even s t ructu ra l damage, is acceptable i n the major event, but not col lapse. F i n a l l y , of course, the v a l i d i t y of E q . 8.19 as a predictor of c rack ing loads needs further invest igat ion . M a n y more exper imenta l d a t a are required in th is respect. 8.4 S u m m a r y In this chapter , the a x i a l behaviour of grouted concrete masonry w i th fu l l bedding has been invest igated. Three possible fa i lure condi t ions have been studied. A fa i lure mode l based on in terna l de format ion compat ib i l i t i es has been proposed. S P E C I M E N u l t . load(kips) crk. load(kips) fmg (ksi) / c * ( k s i ) Pred ic t i on (ksi) S8-1 303.0 120.0 2.52 1.00 1.90 S8-2 264.0 130.0 2.20 1.08 1.90 N13 -3 237.0 155.0 1.97 1.29 1.56 N13 -4 332.0 160.0 2.76 1.33 1.56 M 9 - 1 333.0 250.0 2.77 2.08 1.99 M 9 - 2 333.0 200.0 2.77 1.66 1.99 N12 -3 291.0 220.0 2.42 1.83 1.68 N12-4 254.0 180.0 2.11 1.50 1.68 N14 -3 241.0 187.0 2.00 1.55 1.52 N14-4 289.0 190.0 2.40 1.58 1.52 N10 -3 300.0 180.0 2.49 1.50 1.46 N10 -4 273.0 200.0 2.27 1.66 1.46 P l l - 1 302.0 190.0 2.51 1.58 1.70 P l l - 2 300.0 208.0 2.49 1.73 1.70 T a b l e 8.10 M o d e l P red ic t ion versus C r a c k i n g Loads , Tests by the A u t h o r 153 C H A P T E R I X G R O U T E D M A S O N R Y W I T H F A C E - S H E L L B E D D I N G It is clear by the analysis i n Chapte r V , that under u n i a x i a l compression face-shell bedded masonry w i l l fa i l p remature ly by a deep beam mechan ism at a low load . W h e n face-shell bedded masonry is grouted, the deep beam bending mechan ism w i l l s t i l l be act ivated as the ver t ica l force w i l l be shared by the block shel l and grout. T h i s was shown by the exper iments conducted by the author (see F i g . 9.1). T h e webs of the face-shel l bedded and grouted pr isms cracked ver t ica l ly at a very low load owing to this mechan ism. T h e ver t ica l s t ra in in the block shel l drops m u c h faster than that of the fu l ly bedded counterparts , i m p l y i n g the h ing ing mechan ism of the block shell studied in Chapte r V . However, the author 's tests ind icate that the crack ing of the block shel l due to the beam bending mechan ism w i l l not lead to u l t imate fa i lure of the masonry i f the residual capac i ty of the grout is greater than the crack ing load . T h u s we m a y use fm, = ( 1 - V ) fa 9.1 as a lower bound or as a conservative est imate of the grouted masonry compressive strength. F o r the author 's tests, E q . 9.1 underestimates the p r i s m capac i ty by about 10%, as shown i n T a b l e 9.1, i nd ica t ing a very low grout ing efficiency. A t fai lure, the load was only effectively sustained by the grout, as i m p l i e d by the deformat ion measurement (see F i g . 9.1). E q . 9.1 underestimates the fa i lure loads of the pr isms tested by Drysda le and H a m i d (1983) by a larger m a r g i n , i nd ica t ing a higher grout ing efficiency in their specimens. However , i t seems reasonable i n p ract ica l design to neglect the capaci ty of the block shell since this may not be a rel iable q u a n t i t y in view of the beam bending mechan ism. 154 S P E C I M E N V fg (ksi) P (kips) fmg (ksi) (l-ri)fg N17 0.51 3.70 252 2.09 1.81 A ) N - G R O U T 0.51 3.70 240 2.00 1.81 N - M O R T A R 0.51 3.70 258 2.14 1.81 N B G N 0.56 3.06 123 2.09 1.36 N B G W 0.56 1.99 121 2.05 0.88 N B G S 0.56 5.94 131 2.22 2.64 B ) W B G N 0.56 3.06 93.5 1.59 1.36 S B G N 0.56 3.06 128 2.18 1.36 Q B G N 0.75 3.06 124 2.10 0.76 6 " B G N 0.51 3.06 86.6 1.99 1.50 1 0 " B G N 0.54 3.06 123 1.65 1.41 T a b l e 9.1 G rou ted Masonry w i t h Face -She l l Bedd ing Tests by the author Tests by Drysdale and H a m i d (1983) T h e p rob lem that remains unanswered is whether the crack ing load should be used to govern the design. If so, more exper imenta l work is needed and more at tent ion should be directed to this value, since there have so far been few exper iments mon i to r i ng premature c rack ing . A c c o r d i n g to the author 's tests, face-shell bedded, grouted masonry has a very low grout ing eff iciency, wh ich m a y be even lower i n terms of the crack ing loads. T h i s is because the two const i tuents do not work together properly. It appears that in the early stages of load ing , the b lock un i t takes a b ig share of the load as i m p l i e d by the vert ica l s t ra in measurements (cf. F i g . 9.1 and F i g . 5.3, locat ions 5 and 6). However, after the block shell is cracked, a lmost the whole load is passed to the grout. T h i s is not desirable f r o m a s t ructura l po int of v iew. It is clear that for grouted masonry f u l l bedding is recommended, a l though , as ind icated A ) -B ) -155 above, in pract ice that effective f u l l bedding is sometimes d i f f icu l t to achieve because of the web a l ignment . It is also obvious that the deformat ion properties of the two mater ia ls p lay an i m p o r t a n t role. L o w grout ing efficiency is inevi table unless there is a f undamenta l improvement i n m a t e r i a l design such that the deformat ion properties of grout a n d uni t are more compat ib le . C H A P T E R X 157 G R O U T E D A N D R E I N F O R C E D M A S O N R Y U N D E R E C C E N T R I C L O A D I N G 10.1 Genera l R e m a r k s P r o b a b l y the biggest advantage of concrete masonry over t rad i t i ona l b r ickwork is that the concrete block un i ts are ho l low and can thus be ver t ica l l y reinforced to improve the bending capac i ty . B e n d i n g capac i ty is essential for wal ls designed to sustain eccentric load or ver t ica l force comb ined w i t h latera l ly d is t r ibuted pressure. T h i s is obvious since theoret ical ly the capac i ty of p la in b r ickwork w i l l be drast ica l ly reduced i f the load fal ls outside the kern , a n d the w a l l can not support any load when the eccentr ic i ty reaches ha l f the w a l l depth . W i t h reinforcement, the improved bending capac i ty enables modern masonry structures to become tal ler and thinner , whi le preserving the t rad i t i ona l beauty of these structures. Therefore, eccentr ical ly loaded reinforced concrete masonry , wh ich must be grouted, is of interest. In this chapter , exper imenta l observations are f irst reviewed a n d the f indings i n the preceding chapters are p laced in this context . Some useful relat ions w i l l then be developed. 10.2 E x p e r i m e n t a l Observat ions T o study the basic behaviour of reinforced concrete masonry under compression and bending, 12 grouted pr isms (wi thout reinforcement) were tested under eccentric load ing . T h e fa i lure loads of these specimens are l isted i n T a b l e 10.1 and the deformat ion measurements are p lot ted in F igs . 10.1 a n d 10.2. T h e fai lure process was recorded by a v ideo camera for better observat ion. T h e exper iments ind icate that under eccentric load , the j o i n t cond i t ion and grout ing cond i t ion do not have a s igni f icant influence on the masonry capac i ty (compare also the fa i lure loads of p la in ungrouted concrete masonry under eccentric load in T a b l e 6.1). T h i s is expected since the force shared by the grout d imin ishes w i t h increasing eccentr ic i ty . In other words, the 158 S P E C I M E N e / t 1 2 3 4 A V G C O V N26 ( N J . N G ) 1 /6 178.0 196.0 164.0 200.0 184.5 7 .8% M 2 6 ( M J , N G ) 1 /3 106.0 92.0 82.0 128.5 102.1 17 .1% S25 ( S J . N G ) 1 /3 108.0 93.0 101.0 127.0 107.3 11.7% N G - T y p e N G r o u t ; N J - T y p e N M o r t a r J o i n t , etc. T a b l e 10.1 Fa i l u re Loads of G rou ted P r i s m s under Eccent r ic L o a d (kips) capac i ty of eccentr ical ly loaded masonry is even more strongly governed by the capac i ty of the b lock shel l . T h e fa i lure modes again were character ized by shear, i.e by spa l l ing m i x e d w i th crushing i of the block shel l on the loaded side, as shown in F i g . 2.16. T h i s phenomenon was more obvious for the specimens under larger eccentr icity (e = i / 3 ) . T h e grout d i d not prevent the debonding of the mor ta r j o in ts on the unloaded side, as ind icated by the substant ia l deformat ion measured across the j o i n t ( L V D T #6) for the case of e = i / 3 , a l though i t appears that the ver t ica l cont inu i ty was improved by the grout ing as the opening of the j o in ts was smal ler compared w i t h their ungrouted counterparts . T h e face-shell on the unloaded side d i d not transfer load essentially for the whole load ing range, as shown by the s t ra in measured at locat ion 5, i nd icat ing that debonding took place as soon as the specimen was loaded. Before f i na l fa i lure, no premature ver t ica l cracks were observed du r ing the tests (see also the deformat ion measurements at locations 3 and 4 as shown in F igs . 10.1 and 10.2), w h i c h is in sharp contrast to what was observed for the pr isms under u n i a x i a l load ing , suggesting that the cross-sectional i n c o m p a t i b i l i t y is not a p rob lem for grouted masonry under eccentric load ing . T h i s is another suppor t ing i nd ica t ion that the cont r ibu t ion of the grout to the capac i ty is F I G . 10.1 Measured Deformat ions at Cer ta in Locat ions of G routed P r i s m s under Eccentr ic Compress ion : a) N26 -1 , e = t / 6 ; b) N26-2, e = t / 6 ; c) M26 -2 , e = t / 3 90 o H — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — -7 -5 -3 -1 1 3 5 7 9 11 13 AVERAGE STRAN ( 1/1000 N/N ) AVERAGE STRAN( 1/1000N/N) F I G . 10.2 Measured Deformations at Cer ta in Locat ions of G routed P r i sms under Eccentr ic Compression: a) M26 -3 , e = t / 3 ; b) S25-1, e = t / 3 ; c) S25-1 , e = t / 3 161 re lat ive ly m i n o r when the masonry is under eccentric load ing . These observations are essential ly the same as those for p la in concrete masonry . T h i s encourages us to approach the p rob lem as we d i d for p l a i n concrete masonry under eccentric load ing . T h a t is, fa i lure is assumed to be governed by the block shel l a n d capac i ty es t imat ion is based on the un i t strength rather than on the u n i a x i a l p r i s m strength. T h e force shared by the grout at fa i lure is est imated by considering the ver t ica l deformat ion c o m p a t i b i l i t y . 10.3 Theoret ica l Considerat ions T h e capac i ty of reinforced concrete masonry under eccentric load w i l l be expressed here in terms of the t rad i t i ona l force-moment curve. Such a re lat ionship depends not on ly on the mate r i a l properties of the masonry const i tuents, i nc lud ing block uni t , grout , re inforcing steel and mor tar , but also on i ts geometry, wh ich is further compl icated by var ious bedding and grout ing combinat ions . T o m a k e the s i tua t ion s impler , a t tempts w i l l be made to quant i fy the mate r i a l propert ies, the geometry, the bedding and grout ing condit ions by some parameters, expressed m a i n l y in terms of the modu lus and d imens iona l rat ios. T h e usefulness of such parameters w i l l be i l lus t ra ted . F o r example , i f l inear -e last ic behaviour is assumed, the forces shared by the block shel l , grout , and the reinforcing steel can be calculated based on deformat ion modu lus rat ios. A l inear stress-strain re lat ionship m a y be a good a p p r o x i m a t i o n for concrete masonry as ind icated by var ious exper iments, inc lud ing those by the author , wh ich show that non l inear i ty before fa i lure appears to be rather l i m i t e d . T h e analyses i n the preceding chapters based on this assumpt ion do y ie ld reasonable est imat ions for the masonry capac i ty . Fu r ther , i f l inear s t ra in a long the cross-section (plane sections rema in ing plane) is assumed, the in terna l force P and moment M can be readi ly expressed in terms of the outer fibre stresses ax, <r2 o r * n e crack depth c (depending whether the cross-section is cracked or not) , as 162 shown i n F i g . 