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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 F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R 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  degree at the  thesis  in  University of  partial  fulfilment  of  of  department  this thesis for or  by  his  or  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  representatives.  an advanced  Library shall make it  agree that permission for extensive  scholarly purposes may be her  for  It  is  granted  by the  understood  that  head of copying  my 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-ofplane 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 TABLE OF CONTENTS PAGE ABSTRACT  -ii-  TABLE OF CONTENTS  -iii-  LIST O F T A B L E S . . LIST O F F I G U R E S  •. ,  NOTATION  -vii-ix,.-xvi-  ACKNOWLEDGEMENT DEDICATION  -xx-xxi-  CHAPTER I  II  III  INTRODUCTION  1  1.1  General Remarks  1  1.2  Object and Scope....  2  EXPERIMENTAL WORK  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  SOME B A C K G R O U N D T O COMPRESSION FAILURE OF C O N C R E T E  29  3.1  29  Purpose  - iv C H A P T E R  IV  V  P A G E  3.2  Brittle Failure under U n i a x i a l Compression  30  3.3  M o d e l s of Internal B r i t t l e Failure  35  3.4  Proposed Model  37  3.4.1  Crack Interaction and Critical State  37  3.4.2  S o m e Consequences of the M o d e l : P e a k Stress  40  3.4.3  Relation to Tensile Strength  41  3.5  T h e Stress-Strain C u r v e s for B r i t t l e M a t e r i a l s under C o m p r e s s i o n  44  3.5.1  The Pre-Peak Branch  45  3.5.2  The Post Peak Branch  50  3.5.3  T h e Predicted Stress-Strain Curve  51  3.6  Statistical Consideration  54  3.7  Summary and Corollary  57  PLAIN  M A S O N R Y  WITH FULL BEDDING  58  4.1  T w o Basically Different  4.2  J o i n t Effect  4.2.1  Experimental Results  61  4.2.2  Theoretical Analysis  64  4.2.3  Conclusion on Hilsdorfs Model  74  4.3  Some Comments  75  4.4  Joint Effect on A x i a l C a p a c i t y  76  4.5  Stress i n J o i n t V i c i n i t y  78  4.6  Capacity Eatimation  84  4.7  Summary  88  PLAIN 5.1  M A S O N R Y Introduction  Failure Modes  58  A Revision of HilsdorPs M o d e l  59  on Splitting Failure and Mode Transition Phenomena  WITH FACE-SHELL  BEDDING  89 •  89  - V C H A P T E R  VI  VII  VIII  P A G E  5.2  Experimental Work  89  5.3  Stress A n a l y s i s  94  5.4  Some C o m m e n t s on Joint Effect  97  5.5  Summary  98  PLAIN M A S O N R Y  UNDER  ECCENTRIC  6.1  Failure Mode Transition  6.2  Effect of Joint Conditions  6.3  Summary  R E C O M M E N D E D  COMPRESSION  :  100 •  105 106  DESIGN  A P P R O A C H FOR  PLAIN 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 on the Basis for Design  107 .  7.2  Discussion of the Current Design C o d e  119  G R O U T E D  M A S O N R Y  WITH FULL BEDDING  125  8.1  Introduction  125  8.2  Experimental Observations  126  8.3  Analysis  132  8.4  Summary  152  IX  G R O U T E D  X  G R O U T E D A N D REINFORCED M A S O N R Y UNDER ECCENTRIC  XI  100  M A S O N R Y WITH FACE-SHELL BEDDING  153 C O M P R E S S I O N S  10.1  General Remarks  157  10.2  Experimental Observations  157  10.3  Theoretical Considerations  161  10.4  C o m p a r i s o n of Theory with Experiments  SLENDERNESS OF  C O N R E T E  ...167  M A S O N R Y  11.1  Introduction  11.2  Background Information Review  174 174 '.  ...175  - viCHAPTER  XII  PAGE  11.3  Masonry Characteristics and Some Assumptions  .....178  11.4  Differential Equations Governing Concrete Masonry w i t h Cracked Section  179  11.5  Results and Applications  191  11.6  Usefulness a n d L i m i t a t i o n s  199  11.7  Some Simplifications  202  CONCLUSIONS  207  REFERENCES  209  APPENDIX A  E x p r e s s i o n s for dU a n d dR  B  S o l u t i o n of e q u a t i o n 3.10  C  S o l u t i o n o f e q u a t i o n 4.1  D  Coefficients A ,  E  D e r i v a t i o n o f e q u a t i o n 11.5  226  F  I n t e g r a t i o n of equations 11.14 a n d 11.16  227  G  C o n f i g u r a t i o n of a c o l u m n l o a d e d w i t h d o u b l e c u r v a t u r e b e n d i n g  229  H  Electronic C i r c u i t Used i n Detecting Macroscopic Splitting  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 a n d M o m e n t M a g n i f i e r of  m  B  . Concrete Masonry  m  i n C h a p t e r III  216 219 .  i n stress f u n c t i o n <3> specified b y e q u a t i o n 4.11  (Part)  222 223  ....233  - vii LIST OF  TABLES  TABLE  PAGE  2.1  F a i l u r e L o a d s 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 S t r e n g t h  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 Strength by Standard P r i s m Tests  11  2.6  G r o u t S t r e n g t h b y T e s t s o n Cores T a k e n f r o m 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 a n d C a p a c i t y C h a r a c t e r i s t i c s  23  4.1  F a i l u r e L o a d s of P l a i n P r i s m s w i t h F u l l B e d d i n g  64  5.1  F a i l u r e L o a d s of P l a i n P r i s m s w i t h F a c e - S h e l l B e d d i n g  91  6.1  F a i l u r e L o a d s of P l a i n P r i s m s under E c c e n t r i c L o a d  100  7.1  C o m p a r i s o n w i t h the R e c o m m e n d e d A p p r o a c h : T e s t s 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 R e c o m m e n d e d A p p r o a c h : T e s t s 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 R e c o m m e n d e d A p p r o a c h : T e s t s by H a t z i n i k o l a s et a l  116  7.4  C o m p a r i s o n w i t h the R e c o m m e n d e d A p p r o a c h : T e s t s by D r y s d a l e et a l  117  7.5  F l e x u r a l t o U n i a x i a l S t r e n g t h : T e s t s 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 S t r e n g t h : T e s t s 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 S t r e n g t h : T e s t s b y H a t z i n i k o l a s et a l  123  7.8  F l e x u r a l to U n i a x i a l S t r e n g t h : T e s t s b y D r y s d a l e et a l  124  8.1  F a i l u r e L o a d s 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 i n J o n i t C o n d i t i o n  128  8.2  F a i l u r e L o a d s 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 i n G r o u t  128  8.3  G r o u t e d P r i s m s , T e s t s b y the A u t h o r  144  8.4  Grouted Prisms, 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 , T e s t s by D r y s d a l e H a m i d  146  - viii TABLE  PAGE  8.6  G r o u t e d Prisms, Tests by W o n g and Drysdale  146  8.7  Grouted Prisms, Tests by Priestley and Elder  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 Prisms, Tests by T h u r s t o n  147  8.10  M o d e l P r e d i c t i o n versus C r a c k i n g L o a d s , T e s t s b y the A u t h o r  152  9.1  Grouted Masonry with Face-Shell Bedding  154  10.1  F a i l u r e L o a d s o f G r o u t e d P r i s m s under E c c e n t r i c L o a d (kips)  158  - ix LIST OF  FIGURES  FIGURE  P A G E  2.1  Masonry Unit  5  2.2  Conical Failure of Masonry U n i t  6  2.3  Stress-Strain R e l a t i o n of M a s o n r y U n i t under C o m p r e s s i o n  7  2.4  Stress-Strain Relation of 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 Compressive Strains along B l o c k U n i t s a n d across M o r t a r J o i n t of P l a i n Prisms under Uniaxial Compression  10  2.6  Stress-Strain R e l a t i o n of G r o u t  12  2.7  Testing Device  16  2.8  Loading Platens  17  2.9  Instrumentation:  2.10  Electronic Device Detecting Wire Break Order  20  2.11  D a t a Acquisition Setup  21  2.12  S p l i t t i n g F a i l u r e of P l a i n Concrete M a s o n r y w i t h F u l l B e d d i n g under  L V D T s and Glued Wires  17  Uniaxial  Compression ;  24  2.13  F a i l u r e of P l a i n M a s o n r y w i t h Face-Shell B e d d i n g under U n i a x i a l Compression  25  2.14  F a i l u r e of Face-Shell Bedded, F u l l y C a p p e d M a s o n r y under U n i a x i a l C o m p r e s s i o n  26  2.15  Failure of P l a i n Masonry under Eccentric Compression  27  2.16  Failure of Grouted Masonry under Eccentric Compression  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 in a C o m p r e s s i v e Stress F i e l d , S h o w i n g the  Original  Defect and its Extension  32  3.2  D e p i c t i o n of the Effect of a C r a c k  32  3.3  M o d e l s of M a t e r i a l Defects. (The M i s s i n g Force A c t s on E a c h Side in 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 to A p p l y Forces in the Direction Shown.)  34  3.4  A Series of C r a c k s i n a C o m p r e s s i v e Stress F i e l d : T w o Levels of Idealization  39  3.5  A Series of C r a c k s in a T e n s i l e Stress F i e l d  43  - X -  3.6  Predicted Relation between Tensile strength and Compressive Strength  versus  S i z e / S p a c i n g R a t i o for B r i t t l e M a t e r i a l s 3.7  43  Experimental Stress-Strain Relations of Concrete, under N o r m a l Test  Conditions  ( W a n g 1978) 3.8  "... 4 4  E x p e r i m e n t a l Stress-Strain Relations of Concrete, Specimens 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 versus Displacement 3.9  46  E x p e r i m e n t a l Stress-Strain Relations of some N a t u r a l R o c k s ( W a w e r s i k Fairhurst  and  1970)  47  3.10  Predicted Stress-Strain Relations of B r i t t l e Materials  52  3.11  Depiction of Irregular  54  3.12  Sensitivity of Compressive Strength to Crack Configuration  Cracking Pattern Factor: the  S t r e n g t h P r e d i c t e d b y the M o d e l is P l o t t e d a g a i n s t the C o n f i g u r a t i o n Depends on Crack Configuration  Normalized  Factor,  Which  and Internal Friction  S t r e n g t h Increase P h e n o m e n o n  56  4.1  Apparent  under Eccentric Compression  4.2  M e a s u r e d L a t e r a l Strains in W e b s of M i d d l e Course of P l a i n M a s o n r y  59 Prisms  under  Uniaxial Compression  63  4.3  Detected Orders of Macroscopic Splitting, in T e r m s of 4 Sections along P r i s m s  63  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  Compression;  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  67  4.6  L a t e r a l Interface Shear D i s t r i b u t i o n between M o r t a r J o i n t a n d B l o c k U n i t s .  4.7  D e p i c t i o n of B o u n d a r y  M o r t a r Joint under Lateral T r a c t i o n  Conditions of a W e b  (or F a c e - S h e l l ) u n d e r  69  Action  of Interface Shears  70  4.8  L a t e r a l Tensile Stress along T o p of B l o c k Introduced  by the Lateral Shears  71  4.9  L a t e r a l Tensile Stress along T o p of B l o c k , w i t h V a r i a t i o n in Poisson's R a t i o of J o i n t  71  4.10  L a t e r a l Tensile Stress along T o p of B l o c k , w i t h V a r i a t i o n in J o i n t T h i c h n e s s  72  4.11  L a t e r a l Tensile Stress along T o p of B l o c k , w i t h V a r i a t i o n in D o m a i n A s p e c t R a t i o  72  - xi FIGURE 4.12  PAGE  Lateral Strains Measured along webs and Face-shells of Plain Prisms under Uniaxial Compression  73 79  4.13  A Cross-Sectional View (along the Depth of Block Shells) of Mortar Joint  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  5.2  Detected Orders of Macroscopic Splitting, in Terms of 4 Sections along Prisms. (Face-Shell Bedded Prisms)  5.3  90  90  Measured Deformations at Certain Locations of Face-Shell Bedded Prisms: a) S16-1; b) S16-2; c) M27-1  5.4  Measured Deformations at Certain Locations of Face-Shell Bedded Prisms: a) N15-1; b) S15-3; c) N15-4  5.5  92  93  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  5.6  Forces Acting on a Block with Full Capping and Face-Shell Bedding  5.7  Lateral Stress Distribution in a Web: Full Capping versus Face-Shell Capping;  95 96  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  6.2  101  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  - Xll  -  FIGURE 6.5  S t r a i n D i s t r i b u t i o n i n a S e c t i o n of M a s o n r y under E c c e n t r i c L o a d  7.1  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y the A u t h o r : N 1 8 , N 1 9 , M 2 0 a n d S21  7.2  PAGE 106  110  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y the A u t h o r : N22 (Face-Shell Bedding)  110  7.3  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y F a t t a l a n d C a t t a n e o . . . 111  7.4  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y H a t z i n i k o l a s et a l  7.5  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : Normal Block  7.6  114  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : 6 inch Block  7.11  113  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : 75% Solid Block  7.10  113  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : Light Weight Block  7.9  112  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : Strong Block  7.8  112  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : Weak Block  7.7  Ill  114  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : 10 i n c h B l o c k  115  7.12  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s : S u m m a r y  118  7.13  D e p i c t i o n of R e c o m m e n d e d A p p r o a c h  118  7.14  C u r r e n t D e s i g n Base: U n i a x i a l S t r e n g t h versus F l e x u r a l S t r e n g t h  122  8.1  Grouted P r i s m Strength Versus M o r t a r Strength and Grout Strength  128  8.2  M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s of G r o u t e d P r i s m s under C o n c e n t r i c L o a d : a) M 9 - 1 ; b) M 9 - 2 ; c) S 8 - 1 ; d) S 8 - 2  129  - xm FIGURE 8.3  M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s of G r o u t e d P r i s m s under C o n c e n t r i c L o a d : a) N 1 0 - 3 ; b)  8.4  PAGE  N 1 0 - 4 ; c) N 1 2 - 2 ; d) N 1 2 - 4  130  M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s o f G r o u t e d P r i s m s under C o n c e n t r i c L o a d : a) N 1 3 - 3 ; b) N 1 3 - 4 ; c) N 1 4 - 3 ; d) N 1 4 - 4  130  8.5  A G r o u t e d M a s o n r y P r i s m w i t h Squre C r o s s - S e c t i o n  135  8.6  M o d e l P r e d i c t i o n versus E x p e r i m e n t s , B a s e d o n F a i l u r e 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 E x p e r i m e n t s , B a s e d o n F a i l u r e C o n d i t i o n b)  148  8.8  M o d e l P r e d i c t i o n versus E x p e r i m e n t s , B a s e d o n 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 L o a d s , B a s e d o n F a i l u r e C o n d i t i o n a)  151  9.1  M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s o f G r o u t e d , F a c e - S h e l l B e d d e d P r i s m s under C o n c e n t r i c C o m p r e s s i o n : a) N 1 7 - 3 ; b) N 1 7 - 4  10.1  M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s o f G r o u t e d P r i s m s under E c c e n t r i c C o m p r e s s i o n : a) N 2 6 - 1 , e = t / 6 ; b) N 2 6 - 2 , e = t / 6 ; c) M 2 6 - 2 , e = t / 3  10.2  156  159  M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s of G r o u t e d P r i s m s under E c c e n t r i c C o m p r e s s i o n : a) M 2 6 - 3 , e = t / 3 ; b) S 2 5 - 1 , e = t / 3 ; c) S 2 5 - 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 a n U n c r a c k e d S e c t i o n a n d a C r a c k e d S e c t i o n  163  10.4  A T y p i c a l Section of a Grouted W a l l  164  10.5  Stress D i s t r i b u t i o n a l o n g a S e c t i o n a n d 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 r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s b y the A u t h o r : N 2 6 , M 2 6 , S25  10.7  C o m p a r i s o n of Predicted Interaction Curve w i t h Experiments by Drysdale and H a m i d : Normal Block, Type N Grout  10.8  169  C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : Normal Block, Type W Grout  10.9  169  170  C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : Normal Block, Type S Grout  170  - xiv FIGURE 10.10  PAGE  C o m p a r i s o n of Predicted Interaction Curve w i t h Experiments by Drysdale and H a m i d : Weak Block, Type N Grout  10.11  171  C o m p a r i s o n of Predicted Interaction C u r v e w i t h Experiments by Drysdale and H a m i d : Strong Block, Type N Grout  10.12  171  C o m p a r i s o n of Predicted Interaction Curve w i t h Experiments by Drysdale and H a m i d : 75% Solid Block, Type N Grout  10.13  C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : Full Block  10.14  172  ,  172  C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : 6 inch Block, Type N Grout  10.15  173  C o m p a r i s o n of Predicted Interaction Curve w i t h Experiments by Drysdale and H a m i d : 10 i n c h 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 I n t e r a c t i o n C u r v e a n d L o a d i n g P a t h s of a C o m p r e s s i o n M e m b e r  176  11.2  A C r o s s - S e c t i o n V i e w o f A R e i n f o r c e d C o n c r e t e W a l l under E c c e n t r i c C o m p r e s s i o n  180  11.3  C o l u m n Deflection Curve  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 S e c t i o n o f a P l a i n , S o l i d M e m b e r L o a d e d 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 E c c e n t r i c i t y for a S o l i d S e c t i o n  192  11.6  C r i t i c a l L o a d versus L o a d i n g E c c e n t r i c i t y : X = a/b=0,  194  11.7  C r i t i c a l L o a d versus L o a d i n g E c c e n t r i c i t y : A = 0.5, rap = 0.05, a/b  11.8  C r i t i c a l L o a d versus L o a d i n g E c c e n t r i c i t y : a / f r = 0 . 6 5 , np = 0,  11.9  Theoretical P — M Interaction Curve and Loading Paths C o m p a r e d w i t h Experiments  np  Varies Varies  A Varies  b y H a t z i n i k o l a s et a l : 137 i n c h H i g h W a l l w i t h R e i n f o r c e m e n t 3 # 3 11.10  194 195  197  T h e o r e t i c a 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 E x p e r i m e n t s b y H a t z i n i k o l a s et a l : 137 i n c h H i g h W a l l w i t h R e i n f o r c e m e n t 3 # 3 . T h e P o i n t s S h o w the E x p e r i m e n t a l R e s u l t s w h i l e the C o n t i n u o u s L i n e s S h o w the P r e d i c t i o n  11.11  197  T h e o r e t i c a 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 E x p e r i m e n t s b y H a t z i n i k o l a s et a l : 137 i n c h H i g h W a l l w i t h R e i n f o r c e m e n t 3 # 6  198  - XV FIGURE 11.12  PAGE  T h e o r e t i c a 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 E x p e r i m e n t s b y H a t z i n i k o l a s et a l : 105 i n c h H i g h P l a i n W a l l  198  11.13  M o m e n t M a g n i f i e r versus L o a d for a P l a i n S e c t i o n .  203  11.14  M o m e n t M a g n i f i e r versus L o a d for a R e i n f o r c e d S e c t i o n  203  11.15  M o m e n t M a g n i f i e 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 a n d C r i t i c a l M o m e n t M a g n i f i e r V e r s u s E c c e n t r i c i t y : for P u r p o s e of Design Analysis  204  Al  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  A2  G e o m e t r i c R e l a t i o n between C r a c k O p e n i n g a n d S l i d i n g D i s p l a c e m e n t  218  A3  A C r a c k E x t e n d e d b y a P a i r of S p l i t t i n g Forces  218  A4  A C o l u m n Loaded with Double Curvature Bending  230  - xvi NOTATION  A,  B,  C,  D  A ,A g  c, c  m  n  constants used for  a,  b  =  pre-existing half crack length and half spacing, respectively, in (Chapter  a,  b  =  h a l f w i d t h of h o l l o w core a n d block unit, respectively;  c  =  bedding joint crack depths of a w a l l cross-section, defined in F i g .  =  constants;  =  modulus of elasticity;  =  m o d u l u s of elasticity of block unit, grout, m o r t a r j o i n t , respectively;  =  loading eccentricities;  =  v i r t u a l eccentricities corresponding to different bedding j o i n t crack depths;  =  compressive force;  =  functions of bedding joint cracking defined in A p p e n d i x  =  f r i c t i o n between pre-existing crack surfaces;  i ft  —  compressive a n d tensile strength, respectively;  fmg  —  compressive strength of plain masonry a n d grouted masonry,  —  compressive strength of block unit, grout (prism strength), mortar  E  2  1  Ej,  g  e, e  0  £c,  ej,  e  m  F F,  F  1  2  / f  c  fmp,  At  /gi  strength),  fj  fj i  u  fje  c  f  ut  Gj,  LT  dimensions;  G  2  GJQ  H  = —  u n c o n f i n e d strength of m o r t a r  c  o  n  f i  n  e  d strengths of m o r t a r  joint;  joint;  =  tensile strength of block;  —  functions of bedding joint cracking defined in A p p e n d i x  =  energy release rate of c r a c k e x t e n s i o n a n d its c r i t i c a l v a l u e ;  =  parametric function defined in A p p e n d i x  B;  III);  11.2;  F;  respectively; /•  G  gross area a n d net a r e a of b l o c k u n i t , respectively;  =  , c^, c  E,  = b  E,  u  constants in various contexts;  a,  c  E,  =  F;  respectively; (cube  - xvii =  height of w a l l or specimen;  hj  =  w a l l heights corresponding to different bedding j o i n t crack depths;  h  0  =  height of block unit;  Ig  =  m o m e n t of i n e r t i a of w a l l c o r r e s p o n d i n g to net cross-section a n d gross-section,  =  stress i n t e n s i t y factor at crack tips a n d its c r i t i c a l value;  =  crack configuration factor (in C h a p t e r  2  =  constants;  /, l  =  extending crack (half) length and its i n i t i a l value (in C h a p t e r  =  length of w a l l or specimen;  M  =  n u m b e r of cracks.in specimen, (in C h a p t e r  M  =  bending  m  =  m o d u l u s of W e i b u l l  h  h, c  /,  respectively;  Kj,  K  IC  k  k,  k  k  lt  0  /  m  mi,  2  u  Per,  III);  distribution;  g  a n d E /Ej, u  respectively;  =  m o d u l a r ratio of reinforcing steel to block shell;  P  x  =  tensile s p l i t t i n g force (in C h a p t e r  P  =  applied compressive load;  P).  =  E u l e r l o a d (corresponding to gross section), a n d b u c k l i n g load of w a l l ,  p  =  contact pressure between grout a n d block shell;  "Tf  =  traction  n P,  III);  moment;  m o d u l a r r a t i o o f E /E  =  III);  III);  respectively;  Qii  Qni  Qti  components  on  internal  boundary  elastic b o d y containing cracks, respectively; R  =  energy dissipated by friction;  S  =  shear force;  s  =  contour  T  =  effective crack i n d u c e d shear s l i d i n g force;  length;  and  external  boundary  of  an  - xviii =  thickness of wall;  t  0  =  thickness of m o r t a r  U  =  strain  u, v  =  dispacement  variables;  —  displacement  components  t  u  %i  v  i t "HI  v  t  joint;  energy;  on internal b o u n d a r y a n d external b o u n d a r y of  an  elastic body containing cracks, respectively; V  —  volume;  V  =  work done by external load;  W  =  energy dissipated to f o r m new crack  w  =  specimen w i d t h (in Chapter  =  s u m of the mortared web dimension a n d grout dimension along w a l l length,  x, y  =  variables under different  Z  =  c u m u l a t i v e function of a W e i b u l l  a  =  inclining angle of pre-existing cracks (in C h a p t e r  a  =  release angle of b l o c k inner core;  2  =  external and internal  A,  S  =  displacements;  6,  cij  =  crack opening  e  =  compressive strain;  x  =  extreme  =  compressive strain in block unit, grout and mortar joint,  w,  w  g  III);  respectively;  T  r\,  Chapter  and  context; distribution;  boundaries;  its value at the starting ponit  of t r a n s i t i o n a l i n t e r v a l  III);  c  fiber  strain o n compression side of w a l l ;  e«, £g,  £j  €p  cr j i  =  s t r a i n a n d stress  rj  =  net area to gross area ratio of block unit  t  III);  components;  A /A ; n  g  respectively;  (in  - 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;  v , v3, vj  =  u  £ =  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, a  t  <T ., T j (J  Poisson's ratio of block unit, grout and mortar, respectively;  t  a , CTJ, cr 0  2  =  compressive and tensile stress, respectively;  =  normal and shear stress components, respectively;  =  threshold stress for crack extension, stresses at the starting and finishing  points, of transitional interval (in Chapter III); respectively; a  <7 =  outer fibre stresses of wall;  a,  =  lateral confining stress in joint;  <r , a,  =  compressive stress in masonry (average) and in masonry shell, respectively;  u  2  m  cr , a , a j = u  g  cr  =  ut  compressive stress in block unit, grout and mortar joint, respectively; lateral tensile stress in block unit;  <p = rotation (slope) of wall section; tp , ipj = c  $  =  rotations of wall section corresponding to different bedding joint crack depths; Airy stress function;  <f> = fij, £2  2  =  density function of Weibull distribution;  functions of bedding joint crack depth defined by Eqs. 11.19 and 11.21.  - XX -  ACKNOWLEDGEMENT  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 -  DEDICATION  TO  T H E M E M O R Y  OF  M Y  F A T H E R  1 C H A P T E R  I  INTRODUCTION  1.1  General  Remarks  Masonry  c o n s t r u c t i o n is basically a n  assembly  stone, clay bricks or precast concrete units. T h e y m o r t a r . T h e history! of m a s o n r y masonry  is  still  increasing  appearence of masonry  knowledge  of  blocks can be  natural  are j o i n t e d w i t h cementitious m a t e r i a l called  b u i l d i n g m a y be 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 the interest i n  today  because  of  the  economy  of  construction  and  the  pleasing  the last t w o  decades.  structures.  Serious studies of structural masonry While  of blocks. T h e  the  structural  behaviour  have  been  carried out  of  masonry  has  for  been  greatly  improved,  many  questions still r e m a i n u n a n s w e r e d i n this area, a n d the design rests largely o n a n e m p i r i c a l base. T h i s s t u d y w i l l focus on the failure a n d c a p a c i t y of the concrete m a s o n r y compression  combined  with  out-of-plane  bending.  The  study  extends  investigation of m a t e r i a l failure to a rational analysis of masonry masonry  prisms  with  various  bedding  and  grouting  under  from  a  stability. The  combinations  under  in-plane  background behaviour  of  various  loading  program  is  c o n d i t i o n s is 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 . This reported  in  thesis is o r g a n i z e d Chapter  II;  the  in  the following  results  will  be  manner.  quoted  and  The  experimental  studied  in  detail  in  the  first  subsequent  chapters. T h e b a c k g r o u n d s t u d y on m a t e r i a l failure under 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 be used to  explain  Chapters grouted  IV  some  behaviour  to VII  masonry.  of  focus on The  study  concrete: m a s o n r y  the behaviour on  the  in  later  chapters,  of plain concrete masonry;  slenderness  and  the  stability  follows  in  Chapters of  concrete  Chapter  III.  VIII to X  on  masonry  is  presented in C h a p t e r X I . Finally, C h a p t e r X I I concludes this study. It is h o p e d t h a t the 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 this study w i l l enhance existing knowledge of the failure of concrete masonry,  a n d assist i n  in the  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 EXPERIMENTAL  2.1  Purpose and  pattern  P R O G R A M  Scope  Extensive this program,  II  experimental  however,  of masonry  work  on  concrete masonry  efforts were m a d e  prisms  under  has  conducted  previously.  to observe more closely the deformation  concentric and  and  to re-examine the Hilsdorf m o d e l of m o r t a r  In  fracture  eccentric compression. P r i s m specimens  designed to cover various c o m b i n a t i o n s of bedding a n d grouting In order  been  were  conditions.  expansion, for p l a i n prisms  under  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 lie 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 effect of  joints  on  deformation  and  capacity  of  masonry.  For  grouted  prisms  under  concentric  c o m p r e s s i o n , a t t e n t i o n w i l l be p a i d to the cracks i n d u c e d by the different d e f o r m a t i o n of the masonry cracking.  For  u n i t a n d the grout, a n d the forces shared b y these t w o m a t e r i a l s before a n d masonry  prisms under  eccentric loading, failure modes  j o i n t b o n d o n the u n l o a d e d side w i l l be 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 The concisely  experimental  reported  summarized.  properties  in  results  this  concerning  chapter.  The  the  properties  characteristic  of  results  T h e d e t a i l e d results w i l l be r e p o r t e d a n d s t u d i e d  the for  w i l l be observed  after  and  the  constituents  are  gauges. masonry masonry  specimen  are  in the context of related analysis  in later chapters.  2.2  Materials A l l m a t e r i a l s used i n m a k i n g the test s p e c i m e n s are 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  of those c o m m o n l y  2.2.1  Masonry  used in local construction.  Units  A l l the masonry  prisms tested were built by using 8 i n c h s t a n d a r d concrete block  units  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 A S T M 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 6  the units is about 3.42xlO 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  Unit  1  2  3  4  5  6  7  8  1  169  188  194  196  207  186  213  189  193.0  6.5%  GROUP 2  219  186  232  205  208  187  220  203  208.0  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  (kips)  SPECIMEN GROUP  GROUP  1: H y d r o - s t o n e c a p ;  GROUP  2: F i b e r b o a r d  cap  A V G  COV  6  FIG. 2.2 Conical Failure of Masonry Unit  0  2  1  5  4  3  STRAIN ( 1/1000 IN/IN )  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  2.2.2  Compression  Mortar 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  CSA-179M-  1976. T h e m o r t a r s were 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 during  the construction of the specimens. T h e  m i x proportions  mixer,  are g i v e n i n T a b l e 2.2.  Mortar  cubes were s a m p l e d for every b a t c h . T h e 2 8 - d a y cube strengths are given in T a b l e 2.3. A t same  time,  the stress-strain relationships were measured  for  type  N  and  some  of the type  m o r t a r , as s h o w n i n F i g . 2.4. T h e results i n d i c a t e t h a t the m o r t a r s are m u c h softer t h a n  the S  normal  concrete, a n d t h a t they h a v e very large peak s t r a i n s . T h e i n i t i a l m o d u l i are a b o u t 0 . 