10.3. Note , in the fo l lowing expressions, the cont r ibu t ion of the ver t ica l reinforcement, which p lays an i m p o r t a n t role when the cross-section is cracked, is inc luded. T h i s has been neglected in the analys is for grouted masonry under u n i a x i a l compression, since the cont r ibu t ion is unrel iable unless the steel is t ied against buck l ing . Moreover , for n o r m a l steel rat ios , the cont r ibu t ion i n susta in ing compressive force is s m a l l compared to the concrete mater ia ls , even it is inc luded. T h i s is especial ly true for the case of eccentric load ing . However , in the fo l lowing expressions, the force shared by the reinforcement w i l l be inc luded for cont inu i ty . T h e reinforcing steel is assumed to be p laced in the midd le of the cross-section as is the c o m m o n pract ice. W h e n the eccentr ic i ty e is s m a l l , the cross-section remains uncracked, so the force and the m o m e n t can be expressed as (see F i g . 10.3) where 6 is the ha l f thickness (6=t /2 ) of the masonry , a is the ha l f w i d t h of the inner core of b lock uni t ; a n d / is the length of the w a l l . ax and a2, as have been ment ioned , denote the m a x i m u m and m i n i m u m extreme fiber stresses ( in compression) respectively, p represents the steel ra t io w i t h respect to the gross cross-sectional area, and n stands for the modu la r rat io ; i.e. the elastic modu lus of steel to that of the masonry block shel l . T h e parameter A is int roduced here to characterize the grout ing , bedding condi t ions and cross-sectional geometry i n the transverse d i rect ion : 10.1 10.2 A = J_Wg mg i 10.3 163 where w and wg are the sum of the mortared web d imension and grout d imension along the wal l length, respectively (see F ig . 10.4). mg is the modular rat io : the elastic modulus of the block shell to that of the grout, approx imated as: m, 1 + m U  2 h0 10.4 Reca l l that m j and m 2 are the modular ratios of uni t to grout , and uni t to mortar j o i n t , respectively. h0 is the height of the masonry unit and t0 is the thickness of the mor ta r jo in t . T h u s whether the masonry is fu l ly bedded or not, and whether i t is p la in or fu l ly or par t ia l l y grouted, can be expressed through the parameter A. F o r example , A = 0 corresponds to the case of a sol id section; wg = 0, A = ( 1 — w/l ) stands for the case of ungrouted masonry ; s im i la r l y , w = 0, A = ( 1 — wg/mgl) is for the face-shell bedded masonry; A = 1, when w = wg 2a 2b Uncracked Section Cracked Section 2b 2b P = a dx . M = xcr dx 2b-c 2b-c P = | cr dx M = | xcr dx 0 0 F I G . 10.3 Assumed Stress D is t r i bu t ion of an Uncracked Sect ion and a Cracked Section F I G . 10.5 Stress D is t r i bu t ion along a Section and Its C o m p o s i t i o n 165 = 0, of course, represents the case of the face-shell bedded, ungrouted masonry . B y this means, a l l the combinat ions can be inc luded and the relat ions derived here are general ly useful; they have, inc identa l l y , been app l ied in Chapte r V I for p la in concrete masonry . Eq.10.1 and Eq.10.2 are obta ined based on the pr inc ip le of superposit ion. Due to the difference in de format ion modu lus of the masonry block shel l , grout, and reinforcing steel, the stress d i s t r i bu t ion a long the cross-section must be d iscont inuous at the boundaries of these mater ia ls as depicted i n F i g . 10.5(a). T h i s stress d i s t r i bu t ion , i n an average sense a long the w a l l length, can be decomposed in to the stress d is t r ibut ions as shown i n F i g . 10.5(b), (c) arid (d), where d i s t r i bu t i on (b) corresponds to a sol id section and d is t r ibu t ion (c) represents the difference i n stress d is t r ibut ions between a sol id section and a grouted section. T h e point force depicted in F i g . 10.5(d), of course, stands for the cont r ibut ion of the reinforcing steel. C lea r l y , d i s t r i bu t ion (c) is weighted by parameter A and d is t r ibu t ion (d) by np. These parameters are combined w i th the cross-section factor a/b i n E q . 10.1 and E q . 10.2. T h e same pr inc ip le is used in the der ivat ion of the fo l lowing equations. If the tensile resistance of the cross-section is neglected, the cross-section w i l l crack (by observat ion, cracks a lways occur at the mor ta r jo in ts , see F i g . 10.1 and F i g . 10.2) when cr2/o-1 < 0 (posit ive for compression) . It can be shown that for 0 < c < b — a 10.5 M : (2 - c/b) {(i- + (x)2+ + (x)3)- + ^ )3} 10.6 where c denotes the crack length.(see F i g . 10.3). S i m i l a r l y , for b — a < c < b + a (1 + a/b - c/b)2 2(2 - c/b) + 2np 1 - c/b 2 - c/b 10.7 166 M = 1 + 2t-x 10.8 F i n a l l y , for 6 + a < c < 26 10.10 10.9 A g a i n , i f the masonry un i t strength is used to define the c r i t i ca l state, as for p l a i n concrete masonry under eccentric loading, we readi ly obta in the short w a l l capac i ty curve by P—M curve can be developed by va ry ing cy2l<>'\ f r o m u n i t y to zero, when <r2 > 0; and by stepping c f rom 0 to 26, when the cross-section has cracked. Note that i n the above expressions, the reinforcing steel is i m p l i c i t l y assumed not to reach its y ie ld strength. B y exper imenta l observations, we know that the compressive fai lure good as long as the eccentr ic i ty e is not too s m a l l . F o r concentr ic loads, E q . 10.1 m a y overest imate the fa i lure load , because the steel cou ld y ie ld . However, this part of the capac i ty curve is not of interest here since the concentr ic capaci ty is treated separately, as in Chapte r V I I and VIII. Moreover , as ment ioned earlier, the cont r ibu t ion to the compressive capac i ty of the reinforcing steel is usual ly s m a l l compared to that of the surrounding cross-section. However , i f the crack extends beyond the ha l f depth of the w a l l , the re inforcing steel m a y y ie ld in tension. T h i s m a y happen when the crack depth at the balanced load le t t ing the extreme f ibre stress <r1 be equal to fu, the uni t compressive strength. T h a t is, the s t ra in for concrete masonry is usual ly s m a l l (less than 0.002), so that this assumpt ion m a y be 167 1 + nfu 10.11 is less than that corresponding to the pure moment capaci ty (the c wh ich makes E q . 10.7 or E q . 10.9 vanish) ; fy here is the y ie ld strength of the steel. A l t h o u g h y ie ld ing of the steel is not desirable a n d is not a l lowed i n the current design code, for analys is , we m a y replace the te rm 2 n p ( l — c / 6 ) / ( 2 — c/b) i n E q . 10.7 or E q . 10.9 by 2p/ ! / /<x 1 to include this s i tuat ion . 10.4 C o m p a r i s o n of Theory w i t h Exper iments In s u m m a r y , the capac i ty curve for concrete masonry is determined by the fo l lowing parameters: fu , fy, m 2 , np, a, b, I, w, wg and t0/h0, wh ich can a l l be measured. However, for p ract ica l reasons, the modulus rat ios TOJ and m 2 m a y be related to the corresponding strength rat ios . In the fo l lowing compar ison , the same square root corre lat ion is used as in Chapte r V I I . T h e P—M curves generated for the author 's specimens and those tested by Drysda le and H a m i d (1983) are p lo t ted in F i g . 10.6 to F i g 10.15 w i t h the exper imenta l da ta . T h e plot is nondimensional ized by d i v i d i n g ver t ica l load by P0=/Utl, the n o m i n a l a x i a l capac i ty ; and moments by Mk = P0i/6, the moment capaci ty when P0 is app l ied at the kern eccentr ic i ty of a sol id sect ion. F o r most cases, the agreement is reasonably good. F o r the exper iments by Drysdale and H a m i d , the curves appear to underest imate the bending capaci ty of the specimens tested at the biggest eccentr ic i ty consistent ly , a l though by a s m a l l amount . T h i s is p robab ly caused by the assumpt ion that the cross-section does not resist any tensile force, wh ich is closer to real i ty for p la in masonry than for grouted masonry . N o efforts are made here to compare the results numer ica l l y , since a number of the d a t a 168 points are obta ined at large eccentr ic i ty when the load ing pa th M=eP is a lmost para l le l to the lower boundaries of the capac i ty curves. In this s i tuat ion s m a l l exper imenta l errors can lead to large numer ica l var iat ions in load or moment . T h e mode l based on l inear s t ra in and stress appears to give reasonable predict ions. T h e compar ison aga in supports the assumpt ion that the capac i ty of eccentr ical ly loaded masonry is more closely correlated w i t h the un i t strength than w i t h the concentr ic capac i ty . T h u s i n p ract ica l design, i t m a y be again recommended that the concentr ic capac i ty and the eccentric capac i ty be treated separately, as in the case of p la in masonry . T h e expressions developed here prov ide convenient ways to est imate the eccentric capac i ty of concrete masonry w i t h var ious grout ing a n d bedding condi t ions . F I G . 10.7 Compar i son of Predicted Interact ion C u r v e w i t h Exper iments by Drysdale and H a m i d : N o r m a l B lock , T y p e N G r o u t 170 0. CL a. 0 . 7 0.6 0 . 5 -0 . 4 -0.3 0 . 2 -0 .1 -0 . 4 0 . 6 M / M < F I G . 10.8 Compar i son of Predicted Interact ion C u r v e w i t h Exper iments by Drysdale and H a m i d : N o r m a l B l o c k , T y p e W G r o u t 0 . 9 0 . 3 0 . 2 0 . 4 0 . 6 M / M c F I G . 10.9 Compar ison of Predicted Interact ion Curve w i t h Exper iments by Drysdale and H a m i d : N o r m a l B l o c k , T y p e S G r o u t 0 0 . 2 0 . 4 0 . 6 M / M t F I G . 10.10 Compar ison of Pred icted Interaction Curve w i th Exper iments by Drysdale and H a m i d : Weak B lock , T y p e N G r o u t 0 . 8 - | 0 0 . 2 0 . 4 0 . 6 F I G . 10.11 Compar ison of Predicted Interaction Curve wi th Exper iments by Drysdale and H a m i d : Strong B lock , T y p e N G r o u t 172 0 . 8 0 0 . 2 0 . 4 . 0 . 6 M / M k F I G . 10.12 C o m p a r i s o n of Predicted Interaction C u r v e w i t h Exper iments by Drysdale and H a m i d : 7 5 % Sol id B lock , T y p e N G r o u t a. 0 0 . 2 0 . 4 0 . 6 M / M < F I G . 10.13 Compar i son of Predicted Interaction C u r v e w i t h Exper iments by Drysdale and H a m i d : F u l l B l o c k 0 0 . 2 0 . 4 0 . 6 F I G . 10.14 Compar ison of Predicted Interaction Curve w i t h Exper iments by Drysdale and H a m i d : 6 inch B lock , T y p e N G r o u t 0 . 8 -, : 0 0 . 2 0 . 4 M/Mk F I G . 10.15 Compar i son of Predicted Interact ion Curve w i th Exper iments by Drysdale and H a m i d : 10 inch B l o c k , T y p e N G r o u t 174 C H A P T E R X I S L E N D E R N E S S O F C O N C R E T E M A S O N R Y 11.1 In t roduct ion M o d e r n masonry structures are becoming tal ler , not on ly i n terms of the e levat ion of the bu i l d ing , but also i n terms of storey heights. Besides advances in engineering knowledge, changes i n the masonry const i tuents have cont r ibuted to this development. S t ruc tu ra l behaviour is greatly improved by h igh strength concrete units w i t h steel reinforcement. T a l l , slender concrete masonry can be seen in m a n y places, such as apar tment highrises, department stores, warehouses, supermarkets, gymnas iums and aud i to r iums . T h e benefits of bu i l d ing tal ler and more slender masonry are obvious; besides space savings, m a t e r i a l and const ruct ion costs are reduced. A s the w a l l becomes l ighter, smal ler footings are required and lower seismic forces are induced. These are i m p o r t a n t reasons why modern masonry structures f i nd a place i n today 's compet i t i ve bu i ld ing market . However, the development of t a l l , slender masonry is s t i l l largely hampered by a l i m i t e d understanding of the mechan ica l behaviour , and probab ly by an histor ic prejudice that masonry is not sound when it is t a l l . T h i s is reflected i n the str ingent slenderness requirements in the current masonry design code ( C A N 3 - S 3 0 4 - M 8 4 , 1984). In the last two decades, reinforced slender wal ls have been studied extensively. Some exper iments have shown excellent f lexura l performance; for example , the tests conducted in the early 80's by A C I Southern C a l i f o r n i a Chapter (A they 1982), which lead to some l im i ted re laxat ions of the slenderness requirements i n bu i ld ing codes. However , since fu l l scale w a l l tests are very expensive and t ime consuming , i t is very d i f f icul t to observe the behaviour under var ious load combinat ions . T h e analysis of slenderness effects have so far been largely l i m i t e d to the t rad i t i ona l approach, i.e. the m o m e n t magni f ier method has been app l ied and thus an 175 effective r i g id i t y of the member has had to be assumed. In this chapter, a more ra t iona l analysis w i l l be presented i n the context of these exper imenta l observations, and of the f indings in the preceding chapters, wh ich have been focused on the short w a l l or c o l u m n capac i ty . T h i s w i l l fo l low a brief review of background in fo rmat ion . 