4 x l 0 p s i for 6  type  N  former  mortar, and  and  0.009  for  0 . 5 x l 0 p s i for t y p e S m o r t a r . 6  the  latter. T h e  high  The  peak s t r a i n s are a b o u t  compliance of  the  mortar  was also  0.006 for indicated  the by  8 d e f o r m a t i o n m e a s u r e m e n t s across the j o i n t s of p r i s m specimens. 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 of t h e i n i t i a l m o d u l u s of three t y p e s of m o r t a r to t h a t of the concrete u n i t s is a b o u t  1  to 6-8. T h e d e f o r m a t i o n properties 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 cube tests a n d the u n i t tests are v e r y close to these results.  Mortar Type  Proportion by Cement(Type  III M a s o n r y  N  Sand  Water  1  1  2.5  1  S  1/2  1  3  1  N  —  1  3  0.68  SPECIMEN  S  Fine  M  T a b l e 2.2  M  Cement  Volume  T Y P E T Y P E T Y P E  1  2  M i x Proportions of M o r t a r  3  4  5  4188  4835  4625  5100  3985  4320  3875  4075  3750  1450  1710  1560  1650  1325  T a b l e 2.3  6  1730  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  A V G  COV  4690  7.1%  4000  4.8%  1570  9.2%  (psi)  F I G . 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 : a) T y p e N , b) T y p e S.  10  F I G . 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 across M o r t a r J o i n t of P l a i n P r i s m s under U n i a x a i l C o m p r e s s i o n .  2.2.3 G r o u t T h r e e types of g r o u t w i t h different strengths were designed for the specimens. T h e y are denoted b y G S , G N a n d G W .  T h e m i x proportions are listed i n T a b l e 2.4. T h e water content  was adjusted s l i g h t l y to achieve 3-5 inch s l u m p . F o r every m i x b a t c h , two or three s t a n d a r d prisms were cast a n d cured i n accordance to C S A - 1 7 9 M - 1 9 7 6 . T h e compressive strengths are given i n T a b l e 2.5. T o e x a m i n e the c o r r e l a t i o n between the s t r e n g t h o b t a i n e d b y the s t a n d a r d test a n d t h a t actually  grouted  i n the m a s o n r y ,  20 grout p r i s m s t a k e n f r o m the cores of failed  masonry  specimens were tested. T h e cores were cut by a d i a m o n d saw a n d capped w i t h sulfur before testing. T h e results are s h o w n i n T a b l e 2.6. A s is seen, the strength o f the grout, p r i s m s t a k e n  11 from failed masonry  s p e c i m e n s is 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 of the s t a n d a r d p r i s m s .  This  m a y be p a r t l y d u e t o the different height to w i d t h ratios of the specimens (1.4:1 for the former, 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 were t e s t e d about  a  year  later).  This  suggests that  the strength  obtained  by  the  standard  test is  only  m e a n i n g f u l as a reference p a r a m e t e r . The  deformation  properties  were  measured  on  the  cores t a k e n  from  tested  grouted  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 curves are 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 is 2.8 x l O psi for type S grout, 2 . 6 x l 0  psi for type N grout a n d 1 . 9 x l 0  6  Grout  Proportion by  Type  Cement(Type  III Coarse Sand  GS  1  2.5  GN  1  2  G W  1  5  T a b l e 2.4  6  psi for t y p e W  grout.  Volume Pea Gravel  Water  2.5  0.6  2  0.8  —  1.0  ,  M i x Proportions of G r o u t  SPECIMEN  1  2  3  4  5  S - T Y P E  4720  5445  4890  5320  4705  N — T Y P E  3685  3885  3720  4015  3390  W - T Y P E  3165  3500  3305  3375  3285  6  3425  7  3745  8  3600  9  3815  A V G  C O V  5015  6.1%  3700  5.1%  3325  3.2%  A V G  COV  5590  15.7%  6110  4.9%  4385  5.0%  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 (psi)  SPECIMEN  1  2  3  4  5  S - T Y P E  6305  4170  5530  6690  5245  N - T Y P E  6180  5625  5970  6350  5810  6310  W - T Y P E  4530  4725  4330  4450  4050  4210  6  7  5870  8  6660  9  6215  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 (psi)  6  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 were b u i l t w i t h different to  be  tested  under  different  eccentricities.  Three  bedding, high  and grouting  specimens  were  conditions, chosen  designed  because  b e l i e v e d t h a t t h e e n d effect o f t h e l o a d i n g d e v i c e c a n be 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 a l l the m e a s u r e m e n t s were m a d e . T h e specifications of the specimens are listed i n T a b l e  it  is  where  2.7.  A l l the p l a i n p r i s m s were built by an experienced m a s o n . A l l the m o r t a r j o i n t s were cut 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 specimens were stored  in  the structures laboratory  at  d a y s later (for g r o u t e d p r i s m s ) .  the University  The  of B r i t i s h C o l u m b i a  for  about a year u n t i l they were tested. A few specimens were discarded because of the d e b o n d i n g the mortar joint  (the  debonding  happened  because specimens were m o v e d  e x p e r i m e n t a l activities, d u r i n g the storing period, a n d because of setup F o r the grouted  prisms, hydrostone  once, due  to  other  handling).  was used to finish the top ends prior to testing. A l l  the p r i s m s were transported by using a s m a l l trolley to the testing device a n d then' c a p p e d and bottom) with fibreboard  of  before they were p o s i t i o n e d between the l o a d i n g platens.  (top  Specimen  N o . oi Prisms  Joint  Grouting  Load  Additional  Conditions  Conditions  Eccent.  Description  — — —  0  J o i n t thickness is 3 / 8  0  inch except otherwise  0  specified.  . . .  0  NJ = N  . . .  0  C o n t a c t faces were g r i n d e d  0  jointed b y cement paste.  SI  4  S  N2  4  N  Mortar  M3  4  M  Mortar  N4  4  NJ,  P5  4  G7  4  S8  4  S  M9  4  M  N10  4  NJ,  P l l  4  N12  4  N13  Mortar  t =6/8" 0  t =0 o  4 m m glass plate Mortar  N  Grout  0  Mortar  N  Grout  0  t =6/8"  N  Grout  0  .  N  Grout  0  N  Mortar  S  Grout  0  4  N  Mortar  N  N14  4  N  Mortar  W  N15  4  N J , face-shell  S16  4  S J , face-shell  N17  4  N J , face-shell  N18  4  N  Mortar  . . .  t/6  N19  4  N  Mortar  . . .  t/3  M20  4  M  Mortar  . . .  t/3  S21  4  S  N22  4  N23  4  N  S25  4  S  N26  4  N  M26  4  M  M27  4  0  t =0 o  Grout  N  N J , face-shell  0 0  N J = N  Mortar.  0  S J = N  Mortar.  Grout  0  N J = N  Mortar.  — —  t/3 t/3 0  N  Grout  t/3  Mortar  N  Grout  t/6  Mortar  N  Grout  t/3  M J , face-shell  T a b l e 2.7  Mortar.  0  Grout  Mortar  N  NJ = N  — . . .  Mortar  Mortar  Grout  Mortar.  . . .  0  P r i s m Specimens  Half block  M J = M  Mortar.  15 2.4 T e s t i n g D e v i c e Since it was expected t h a t for the g r o u t e d p r i s m s the f a i l u r e loads w o u l d be higher t h a n the c a p a c i t y o f the e x i s t i n g t e s t i n g facilities i n the structures l a b o r a t o r y (up t o 400 k i p s ) , a l o a d i n g device was b u i l t as s h o w n i n F i g . 2.7. It was f o r m e d b a s i c a l l y w i t h a girder s e r v i n g as a lever, w i t h a p p r o p r i a t e s u p p o r t i n g m e m b e r s ; it h a d a m e c h a n i c a l a d v a n t a g e of 2. T h e device was connected t o a h y d r a u l i c j a c k w i t h 400 k i p s c a p a c i t y , a n d so t h a t it c o u l d a p p l y a l o a d up to 800 k i p s . It was" c a l i b r a t e d u p t o 600 k i p s , b u t , i n the event, the f a i l u r e loads of the specimens never exceeded 400 k i p s . T h e specimens were designed to be compressed w i t h p i n - e n d e d c o n d i t i o n s . T h e t o p a n d b o t t o m l o a d i n g platens, therefore, were designed w i t h c y l i n d r i c a l bearings, as s h o w n i n F i g s . 2.7 a n d 2.8. T h e p l a t e n s were designed for three l o a d i n g eccentricities, i.e. e=0,  e=t/6  and  e=i/3.  T h e s u p p o r t i n g devices were b u i l t w i t h h i g h s t r e n g t h steel. T h e h y d r a u l i c j a c k was c o n t r o l l e d by a n M T S c o n t r o l console ( M o d e l 483.02), w i t h force c o n t r o l m o d e ( d i s p l a c e m e n t c o n t r o l not a v a i l a b l e ) . T h e l o a d was set t o increase a u t o m a t i c a l l y so t h a t a s p e c i m e n w o u l d f a i l i n a b o u t 3 m i n u t e s , for p l a i n specimens, a n d 5 m i n u t e s for g r o u t e d ones. T h e l o a d was read t h r o u g h a n electronic l o a d cell m o u n t e d i n the j a c k .  2.5 I n s t r u m e n t a t i o n To  measure  the  deformations  of  the  prisms,  about  half  of  the  specimens  i n s t r u m e n t e d w i r h s i x q u a r t e r - i n c h linear v a r i a b l e d i f f e r e n t i a l t r a n s f o r m e r s ( L V D T ,  were  Trans-Tek  Series 240). T h e a r r a n g e m e n t a n d l o c a t i o n s of the L V D T s were different for c o n c e n t r i c a n d for eccentric c o m p r e s s i o n c o n d i t i o n s , a n d are denoted b y 1 t o 6 i n the figures s h o w i n g the m e a s u r e d curves (cf. F i g . 4.3, for e x a m p l e ) . T h e L V D T across the m o r t a r j o i n t h a d a gauge l e n g t h o f 1.8 inches (45 m m ) , w h i l e a l l rest were 5 i n c h (125 m m ) . T h e L V D T s were c l a m p e d t o a l u m i n u m s u p p o r t s w h i c h were then m o u n t e d o n s m a l l disc screw nuts glued i n a d v a n c e b y fast s e t t i n g e p o x y , t y p i c a l l y as s h o w n i n F i g . 2.9.  16  F I G . 2.7  Testing Device  F i g . 2.9 I n s t r u m e n t a i o n : L V D T s a n d G l u e d W i r e s  18 Because of the destructive nature of the experiments, it was expected that the specimen w o u l d fail in a sudden, explosive pattern, especially w i t h the load controlled testing machine. reduce the i m p a c t of the failing specimen, the L V D T s  were surrounded  To  w i t h p l e x i g l a s s sleeves  a n d sponges as c a n be seen i n F i g . 2.9. T o m o n i t o r t h e effect o f t h e i m p a c t , t h e y were c h e c k e d for  normal  functioning  after  every  test a n d  calibrated against gauge  thickness for  every  two  tests, or whenever the c e n t r a l core of a transducer was bent (this h a p p e n e d several t i m e s  during  the tests, s t r a i g h t e n i n g was often necessary). F o r t u n a t e l y , the outer coils s u r v i v e d for the  whole  testing program, although the l i n k i n g wires broke several times. A c c o r d i n g to the manufacturer,  the L V D T s  have an infinite resolution. However,  t h e d i s p l a c e m e n t m e a s u r e d is too s m a l l , the readings m a y  when  b e b u r i e d i n t h e n o i s e . It t u r n e d  out  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 given gauge  length),  this was not  a big problem,  and  the readings were satisfactory for  most  cases. For the  plain prisms under  axial compression, an electronic circuit was designed  macroscopic splitting of plain  w i r e s ( g a u g e 4 2 , d> 0 . 0 8 m m )  prisms under  uniaxial compression. Four  very  to  thin  study copper  were glued w i t h epoxy to different locations on the prisms.  These  wires served 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 electrical pulses w h e n t h e y b r e a k . S i n c e t h e wires were fully surrounded  by  the hardened glue a n d adhering to the surface of the specimen,  they  were supposed to break when the specimen split. B y detecting the order of the wires breaking, we o b t a i n the r u n n i n g direction of a crack w h i c h runs across the wires. T h e crack speed i n c o n c r e t e is a b o u t through  the  block  height  180m/sec (Bhargava and Rehnstorm in  about  0.001  second. T h e  1975) so t h a t a s p l i t w o u l d  e l e c t r o n i c c i r c u i t (see  designed by an experienced electrician in the civil engineering department, detecting  the  break  order  electrical pulses given by  for  intervals  less  open circuits due  than  5 x l 0  -  6  propagation  second.  appendix)  run was  w h i c h was capable of  B a s i c a l l y , it  recorded  to breakages of the wires in an ordered  way.  the The  circuit was built a n d then tested i n the electrical engineering d e p a r t m e n t at TJBC. T o gain m o r e  19 confidence w i t h the m e t h o d , it was first used i n face-shell bedded p r i s m s a n d gave consistent results. It w a s t h e n t r i e d w i t h f u l l y bedded m a s o n r y w i t h lines glued to the face-shells as w e l l as the webs, a n d a g a i n gave consistent results (webs s p l i t i n contrast to face-shells). T h e wires were t h e n a l l g l u e d to the webs, where s p l i t t i n g a l w a y s o c c u r r e d . T h e device is s h o w n i n F i g . 2.10 (also see F i g . 2.9 for g l u e d wires), a n d it is seen t h a t the b r e a k i n g order is i n d i c a t e d b y four rows of l i g h t e m i s s i o n diodes ( L E D ) . T o give better i n s i g h t i n t o the failure processes, a V H S s t a n d a r d v i d e o c a m e r a was used to record m o s t o f the tests. T h i s was f o u n d very useful for later o b s e r v a t i o n since, as the t e s t i n g m a c h i n e was l o a d c o n t r o l l e d , m a n y specimens were t o t a l l y destroyed  (often i n a n e x p l o s i v e  m a n n e r ) as soon as u l t i m a t e l o a d was reached. T h e c a m e r a was i n s t a l l e d to face one of the webs, because fractures were m o r e often observed to occur i n webs t h a n i n face-shells ( c o m p a r e  the  d e f o r m a t i o n s measured at l o c a t i o n s 3, 4 w i t h those at l o c a t i o n s 1, 2 g i v e n i n f o l l o w i n g chapters). T h e c a m e r a was able to record v i s i b l e cracks o n the web faces, u s u a l l y  immediately  before final f a i l u r e . H o w e v e r , L V D T s were m o r e sensitive to s m a l l e r cracks o c c u r r i n g at earlier stages (as inferred b y a sudden increase i n m e a s u r e d d i s p l a c e m e n t ) . F o r the m a j o r i t y o f the specimens, the o v e r a l l f a i l u r e p a t t e r n c o u l d be e x a m i n e d by slow p l a y b a c k of the recorder. F o r a few specimens t h a t f a i l e d i n a h i g h l y explosive m a n n e r , however, the r e c o r d i n g was n o t  very  satisfactory.  was  U s u a l l y there was no  warning,  s u c h as c r a c k i n g or s p a l l i n g , t h a t f a i l u r e  a p p r o a c h i n g i n these specimens.  2.6 D a t a A c q u i s i t i o n T h e d a t a , i.e. s i x d i s p l a c e m e n t s a n d one l o a d , were read b y a n O p t i l o g s y s t e m , a n electronic d a t a a c q u i s t i o n u n i t , w h i c h is b a s i c a l l y a microprocessor d i g i t i z i n g a n d r e c o r d i n g the a n a l o g signals. It was c o n t r o l l e d b y a n I B M personal c o m p u t e r w i t h O p t i l o g software. T h e w h o l e setup is s h o w n i n 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 of Characteristic Results Since the p r i s m specimens cover a wide range a n d each g r o u p has its o w n emphasis,  may  be i n a p p r o p r i a t e  to give all the detailed results at this stage. Therefore,  the results w i l l  it be  reported a n d studied in the context of analysis in the related chapters. In order t o h a v e a n o v e r a l l v i e w of the test results, we give a short, d e s c r i p t i v e of some of the i m p o r t a n t  experimental characteristics. T h e y  m o d e a n d c a p a c i t y , w h i c h are of c o m m o n different  are o u t l i n e d i n t e r m s of the failure  interest b u t w h i c h are yet generally d i s t i n c t  specimens  may  loading  be r o u g h l y  characterized into  conditions, as w e l l as their  6 major  groups  bedding,  grouting,  masonry  with full bedding under concentric compression; plain masonry  concentric compression;  compression; grouted masonry with  between  specimens. The  under  summary  face-shell bedding  under  plain  masonry  (with  according  failure characteristics. T h e y  both  bedding  to  their  are:  plain  w i t h face-shell  conditions)  under  w i t h full bedding under concentric compression; grouted concentric compression;  grouted  masonry  (with  bedding eccentric masonry  both  bedding  conditions) under eccentric compression. The modes.  summary  is o r g a n i z e d  in Table  2.8.  F i g s . 2.12  t o 2.16  give some  typical  failure  23 SPECIMEN  '  FAILURE  MODE  CAPACITY  CHARACTERISTICS  1) P l a i n m a s o n r y w i t h  O f t e n one m a j o r s p l i t r a n t h r o u g h J o i n t c o n d i t i o n s h a d a s i g n i f i c a n t  f u l l b e d d i n g under  s p e c i m e n w i t h i n m i d d l e t h i r d of  c o n c e n t r i c c o m p r e s s i o n . webs, i m m e d i a t e l y before  influence o n the c a p a c i t y .  final  f a i l u r e . S p l i t s d i d not c o n s i s t e n t l y i n i t i a t e f r o m the m o r t a r j o i n t . S p l i t s were c o n t i n u o u s . 2) P l a i n m a s o n r y w i t h  O n e or t w o splits i n webs occured  J o i n t strength h a d a r e l a t i v e l y  face-shell b e d d i n g  at or i m m e d i a t e l y before  s i g n i f i c a n t influence o n the  under c o n c e n t r i c  failure. S p l i t s c o n s i s t e n t l y  capacity.  compression.  i n i t i a t e d f r o m j o i n t s , at l o c a t i o n s  Capping conditions had a  final  near t w o j o i n t ends a n d wandered s u b s t a n t i a l influence on the afterwards.  capacity.  S p l i t s were d i s c o n t i n u o u s at j o i n t s . 3) P l a i n m a s o n r y under  F a i l u r e was characterized b y  Joint strength and bedding  eccentric c o m p r e s s i o n .  shear, i.e. b y s p a l l i n g a n d c r u s h i n g  pattern had a relatively minor  on the l o a d e d side; a n d was often  effect o n the c a p a c i t y .  l o c a l i z e d i n p a r t of the s p e c i m e n . J o i n t s on u n l o a d e d side d i d not effectively transfer tension. 4) G r o u t e d  masonry  w i t h f u l l b e d d i n g under  S p l i t s b o t h i n webs a n d face-shells  J o i n t strength a n d grout s t r e n g t h  were observed well before  h a d a r e l a t i v e l y m i n o r effect o n  final  c o n c e n t r i c c o m p r e s s i o n . failure, some at as l o w as 4 0 % of  the c a p a c i t y .  f a i l u r e loads. B l o c k shells s t i l l c a r r i e d s u b s t a n t i a l l o a d after c r a c k i n g . F i n a l failure b r o u g h t  by  s p a l l i n g of the shells, f o l l o w e d b y c r u s h i n g of grout at t h e m i d h e i g h t . 5) G r o u t e d p r i s m w i t h  S p l i t s i n webs occured well before  face-shell b e d d i n g under f i n a l faiure. B l o c k shells c a r r i e d  C a p a c i t y was not m u c h higher t h a n t h a t of g r o u t alone.  c o n c e n t r i c c o m p r e s s i o n . l i t t l e l o a d after c r a c k i n g . 6) G r o u t e d  masonry  u n d e r eccentric c o m p r .  A s described i n 3).  B o t h grout a n d j o i n t have a m i n o r effect on the c a p a c i t y .  T a b l e 2.8 A S u m m a r y of F a i l u r e a n d C a p a c i t y C h a r a c t e r i s t i c s  24  FIG.  2.12  Splitting Failure of P l a i n Concrete M a s o n r y F u l l Bedding under U n i a x i a l  Compression  with  FIG. 2.13  Failure of Plain Masonry with Face-Shell Bedding under Uniaxial Compression  26  F I G . 2.14  Failure of Face-Shell Bedded, F u l l y C a p p e d M a s o n r y under U n i a x i a l Compression  FIG.2.15  Failure of P l a i n M a s o n r y under Eccentric Compression  28  F I G . 2.16 F a i l u r e of 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 SOME  3.1  B A C K G R O U N D  TO  III  COMPRESSION  FAILURE  OF  C O N C R E T E  Purpose  Concrete  masonry  is  basically  a  concrete  member  with  discontinuity  in  material  p r o p e r t i e s . In s t r u c t u r a l d e s i g n , it is u s u a l l y used to s u s t a i n c o m p r e s s i v e force. In  traditional  analysis for  used: experimentally  observed  applied.  been  Failure  has  has  been  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 tests, h a v e  been  defined  concrete structures, a phenomenological  as t h e stress o r  strain in  the  member  approach  w h i c h reaches  some  c r i t i c a l v a l u e (strength or u l t i m a t e strain), w h i c h is o b t a i n e d f r o m u n i a x i a l tests a n d a s s u m e d  to  be c o n s t a n t i n a general stress state. However,  t h i s a p p r o a c h is subject 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 of 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 is c h a r a c t e r i z e d b y very  meaningful  parameter.  It  varies w i t h the stress state due  effect, a p h e n o m e n o n w h i c h is m o r e o b v i o u s  to the so-called strain  for concrete m a s o n r y .  give a n e x p l a n a t i o n for the s p l i t t i n g failure m e c h a n i s m well as concrete under  b r i t t l e c l e a v a g e f r a c t u r e , is not  The  often observed  approach  a  gradient  also fails to  in concrete masonry  as  u n i a x i a l compression. These problems have been partially recognized  but  never been fully explained. T h e s t u d y presented i n this chapter w i l l a t t e m p t to raise the question a n d to shed light on the problems  by examining brittle materials under  uniaxial compression. T h e  some  intention  is 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 these 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 the f u n d a m e n t a l  level.  T h e p r i n c i p l e s of t h i s s t u d y w i l l be used to e x p l a i n s o m e b e h a v i o u r and help  to support to  develop  at  an alternative approach a  better  understanding  splitting failure phenomenon commonly  i n l a t e r c h a p t e r s . It is h o p e d of  the  nature  of  the  strain  that  of concrete this study  gradient  effect  masonry will  also  and  the  exhibited in brittle m a t e r i a l testing, a n d lead to  more  30 general a n d consistent f a i l u r e c r i t e r i a for these m a t e r i a l s .  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 C o m p r e s s i o n A l t h o u g h concrete m a y e x h i b i t h i g h n o n l i n e a r i t y at w o r k i n g c o m p r e s s i v e stress, it is essentially a b r i t t l e m a t e r i a l . T h i s is so m a i n l y because the failure of concrete is c h a r a c t e r i z e d b y b r i t t l e cleavage f r a c t u r e a n d the p l a s t i c d e f o r m a t i o n due to viscous b e h a v i o u r of the  hardened  cement is r a t h e r l i m i t e d ( H s u et a l 1963, Ziegeldorf 1983). E x t e n s i v e research work at b o t h s t r u c t u r a l a n d p h e n o m e n o l o g i c a l levels has i n d i c a t e d that  under  initiation  of  compression, the cracks;  concrete experiences three slow stable crack g r o w t h  d i s t i n c t stages before accompanied  by  its f i n a l  failure:  crack arrest; a  critical  c o n d i t i o n c h a r a c t e r i z e d b y unstable crack p r o p a g a t i o n a n d a n extensive crack n e t w o r k f o r m a t i o n (for e x a m p l e , see M i n d e s s 1983). T h e c r a c k i n g process is reflected i n the g l o b a l n o n l i n e a r i t y of the m a t e r i a l , w h i c h appears i n spite of the fact t h a t b o t h aggregate a n d hardened cement paste are, i n d i v i d u a l l y , essentially linear u p to the failure stress of the concrete. A n o t h e r i n d i c a t i o n is the a p p a r e n t v o l u m e increase of concrete under c o m p r e s s i o n . A n i m p o r t a n t feature is t h a t g l o b a l l y , the fractures i n the m a t e r i a l c o i n c i d e w i t h the d i r e c t i o n o f the m a x i m u m p r i n c i p a l compressive stress ( K o t s o v o s 1979). F o r the case o f u n i a x i a l c o m p r e s s i o n , t h i s corresponds to the w e l l k n o w n s p l i t t i n g f a i l u r e o c c u r r i n g o v e r w h e l m i n g l y  in  careful e x p e r i m e n t s . T h e c o n i c a l failure m o d e frequently observed i n concrete c o m p r e s s i o n tests is due to the l a t e r a l c o n f i n i n g effect of the l o a d i n g p l a t e n ; it w i l l change to a s p l i t t i n g m o d e if the f r i c t i o n between the s p e c i m e n a n d the l o a d i n g p l a t e n is reduced. A l t h o u g h shear stress does develop o n i n c l i n e d planes i n u n i a x i a l c o m p r e s s i o n tests, the classical theories based o n shear fracture proposed b y C o u l o m b ,  N a v i e r , a n d M o h r are s i m p l y  not b o r n out b y e x p e r i m e n t . F r a c t u r e m e c h a n i c s based on G r i f f i t h ' s theory provides a p o w e r f u l methodology  in brittle failure analysis. Unfortunately,  its success i n a p p l i c a t i o n to concrete,  c o m p a r e d w i t h m e t a l s , is rather m o d e r a t e . T h i s is largely because, as a c e m e n t - b a s e d c o m p o s i t e ,  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 , no clearly defined confined  front for a m a j o r  to the surface energy.  anisotropic, heterogeneous,  crack a n d the energy  Direct  recent developments  been m o r e encouraging. in  which  a p p l i c a t i o n of the single crack m o d e l  Bazant  (1985). T h i s  (in  average  an  c r i t e r i o n for essentially  at  the  fracture  front  sense over  e l a s t i c a l l y . It  front  flux  energy  have into  a smeared  crack  structure. A l t h o u g h  is f o u n d  that  band)  the  l i t t l e effect g l o b a l l y , the these  merely  in Linear  Elastic  results.  nonlinear  is t a k e n  f o r m of fracture  into  account, for  have  mechanics  example  see  zone can model strain-localization due to strain softening of concrete at  the  crack extension. Concrete outside of this  fracture of  finite  zone  is  is n o t  in the application of fracture mechanics to concrete  T h i s i n v o l v e s the use of a p r o p e r ,  a finite nonlinear  system. There  dissipating mechanism  F r a c t u r e M e c h a n i c s does not lead to satisfactory q u a n t i t a t i v e However,  multiphase  fracture findings  detailed  finite  zone,  zone  provide  an  energy  can be considered  to  behave  distributions  since fracture  process  crack front  which  of  propagation is a  global  and  stress a n d  depends  strain  at  the  essentially on  the  characteristic of  were o b t a i n e d i n the study of concrete under  the  entire  tension, some  of  the basic p r i n c i p l e s s h o u l d also be a p p l i c a b l e to the case of u n i a x i a l c o m p r e s s i o n . In  the case of u n i a x i a l c o m p r e s s i o n ,  ceramics, glass a n d  especially on  natural  experiments  rocks, have  m o d e a n d a s i m i l a r stable-unstable failure process. F o r  on  again  different revealed  the same  relatively homogeneous  o n l y one or a f e w s p l i t s are o b s e r v e d , w h i l e for less h o m o g e n e o u s cracks are f o u n d to a c c o m p a n y  b r i t t l e m a t e r i a l s s u c h as  the m a i n splitting. (Seldenrath  splitting materials,  level, i n these areas. T h e  most frequently  often  materials, more visible vertical et a l 1958,  Fairhurst  and  1966, B r a c e a n d B y e r l e e 1966, P a t e r s o n 1978). T h i s has led t o r e l a t i v e l y extensive m o d e l at a f u n d a m e n t a l  failure  Cook  studies,  studied models are g r o u n d e d in  the  idea that frictional sliding of a pre-existing crack produces, at the crack tips, tension cracks that g r o w i n the 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. It 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 of the effect o f the presence 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 the c r a c k a p p e a r e d , the stress  32  cr  mn  F I G . 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 Stress F i e l d , S h o w i n g the O r i g i n a l Defect a n d  its  Extension  t I It t  a  1 llll  w i n  \ %  \ \ t ft  I M M  (a) F I G . 3.2 D e p i c t i o n of the Effect of a C r a c k  (b)  33 field  was such t h a t a p a i r of n o r m a l forces N a n d shear forces T were transferred across the space  n o w o c c u p i e d by the c r a c k . W h e n the crack f o r m s , N is s t i l l transferred, b u t T c a n no longer be c a r r i e d . T h u s the effect of the crack o n the o r i g i n a l stress field is the same as the a p p l i c a t i o n of t w o opposite shear forces T o n the crack zone, as d e p i c t e d i n F i g . 3.2(b) a n d F i g . 3.1 (i.e. the 1  r e m o v a l of 7). S i m i l a r a r g u m e n t m a y also a p p l y for m a t e r i a l defects w i t h other c o n f i g u r a t i o n s . A l t h o u g h t h i s m o d e l is a r a d i c a l i d e a l i z a t i o n of r e a l i t y , it does c a p t u r e some of the basic features of the observations  m a d e at the m i c r o s c o p i c level o n rocks a n d concrete. F r i c t i o n a l  s l i d i n g does occur a l o n g the p r e - e x i s t i n g interface cavities or cleavage cracks, a n d for concrete t h i s often takes place at the m a t r i x - a g g r e g a t e interface. T h e s l i d i n g - i n d u c e d tension c r a c k s t e n d to grow i n the d i r e c t i o n of the c o m p r e s s i o n , i n spite of l o c a l i n h o m o g e n i e t y ,  i n a n i n i t i a l l y stable  manner. A l t h o u g h m a t e r i a l defects are also f o u n d i n the f o r m of c a v i t i e s w i t h o u t c o n t a c t faces, i t has been observed t h a t the i n d u c e d tension c r a c k i n g has a m u c h lower tendency to g r o w does the s l i d i n g case (Ziegeldorf 1983). T h i s Panasjuk  (1976), Zaitsev(1983),  or  Sammis  can and  also  be  Ashby  inferred  than  f r o m the a n a l y t i c a l w o r k of  (1986), w h i c h i n d i c a t e s t h a t  under  compressive stress cr, the energy release rate for a crack w i t h extended l e n g t h / is i n the order of <r / 2  3.3).  / i f the defect is a n i n c l i n e d p r e - e x i s t i n g c r a c k , a n d c r / I 2  5  if the defect is a v o i d (see F i g .  