11.2 B a c k g r o u n d In fo rmat ion Rev iew T h e slenderness effects discussed here refer to masonry under eccentric load . W a l l s under concentr ic load ing are not of p ract ica l concern since a m i n i m u m eccentr ic i ty has a lways to be assumed (0.12 or 25 m m , specified by the current design code ( C A N 3 - S 3 0 4 - M 8 4 , 1984)) to take account of member imperfect ions and a l ignment error. W h e n a slender member carries an eccentric load , i t is i m p o r t a n t to bear in m i n d that it m a y suffer loss of capac i ty either due to mate r ia l fa i lure or by i ns tab i l i t y . T h i s par t icu la r point has been clear ly expla ined by N a t h a n (1977). F i g . 11.1 shows the in teract ion curve for a c o l u m n subject to a compressive load w i t h equal end eccentricit ies. T h e l ine 0 — A defines the re lat ionship between load and end moment . However, due to the slenderness, the midhe ight m o m e n t is magni f ied by the member def lection, and the corresponding l oad -moment path is defined by 0 — B . M a t e r i a l fa i lure occurres at point B , when the end condi t ions are as ind icated at po int C . Theoret ica l l y , i f the m o m e n t — c u r v a t u r e ra lat ionsh ip of the beam c o l u m n remains l inear , mate r i a l fa i lure a lways governs the behaviour , since the m idspan def lection w i l l be unbounded when the Eu ler load is approached. T h e moment magni f ier method perfectly predicts this fa i lure mode. W h e n the member develops some nonl inear i ty i n its m o m e n t — c u r v a t u r e re lat ionship , the method is s t i l l a v a l i d a p p r o x i m a t i o n prov ided an appropr iate effective cross-sect ional r i g id i t y is used. However, when the cross-section has developed substant ia l non l inear i ty , usua l ly at greater eccentricit ies, the m idspan moment increases w i t h def lection to a po int such as D i n F i g . 11.1, and the member becomes unstable i n the sense that equ l i b r i um cannot be ma in ta ined even though the mate r ia l of the cross-section is s t i l l sound. 0 M O M E N T F I G . 11.1 A L o a d — M o m e n t Interaction Curve and L o a d i n g P a t h s of a Compression Member 177 T h e member w i l l f a i l at this point , corresponding to end cond i t ion E , unless the load can be shed by other means, to lead to mate r i a l fa i lure at po int F. T h e m o m e n t magni f ier procedure, in w h i c h the design m o m e n t is compared w i t h the short c o l u m n moment , no longer applies r igorously to th is s i tuat ion . T h e procedure adopted i n the current code ( C A N 3 - S 3 0 4 - M 8 4 , 1984) is, at best, an a r t i f i c ia l emp i r i ca l device for the member governed by i ns tab i l i t y . It is clear that for the moment magni f ier method (albeit i n an a r t i f i c ia l way) to be app l ied succesful ly to the design p rob lem, the key issue is how to est imate the nonl inear development of the cross-section. A s i n a concrete c o l u m n , the nonl inear development of a masonry member is due to mate r ia l nonl inear i ty as wel l as to the crack ing of the cross-section. T o est imate these effects accurately is d i f f icul t since they are coupled w i t h the magn i tude as wel l as the eccentr ic i ty of the load . Therefore it is not surpr is ing that the current design process is subject to m a n y l im i ta t i ons , since these effects cannot be inc luded in a single assumed "effective cross-sectional r i g i d i t y " . Fur ther , after the cross-section has cracked, the r i g id i t y is a var iab le a long the member height rather than a single constant represented by the "effective r i g i d i t y " ; the phys ica l p icture of the s imp l i f i ca t ion is not clear. T o inc lude a l l the nonl inear effects, a ra t iona l analysis w i t h some numer ica l procedures is often necessary. F o r analysis of concrete beam columns, N a t h a n (1985) has developed a computer p rog ram based on some wel l establ ished pr inciples. B y numer ica l in tegrat ion , i t first finds fo rce -moment -curvature relat ionships for any cross-sectional geometry, and for mater ia ls w i t h any const i tu t ive law. A n i terat ion scheme is then used to give the c o l u m n deflection curve wh ich matches to any boundary condi t ions . It is of course very general and useful , and m a y be app l icab le to concrete masonry w i t h a few modi f icat ions . O n the other hand , Suwa lsk i and Drysda le (1986) have used a finite element mode l to d i rect ly analyze the slenderness influence of the capac i ty of concrete masonry . These approaches are useful in the sense that they m a y include a l l the factors wh ich affect the behaviour . However, at the same t ime , they require more input parameters, wh ich , in 178 design pract ice, often must be assumed rather than measured. It appears that w i t h these approaches, the wal ls must be studied i nd i v idua l l y , and i t is d i f f icul t to perform a parametr ic s tudy wh ich m a y y ie ld some s impl i f ied relat ions governed by some key factors. In the fo l lowing analys is , some assumptions w i l l be made based on the observed character ist ics of concrete masonry . W i t h these assumptions, some a n a l y t i c a l relat ions w i l l be developed to exp l i c i t l y reveal some key factors representing the masonry slenderness effect. T h i s w i l l be shown to lead to a re lat ively s imple but yet ra t iona l approach to the p rob lem. T h i s approach w i l l be shown to be easily adapted to design analysis . T h e usefulness and l i m i t a t i o n s of th is approach w i l l then be discussed. 11.3 Mason ry Character is t ics and Some Assumpt ions C o m p a r e d w i t h concrete co lumns, concrete masonry is more prone to crack when the cross-section is subjected to tension because of the mate r ia l d iscont inu i ty at the mor ta r j o in t . T h i s is c lear ly evident f r o m the deformat ion measurements across the jo in ts , (see F igs . 10.1, 10.2, also see F igs . 6.1 and 6.2 for p la in masonry) . S i m i l a r observations were also reported by F a t t a l et a l (1976) and by Hatz in iko las et a l (1978). T h u s for a l l p ract ica l purposes this tensile bond can be assumed to be zero. A n d since the bed jo in ts are evenly spaced, i t is reasonable to treat the p rob lem i n an average sense, wh ich is necessary to lead to a cont inuous fo rmu la t i on a long the member height. A n o t h e r s igni f icant observat ion, ment ioned m a n y t imes earl ier, is that mate r i a l non l inear i ty is rather l i m i t e d up to the fa i lure stress, and the l inear stress-strain mate r ia l re lat ion is a v a l i d a p p r o x i m a t i o n (also see Y o k e l and D ikkers 1971, Hatz in iko las et a l 1978, W a r w a r u k et a l 1986). T h e l inear mate r i a l and the zero tensile bond assumpt ions are equivalent to supposing that the nonl inear i ty i n the moment —curvatu re re lat ionship of a concrete masonry member is m a i n l y due to the crack ing of the cross-section. Indeed, the cross-sectional r ig id i ty , wh ich is p ropor t iona l to the t h i r d power of the section depth , w i l l decrease drast ica l ly as the 179 depth is reduced by crack extension. T h e t h i r d assumpt ion is that of the plane section rema in ing plane, corresponding to a l inear s t ra in d i s t r i bu t ion across the section. T h i s is a c o m m o n l y accepted assumpt ion , however r igorously speaking, i t impl ies , i n the context of the f irst assumpt ion , an i n f i n i tes ima l c rack ing spacing when the side of a cross-section is subject to tension. Since the tension cracks occur on ly at the mor ta r j o in ts , the mater ia ls between two cracked jo in ts must transfer some tension force and thus alter the plane sections. Therefore the l inear s t ra in d i s t r i bu t ion m a y be a good a p p r o x i m a t i o n only when the crack depth is not b ig compared to the un i t height. W i t h these m a i n assumptions, i t is possible to establ ish re lat ive ly compact re lat ionships governing the mechan ica l behaviour of a masonry member under var ious load ing condi t ions , and thus i t is easier to perform some parametr ic studies on slenderness effects. T h e equations governing the cross-sectional behaviour der ived in the preceding chapter are s t i l l v a l i d and w i l l be quoted wi thout further comments . 11.4 Di f ferent ia l Equat ions Govern ing Concrete Mason ry w i t h C r a c k e d Sect ion E q u a l end eccentricit ies w i l l f irst be invest igated, and the approach w i l l then be extended to more general load ing condit ions. Different di f ferential equations are used to describe the behaviour , depending on whether the cross-section has cracked and how deep the crack extends. F i g . 11.2 depicts the most general case: a concrete masonry member under eccentric load w i t h uncracked sections at the two ends and , due to def lect ion, a cracked section i n the m idd le range. Note , c represents the crack length or cracked sectional depth . M , F , and C denote the cross-section at m i d s p a n , the cross-section at wh ich the crack extends to the f lange ( face-shell) depth a n d the cross-section at wh ich the crack begins to extend, respectively. T h e var iables subscr ipted w i t h these letters ( in lower case) s tand for the corresponding values at these cross-sections. Stress Enlarged Diagram compression side c ' l s boundary of cracked zone i FIG. 11.2 A Cross-Section View of A — 2b-c—»-j Reinforced Concrete Wall under Eccentric Compression 181 B y symmet ry , we need only study the upper ha l f of the masonry wa l l . F o r the end por t ion of the w a l l , c rack ing does not take place. Referr ing to the selected coordinates in which the x ax is coincides w i t h the thrust l ine and y lies through the s y m m e t r i c section, we have, for the curve def in ing the compression face EI^X- P(b-y) = 0 h/2 < x < h/2 11.1 dx w i t h boundary condi t ions : y(h/2) = b - t0 11.2 at the end; and y(hc/2) = b - ec w i t h dx\2) ~ ^ at the C — c r o s s - s e c t i o n ; where ec is the v i r t u a l load ing eccentr ic i ty and <pc is the ro ta t ion at this cross-section. Note , for this load ing cond i t ion , the end eccentr ic i ty e0 is smal ler than the crack ing eccentr ic i ty ec. W h e n E q . 11.1 is integrated and matched to the boundary condi t ions (see appendix ) , we obta in 11.3 11.4 11.5 182 where Pcr is the Eu le r load corresponding to the gross section: p " 7T Hill 11 c Pcr ~ ~ir ~ w 1 L 6 a n d cj is the rat io of the moment of i n te r ia of the net cross-section to that of the gross-section i n w h i c h A and a/b are defined as in the preceding chapter. F o r a given cross-section, ec can be wr i t ten as 3 ( l - X-f + np) ec = Mc = —, T b 11.9 i n wh ich Mc and Pc are expressed through Eqs . 10.1 and 10.2 w i t h c 2 being set equal to zero by neglecting the tensile resistance of the cross-section. E q . 11.5 gives the re lat ion between load P and two unknowns, namely hc and <pc, wh ich w i l l be found by the equations governing the cracked section as shown further below. T h e di f ferent ia l equat ion for the cracked section can be der ived by f irst consider ing the geometr ic re lat ion . A s shown by the enlarged d iag ram i n F i g . 11.2, the cross-section w i l l rotate due to the uneven compression wh ich produces the outer fiber s t ra in e x at the compression face but zero at the boundary between cracked and uncracked zones. T h e change of the ro tat ion of a s m a l l sect ion, therefore, can be app rox imated as e1ds e1dx = -2b-=T * -W=T 1L1° B y recogniz ing 183 11 .11 i t f o l l o w s t h a t T h e a s s u m e d l i n e a r s t r e s s - s t r a i n r e l a t i o n a l l o w s us t o w r i t e f l = -j 11.13 F i n a l l y , cr1 c a n b e e x p r e s s e d i n t e r m s o f t h e l o a d P a n d t h e c r o s s - s e c t i o n a l p a r a m e t e r s b y t h e e q u i l i b r i u m c o n d i t i o n , e i t h e r t h r o u g h E q . 10 .5 o r E q . 10 .7 , d e p e n d i n g o n w h e t h e r t h e c r a c k h a s e x t e n d e d b e y o n d t h e f l a n g e . T h u s f o r 0 < c < b— a, w e h a v e d2y P 1 - 1 1 . 1 4 dxz 2Eb ' '(('-6)'- +)) w i t h y b e i n g r e l a t e d t o t h e p a r a m e t r i c v a r i a b l e c b y t h e g e o m e t r i c r e l a t i o n (see F i g . 11.2): y = h - e = b V 2b_J_\ b_J \JJ b u . i 5 b y r e c o g n i z i n g e=M/P a n d r e l a t i o n s g i v e n b y E q s . 10 .5 a n d 1 0 . 6 . S i m i l a r l y , f o r b—a< c < 6 + a w e o b t a i n d2y dx" 2Eb2l 11 .16 184 and y = ( ' - & ) ' ( • + 1 ) - {(f)'- i O - f - f )) 11.17 in view of Eqs. 10.7 and 10.8. It is not intended to include the case of b+a< c < 26, since it is of little practical significance when the crack extends so deep; although it poses no further difficulties. With the relations given by Eqs. 11.15 and 11.17, Eq. 11.14 and Eq. 11.16 can be integrated, by some manipulations presented in the appendix, in closed form to give the slope for 0 < c < 6—a, in which C± is a constant of integration and is a function of c expressed as dy _ [ "FT dx ~ \Ebl \ 11.18 4( 1 ~ A)' - A ( t ) 3 - 3 ( A t - "'X1 - t)' 6 ( ( i - 2 i ) 2 - ( n - " X 1 - 1 ) ) 2 11.19 And similarly for 6—a< c < 6+a dy _ \P_ ( dx ~ iEbl \ C2- £l2(c)) 11.20 with 11.21 185 T h e constants of in tegrat ion C i and C 2 can be determined by m a t c h i n g to the k n o w n condi t ions on the ro tat ion . B y symmet ry , we have 11.22 wh ich leads to C2 — Q2(c">) 11.23 where c m denotes the m idspan crack length. T h u s the ro tat ion at section F is 11.24 wh ich also leads, by cont inu i ty of the rotat ion , to an expression for C x C i = fi2(cm) - fi2(C/) + il^cj) 11.25 where Cj = b—a, the crack length at section F. Eqs . 11.18 a n d 11.20 can then be rearranged and integrated a long the w a l l height to give p hc/h—hj/h \|Pc7 2 Cc 7T = dy 11.26 for 0 < c < b—a, where c c = 0, the crack length at section C ; and hf/h p_ -7T = _3_ N 2 dy Cm b\C2- fi2(c) 186 11.27 for b—a< c < 6+a . <fy can be expressed i n terms of dc by d i f ferent iat ion of E q . 11.15 or E q . 11.17 w i t h i n their specified domains . T h u s , for given a cm the r ight hand sides of Eqs . 11.26 and 11.27 can be integrated numer ica l l y . Fu r ther , the ro ta t ion at section C , wh ich is conta ined i n E q . 11.5, can be readi ly obta ined i n v iew of E q . 11.18 fc = ^ ( fi2(cm) - J 2 2 ( C / ) +ni(Cf) - fi^Ce)) 11.28 F i n a l l y , by s u m m i n g Eqs. 11.26, 11.27 and 11.5, a def in i t ive re lat ion between the appl ied load P a n d the m idspan crack depth c m is reached P Per 3_ N 2 c./b Cm/b dy/ dc •\ ^ 2 ( c m ) - ^ 2 ( c ) <+) + cc/b c./b dy/dc + H ^ s i n " 1 e c / 6 — sin e0/b 11.29 T h i s equat ion i m p l i c i t l y defines the force—def lect ion re lat ion of a concrete masonry wa l l 187 through the paramet r ic var iab le c m , wh ich can be used to s tudy the slenderness effects and the s tab i l i t y of the w a l l . B y e x a m i n i n g E q . 11.29, one finds that the three terms on the r ight hand side actua l ly represent the capac i ty cont r ibut ions of three sections of the masonry w a l l , namely , the cracked section i n wh ich the crack has extended into the grout core, the cracked section in wh ich the crack extends w i t h i n the face-shel l , and the uncracked section. Therefore, by add ing or subt ract ing the cont r ibut ions , the results can be extended to more general load ing cases. F o r (equal) end eccentricit ies e0 larger than c rack ing eccentr ic i ty e c , the cracked zone w i l l extend over the entire height, and E q . 11.29 reduces to where c0 is the end c rack ing corresponding to e 0 , found through Eqs. 10.5 and 10.6. If t0 is greater than the flange crack ing eccentr ic i ty t., E q . 11.30 further reduces to where c0 is determined through Eqs. 10.7 and 10.8. S i m i l a r l y , when the m idspan cracking cm is less than the f lange c rack ing Cy, wh ich may happen when e0 is less than t,, E q . 11.29 and E q . 11.30 become Cj/b 11.30 11.31 cc/b 188 + f t s i n " 1 e ^ l L . 'olh - sin"1 , )• 11.32 and c0/b P _ 6 P — 2 dy/dc C m / O < + ) 11.33 respectively. T h u s , a l l the possible combinat ions for equal eccentr ic i ty load ing are inc luded . B y the same pr inc ip le , the results can also be extended to the case of unequal eccentr ic i ty load ing . A c c o r d i n g to N a t h a n (1972), the conf igurat ion of a c o l u m n loaded w i t h a rb i t ra ry eccentricit ies can be represented by a por t ion of a wave of an imag inary , in f in i te ly long c o l u m n under the act ion of the a x i a l load , as shown in F i g . 11.3. W i t h o u t loss of general i ty , we assume that the magn i tude of the b o t t o m eccentr ic i ty is not less than that of the top one. T h u s the m a x i m u m deflection f r o m the thrust l ine a lways lies i n the lower por t ion of the c o l u m n . T h i s po int , at w h i c h dy/dx=0, corresponds to the midspan of the case of equal eccentr icity load ing studied above. T h e c o l u m n loaded w i t h arb i t rary eccentricit ies then is actua l ly composed of a por t ion s y m m e t r i c a l about the m a x i m u m deflection po int , w i t h an extension at the top end as far as the appropr iate value of eccentr ic i ty . F r o m this v iewpoint , the corresponding capac i ty cont r ibut ions can easily be evaluated and s u m m e d to give the f o r c e — c r a c k i n g re lat ion . It should be ind icated that for the case of double curvature load ing (et/eb negative), there are apparent ly two possible equ i l i b r i um conf igurat ions depicted by sections A B and A C in F i g . 11.3. However , as far as the lowest buck l ing load is concerned, the conf igurat ion A B w i l l be under considerat ion. T h i s conf igurat ion should also be realized for the case of a n t i - s y m m e t r i c load ing (et/eb = — l). T h i s has been shown by the experiments (Hatz in iko las et a l , 1978; F a t t a l F I G . 11.3 C o l u m n Deflection Curve 190 et al 1976), and a theoretical explanation will be presented in appendix. Consider two most general cases. First 0 < et < eb < ec; the corresponding relation is _P_ Per _ 4 r 2 1 > Cj/b T 2 J \ V2(cm) - Jl2(c) Cm/b dy/ dc if) + cc/b _E Cj/b ^ dy/ dc if) + ft I s i n - 1 - ^ 1 (( ec/b sin - l N(T) + l H ("2(Cm) ~ a M + f ii( c/) - ni(<=«)) + - ± - ^ ^ sin"1-sin - l et/b ( l ) 2 + i N ( f i 2(Cm) - fi2(C/) + « 1 ( C / ) - ^ ( C c ) ) 11.34 The second case is when eb > tj > 0, et < 0, but e6 > |e t| > e^ , the relation becomes P_ Per _ 4 r 2 ^ c„/b Cm/b dy/ dc A| ^ 2 ( c m ) - ft2(c) <+) cj/b + dy/ dc ,Jfi 2 (c m ) - fi2(c) < f ) + + c./b 191 dy/ dc ]tt2(cm) - 0 2( c) < + ) + cc/b Cj/b dy/dc <-fr) s in 1 , e c / 6 0 2 (c m ) - fi2(C/) -rfiiCc/) - fi^cc)) 11.35 where c 6 and ct are the crack depths corresponding to eb and e ( , respectively. In these two load ing condi t ions , the crack ing c m is assumed to be greater than Cj. These results are readi ly generalized to any other load combinat ions for unequal end ^ eccentr ic i ty load ing . A computer p rogram wr i t ten i n F O R T R A N - G was developed based on the equations der ived above. A l i s t ing of the program is given i n an appendix . 11.5 Resul ts and A p p l i c a t i o n s T h e a l g o r i t h m developed above w i l l be used to study two m a i n aspects of concrete masonry w a l l behaviour , namely the s tab i l i t y , a n d the force—def lect ion re lat ion ; the latter also affects the w a l l capac i ty . T h e focus w i l l be on the case of equal eccentr ic i ty load ing . A c c o r d i n g to the model , the buck l i ng load of concrete masonry can be found by stepping c m f r o m zero to some c r i t i ca l depth , at wh ich the load P reaches a m a x i m u m . T h i s is i l lust rated by the cm — P re lat ionship for a p la in , so l id section (X = a/b=np = 0) loaded at the kern eccentr ic i ty (e0 = t/6), as shown i n F i g . 11.4. O n the hor i zonta l ax is , the value cm/t=0 represents an undeflected member . T h u s this corresponds to no a x i a l load or P/Pcr=0. Obv ious ly , no load cou ld be ma in ta ined i f the whole cross-section were cracked, cm/i=l. T h u s 0.3 0.28 -.0.26 -0.24 -0.22 -0.2 -u 0.18 _ o fx, 0.16 -\ cu 0.14 0.12 -0.1 -0.08 -0.06 -0.04 -0.02 -0 192 C m / t FIG. 11.4 Critical Load versus Crack Depth at Middle Section of a Plain, Solid Member Loaded at e=t/6 o a, l 0.9 0.8 -0.7 -0.6 -0.5 0.4 0.3 -0.2 -0.1 0 0 0.1 P R E S E N T R E S U L T Y O K E L 0.5 FIG. 11.5 Critical Load versus Loading Eccentricity for a Solid Section 193 this po int also corresponds to the value P/Pcr = Q. W h e n P is app l ied and increased, deflection w i l l increase together w i t h the crack depth . A s s u m i n g there is no compression fa i lure dur ing the load ing stage, the load P w i l l reach a m a x i m u m corresponding to some crack depth ( c m / z s s 0 . 4 a n d PjPer~0.28 for this case). A n y further increase of P beyond this po int w i l l cause the member to collapse. It is clear that the re lat ion before this c r i t i ca l po int , dP/dcm > 0, represents stable equ i l i b r i um. Beyond this po int , dP/dcm < 0 represents unstable equ i l i b r i um. A t the po int , dP/dcm = 0; P=Pmax of course, stands for the buck l i ng load . Obv ious l y , the cross-sectional c rack ing of a member w i l l depend heav i ly on the load ing eccentr ic i ty a n d so, therefore, w i l l the buck l i ng load . F o r a p la in , so l id sect ion, the buck l i ng load is p lo t ted against the eccentr ic i ty in discrete f o r m in F i g . 11.5. A t the point where eo/l = 0, when the member is loaded concentr ica l ly , P / P c r = l , and the buck l ing load coincides w i t h the Euler load . T h e b u c k l i n g load decreases drast ica l ly w i t h increase i n eccentrci ty . W h e n eo/t = 0.5, P/PCr = 0, i.e., no load can be sustained i f the load is appl ied at the edge of a member w i t h no tension resistance. T h e classic p rob lem of the buck l i ng of a p l a i n , so l id member w i t h no tension resistance was f i rst invest igated by R o y e n (1937). T h e prob lem a n d its app l icat ion to b r ickwork have been subsquent ly studied by C h a p m a n and S lat ford (1957), Sah l i n (1971), Y o k e l (1971), Hatz in iko las (1978). In F i g . 11.5, the results obta ined by the a l g o r i t h m are compared w i t h a closed fo rm so lut ion for load ing eccentr ic i ty larger than t / 6 given by Y o k e l . F o r the range compared, the results are essential ly ident ica l . T h e fo l lowing are some of the interest ing predict ions given by the a l g o r i t h m . A s shown by F i g . 11.6 for the sol id section ( A = a / 6 = 0 , np varies) , whi le the buck l ing loads corresponding to s m a l l eccentricit ies are essential ly unaffected, the s tab i l i t y of the w a l l is great ly improved w i t h increase of the reinforcement rat io at large eccentricit ies. T h e rather f lat ta i ls at large eccentricit ies i m p l y that the capaci ty of reinforced wal ls is largely governed by the bending r i g id i t y . 