T h u s defects i n the a p p r o x i m a t e f o r m of s l i d i n g cracks w i l l d o m i n a t e the c r a c k extension  unless the d i s t r i b u t i o n of defects i n other f o r m s is o v e r w h e l m i n g . Additionally,  the i d e a l i z a t i o n of m a t e r i a l defects appears to be necessary i f we are to  reach a n a n a l y t i c a l l y m a n a g e a b l e  approach. Probably  for a l l these reasons, since it was  first  proposed by M c C l i n t o c k a n d W a l s h (1963), the m o d e l of a s l i d i n g crack w i t h k i n k s has received considerable a t t e n t i o n . It has been s t u d i e d b o t h a n a l y t i c a l l y a n d by m o d e l e x p e r i m e n t ; the l a t t e r is often a c h i e v e d b y c a r r y i n g out tests o n some s y n t h e t i c b r i t t l e m a t e r i a l w i t h m a n - m a d e s l i d i n g c r a c k ( s ) . T h e m o s t n o t a b l e w o r k includes B r a c e a n d 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), S a n t i a g o a n d H i l s d o r f (1973), K a c h a n o v (1982), N e m a t - N a s s e r a n d H o r i i (1982), Zaitsev  34  o  a  n m  f t t t t  a  FIG.  3.3 M o d e l s of M a t e r i a l Defects. (The  n m  .  ft t t •  a  M i s s i n g F o r c e A c t s on E a c h Side i n a  Direction  Opposite to that S h o w n . T h e Effect of the Defect is Therefore to A p p l y Forces i n the D i r e c t i o n Shown.)  (1983), Steiff (1984), H o r i i a n d N e m a t - N a s s e r (1985), A s h b y a n d C o o k s l e y (1986), a n d H o r i i a n d N e m a t - N a s s e r (1986). Because of the c o m p l e x i t y of the p r o b l e m , a l m o s t a l l the a n a l y t i c a l studies have been based on p l a i n e l a s t i c i t y . T h i s is, of course, closer to the behaviour  of homogeneous  brittle  materials such as ceramics a n d glass t h a n to rocks a n d concrete. Nevetherless, they appear give reasonable e x p l a n a t i o n s of some of the characteristics of these inhomogeneous  to  materials.  In this chapter, some aspects of previous m o d e l studies w i l l be briefly reviewed, a n d a  35 s i m p l i f i e d m o d e l based on i n t e r a c t i o n of the s l i d i n g cracks w i l l be presented. T h e focus w i l l be o n the t r a n s i t i o n f r o m s t a b l e to u n s t a b l e c r a c k i n g under u n i a x i a l c o m p r e s s i o n ; a n d the l a t t e r w i l l be s h o w n to m a n i f e s t the w e l l k n o w n s p l i t t i n g failure. T h e m o d e l w i l l be s h o w n to reveal the  c h a r a c t e r i s t i c s o f the compressive s t r e n g t h ,  and  of the s t r e s s - s t r a i n r e l a t i o n of  brittle  m a t e r i a l s under u n i a x i a l c o m p r e s s i o n . T h e m o d e l w i l l be based o n plane e l a s t i c i t y a n d a n i d e a l i z e d c r a c k p a t t e r n . W h e n i t is a p p l i e d to the b e h a v i o u r of concrete a n d rock, it m a y be subject to the same l i m i t a t i o n s as the previous a n a l y t i c a l w o r k , b u t , i n v i e w of the l i m i t e d p l a s t i c d e f o r m a t i o n of these m a t e r i a l s under u n i a x i a l c o m p r e s s i o n , a n d of the successful a p p l i c a t i o n of f r a c t u r e m e c h a n i c s to concrete under t e n s i o n , t h i s a p p r o a c h s h o u l d reveal some of the basic features of c o m p r e s s i o n . H o w e v e r , as indicated  above,  local nonlinear  behaviour  must  be  included  to give  correct  quantitative  p r e d i c t i o n s for concrete. T h u s , a l t h o u g h some q u a n t i t a t i v e conclusions d r a w n f r o m the proposed m o d e l w i l l be presented, the basic objective is to i l l u s t r a t e rather t h a n q u a n t i f y . It is h o p e d , t h a t t h i s t h e o r e t i c a l t r e a t m e n t , based o n a hypothesis for the failure m e c h a n i s m , w i l l shed some l i g h t o n the a c t u a l f a i l u r e process, a n d lead to a better u n d e r s t a n d i n g of the p r o b l e m .  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 B r i t t l e f a i l u r e under u n i a x i a l c o m p r e s s i o n is d i s t i n c t f r o m t h a t i n tension i n t h a t there exists  a  stable  cracking  process  before  final  unstable  fracture.  This  has  been  observed  e x p e r i m e n t a l l y (for i n s t a n c e , as we reviewed for concrete i n the i n t r o d u c t i o n ) , a n d has been i d e n t i f i e d f r o m the f r a c t u r e m e c h a n i c s p o i n t of v i e w (for e x a m p l e , see K o s t o v o s a n d  Newman  1981). W i t h c r a c k g r o w t h , the strain-energy c o n c e n t r a t i o n at a crack f r o n t tends to increase i n the case o f t e n s i o n , b u t decrease i n the case of c o m p r e s s i o n . T h u s i n c o m p r e s s i o n , the fracture occurs i n i t i a l l y i n a discrete, s t a b l e m a n n e r w i t h increasing l o a d ; failure occurs when the stable c r a c k i n g reaches a c e r t a i n extent, b u t not at the i n i t i a t i o n of these cracks. O v e r l o o k i n g  the  s u b c r i t i c a l c r a c k g r o w t h w i l l lead to erroneous results, s u c h as G r i f f i t h ' s p r e d i c t i o n of 1:8 for the  36 r a t i o of tensile to compressive strength - a s u b s t a n t i a l u n d e r e s t i m a t e for m a n y b r i t t l e m a t e r i a l s ( O b e r t 1972). (In  the case of concrete under  tension, fracture m a y  appear to be  temporarily  s t a b i l i z e d ; b u t t h i s is due to arrest b y the aggregate rather t h a n the release of the s t r a i n - e n e r g y concentration.) T h e m o d e l of a single crack w i t h k i n k s c e r t a i n l y e x h i b i t s t h i s stable feature. R e f e r r i n g to F i g . 3.1, the s l i d i n g shear force, w h i c h represents the effect of a n i n c l i n e d crack w i t h l e n g t h 2a i n a n otherwise c o m p r e s s i v e stress f i e l d , is the resultant of the d r i v i n g shear stress a l o n g the c r a c k (for e x a m p l e , see Z a i t s e v 1983)  T =  2a a ( s i n a c o s a — u s i n a )  3.1  2  where u is the coefficient of f r i c t i o n of the m a t e r i a l . W h e n the extended crack l e n g t h 21 is l o n g c o m p a r e d w i t h 2 a, the h o r i z o n t a l c o m p o n e n t s of these shear forces m a y , as far as s p l i t t i n g is concerned, be considered as a p a i r of tension forces of m a g n i t u d e  T s i n a . A s the c r a c k  P=  extends i n the d i r e c t i o n of the a p p l i e d stress, these forces r e m a i n a p p r o x i m a t e l y c o n s t a n t , a n d the w e l l - k n o w n fracture m e c h a n i c s s o l u t i o n ( B r o e k  1978)  i n t e n s i t y at the crack t i p attenuates w i t h extension ( Kj  for t h i s case shows t h a t the stress  =  P/-fJTl  ).  It is for t h i s reason t h a t  the crack is i n i t i a l l y stable. A n exact f o r m u l a t i o n of the p r o b l e m has been g i v e n by H o r i i a n d Nemat-Nasser  (1985), w h i c h gives results very  extension does not appears to be  lead to i m m e d i a t e  close to t h i s a p p r o x i m a t i o n .  f a i l u r e , the d e t a i l e d t u r n i n g  Since the stable  p a t h of the w i n g cracks  unimportant.  Model  experiments  o n b r i t t l e m a t e r i a l s w i t h a m a n - m a d e s l i d i n g crack have  indeed  i n d i c a t e d t h i s stable, tensile crack extension, t u r n i n g i n t o the d i r e c t i o n of the l o a d i n g ( B r a c e a n d Bombolakis  1963, H o e k a n d B i e n i a w s k i 1965, S a n t i a g o a n d H i l s d o r f 1973, N e m a t - N a s s e r  and  Horii  Horii  the  1982,  and  Nemat-Nasser  1985).  This  direction  is f a v o r e d  because  o r i e n t a t i o n i n w h i c h the least w o r k is required to open the c r a c k . A l t h o u g h  t h i s is  Mode  II stress  37 i n t e n s i t y appears i n the crack t i p s , shear f r a c t u r e i n the plane of the prepared s l i d i n g c r a c k was never observed unless the w i d t h of the s p e c i m e n was close to the crack l e n g t h . However,  since f i n a l f a i l u r e is b r o u g h t a b o u t  b y u n s t a b l e f r a c t u r e , there m u s t be a  t r a n s i t i o n f r o m s t a b i l i t y to i n s t a b i l i t y i n the c r a c k i n g . R e c o g n i z i n g  this point,  Ashby  and  C o o k s l e y (1985) developed a m o d e l based o n the w i n g c r a c k i n t e r a c t i o n . T h e y h y p o t h e s i z e t h a t when  stable cracks are r e l a t i v e l y l o n g ,  the  branches  between  cracks t e n d  to b e n d ,  which  intensifies the stress c o n c e n t r a t i o n s at the crack t i p s a n d leads to i n s t a b i l i t y . H o w e v e r , bending  interaction mechanism  this  appears to be insufficient to e x p l a i n a n u n s t a b l e s p l i t i n  a  r e l a t i v e l y short s p e c i m e n . It m a y be w o r t h m e n t i o n i n g t h a t K e n d a l l (1978) has also developed a s i m i l a r b e a m b e n d i n g m o d e l to e x p l a i n a x i a l s p l i t t i n g ; b u t t h i s one requires a n i n d e n t e d (i.e. a l o a d w h i c h does not cover the outer edges of the l o a d e d face) compressive l o a d a c t i n g o n a v e r t i c a l c r a c k , f o r c i n g the t w o s t r u t s separated b y the crack to bend o u t w a r d s . O b v i o u s l y , t h i s m o d e l fails to give a n e x p l a n a t i o n when the g l o b a l compressive stress is u n i f o r m . B a s e d on their a n a l y t i c a l w o r k , H o r i i a n d N e m a t - N a s s e r (1982, 1985) c o n c l u d e d t h a t the s l i d i n g - i n d u c e d tensile c r a c k is very sensitive to l a t e r a l stress. T h e crack extension soon becomes u n s t a b l e i f a s m a l l l a t e r a l tension exists. T h e i r m o d e l e x p e r i m e n t s o n a b a r r e l - s h a p e d s p e c i m e n gave a n excellent i l l u s t r a t i o n of t h i s p o i n t . H o w e v e r , a n e x p l a n a t i o n is s t i l l needed for the case o f u n i a x i a l c o m p r e s s i o n c o r r e s p o n d i n g to zero l a t e r a l stress.  3.4 P r o p o s e d M o d e l It is clear t h a t the m o d e l of a single crack w i t h k i n k s o n l y p r o v i d e s the source of the s p l i t t i n g . O t h e r effects m u s t be i n c l u d e d to e x p l a i n the u n s t a b l e t r a n s i t i o n . W e n o w present a r e l a t i v e l y s i m p l e m o d e l to show t h a t this t r a n s i t i o n c a n be a consequence of the extension o f a g r o u p of stable cracks. S o m e i n t e r e s t i n g results w i l l f o l l o w i m m e d i a t e l y .  3.4.1 C r a c k I n t e r a c t i o n s a n d C r i t i c a l S t a t e  38 Since, as i n d i c a t e d above, crack extension i n c o m p r e s s i o n is i n i t i a l l y stable, there is a h i g h p r o b a b i l i t y t h a t , w i t h increasing stress, cracks w i l l e x t e n d f r o m a l l defects w i t h s i m i l a r c o n f i g u r a t i o n s . A s a result, compressive failure is u s u a l l y not governed b y a n y i n d i v i d u a l defect; t h i s c o n t r a s t s w i t h tension failure, w h i c h is governed b y the defect w i t h c r i t i c a l c o n f i g u r a t i o n , a n d i n w h i c h f r a c t u r e is h i g h l y l o c a l i z e d . T h u s it appears necessary to consider a l l the defects w h i c h govern  the b e h a v i o u r .  B y the same a r g u m e n t ,  it m a y  also be reasonable, as w i l l be.  discussed l a t e r , to treat the c r a c k i n g process i n a n average sense. B y u s i n g the m o d e l o f the s l i d i n g c r a c k w i t h k i n k s a n d the described a p p r o x i m a t i o n , every defect i n a m a t e r i a l corresponds to a p a i r o f s p l i t t i n g forces  P  {  =  a  {  3.2  cr  where k  t  Note that P  {  k  {  =  2 ( sinaj-cosa,- — / i s i n a - ) sina,2  3.3  8  depends o n the i n i t i a l , i n c l i n e d , l e n g t h of the crack a n d not on the extended l e n g t h .  a,- takes a c c o u n t o f the c o n f i g u r a t i o n of the crack. F o r concrete, a  t  m a y be i n the  order o f the aggregate p a r t i c l e size; the coefficient of f r i c t i o n /z is a b o u t 0.36 ( T r o x e l l et a l . 1968),  so t h a t k for the worst angle is about 0.45 (i.e. the angle c o r r e s p o n d i n g to the biggest  force). L e t us e x a m i n e a n i d e a l i z e d case where a series of defects lies i n a line as s h o w n i n F i g . 3.4(a). B y the s t a t e d a p p r o x i m a t i o n s , the s i t u a t i o n i n F i g . 3.4(a) is e q u i v a l e n t to F i g . 3.4(b): a series of c r a c k s w i t h a n average l e n g t h of 21 a n d a n average s p a c i n g 26 a c t e d on b y p a i r s o f p o i n t forces. For  an infinite medium,  the p r o b l e m  has been s t u d i e d by I r w i n  (1957). T h e stress  i n t e n s i t y f a c t o r at the crack t i p s for t h i s case is a v a i l a b l e f r o m the W e s t e r g a a r d stress f u n c t i o n given by h i m :  39  21  2b  P —  21  2b  21  (a)  (b)  F I G . 3.4 A Series of C r a c k s i n a C o m p r e s s i v e Stress 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  1  3.4  'bsin(irl/b)  or, i n t e r m s o f energy release rate for plane s t r a i n c o n d i t i o n s :  .  _  'I  2  P\l-v ) Eb sin ( W / 6 )  3.5  where E = Y o u n g ' s m o d u l u s ; v = Poisson's r a t i o . C r a c k s extend when  Eqs. 3.4 or 3.5 reach some c r i t i c a l v a l u e , w h i c h is a m a t e r i a l  constant. T h e solution indicates that, when l/b < 1 / 2 , dP/dl p r o p a g a t i o n becomes unstable when l/b  > 1 / 2 , dP/dl  > 0, cracks propagate s t a b l y ; the  < 0. O n c e l/b  reaches 1 / 2 , cracks w i l l  40 p r o p a g a t e e x t e n s i v e l y , a n d one or m o r e w i l l r u n t h r o u g h the m a t e r i a l i m m e d i a t e l y . T h i s  point  m a y therefore be defined as the c r i t i c a l state. T h i s r e l a t i v e l y s i m p l e m o d e l c l e a r l y reveals the c h a r a c t e r i s t i c s of the s t a b l e - u n s t a b l e fracture process. It shows that c r i t i c a l i n s t a b i l i t y c a n be the result of the stable c r a c k g r o w t h itself. In  r e a l i t y , p r e - e x i s t i n g defects m a y  experiments  by  Horii  and  Nemat-Nasser  rarely  exist e x a c t l y c o l i n e a r l y .  However,  (1985), i n w h i c h plates of C o l u m b i a  resin  model CR39  c o n t a i n i n g a n u m b e r of p r e - e x i s t i n g s l i d i n g cracks were tested under u n i a x i a l c o m p r e s s i o n , have i n d i c a t e d t h a t the v e r t i c a l l y d i s t r i b u t e d cracks do indeed tend to j o i n each other to f o r m the f i n a l f r a c t u r e , even t h o u g h they are not i n a v e r t i c a l line. F o r a r e l a t i v e l y homogeneous  specimen ( w i t h o u t  m a n - m a d e c r a c k s ) , surface cracks at  the top a n d b o t t o m w i l l be l i k e l y to govern the behaviour, i.e. v e r t i c a l cracks w i l l i n i t i a t e f r o m t o p a n d b o t t o m i n s t e a d of f r o m inside of m a t e r i a l (this can easily be verified by t e s t i n g , say, a plexiglass s t r u t ) . T h e m o d e l s t i l l applies i f we consider the s p e c i m e n height as 2b,  referring to  F i g . 3.4(b), a n d recognize the fact t h a t the s o l u t i o n is s y m m e t r i c w i t h respect to every line of s p l i t t i n g forces. P h y s i c a l l y , it means t h a t the equivalent s p l i t t i n g forces are a p p l i e d at the t o p a n d the b o t t o m of the s p e c i m e n , a n d the fractures become unstable w h e n the v e r t i c a l cracks i n i t i a t i n g f r o m the t o p a n d the b o t t o m b o t h reach a p p r o x i m a t e l y  one quarter of the s p e c i m e n  height. A compared  similar  argument  applies  in  the  case where  the  height  of a s p e c i m e n  to the size of a p r e - e x i s t i n g crack inside the m a t e r i a l , so t h a t the  final  is s m a l l  f r a c t u r e is  governed b y a s p l i t f r o m t h i s defect. In t h i s case, the s p e c i m e n height c a n be s t i l l considered as 2b,  b u t the p a i r of s p l i t t i n g forces is a p p l i e d inside. T h e fracture becomes unstable w h e n the s p l i t  reaches a p p r o x i m a t e l y  h a l f of the specimen height. T h u s it appears t h a t the m o d e l is useful i n  m a n y cases.  3.4.2 S o m e Consequences of the M o d e l : P e a k Stress  41 P u t t i n g E q . 3.2 i n t o E q . 3.4,  w i t h l/b  =  1 / 2 , a n d s o l v i n g for a, we c a n e s t i m a t e the  f a i l u r e stress (or the s o - c a l l e d compressive strength) of concrete, or any other b r i t t l e m a t e r i a l , as  fc =  3.6  where KJQ= c r i t i c a l stress i n t e n s i t y factor; k = s p a c i n g of defects; £  =  average  value  of  f u n d a m e n t a l m a t e r i a l constants. N o t e that f  c  average c o n f i g u r a t i o n factor; 6 =  a/b.  These q u a n t i t i e s  average  are a l l considered to  half be  is used here to denote the f a i l u r e stress of a b r i t t l e  m a t e r i a l , not necessarily concrete. N o t e t h a t a l l the terms on the r i g h t h a n d side of E q . 3.6 s h o u l d be u n d e r s t o o d i n a n effective sense w h e n they are not clearly defined b y m i c r o s c o p i c o b s e r v a t i o n . F o r concrete a n d rocks, the t e r m K  JC  or G  IC  s h o u l d be u n d e r s t o o d as the energy  d i s s i p a t e d b y a l l the m e c h a n i s m s w h e n a crack propagates, not merely the surface energy. It c a n be easily s h o w n t h a t , based o n this m o d e l , the stress i n t e n s i t y at the c r a c k t i p s w i l l be d r a s t i c a l l y reduced even if a s m a l l l a t e r a l compressive stress is present. T h u s s u c h a stress w i l l lead to a different f a i l u r e m o d e corresponding a higher f a i l u r e stress. T h i s m a y e x p l a i n the shear failure m o d e , w h i c h is a c c o m p a n i e d by a significant increase i n s t r e n g t h , t h a t is e x h i b i t e d i n a c o m p r e s s i o n test o n a confined specimen. In practice, the l a t e r a l stress is often i n t r o d u c e d b y the l o a d i n g p l a t e n i n u n i a x i a l compression tests. E q u a t i o n 3.6 m a y need m o d i f i c a t i o n for specimens of f i n i t e size a n d for the i n t e r l o c k a n d c r a c k arrest m e c h a n i s m s t h a t are present i n concrete a n d rocks; a n d for concrete, i n c l u s i o n of the n o n l i n e a r  behaviour  at the crack front appears to be necessary for precise a n a l y s i s .  Nevertheless, the e q u a t i o n s h o u l d give a reasonable e s t i m a t e of the compressive s t r e n g t h .  3.4.3 R e l a t i o n to T e n s i l e S t r e n g t h B r i t t l e tension failure is r e l a t i v e l y well understood. It is governed d i r e c t l y by the pree x i s t i n g cracks, because extension is unstable under tensile l o a d i n g . T h e tensile s t r e n g t h c a n be  42 e s t i m a t e d , based o n the p r e - e x i s t i n g crack c o n f i g u r a t i o n a n d d i s t r i b u t i o n (see F i g . 3.5) u s i n g the solution (Broek  1978)  Kj  where  the t e r m  =  [<r JWa] t  [(2b/w a) t a n ( 7 r a / 2 6 ) ]  3.7  1 / 2  i n the first b r a c k e t is the w e l l - k n o w n s o l u t i o n for a n  i s o l a t e d crack i n  a  b a c k g r o u n d tensile stress f i e l d ; the t e r m i n the second b r a c k e t is i n c l u d e d to p r o v i d e a n e s t i m a t e o f the effect of a d j a c e n t c r a c k s . A l t h o u g h c r a c k s w o u l d rarely exist i n the c o n f i g u r a t i o n of F i g . 3.5, t e n s i o n f a i l u r e is governed b y a single crack w i t h c r i t i c a l c o n f i g u r a t i o n , so t h a t E q . 3.7 need o n l y h o l d 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 s t r e n g t h as  ft  • \2b  -  K  j  3.8  C  tan(7r£/2)  so t h a t , i n v i e w of E q . 3.6  A fe  =  39  tan(7r£/2)  ft  T h i s i m p l i e s t h a t the r a t i o of tensile to c o m p r e s s i v e s t r e n g t h of a b r i t t l e m a t e r i a l is solely dependent o n the c o n f i g u r a t i o n a n d d i s t r i b u t i o n o f the p r e - e x i s t i n g defects, a n d the i n t e r n a l f r i c t i o n of the m a t e r i a l . T h i s r a t i o for concrete is p l o t t e d against £ i n F i g . 3.6. T h e r a t i o for the e x t r e m e case, u =  0 (e.g. for some ceramics) is s h o w n as w e l l .  E x p e r i m e n t s h o w s t h a t f /f t  c  ranges f r o m 0.06 t o 0.13 for concrete, suggesting a range of  £ f r o m 0.05 t o 0.25. T h e m o d e l also suggests t h a t the lower s t r e n g t h r a t i o is associated w i t h a s m a l l e r £, w h i c h tends t o i n d i c a t e a higher s t r e n g t h m a t e r i a l . T h i s agrees w i t h the w e l l - k n o w n non-proportional  relationship  between  tensile s t r e n g t h  and  compressive strength  o f concrete  F I G . 3.6  Predicted Relation between Tensile Strength a n d Compressive versus S i z e / S p a c i n g R a t i o for B r i t t l e M a t e r i a l s  Strength  44 ( P a r k a n d P a u l e y 1975),. T h e m o d e l also predicts an upper b o u n d of about 0.16 for this r a t i o . T o the author's knowledge, this extreme case has never been surpassed. A n e x p l a n a t i o n is p r o v i d e d for the w i d e range in strength ratio observed i n other b r i t t l e materials. R o c k s , for example, e x h i b i t values f r o m 0.02 to 0.10 ( O b e r t 1972), w h i c h are covered by the m o d e l when the p r e - e x i s t i n g cracks range f r o m short to long relative to their average spacing. T h e m o d e l also predicts t h a t no b r i t t l e m a t e r i a l can have a tensile strength exceeding 2 8 % of its compressive strength.  3.5 T h e Stress-Strain Curves for B r i t t l e M a t e r i a l s under U n i a x i a l C o m p r e s s i o n W e first review e x i s t i n g knowledge of the f o r c e - d e f o r m a t i o n relationship, w h i c h m a y be d i v i d e d into two parts: the pre- a n d post-peak branches. A l t h o u g h the word b r i t t l e implies l i m i t e d d e f o r m a t i o n even for 120  before failure, it appears very  that,  b r i t t l e m a t e r i a l s , there is  some n o n l i n e a r i t y i m m e d i a t e l y before the NORMAL  WEIGHT  peak stress. F o r less b r i t t l e m a t e r i a l s such  10.0  as concrete, there is an essentially linear 8.0  response up to a certain stress level, then ^  6.0  40 E  nonlinearity  becomes  apparent,  and  increasingly so as the m a t e r i a l approaches £  4.0  in  failure 2.0  (Wawersik  and  Fairhurst  O b e r t 1972; M i n d e s s 1983; B r a d y The 0.001  0.002  0.003  0.0O4  0.005  post-peak  behaviour  1970;  1985). is s t i l l  0.006  STRAIN  F I G . 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 of C o n c r e t e , under N o r m a l Test C o n d i t i o n s ( W a n g 1978)  more c o m p l i c a t e d . Less b r i t t l e m a t e r i a l s , such  as concrete, e x h i b i t  a  pronounced  long  t a i l ( W a n g 1978) i n the stress-strain  45 c u r v e under n o r m a l testing c o n d i t i o n s , as s h o w n i n F i g . 3.7. H o w e v e r , K o t s o v o s (1983) i n d i c a t e s t h a t t h i s w i d e l y h e l d v i e w m a y be m i s l e a d i n g . H i s e x p e r i m e n t s show t h a t end c o n d i t i o n s s i g n i f i c a n t l y affect the post-peak b e h a v i o u r , especially for h i g h s t r e n g t h concrete. He p l a c e d v a r i o u s " a n t i - f r i c t i o n " m e d i a between the s p e c i m e n a n d l o a d p l a t e n , a n d f o u n d very  different  behaviour  (Fig.  3.8).  He concludes t h a t , i f the f r i c t i o n a l  restraint is e l i m i n a t e d , the m a t e r i a l w i l l suffer a c o m p l e t e a n d i m m e d i a t e loss o f l o a d - c a r r y i n g c a p a c i t y . H i s results show a n a p p a r e n t recovery of compressive s t r a i n after the peak l o a d , b u t he does not c o m m e n t at a n y l e n g t h o n t h i s s u r p r i s i n g p h e n o m e n o n . It is s e l d o m observed, even i n tests of m o r e b r i t t l e m a t e r i a l s , since r e c o r d i n g i n t h i s range is very d i f f i c u l t w i t h o u t special arrangements. W a w e r s i k a n d F a i r h u r s t (1970), u s i n g very careful test procedures, were a b l e to f o l l o w , i n p a r t , the  descending  branch  for  some f i n e - g r a i n e d rocks. S o m e s t r a i n was c l e a r l y recovered  as t h e l o a d was reduced b e y o n d the peak stress, a n d the s t r e s s - s t r a i n curve t u r n e d t o w a r d s the o r i g i n ( F i g . 3.9). T h e l o n g t a i l , w i t h decreasing stress a c c o m p a n i e d by increasing s t r a i n ( a s s u m i n g t h a t it is not m e r e l y a result of i m p r e c i s e test procedures),  is k n o w n as class I response. T h e a l t e r n a t i v e  o b s e r v a t i o n , w h e n the s t r a i n is recovered, is k n o w n as class II response; i t has the  defining  c h a r a c t e r i s t i c s t h a t " t h e f r a c t u r e process is unstable or s e l f - s u s t a i n i n g ; to c o n t r o l f r a c t u r e , energy m u s t be e x t r a c t e d f r o m the m a t e r i a l "  ( B r a d y 1985).  T h e c l a s s i f i c a t i o n has been based entirely  o n e x p e r i m e n t a l o b s e r v a t i o n , a n d it is hoped t h a t the f o l l o w i n g a n a l y s i s , based on the proposed model,  will  shed some  light  on  the observed  phenomena.  3.5.1 T h e P r e - P e a k B r a n c h The  initial deformation,  before  the cracks begin to e x t e n d , c a n be c a l c u l a t e d f r o m  Y o u n g ' s m o d u l u s . A second phase, w h i c h w i l l now be s t u d i e d , is entered when crack p r o p a g a t i o n begins.  46  FIG.  3.8  Experimental  S t r e s s - S t r a i n R e l a t i o n s of C o n c r e t e , Specimens 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 1983): (a) Stress versus S t r a i n M e a s u r e d 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 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 some N a t u r a l R o c k s ( W a w e r s i k  and Fairhurst  1970)  C o n s i d e r a rectangular region o f height  h and width  w under  assume the c r a c k i n g process is q u a s i - s t a t i c . W h e n the cracks e x t e n d  uniaxial  compression;  dl we have, for energy  conservation,  dV -  where dV  =  3.10  dU = dW + dR  work done b y external l o a d ; dU  d i s s i p a t e d to f o r m new crack extensions; dR  =: increase o f the s t r a i n energy;  dW  =  energy  = <energy d i s s i p a t e d b y the f r i c t i o n between t h e  48 c o n t a c t surfaces o f p r e - e x i s t i n g cracks. Clearly  3.11  dV = FdA  where F =  external load; A  = associated d i s p l a c e m e n t .  S i n c e i n a b r i t t l e m a t e r i a l the p l a s t i c d e f o r m a t i o n  is l i m i t e d , the m a t e r i a l w i l l  essentially l i n e a r l y e l a s t i c regardless o f the c r a c k i n g . It c a n be s h o w n (see  remain  a p p e n d i x ) t h a t as  l o n g as o v e r a l l f r a c t u r e does n o t occur, so t h a t the region is s t i l l c o n n e c t e d , the s t r a i n energy c a n be expressed i n t e r m s o f the e x t e r n a l l o a d a n d the associated d i s p l a c e m e n t , as  U = 1/2  3.12  FA  If t h e f r i c t i o n between the p r e - e x i s t i n g crack surfaces is i n c l u d e d , t h i s expression becomes  3.13  where M is t h e n u m b e r o f p r e - e x i s t i n g c r a c k s i n the region, l  0  m a y be c a l l e d the effective i n i t i a l  e x t e n d i n g c r a c k l e n g t h , w h i c h is a f u n c t i o n o f the c r a c k c o n f i g u r a t i o n ,  and A  is a c o n s t a n t  expressed as  2  8(l-v )  2  nka sma  3.14  2  irEw  T h u s dU is r e a d i l y a v a i l a b l e b y d i f f e r e n t i a t i o n o f this expression:  dU  = 4- FA'+  tan(7r//26)  F A - 2MAFF log (t a n ( 7 r / o / 2 6 ) ) - MAF -^c o s e c ^ J  1  2  dl  3.15  49 T h e new c r a c k surface energy c a n be expressed as  dW = 2M G  3.16  dl  Tn  F i n a l l y , the energy d i s s i p a t e d b y the f r i c t i o n c a n be a p p r o x i m a t e d as (see a p p e n d i x )  dR «  M  8uG sma rr  1 +  k  tan(7r//26)  -k- cos|  3.17  tan(7r/ /26) 0  T h e s e e q u a t i o n s are v a l i d w h e n cracks are e x t e n d i n g , i.e., w h e n Kj  or Gj  defined  E q s . 3.4 or 3.5 have reached the c r i t i c a l values. T h e y give the r e l a t i o n s h i p between l o a d F the d i s p l a c e m e n t A  by and  i n t e r m s of the independent p a r a m e t e r /, the c r a c k l e n g t h . F o r a g i v e n l o a d ,  the e q u a t i o n s c a n be s o l v e d for A  b y s u b s t i t u t i n g E q s . 3.11, 3.15, 3.16 a n d 3.17 i n t o  E q . 3.10,  a n d u s i n g the i n i t i a l c o n d i t i o n  A  where a  =  (Toh/E  when  / =  l  m a y be c a l l e d the threshold stress for the crack e x t e n s i o n ; the r e l a t i o n to l  0  3.18  0  0  is o b t a i n e d  i n v i e w o f E q s . 3.4 a n d 3.6:  3.19  N o w , the l o a d F =  A  where e =  =  aw  eh  longitudinal strain.  is related t o / b y E q s . 3.2, a n d 3.4; a n d  3.20  50 T h u s we are able t o e x t r a c t a n expression f o r s t r a i n i n t e r m s o f stress. A s s u m i n g the average s p a c i n g o f the cracks is the same h o r i z o n t a l l y a n d v e r t i c a l l y ,  M  _  1 (2 b)  3.21  2  wh  we get  2/^sina  e =  where 0  O  k  = arc sin(<x /y£) ; 0  2  0 = arc  log  tan(8/2) tan(9o/2)  +  , fc\ E ]  a f  3.22  c  2  sin(a/f ) . c  P h a s e II o f the pre-peak b r a n c h covers the range a  0  < a < f , with Q c  0  < 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 m o d e l discussed above is f o u n d t o represent b e h a v i o u r o f class II m a t e r i a l s i n t o t h e p o s t - p e a k b r a n c h . W h e n \x =  0, E q . 