1 0.9 H 0.8 0.7 0.6 -0.5 -0.1 0.3 -0.2 -0.1 0 "X =0.0 a/b=0.0 np = 0.05 0.025 0.005 0.0 0 0.2 0.4 0.6 Eo / T FIG. 11.6 Critical Load versus Loading Eccentricity: X = a/b=0, np Varies 0.8 0.7 0.4 H 0.3 H 0.2 0.1 0 \ a/b=0 \ \V^ - a/b=0.65 1 - 0 . 5 np =0.05 a/b=0.75 \ \ 0.2 0.4 o.e Eo/T FIG. 11.7 Critical Load versus Loading Eccentricity: A = 0.5, np = 0.05, a/b Varies 195 0 0 . 2 0 . 4 E o / T F I G . 11.8 C r i t i c a l Load versus Load ing Eccentr ic i ty : a /6 = 0.65, 71/9 = 0, A Varies In contrast, the var ia t ion of a/b only affects the s tab i l i t y at s m a l l eccentricities as shown by F i g . 11.7 (A = 0.5, corresponding to a par t ia l l y grouted wa l l ; 71/9 = 0.05; a / 6 = 0 , 0.65, 0.75). T h e effect of changes in A is i l lustrated through an example compar ing different bedding condit ions. F i g . 11.8 shows the buck l ing loads for a typ ica l 8 inch p la in section ( a / 6 = 0 . 6 5 ; n p = 0 . A = l for face-shell bedding and A = 0.75 for fu l l bedding; A = 0 represents a sol id , or fu l ly grouted section, included here as a reference). A l t h o u g h the buck l ing load of the sol id section is higher at very sma l l eccentricities, i t drops rapid ly as the eccentr icity increases and soon becomes the lowest. Face-shel l bedded masonry, on the contrary , has lower buck l ing loads at smal l eccentricities but remains relat ively higher at larger eccentricit ies. A ful ly bedded section falls in between. Since buck l ing usual ly only governs fai lure at larger eccentricit ies, one may conclude that in terms of s tab i l i t y , face-shell bedded masonry is better than its ful ly bedded counterpart which is in turn better than a sol id section. T h i s is not surpr is ing considering that 196 face-shel l bedded masonry is least prone to crack under eccentric load ing . W e m a y infer, i n the context of the strength studies presented i n the preceding chapters, that face-shel l bedded masonry is more l ike ly to be governed by strength than by s tab i l i t y . T h e present approach is compared w i t h some of the exist ing d a t a obta ined f rom fu l l scale concrete w a l l tests. These include eleven 137 inch and 105 inch h igh wal ls ( 8 x 4 0 x 1 2 8 inch and 8 x 4 0 x 9 6 i nch nomina l ) w i t h different reinforcement tested under equal end eccentricit ies by Ha tz in iko las et a l (1978): A s discussed at the begining of the chapter, t a l l masonry wal ls m a y lose strength either by mate r i a l fa i lure or by instabi l i ty . T h e examinat ion of the 137 inch h igh w a l l w i th 3#3 reinforcing steel (np=0.027) provides an excellent i l lust rat ion of this point . In F i g . 11.9, the load — m o m e n t interact ion curve is developed by the equations given in C h a p t e r X . T h e stra ight lines rad ia t ing f rom the or ig in define the end condi t ions for different load ing eccentr icit ies. These exper imenta l lines are terminated by the d a t a points and paired w i t h the predicted curves representing the l o a d — m o m e n t relat ionships at midhe ight . It is clear that the moment is magni f ied due to the slenderness. W h e n the cross-section remains uncracked, usual ly under s m a l l eccentricit ies w i t h low load magni tude , the magni f ier is given by the l inear so lut ion : 11.36 R e c a l l that Pcr represents the Eu ler load for the gross section wh ich must be adjusted by £ for par t icu la r condi t ions . W h e n the cross-section is cracked, the magni f ier 6 = em/e0 is ca lcu lated , by the a l g o r i t h m , for every midspan crack depth cm. It is interest ing to note that for the case of the smal lest eccentr ic i ty e0 = i / 6 , as depicted by the lines w i t h the steepest i n i t i a l slope, the po int def in ing the end condi t ions is w i t h i n the P—M capac i ty curve whi le the corresponding point representing the m i d s p a n condit ions (moment has been magni f ied) is outside but fa i r ly 197 MOMENT (KP-W) FIG. 11:9 Theoretical P —M Interaction Curve and Loading Paths Compared with Experiments by Hatzinikolas et al: 137 inch High Wall with Reinforcement 3#3 1 -I : 1 0 . 0 -— CL 0 0 . 2 0 . 4 E o / T FIG. 11.10 Theoretical Load —Eccentricity Curve Compared with Experiments by Hatzinikolas et al: 137 inch High Wall with Reinforcement 3#3. The Points Show the ExperimentarResults while the Continuous Lines Show the Prediction 198 0.9 -0.8 -0.7 -0.6 -0.5 - \ \ STABLITY GOVERNS 0.4 -0.3 -0.2 -MATERIAL GOVERNS — 0.1 -0 - 1 r— 1 1 0 0.2 0.4 Eo / T F I G . 11.11 Theoret ical Load — E c c e n t r i c i t y Curve C o m p a r e d w i t h Exper iments by Hatz in iko las et a l : 137 inch H igh W a l l w i th Reinforcement 3#6 \ • STABUTY FALURE >v O MATERIAL FAILURE N . STABLITY GOVERNS MATERIAL GOVERNS > s 0 I i 0.2 1 1 0.4 Eo/T F I G . 11.12 Theoret ical L o a d — E c c e n t r i c i t y Curve C o m p a r e d w i t h Exper iments by Hatz in iko las et a l : 105 inch H igh P l a i n W a l l 199 close to the curve. T h i s indicates mate r ia l fai lure since the cross-sectional capac i ty is reached at m idspan . However , for the cases of larger eccentricit ies (e 0 = </3, e 0 = 3 i n and e 0 = 3.5 in) , a l l the end po ints are wel l w i t h i n the capaci ty curve. T h e m a x i m u n force for e q u i l i b r i u m is reached whi le the cross-sectional capaci ty is not exceeded as shown by the curves def in ing the m idspan condi t ions . It is clear that these are the cases of i ns tab i l i t y fai lure. ( A t i ns tab i l i t y , the midhe ight l oad pa th should reach a hor i zonta l tangent. It is seen that this is app rox imate l y true of the predicted curves.) T h e compar ison in terms of the fai lure loads m a y be better i l lust rated by p l o t t i n g P/Pcr against e0/t as shown in F i g . 11.10. T h e plot includes two curves, one of wh ich represents i ns tab i l i t y fa i lure generated by the a lgo r i thm s imi la r to the curve in F i g . 11.5. T h e other defines mate r i a l fa i lure , w h i c h is converted and shrunk (by the slenderness effect) f rom the P—M capac i ty curve g iven i n F i g . 11.9. It is clear that when load ing eccentr ic i ty is s m a l l , the w a l l is governed by mate r i a l fa i lure. W h e n the eccentr ic i ty is great, the w a l l w i l l f a i l by i ns tab i l i t y . T h e agreement w i t h the exper iments in terms of the fa i lure loads is very good. A s expected, an increase of the reinforcement w i l l overcome the britt leness of the w a l l and prevent i ns tab i l i t y fa i lure. F i g . 11.11 shows the P—e re lat ionship for wal ls of the same conf igurat ion as the ones studied above but w i t h heavier reinforcement (3#6, np = 0.108). M a t e r i a l fa i lure governs for the whole eccentr ic i ty range as i l lust rated i n the plot . W h e n wal ls are lower, mate r ia l fai lure w i l l again govern the behaviour , as shown in F i g . 11.12 for the case of the 105 inch h igh p la in concrete w a l l w i t h smal ler eccentr icit ies. W e see aga in , by compar ing F igs . 11.10, 11.11 and 11.12, that behaviour under large eccentricit ies is s ign i f icant ly enhanced by an increase in the reinforcement. 11.6 Usefulness and L i m i t a t i o n s T h e analysis presented leads to a very at t ract ive approach to the slenderness of concrete masonry . F o r a g iven w a l l , i.e. when the dimensions a n d the parameters fu , E, A, a/b and np of 200 the w a l l are k n o w n , the P—M cross-sectional capac i ty curve and the curve def in ing the re lat ionship between buck l i ng load and eccentr ic i ty (such as the one in F i g . 11.5) can be developed. T h e designer must first ensure that the design load at the design eccentr ic i ty does not exceed the b u c k l i n g value. He is then required to make sure that the design load and the design moment at m i d s p a n (or the point of m a x i m u m deflection for unequal eccentricit ies) l ie inside the P—M capac i ty curve so that mate r ia l fa i lure w i l l not happen. T h e end m o m e n t is magni f ied to give the m i d s p a n moment . T h e magni f ier , wh ich varies w i t h P/Pcr, is a byproduct of the der ivat ion of the buck l i ng curve. T h e attractiveness of the approach lies in the fact that the buck l i ng curve as a funct ion of the load ing eccentr ic i ty is uncoupled f r o m the specif ic mate r ia l properties and d imensions of a w a l l . T h e curve is dependent on ly on the three cross-sectional parameters: namely , A, the extent of the grout and the bedding; a/b, the hollowness of the block uni t ; and r i p , the reinforcement parameters. T h u s , for any combinat ions of these parameters, the curve m a y be pre-prepared. A designer is then only required to work w i t h these prepared curves and the P—M cross-sectional capac i ty bound , w h i c h can be developed for a specific w a l l by equations g iven i n C h a p t e r X or by any other s imp l i f ied means, to determine i f the w a l l is adequate. W i t h o u t per forming a specia l , cost ly analys is for an i n d i v i d u a l w a l l , the designer is able to approach the p rob lem w i t h assured accuracy . T h i s approach, the author believes, is m u c h more ra t iona l than the current design analys is at the cost of very l i m i t e d add i t i ona l effort. T h e independence of the buck l ing load f rom the specific mate r i a l properties and d imensions of a w a l l arises f rom the assumpt ion of l inear mate r ia l re lat ionships. Fur ther , the v a l i d i t y of the approach is also based on the assumpt ion that plane sections rema in plane. T h e approach is good as long as these assumptions are s t i l l close to real i ty ; otherwise it is subject to l i m i t a t i o n s . Substant ia l nonl inear i ty m a y be caused by the y ie ld ing of the reinforcing steel in tension. T h i s m a y happen when the necessary cond i t ion specified in Chapte r X (see the context 201 of E q . 10.11) is sat isf ied, wh ich usual ly corresponds to a low steel rat io . A l t h o u g h the steel y ie ld ing can be incorporated in the a l g o r i t h m wi thout m u c h d i f f icu l ty , by changing the np va lue for appropr iate sections at wh ich the y ie ld s t ra in is exceeded, the advantage of s i m p l i c i t y is lost. If th is happens i t appears that the w a l l must be studied i n d i v i d u a l l y . However , further invest igat ion indicates that when the steel ra t io is low, the behaviour of the w a l l w i l l be governed m a i n l y by the sur rounding concrete. B u c k l i n g usual ly takes place before the y ie ld s t ra in is reached, as in the 3 # 3 reinforced wal ls studied above. Indeed, y ie ld ing of the steel was never observed in the experiments (Hatz in iko las et a l 1978), a n d the proposed procedures do give very good predict ions, as shown above. Fur ther , for very low steel rat ios, the changing of np i n the a l g o r i t h m makes very l i t t le difference i f the load ing eccentr ic i ty is not too large. A n y h o w , the steel y ie ld ing in tension corresponding to large deflections is unfavorable and m a y be prevented th rough design requirements. T h e presented procedure tends to overestimate the deflections for wal ls w i t h heavy reinforcement, as is seen w i t h the 3#9 (np=0.245) reinforced wal ls tested by Ha tz in iko las et a l (1978). T h i s is bel ieved to be m a i n l y caused by the v io la t ion of the plane section assumpt ion . W i t h heavy reinforcement, a w a l l tends to a l low development of deeper crackings in i ts m idspan region. Since the cracks occur usual ly on ly at the bed jo ints , the compressive strains between two jo in ts , i.e., w i t h i n a b lock un i t , w i l l depart correspondingly f r o m the l inear d i s t r i bu t ion as the crack depths increase. A s ind icated , the mode l assumes a l inear s t ra in d i s t r ibu t ion corresponding to an in f in i tes ima l c rack ing spacing, wh ich , of course, underest imates the r ig id i ty of the cross-section. T h e underest imat ion m a y be substant ia l when crack depth is large compared to crack spacing (unit height) , leading to erroneous results. v. . T h e non l inear i ty of concrete m a y also affect the accuracy of the approach. However , the assumpt ion of l inear mate r i a l tends to overest imate the r ig id i ty of the cross-section. F o r the cases compared , the approach gives good results for reinforcement up to 202 n/? = 0.108, which corresponds to a steel ratio up to about 1% with respect to the gross cross-sectional area. This covers most of the normal design reinforcement range. Thus the approach will be useful for many design cases without major modifications. 11.7 Some Simplifications To examine material failure, the method uses the moment magnifier which is produced by the algorithm during generation of the buckling load curve. For a given cross-section, the magnifier is a function of the loading eccentricity as well as the magnitude of the load. For a plain solid section (A = a/&=n/9 = 0), the relationship is plotted in Fig. 11.13. In the figure, the two curves running from the origin through the upper right part represent the linear solutions for a member with an uncracked section; the lower one is exact and the upper one is the commonly adopted approximation (see Eq. 11.36). The four lower curves define the magnifier for four different eccentricities. For the smallest eccentricity (e0 = l/9t), the curve coincides with the linear solution when the load is small. It begins to depart therefrom at about P/Per = 0.32, indicating that the cross-section has started to crack and that nonlinearity in moment-curvature has started to develop at this point. Three other curves, which correspond to loading eccentricities equal to or larger than the kern eccentricity (ek = l/6t), depart from the linear solution at the origin. This indicates that the cross-section begins to crack as soon as the member is loaded. As expected, the nonlinearity leads to a larger magnifier, as clearly illustrated in the plot. Due to the nonlinearity, the member, if it does not fail materially first, will eventually buckle, and therefore these curves are terminated at the buckling load Pk. Fig. 11.14 shows a similar plot for a member with reinforcement (A = a/6=0, n/> = 0.05). It appears that the ductility of the member at the greater eccentricities, in the sense of the deflection development before instability, is greatly improved by the reinforcement. For design purposes, the procedure can be simplified. The magnifier curve for a given eccentricity may be characterized by two parameters, the buckling load, Pk, and the 203 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 0 .1 Linear, Exact Linear, Approxiiiated Eo/T=1/9 / Eo/T-1/6 Eo/T=2/9 ^ — Eo/T-5/18 1.2 1 .4 1.6 1.8 MAGNRER 2 . 2 2 . 4 F I G . 11.13 M o m e n t Magni f ier versus L o a d for a P l a i n Section 0 . 6 0 . 5 -0 . 4 0 . 3 0 . 2 0 .1 np =0.05 Linear Solution 2 . 2 2 . 4 MAGMFER F I G . 11.14 Moment Magni f ier versus L o a d for a Reinforced Section 204 CL 1 1.2 1.4 1.6 1.8 2 2.2 , 2.4 MAGNFER F I G . 11.15 M o m e n t Magnif ier : E x a c t versus A p p r o x i m a t i o n F I G . 11.16 C r i t i c a l L o a d and C r i t i c a l M o m e n t Magni f ier Versus Eccent r ic i ty : for Purpose of Design Ana lys i s 205 corresponding magni f ier Sk. If the curve can be constructed by some means to end w i t h a hor i zonta l tangent at this po int , i t m a y be accurate enough for design. F o r th is purpose we int roduce the f o r m wh ich passes through the or ig in and reaches the end point w i t h zero slope. T h i s gives sat isfactory f i t t i ng for the p la in section case, as shown in F i g . 11.15. F o r a reinforced cross-section at large eccentr ic i ty , the curve m a y be t runcated at the beginning of the f lat p lateau to y ie ld a good f it (cf. F i g . 11.14). T h i s w i l l lead to l i m i t e d errors since usual ly values wel l below the b u c k l i n g load are of interest. T h r o u g h th is s imp l i f i ca t ion , we only need to know the b u c k l i n g load Pk and the corresponding magni f ier 8k for a given cross-section at given eccentr ic i ty . These two parameters can be pre -determined and exhib i ted i n the f o r m of tables, or graphs such as the one shown in F i g . 11.16. T h e f igure is for a p la in section w i t h different cross-sectional factors A. Note that two scales are used for the ordinate so that Pk and 6k can be p lot ted i n the same graph . F o r commerc ia l l y ava i lab le b lock units , the range of va r ia t i on i n a/b is s m a l l . T h e parameter A varies f r o m about 0 up to 1; np also has an upper l i m i t (of 0.1 for the t i m e being). T h u s the combinat ions of these three parameters are l i m i t e d and it is not i m p r a c t i c a l to prepare tables or graphs of Pk and 6k for design purposes. F i n a l l y , i t m a y be wor th repeating the approach wh ich has been developed and wh ich is strongly recommended for design purposes: 1) Select the w a l l cross-section, and , using the mate r ia l properties and d imensions construct the P—M cross-sectional capac i ty curve (or choose a pre-prepared one). T h i s is a wel l developed procedure except that i t is recommended that the curve be based on the un i t strength. (The equations i n Chapte r X m a y be used.) 11.37 206 2) Ca lcu la te the Eu le r l oad for the gross section PCr-3) Determine parameters A, a/b and np. (these m a y have been determined in the first step) A c c o r d i n g to these three parameters, choose an appropr iate pre-prepared buck l i ng load graph (such as the one shown i n F i g . 11.16) or table. E x a m i n e the s tab i l i t y by checking whether the design load (P/PCr) is below the buck l i ng load (P^/Pcr) at the design eccentr ic i ty (e 0 /<). If not, repeat f r o m step 1. 4) T o check mate r ia l fa i lure, read Pk and 6k at the design eccentr ic i ty ( in terpo lat ion often necessary), a n d calculate the m o m e n t magni f ier 6 by using E q . 11.37. 5) M a g n i f y the design end moment by S and ensure that mate r ia l fa i lure w i l l not occur by checking i f th is moment combined w i t h the design load fal ls w i t h i n the P—M cross-sectional curve. If not, repeat f r o m step 1. T h e recommended design approach, the author believes, can be extended to the case of unequal eccentricit ies (which is inc luded i n the a lgor i thm) w i thout m u c h d i f f icu l ty . 207 C H A P T E R X I I S U M M A R Y A N D C O N C L U S I O N S 1) T h e mechan ica l properties of concrete masonry subject to a x i a l compression and out plane bending have been invest igated exper imenta l ly , by testing block pr isms w i t h var ious bedding and grout ing condi t ions under var ious eccentricit ies. 2) S p l i t t i n g fa i lure has been examined and H i l s d o r f s mode l has been revised in the l ight of both exper imenta l and ana l y t i ca l work. It is concluded that the sp l i t t i ng fa i lure mode of concrete masonry under a x i a l compression cannot s imp ly be a t t r ibuted to the lower stiffness of the mor ta r j o in ts . 3) B r i t t l e fa i lure under u n i a x i a l compression has been invest igated at the fundamenta l level . A qua l i ta t i ve mode l was proposed to exp la in the sp l i t t i ng fai lure, and to reveal some of the character ist ics of concrete and other b r i t t le mater ia ls under ax ia l compression. 4) T h e j o in t effect on masonry strength can be a t t r ibu ted to the d is tor t ion of the un i fo rm compressive stress in the v ic in i t y of the jo in t . 5) T h e deep beam bending model proposed by Shr ive for fa i lure of face-shell bedded masonry under a x i a l compression has been reviewed and verif ied exper imenta l ly . 6) Based on the fa i lure mechan ism and j o in t effect study, i t is concluded that concentr ic and eccentric capacit ies should be treated dif ferently. It is shown that eccentric capac i ty can be sat is factor i ly predicted on the basis of masonry uni t compressive strength. 7) T h e behaviour of grouted masonry is h igh ly governed by the deformat ion properties of the masonry const i tuents. P remature c rack ing is caused by the i n c o m p a t i b i l i t y between block shell and grout . T h e u l t imate capac i ty is more strongly governed by the strength of the block shel l . 8) Based on the above observations, an ana ly t i ca l mode l considering ver t ica l as wel l as cross-sect ional de format ion in teract ion has been presented wh ich gives satisfactory predict ions for u l t i m a t e capac i ty and crack ing loads. 208 9) Based on the observations and studies on the masonry p r i s m character ist ics, a theoret ical mode l has been developed to s tudy the slenderness and the s tab i l i t y of concrete masonry wal ls . C o m p a r e d w i t h exper iments, the model gives very good predict ions for low and moderate reinforcement rat ios . 10) T h e geometry, g rout ing , and bedding condit ions a n d the reinforcement are quant i f ied by a few parameters, a n d the mode l is presented i n a re lat ively s imple f o rm . 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A general re lat ion for strengths of concrete specimens of different shapes and sizes, J. of A.C.I. Oct., 1095-1109. Obert , L . (1972). " B r i t t l e f racture of rocks." Fracture VII H . L iebowi t z (Ed . ) , A c a d e m i c Press, New Y o r k and L o n d o n , 93-155. Panas juk , V . V . (1976). Stress d i s t r ibu t ion around cracks i n plate andshells. Naukova Dumka, K i e v , 444. P a r k , R. and Pau ley , T . (1975). Reinforced concrete structures. W i l e y , New Y o r k . Paterson , M . S. (1978) Experimental rock deformation — the brittle field. Springer, B e r l i n . Pr iest ley , M . J. N . a n d M c G . E lder , D. (1982). Seismic behaviour of slender concrete masonry shear wal ls . Research report ISSN 0110-3326, Depar tment of C i v i l Engineer ing . Un ive rs i t y 213 of Cante rbu ry . Pr iest ley , M . J . N . and H o n , C . Y . (1983). P red ic t ion of masonry compressive strength f rom const i tuent properties. Research Report, Depar tment of C i v i l i Engineer ing , Un ive rs i t y of Cante rbu ry . Pr iest ley , M . J . N . , a n d E lder , D. M . (1985). Stress st ra in curves for unconf ined and confined concrete masonry , / . of A. C.L, V.80, No.3, May/June, 192-201. R o y e n , N . (1937). Kn ick fest igke i t Exzent r i sch Beanspruchter Sau len A u s Baustoff , der Nur Gegen Druck Widerstandsfdhig 1st. Der Bauingenieur, vol. 18, p444. Sah l in , S. (1971). Structural masonry. Eng lewood C l i f fs , N . J . , Prent ice H a l l . S a m m i s , C . G . and A s h b y , M . F. (1986) " T h e fa i lure of br i t t le porous solids under compressive stress states." Acta metallVol. 34, N o . 3, 511-526. Sant iago, D. S. a n d Hi lsdorf , H . K . (1973). " F r a c t u r e mechanics of concrete under compressive l oad . " Cement and Concrete Research. V o l . 3 , 363-388. Seldenrath , I. T . R. and G r a m b e r g , I. J . (1958). "St ress -s t ra in relations and breakage of rocks ." 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Steif, P . S. , (1984). " C r a c k extension under compressive load ing . " Engng. Fract. Mech. 20, 463-473. Sturgeon, G . R . a n d L o n g w o r t h , J . (1985). Reinforced concrete block masonry co lumns, Proc. Third North American Masonry Conference. 20-1, 20-16. Suwa lsk i , P . D. and Drysdale , R. G . (1986). Influence of slenderness of the capac i ty of concrete block wal ls . Proc. of 4th Canadian Masonry Symposium. 122-135. T h u r s t o n , S. J . , C y c l i c R a c k i n g (1981) Tests of reinforced concrete masonry shear wal ls , M. W. D. Central Labs. Report No. 5 -81 /8 . T i m o s h e n k o , S. and Kr ieger , S. W . (1959), Theory of Plates and Shells Sec. E d t . M c G R A W -H I L L book company . T r o x e l l , G . E , Dav is , H . E . and K e l l y , J . W . (1968) Composition and properties fo concrete, 2nd edt. M c g r a w - H i l l C i v i l E n g . Series. T u r k s t r a , C . J , a n d T h o m a s , G . R. (1978). S t ra in gradient effects i n masonry , Research Report, Depar tment of C i v i l Engineer ing , M c G i l l Un ivers i ty . W a n g , P . T . , Shah , S. P . and N a a m a n , A . E . (1978). Stress-stra in curves of n o r m a l and l ightweight concrete in compression. ACI J. Nov . 