3.22 is v a l i d f o r t h e f u l l range 0  O  < 0 < 7r. W h e n f r i c t i o n  is i n c l u d e d , however, t h e e q u a t i o n applies o n l y u n t i l t h e cracks stop o p e n i n g somewhere i n t h e descending b r a n c h . T h e r e is then a c o m p l i c a t e d s i t u a t i o n as t h e cracks begin t o close a n d the f r i c t i o n t o change d i r e c t i o n ; a m o r e elaborate t r e a t m e n t is g i v e n i n t h e a p p e n d i x . A f t e r a n i n t e r v a l , t h e c r a c k w i d t h s decrease a n d t h e f r i c t i o n force is r e a c t i v a t e d i n t h e opposite d i r e c t i o n ; Eq.  3.22 is a g a i n a p p l i c a b l e , b u t w i t h opposite sign o n the terms c o n t a i n i n g u i n t h e b r a c k e t .  T h e r e is also a different constant o f i n t e g r a t i o n i n this range. T h i s a p p l i c a t i o n t o class II b e h a v i o u r  is p r e d i c a t e d o n t h e a s s u m p t i o n t h a t  cracks  e x t e n d v e r t i c a l l y i n i s o l a t i o n f r o m each other, so t h a t the region is s t i l l connected. F u r t h e r , i t is a s s u m e d t h a t t h e cracks are regular, so t h a t h o r i z o n t a l fracture does n o t occur, a n d t h a t t h e crack surfaces are r e l a t i v e l y s m o o t h , so t h a t they close d u r i n g t h e descending b r a n c h w i t h o u t i n t e r l o c k i n g . T h e s e a s s u m p t i o n s , necessary f o r c o n t i n u e d a p p l i c a t i o n o f E q . 3.22, are good f o r a  51 r e l a t i v e l y homogeneous b r i t t l e m a t e r i a l . For  less  homogeneous  materials  s u c h as concrete a n d  coarse g r a i n e d  rocks  these  a s s u m p t i o n s m a y be expected to be a p p r o x i m a t e l y f u l f i l l e d d u r i n g the l o a d i n g stage, w h e n the cracks are less extensive, a n d s t i l l o p e n i n g . O n the descending b r a n c h , however, the cracks tend to p r o p a g a t e t h r o u g h weak g r a i n boundaries or aggregate-cement m a t r i x bonds (Ziegeldorf 1983; B r a d y 1985), a n d the z i g - z a g crack p a t h s have a tendency to i n t e r c o n n e c t p a r a l l e l fractures. T h i s leads t o t y p e I response as w i l l be discussed below. B u t , even i n t h i s case, the e x p e r i m e n t s of K o t s o v o s (1983) suggest t h a t the t y p e  I  response m a y m e r e l y be due to the end f r i c t i o n a l c o n s t r a i n t s i n h i b i t i n g v e r t i c a l c r a c k extension and  l e a d i n g t o t h i s t y p e of failure. W i t h  " a n t i - f r i c t i o n " c a p p i n g , he observed  that vertical  c r a c k i n g of the higher strength specimens a l w a y s extended i n b o t h d i r e c t i o n s , w h i l e , for  the  lower s t r e n g t h ones, i t extended i n one d i r e c t i o n o n l y , i n d i c a t i n g t h a t the r e s t r a i n i n g a c t i o n of at least one end zone was s t i l l present. T h u s i t appears t h a t the v a l i d i t y of the a s s u m p t i o n s of E q . 3.22 depends u p o n the m a t e r i a l properties a n d the l o a d i n g c o n d i t i o n s .  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 E q . 3.22 is p l o t t e d for a // 0  c  =  0.3 , w i t h u =  0 and a =  0.36, i n F i g . 3.10. T h e shape  of the T y p e II curve is c h a r a c t e r i z e d b y 4 p o i n t s as i n d i c a t e d on the figure. F r o m O to A , below the t h r e s h o l d stress, I =  the cracks d o not e x t e n d , a n d d i s p l a c e m e n t is due solely to the  l, 0  l i n e a r elastic response. In r e a l i t y , o f course, the onset o f stable crack extension is d i f f i c u l t to i d e n t i f y ; i t is a g r a d u a l process r a t h e r t h a n a sudden one, because o f the v a r i e t y of defect c o n f i g u r a t i o n s . T h u s there is a t r a n s i t i o n rather t h a n a well-defined p o i n t A . From  A  to B , a d d i t i o n a l d e f o r m a t i o n  becomes m o r e n o n - l i n e a r as f  c  occurs due to crack e x t e n s i o n , a n d the curve  is a p p r o a c h e d . I n c l u s i o n of f r i c t i o n increases b o t h peak stress a n d  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 of B r i t t l e M a t e r i a l s  53 s t r a i n . G r e a t e r n o n - l i n e a r i t y appears due to energy d i s s i p a t i o n t h r o u g h f r i c t i o n . A f t e r the peak, the cracks c o n t i n u e to propagate as the a p p l i e d stress decreases,  in an  u n s t a b l e e x t e n s i o n process. A t p o i n t C the s t r a i n reaches i t s m a x i m u m v a l u e , a n d t h e n begins to reduce; the w o r k required for the c r a c k i n g process b e y o n d C is p r o v i d e d b y p a r t o f the s t r a i n energy released f r o m the m a t e r i a l , b u t s u r p l u s energy m u s t be e x t r a c t e d b y the l o a d i n g device. T h e a r e a enclosed b y the c o m p l e t e curve is, of course, equal to the energy d i s s i p a t e d i n c r e a t i n g new crack surfaces. A " l e s s b r i t t l e " m a t e r i a l m a y have a lower t h r e s h o l d stress for crack e x t e n s i o n , a n d a higher coefficient of f r i c t i o n . A " m o r e b r i t t l e " m a t e r i a l , o n the c o n t r a r y , t h r e s h o l d a n d l o w f r i c t i o n coefficient,  m a y have a very h i g h  so t h a t i t gives the appearance of a linear s t r e s s - s t r a i n  curve. B u t the m o d e l i m p l i e s t h a t , since failure is caused b y c r a c k i n g , there m u s t a l w a y s be some n o n - l i n e a r i t y before it occurs. N o t e t h a t f a i l u r e is f i n a l l y b r o u g h t a b o u t b y the u n s t a b l e crack e x t e n s i o n s p l i t t i n g the s p e c i m e n i n t o pieces w h i c h are i n d i v i d u a l l y u n s t a b l e , thus r e d u c i n g the l o a d c a p a c i t y . With  conventional  test a r r a n g e m e n t s ,  a load controlled testing machine  w i l l cause  m a t e r i a l f a i l u r e at p o i n t B ; d i s p l a c e m e n t c o n t r o l w i l l lead to failure at C i f the m a c h i n e is s t i f f e n o u g h . F a i l u r e w i l l be e x p l o s i v e because of the sudden release of s t r a i n energy. M o s t results for m o r e b r i t t l e m a t e r i a l s are i n c o m p l e t e i n t h i s sense, b u t W a w e r s i k  and  reported Fairhurst  showed c o m p l e t e curves t h a t agreed, q u a l i t a t i v e l y , w i t h the m o d e l p r e d i c t i o n , as d i d the postpeak curves for h i g h strength concrete o b t a i n e d b y K o t s o v o s . N o t e t h a t , d u r i n g unstable crack e x t e n s i o n , one or a few cracks w i l l propagate preferentially, so t h a t the m o d e l m a y lose some validity. H o w e v e r , for less homogeneous m a t e r i a l s , such as concrete a n d c o a r s e - g r a i n e d rocks, the a s s u m p t i o n o f regular v e r t i c a l c r a c k extension m a y  not be v a l i d i n the p o s t - p e a k range, as  discussed earlier. T h i s w o u l d e x p l a i n the frequently observed class I response. W a w e r s i k Fairhurst  (1970)  found  that,  in  class  I  behaviour  of  rocks,  vertical fracture  is,  and  indeed,  54 a c c o m p a n i e d by g r a d u a l d e v e l o p m e n t fracture  in  the  post-peak  range;  of shear a  similar  d e s c r i p t i o n is given by K o t s o v o s for his lower strength specimens. During  this  range,  Fig.  3.11  the  ratchet-like  mechanism  of  forms,  vertical  deformation  involves the wedges being  driven  i n t o each other, a n d the energy of the l o a d is converted to s t r a i n energy i n the wedges, a n d f r i c t i o n losses between t h e m , as well as surface energy i n new cracks. T h e first two are c l e a r l y  F I G . 3.11 D e p i c t i o n of Irregular C r a c k i n g Pattern  nonlinear,  requiring  decrease  less  deformation, curves  to  that  rapidly  the with  vertical  load  increasing  a n d c a u s i n g the l o a d - d e f o r m a t i o n be  concave  upward.  Hence  the  inflection point observed by W a n g et a l . (1978) a n d the l o n g t a i l thereafter. From  the above analysis one m a y conclude t h a t the fracture p a t t e r n determines  the  post-peak stress-strain r e l a t i o n . V e r t i c a l fracture t h r o u g h the m a t e r i a l w i l l lead to " m o r e b r i t t l e " failure, w h i l e the development of shear faults w i l l give the appearance of m o r e d u c t i l e b e h a v i o u r . T h e fracture p a t t e r n , i n t u r n , m a y often be governed by the l o a d i n g c o n d i t i o n s ; f r i c t i o n i n the l o a d i n g p l a t e n , for example, m a y cause shear cracks a n d , more d u c t i l e behaviour.  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 s W e now consider briefly the s e n s i t i v i t y of the m o d e l predictions to s t a t i s t i c a l v a r i a t i o n s i n the parameters, w h i c h have hitherto been treated i n a n average sense. A s s u m e that the defects are u n i f o r m l y d i s t r i b u t e d s p a t i a l l y but t h a t the c o n f i g u r a t i o n s  55 have a r a n d o m character described b y some p r o b a b i l i t y d i s t r i b u t i o n , w i t h d e n s i t y f u n c t i o n  $ = $(ka)  3.23  and cumulative function  Z =  Z(ka)  3.24  w i t h ka i n some range  ( )min  <  ka  ka <  (ka) ax  3.25  m  E q u a t i o n 3.6 c a n be m o d i f i e d b y these a s s u m p t i o n s to give  1/4  K (b  Fa ( l-Z(ka) ka I l-Z(ka)  IC  ka[l-Z(ka)}  ~  1/4  where  b  0  is the average  ka  h a l f s p a c i n g of t o t a l defects under  3.26  consideration, and  the  barred  q u a n t i t i e s denote the m e a n values. Assuming  a normal  d i s t r i b u t i o n for  defect c o n f i g u r a t i o n ,  and  approximating  by  a  W e i b u l l d i s t r i b u t i o n , we w r i t e  Z(ka) = 1 -  The Weibull modulus m =  exp [  -(ka/6) ]  3.27  m  3.6 best represents a n o r m a l d i s t r i b u t i o n ; 8 is t a k e n c o r r e s p o n d i n g to  a coefficient of v a r i a t i o n of ka of a b o u t 0.32.  the second t e r m of E q . 3.26, w h i c h a c c o u n t s for  the s t a t i s t i c a l c o n s i d e r a t i o n s , is p l o t t e d i n F i g . 3.12, a n d shows the v a r i a t i o n of f  t  as p r e d i c t e d  56 b y the m o d e l , against the v a r i a t i o n in crack c o n f i g u r a t i o n . -  If o n l y a few defects w i t h larger c o n f i g u r a t i o n factor ka (i.e. w i t h wider crack spacing)  have extended to govern specimen behaviour, the failure stress w i l l be higher. T h i s is because these defects are sparsely d i s t r i b u t e d , the stress required to b r i n g t h e m to interact u p o n each other to reach the c r i t i c a l state w i l l be h i g h , as i n d i c a t e d at the r i g h t e n d of the g r a p h . Although  defects w i t h s m a l l e r c o n f i g u r a t i o n are more densely d i s t r i b u t e d , the stress  needed to cause them to extend w i l l s t i l l be higher, as s h o w n at the left end of the g r a p h . T h e m i n i m u m value is reached when ka is close to the m e a n value, a r o u n d w h i c h the v a r i a t i o n is s m a l l for a wide range of ka. T h u s the average p a r a m e t e r s do y i e l d a reasonable a p p r o x i m a t i o n . T h i s conclusion s h o u l d be v a l i d as l o n g as the d i s t r i b u t i o n is not  extremely  distorted.  i0 -I 0  1  I  1  0.2  0.4  0.8  I—  1 1.2  1  0.8  1  ka /  T 1-4  1  I  1.8  1.8  I 2  ka  F I G . 3.12 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 to C r a c k C o n f i g u r a t i o n F a c t o r : the 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 by the M o d e l is P l o t t e d against the C o n f i g u r a t i o n F a c t o r , W h i c h Depends o n C r a c k C o n f i g u r a t i o n a n d Internal F r i c t i o n  57 3.7  Summary and Corollary The  failure  mechanism  of  brittle  materials  under  uniaxial  compression  has  been  e x a m i n e d at the f u n d a m e n t a l level. A f a i l u r e m o d e l based o n the i n t e r n a l m e c h a n i s m has been proposed to reveal the c h a r a c t e r i s t i c s of the compressive s t r e n g t h , a n d s t r e s s - s t r a i n r e l a t i o n of these m a t e r i a l s . T h e observed s p l i t t i n g failure has been s h o w n to be the result o f the c u m u l a t i v e , s u b c r i t i c a l , stable c r a c k g r o w t h . T h e u n d e r l y i n g concepts of the m o d e l h a v e been j u s t i f i e d b y r e p o r t e d observations. It  may  be further  inferred f r o m the s t u d y  that  the f a i l u r e stress, or  the s o - c a l l e d  c o m p r e s s i v e s t r e n g t h of a b r i t t l e m a t e r i a l is closely dependent on the i n t e r n a l f a i l u r e m e c h a n i s m . T h e i n t e r n a l m e c h a n i s m , however, depends not o n l y o n the m a t e r i a l t e x t u r e , b u t is also affected b y the testing or l o a d i n g c o n d i t i o n s . S p l i t t i n g failure corresponds to the lowest failure stress. A n y c o n d i t i o n s w h i c h prevent this failure m o d e f r o m being realized w i l l lead to a n a p p a r e n t l y higher f a i l u r e stress. These c o n d i t i o n s m a y be l a t e r a l c o n f i n e m e n t such as t h a t i n t r o d u c e d b y the end f r i c t i o n , or a s t r a i n g r a d i e n t w h i c h causes u n e q u a l c o m p r e s s i o n i n the m a t e r i a l . Although determine, mechanisms  the compressive s t r e n g t h as a f u n c t i o n of these c o n d i t i o n s is d i f f i c u l t  one m a y are  to  expect the failure stresses to be better correlated if the i n t e r n a l f a i l u r e  similar.  T h i s , is of  p r a c t i c a l significance. In  the  later  chapters,  separate  t r e a t m e n t s for concrete m a s o n r y under different l o a d i n g c o n d i t i o n s w i l l be proposed a n d i t w i l l be seen t h a t t h i s leads to better correlations i n t e r m s of the failure stresses.  58 CHAPTER  IV  PLAIN MASONRY WITH FULL BEDDING  4.1 T w o B a s i c a l l y Different F a i l u r e M o d e s In concrete m a s o n r y compression tests, the specimens f a i l b a s i c a l l y i n t w o modes. O n e is s p l i t t i n g i n the d i r e c t i o n of the l o a d ; the other shows c o n i c a l failure planes (see C h a p t e r II). significance of these t w o different m e c h a n i s m s arises f r o m the f a c t t h a t the different  The  failure  modes y i e l d different a p p a r e n t strengths, as i n d i c a t e d i n the preceding c h a p t e r . It has been f o u n d repeatedly i n previous e x p e r i m e n t a l studies (for e x a m p l e , F a t t a l a n d C a t t a n e o 1976; T u r k s t r a a n d T h o m a s 1978) t h a t when the e c c e n t r i c i t y of the l o a d o n m a s o n r y specimens is increased, there is a s i g n i f i c a n t a p p a r e n t increase i n compressive s t r e n g t h .  This  p h e n o m e n o n is also revealed i n the tests c o n d u c t e d b y the a u t h o r , as d e p i c t e d i n F i g . 4.1 b y comparing a theoretical load-moment results. A l t h o u g h  i n t e r a c t i o n curve for a m a s o n r y p r i s m w i t h e x p e r i m e n t a l  this s t r e n g t h increase was a t t r i b u t e d i n some earlier studies ( T u r k s t r a  and  T h o m a s 1978) to the stress gradient effect, it is n o w generally accepted t h a t it is essentially due to a difference i n the f a i l u r e m e c h a n i s m . W h e n masonry masonry  is under u n i a x i a l 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 d o m i n a t e s , whether  the  is f u l l y or face-shell bedded. T h e failure m o d e changes to the shear t y p e w h e n  the  m a s o n r y is under eccentric l o a d i n g . Two mechanisms?  obvious 2)  what  questions  arise:  1)  what  is the  cause of  these  is the i m p l i c a t i o n o f these failure m e c h a n i s m s  two for  different the  failure  compressive  s t r e n g t h , the p a r a m e t e r o f m o s t p r a c t i c a l concern. In t h i s s t u d y , the f a i l u r e m e c h a n i s m is carefully r e - e x a m i n e d , a n d some of the e x i s t i n g theory is revised, i n the l i g h t of b o t h e x p e r i m e n t a l a n d a n a l y t i c a l w o r k . W e s t a r t w i t h the case o f f u l l y bedded p l a i n concrete m a s o n r y under u n i a x i a l c o m p r e s s i o n .  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 Increase P h e n o m e n o n  Splitting failure under  under Eccentric Compression  u n i a x i a l compression has been  experiments, a n d the experiments conducted by the author  indicated by  numerous  previous  have also revealed this phenomenon  (see F i g . 2.12). Under field  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  is t h e j o i n t .  We  shall discuss the  effect o f  disturbance in the u n i a x i a l compression  the joint  on  the strength  of  masonry,  and  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  Joint E f f e c t — A Revision of Hilsdorf s Model 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 to 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  water  tightness, as w e l l as a r c h i t e c t u r a l effect. T h e j o i n t s c a n be in 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 bond  or stack b o n d ,  and  they  c a n be r a k e d or f l u s h . H o w e v e r ,  for  reasons of s i m p l i c i t y ,  the  60 s t u d y i n t h i s c h a p t e r is confined to stack b o n d e d , f u l l y bedded m a s o n r y w i t h u n r a k e d j o i n t s . It is believed t h a t (Maurenbrecher  the b o n d p a t t e r n  does not  have  a s i g n i f i c a n t effect o n m a s o n r y  strength  1980; S h r i v e 1982), a n d t h a t the a n a l y s i s to be presented is generally a p p l i c a b l e .  T o achieve cohesiveness a n d w o r k a b i l i t y , m o r t a r c o n t a i n s c e r t a i n p r o p o r t i o n s of cement, s a n d a n d l i m e . M e c h a n i c a l l y , i t is u s u a l l y weaker a n d less s t i f f t h a n the s u r r o u n d i n g concrete u n i t s (see C h a p t e r  II).  It is w i d e l y accepted t h a t the m o r t a r j o i n t s affect the m a s o n r y s t r e n g t h , stronger m o r t a r m a k i n g stronger m a s o n r y . Hilsdorf  (1969).  His  T h e m o s t i n f l u e n t i a l theory for the j o i n t effect was proposed  theory  postulates  that  when  masonry  prisms  are  under  by  uniaxial  c o m p r e s s i o n , the less s t i f f m o r t a r has a tendency to e x p a n d l a t e r a l l y ; t h i s l a t e r a l e x p a n s i o n of the m o r t a r is c o n f i n e d b y the m a s o n r y  u n i t s , g i v i n g rise to l a t e r a l compressive stress i n the  m o r t a r a n d to l a t e r a l tensile stresses i n the u n i t s , thereby c a u s i n g tensile s p l i t t i n g failure of the b l o c k s . U s i n g the C o u l o m b - N a v i e r f a i l u r e c r i t e r i o n a n d some rather coarse a s s u m p t i o n s  about  e q u i l i b r i u m a n d c o m p a t i b i l i t y , H i l s d o r f d e r i v e d a n e q u a t i o n r e l a t i n g the c o m p r e s s i v e s t r e n g t h of masonry  to the strengths of u n i t  and  mortar.  T h i s , of course, is very  p r a c t i c a l , since the  strengths of the u n i t a n d the m o r t a r are c o m p a r a t i v e l y easy to measure. H o w e v e r , there has been a l o t o f c o n t r o v e r s y a b o u t the correctness of H i l s d o r f s m o d e l i n the subsequent l i t e r a t u r e . O n the one h a n d , H a t z i n i k o l a s et a l (1978) m a d e a n u m e r i c a l a n a l y s i s based o n H i l s d o r f s m o d e l a n d c o n c l u d e d t h a t the m a g n i t u d e of the tensile stress i n the b l o c k u n i t s due to the l a t e r a l e x p a n s i o n of the m o r t a r was sufficient to exceed the tensile s t r e n g t h o f concrete b l o c k s a n d thus was responsible for the s p l i t t i n g f a i l u r e of concrete m a s o n r y . P r i e s t l e y et a l  (1983) extended  agreement  Hilsdorfs  equation  to g r o u t e d  concrete m a s o n r y  and  claimed  good  (in t e r m s of m a s o n r y strength) w i t h the e x i s t i n g e x p e r i m e n t a l d a t a . M o s t recently,  B i o l z i (1988) a p p l i e d the failure m o d e l i n a n a p p r o x i m a t e p l a s t i c a n a l y s i s for b r i c k m a s o n r y .  On  the other h a n d , S h r i v e (1980, 1983) s t r o n g l y opposed the n o t i o n t h a t the l a t e r a l e x p a n s i o n of • m o r t a r was the m a i n cause o f s p l i t t i n g failure. H e n o t e d t h a t 1) s p l i t t i n g f a i l u r e of c o m p r e s s i o n  61 specimens is not u n i q u e to m a s o n r y .  2) the tensile stress caused b y m o r t a r e x p a n s i o n iri the  b l o c k is t o o s m a l l to exceed the tensile strength of the block. T h e l a t t e r c o n c l u s i o n was based on the n u m e r i c a 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 S h r i v e h i m s e l f w i t h Jessop (1980), w h i c h i n d i c a t e t h a t the tensile stress is m u c h less t h a n t h a t required to  break  the  tensile b o n d s the  lateral  i n the  block. Drysdale  expansion  of  mortar  and  mechanism  of  needed  experiments  the m o r t a r j o i n t s h a d a r e l a t i v e l y m i n o r  Hamid  (1979) suggested  reconsideration  because  that in  the their  influence o n the c a p a c i t y o f concrete  masonry. The  emergence  of these controversies is not s u r p r i s i n g , since some p o i n t s were  not  c l a r i f i e d i n the p r e v i o u s studies. W h e n p o s t u l a t i n g a tensile stress w h i c h w i l l i n i t i a t e a c r a c k , it is i m p o r t a n t to i n d i c a t e the l o c a t i o n where it w i l l occur. T h i s provides a l o g i c a l w a y to check the correctness  of  the  model  by  examining  e x p e r i m e n t a l studies. T h i s was somehow  whether  the  location  is  correctly  o v e r l o o k e d i n the previous w o r k . T h e  predicted  in  arbitrariness  i n v o l v e d i n the a s s u m p t i o n of the m a t e r i a l constants used i n n u m e r i c a l analyses m a y also have c o n t r i b u t e d to the c o n t r o v e r s i a l nature of some previous f i n d i n g s . A n d so far, there has been no direct e x p e r i m e n t a l evidence w h i c h c o u l d lead to a conclusive assessment o f the m o d e l . Because of t h a t , some e x p e r i m e n t a l a n d a n a l y t i c a l w o r k is d i r e c t e d here to e v a l u a t i o n o f t h i s theory. It s h o u l d be i n d i c a t e d t h a t a l l the previous w o r k i m p l i c i t l y takes one n o t i o n for g r a n t e d : t h a t f a i l u r e is a l o c a l i z e d effect. W h e t h e r m a s o n r y fails depends on whether tensile stress at some p o i n t exceeds the tensile bonds of the m a t e r i a l . In the l i g h t of the s t u d y i n C h a p t e r III, we k n o w t h a t i n the case o f c o m p r e s s i o n , l o c a l tensile c r a c k i n g is o n l y a necessary c o n d i t i o n for g l o b a l f a i l u r e ; i t m a y not be sufficient. In our a p p r o a c h , we w i l l consider b o t h the causes of tensile c r a c k i n g , a n d whether t h i s is t a n t a m o u n t to failure.  4.2.1 E x p e r i m e n t a l R e s u l t s • T h e e x p e r i m e n t a l "results i n d i c a t e :  62 1) for a l l f u l l y bedded p l a i n concrete m a s o n r y p r i s m s tested under u n i a x i a l c o m p r e s s i o n , v e r t i c a l s p l i t t i n g ( p a r a l l e l to the d i r e c t i o n of l o a d i n g ) was the p r e d o m i n a n t f a i l u r e m o d e . T h e s p l i t t i n g occurs i n the m i d d l e t h i r d of the web, c o n t i n u o u s l y runs t h r o u g h the s p e c i m e n , as t y p i c a l l y i l l u s t r a t e d i n F i g . 2.12. S i m i l a r observations were reported i n previous e x p e r i m e n t a l w o r k  (for  e x a m p l e , H a t z i n i k o l a s et a l 1978). 2)  There  is  a  lateral  expansion  effect  due  to  the  mortar,  as  evidenced  by  the  strain  m e a s u r e m e n t s o n the block u n i t s . F i g 4.2 shows some t y p i c a l results recorded i n e x p e r i m e n t s . T h e average l a t e r a l s t r a i n a t l o c a t i o n # 3 ,  w h i c h is closer to the m o r t a r j o i n t , is a p p r e c i a b l y  larger t h a n t h a t m e a s u r e d at l o c a t i o n # 4 , w h i c h is at the m i d - h e i g h t of the web. 3) H o w e v e r , there is some randomness i n where the m a c r o c r a c k is i n i t i a t e d . B y d e t e c t i n g the order o f b r e a k i n g of the wires crossing splits ( A d e t a i l e d d e s c r i p t i o n of t h i s procedure was g i v e n i n C h a p t e r II), cracks were f o u n d to i n i t i a t e at a l o c a t i o n close to the m o r t a r j o i n t o n l y i n a b o u t two  t h i r d s of the f u l l y  bedded  specimens, as d e p i c t e d i n F i g . 4.3. T h i s does not  support  H i l s d o r f s m o d e l , since the m o d e l suggests t h a t crack s h o u l d i n i t i a t e c o n s i s t e n t l y f r o m the j o i n t . If we assume t h a t a crack w h i c h i n i t i a t e d f r o m t h i s l o c a t i o n is a r a n d o m event a n d  relax  H i l s d o r f s h y p o t h e s i s s u c h t h a t there is o n l y a 9 0 % chance of t h i s event o c c u r r i n g , then this h y p o t h e s i s is rejected at a 0.1 level of significance. W e  also note the test result is in sharp  c o n t r a s t to t h a t observed i n face-shell bedded p l a i n concrete m a s o n r y ,  where cracks i n i t i a t e d  c o n s i s t e n t l y at a l o c a t i o n close to the j o i n t , (cf. F i g . 5.2) T h i s strengthens the assertion t h a t the test result does not  support  Hilsdorfs postulation  that  the s p l i t t i n g is due  to the l a t e r a l  e x p a n s i o n o f the m o r t a r j o i n t . 4) T h e j o i n t c o n d i t i o n s have a b i g influence, on the c a p a c i t y of a m a s o n r y p r i s m (see T a b l e 4.1). It is n o t e d t h a t stronger m o r t a r does not necessarily m a k e a stronger p r i s m , as i n d i c a t e d b y c o m p a r i n g the f a i l u r e loads of the p r i s m s w i t h type S m o r t a r w i t h those w i t h t y p e M  mortar.  T h e lower failure, loads of the p r i s m s w i t h t y p e M m o r t a r are believed to be due to the poorer a d h e s i o n of t h a t t y p e of m o r t a r ,  w h i c h appeared d u r i n g the e x p e r i m e n t s . T h e effect of the  63  LATERALSTRAN (1/1000 N/N)  F I G . 4.2  Measured  LATERAL STRAN (1/1000 N/N)  Lateral Strains in W e b s of M i d d l e Courses of P l a i n  Masonry  P r i s m under U n i a x i a l  1  III  I  III  - II  -  -1  Ul  II  111  II  - iv 1  - i vV  V  M3-1  1  N4-3  II  It  II  II - Ill  -1 III  1  III  1  - iv -  F I G . 4.3  -  S1-3  -  N2-1  Compression  - i vV 1  N2-2  N2-3  Detected Orders of Macroscopic Splitting, i n T e r m s of 4 Sections along  Prisms.  64 SPECIMEN M3  (M-MORTAR)  1  2  3  4  AVG  COV  187.0  147.0  123.0  140.0  149.0  15.7%  SI  (S-MORTAR)  204.0  194.0  178.0  168.0  186.0  7.5%  N2  (N-MORTAR)  125.0  140.5  143.0  164.0  143.0  9.7%  N4  (t  120.0  105.0  133.0  119.0  9.6%  P5  (to  103.0  123.0  112.0  110.0  7.5%  0  =  3/4in)  = 0 in )  103.0  T a b l e 4.1 F a i l u r e L o a d s o f P l a i n P r i s m s w i t h F u l l B e d d i n g (kips)  adhesion o n m a s o n r y s t r e n g t h w i l l be discussed later. 5)  V e r t i c a l strain measurements  indicate that  the m o r t a r j o i n t s are m u c h  softer t h a n the  concrete u n i t s . T h e r a t i o o f the i n i t i a l m o d u l u s o f concrete t o t h a t of three m o r t a r types is about 6-8 t o 1. (See F i g s . 2 . 3 , 2.4 a n d 2.5 i n C h a p t e r II.) T o s t u d y j o i n t effects, some o f the p r i s m s were b u i l t w i t h zero j o i n t thickness, a n d one g r o u p w i t h glass p l a t e . T h e glass w a s chosen because of i t s h i g h m o d u l u s r e l a t i v e l y l o w r u p t u r e strength ( £ = 8 x l 0 p s i , 6  f  rup  =  of elasticity a n d  5000 p s i , o b t a i n e d b y e x p e r i m e n t ) . It was  expected t h a t t h e glass p l a t e f i l l e d j o i n t s w o u l d m i n i m i z e the P o i s s o n ' s effect a n d a t the same t i m e p r o v i d e l i t t l e l a t e r a l c o n f i n e m e n t . H o w e v e r , these specimens w i t h o u t m o r t a r bedded j o i n t s f a i l e d a t r e l a t i v e l y l o w loads (see T a b l e s 4.1, a n d 8.1 f o r grouted p r i s m s ) ; t h e e x p e r i m e n t s were not c o n c l u s i v e . T h e l o w f a i l u r e loads are believed t o have been caused b y stress c o n c e n t r a t i o n s i n the v i c i n i t y o f t h e j o i n t s w i t h o u t a m o r t a r c u s h i o n . T h e i n d i c a t i o n s are t h a t v e r t i c a l cracks o c c u r r e d d u r i n g t h e l o a d i n g stage o f these p r i s m s ; a n d for the specimens w i t h glass plates, the c r a c k i n g noise o f the glass was also heard.  4.2.2 T h e o r e t i c a l A n a l y s i s We  proceed n o w t o revise H i l s d o r f s m o d e l i n t h e l i g h t of a stress a n a l y s i s . A l t h o u g h  n u m e r o u s stress analyses ( m e n t i o n e d above) i n c l u d i n g some based o n 3 d i m e n s i o n a l (Hamid and Chukwunenye,  modeling  1986) o f this p r o b l e m have been c o n d u c t e d , they were a l l based o n  65  numerical approaches. T o 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 faceshell 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  t , the variation of the vertical displacements 0  with x must also be small. Therefore we assume  throughout the region u — u(x,y)  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  +  with boundary conditions  0 < x < a/2 0 < y <  to  4.1  0  a A  -V  V  -X-  q=y<T  -x-  — V  -V  0" (a) F I G . 4.4  (b)  (c)  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) under A x i a l C o m p r e s s i o n ; b) under B i a x i a l C o m p r e s s i o n c) under L a t e r a l T r a c t i o n .  