603-611. W o n g , H . E . and Drysdale , R. G . (1983) Compress ion character ist ics of concrete block masonry pr isms ASTM. STP 871, 167-177. W a r w a r u k , J . ; L o n g w o r t h , J . and Feeg, C . (1986). Response of masonry c o l u m n using s tandard w a l l un i ts . Proc. of 4th Canadian Masonry Symposium. 894-909. Wawers ik , W . H . and Fa i rhurs t , C . (1970). A study of br i t t le rock fracture i n laboratory compression exper iments. Int. J. Rock Mech. Min. Sci. 7, 561-575. X u , Z . (1979). A concise course of elasticity, 1979. Q i n g h u a Un ivers i ty ( in Chinese) . Y o k e l , F . Y . a n d D ikkers , R. D. (1971). Strength of load bear ing masonry wal ls . Journal of the Structural Division, Proceedings of the ASCE M a y , 1593-1609. 215 Y o k e l , F . Y . (1971) S t a b i l i t y and load capaci ty of member w i t h no tensile strength. Journal of the Structural Division, Proceedings of the ASCE July, 8253-1925 Zaitsev, Y . (1983) " C r a c k propagat ion i n a composite mate r i a l . " Fracture Mechanics of Concrete F. H. Wittmann {Ed.), Elsevier , 251-299. Ziegeldorf, S, (1983). "Phenomeno log ica l aspects of the fracture of concrete" Fracture Mechanics of Concrete F . H . W i t t m a n n (Ed. ) , E lsevier , 31-41. 216 APPENDICES APPENDIX A. Expressions for dU and dR in Chapter III The strain energy for a linear-elastic body with volume T is u= 4- <rij(ij dr A l For a cracked body, as long as the cracks have not gone through the body, so that the region is still connected, general energy relations should hold as for a solid body. Without loss of generality, consider an elastic body containing a single crack, as shown in Fig. A . l . At the equilibrium state, we have f •!>,. ds + \ QtVi ds r aij£ij dr = 2U ( in view of Eq. Al) A2 FIG. A l An Elastic Body Containing a Single Crack as a result of the application of the divergence theorem, equilibrium and compatibility conditions; where T{, u,- and Qt, vt denote the tractions and associated displacements on the external boundary Tx and internal T 2 (crack surface), respectively. Note the integral path we have chosen; the repeated path does not contribute. If the surface of the crack is free, then Qt = 0. If the opposite surfaces of the crack 217 slide against each other, as is the case in the model, then Qn ^ 0 and vt ^ 0, but Qt = vn = 0, so that Qivi = 0; where subscripts n and t denote normal and tangential components respectively. Therefore, it is always true that !>,• ds A3 regardless of how this crack extends within the material. Of course, for our model this is U = - - i - FA . A4 If friction between the crack surfaces is included, the situation becomes more complicated. Restricting attention to our model, we have Qt = / , the friction force, which can be related to the applied force when crack surfaces are sliding against each other: • 9 r *> f = au sin a = -^j-u sin a A5 (Recall that w is the specimen width and F is the applied force). vt may be approximated by the geometric relation between the crack opening and the sliding displacement, as shown in Fig. A2 vt « 6/s'mct A6 In view of Eq. A2, the expression for the strain energy becomes U = - i - ( FA - 2a MfS/sina ) A7 The negative sign preceding the second term indicates that the friction force is in the opposite 218 (5/s ma 21 F I G . A 2 Geometr ic Re lat ion between C r a c k . Opening and S l id ing Displacement F I G . A 3 A C rack Extended by a P a i r of Sp l i t t i ng Forces d i rect ion to that of the s l id ing displacement. A n expression for the crack opening 6 is s t i l l needed. Consider a crack w i th i n i t i a l length 2l0 being extended to 21 under the act ion of a pair of forces P, as shown in F i g . A 3 . A c c o r d i n g to the energy theorems concerning the format ion and extension of cracks in the elastic sol id (Goodier 1968), we have I Gj dl A 8 Rearranging the equation and not ing E q . 3.5, we obta in 219 rl 4P(1-^ 2) _ AP(l-v2) ( tan(7r//26) \ 8 = Eb sin (rrl/b) TTE \ tan(7r/0/26) J A9 Eq. 3.13 follows when Eq. A5 and Eq. A9 are subsituted into Eq. A7 It should be noted that in a later stage of the post-peak branch, when the crack opening width is decreasing, the friction force will change direction, and the sign preceding u contained in the expression should be changed. When crack surfaces are sliding against each other, the energy dissipated by friction is Eq. 3.17 follows when Eq. A9 is differentiated and substituted into this expression. APPENDIX B. Solution of equation 3.10 After making the appropriate substitutions and rearranging, some cancellation occurs and Eq. 3.10 reduces to which is solved by the method of variation of parameters, as shown further below. Letting A = H(J)F(t) and substituting, it follows that dR = 2a M f dvt = 2a M fia sina d8 A10 A l l ( recall F= aw) A12 220 in w h i c h relat ions g iven by E q . 3.2 and E q . 3.4 are app l ied . E q . A 1 2 is then integrated and matched to the i n i t i a l cond i t ion g iven by E q . 3.18. T h e so lut ion takes the f o r m as given by E q . 3.22 when the relat ions defined by Eqs. 3.6, 3.19, 3.20 and 3.21 are used. T h e so lut ion is v a l i d for the whole range except the f r ic t ion t rans i t iona l in terva l , i n wh ich the f r ic t ion force is changing magn i tude as wel l as d i rect ion ; the re lat ion given by E q . A 5 does not then ho ld . C e r t a i n l y , after the t rans i t iona l in terva l , the sign preceding u should be changed. T h e s ta r t ing point of this t rans i t iona l in terva l m a y be found by sett ing dS equal to zero. T h i s cond i t ion fo l lows by di f ferent iat ing E q . A 9 : (recall 0 = arc sm(<r/fc)2, 0 O = arc sin(<r0/'fc)2, and recognize also arc sm(a/fc)2= irl/b ) W e define this s ta r t ing point by 0 X or ax (quantit ies at this po int denoted w i t h subspr ipt 1). D u r i n g the t rans i t iona l i n te rva l , the expression for the tensile sp l i t t i ng force becomes P = 2a ( i r s i n a cosa — / ) s i n a A 1 4 T h e crack extension cond i t ion is s t i l l governed by E q . 3.4. Recogn iz ing that dur ing this in te rva l the crack opening remains constant , we have an ext ra cond i t ion *W-"2) log ptglM = 8, = constant A 1 5 TTE 6 t a n ( 0 o / 2 ) T w o cases need to be discussed. F i rs t we assume / increases dur ing the in terva l . It is obvious then that E q . 3.4 and E q . A 1 5 can not be satisf ied s imul taneous ly . Fu r the r inspect ion indicates that the force defined by E q . A 1 5 is a lways higher than that def ined by E q . 3.4. T h i s 221 actua l l y impl ies that the crack w i l l extend immed ia te l y and the mate r i a l w i l l f a i l a lmost at the instant the s ta r t ing point is reached. In the second case, i f we assume that the app l ied load retreats so fast that / or P remains unchanged du r ing the in terva l , then both E q . 3.4 and E q . A 1 5 are sat isf ied, and the f r ic t ion force / can be found by using the re lat ion given by E q . A 1 4 . In view of E q . A 7 , the s t ra in energy is then expressed as FA — 2a M [ crs inacosa — — ' 2a s i n a / s i n a A 1 6 Fu r the r since P = Px remains constant , the f in ish point , defined as <7 2 , can be found by equat ing the expression for P at the beginning point to that at the f in ish point , as cosa — zzsina . , _ r, = — — A 1 7 ' cosa + ^ s i n a 1 Since db = dl = 0, the di f ferent ial equat ion of the energy re lat ion reduces to dV - dU = 0 al < a < cr2 A 1 8 A f t e r subst i tu t ing E q . 3.11 and E q . A 1 6 , the so lut ion of Eq. ' A 1 8 turns out to be a l inear re lat ion between stress a n d s t ra in : a 6, cosa • _ e - L_5 (- C7cr A 1 9 2b2 where C is an integra l constant found by the condit ions at the s tar t ing point . T h i s re lat ion is used in p lo t t i ng F i g . 3.9. 222 A P P E N D I X C . So lu t ion of equat ion 4.1 A series so lut ion can be formed by the eigen funct ions of the p rob lem: oo u(x,y) = ^2 Xn(x) sm(any) A 2 0 n = l w h i c h satisfies boundary condit ions specified by Eqs. 4.3 and 4.4; where Xn(x) is a funct ion of x and a n = mr/t0. W h e n A 2 0 is subst i tuted in to E q . 4.1, i t fol lows that Xn(x) - k2al Xn{x) - 0 n = l , 2 oo A 2 1 R e c a l l that K — ^ {\-v)/2. A 2 1 is integrated and subst i tuted back in to A 2 0 , w h i c h , when boundary cond i t ion E q . 4.2 is app l ied , reduces to oo u(x,y) = ^2 An sinh(/can£) sin(a„j/) A 2 2 71=1 where An is a constant , wh ich is then found by m a t c h i n g A 2 3 to the boundary cond i t ion E q . 4.5: oo ^^AnKctn cosh^^,"" j sin(a„2/) = 1 " q 71=1 to t0KCX„ C O S h ( ^ ) J b 4(l-V2)qto . . . . when n is odd ^ T T ^ C O S ^ ^ ) A 2 3 when n is even 223 E q . 4.6 fo l lows when A 2 3 is substuted into A 2 2 . A P P E N D I X D. Coeff icients Am, Bm i n stress funct ion 4> specified by equat ion 4.11 E q . 4.11 is actua l l y a s u m m a t i o n of Levy 's type solut ions for plate bending (T imoshenko a n d Kr ieger , 1959), so i t is clear that V 4 $ = 0 A 2 4 A n d i t is also obvious that E q . 4.11 is constructed acoording to the d iamet r ica l l y s y m m e t r i c properties of the p rob lem, so that on ly boundary condit ions at x=0 and ?/=0 need to be considered. Referr ing to F i g . 4.7, i f we integrate the boundary d a t a we have <3> = cxy + c 2 A 2 5 w i t h ® x = c 3 A 2 6 at the boundary x = 0; and <& = c4x + c5 A 2 7 w i t h •*v(*) = TXy(xfi) dx + C 6 4qt0 ^ c o s h [ ( 2 n - l ) / c 7 r ( : E - a / 2 ) / < 0 ] ^ ^ I " 2 ^ ( 2 n - l ) 2 c o s h [ ( 2 w - l ) K 7 r a / 2 * 0 ] ° 6 224 at y = 0; where cx to c 6 are constants of in tegrat ion . Note in A 2 8 , rxy is specif ied by E q . 4.7 but w i t h the shifted or ig in , q and c 4 must van ish for symmet ry of the p rob lem. T h u s 4> is constant a long the boundaries, and E q . 4.11 assumes c 2 = c 5 = 0 by the fact that i t is i m m a t e r i a l to a d d a constant to <£. Fur ther , the slopes defined by A 2 6 and A 2 8 must also van ish at the or ig in (r=2/=0) because of the constancy. T h i s leads to c 3 = 0 and c f i = - 2*2 A 2 9 W h e n E q . 4.11 is dif ferentiated and set equal to the boundary condi t ions , we ob ta in two equations: oo i — _ \Am<Vm(y) + -fBmbm s i n ^ y j = 0 A 3 0 TO=1 and 53 [ -TTAmam s i n ( ^ z ) + 5mSGm(x)~| = $ 9 ( i ) A 3 1 m = l where S m ( « ) = !p s inh !22E(W2) - ^ t a n h ^ c o s h ^ ^ j = ^ sinh - a m t a n h a m c o s h ^ % ^ am = ^ ((<*mtanha: m — l ) s i n h a m — a m c o s h a m j 6 m = nm. ( ( / ? m t a n h / ? m - l ) s i n h / ? m - / ? m c o s h / ? m ) and <&y(x) is g iven by A 2 8 . B y or thogonal i ty , A 3 0 and A 3 1 become two l inear system equations, 225 wh ich can be wr i t ten symbo l i ca l l y where A 3 2 Am "f" kmnBn — On m, n = 1, 2 oo A 3 3 kmn ni / \ • miry . kb - -L S G n ( i ) sin^P (fe fcy(x) sin^P <fe Therefore, for f in i te size iV, i t is a lways possible to solve for Am and Bm, m = l , 2, N: {A} =[[!}- [kb)[ka]j\c} A 3 4 {5} = -[ *»] {^ } A 3 5 in view of A 3 2 and A 3 3 ; where [ I ] is an ident i ty m a t r i x of size N, and hence to obta in an a p p r o x i m a t i o n for In Chapter I V , N=8 was used. 226 A P P E N D I X E . D e r i v a t i o n of equat ion 11.5 T h e general so lut ion for E q . 