T  x,u  i  q =  •a/2-  y-.v  F I G 4.5  A Mortar Joint under Lateral Traction  va  67 u(0,y) = 0  0  < 2/ <  u(x,0) = 0  0  < x < a/2  4.3  u(x,l ) = 0  0  < x < a/2  4.4  0  < y < U  4.5  0  ^ - g u (a/2,y) x  = q  l-v  <o  (by symmetry)  4.2  where E is Y o u n g ' s m o d u l u s a n d v is P o i s s o n ' s r a t i o . A series s o l u t i o n f o r t h i s b o u n d a r y v a l u e p r o b l e m c a n be f o u n d (see a p p e n d i x )  4(l-t/ )gt 2  K KE  where  0  £2, sinh[(2n-l)K7T2:/i ] 0  ^  2  sin[(2n-l)7ry/<  0  4.6  (2n-l) cosh[(2n-l)«:7ra/2< ,] 2  <  K= ^(1-U)/2  F r o m w h i c h w e deduce the shear stress a l o n g the j o i n t  xy  —  2$k)  _  4nq sr^  w  (  " *'  0  )  s i n h [(2 n-1) K TT X/ t ]  4.7  0  ^  (2n-l)cosh[(2n-l)K7ra/2/ ] 0  w i t h t h e r e s u l t a n t force  a/2  S =  T xy  dx — 2  *  Since  a ^> t , 0  h  ( -D 2  2  L  cosh[(2n-l)«;7ra/2f ]  c o s h [ ( 2 n - l ) « 7 r a / 2 i ] 3> 1, the second t e r m 0  4.8  0  i n the b r a c k e t o f E q . 4.8, w h i c h  represents t h e v a n i s h i n g l y s m a l l force transferred b y t h e m i d d l e o f t h e j o i n t , c a n be neglected. F u r t h e r , b y n o t i n g t h a t q = va a n d  68  oo  1 (2n-l) 2  n=l  2  8  '  we o b t a i n S  veto 2  =  4.9  T h e p o i n t of a c t i o n of t h i s resultant is a/2  J" X  Ty  dx  x  0  a 2(2n-l)  to 2  K7r(2n-1)  a 2  3  _  to_  4.10  K 7 r  w h i c h lies near the end o f the j o i n t . B y i n s p e c t i n g E q s . 4.7 a n d 4.10, it is c o n c l u d e d t h a t the interface shear m u s t be h i g h l y c o n c e n t r a t e d near t w o ends of the j o i n t . It is also clear i n v i e w of E q . 4.9, t h a t this shear is d i r e c t l y p r o p o r t i o n a l to the a p p l i e d compressive stress, the thickness a n d the P o i s s o n ' s r a t i o o f the m o r t a r j o i n t . It is these shear forces a c t i n g l i k e l a t e r a l p o i n t loads w h i c h i n t r o d u c e  the  tensile stresses i n the web a n d face-shell. E q . 4.7 is c o m p a r e d w i t h a n u m e r i c a l s o l u t i o n using the b o u n d a r y element m e t h o d . T h e c o m p u t e r p r o g r a m ( T W O F S ) is g i v e n b y C r o u c h et a l (1983), a n d 67 elements were used for t h i s p r o b l e m . F o r i / = 0.3 a n d t / o = 3 / 6 4 , the s o l u t i o n s are p l o t t e d i n F i g . 4.6, together w i t h a 0  d e p i c t i o n of h o w these shears act o n a web. T h e a n a l y t i c a l s o l u t i o n is i n g o o d agreement the  numerical  one,  which supports  the a s s u m p t i o n  that  the  vertical displacement  with  can  be  neglected. W e proceed to p e r f o r m stress a n a l y s i s for a web or a face-shell under the a c t i o n of these shear forces. A s s h o w n i n F i g . 4.7, t h i s is a plane p r o b l e m i n a r e c t a n g u l a r d o m a i n w i t h stress specified b o u n d a r y c o n d i t i o n s . T h e shear d i s t r i b u t i o n s on the boundaries are g i v e n b y 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 =-T  d> Y  xy  xy a  h  =0  b <t =0  4  V ^ =0  xy  <t) o xx=  (t> = T xy  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 a p p r o a c h for t h i s k i n d o f p r o b l e m is t o f i n d a stress f u n c t i o n ( A i r y stress f u n c t i o n ) . F o r this p a r t i c u l a r case we c a n w r i t e the stress f u n c t i o n i n a series f o r m as  *  =  £  A  m  m=l  a  m  boundary  s  =  z  i  n  -  g  tanha  m  cosh-y/ >)  . S  l  /?m  n  mir(x-a/2)  a  ^ <  a  n  d i  mir(x-a/2)\  — /?mtanh/3 cosh  n  v  m  b  = = !  s i n ^  2  m  /  b  ^»  h  V  frmr(x-a/2)  I  B  + m  where  (  r  a  , - S  m  are d e t e r m i n e d  f e  ' '  s  b y having  . miry i  n  —  4  .  1  .  1  E q . 4.11 satisfy t h e  c o n d i t i o n s d e p i c t e d i n F i g . 4.7 (for d e t a i l e d d e r i v a t i o n see a p p e n d i x ) . B y i n s p e c t i o n ,  the m a x i m u m  tensile stress w i l l occur a t t h e t o p a n d b o t t o m b o u n d a r i e s  o f the d o m a i n . S o  f i n a l l y , the tensile stress d i s t r i b u t i o n we are interested i n is  <r — ®yy x  =  2£  A  m  (Vf)  cosha  m  s i n ^  4.12  m=l  F o r a square d o m a i n , as i n the g e o m e t r y o f the web, the series s o l u t i o n is p l o t t e d i n F i g . 4.8, together (yj/Ej)/(v /E ) u  xl  w i t h a n u m e r i c a l s o l u t i o n . A n u m e r i c a l s o l u t i o n f o r t h e m o r e r e a l i s t i c case o f = §, where s u b s c r i p t s j a n d u denote m o r t a r j o i n t a n d b l o c k u n i t respectively,  is also i n c l u d e d i n the g r a p h . T h e changes o f t h i s d i s t r i b u t i o n due t o v a r i a t i o n s i n P o i s s o n ' s r a t i o a n d t h e thickness o f the j o i n t , the aspect r a t i o o f the r e c t a n g u l a r d o m a i n ( c o r r e s p o n d i n g t o a web a n d a face-shell) are p l o t t e d i n F i g . 4.9, F i g . 4.10 a n d F i g . 4.11 respectively. T h e above stress a n a l y s i s c l e a r l y i n d i c a t e s t h a t the tensile stress reaches its m a x i m u m a t a l o c a t i o n close t o the t w o ends o f the t o p o r b o t t o m edge o f the d o m a i n a n d a m i n i m u m i n the middle  o f t h e edge; c h a n g i n g  t h e p a r a m e t e r s i n t h e stress a n a l y s i s does n o t alter t h e basic  features o f t h i s stress d i s t r i b u t i o n . T h i s is n o t s u r p r i s i n g i n v i e w o f the p o i n t l o a d l i k e shear  o.s  i  0  i  i  0.2  i  i  i  i  0.6  0.4  1  r~  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/0.5a  F I G 4.10 L a t e r a l T e n s i l e Stress 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/0.5a FIG  4.11 L a t e r a l T e n s i l e Stress 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  Ratio  73  LATERAL STRAW ( 1 / 1 0 0 0 N/N)  LATERAL STRAN ( 1 / 1 0 0 0  N/N)  F I G 4.12 L a t e r a l S t r a i n s M e a s u r e d a l o n g webs a n d F a c e - s h e l l s o f P l a i n P r i s m s under U n i a x i a l C o m p r e s s i o n  d i s t r i b u t i o n specified b y E q . 4.7. A n d i t is consistent w i t h the e x p e r i m e n t a l observation t h a t the strains p a r a l l e l t o the j o i n t were higher at the ends t h a n i n the m i d d l e o f the j o i n t , as i n d i c a t e d by  gauges i n these locations. F i g . 4.12 gives the t y p i c a l results o f this measurement. N o  appreciable s t r a i n was measured at l o c a t i o n # 1 or at l o c a t i o n # 2 , w h i c h are i n the m i d d l e o f a face-shell a n d does not cover t w o ends o f the j o i n t . T h i s is i n c o n t r a s t to those measured i n locations # 3 a n d # 4 , w h i c h cross the whole length o f the web. In other words, the tensile s t r a i n is h i g h l y concentrated near t w o ends of the j o i n 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 o n H i l s d o r P s M o d e l B a s e d o n above s t u d y , the c o n c l u s i o n is o b v i o u s , t h a t s p l i t t i n g i n m a s o n r y c a n not be s i m p l y a t t r i b u t e d to the l a t e r a l e x p a n s i o n of the m o r t a r . F i r s t , if l a t e r a l e x p a n s i o n o f ' t h e m o r t a r were the m a i n cause of the s p l i t t i n g , it w o u l d occur somewhere near t w o edges of a web or a face-shell 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 o f the face-shell a n d the web i n the f i l l e t s near the corner of a u n i t are t a k e n i n t o a c c o u n t , one m a y c o n c l u d e t h a t s p l i t t i n g w o u l d occur somewhere near the shell j o i n t ,  a c o n c l u s i o n w h i c h is not  supported  by  experimental  web —  observation.  face-  Second,  the  interface shear, w h i c h is responsible for the tensile stresses i n the web a n d the f a c e - s h e l l , is a m o n o t o n i c a l l y increasing f u n c t i o n of the m o r t a r j o i n t l e n g t h a, as c l e a r l y i n d i c a t e d b y E q . 4.8. T h u s the shear forces a l o n g the face-shell are not less t h a n those a l o n g the web. If  mortar  e x p a n s i o n were the m a i n cause for the s p l i t t i n g of a m a s o n r y p r i s m , the s p l i t t i n g w o u l d be m o r e l i k e l y to occur or at least have a n equal p r o b a b i l i t y of o c c u r r i n g , i n the f a c e - s h e l l , a c o n c l u s i o n w h i c h c o n t r a d i c t s the e x p e r i m e n t a l observations. T h i r d , the e x p e r i m e n t a l m o n i t o r i n g of c r a c k i n i t i a t i o n , as we have s h o w n earlier, indicates t h a t the l a t e r a l e x p a n s i o n of the m o r t a r being the cause of the s p l i t t i n g m e c h a n i s m is not acceptable. These p o i n t s alone are sufficient to rule out the correctness of H i l s d o r f s m o d e l , since even the necessary c o n d i t i o n s for f a i l u r e c a n not be j u s t i f i e d b y his m o d e l . Moreover,  r i g o r o u s l y , s p e a k i n g , the u n d e r l y i n g  concepts of H i l s d o r f s m o d e l  may  be  m i s l e a d i n g . A s i n d i c a t e d at the b e g i n n i n g , it is not sufficient to focus o n a l o c a l tensile event i n the case o f c o m p r e s s i o n . E v e n i f the v e r t i c a l s p l i t t i n g were i n i t i a t e d b y j o i n t e x p a n s i o n , for t h i s to lead to d i r e c t c a t a s t r o p h i c failure o f the m a s o n r y needs f u r t h e r j u s t i f i c a t i o n . F r o m a f r a c t u r e m e c h a n i c s p o i n t o f v i e w , the energy required to open t h i s crack w o u l d c o m e f r o m the s t r a i n energy released i n the m o r t a r j o i n t , as a result of p a r t i a l r e l a x a t i o n of the l a t e r a l c o n f i n i n g stress. It c a n be s h o w n t h a t the a m o u n t of t h i s energy is l i m i t e d , so t h a t the c r a c k w o u l d J s t a b i l i z e . E v e n i f we assume t h i s crack c o u l d r u n t h r o u g h a m a s o n r y b l o c k , the l a t t e r w o u l d 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) T h e 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  different, and a one parameter failure criterion will not be satisfactory.  will be  76 4.4 J o i n t Effect o n A x i a l C a p a c i t y It has been d e m o n s t r a t e d t h a t a p l a i n m o r t a r j o i n t is not the g o v e r n i n g f a c t o r for the f a i l u r e p a t t e r n o f concrete m a s o n r y . H i l s d o r f s m o d e l is not a p p r o p r i a t e for assessing the effect of the j o i n t s o n m a s o n r y s t r e n g t h . A v a i l a b l e e x p e r i m e n t a l results on  the j o i n t  effect, i n c l u d i n g the tests done b y  the  a u t h o r , a p p e a r t o be scattered. A possible w a y t o assess t h i s effect w o u l d be t o c o m p a r e tests o n prisms w i t h mortar joints a n d w i t h dry joints  not a very p r a c t i c a l a p p r o a c h .  The masonry  u n i t s t r e n g t h is not a good reference, since under s t a n d a r d t e s t i n g c o n d i t i o n s , it w i l l e x h i b i t a c o n i c a l f a i l u r e m e c h a n i s m as a result of the end f r i c t i o n , w i t h a s u b s t a n t i a l l y higher  apparent  s t r e n g t h . Because o f these d i f f i c u l t i e s , i n m o s t e x p e r i m e n t a l w o r k , the j o i n t effect has been e x a m i n e d b y v a r y i n g the j o i n t c o n d i t i o n s . S o m e e x p e r i m e n t a l observations m a y be w o r t h r e v i e w i n g , a) U s u a l l y the c o m p r e s s i v e s t r e n g t h of u n i t is higher t h a n t h a t of p r i s m , w h i c h is i n t u r n higher t h a n t h a t o f m o r t a r . H o w e v e r , a l t h o u g h the m o r t a r s t r e n g t h is lower t h a n the p r i s m s t r e n g t h ( c a l c u l a t e d o n the m o r t a r e d area), j o i n t f a i l u r e has never been observed. It s h o u l d be n o t e d t h a t , w h e n t a l k i n g a b o u t m o r t a r s t r e n g t h , we i m p l i c i t l y assume the u n c o n f i n e d compressive s t r e n g t h . T h e s t r e n g t h o b t a i n e d b y the s t a n d a r d cube test is a c t u a l l y p a r t l y c o n f i n e d since its height t o w i d t h r a t i o is s m a l l . E x p e r i m e n t s by H a t z i n i k o l a s et a l (1978) h a v e s h o w n t h a t the u n c o n f i n e d s t r e n g t h c a n be as l o w as 6 3 % of the cube s t r e n g t h . T h e m o r t a r i n the j o i n t c o u l d h a v e even lower s t r e n g t h due t o poorer c u r i n g c o n d i t i o n s . T h i s o b s e r v a t i o n is also revealed b y the a u t h o r ' s tests. T h e average u n i t s t r e n g t h f  u  is  3250 p s i . F o r m o s t p r i s m s N type m o r t a r was used, w h i c h has an average cube s t r e n g t h o f 1570 p s i . T h e average f a i l u r e l o a d of m a s o n r y p r i s m s w i t h t h i s m o r t a r is 143 k i p s , c o r r e s p o n d i n g to an f  m  of a b o u t 2320 p s i . (cf. T a b l e 4.1) A l t h o u g h the exact c o r r e l a t i o n between the s t r e n g t h of  the m o r t a r cube a n d t h a t o f the m o r t a r i n the j o i n t is u n k n o w n , a n d t y p e M a n d S m o r t a r s a p p e a r t o h a v e very h i g h cube strengths, it seems reasonable, to accept t h a t f  u  >  fm  >  fj , u  77 where / -  denotes the u n c o n f i n e d m o r t a r s t r e n g t h . F o r a l l the p r i s m s tested, no j o i n t f a i l u r e was  u  observed. T h i s is also evidenced b y the d e f o r m a t i o n measured across the j o i n t (see F i g . 2.5 i n Chapter  II).  b) B o t h m o r t a r t y p e a n d j o i n t thickness have a n influence on the m a s o n r y s t r e n g t h , a l t h o u g h there is s t i l l a n u n c e r t a i n t y a b o u t the degree a n d nature of t h i s influence. T h i s is reflected i n t h a t the tests done b y D r y s d a l e a n d H a m i d (1979) have s h o w n the influence to be r e l a t i v e l y m i n o r , w h i l e i n the a u t h o r ' s tests the influence is s i g n i f i c a n t (see T a b l e 4.1 a n d F i g . 4.16); a n d a l t h o u g h the a v a i l a b l e test d a t a t e n d to i n d i c a t e t h a t stronger m o r t a r m a k e s stronger  masonry,  b o t h e x p e r i m e n t s have i n d i c a t e d t h a t t h i s is not a l w a y s true. c) T h i s influence becomes r e l a t i v e l y m i n o r w i t h increase of l o a d i n g e c c e n t r i c i t y , (see T a b l e 6.1 in Chapter  VI)  d) R e i n f o r c e m e n t b y m e t a l plates enhances b o t h the c a p a c i t y a n d d u c t i l i t y of m a s o n r y ( P r i e s t l e y a n d E l d e r 1982), w h i l e reinforcement b y steel bars i n the j o i n t reduces the s t r e n g t h ( H a t z i n i k o l a s 1978). It c a n be c o n j e c t u r e d t h a t the m o r t a r j o i n t affects the s t r e n g t h of m a s o n r y b a s i c a l l y i n t h a t the j o i n t i n t r o d u c e s d i s c o n t i n u i t i e s i n the m a t e r i a l properties, such as s t r e n g t h a n d stiffness. These d i s c o n t i n u i t i e s w i l l c o m p l i c a t e the stress d i s t r i b u t i o n i n the v i c i n i t y of the j o i n t a n d thus affect the v e r t i c a l l o a d transfer a b i l i t y . It m a y  be reasonable to assume t h a t as l o n g as the j o i n t c o n d i t i o n s d o not  l a t e r a l c o n f i n e m e n t to prevent s p l i t t i n g f a i l u r e , as i n the case of plate reinforcement  provide  (observation  d ) , the j o i n t w i l l generally have a negative effect o n the m a s o n r y s t r e n g t h i n the presence o f uniaxial  compression. T h i s  is because the j o i n t  w i l l generally  alter the otherwise  uniform  c o m p r e s s i v e stress i n its v i c i n i t y , a n d thus the force is effectively transferred b y a s m a l l e r area. T h i s c a n be i l l u s t r a t e d b y f o l l o w i n g a n a l y s i s .  78 4.5 Stress i n J o i n t V i c i n i t y As  shown  by  the stress a n a l y s i s i n section 4.2.2,  when  masonry  is under  uniaxial  c o m p r e s s i o n , the less s t i f f m o r t a r j o i n t is subject to v e r t i c a l compressive force as w e l l as l a t e r a l interface shear. A l t h o u g h the p r o b l e m was solved i n a plane c o i n c i d i n g w i t h webs or face-shells, the p r i n c i p l e c a n be extended to the p e r p e n d i c u l a r plane representing the cross-section of webs or face-shells. T h u s the m o r t a r is a c t u a l l y c o n f i n e d b i l a t e r a l l y ; a n d because of t h a t , the a p p a r e n t s t r e n g t h (the c o n f i n e d strength) is increased. T h i s e x p l a i n s w h y j o i n t f a i l u r e is not observed i n tests a l t h o u g h f  m  >  /. JU  A n i n d i c a t i o n of t h i s c o n f i n e m e n t is f o u n d i n the v e r t i c a l d e f o r m a t i o n curves o f m o r t a r j o i n t s recorded i n the tests, w h i c h reveal t h a t the j o i n t s became stiffer w i t h increase of l o a d (cf. Fig.  2.5).  A s a consequence of this c o n f i n e m e n t ,  the otherwise u n i f o r m  c o m p r e s s i v e stress  d i s t r i b u t i o n i n the v i c i n i t y of the j o i n t is changed. F i g . 4.13 d e p i c t s a cross-sectional view of a m o r t a r j o i n t a n d a free b o d y d i a g r a m of the j o i n t . It is o b v i o u s t h a t l a t e r a l c o n f i n i n g stress results f r o m the interface shear a n d t h a t i t reaches a m a x i m u m  i n the m i d d l e  of the j o i n t .  distribution,  the  approach  we  use  simplified  A s a n e s t i m a t i o n of the c o n f i n i n g stress  as presented  in  section 4.2.2.  The  problem  a p p r o x i m a t e s p l a n e s t r a i n c o n d i t i o n s , since the j o i n t is a l m o s t f u l l y confined i n the d i r e c t i o n a l o n g its l e n g t h . R e c a l l i n g the f o r m of s o l u t i o n for the l a t e r a l d i s p l a c e m e n t u as g i v e n i n E q . 4.6, we m a y w r i t e the c o n f i n i n g stress as (referring to the cross-sectional plane s h o w n i n F i g . 4.13):  4.13  T a k i n g the average of t h i s stress over the j o i n t thickness leads to  79  F I G 4.14  C o m p r e s s i v e Stress, L a t e r a l C o n f i n i n g Stress a n d C o n f i n e d Strength i n M o r t a r J o i n t  80 to to  ^ f l - J - V  =  V  n- ^  cosh[(2n-l)«7rr/< ]  \  0  (2n-l) cosh[(2n-l)K7ra /2V| /  2  2  0  where t a n d a are t h e thickness a n d the w i d t h (the transverse d i m e n s i o n o f t h e b l o c k face-shell 0  0  or w e b ) o f a j o i n t . T o c o r r e s p o n d t o plane s t r a i n c o n d i t i o n s , q a n d K become  4.15  l - i /  ~ > 2(1-1/) l  For  2  4.16  v  z / a « 1/4, t y p i c a l of the geometry 0  o f concrete m a s o n r y  conditions, a n d v =  0.3, this  c o n f i n i n g stress d i s t r i b u t i o n i s p l o t t e d i n F i g . 4.14. It is clear t h a t t h e j o i n t i s n o t u n i f o r m l y confined.  Under  this  non-uniform  confinement,  the joint  will  develop  a varying  confined  s t r e n g t h . Since t h e j o i n t is m o r e c o n f i n e d i n t h e d i r e c t i o n a l o n g i t s l e n g t h , the increase i n m o r t a r compressive s t r e n g t h w i l l m a i n l y depend o n the c o n f i n i n g stress i n the j o i n t w i d t h d i r e c t i o n (the x d i r e c t i o n i n F i g . 4 . 1 3 ) . T h u s , we m a y use t h e k n o w n e m p i r i c a l r e l a t i o n ( P a r k a n d P a u l e y (  1975)  fjc  w i t h cr, =  =  /  j o  + 4.1<T,  4.17  C|(x) here. T h e c o n f i n e d c o m p r e s s i v e strength / -  c  o f t h e j o i n t , based o n the c o n f i n i n g  stress g i v e n b y E q . 4.14, is also p l o t t e d i n F i g . 4.14. T h i s m a y u n d e r e s t i m a t e t h e s t r e n g t h s o m e w h a t i n v i e w o f t h e f u l l c o n f i n e m e n t a l o n g t h e l e n g t h o f the j o i n t . When  the a p p l i e d  compressive  stress  is s m a l l  compared  to / . , u  the  unconfined  c o m p r e s s i v e s t r e n g t h o f m o r t a r , a good a p p r o x i m a t i o n f o r t h e c o m p r e s s i v e stress d i s t r i b u t i o n i n  81 the j o i n t w i l l be g i v e n b y the elastic case :  cr  •  ~  cr —  =  (  -3— dx  =  l-u  2  ( l _ ^ V  T  V  TV ^ 2  cosh[(2n-l)K7rx/to]  \  (2n-l) cosh[(2n-l)K7ra /2io] 2  0  4  /  w h i c h d e p a r t s s l i g h t l y f r o m u n i f o r m d i s t r i b u t i o n , as p l o t t e d i n F i g . 4.14. W h e n cr exceeds / - , the end p a r t of the j o i n t w i l l " y i e l d " , because the c o n f i n e d s t r e n g t h u  of  the end part  i s less  than  cr. T h e stress d i s t r i b u t i o n  i n the j o i n t  is then  dramatically  c o m p l i c a t e d . T h e l a t e r a l c o n f i n i n g stress g i v e n b y E q . 4.14 is no longer v a l i d since the end p a r t s of the j o i n t h a v e developed s u b s t a n t i a l n o n l i n e a r i t y . A precise stress a n a l y s i s f o r the j o i n t i s d i f f i c u l t , b u t we s h a l l give a n a p p r o x i m a t e a p p r o a c h to t h i s p r o b l e m . Because the inner p a r t o f the j o i n t is m o r e c o n f i n e d ,  i t develops higher s t r e n g t h  and  therefore transfers m o r e stress. T h u s we m a y d i v i d e the j o i n t i n t o t w o p a r t s w i t h the d i v i d i n g point x , 0  w i t h i n w h i c h the m a t e r i a l r e m a i n s elastic i n the sense t h a t i t does not f a i l o r develop  s u b s t a n t i a l n o n l i n e a r i t y , as s h o w n i n F i g . 4.13. W e assume, as i s suggested b y F i g . 4.14,  that  the c o m p r e s s i v e stress is a p p r o x i m a t e l y u n i f o r m l y d i s t r i b u t e d w i t h i n this range. Since the end p a r t is subjected to compressive stress as w e l l as h i g h l y c o n c e n t r a t e d shear force a n d l a t e r a l c o n f i n i n g force as i n d i c a t e d i n the preceding stress a n a l y s i s , i t w i l l f a i l under the c o m b i n a t i o n o f these forces. T h e k n o w n e m p i r i c a l f a i l u r e curve o f concrete under shear a n d c o m p r e s s i o n , as s h o w n i n F i g . 4.15 ( P a r k a n d p a u l e y , 1975), m a y serve as a g o o d a p p r o x i m a t i o n o n l y for the v e r y end of the j o i n t , where the l a t e r a l c o n f i n i n g stress is negligible. In the presence o f large l a t e r a l c o n f i n i n g stress, we m a y m o d i f y  the f a i l u r e curve b y  a s s u m i n g t h a t i t i s c h a r a c t e r i z e d b y the c o n f i n e d s t r e n g t h i n s t e a d o f 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 v a l u e w o u l d be s l i g h t l y higher g i v e n b y E q . 4.18.  t h a n the average stress cr  82  a 0.3  - O I - * —  i  £ 0.2  \  O.I  0  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)  a = width of the joint 0  x = half width of the middle part of the joint 0  x = width of the end part of the joint e  to = thickness of the joint fj  •fj  fj  = unconfined strength of the mortar joint  u  t  e  = confined strength of the mortar joint in the centre part = confined strength of the mortar joint in the end part  cr = average compressive stress in entire joint a, = lateral confining stress in joint aj  = average compressive failing stress in the end part  Tj  = average shear failing stress in the end part  e  t  Thus the failure criterion for the end part is  .22/,-  for  cr /f je  je  <  0.6.  /  <r  I  fj  e  A n d for  i  e  /  J  j  . - .D g  u  0.4  cr /f je  > 0.6  je  y  )  /  r  y  je  0-22/ V 0.22/,,  i£  +  )  _  4.196  ~  can be written  /;,=  t t L t ^ .  4.20  We assume a loading path by noting the relation given by E q . 4.9 and 5 ~ x Tj , e  e  from which it  follows that, at failure  Vertical equilibrium requires  <xa = 2 0  x  ofj {x?) + c  X (T e  je  4.22  84  where fj is defined by c  fjc  =  fju  + 4.1 cr,  4.23  and <r can be found by lateral equilibrium of the end part t  4.24  a, t = 2r x 0  je  e  Eq. 4.19 to Eq. 4.24 together with the relation that x + x = a /2 can be used to find 0  e  0  the 7 unknowns; namely, x , x , /, , a,, f , a and r . e  0  c  je  je  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 f . This may ju  underestimate the distortion of the uniform stress, since the actual value for / - is even lower. u  At failure of the N type masonry, a= f = 0.84, fj /<r =1.15, c  Cje/v  = 0.27.  m  = 2320  psi, then fj /<r = 0.677, leading to x /(a /2) u  0  o  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 H o w e v e r , if. one wishes t o generalize the j o i n t effect o n m a s o n r y s t r e n g t h by c o r r e l a t i n g the m o r t a r cube s t r e n g t h a n d the u n i t strength w i t h the p r i s m s t r e n g t h ,  as is often desired i n  p r a c t i c e , some u n c e r t a i n t i e s h a v e t o be recognized: 1) T h e c o r r e l a t i o n between m o r t a r cube s t r e n g t h a n d the s t r e n g t h of the m o r t a r p l a c e d i n the joint. 2) T h e c o r r e l a t i o n between u n i t s t r e n g t h a n d the s t r e n g t h o f " a p r i s m w i t h d r y j o i n t s " , a desired reference p a r a m e t e r . 3)  U n c e r t a i n t i e s i n f a i l u r e c r i t e r i a o f concrete.  4)  M a t e r i a l properties such as those g o v e r n i n g  deformation  and  adhesion, w h i c h are  often  a s s u m e d b u t not m e a s u r e d . T h e y are b y no means u n i m p o r t a n t to m a s o n r y s t r e n g t h . B a s e d o n above a n a l y s i s a n d a r g u m e n t s , we present here a s e m i - e m p i r i c a l a p p r o a c h . Since no better f a i l u r e c r i t e r i o n is a v a i l a b l e , we assume t h a t the m a s o n r y u n i t w i l l f a i l w h e n the average c o m p r e s s i v e stress i n the m i d d l e p a r t of the m o r t a r j o i n t , w h i c h w i l l be higher t h a n the average stress i n the m a s o n r y , reaches some c r i t i c a l v a l u e ; a n d this c r i t i c a l v a l u e m a y be l i n e a r l y correlated to the m a s o n r y u n i t s t r e n g t h . T h u s we m a y w r i t e the f a i l u r e c o n d i t i o n  f (x ) jc  =  0  0 <  kJ*  *  x  < 1  4.25  F u r t h e r , we assume t h a t the u n c o n f i n e d s t r e n g t h of the m o r t a r placed i n j o i n t s is p r o p o r t i o n a l 1  t o the cube m o r t a r s t r e n g t h  f  ju  =  0 <  hfj  where f- denotes the cube m o r t a r s t r e n g t h , a n d k  l  and  k  2  k  2  < 1  4.26  are some assumed c o r r e l a t i o n factors.  A t f a i l u r e , e q u a t i o n s f r o m 4.19 to 4.24 are a s s u m e d t o be satisfied. T h u s the m a s o n r y s t r e n g t h , i.e. the average f a i l u r e stress i n m a s o n r y , is a c t u a l l y expressed i n E q . 4.22, f r o m w h i c h  I  86 we c a n w r i t e  where  <7  j e  and  r  j e  c a n be e x p l i c i t l y expressed i n t e r m s of fj a n d f  4.21, 4.23 a n d 4.24. F o r k  x  =  0.95, k  2  = 0.75, i/a  0  -  u  i n v i e w of E q s . 4.19  —  0.25 a n d P o i s s o n ' s r a t i o v = 0.3, E q . 4.27  is p l o t t e d i n F i g . 4.16 a n d c o m p a r e d w i t h the e x p e r i m e n t a l d a t a . T h e v a l u e of the c o r r e l a t i o n factor k  2  is very close to the c o n v e r s i o n f a c t o r between concrete c y l i n d e r s t r e n g t h a n d  cube  s t r e n g t h r e c o m m e n d e d b y L ' H e r m i t e ( N e v i l l e 1965). F o r the cube strength r a n g i n g f r o m 2000 psi to 3000 p s i , t h i s f a c t o r is between 0.73 a n d 0.76. T h e m o d e l curve is essentially i d e n t i c a l to a l i n e a r regression curve of the d a t a , i.e.  i  f = 0.68 + 0.19  /»  4.28  fu  T h e f o u r p o i n t s o n the r i g h t have been e x c l u d e d f r o m this a n a l y s i s ; they were type M  mortar,  a n d are believed to represent a different p h e n o m e n o n — failure of the adhesion between b l o c k mortar. T h e m o d e l also gives a reasonable c o r r e l a t i o n w i t h the l i m i t e d d a t a o n m a s o n r y c a p a c i t y w h e n the m o r t a r j o i n t is d o u b l e d ( t = 3 / 4 i n c h ) , as s h o w n i n F i g . 4.17. T h e m o d e l m a y be used to e s t i m a t e the m a s o n r y s t r e n g t h . H o w e v e r , as i n d i c a t e d before, some u n c e r t a i n t i e s heed further i n v e s t i g a t i o n . O n e of t h e m , is of course, the c o r r e l a t i o n between the cube m o r t a r  s t r e n g t h a n d the strength of m o r t a r  placed i n the j o i n t , since the  curing  c o n d i t i o n s are so different. T h e other is the effect of the j o i n t adhesion. We  may  conclude f r o m the above a n a l y s i s t h a t since the l o a d transfer c a p a b i l i t y of  m o r t a r j o i n t s depends largely o n the existence of the l a t e r a l c o n f i n i n g stress, i n t r o d u c e d b a s i c a l l y b y the interface shear between the j o i n t a n d block u n i t , t h a t the adhesion between j o i n t a n d u n i t  1.2  1.1 H 1 0.9 0.8 0.7 -\ 0.6 0.5 in  2  0.4 -  AUTHOR  0.3 0.2 0.1  MODEL  DRYSDALE&HAME  Lt£AR REGRESSION  HATZNKOLAS  -  0  1~  0.2  i  0.4  0.6  r  — I —  0.8  1  1.2  1.4  MORTAR CUBE STRENGTH / UNIT STRENGTH  F I G . 4.16 P r i s m S t r e n g t h versus M o r t a r C u b e S t r e n g t h  i  0.9 (to-3/4 in) .0.8  -  0.7  -  0.8  -  0.5 0.4 MODEL  0.3  -  0.2  -  0.1  -  0 0.2  "i  0.4  r  •  AUTHOR  •  DRYSOALE&HAMD  — I —  0.6  0.8  1  1.2  1.4  MORTAR CUBE STRENGTH / IMT STRENGTH  F I G . 4.17 P r i s m S t r e n g t h versus M o r t a r C u b e S t r e n g t h , w i t h J o i n t T h i c k n e s s D o u b l e d  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 CHAPTER  V  PLAIN MASONRY WITH FACE-SHELL  BEDDING  5.1 I n t r o d u c t i o n In  N o r t h A m e r i c a concrete m a s o n r y  is often m o r t a r e d o n l y o n the face-shells. E v e n  w h e n a m a s o n a t t e m p t s to a p p l y m o r t a r to the cross-webs as w e l l , he m a y not be able to ensure v e r t i c a l a l i g n m e n t so t h a t the webs c a n t r a n s m i t force effectively. T h e m e c h a n i c a l properties of face-shell bedded m a s o n r y , therefore, have been s t u d i e d e x t e n s i v e l y . T h e s p l i t t i n g f a i l u r e o f face-shell bedded m a s o n r y is r e l a t i v e l y w e l l u n d e r s t o o d .  Shrive  (1982) c o n c l u d e d t h a t tensile stress is developed at the centre of the webs, b y a m e c h a n i s m s o m e w h a t analogous to deep b e a m b e n d i n g , i.e. the t o p a n d b o t t o m halves of the web are t a k e n as deep beams, b e n d i n g  u p a n d d o w n respectively (see F i g . 5.1),  thus c a u s i n g the s p l i t t i n g  f a i l u r e i n face-shell bedded m a s o n r y . T h e a u t h o r is i n f u l l agreement w i t h the reasoning i n S h r i v e ' s paper. T h e present s t u d y of f a c e - s h e l l bedded m a s o n r y was i n t e n d e d to c o n f i r m his m o d e l e x p e r i m e n t a l l y , to s t u d y  the  t r a n s i t i o n to a f a i l u r e m e c h a n i s m for e c c e n t r i c a l l y l o a d e d specimens, to explore the r e l a t i o n s h i p to f u l l y bedded m a s o n r y , a n d t o develop some r e l a t e d q u a n t i t a t i v e results.  5.2 E x p e r i m e n t a l W o r k S i x t e e n face-shell bedded  p r i s m s were b u i l t a n d  twelve of t h e m  were tested  under  u n i a x i a l c o m p r e s s i o n i n c l u d i n g t w o w i t h f u l l c a p p i n g . T a b l e 5.1 s u m m a r i z e s the f a i l u r e loads of these specimens. U n d e r u n i a x i a l c o m p r e s s i o n , s p l i t t i n g of the webs was a g a i n revealed b y the tests. B y observing  the  wire  breaking  order  as p r e v i o u s l y  described,  cracks  were  found  to  initiate  c o n s i s t e n t l y f r o m the t o p or b o t t o m of the webs i n m i d d l e course (see F i g . 5.2). T h i s s u p p o r t s Shrive's  model.  Both  the  failure  process recorded  on  video  and  the  lateral  deformation  90  tension compression  tlttlHMIHI IHHHHIIII  deep beam mechanism  compression tension  F I G . 5.1  D e p i c t i o n of D e e p B e a m M e c h a n i s m  11  HI  II  III  - 1  - 1  III  ll  III - 1IVV  M27-2  N15-1  II -IV 1 V  -  -  S16-1  - IV " III  -  -1  -  - II  -  IV  S16-2  - II -1  - Ill - IV  N15-3  -  IV  N15-4  F I G . 5.2 Detected Orders of 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 Sections along P r i s m s . ( F a c e - S h e l l B e d d e d P r i s m s )  91 SPECIMEN M27(M-  MORTAR)  1  2  4  3  AVG  cov  118.0  99.0  86.0  75.0  94.5  19.9%  140.0  109.0  123.8  9.2%  107.5  7.0%  50.5  5.0%  S16  (S-MORTAR)  119.0  127.0  N15  (N-MORTAR)  100.0  115.0  N15  (N-MORTAR)  53.0*  48.0*  T a b l e 5.1 F a i l u r e L o a d s of P l a i n P r i s m s w i t h F a c e - S h e l l B e d d i n g (kips)  * Tested with full capping  m e a s u r e m e n t s i n d i c a t e d t h a t s p l i t t i n g occurs at or i m m e d i a t e l y before f i n a l f a i l u r e . 5.4  give the d e f o r m a t i o n  curves. A  deep b e a m  mechanism  difference i n the d e f o r m a t i o n measurement at l o c a t i o n s # 3  is suggested b y  F i g s . 5.3,  the s u b s t a n t i a l  # 4 ; s p l i t t i n g is c l e a r l y evidenced by  the j u m p s i n these curves. T h e final f a i l u r e is c h a r a c t e r i z e d b y peeling off ( f u l l y or p a r t l y ) of the face-shells, as s h o w n i n F i g . 2.13 i n C h a p t e r II. H o w e v e r , for m o s t o f the specimens w i t h face-shell c a p p i n g cracks were f o u n d to i n i t i a t e i n the web somewhere near t w o ends of the m o r t a r j o i n t s , a n d tended to w a n d e r a f t e r w a r d s , as t y p i c a l l y i l l u s t r a t e d i n F i g . 2.13. T h i s appears s o m e w h a t different f r o m w h a t one w o u l d expect b y the deep b e a m b e n d i n g m o d e l , w h i c h suggests t h a t s p l i t t i n g w o u l d occur i n the centre of the web. S p l i t t i n g i n the centre of webs was f o u n d i n the specimens tested w i t h f u l l c a p p i n g (see F i g . 2.14), u s u a l l y o c c u r r i n g i n the top course. These specimens f a i l e d at very l o w loads ( a b o u t 5 0 % o f t h a t o f the face-shell c a p p i n g , see T a b l e 5.1), i m m e d i a t e l y after web s p l i t t i n g ; the t w o halves of b l o c k s c o l l a p s e d b y h i n g i n g about the inside toes of the m o r t a r j o i n t s . T h e  hinging  m e c h a n i s m is i m p l i e d b y the v e r t i c a l d i s p l a c e m e n t measured across the outside of the j o i n t of s p e c i m e n N 1 5 - 3 , w h i c h c o n t r a c t e d first because of c o m p r e s s i o n then tended to open due to the joint rotation.  FIG. 5.3 Measured Deformations at Certain Locations of Face-Shell Bedded Prisms  F I G . 5.4  M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s of F a c e - S h e l l B e d d e d P r i s m s  94 5.3 Stress A n a l y s i s Shrive did a 3-dimensional u s i n g the f i n i t e element  stress a n a l y s i s b y m o d e l i n g a 2 - h i g h face-shell bedded p r i s m  method.  However,  the a n a l y s i s was o n l y for the case of u n i a x i a l  c o m p r e s s i o n a n d the results g i v e n i n his paper are l i m i t e d to c e r t a i n l o c a t i o n s . T h e r e f o r e , some a d d i t i o n a l n u m e r i c a l stress a n a l y s i s is p e r f o r m e d here. We  m o d e l a web as a p l a n e elastic p r o b l e m for s i m p l i c i t y . T h e a u t h o r believes the  2—dimensional  m o d e l has some v a l u e , a l t h o u g h  this is a c t u a l l y a 3 — d i m e n s i o n a l  problem  r e q u i r i n g the e x a c t geometry of the p r i s m . T h e stress f i e l d was s o l v e d b y u s i n g the b o u n d a r y element m e t h o d ( C r o u c h 1983). T h i r t y four elements per edge l e n g t h were used, a n d the results g i v e n o n the b o u n d a r i e s i n the f o l l o w i n g figures are the stresses e v a l u a t e d at the centre o f each element. T h e stress d i s t r i b u t i o n s d e t e r m i n e d for c e r t a i n l o c a t i o n s i n the face-shell l o a d e d web are g i v e n i n F i g . 5.5.  It is i n t e r e s t i n g to note t h a t l a t e r a l tensile stress i n the t o p of the web  r e m a i n s a p p r o x i m a t e l y c o n s t a n t w i t h i n the m i d d l e range a n d reaches its m a x i m u m at a b o u t the q u a r t e r p o i n t s i n s t e a d of i n the centre. T h e h i g h l a t e r a l tension at the q u a r t e r p o i n t s c a n be a t t r i b u t e d t o the l o c a l stress c o n c e n t r a t i o n a r i s i n g f r o m the compressive forces i n the f a c e - s h e l l , w h i l e the centre p a r t is stressed i n tension because of the b e a m b e n d i n g m e c h a n i s m (cf. F i g . 5.1).  This  is  implied  by  the  tensile stress d i s t r i b u t i o n s  along  the  depth  at  these  two  c o r r e s p o n d i n g l o c a t i o n s ; the f o r m e r has a m u c h sharper stress g r a d i e n t , as s h o w n i n F i g . 5.5. T h e e l a s t i c a n a l y s i s gives the a s t o n i s h i n g l y h i g h v a l u e of the m a x i m u m tensile stress, read as 4 9 % o f the v e r t i c a l c o m p r e s s i v e stress a c t i n g o n the f a c e - s h e l l . T h i s result is c o m p a r a b l e t o t h a t given by  S h r i v e (1982). H o w e v e r ,  we m a y  argue t h a t , since n o n l i n e a r d e v e l o p m e n t s i n the  concrete a l l o w some degree of stress r e d i s t r i b u t i o n , the tensile stress m a y be expected not t o reach s u c h a h i g h v a l u e at the m o m e n t of f a i l u r e . T h e stress a n a l y s i s suggests t h a t the t w o sides of the web are not o n l y c a r r y i n g higher l o c a l tensions t h a n the centre p a r t , but are also under a c o m p l e x stress state, i.e, under tension,  95 0.6  > to "5  .5  -0.6  x/O.Sa  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  F I G . 5.6  F I G . 5.7  Forces Acting on a Block with Full Capping and Face-Shell Bedding  Lateral Stress Distribution in a Web: Full 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  f  m  =  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 H o w e v e r , e x p e r i m e n t s c o n d u c t e d b y b o t h S h r i v e (1982) a n d b y the a u t h o r b y  varying  m o r t a r s t r e n g t h have i n d i c a t e d t h a t the effect of m o r t a r t y p e m a y not be t o t a l l y neglected (cf. T a b l e 4 . 1 ; the v a r i a t i o n i n the m a s o n r y c a p a c i t y w i t h m o r t a r s t r e n g t h is also reflected i n the scatter of the a u t h o r ' s d a t a i n F i g . 5.8, w h i c h includes p r i s m s w i t h three different types of m o r t a r ) . A g a i n , it is n o t e d t h a t the stronger m o r t a r does not necessarily m a k e stronger m a s o n r y . I n the tests c o n d u c t e d b y the a u t h o r , the strongest m o r t a r m a d e the weakest m a s o n r y p r i s m s . W e m a y argue t h a t a l t h o u g h the deep b e a m m e c h a n i s m d o m i n a t e s the f a i l u r e , p a r t i a l failure o f the m o r t a r j o i n t m a y s t i l l occur at the failure based o n m o r t a r e d  stress. T h i s is because the failure stress  area is s t i l l h i g h c o m p a r e d w i t h t h a t of the f u l l y bedded m a s o n r y .  The  o u t s i d e edges of the m o r t a r tend to f a i l a n d s p a l l out, l e a v i n g a n a r r o w s t r i p of m o r t a r d o w n the centre o f the f a c e - s h e l l . T h i s p a r t i a l j o i n t f a i l u r e w i l l not o n l y cause a l o c a l  stress c o n c e n t r a t i o n  i n the v i c i n i t y o f the j o i n t , as s t u d i e d i n d e t a i l i n the preceding chapter, b u t m a y also change the j o i n t essentially f r o m a f l a t - b a s e to a h i n g e - l i k e s u p p o r t ,  which provides little  rotation  c o n s t r a i n t . T h e deep b e a m b e n d i n g m e c h a n i s m m a y be intensified by t h i s s u p p o r t change. T h e above a r g u m e n t suggests t h a t the adhesion of m o r t a r j o i n t to block u n i t is i m p o r t a n t to faceshell m a s o n r y c a p a c i t y as w e l l . E q . 5.2 gives a n e s t i m a t e of m a s o n r y  c a p a c i t y based o n the u n i t s t r e n g t h .  Further  i n v e s t i g a t i o n is needed to i n c l u d e the j o i n t effect q u a n t i t a t i v e l y .  5.5 S u m m a r y The  behaviour  of p l a i n concrete m a s o n r y  w i t h face-shell b e d d i n g  under  concentric  c o m p r e s s i o n has been s t u d i e d . T h e deep b e a m b e n d i n g m o d e l for s p l i t t i n g i n webs proposed b y S h r i v e has been verified b y e x p e r i m e n t s . T h e effect of c a p p i n g c o n d i t i o n s on c a p a c i t y a n d f a i l u r e m o d e has been i n v e s t i g a t e d . 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 CHAPTER  VI  PLAIN MASONRY UNDER ECCENTRIC  COMPRESSION  6.1 F a i l u r e M o d e T r a n s i t i o n W h e n p l a i n m a s o n r y (whether f u l l y m o r t a r e d or face-shell m o r t a r e d ) is under eccentric c o m p r e s s i o n , it f a i l s i n a rather different m o d e a n d at a higher a p p a r e n t stress t h a n i t does under u n i a x i a l c o m p r e s s i o n . 5 groups of p l a i n p r i s m s were tested under eccentric c o m p r e s s i o n . M o s t o f the specimens e x h i b i t e d shear t y p e f a i l u r e , i.e. failure is r o u g h l y c h a r a c t e r i z e d b y a n i n c l i n e d f r a c t u r e p l a n e (or m o r e precisely, a f r a c t u r e zone i n w h i c h m a t e r i a l is h i g h l y c r a c k e d or crushed) s e p a r a t i n g the m a t e r i a l . Because of t h i s m o d e , the f a i l u r e appeared to be r e l a t i v e l y s u d d e n . A l l specimens f a i l e d o n the l o a d e d c o m p r e s s i o n side, a n d f a i l u r e was often l o c a l i z e d i n some p a r t o f the p r i s m . F i g . 2.15 i n C h a p t e r II i l l u s t r a t e s the t y p i c a l f a i l u r e p a t t e r n . T a b l e 6.1 s u m m a r i z e s the f a i l u r e loads. F i g s . 6.1, 6.2 give the m e a s u r e d d e f o r m a t i o n curves. T h e a p p a r e n t increase i n s t r e n g t h p h e n o m e n o n  is d e p i c t e d b y c o m p a r i n g a t h e o r e t i c a l P — M i n t e r a c t i o n curve  I  SPECIMEN1  e/t  1  2  3  4  AVG  COV  N18 ( N - M O R T A R )  1/6  150.5  107.0  120.0  121.0  124.6  12.8%  M20 ( M - M O R T A R ) 1/3  77.5  79.0  86.5  95.0  84.5  8.2%  S21  1/3  96.0  90.0  100.0  93.0  94.8  3.9%  1/3  83.0  81.0  85.0  69.0  79.5  7.8%  ( N - M O R T A R ) * 1/3  64.0  78.0  69.0  73.0  71.0  7.3%  (S-MORTAR)  N19 ( N - M O R T A R ) N22  T a b l e 6.1 F a i l u r e L o a d s of P l a i n P r i s m s under E c c e n t r i c L o a d (kips)  * Face-shell Bedded  160  130  150  120  140  110  130  100  120 110  90  100 -  80  90 eo 70 80 50 40 -  (\ 1 t  -4-  (\ I 1  2  70  o <  60  s  50 40 30  30 20  20 Loaded Side  Unloaded Side  10  10 -  0  0  Loaded Side  -4 AVERAGE STRAIN ( 1/1000 IN/IN )  (a)  -2  -2  AVERAGE STRAIN ( 1/1000 IN/IN )  (b)  2  AVERAGE STRAIN ( 1/1000 IN/IN )  (c)  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 ) (a)  AVERAGE STRAIN ( 1/1000 IN/IN ) (C)  AVERAGE STRAIN ( 1/1000 IN/IN ) (b)  AVERAGE STRAIN ( 1/1000 IN/IN ) (d)  FIG. 6.2 M e a s u r e d D e f o r m a t i o n s a t C e r t a i n L o c a t i o n s o f P l a i n P r i s m s u n d e r E c c e n t r i c L o a d : a) S21-4, e = t / 3 ; b ) S 2 1 - 3 , e = t / 3 ; c) N 2 2 - 2 , e = t / 3 ; d ) N 2 2 - 4 , e = t / 3  o  103 based on the uniaxial compressive strength with the eccentric compression test data, as shown by F i g . 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,  importance of sound adhesion in the joints.  it again implies the  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 Stress a l o n g T o p of a W e b  with Face-Shell Bedding under Eccentric  Load  105 tensile stress due t o t h i s m e c h a n i s m is s u b s t a n t i a l l y reduced w i t h i n c r e a s i n g e c c e n t r i c i t y , as s h o w n i n F i g . 6.4. H o w e v e r , t o f u l l y e x p l a i n the preference o f the shear f a i l u r e m o d e w h e n m a s o n r y is under eccentric l o a d i n g needs a t h o r o u g h u n d e r s t a n d i n g of the f a i l u r e o f a c o n c r e t e - l i k e b r i t t l e m a t e r i a l under v a r i o u s c o n d i t i o n s . In C h a p t e r III we h a v e proposed a f a i l u r e m o d e l e x p l a i n i n g the s p l i t t i n g f a i l u r e under u n i a x i a l c o m p r e s s i o n . H o w e v e r , it appears no easy e x t e n s i o n c a n be m a d e w h e n the m o d e l m e c h a n i s m is under a compressive stress w i t h g r a d i e n t . the  uneven  compression  due  to  the  stress gradient  intensifies the  It c o u l d be t h a t  friction and  interlock  m e c h a n i s m between c r a c k surfaces a n d thus prevents the s p l i t t i n g m o d e f r o m o c c u r r i n g .  6.2 Effect o f J o i n t C o n d i t i o n s A s s h o w n i n T a b l e 6.1, under large e c c e n t r i c i t y , change of m o r t a r s t r e n g t h a p p a r e n t l y has a r e l a t i v e l y m i n o r effect o n the c a p a c i t y of the p r i s m . T h i s m a y be e x p l a i n e d as f o l l o w s . W h e n p l a i n concrete m a s o n r y is under h i g h l y eccentric c o m p r e s s i o n , the compressive force is m a i n l y transferred b y the face-shell o n the l o a d e d side. Since there is a s t r a i n g r a d i e n t across the f a c e - s h e l l , the stress d i s t r i b u t i o n across i t , at a p o i n t r e m o t e f r o m the j o i n t , m u s t be h u m p s h a p e d because of the n o n l i n e a r i t y of the m a t e r i a l . T h i s is q u i t e different f r o m t h a t under u n i a x i a l c o m p r e s s i o n , where the stress w o u l d a l w a y s be u n i f o r m l y d i s t r i b u t e d i n the absence of the j o i n t , regardless o f the d e v e l o p m e n t of m a t e r i a l n o n l i n e a r i t y . T h e h u m p shaped stress d i s t r i b u t i o n suggests t h a t the force w o u l d be largely transferred b y the m i d d l e p a r t o f the face-shell. W e k n o w b y the a n a l y s i s of the preceding sections t h a t the m o r t a r j o i n t c a n develop r e l a t i v e l y h i g h s t r e n g t h i n its m i d d l e p a r t (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 i s t r i b u t i o n as m u c h as the j o i n t under  u n i a x i a l c o m p r e s s i o n w i l l do. M o r e o v e r , because o f the eccentric l o a d i n g , the outer fiber of the l o a d e d face-shell w i l l d e f o r m m o r e t h a n the i n n e r fibre w i l l do. F o r l o a d i n g w i t h e c c e n t r i c i t y e q u a l t o one t h i r d of the  106 width  of  the  prism,  depicted i n F i g . 6.5.  e  =  i  0.64e ,  as  o  Here we neglect the  tensile strength of concrete a n d assume e,-  =  0 i n the m i d d l e of the cross-section after the tensile  part  of  the  prism  has  debonded.  W h e n t h i s s t r a i n is i m p o s e d o n the j o i n t , the joint  is a c t u a l l y  under  a  combination  of  u n i a x i a l compression a n d bending. A stress analysis shows t h a t the b e n d i n g stresses w i l l lend  additional  l a t e r a l confinement  to  m o r e compressed side of the j o i n t a n d  the thus  enhance the j o i n t s t r e n g t h i n t h a t p a r t , i.e., under F I G . 6.5 S t r a i n D i s t r i b u t i o n i n a Section of M a s o n r y under E c c e n t r i c L o a d  eccentric  compressed  part  compression o f the j o i n t  the will  more develop  m o r e strength. T h i s ensures t h a t the j o i n t does not f a i l d u r i n g the l o a d i n g to the final stress d i s t r i b u t i o n discussed above. T h u s , i n practice we m a y neglect the j o i n t c o n d i t i o n s in designing walls under eccentric l o a d i n g . T h i s a p p r o a c h is further studied i n the next chapter.  6.3 S u m m a r y In t h i s chapter, the behaviour of plain m a s o n r y under eccentric compression has been investigated.  The  eccentric behaviour  differs  from  the  concentric one  not  only  in  failure  m e c h a n i s m b u t also i n the j o i n t effect on the strength. In the f o l l o w i n g chapter, we w i l l propose a design a p p r o a c h based on these findings, a n d conclude the s t u d y on p l a i n concrete masonry .  107 CHAPTER RECOMMENDED  VII  DESIGN A P P R O A C H FOR PLAIN  MASONRY  7.1 R e c o m m e n d a t i o n s o n the B a s i s for D e s i g n It has been d e m o n s t r a t e d t h a t under different l o a d c o n d i t i o n s p l a i n m a s o n r y w i l l f a i l b y different modes. U n d e r u n i a x i a l c o m p r e s s i o n , m a s o n r y w i l l f a i l by v e r t i c a l s p l i t t i n g , but not due to the m e c h a n i s m proposed b y H i l s d o r f . F o r f a c e — s h e l l bedded m a s o n r y , s p l i t t i n g c a n be a t t r i b u t e d to a m e c h a n i s m s i m i l a r to deep b e a m b e n d i n g . U n d e r eccentric l o a d i n g , m a s o n r y tends to f a i l i n a m o d e a p p r o x i m a t i n g shear f a i l u r e . T h e s e t w o different f a i l u r e modes w i l l y i e l d different a p p a r e n t strengths. T h e j o i n t c o n d i t i o n s w i l l affect the c a p a c i t y of the m a s o n r y to a different extent under each of these t w o basic l o a d patterns. In  p r a c t i c e , one  compressive s t r e n g t h a n d  wishes to e s t i m a t e the c a p a c i t y of m a s o n r y the m o r t a r  from  the  block  unit  c u b i c s t r e n g t h , since the l a t t e r are r e l a t i v e l y easy  to  measure e x p e r i m e n t a l l y . T h e c o r r e l a t i o n g i v e n b y E q s . 4.27 or 4.28 a n d E q . 5.2 m a y serve t h i s purpose. H o w e v e r , w h e n u s i n g these relations, one m u s t keep i n m i n d t h a t some u n c e r t a i n t i e s are i n v o l v e d as was i n d i c a t e d i n the d e v e l o p m e n t of the equations. In p a r t i c u l a r , we have u n c e r t a i n t i e s i n the f a i l u r e c r i t e r i a of the m a t e r i a l itself, i n the m a t e r i a l properties other t h a n s t r e n g t h , i n the c o r r e l a t i o n between strengths, a n d last b u t  not  least, i n the w o r k m a n s h i p . These u n c e r t a i n t i e s are reflected i n the scattered d a t a of n u m e r o u s experiments. Therefore, i t is r e c o m m e n d e d t h a t i n p r a c t i c e either we use the relations s u c h as g i v e n b y E q s . 4.27 or 4.28 a n d E q . 5.2 i n a conservative m a n n e r or we r e t a i n the m a s o n r y p r i s m test to estimate f , m  the design base of p l a i n concrete masonry, under u n i a x i a l c o m p r e s s i o n .  H o w e v e r , for the case of e c c e n t r i c a l l y l o a d e d m a s o n r y , a n a p p r o a c h w h i c h differs f r o m  108 the t r a d i t i o n a l one w i l l be r e c o m m e n d e d . In t h e t r a d i t i o n a l a p p r o a c h , the eccentric c a p a c i t y e s t i m a t i o n is also based o n the v a l u e fm associated w i t h c o n c e n t r i c l o a d i n g . T h e a p p a r e n t increase i n s t r e n g t h is t a k e n i n t o a c c o u n t b y a ( s t r a i n g r a d i e n t ) f a c t o r . T h i s factor as a f u n c t i o n o f e c c e n t r i c i t y has been f r e q u e n t l y s t u d i e d t h r o u g h e x p e r i m e n t s (for e x a m p l e , T u r k s t r a a n d T h o m a s 1978; D r y s d a l e a n d H a m i d 1983). I n the c u r r e n t  design code (CAN3-S304-M84  1984)  the factor is g i v e n as a  fixed  value  (1.3,  reflected i n the e c c e n t r i c i t y coefficient Ce). The  usefulness o f this a p p r o a c h  depends  o n a n a s s u m e d close a n d  fixed  correlation  between the c o n c e n t r i c c a p a c i t y a n d the eccentric c a p a c i t y . I n the l i g h t o f preceding studies, we k n o w t h a t t h i s c o r r e l a t i o n is questionable since different failure m e c h a n i s m s are i n v o l v e d . I n view o f the f a i l u r e m e c h a n i s m s , the eccentric c a p a c i t y o f concrete m a s o n r y m a y be better correlated w i t h the u n i t compressive strength f  u  instead of f . m  A s s h o w n i n F i g s . 2.2 a n d  2.15 i n C h a p t e r II, the f a i l u r e p a t t e r n o f the u n i t block is very s i m i l a r to t h a t o f p r i s m s under eccentric c o m p r e s s i o n . T h u s i t is r e c o m m e n d e d here t h a t t h e eccentric c a p a c i t y e s t i m a t i o n be d i r e c t l y based o n the u n i t compressive s t r e n g t h f , u  w h i l e the c o n c e n t r i c c a p a c i t y is based o n the p r i s m s t r e n g t h  fm. T h e j o i n t effect is neglected since a p p a r e n t l y i t is r e l a t i v e l y m i n o r for the case o f eccentric l o a d i n g . A l t h o u g h the a p p a r e n t compressive strength o f m a s o n r y m a y v a r y w i t h the e c c e n t r i c t y , the v a r i a t i o n is ignored for p r a c t i c a l reasons. It is believed t h a t t h i s a p p r o a c h w i l l y i e l d better c o r r e l a t i o n s since i t is based o n r e c o g n i t i o n o f the failure m e c h a n i s m s . F u r t h e r , o f course, t h i s r e c o m m e n d e d a p p r o a c h considers the fact t h a t i t is n o t p r a c t i c a l to test p r i s m s under eccentric l o a d i n g t o assess the c a p a c i t y . T h e t r a n s i t i o n a l p o i n t where the failure m o d e changes f r o m s p l i t t i n g to shear f a i l u r e needs t o be i n d e n t i f i e d . It is suggested b y the a v a i l a b l e e x p e r i m e n t a l w o r k t h a t t h i s occurres a t a s m a l l e c c e n t r i c i t y (e <  t/6).  T h i s i m p l i e s t h a t the cross-section c a p a c i t y curve is d i s c o n t i n u o u s  somewhere between e = t/6  a n d e = 0. T h e d e t a i l e d b e h a v i o u r o f the cross-section c a p a c i t y i n  109 t h i s range needs f u r t h e r i n v e s t i g a t i o n . A t t h i s p o i n t we r e c o m m e n d t h a t t h i s p a r t of the curve be i n t e r p o l a t e d between the c a p a c i t i e s at zero e c c e n t r i c i t y a n d at </6 e c c e n t r i c i t y , but not to exceed the v e r t i c a l l o a d c a p a c i t y o f the zero e c c e n t r i c i t y case. T h i s is on the conservative side, as w i l l be s h o w n l a t e r , since the c a p a c i t y at z / 1 2 is also w e l l correlated w i t h the u n i t s t r e n g t h . T o e x a m i n e t h i s p r a c t i c a l a l t e r n a t i v e of b a s i n g the e c c e n t r i c a l l y l o a d e d c a p a c i t y o n the unit  strength,  we c o m p a r e  a v a i l a b l e test d a t a w i t h the r e c o m m e n d e d  c a p a c i t y curve.  The  c a p a c i t y curves are generated b y a c o n v e n t i o n a l m e t h o d , i.e. linear elastic b e h a v i o u r a n d plane cross-section are a s s u m e d a n d the extreme fibre stress is set equal to the u n i t s t r e n g t h .  The  general expressions based o n t h i s m e t h o d (for b o t h grouted a n d u n g r o u t e d m a s o n r y ) are d e r i v e d i n C h a p t e r X . These expressions were checked against a c o m p u t e r p r o g r a m developed b y N a t h a n (1985), w h i c h performs a r a t i o n a l a n a l y s i s for b e a m c o l u m n s based o n m a t e r i a l properties a n d cross-section geometry. Since, under bedding  large e c c e n t r i c i t y , m a s o n r y  c o n d i t i o n s , e x p e r i m e n t a l d a t a for  both  fails i n a s i m i l a r p a t t e r n bedding  conditions  regardless of  (full a n d  face-shell)  the are  i n c l u d e d . T h e c o m p a r i s o n is i l l u s t r a t e d i n F i g s . 7.1 to F i g . 7.11, w h i c h i n c l u d e the tests done b y the a u t h o r , b y F a t t a l a n d C a t t a n e o (1976), b y H a t z i n i k o l a s et a l (1978) a n d b y D r y s d a l e a n d H a m i d (1983). T a b l e s 7.1 to 7.4 s u m m a r i z e the n u m e r i c a l results. F o r the 58 cases c o m p a r e d , the average v a l u e of the r a t i o of f a i l u r e l o a d to p r e d i c t e d l o a d is 1.026 w i t h a coefficient of v a r i a t i o n  of 11.36%, c o r r e s p o n d i n g to a n expected r a t i o of  -1.026 w i t h 9 5 % confidence l i m i t s equal to 0.996 a n d 1.056. T h e agreement is e x t r e m e l y good c o n s i d e r i n g the scatter of the d a t a a n d the e r r a t i c nature of the m a t e r i a l . F i g . 7.12 s u m m a r i z e s the c o m p a r i s o n of the failure loads p r e d i c t e d b y the r e c o m m e n d e d m e t h o d w i t h the e x p e r i m e n t a l d a t a . T h e coefficient o f c o r r e l a t i o n is 0.956 a n d the m a j o r i t y  of  d a t a p o i n t s lie w i t h i n the 99 percent confidence l i m i t s , w h i c h is h i g h l y s i g n i f i c a n t . T h e r e c o m m e n d a t i o n s for design are concisely d e p i c t e d i n F i g . 7.13 by a P— M c a p a c i t y curve.  Curve  O  —  B  lt  the  masonry  c a p a c i t y under  eccentric l o a d , s h o u l d  be  determined  110 200  280  M(KP-N) F I G . 7.1  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y the A u t h o r : N 1 8 , N 1 9 , M 2 0 a n d 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 R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y the A u t h o r : N22 (Face-Shell Bedding)  Ill  F I G . 7.3  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s by Fattal and Cattaneo  2  400 M(KP-N)  F I G . 7.4  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y H a t z i n i k o l a s 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 R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s by Drysdale and H a m i d : N o r m a l Block  M (KN-M)  F I G . 7.6  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s by Drysdale and H a m i d : W e a k Block  113 490  M(KN-M)  F I G . 7.7  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s by D r y s d a l e a n d H a m i d : S t r o n g B l o c k  2"  M (KN-M)  F I G . 7.8  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y Drysdale a n d H a m i d : L i g h t W e i g h 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 R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s by Drysdale and H a m i d : 7 5 % Solid Block  280  -r  v  1  i  0  i  i.  2  i  4  ;  i  6  M(KN-M)  F I G . 7.10  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s by Drysdale and H a m i d : 6 inch Block  400  a.  0  2  4  6  10  B  12  14  16  M (KN-M)  F I G . 7.11 C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : 10 i n c h B l o c k  N 1 8 (e== l / 6 t )  N 1 9 (e== l / 3 t )  P =121.7kips  P =  0  AVG COV  0  84.4kips  M 2 0 (e:= l / 3 t ) P = 0  84.4kips  (predicted)  (predicted)  (predicted)  P - k i p s P/Po  P - k i p s P/Po  P - k i p s P/Po  S21 (e=: l / 3 t ) P = 0  84.4kips  (predicted) P-kips  P/Po  N 2 2 (c== l / 3 t ) P = 0  70.1kips  (predicted) P - k i p s 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  124.6  1.02  79.5  0.94  84.5  1.00  94.8  1.12  71.0  1.01  12.8%  7.8%  8.2%  3.9%  T a b l e 7.1 C o m p a r i s o n w i t h the R e c o m m e n d e d A p p r o a c h : T e s t s by A u t h o r  7.3%  e=l/12't P  0  = 116.3 k i p s  P(kips)  AVG  P  P/Po  0  e= l/3t  e=l/4b  e=l/6t = 97.1 k i p s  P  0  = 83.3 k i p s  P(kips)  P/Po  P(kips)  P/P  0  P  0  = 72.9 k i p s  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  122.6  1.05  113.7  1.17  83.1  1.00  69.1  0.95  COV  3.1%  24.1%  1.1%  T a b l e 7.2 C o m p a r i s o n w i t h the R e c o m m e n d e d  A p p r o a c h : T e s t s b y F a t t a l et a l  e= l/6t P P  AVG COV  0  e=l/3t  = 185.4 k i p s  (kips)  8.8%  P P/Po  P  0  = 138.7 k i p s  (kips)  P/Po  180.0  0.97  119.3  0.86  196.0  1.06  158.7  1.14  150.1  0.81  175.4  0.95  139.0  1.00  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 R e c o m m e n d e d A p p r o a c h : T e s t s b y H a t z i n i k o l a s et a l  SPECIMEN  e/t  n o r m a l block  1/6  247  55.5  54.4  1.02  (NB)  1/3  206  46.3  40.8  1.13  5/12  158  35.5  36.1  0.98  1/6  171  38.4  37.1  1.04  1/3  133  29.9  28.0  1.07  5/12  99  22.2  22.5  0.99  s t r o n g block  1/6  301  67.7  60.5  1.12  (SB)  1/3  236  53.1  45.4  1.17  5/12  194  43.6  40.2  1.09  lightweight block  1/6  228  51.3  43.8  1.17  (LB)  1/3  169  38.0  32.8  1.16  5/12  149  33.5  29.1  1.15  1/6  258  58.0  62.8  0.92  1/3  190  42.7  45.5  0.94  5/12  100  22.5  22.2  1.01  1/6  185  41.6  40.7  1.02  1/3  137  30.8  30.3  1.02  5/12  94  21.1  21.4  0.99  1/6  200  45.0  54.7  0.82  1/3  172  38.7  41.5  0.93  5/12  132  29.7  28.7  1.03  weak b l o c k (WB)  7 5 % solid (QB)  6 inch block (6"B)  10 i n c h b l o c k (10"B)  P  (kN)  P  (kips)  Po(kips)  P/Po  AVG  1.04  COV  8.7%  T a b l e 7.4 C o m p a r i s o n w i t h the R e c o m m e n d e d A p p r o a c h : T e s t s by D r y s d a l e et a l  118  200 180  —  s  AUTHOR  n  '  A  s s  HATZlhJKOLAS  A 160  A  140  v  DRYSDALE  0  FATTAL  t  •  120 s  x  y  60  •  Y'  •V  w V  40 s  ft  20 r  y  yy / I/ l V '1  —Ar-  'Y  99% COhFIDENCE LIMFTS  LINEAR REGRESSION  s  0  s  \\  80  UJ  1  1  •  100  .