11.1 is y = A s in + B cos P_ EI A 3 6 B= — t0 for the boundary cond i t ion at the top. W h e n the boundary condi t ions at section C are app l ied , we o b t a i n fo l lowing two relat ions A s in + t0 C O S P_ (h—h EI A 3 7 P_ A cos ( ^ ) ) - ( ^ ) ) A 3 8 wh ich lead to A2 + e2 e c + fc -p A 3 9 ( o o \—1/2 yl + e 0 1 , i t can be rewr i t ten sin U P , p /l-hc/h\ . _! -— ' — w + sin (—2 )* ^ 2 + e 2 J .4 2 + e 2 A 4 0 - l e c - l e„ ^ 2 + e 2 -JA 2 + e 2 A 4 1 E q u a t i o n 11.5 fo l lows when the re lat ion given by A 3 9 is used. 227 A P P E N D I X F . Integration of equations 11.14 and 11.16 By letting jC1 - ft)'(1 + + )•- *(+)*] and <h(c) = ( 1 - i-b f - ( A f - nP)(l - f) A42 A43 Eqs. 11.14 and 11.15 can be written as d2y = P dx2 2Eb2l G x ( c ) A44 and y _ i ^ i (c ) b ~ Gl(c) A45 A45 defines the relationship between y and c, which must be used in integration of A44. By recognizing 4-(Q\2 = 2 d2i dx\dx) dx1 A46 A44 becomes \dx) Ebl J G i (c ) V 6 J A47 A45 is then substituted into A47, which becomes, after integration by parts 'dy\2 <dx) 2Ebl EM G\{c) + Gl(c) dc 228 A 4 8 where * M = ^ = A[(i)S-2(1-)] A 4 9 i n v iew of A 4 2 . N o w the p rob lem becomes to integrate J G{{c) A 5 0 It is seen that the integrand is a rat iona l funct ion of c. T h e denominator is fo rmed by the square of the Gx funct ion , w h i c h , i n view of A 4 3 , is quadrat ic i n c. T h e numerator can be broken into two terms. T h u s the integral can be carr ied out by any s tandard approach, for example , see C R C S tandard M a t h e m a t i c a l Tab les 27th E d . p245 (Beyer 1986). A f te r appropr iate ca lcu la t ion , the result turns out to be rather s imple : _ W 6 1 _ Gi(c) A 5 1 W h e n A 5 1 is subst i tuted back into A 4 8 w i t h a constant of in tegrat ion , we obta in (2)' = A (»'«>) A 5 2 where fi^c) is def ined by E q . 11.19. E q . 11.18 takes the posit ive square root of A 5 2 referring to F i g . 11.2. A s im i la r approach is used to integrate E q . 11.16. However , the equivalent F and G funct ions become 229 T i1- 2 l ) 2 ( 1 + f ) ~ A ( ( f ) 3 - - f )2(1 + 2 f - f ) ) ] A 5 3 and Ga(c) = ( 1 - ih )2 - i( 1 a c 6 6 y ) 2 + n K A 5 4 T h e integrand of the equivalent integral _ f ^2 (C) A 5 5 2 " J G2(c) is also a ra t iona l funct ion . However, F (c) contains three terms by di f ferent iat ion of A 5 3 : E q . 11.20 fo l lows after appropr iate subst i tut ions. A P P E N D I X G . Con f igu ra t ion of a c o l u m n loaded w i t h double curvature bending W e t ry to shed some l ight on the p rob lem by invest igat ing an elastic c o l u m n . F i g . A 4 shows a c o l u m n loaded w i t h top eccentr ic i ty et and b o t t o m eccentr ic i ty eh, wh ich w i l l deflect accord ing to H (c) = ij ( 1 - A ) ( - | - ) 2 - 2 ( 1 - A ) ( i - ) - A ( 1 - ( t ) 2 ) A 5 6 A f te r a lengthy but contro l lab le ca lcu la t ion , i t turns out again in s imi la r form to A51 1-c/b G2(c) A 5 7 F I G . A 4 A C o l u m n Loaded w i t h Double Curvatu re Bend ing e, — ehcoskl . y = sinfc/ smkx + ebcoskx 230 A 5 8 where k = \P/EI. A 5 8 must vanish at the inf lect ion point x0, wh ich leads to tan fcr0 - sinfc/ coskl — et/eb A 5 9 It is clear that when et/eb = — l, i.e. the column is loaded an t i - symmet r i ca l l y , x0 = l/2. However, we w i l l show that this conf igurat ion is not stable when the Euler load is approached. F o r this purpose we introduce a smal l per turbat ion e to the load ing condit ions e « / e t = - ( l - 0 (*>0) A 6 0 and examine the sensit iv i ty of the deflected conf igurat ion. W h e n A 6 0 is subst i tuted into A 5 9 , we obta in F = (coskl + 1 — e)tankx0 — sinkl = 0 A61 T h e sensit iv ity of the conf igurat ion to the perturbat ion is reflected in the der ivat ive of x0 w i t h respect to e 231 dxo de Fe _ sin2fcE0 sinfc/ A 6 2 Fx0e_Q 2k(l+coskl) 2k(l + coskl) It is seen that when the load P is re lat ively low, the der ivat ive w i l l be s m a l l . However , when P approaches the Eu le r load , kl -» IT, i t becomes unbounded (note A 6 2 is i n an indeterminate fo rm, L ' H o s p i t a l ' s rule has been app l ied once). T h e h igh sensi t iv i ty is obvious. T h a t is , when the Eu le r l oad is approached, the c o l u m n w i l l have a very h igh tendency to depart f r o m its o r ig ina l a n t i -s y m m e t r i c conf igurat ion . In real i ty , i t is a lways reasonable to assume some imperfect ion reflected i n the s m a l l q u a n t i t y e. T h u s A 6 1 can be rewr i t ten as T h a t is, the c o l u m n w i l l assume its lowest buck l ing conf igurat ion , for any s m a l l imperfect ion e, when the Eu le r load is approached. F o r a nonl inear c o l u m n , the s i tuat ion becomes m u c h more compl icated . It appears that a s i m i l a r tendency w o u l d cont ro l the behaviour . F o r design purposes, i t is reasonable to assume, conservat ively , that this wou ld happen. A 6 3 It is seen by the second equat ion of A 6 3 that A 6 4 A P P E N D I X H . Electronic C i rcu i t Used in Detect ing Macroscop ic Sp l i t t i ng (Par t ) 2 3 2 Vcc 74LS74 O T v v D Q Vcc 0 74LS74 D Q > V V DATA TJ DATA TJ 74LS373 n v LE-OE 74LS04 74LS30 LEDs • c t 74LS32 100K Vcc 74LS373 V LE OE : LEDs 100K Vpf 4017 1234 56 V. 74LS02 74LS32 v >— 74LS02 Vcc NO / 233 A P P E N D I X J. Computer Program Calculating Buckling Load and Moment Magnifier of Concrete Masonry C C PROGRAM TO EVALUATE MAXIMUM BUCKLING LOAD OF REINFORCED MASONRY C EXTERNAL F1,F2,F3,F4 COMMON R0,RC,RS,E0,EC,EF,EE,D1,D2 DIMENSION TITLE(20) C C * * c C NOTATION OF VARIABLES C C RO = CROSS-SECTIONAL FACTOR C RC = A/B CORE RATIO C RS = STEEL RATIO C EO = (EQUAL) END ECCENTRICITY C EA = SMALLER END ECCENTRICITY C EB = LARGER END ECCENTRICITY (EB.GE.ABS(EA)) C EC = CRACKING ( KERN ) ECCENTRICITY C EF = ECCENTRICITY CORRESPONDING TO FLANGE CRACKING C D1.D2 = INTERGAL CONSTANTS C C NOTE: ALL ECCENTRICITIES ARE TAKEN AS RATIOS TO HALF DEPTH OF CROSS - SECTION C C **** * * * *** C ' C C DEFINE PARAMETERS C READ(5,1)(TITLE(I),1=1,20) WRITE(6,2)(TITLE(I),1=1,20) 1 FORMAT(20A4) 2 FORMAT(/ / ,1H1, / ,24X,20A4, / ) READ(5,3)RO,RC,RS,EA,EB 3 FORMAT(5F10.5) IF(RC.LT.O.O.OR.RS.LT.0.O.OR.EB.LT.ABS(EA))STOP 1 C PI=4.* ATAN(1.) DRT=SQRT(1.5) RST=1.+RS CF=1.-RC CU=1.+RC IF(RC.EQ.0.0.OR.RO.EQ.0.0)CU=2. CV=(RST-0.25*R0*CU*CU)/(RST-0.5*R0'CU) IFIRO.NE.1.0)CV=2./(1.-RO)*(RST-0.5•RO*CU-SQRT(RST*(RS-.RO*RC)+0.25*R0'CU"CU)) CMU=AMIN1((CV-0.001),CU) EC=E1(0.) EF=E1(CF) EU=E2(CV-0.001) IF(CV.GT.CU)EU=E2(CU) C WRITE(6,4)RO,RC,RS,EC,EF.EU 4 FORMAT!/,T25,'CROSS-SECTIONAL P R O P E R T I E S ' , / / , ' S E C . FACTOR = ' , . F8 . 3,5X,'CORE RATIO = ' ,F8 .3 ,5X, 'STEEL RATIO = ' , F 8 . 3 , / / / , ' . ' E C / B = ' , F 8 . 3 , 5 X , ' E F / B = ' , F 8 . 3 . 5 X , ' E U / B = \ F 8 . 3 ) C WRITE(6,5)EA,EB 5 F0RMAT(/,T25,'LOADING CONDITIONS' , / / , 'EA/B = ' , F 8 . 3 , 5 X , ' E B / B = ' . , F 8 . 3 , / / / , T 2 5 , 'CM/B' ,11X, 'P/PCR' , 10X , 'EM/EO' . 10X, 'ERROR') EO=EB C C EVALUATE BUCKLING LOAD C PM=0.0 DM=0.0 IF(EO.GT.EC)GO TO 200 C C END ECCENTRICITY EB LESS THAN CRACKING ECCENTRICITY C CD=CMU/30. CM=6.0 DO 500 1=1,29 CM=CM+CD IF(CM.GT.CF)GO TO 110 D1=01(CM) TH=D1-01(0.) CML=CM-5.E-6 C C SUB-FUNCTION CADRE PERFORMS NUMERICAL INTEGRATION C S1=CADRE(F1,CML,0.0,0.0001,0..ERROR) SQ=Q(TH,EC,EO) IF (EA . NE . EO)SQ=SQ+0 . 5*0.( TH , EO, EA ) P=4./(PI'PI) * (DRT'SUSQ) * *2 IF(P.LE.PM)GO TO 1000 PM=P IF(EO.NE.0.0)DM=E1(CM)/EO GO TO 505 110 IFUCM-CD) .LE.CF)PM=0,0 D1=02(CM)-02(CF)+01(CF) D2=02(CM) TH=D1-01(0.) CML=CM-2.E-6 S1=CADRE(F1,CF,0.0,0.0001,0..ERROR) S2=CADRE(F2,CML,CF,0.0001,0..ERROR) SQ=Q(TH,EC,EO) IF(EA.NE.EO)SQ=SQ+0.5 * Q(TH,EO,EA) P=4./(PI*PI)*(DRT*(S1+S21+SQ) " 2 IF(P.LE.PM)GO TO 1000 PM=P IF(E0.NE.0.0)DM=E2(CM)/E0 505 WRITE(6,470)CM,P,DM,ERROR 500 CONTINUE WRITE(6,555) GO TO 1000 200 IF(EO.GE.EF)GO TO 300 C C END ECCENTRICITY EB LARGER THAN CRACKING ECCENTRICITY C BUT LESS THAN FLANGE CRACKING ECCENTRICITY C C0=0.0 C1=0.0 CL=CF ER=0.0001 EE=EO C 235 C SUBROUTINE ROOT FINDS CO FOR GIVEN EO C CALL ROOT(CO,CL,F3.ER) EE=ABS(EA) IF( EA . NE . EO . AND . E E . G T . E O C A L L ROOT ( C 1 , CL , F3 , ER) CD=(CMU-CO)/30. CM=CO DO 510 J=1,29 CM=CM+CD IF(CM.GT.CF)GO TO 210 D1=01(CM) IF(EA.LE.EC)TH=D1-O1(0.) CML=CM-2.E-6 S1=CADRE(F1,CML,CO,0.0001,0..ERROR) IF(EA.NE.EO.AND.EA.GT.EC)S1=S1+0.5*CADRE(F1,CO,C1,0.0001,0.,ERR) IFtEA.LE.EC.AND.EA.GE. - EC)S1=S1+0.5*CADRE(F1,CO,0.,.0001,0. ,ERR) .•0.5/DRT*Q(TH,EC,EA) IF(EA.LT.-EC)S1=S1+0.5*CADRE(F1,C0,0..0.0001,0.,ERR)+0.5/DRT* . Q(TH, EC,-EC)+0.5*CADRE(F1,C1,0.0,0.0001,0.0,ERR) P=6./(PI*PI)*S1*S1 IF(P.LE.PM)GO TO 1000 PM=P DM=E1(CM)/E0 GO TO 515 210 IF((CM-CD).LE.CF)PM=0.0 DT=02(CM)-02(CF)+01(CF) D2=02(CM) IF(EA.LE.EC)TH=D1-01(0.) CML=CM-2.E-6 S1=CADRE(F1,CF,CO,0.0001,0..ERROR) S2=CADRE(F2,CML,CF,0.0001,0..ERROR) SS=S1+S2 IF(EA.NE.EO.AND.EA.GT.EC)S'S=SS+0.5*CADRE(F1,CO,C1,0.0001,0..ERR) IFtEA.LE.EC.AND.EA.GE. - EC)SS=SS+0.5*CADRE(F1.CO.O.,.0001,0.,ERR) .+0.5/DRT*Q(TH.EC,EA) I F ( E A . L T . - EC)SS=SS+0.5*CADRE(F1,CO,0.,0.0001,0.,ERR)+0.5/DRT* .Q(TH,EC,-EC)+0.5* CADRE(F1,C1,0.0,0.0001,0.0.ERR) P=6./(PI*PI)*SS'.*2 IF(P.LE.PM)GO TO 1000 PM=P DM=E2(CM)/E0 515 WRITE(6,470)CM,P,DM,ERROR 510 CONTINUE WRITE(6,555) GO TO 1000 300 IF(E0.GE.EU)G0 TO 900 C C END ECCENTRICITY EB LARGER THAN FLANGE GRACKING ECCENTRICITY C CO=CF C1=CF C2=0.0 CL=CMU CL2=CF ER=0.0001 EE=E0 CALL R00T(C0,CL,F4,ER) EE=ABS(EA) 236 470 520 555 900 920 1000 C C C C C +0.5*CADRE(F1,CF,C2,0. IF(EA.LT.EG.AND.EA.GE. IF(EA.NE.E0.AND.EE.GT.EF)CALL ROOT(C1,CL,F4,ER) IF(EE.LE.EF.AND.EE.GT.EC)CALL ROOT(C2,CL2,F3,ER) CD=(CMU-C0)/30.0 CM=CO DO 520 K=1,29 CM=CM+CD IF(EA . LE . EF)D1=02(CM)-02(CF)+01(CF) D2=02(CM) IF(EA.LE.EC)TH=D1-01(0.) CML=CM-2.E-6 S2=CADRE(F2,CML,CO,0.0001,0..ERROR) IF(EA.NE.EO.AND.EA.GE.EF)S2=S2+0.5* CADRE(F2,CO,C1,0.0001.0.,ERR) IF ( EA.LT.EF.AND.EA.GE.EC)S2=S2 + 0.5 * CADRE(F2,CO,CF,0.0001,0. ,ERR) 0001,0.,ERR) -EC)S2=S2+.5 * CADRE(F2,CO,CF,0.0001,0. ,ERR) +0.5*CADRE(F1,CF,0.,0.0001,0.,ERR)+0.5/DRT•Q(TH,EC,EA) IF(EA . LT . - EC.AND.EA.GE.-EF)S2=S2+.5* CADRE(F2,CO,CF,.0001,0. ,ERR) + 0.5*CADRE(F1,CF,0. ,0.0001,0. ,ERR)+0.5/DRT*Q(TH,EC,- EC) +0.5*CADRE(F1,C2,0.,0.0001,0.,ERR) IF(EA.LT.-EF)S2=S2 + 0.5* CADRE(F2,CO,CF,0.0001,0. ,ERR)+CADRE(F1,CF ,0.0,0.0001,0. ,ERR)+0.5/DRT*Q(TH,EC,- EC)+0.5 * CADRE(F2,C1.CF, 0.0001 ,0. ,ERR) P=6./(PI*PI)*S2*S2 IF(P.LE.PM)GO TO 1000 PM=P DM=E2(CM)/E0 WRITE(6,470)CM,P,DM,ERROR F0RMAT(T20,3(F10.3,5X),G12.3) CONTINUE WRITE(6,555) F0RMAT(T20,'(MAXIMUM LOAD NOT REACHED FOR THE CRACKING RANGE)') GO TO 1000 WRITE(6,920) F0RMAT(T20,'THE ECCENTRICITY IS TOO BIG FOR THE CROSS - SECTION') STOP END FUNCTION DEFINING INTERGRAND 1 FUNCTION F1(C) COMMON R0,RC,RS,E0,EC,EF,EE,D1,D2 T1 = 1 .-0.5*C T2=1.-C T3=R0*RC-RS T4=T1*T1-T2*T3 YP=(0.5"C*T1-(T1-T3)*E1(C))/T4 DEN=D1 -01(C) IF(DEN.LE.0.)DEN=D1* 1.E-6 F1=YP/SQRT(DEN) RETURN END FUNCTION DEFINING INTERGRAND 2 FUNCTION F2(C) COMMON R0,RC,RS,E0,EC,EF,EE,D1,D2 237 T1=1.-0.5'C T2=1.-C T3=1.+RC-C T4=T1*T1 -0.25'R0*T3*T3+RS'T2 YP=(0.5*(1.-RO)* T1*C + 0.25'RO'(1.-RC'RC) . - (T1-0. 5'R0*T3 + RS)*E2(C))/T4 DEN=D2-02(C) IF(DEN.LE.0.)DEN=D2'1.E-6 F2=YP/SQRT(DEN) RETURN END C C FUNCTION DEFINING CRACKING ECCENTRICTY 1 C FUNCTION F3(C) COMMON RO,RC,RS,EO,EC,EF,EE,D1,D2 F3=EE-E1(C) RETURN END C C FUNCTION DEFINING CRACKING ECCENTRICTY 2 C FUNCTION F4(C) COMMON R0,RC,RS,E0.EC,EF,EE,D1,D2 F4=EE-E2(C) RETURN END C C FUNCTION DEFINING THE TERM IN INTERGRAND 1 C FUNCTION 01(C) COMMON R0,RC,RS,E0,EC,EF.EE,D1,D2 T1=1.-0.5*C T2=1 .-C T 3=R0 * RC ~ RS 01=(4.*T1*T1*T1-R0*RC*RC*RC-3.*T3*T2'T2) ./6./(T1*T1-T3*T2)**2 RETURN END C C FUNCTION DEFINING THE TERM IN INTERGRAND 2 C FUNCTION 02(C) COMMON RO , RC . RS . EO , EC , EF . EE , D 1 , D2 Tl = 1.-0.5'C T2= 1 . -C T3=1 .-RC-C T4=1.+RC-C T5=R0 * RC-RS 02=(4.*T1 "T1'T1-R0*RC'RC*RC-0.5'RO'T3'T3*T3- 3.«T5*T2*T2) ./6./(T1*T1-0.25'RO*T4*T4 + RS'T2)"2 RETURN END C C INVERSING SIN FUNCTIONS C FUNCTION Q(TH,EV,EU) COMMON RO,RC,RS,EO.EC,EF.EE,D1,D2 238 T=1 .-R0*RC*RC*RC TD=SQRT(EC*EC+2./3.*T*TH) Q=SQRT(T)*(ASIN(EV/TD)-ASINIEU/TD)) RETURN END C C FUNCTION DEFINING CRACKING ECCENTRICITY 1 C FUNCTION E H C ) COMMON RO,RC,RS,EO,EC,EF,EE,D1,D2 T1=(1 . -0 .5 'C) ' (1 . -0 .5 'C) E1=(T1*(1.+C)-RO*RC*RC*RC)/3./(T1-(RO'RC-RS)'(1.-C)) RETURN END C C FUNCTION DEFINING CRACKING ECCENTRICITY 2 C FUNCTION E2(C) COMMON RO,RC,RS,EO,EC,EF,EE.D1 ,D2 . T1=(1.-0.5*C)*(1.-0.5'C) T2=( 1. -RC-C)*(1.-RC-C) T3=(1,+RC-C)'(1.+RC-C) E2=(T1 *(1.+C)-RO*(RC'RC'RC-O.25*T2"(1 .+2.*RC-C)) ) . /3./(T1-0.25*RO*T3+RS*(1 .-C)) RETURN END C C SUBROUTINE FINDING ZERO OF FUNCTION F C SUBROUTINE ROOT(A,B,F,TL) Y1=F(A) Y2=F(B) IF(Y1*Y2.GT.O.OR.Y1-Y2.EQ.O.OR.TL.LE.O.OR.A.GE.B)STOP 2 20 X=0.5*(A+B) Y=F(X) IF(Y'Y1.GT.O.)A=X IF(Y*Y1.LE.O.)B=X IF( (B-A) .GT.TDGO TO 20 A=X RETURN END 

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