•f  V  s y  /  S "' s  20  40  60  80  100  120  140  160  180  200  PREDICTIONS (KPS )  F I G . 7.12  C o m p a r i s o n between R e c o m m e n d e d A p p r o a c h a n d E x p e r i m e n t s : S u m m a r y  F I G . 7.13  D e p i c t i o n of R e c o m m e n d e d A p p r o a c h  119 based o n the m a s o n r y u n i t s t r e n g t h f the k e r n e c c e n t r i c i t y c a p a c i t y (e = s h o u l d be d e t e r m i n e d strength f A — Bi~O  or b y A  :  2  a n d the geomerty of the cross-section, where B  denotes  x  A.± or A , w h i c h stands for the c o n c e n t r i c c a p a c i t y ,  t/&).  2  either b y the p r i s m test or b y the c o r r e l a t i o n w i t h the m a s o n r y  a n d the m o r t a r  u  u  strength fj . T h e  unit  w h o l e c a p a c i t y curve is represented either  by  — B — O d e p e n d i n g whether the c o n c e n t r i c c a p a c i t y is greater t h a n the kern 2  eccentricity capacity.  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 T h e c u r r e n t design code ( C A N 3 - S 3 0 4 - M 8 4 ,  1984) p e r m i t s t w o w a y s of d e s i g n i n g w a l l s  for c a r r y i n g i n - p l a n e a x i a l c o m p r e s s i o n a n d out of p l a n e b e n d i n g  due to e c c e n t r i c i t y of the  v e r t i c a l l o a d . T h e y are the s o - c a l l e d "coefficient m e t h o d " a n d the " l o a d d e f l e c t i o n m e t h o d " .  The  f o r m e r gives a n a d d i t i o n a l a l t e r n a t i v e to d e t e r m i n e the e c c e n t r i c i t y coefficient. Therefore,  one  a c t u a l l y c o u l d develop three different P — M  design curves for the same m e m b e r . W e  denote  t h e m b y m e t h o d 1, 2 a n d 3 for ease of discussion. The  basic i n f o r m a t i o n  needed for design is f ,  the u l t i m a t e compressive s t r e n g t h  m  m a s o n r y . T h e code specifies t w o m e t h o d s to d e t e r m i n e f . m  five prisms in u n i a x i a l compression. f  m  of  M e t h o d A requires testing of at least  is then t a k e n as the average s t r e n g t h m i n u s 1.5 t i m e s the  s t a n d a r d d e v i a t i o n of the s a m p l e . ( T h i s v a l u e m a y be reduced b y a factor d e p e n d i n g on whether the specified u n i t s t r e n g t h is consistent w i t h the tested u n i t strength.) M e t h o d B a l l o w s one to test u n i t  and  adequate) a n d f  mortar  m  separately  (the  l a t t e r is to ensure t h a t  the specified m o r t a r  type  is  is t a k e n f r o m the t a b l e d v a l u e based on u n i t s t r e n g t h a n d m o r t a r t y p e .  T h e m o s t o b v i o u s object of a design code is to ensure consistent r e l i a b i l i t y i n structures. F o r f l e x u r a l design o f m a s o n r y required.  The  current  eccentric  loading,  and  w a l l s , consistent r e l i a b i l i t y for v a r i o u s l o a d c o m b i n a t i o n s is  code recognizes the a p p a r e n t some  modifications  are  s t r e n g t h increase when  included  in  a p p r o a c h . T h i s is reflected i n m e t h o d 1 b y a n increase i n C , e  the  w a l l s are  conventional  under  beam-column  the e c c e n t r i c i t y coefficient, of 3 0 %  120 (this  is  fairly  reasonable  when  compared  with  the  author's  experiments)  when  loading  e c c e n t r i c i t y is greater t h a n r / 2 0 . F o r m e t h o d s 2 a n d 3, t h i s is reflected i n the different a l l o w a b l e stresses t h a t are used (compressive, f l e x u r a l ) i n d e v e l o p i n g the P—M  i n t e r a c t i o n curve. A l t h o u g h  the a u t h o r has d i f f i c u l t y i n u n d e r s t a n d i n g w h y the code a l l o w s quite different results b y the three different m e t h o d s for w a l l s l o a d e d w i t h equal eccentricities, we w i l l consider here the i m p l i c a t i o n s o f present research w i t h respect to these p r o v i s i o n s . F i r s t of a l l , i t was c o n c l u d e d t h a t the p r i s m s t r e n g t h does not correlate w e l l w i t h the strengths o f m o r t a r a n d u n i t . T h i s suggests that there is considerable u n c e r t a i n t y i n the use of m e t h o d B to a v o i d m a k i n g a x i a l p r i s m tests. F o r the f u l l y bedded p r i s m s tested b y the a u t h o r , m e t h o d A w o u l d give f  m  e q u a l to 1760, 2625 a n d 1930 p s i for p r i s m s w i t h M , S, a n d N type  m o r t a r respectively. M e t h o d B , w h i c h is based o n the c o r r e l a t i o n between p r i s m s t r e n g t h a n d strengths of u n i t a n d m o r t a r , w o u l d give f  m  e q u a l to 1855 psi for p r i s m s w i t h M a n d S t y p e  m o r t a r a n d 1430 p s i for p r i s m s w i t h t y p e N m o r t a r . T h i s is very i n c o n s i s t e n t , especially i n 2  t e r m s o f the p r o b a b i l i t y of non-exceedence of the s t r e n g t h v a l u e . f  m  determined by method  A  corresponds t o a non-exceedence p r o b a b i l i t y of a b o u t 6 . 7 % for a l l three t y p e m o r t a r p r i s m s , while f  d e t e r m i n e d b y m e t h o d B gives the non-exceedence p r o b a b i l i t y o f a b o u t 1 2 % for M t y p e  m  m o r t a r p r i s m , 0 . 0 0 1 % for S type a n d 0 . 0 3 % for N type m o r t a r p r i s m s . T h e c o n c l u s i o n is clear: m e t h o d B c a n n o t be r e c o m m e n d e d , or it s h o u l d be used very c o n s e r v a t i v e l y . T h e c u r r e n t code m a y a l r e a d y be o n the very c o n s e r v a t i v e side i n m o s t cases, b u t it can not prevent u n f a v o r a b l e results i n some p a r t i c u l a r cases, such as the t y p e M p r i s m i n the above e x a m p l e . S e c o n d , the code requires t h a t the design be based o n f . m  masonry  under  However, f  m  is the s t r e n g t h o f  u n i a x i a l c o m p r e s s i o n . It was c o n c l u d e d t h a t , because of the different f a i l u r e  m e c h a n i s m s , t h i s s t r e n g t h does not correlate w e l l w i t h the 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 , w h i c h  2  H e r e we o n l y have a s a m p l e size of 4 a n d we relax the 1 5 % r e s t r i c t i o n o n coefficient o f  v a r i a t i o n for M t y p e m o r t a r m a s o n r y p r i s m .  121 is m u c h less d e p e n d e n t of the apparent under  on the j o i n t c o n d i t i o n a n d bedding pattern. T h e code only takes account  increase in strength  by  allowing an  increase i n the design stress for  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 and 20% in method  masonry 2 and 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 between uniaxial strength and flexural strength of masonry and  mortar  type.  In  other  words,  w h i c h is i n d e p e n d e n t  since u n i a x i a l compression strength  of joint  depends  pattern  largely on  the  joint condition a n d bedding pattern, the strength under eccentric loading also depends largely  on  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 not the case. T h e s t r e n g t h of m a s o n r y u n d e r eccentric c o m p r e s s i o n m a y be better 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 the 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 the current code m a y be expected. F o r e x a m p l e , for the 58 e x p e r i m e n t a l cases 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 of w a l l sections tested b y the a u t h o r a n d others, the f l e x u r a l c o m p r e s s i v e stress at failure w a s c a l c u l a t e d (based o n the a s s u m p t i o n t h a t stress is l i n e a r l y d i s t r i b u t e d i n the c r o s s - s e c t i o n ) , and  listed in T a b l e  7.5  to Table  7.8  and  summarized  i n F i g . 7.14.  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 to  average  ratio of  w i t h a coefficient of v a r i a t i o n  the  equal  13.3%. Although  the  average  21  percent  higher  flexural compressive strength  eccentric coefficient specified by the code ( m e t h o d 2 a n d m e t h o d is  The  higher  than  that  of  the  recommended  method  (11.4%).  is c l o s e t o  3), the coefficient o f  Fig.  7.14  shows  the  the  variation  comparison  between a x i a l a n d f l e x u r a l strengths for the reported tests, w h i c h has a coefficient of c o r r e l a t i o n of  only  0.875,  superiority  comparing  unfavorably  of the r e c o m m e n d e d  F i g s . 7.12 a n d  7.14.  with  method  that  over  of  0.956  for  the  recommended  the code specified m e t h o d  method.  is o b v i o u s  The  in view  of  o  -Y0  1  2  3  4  5  UNIAXAIL STRENGTH ( KSI )  FIG. 7.14 Current Design Base: Uniaxial Strength versus Flexural Strength  AVG COV  N18 (e==l/6t)  N19 (e==l/3t)  M20 (e:=l/3t)  S21 (e=: l / 3 t )  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)  f'e(ksi) f'e/f'm  fe(ksi)  il(ksi)  f'e/f'm  fl(ksi)  f'e/f'm  fl(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  3.32  1.43  3.05  1.31  3.24  1.34  3.63  1.21  3.28  1.30  12.8%  7.8%  8.2%  3.9%  Table 7.5 Flexural to Uniaxial Strength: Tests by Author  f'e/f'm  7.3%  e=  4 it  AVG  =  e=l/6t  l/12t  4  1.89 k s i  (ksi)  it  ft/4  =  (ksi)  e=  1.89 k s i  it/4  4 it  =  (ksi)  l/4t  e=  4  1.89 k s i  ft  it/4  =  (ksi)  l/3t 1.89 k s i  fe/4  2.28  1.20  2.61  1.38  2.18  1.15  1.88  0.99  1.67  0.88  2.47  1.31  2.23  1.18  2.33  1.23  3.04  1.61  2.66  1.41  2.18  1.15  2.05  1.09  2.33  1.23  2.58  1.37  2.20  1.16  2.09  1.10  COV  3.1%  24.1%  8.8%  1.1%  T a b l e 7.6 F l e x u r a l to U n i a x i a l S t r e n g t h : T e s t s by F a t t a l et a l  e=  4 it  AVG COV  =  (ksi)  l/6t  e=  4  1.96 k s i  fe/4  fe  =  (ksi)  l/3t 1.96 k s i  f'e/4  2.28  1.17  2.20  1.03  2.49  1.28  2.68  1.37  1.90  0.97  2.22  1.14  2.35  1.20  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 S t r e n g t h : T e s t s by H a t z i n i k o l a s et a l  SPECIMEN  e/t  n o r m a l block  1/6  28.0  4.06  1.12  (NB)  1/3  27.8  4.03  1.12  5/12  27.0  3.91  1.08  1/6  19.4  2.81  1.08  1/3  19.9  2.88  1.11  5/12  16.8  2.43  0.93  s t r o n g block  1/6  34.0  4.93  1.14  (SB)  1/3  35.1  5.09  1.17  5/12  32.9  4.77  1.10  l i g h t w e i g h t block  1/6  25.8  3.74  1.24  (LB)  1/3  25.1  3.64  1.21  5/12  25.3  3.67  1.22  1/6  19.1  2.77  1.17  17.6  2.55  1.08  5/12  21.1  3.06  1.29  1/6  26.5  3.84  1.11  1/3  26.1  3.78  1.10  5/12  23.6  3.42  0.99  1/6  20.8  3.01  0.96  1/3  23.4  3.39  1.08  5/12  22.5  3.26  1.04  fm  = 24.9  mpa  weak b l o c k (WB) fm = 18.0  fm = 29.9  f  m  mpa  mpa  = 20.8 m p a  7 5 % solid (QB) fm = 16.3  1/3 mpa  6 inch block (6"B) fm = 23.8  mpa  10 i n c h block (10"B) fm  = 21.6 m p a  ft  .  (mpa)  fe  .  (ksi)  fe/4  AVG  1.11  COV  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 S t r e n g t h : Tests by D r y s d a l e et a l  125 C H A P T E R VIII GROUTED  MASONRY WITH FULL  BEDDING  8.1 I n t r o d u c t i o n In the west coast a r e a of C a n a d a , where e a r t h q u a k e resistance is a m a i n concern i n s t r u c t u r a l design, m a s o n r y  w a l l s a n d c o l u m n s are required to be g r o u t e d  a n d reinforced  to  improve structural continuity and ductility. T h e a x i a l c a p a c i t y o f g r o u t e d a n d reinforced m a s o n r y is of interest not o n l y because it d i r e c t l y d e t e r m i n e s the design thickness of a w a l l p r o v i d i n g a x i a l a n d l a t e r a l l o a d resistance i n a m u l t i — s t o r y b u i l d i n g , b u t also, because it is closely related to the design d u c t i l i t y ( P r i e s t l e y a n d H o n 1983). The  methods  for d e t e r m i n i n g  the compressive s t r e n g t h of g r o u t e d  specified i n the current code ( C A N 3 - S 3 0 4 - M 8 4 ,  concrete  masonry  1984) are essentially the same as those for p l a i n  concrete m a s o n r y , as reviewed i n the preceding chapter. M e t h o d A , w h i c h requires a p r i s m test, is not very p r a c t i c a l . Since the f a i l u r e l o a d of a s t a n d a r d 8 i n c h g r o u t e d p r i s m w i l l u s u a l l y be w e l l a b o v e 300 k i p s , testing facilities w i t h adequate c a p a c i t y are e x t r e m e l y l i m i t e d . M e t h o d  B,  w h i c h estimates the c o m p r e s s i v e s t r e n g t h based o n the u n i t s t r e n g t h a n d m o r t a r t y p e , tends to be excessively conservative due to the uncertainties i n v o l v e d . T h e code does not correlate the m a s o n r y c o m p r e s s i v e s t r e n g t h w i t h g r o u t s t r e n g t h , b u t m e r e l y requires t h a t the g r o u t s t r e n g t h be at least e q u a l to t h a t o f the b l o c k shell. O n the one h a n d , t h i s does not a l l o w one to take a d v a n t a g e of h i g h s t r e n g t h g r o u t i n g , a n d o n the other h a n d , it is a d i f f i c u l t s p e c i f i c a t i o n to meet since the u n i t s t r e n g t h is often m u c h higher t h a n the specified v a l u e (cf. T a b l e s 8.3 — 8.9). If the f a i l u r e m e c h a n i s m is dependent o n the r e l a t i v e strengths, it w i l l be c o r r e s p o n d i n g l y u n c e r t a i n . T h e a x i a l b e h a v i o u r o f grouted concrete m a s o n r y has been s t u d i e d b o t h e x p e r i m e n t a l l y a n d a n a l y t i c a l l y . D r y s d a l e a n d H a m i d (1979) first addressed the c o m p a t i b i l i t y p r o b l e m  between  m a s o n r y u n i t a n d g r o u t based o n their e x p e r i m e n t a l observations. A k i o B a b a a n d O s a m u (1986) proposed  the concept of the g r o u t efficiency, a n d f o u n d it v a r i e d c o n s i d e r a b l y  Senbu with  126 different c o m b i n a t i o n s of m a s o n r y u n i t a n d g r o u t they tested. A few failure m o d e l s have been suggested to predict the u l t i m a t e s t r e n g t h c o n s i d e r i n g the i n t e r a c t i o n between u n i t , g r o u t a n d mortar  ( A h m a d and Drysdale  1979, P r i e s t l e y a n d H o n 1983). H o w e v e r , the i n t e r n a l forces o f  these m o d e l s were e n t i r e l y based o n a state t h a t a l l the three m a t e r i a l s reach some a s s u m e d f r a c t u r e c r i t e r i a . T h i s is not a l w a y s a realistic d e s c r i p t i o n . F u r t h e r , the f r a c t u r e is not necessarily e q u i v a l e n t to u l t i m a t e state. It is clear t h a t a better u n d e r s t a n d i n g of the m e c h a n i c a l b e h a v i o u r is needed a n d a m o r e a c c u r a t e e s t i m a t e for m a s o n r y strength is desirable. In t h i s s t u d y , the e x p e r i m e n t a l b e h a v i o u r o f g r o u t e d m a s o n r y p r i s m s is c a r e f u l l y e x a m i n e d a n d a better c o r r e l a t i o n of the m a s o n r y  strength  w i t h the u n i t s t r e n g t h , g r o u t s t r e n g t h a n d m o r t a r s t r e n g t h is sought.  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 23 grouted  p r i s m s were tested under  uniaxial compression, w i t h various joint  and  g r o u t i n g c o n d i t i o n s . T h e f a i l u r e loads of the p r i s m s are s u m m a r i z e d i n T a b l e s 8.1 a n d 8.2. T h e g r o u t strengths, e v a l u a t e d b y testing i n accordance w i t h C S A S t a n d a r d ( A 1 7 9 — 1 9 7 6 ) ,  are l i s t e d  i n T a b l e 2.5. T h e experiments indicate: 1) B o t h m o r t a r  strength and  g r o u t strength affect the p r i s m s t r e n g t h . A p p a r e n t l y  m o r t a r a n d g r o u t m a k e stronger m a s o n r y masonry  c a p a c i t y is m i n o r ,  stronger  (see T a b l e s 8.1 a n d 8.2). H o w e v e r , the increase i n  even w i t h a s u b s t a n t i a l increase i n the c o n s t i t u e n t strengths, as  d e p i c t e d i n F i g . 8.1. T h i s is especially true for grout, suggesting t h a t the c o n t r i b u t i o n o f g r o u t a n d b l o c k shell ( i n c l u d i n g  m o r t a r j o i n t ) to the c a p a c i t y of m a s o n r y  is a f u n c t i o n of their  c o m p a t i b l e d e f o r m a t i o n s , a n d is not s i m p l y g i v e n b y s u p e r p o s i t i o n of their i n d i v i d u a l c a p a c i t i e s . T h i s o b s e r v a t i o n c o n f i r m s t h a t b y D r y s d a l e a n d A h m a d (1979). 2) D e f o r m a t i o n m e a s u r e m e n t s a n d direct o b s e r v a t i o n (recorded b y a video c a m e r a ) i n d i c a t e t h a t a l m o s t a l l the p r i s m s were c r a c k e d before f i n a l f a i l u r e . C r a c k s were f o u n d i n the webs as w e l l as  127 i n the f a c e - s h e l l , o c c u r r i n g at loads as l o w as 4 0 % of the f a i l u r e value, as evidenced b y recorded d e f o r m a t i o n s  the  i n F i g s . 8.2, 8.3 a n d 8.4. S i m i l a r observations have been reported  Sturgeon and Longworth  (1985). U s i n g a c o u s t i c m e a s u r e m e n t ,  A k i o B a b a and Osamu  by  Senbu  (1986) have also observed m o r e d e t a i l e d c r a c k i n g process w e l l before u l t i m a t e state for grouted p r i s m s w i t h b o n d b e a m concrete u n i t . T h i s p r e m a t u r e c r a c k i n g m a y have been caused b y the i n c o m p a t i b i l i t y between the g r o u t a n d the block shell, as w i l l be f u r t h e r s t u d i e d below. H o w e v e r , closer  inspection  shows  that  the  block  shell s t i l l  carried substantial load  after  cracking,  i n d i c a t i n g t h a t p r e m a t u r e c r a c k i n g is not necessarily equivalent to failure. W e k n o w t h i s m a i n l y from  t w o facts: a) T h e v e r t i c a l d e f o r m a t i o n  measurement  of the b l o c k shell shows t h a t  the  c o m p r e s s i v e s t r a i n r e m a i n e d at a h i g h level after c r a c k i n g h a d o c c u r r e d (cf. F i g s . 8.2-8.4).  b)  T h e c a p a c i t y o f the p r i s m is u s u a l l y s u b s t a n t i a l l y greater t h a n t h a t of g r o u t i n g concrete alone. 3) B l o c k shells are stiffer t h a n concrete grout w h i c h i n t u r n are stiffer t h a n m o r t a r . T h e peak s t r a i n is between 0.0015 to 0.002 for the b l o c k u n i t a n d 0.0025 to 0.003 for three types o f grout (see "Figs. 2.3, 2.4 a n d 2.6 i n C h a p t e r II).  A similar phenomenon  has been i n d i c a t e d b y  research o n concrete m a s o n r y i n N e w Z e a l a n d ( P r i e s t l e y a n d E l d e r 1985). T h e Y o u n g ' s of various  concrete u n i t s  (used  in Japan)  a c c o r d i n g to A k i o B a b a a n d O s a m u  are also a p p e a r e n t l y  Senbu  (1986). T h e  mortar  higher  than  that  the  modulus of  even e x h i b i t e d higher  grout peak  s t r a i n s , w h i c h exceeded 0.005 (measured d u r i n g the cube strength testing). T h e difference i n the d e f o r m a t i o n properties is p r o b a b l y due to different m a t e r i a l textures a n d c u r i n g c o n d i t i o n s . T h i s o b s e r v a t i o n s u p p o r t s the v i e w t h a t c o m p a t i b i l i t y p l a y s a n i m p o r t a n t role i n concrete m a s o n r y c a p a c i t y . T h e v e r t i c a l d e f o r m a t i o n measurements, indeed, i n d i c a t e d t h a t the block shells c a r r i e d a larger  share o f l o a d relative to the grout  before  they  c r a c k e d ; after c r a c k i n g , the shell  c o n t i n u e d to c a r r y a s u b s t a n t i a l p o r t i o n of the l o a d , a l t h o u g h i n some cases there was a slight decrease.  1  2  3  M9 (NG, MJ)*  333.0  333.0  S8 (NG, SJ)  303.0  264.0  SPECIMEN  N13 (NG, NJ) N10 (NG, t =6/8") 0  P l l (NG, t = 0)  274.0  0  N17(NG,Face-Shell)  4  AVG  COV  310.5  325.5  3.3%  321.0  296.0  8.0%  237.0  332.0  284.5  16.7%  302.0  300.0  273.5  291.8  4.5%  234.0  312.5  273.5  11.7%  252.0  240.0  250.0  3.0%  258.0  Table 8.1 Failure Loads of Grouted Prisms (kips), with Variation in Joint Condition * NG - Type N Grout; NJ - Type N Mortar Joint, etc.  1  SPECIMEN N12 (SG, NJ)  2  3  4  316.0  291.0  N13 (NG, NJ) N14 (WG, NJ)  257.0  AVG  COV  254.0  287.0  8.9%  237.0  332.0  284.5  241.0  289.0  262.3  16.7% 7.6%  Table 8.2 Failure Loads of Grouted Prisms (kips), with Variation in Grout  0.9  •  •  I  •  0.8  •  • •  •  0.7  •  1  •  I  •  1 I  O.S  0.4  0.3  1  0.2 0.2  0.4  0.6  0.8  1— 1 1 1 1  1.2  •  GROUT  •  MORTAR  1 — 14  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 i n the s t r e n g t h d a t a o b t a i n e d by the a u t h o r  and  i n n u m e r o u s previous studies, one c o n c l u s i o n c a n be d r a w n w i t h c e r t a i n t y : t h a t the c a p a c i t y of the b l o c k u n i t a n d t h a t of the g r o u t are not s i m p l y a d d i t i v e . T h i s o b v i o u s l y results f r o m the difference i n d e f o r m a t i o n  properties of the m a t e r i a l s , as discussed i n the preceding  paragraph.  W e consider t w o aspects of t h i s d e f o r m a t i o n c o m p a t i b i l i t y p r o b l e m : F i r s t , v e r t i c a l c o m p a t i b i l i t y . Since the g r o u t a n d the b l o c k u n i t u s u a l l y have  different  peak s t r a i n s , as s h o w n b y e x p e r i m e n t , they are not able to reach their f u l l c a p a c i t i e s at the same strain.  From  this viewpoint  efficiency of the g r o u t i n g  it is o b v i o u s  w i l l depend  on  that how  simple capacity addition close the d e f o r m a t i o n  is not  valid.  The  properties  of the  two  m a t e r i a l s are. S e c o n d , h o r i z o n t a l (or cross-sectional) c o m p a t i b i l i t y . T h i s includes t w o p a r t s . O n e is due to the different P o i s s o n ' s effect of g r o u t a n d b l o c k shell. T h e other is due to the g e o m e t r y : manufacturing  for  reasons, the h o l l o w core of a concrete m a s o n r y block is a c t u a l l y tapered w i t h a  release angle 1° — 3 ° .  T h i s m a y i n t r o d u c e a n a d d i t i o n a l cross-sectional c o m p a t i b i l i t y  problem  when g r o u t a n d b l o c k shell undergo different v e r t i c a l strains. The  premature  c r a c k i n g observed  in experiments  is c e r t a i n l y caused by  these cross-  sectional incompatibilities. T h u s , a f a i l u r e m o d e l of g r o u t e d m a s o n r y  based o n d e f o r m a t i o n  c o m p a t i b i l i t y w i l l be  closer to r e a l i t y t h a n one based o n s t r e n g t h s u p e r p o s i t i o n . T h i s w i l l serve as a guideline for the following model  development.  It m a y be useful to list a l l the n o t a t i o n a p p l i e d i n the m o d e l :  Ag,  Eu,  gross a r e a a n d net a r e a of b l o c k u n i t , respectively;  An  —  2a, 26  =  w i d t h of b l o c k inner core a n d b l o c k u n i t , respectively;  Eg,  =  m o d u l u s of e l a s t i c i t y of block u n i t , grout, m o r t a r j o i n t , respectively;  Ej  133 fmp, fu,  fmg  =  compressive s t r e n g t h of 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 ,  respectively;  fj  =  compressive s t r e n g t h of b l o c k u n i t , g r o u t ( p r i s m strength),  mortar  f, g  (cube  s t r e n g t h ) , respectively; f  ut  h m  0  m  1 ?  2  p t  0  a e, u  e,  €j  g  =  tensile strength of b l o c k u n i t ;  =  height of block u n i t ;  =  m o d u l a r r a t i o of E /E  =  c o n t a c t pressure between g r o u t a n d b l o c k shell;  =  thickness of m o r t a r j o i n t ;  =  release angle of b l o c k inner core;  =  compressive s t r a i n i n block u n i t , g r o u t a n d m o r t a r j o i n t , respectively;  77 = Vn-, Vg, vj <r ,  CT  m  S  <7 , (T , (Tj U  g  a  ut  u  g  a n d E /Ej, u  respectively;  net area to gross a r e a r a t i o of b l o c k u n i t  A /A ; n  g  =  P o i s s o n ' s r a t i o of block u n i t , grout a n d m o r t a r , respectively;  =  compressive stress i n m a s o n r y (average) a n d i n m a s o n r y shell, respectively;  =  compressive stress 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 , respectively;  =  l a t e r a l tensile stress i n b l o c k u n i t ;  In general, the c a p a c i t y of grouted m a s o n r y depends o n m a n y factors, m o s t * the s t r e n g t h of the m a t e r i a l s f , u  f, ut  f, g  importantly:  fj  * the d e f o r m a t i o n properties of the m a t e r i a l s E«, E , g  Ej,  v, u  v, g  vj  * g e o m e t r i c properties such as the shape of the block u n i t s , the thickness of the m o r t a r j o i n t , b o n d p a t t e r n , etc. * w o r k m a n s h i p , test m e t h o d  T o m a k e the m o d e l p r a c t i c a l l y useful, we neglect those effects w h i c h are not easy to q u a n t i f y , such as w o r k m a n s h i p or test m e t h o d . W e w i l l also t r y to a v o i d e x p l i c i t l y i n c l u d i n g the d e f o r m a t i o n properties i n the m o d e l , since they are d i f f i c u l t to measure. F u r t h e r , 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. W e 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  r,  8.2  or  -f  = J 1 -  In the derivation, an expression for {b—a)/a is needed. E q . 8.1 can be rewritten as  b - a b  _  V 1 + a/b  When a/b is expanded as 1— n/2 in view of E q . 8.2,  the above expression becomes  8.3  which gives good approximations even when r\ is as large as 0.6. Based on E q . 8.3, it is easy to obtain  o  »  2  . \ 4 — 3n  8.4  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 i m p l i c i t y a n d p r a c t i c a l i t y , we a d o p t a q u a s i - e l a s t i c a p p r o a c h .  T h a t is, we use  the theory of elasticity a n d i m p l i c i t l y assume that the d e f o r m a t i o n properties i n v o l v e d are t a k e n as secant or effective values. F u r t h e r ,  we assume that stress a n d s t r a i n i n the m a t e r i a l s are  u n i f o r m , or, i n other words we treat the stress a n d s t r a i n i n a n average sense i n each m a t e r i a l . F o r the p r i s m shown i n F i g . 8.5, the f o l l o w i n g relations c a n be w r i t t e n . A ) In V e r t i c a l D i r e c t i o n Equilibrium:  <Tm = r]<J, + (l-Tj)crg  If the shear force between the block shell a n d grout is neglected, we have  8.5  136 Ca  =  cr  =  u  cr j  Compatibility:  ~  ~  Cu  h + U 0  + U , J> C  8.6  Stress — S t r a i n R e l a t i o n : F o r g r o u t we h a v e  e  g  =  '  y  r  8.7  f  F o r the b l o c k s h e l l , a n expression for the v e r t i c a l s t r a i n due t o the c o n t a c t pressure p is needed. T h i s can be o b t a i n e d by B e t t i ' s l a w . R e f e r r i n g to F i g . 8.5, we h a v e  A(2a)h p 6(cr ) = A cr ho 0  s  n  s  e (p) u  where S(cr ) is the l a t e r a l d i s p l a c e m e n t due to the v e r t i c a l stress, expressed as s  CT v a s  u  E  u  and e (p) u  is the v e r t i c a l s t r a i n i n the u n i t due t o the l a t e r a l pressure p. U p o n s u b s t i t u t i o n a n d  r e a r r a n g i n g , one f i n d s  _  2pi/ (l-I?) u  T h u s the v e r t i c a l s t r e s s — s t r a i n r e l a t i o n of the b l o c k shell is  137  _ tu —  v> + 2 p i / ( l - » ? ) / t ? u  o.o  o  The stress—strain relation for the mortar joint is  B) In Lateral Direction Equilibrium:  2ap = 2(6—a) cr  ut  or ap  (4-3r/)p  w  6—a  277  ~  8.10  in which the relation given by Eq. 8.4 has been used. Compatibility:  (u cr -p)a _ g  g  (<T  ui  +  Eg  <Tsl'u)a E U  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  'T  P)  =  8.11  138 T h e a b o v e seven equations c a n be used to d e t e r m i n e the stress a n d s t r a i n state i n a  grouted  m a s o n r y p r i s m when a v e r t i c a l stress or s t r a i n is i m p o s e d . C ) Failure Conditions T h e r e are several possible w a y s for a g r o u t e d concrete m a s o n r y a s s e m b l y to f a i l . i n c l u d e c h i e f l y : a)  Premature  s p l i t t i n g of the block shell due  to the i n c o m p a t i b l e  They  material  properties of the g r o u t a n d the b l o c k , w h i c h give rise to t e n s i o n s . i n the shell, b) If the assembly survives this condition, failure m a y reaches a m a x i m u m ,  occur w h e n  at a d e f o r m a t i o n  the s u m of the shell a n d  g r o u t resistances  between their respective peak strains, c) T h e  assembly  m a y f a i l w h e n the g r o u t reaches its f u l l c a p a c i t y ; at t h i s p o i n t the s u b s t a n t i a l v o l u m e increase due to the i n t e r n a l c r a c k i n g of the g r o u t causes the b l o c k shell to f a i l . W h i c h e v e r the case, the lower b o u n d of the m a s o n r y s t r e n g t h s h o u l d be a l w a y s satisfied:  fmg  >  ( 1 -  r, ) f  8.12  g  w h i c h corresponds to the f a i l u r e l o a d being c a r r i e d b y the g r o u t alone. F o r n o r m a l range of n, failure c o n d i t i o n c) requires t h a t the peak s t r a i n of the g r o u t be reached f i r s t . H o w e v e r t h i s is u n l i k e l y to be the case, i n view not o n l y of the e x p e r i m e n t s by the a u t h o r a n d of those done i n N e w Z e a l a n d , w h i c h have i n d i c a t e d t h a t the concrete block u n i t s were stiffer t h a n grout, b u t also of the e x p e r i m e n t a l observations by several other t h a t the g r o u t core was i n t a c t even after m a s o n r y specimens h a d f a i l e d ( D r y s d a l e  researchers and  Hamid  1979; H a t z i n i k o l a s et a l 1978). F a i l u r e c o n d i t i o n a), of course, is governed b y the b l o c k shell. H o w e v e r , to d e t e r m i n e the f a i l u r e l o a d for c o n d i t i o n b ) , a knowledge of the d e f o r m a t i o n entire s t r a i n range  is needed.  Since t h i s i n f o r m a t i o n  properties of the m a t e r i a l s over  is d i f f i c u l t to e s t a b l i s h , as a p r a c t i c a l  a l t e r n a t i v e , we m a y i n q u i r e w h i c h m a t e r i a l , b l o c k shell or grout, is closer to its f u l l c a p a c i t y at the p o i n t of failure of the assemblage. In the l i g h t of the s t u d y i n C h a p t e 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 masonry  capacity  is much more closely  conducted by the author) indicates that  correlated to  the  block unit strength.  the  Thus  the  assumption is supported by the statistical implications. O f 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 CoulombNavier failure criterion is assumed, one writes  +  mp  E q . 8.5 to E q . 8.11 with  €j, <Ts, a , g  strength f , mg  <r , V m  a  n  d  = 1  8.13  E q . 8.13 can be used to find the 8 unknowns; namely C u , £  The capacity of the grouted masonry, in terms of the masonry  is equal to <r , since once E q . 8.13 is included, the group of equations is actually m  solved at the critical condition. Upon substituting and neglecting higher order terms, we get  140 [77 + ( l - 7 ) ( l + m ^ ) i ] / 2  7  fmg  m  p  8.14  —  fmp fut  It is interesting to e x a m i n e the p h y s i c a l i n t e r p r e t a t i o n of this s o l u t i o n . T h e represents  the  compatibility  capacity  o f grouted  o f the d e f o r m a t i o n  masonry  properties.  determined  b y considering  I t means t h a t w i t h o u t  only  numerator  the  vertical  the difference i n l a t e r a l  properties, w h i c h w i l l lead to failure c o n d i t i o n b), the failure l o a d of grouted m a s o n r y w o u l d be c o m p o s e d of t w o parts. O n e is the p l a i n m a s o n r y strength t i m e s the net area of the block u n i t . T h e other is the p r o d u c t of the stress i n the grout, w h i c h , due to the difference i n stiffness, reaches 1/TOJ  t i m e s the stress i n the b l o c k shell, w i t h the grouted area. T h e factor, m to/h 2  0  takes  a c c o u n t of the "softer" m o r t a r j o i n t , i n d i c a t i n g t h a t the latter tends to increase the stress i n the grout. The denominator masonry  strength  f  mg  accounts for the cross-sectional c o m p a t i b i l i t y . I t c a n be seen t h a t the i s a n increasing  i n c o m p a t i b i l i t y becomes less i m p o r t a n t  function  o f 77, i n d i c a t i n g  that  the  cross-sectional  as the thickness o f the b l o c k shell increases. T h e  square bracket c o n t a i n s some very s m a l l q u a n t i t i e s . T h e t e r m (v — v ), g  u  big  which is implicitly  assumed to be greater t h a n or equal to zero i n the d e r i v a t i o n , represents the i n c o m p a t i b i l i t y due to the difference i n P o i s s o n ' s effect of the t w o m a t e r i a l s . T h e t e r m m a(t /a) 0  2  i n c o m p a t i b i l i t y caused by the t a p e r i n g of the core. T h e t e r m m v (t /'h ) 2  the i n c o m p a t i b i l i t y of the softer m o r t a r j o i n t , Since  the  mortar  is u s u a l l y  much  which softer  g  0  0  accounts for the  i m p l i e s the effect o n  needs m o r e detailed e x p l a n a t i o n . than  the  block  units,  as i n d i c a t e d b y  e x p e r i m e n t , the grout is a c t u a l l y strained m o r e t h a n the block u n i t i n the v e r t i c a l d i r e c t i o n , due to the presence of the j o i n t . T h u s even i f the block u n i t a n d grout have the same value o f P o i s s o n ' s r a t i o , the grout w i l l e x p a n d more, l a t e r a l l y , t h a n the block shell, c a u s i n g a d d i t i o n a l cross-sectional i n c o m p a t i b i l i t y . I t is clear t h a t the m a s o n r y strength f of these 3 terms a p p e a r i n g i n the square bracket of the  denominator.  mg  is a, decreasing f u n c t i o n  141 O b v i o u s l y , the d e n o m i n a t o r w o u l d reduce to u n i t y i f there were no l a t e r a l c o m p a t i b i l i t y effect. I n other words, the n u m e r a t o r predicts the u l t i m a t e failure l o a d of f a i l u r e c o n d i t i o n b ) w h e n the p r i s m survives f a i l u r e c o n d i t i o n a).  (  T o m a k e the m o d e l p r a c t i c a l l y useful, some s i m p l i f i c a t i o n s are necessary. Since the t e r m m t /h 2  0  is u s u a l l y s m a l l c o m p a r e d w i t h 1, e.g., for s t a n d a r d 8 i n c h b l o c k u n i t s , i /h  0  0  is s m a l l e r  0  t h a n 0.05 a n d m , the secant m o d u l a r r a t i o is a r o u n d 3 a c c o r d i n g to the a u t h o r ' s e x p e r i m e n t s 2  (see F i g . 2.5 i n C h a p t e r I I ) , we m a y numerator  neglect i t s v a r i a t i o n b y a s s u m i n g (l + m t /h ) 2  t o be a constant s l i g h t l y bigger  than  unity.  0  i n the  0  B y a similar argument,  the  term  ?7(3 — 2 m ) i n the d e n o m i n a t o r c a n be neglected since i t is s m a l l c o m p a r e d w i t h the t e r m 4. T h e x  v a r i a t i o n of the t e r m at /a  m a y also be ignored since i t is s m a l l c o m p a r e d w i t h is (t /h );  0  g  0  for the geometry of a s t a n d a r d 8 i n c h b l o c k these t w o terms give a p p r o x i m a t e d 0.015. it  appears  that  the cross-sectional i n c o m p a t i b i l i t y  would  be m a i n l y  0  and  Although  caused b y the  lateral  e x p a n s i o n due to P o i s s o n ' s effect on the grout, as believed by some researchers ( D r y s d a l e  and  H a m i d 1979), we m a y neglect the v a r i a t i o n of the t e r m (y — u ) by r e p l a c i n g it w i t h a c o n s t a n t g  u  t , say, not exceeding 0.1. D u e to the difference i n the stiffness between grout a n d block shell, the block shell tends to be m o r e stressed a t failure. T h i s view is also s u p p o r t e d by the s t a t i s t i c a l a r g u m e n t s t a t e d a b o v e t h a t m a s o n r y c a p a c i t y is governed m o r e by the b l o c k shell. T h u s the effective v  g  w i l l not increase as m u c h as v'u a r o u n d the c r i t i c a l state due to i n t e r n a l c r a c k i n g .  T h e last t e r m fmplf u the r a t i o of the b l o c k shell strength to b l o c k u n i t tensile s t r e n g t h , m a y be u  a s s u m e d to be a c o n s t a n t £ i n the order of 10. T h u s for the geometry of a s t a n d a r d 8 i n c h block, E q . 8.14 c a n be s i m p l i f i e d to  f  ( n + ( 1 - ^ ) 4 )fmp  _  mg  —  1 +  4 _ Q  C ( * + 0 . 0 1 5 m ) ?—p-  n  8.15  2  E q . 8.15 is based o n fracture of the b l o c k shell, w h i c h m a y or m a y not lead to collapse of the m a s o n r y  assemblage, as discussed earlier. T h u s E q . 8.15 p r e d i c t s the l o a d for f a i l u r e  142 c o n d i t i o n a): f r a c t u r e of b l o c k shell leads to f i n a l f a i l u r e . H o w e v e r i f the a s s e m b l y survives t h i s c o n d i t i o n , the u l t i m a t e f a i l u r e l o a d is g i v e n b y the n u m e r a t o r of E q . 8.15 b y neglecting crosss e c t i o n a l i n c o m p a t i b i l i t y . E q . 8.15  t h e n corresponds to m e r e l y the c r a c k i n g l o a d of the b l o c k  shell. F r o m the e x p e r i m e n t s c o n d u c t e d b y the a u t h o r a n d b y n u m e r o u s other researchers, it appears t h a t either f a i l u r e c o n d i t i o n c a n occur. T h i s poses the p r o b l e m i n p r a c t i c e as to w h i c h s o l u t i o n s h o u l d be used i n p r e d i c t i n g m a s o n r y  capacity. T h i s question, again, m a y  only  be  answered i n a s t a t i s t i c a l sense. W e m a y e x a m i n e the a v a i l a b l e e x p e r i m e n t a l d a t a to see whether one of the t w o f a i l u r e c o n d i t i o n s has a p r o b a b i l i t y o f occurrence h i g h e n o u g h to d o m i n a t e the failure mode. To  correlate the a v a i l a b l e e x p e r i m e n t a l  d a t a , expressions for fmp,  m  1  and  m  2  needed. In v i e w of the s t u d y i n C h a p t e r III, for n o r m a l j o i n t thickness, the expression for  are f  mp  m a y be t a k e n i n the f o r m as g i v e n b y E q . 4.28. i.e.  fmp  = cj  u  + c fj 2  8.16  F u r t h e r , the m o d u l a r ratios m a y be related to the s t r e n g t h values as  8.17 and 8.18  where c  l 5  c, 2  k  x  and k  2  are constants. T h e square root r e l a t i o n between the s t r e n g t h a n d the  m o d u l u s o f e l a s t i c i t y is a d o p t e d b y m a n y b u i l d i n g codes. W e proceed to give a n e s t i m a t e for the constants i n v o l v e d i n these relations. W e w i l l do t h i s based o n the 77 a v a i l a b l e d a t a p o i n t s f r o m 7 different sources (Presents tests; H a m i d  and  143 D r y s d a l e 1978; B o u l t 1979; D r y s d a l e a n d H a m i d 1979; T h u r s t o n 1981; P r i e s t l e y a n d E l d e r 1982, 1985; W o n g a n d D r y s d a l e 1983). These a c t u a l l y i n c l u d e m a n y m o r e t h a n 77 specimens because several of these d a t a p o i n t s were reported as average values. T h e d a t a are s u m m a r i z e d i n T a b l e s 8.3 — 8.9. T h e N e w Z e a l a n d results i n c l u d e fj  values based o n p r i s m s . These have been c o n v e r t e d  to e q u i v a l e n t cube s t r e n g t h u s i n g the T H e r m i t e e q u a t i o n ( N e v i l l e 1965). The values o f m  k values i n E q . 8.17 a n d E q . 8.18 s h o u l d m a k e the equations y i e l d the average and m  1  2  w h e n the f ,  a n d fj  f  u  g  take their m e a n values. A c c o r d i n g to the d a t a , the  r a t i o s of the average u n i t strength to grout strength a n d to m o r t a r s t r e n g t h are 1.04 a n d respectively. T h e average v a l u e of m  u  a c c o r d i n g to  the e x p e r i m e n t a l  results b y  b y the N e w Z e a l a n d researchers, m a y be t a k e n as 1.32. T h e m e a n v a l u e of m be 3 as m e n t i o n e d earlier. T h i s leads to k  x  =  1.29 a n d k  2  =  2.54. F u r t h e r  the a u t h o r a n d m a y be t a k e n to  2  c  1.5  1  and c  may  2  be  a w a r d e d the values g i v e n b y E q . 4.28. For  failure  condition  a)  Eq.  8.15  is used  w h i l e for  failure condition  b)  only  the  n u m e r a t o r o f the e q u a t i o n is a p p l i e d . T h e results are also s u m m a r i z e d i n T a b l e 8.3 — 8.9. It  appears  underestimate  that  p r e d i c t e d failure  those o b t a i n e d  by  loads based o n  experiments.  The  failure  condition  a)  substantially  results based o n c o n d i t i o n b),  however,  correlate w e l l w i t h the e x p e r i m e n t a l d a t a , a l t h o u g h it appears t h a t they o v e r e s t i m a t e s t r e n g t h i n the lower range. T h e c o r r e l a t i o n coefficient for the f o r m e r is 0.894, w h i l e for the l a t t e r is 0.918. These results are p l o t t e d i n F i g s . 8.6 a n d 8.7 as p r e d i c t i o n s versus e x p e r i m e n t s . It is clear t h a t the difference between the t w o m e t h o d s is s i g n i f i c a n t . T h u s , i t m a y be c o n c l u d e d , based on the above s t u d y a n d on the a v a i l a b l e e x p e r i m e n t a l d a t a f r o m v a r i o u s sources t h a t , under n o r m a l c o n s t r u c t i o n c o n d i t i o n s , the s t r e n g t h of grouted m a s o n r y is m a i n l y governed by the v e r t i c a l c o m p a t i b i l i t y of the g r o u t a n d block shell. F u r t h e r , since the b l o c k shell is stiffer i n the pre-peak range of s t r a i n , a n d less d u c t i l e i n the post-peak range t h a n grout,  the m a s o n r y  w i l l t e n d to f a i l when the f u l l c a p a c i t y of the b l o c k shell is  reached; thus the c a p a c i t y of m a s o n r y is m o r e closely correlated w i t h b l o c k u n i t strength t h a n  144 w i t h grout strength. It s h o u l d be n o t e d t h a t the above c o n c l u s i o n does not e l i m i n a t e the p o s s i b i l i t y t h a t the f a i l u r e m a y occur i n the f o r m o f c o n d i t i o n c), a n d , especially c o n d i t i o n a). It o n l y means t h a t f a i l u r e c o n d i t i o n b) has a p r e d o m i n a n t p r o b a b i l i t y of governing.  4 1.57  3.70  1.97  1.57  2.30  2.22  1.57  3.70  2.76  1.57  2.30  2.22  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  V  fu  0.51  3.25  0.51  3.25  0.51  fmg  a  b  c  T a b l e 8.3 G r o u t e d P r i s m s , T e s t s b y the A u t h o r  fmg  — E x p e r i m e n t a l v a l u e of p r i s m s t r e n g t h  a — T h e o r e t i c a l p r e d i c t i o n of p r i s m s t r e n g t h b y f a i l u r e c o n d i t i o n a) b — T h e o r e t i c a l p r e d i c t i o n o f p r i s m strength b y failure c o n d i t i o n b) c — T h e o r e t i c a l p r e d i c t i o n of p r i s m s t r e n g t h b y failure c o n d i t i o n c) ( A l l i n k s i , same for the f o l l o w i n g tables)  145  0.62  n  A  a  fmg  b  c  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 s m s , T e s t s b y H a m i d a n d D r y s d a l e  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 r o u t e d P r i s m s , T e s t s b y D r y s d a l e H a m i d  V  fu  0.51  2.78  0.51  2.78  u  a  b  c  2.16  1.75  2.45  2.35  2.10  1.75  2.45  2.35  /,  fmg  2.72  4.93  2.72  4.93  T a b l e 8.6 G r o u t e d P r i s m s , T e s t s b y W o n g a n d D r y s d a l e  147  a  b  c  3.91  2.43  3.55  3.47  4.03  3.77  2.43  3.55  3.47  2.24  4.03  3.93  2.43  3.55  3.47  1.70  5.30  3.90  2.54  3.70  3.59  fmg  fu  h  0.55  5.54  2.24  4.03  0.55  5.54  2.24  0.55  5.54  0.61  5.54  V  T a b l e 8.7 G r o u t e d P r i s m s , T e s t s b y P r i e s t l e y a n d E l d e r  a  b  c  2.61  2.28  3.34  3.07.  2.25  2.77  2.28  3.34  3.07  2.20  2.25  2.99  2.28  3.34  3.07  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  a  b  c  1  fmg  fu  1  0.55  5.80  2.20  2.25  0.55  5.80  2.20  0.55  5.80  0.48  T a b l e 8.8 G r o u t e d P r i s m s , T e s t s b y B o u l t  fmg  V 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 , T e s t s b y 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 P r e d i c t i o n versus E x p e r i m e n t s , Based o n F a i l u r e 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 P r e d i c t i o n versus E x p e r i m e n t s , Based o n F a i l u r e C o n d i t i o n b)  149 Therefore, i t is not s u r p r i s i n g t o see t h a t the p r e d i c t i o n based on c o n d i t i o n b) appears t o overestimate  the masonry  c a p a c i t y i n t h e lower  strength  range,  since i f m a s o n r y  fails i n  c o n d i t i o n a ) , i.e. as t h e result o f the p r e m a t u r e failure o f b l o c k s h e l l , i t w i l l l e a d t o a lower failure load. It is clear t h a t a l t h o u g h we c o u l d use the n u m e r a t o r o f E q . 8.15 d i r e c t l y t o e s t i m a t e the grouted  masonry  r a t i o , some  c a p a c i t y based o n u n i t s t r e n g t h , g r o u t s t r e n g t h , m o r t a r  strength a n d area  d i s c r e p a n c y s h o u l d be expected since o c c a s i o n a l l y f a i l u r e c o n d i t i o n s other  c o n d i t i o n b ) m a y o c c u r . M o r e o v e r , i t is n o t desirable i n p r a c t i c e t o o v e r e s t i m a t e the  than  masonry  c a p a c i t y . Therefore the e q u a t i o n m a y need e m p i r i c a l m o d i f i c a t i o n . One  m o d i f i c a t i o n m a y be t o adjust  t h e coefficient i n t h e e q u a t i o n  t o best f i t t h e  a v a i l a b l e e x p e r i m e n t a l d a t a . S u b s t i t u t i n g E q s . 8.16-8.18 a n d neglecting s m a l l q u a n t i t i e s , t h e n u m e r a t o r o f E q . 8.15 m a y be e x p a n d e d i n the f o r m o f  fmg = Anfu + B(\-r))\fgf  u  + Cnfj + D  (in ksi)  8.19  A m u l t i p l e regression a n a l y s i s o f 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 b o u n d given b y E q . 8.12 m a y be used t o give a n e s t i m a t e for the u l t i m a t e c a p a c i t y o f g r o u t e d m a s o n r y . T h i s e s t i m a t e i s also l i s t e d i n T a b l e 8.3 — 8.9 a n d p l o t t e d i n F i g . 8.8 versus the d a t a base. T h e r e l a t i o n has a c o r r e l a t i o n coefficient o f 0.934, w h i c h is s i g n i f i c a n t . H o w e v e r , the d a t a used t o evaluate p a r a m e t e r s c e r t a i n l y d o not a l l refer t o failure  150 c o n d i t i o n b ) , a n d the c o r r e l a t i o n is affected by a d d i t i o n a l uncertainties s u c h as and  test m e t h o d ;  and  this is p r o b a b l y  why  workmanship  a n u m b e r of p o i n t s f a l l outside the 99  percent  confidence l i m i t (see F i g . 8.8). T h e m o d e l e q u a t i o n clearly reflects the fact t h a t m a s o n r y c a p a c i t y is not very sensitive to the g r o u t s t r e n g t h , as observed b y D r y s d a l e a n d H a m i d (1979) a n d by the a u t h o r (see T a b l e s 8.3, 8.5). T h e m a s o n r y s t r e n g t h is better correlated w i t h the square root of the g r o u t s t r e n g t h , based o n the d e f o r m a t i o n  c o m p a t i b i l i t y . Indeed, l i n e a r regression o n the basis of E q . 8.19  w h i c h \ f fu  g  g  is replaced by f ,  in  a f o r m often seen i n l i t e r a t u r e , i n d i c a t e s t h a t it leads not o n l y to a  lower c o r r e l a t i o n coefficient of 0.907 b u t also to a m u c h higher D v a l u e , w h i c h is not desirable. T h e a b o v e a n a l y s i s suggests t h a t E q . 8.15 m a y be used to e s t i m a t e the c r a c k i n g l o a d of concrete g r o u t e d  masonry.  Unfortunately,  no  experimental  data  are  available  in  terms  of  c r a c k i n g loads except those recorded b y the a u t h o r . F o r these very l i m i t e d d a t a , the c o m p a r i s o n is listed i n T a b l e 8.10 a n d p l o t t e d i n F i g . 8.9, i n w h i c h E q . 4.16 is scaled d o w n by a factor of 0.92. T h e c r a c k i n g loads for the specimens w i t h v a r y i n g j o i n t thickness are also i n c l u d e d . E x c e p t for t w o d a t a p o i n t s (S8) the agreement is reasonable, c o n s i d e r i n g the c r a c k i n g l o a d is a rather r a n d o m event. T h e c o r r e l a t i o n coefficient for t h i s case is 0.618, w h i l e p r e d i c t i o n of f a i l u r e loads it is 0.563, i n d i c a t i n g t h a t the l o a d e s t i m a t e d by E q . 8.15 is indeed m o r e closely correlated w i t h the c r a c k i n g l o a d t h a n w i t h the u l t i m a t e l o a d . O n e p r a c t i c a l i m p l i c a t i o n of the above s t u d y is t h a t one s h o u l d consider the c r a c k i n g l o a d e s t i m a t e d by E q . 8.15 as the lower l i m i t l o a d i n design, since block shell c r a c k i n g is, i n any case, not a desired event under n o r m a l service c o n d i t i o n s . T h i s u s u a l l y c a n be a c h i e v e d , since, i n most small masonry  b u i l d i n g s , the a x i a l l o a d levels are l o w a n d therefore the a c t u a l value of  a l l o w a b l e stress is not c r i t i c a l . (It  is noted t h a t the c r a c k i n g l o a d e s t i m a t e d by E q . 8.15 is  a r o u n d 7 0 % of the u l t i m a t e l o a d e s t i m a t e d by E q . 8.19 or by the n u m e r a t o r  of E q . 8.15.)  H o w e v e r , the u l t i m a t e load e s t i m a t e d by the m o d e l can be used as the final l i m i t load  under  severe service c o n d i t i o n s . F o r e x a m p l e , under e a r t h q u a k e l o a d i n g , the a x i a l c a p a c i t y of m a s o n r y  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  1.2  M o d e l P r e d i c t i o n versus E x p e r i m e n t s , B a s e d o n M o d i f i e d  1.4  1.6  1.8  PREDICTION BASED ON CONDITION a)  F I G . 8.9  2  2.2  Equation  2.4  (KSI)  M o d e l P r e d i c t i o n versus C r a c k i n g L o a d s , Based o n F a i l u r e C o n d i t i o n a)  152 c a n b e c o m e c r i t i c a l not o n l y because of the d u c t i l i t y r e q u i r e m e n t b u t also because of the i n e r t i a force itself. O n e m a y t h e n t a k e a d v a n t a g e of the higher u l t i m a t e s t r e n g t h b y a l l o w i n g a higher a l l o w a b l e stress based o n  Eq.  8.19.  This  is e c o n o m i c a l a n d  c e r t a i n l y agrees w i t h the risk  p h i l o s o p h y c o m m o n l y a d o p t e d i n the e a r t h q u a k e engineering design, t h a t some d a m a g e ,  even  s t r u c t u r a l d a m a g e , is a c c e p t a b l e i n the m a j o r event, b u t n o t c o l l a p s e . 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 p r e d i c t o r of c r a c k i n g l o a d s needs f u r t h e r i n v e s t i g a t i o n . M a n y m o r e e x p e r i m e n t a l d a t a are required i n t h i s respect.  8.4  Summary In t h i s c h a p t e r , the a x i a l b e h a v i o u r of g r o u t e d concrete m a s o n r y w i t h f u l l b e d d i n g has  been i n v e s t i g a t e d . T h r e e possible f a i l u r e c o n d i t i o n s have been s t u d i e d . A f a i l u r e m o d e l based o n i n t e r n a l d e f o r m a t i o n c o m p a t i b i l i t i e s has been p r o p o s e d .  SPECIMEN  ult. load(kips)  crk. l o a d ( k i p s )  fmg  (ksi)  /c*(ksi)  P r e d i c t i o n (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  M9-1  333.0  250.0  2.77  2.08  1.99  M9-2  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  Pll-1  302.0  190.0  2.51  1.58  1.70  Pll-2  300.0  208.0  2.49  1.73  1.70  T a b l e 8.10 M o d e l P r e d i c t i o n versus C r a c k i n g L o a d s , T e s t s b y the A u t h o r  153 CHAPTER GROUTED  It is clear b y  IX  MASONRY WITH FACE-SHELL BEDDING  the a n a l y s i s i n C h a p t e r V , t h a t under  u n i a x i a l c o m p r e s s i o n face-shell  bedded m a s o n r y w i l l f a i l p r e m a t u r e l y b y a deep b e a m m e c h a n i s m at a l o w l o a d . W h e n face-shell bedded m a s o n r y is g r o u t e d , the deep b e a m b e n d i n g m e c h a n i s m w i l l s t i l l be a c t i v a t e d as the v e r t i c a l force w i l l be shared b y the b l o c k shell a n d grout. T h i s was s h o w n b y the e x p e r i m e n t s c o n d u c t e d b y the a u t h o r (see F i g . 9.1). T h e webs of the f a c e - s h e l l bedded a n d g r o u t e d p r i s m s c r a c k e d v e r t i c a l l y at a very l o w l o a d o w i n g to t h i s m e c h a n i s m . T h e v e r t i c a l s t r a i n i n the block shell drops m u c h faster t h a n t h a t of the f u l l y bedded c o u n t e r p a r t s , i m p l y i n g the h i n g i n g m e c h a n i s m of the block shell s t u d i e d i n C h a p t e r V . H o w e v e r , the a u t h o r ' s tests i n d i c a t e t h a t the c r a c k i n g of the b l o c k shell due to the b e a m b e n d i n g m e c h a n i s m w i l l not lead to u l t i m a t e f a i l u r e of the m a s o n r y if the r e s i d u a l c a p a c i t y of the g r o u t is greater t h a n the c r a c k i n g l o a d . T h u s we m a y use  fm,  =  ( 1 -  9.1  V ) fa  as a lower b o u n d or as a conservative e s t i m a t e of the g r o u t e d m a s o n r y compressive s t r e n g t h . F o r the a u t h o r ' s tests, E q . 9.1 underestimates the p r i s m c a p a c i t y b y a b o u t  1 0 % , as  s h o w n i n T a b l e 9.1, i n d i c a t i n g a very l o w g r o u t i n g efficiency. A t failure, the l o a d was o n l y effectively s u s t a i n e d b y the grout, as i m p l i e d b y the d e f o r m a t i o n m e a s u r e m e n t (see F i g . 9.1). E q . 9.1 underestimates the f a i l u r e loads of the p r i s m s tested b y D r y s d a l e a n d  Hamid  (1983) b y a larger m a r g i n , i n d i c a t i n g a higher g r o u t i n g efficiency i n their specimens. H o w e v e r , it seems reasonable i n p r a c t i c a l design to neglect the c a p a c i t y of the block shell since this m a y not be a r e l i a b l e q u a n t i t y i n view o f the b e a m b e n d i n g m e c h a n i s m .  154  A)  B)  SPECIMEN  V  N17  0.51  3.70  252  2.09  1.81  N-GROUT  0.51  3.70  240  2.00  1.81  N-MORTAR  0.51  3.70  258  2.14  1.81  fg (ksi)  P  (kips)  fmg (ksi)  (l-ri)fg  NB  GN  0.56  3.06  123  2.09  1.36  NB  GW  0.56  1.99  121  2.05  0.88  NB  GS  0.56  5.94  131  2.22  2.64  WB  GN  0.56  3.06  93.5  1.59  1.36  SB  GN  0.56  3.06  128  2.18  1.36  QB  GN  0.75  3.06  124  2.10  0.76  6"B  GN  0.51  3.06  86.6  1.99  1.50  10"B  GN  0.54  3.06  123  1.65  1.41  T a b l e 9.1 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  A)  - T e s t s by the a u t h o r  B)  - T e s t s b y D r y s d a l e a n d H a m i d (1983)  T h e p r o b l e m t h a t r e m a i n s unanswered is whether the c r a c k i n g l o a d s h o u l d be used to govern the design. If so, m o r e e x p e r i m e n t a l w o r k is needed a n d m o r e a t t e n t i o n s h o u l d directed to t h i s value, since there have so far been few e x p e r i m e n t s m o n i t o r i n g  be  premature  cracking. A c c o r d i n g to the a u t h o r ' s tests, face-shell bedded, grouted m a s o n r y  has a very  low  g r o u t i n g efficiency, w h i c h m a y be even lower i n terms of the c r a c k i n g loads. T h i s is because the t w o c o n s t i t u e n t s d o not w o r k together properly. It appears t h a t i n the early stages of l o a d i n g , the b l o c k u n i t takes a b i g share of the load as i m p l i e d b y the v e r t i c a l s t r a i n m e a s u r e m e n t s (cf. F i g . 9.1 a n d F i g . 5.3, l o c a t i o n s 5 a n d 6). H o w e v e r , after the b l o c k shell is c r a c k e d , a l m o s t the w h o l e l o a d is passed to the grout. T h i s is not desirable f r o m a s t r u c t u r a l p o i n t of v i e w . It is clear t h a t for grouted m a s o n r y f u l l b e d d i n g is r e c o m m e n d e d , a l t h o u g h , as i n d i c a t e d  155 above, i n p r a c t i c e t h a t effective f u l l b e d d i n g is s o m e t i m e s d i f f i c u l t to achieve because of the web alignment.  It is also o b v i o u s t h a t the d e f o r m a t i o n  properties of the t w o m a t e r i a l s p l a y  i m p o r t a n t role. L o w g r o u t i n g efficiency is i n e v i t a b l e unless there is a f u n d a m e n t a l  an  improvement  i n m a t e r i a l design s u c h t h a t the d e f o r m a t i o n properties o f g r o u t a n d u n i t are m o r e c o m p a t i b l e .  157 CHAPTER GROUTED  AND REINFORCED  X  MASONRY UNDER ECCENTRIC  LOADING  10.1 G e n e r a l R e m a r k s P r o b a b l y the biggest a d v a n t a g e of concrete m a s o n r y over t r a d i t i o n a l b r i c k w o r k is t h a t the concrete b l o c k u n i t s are h o l l o w a n d can thus be v e r t i c a l l y reinforced to i m p r o v e the b e n d i n g c a p a c i t y . B e n d i n g c a p a c i t y is essential for w a l l s designed to s u s t a i n eccentric l o a d or v e r t i c a l force c o m b i n e d  with  laterally distributed  pressure.  This  is o b v i o u s  since t h e o r e t i c a l l y  the  c a p a c i t y o f p l a i n b r i c k w o r k w i l l be d r a s t i c a l l y reduced if the l o a d falls outside the k e r n , a n d the wall  c a n not  reinforcement,  support  any  the i m p r o v e d  load  when  the  e c c e n t r i c i t y reaches h a l f  the  b e n d i n g c a p a c i t y enables m o d e r n m a s o n r y  wall depth.  s t r u c t u r e s to  With become  t a l l e r a n d t h i n n e r , w h i l e preserving the t r a d i t i o n a l b e a u t y of these s t r u c t u r e s . Therefore, e c c e n t r i c a l l y loaded reinforced concrete m a s o n r y , w h i c h m u s t be g r o u t e d , is of interest. I n t h i s c h a p t e r , e x p e r i m e n t a l observations are first reviewed a n d the f i n d i n g s i n the preceding chapters are p l a c e d i n t h i s c o n t e x t . S o m e useful relations w i l l then be developed.  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 T o s t u d y the basic b e h a v i o u r bending,  12 g r o u t e d p r i s m s ( w i t h o u t  of reinforced concrete m a s o n r y reinforcement)  under c o m p r e s s i o n  and  were tested under eccentric l o a d i n g .  The  f a i l u r e loads of these specimens are l i s t e d i n T a b l e 10.1 a n d the d e f o r m a t i o n m e a s u r e m e n t s are p l o t t e d i n F i g s . 10.1 a n d 10.2. T h e failure process was recorded b y a v i d e o c a m e r a for better observation. T h e e x p e r i m e n t s i n d i c a t e t h a t under eccentric l o a d , the j o i n t c o n d i t i o n a n d  grouting  c o n d i t i o n do not have a s i g n i f i c a n t influence o n the m a s o n r y c a p a c i t y ( c o m p a r e also the f a i l u r e loads o f p l a i n u n g r o u t e d concrete m a s o n r y under eccentric l o a d i n T a b l e 6.1). T h i s is expected since the force shared b y the g r o u t d i m i n i s h e s w i t h increasing e c c e n t r i c i t y . In other words, the  158 SPECIMEN  e/t  1  2  3  4  AVG  COV  N26 ( N J . N G )  1/6  178.0  196.0  164.0  200.0  184.5  7.8%  M26 ( M J , N G ) 1/3  106.0  92.0  82.0  128.5  102.1  17.1%  1/3  108.0  93.0  101.0  127.0  107.3  11.7%  S25 ( S J . N G )  NG -  Type N Grout;  NJ -  T y p e N M o r t a r J o i n t , etc.  T a b l e 10.1 F a i l u r e L o a d s o f G r o u t e d P r i s m s under E c c e n t r i c L o a d (kips)  c a p a c i t y of e c c e n t r i c a l l y loaded m a s o n r y is even m o r e s t r o n g l y governed b y the c a p a c i t y of the b l o c k shell. T h e f a i l u r e modes a g a i n were c h a r a c t e r i z e d b y shear, i.e b y s p a l l i n g m i x e d w i t h c r u s h i n g i  of the b l o c k s h e l l o n the loaded side, as s h o w n i n F i g . 2.16. T h i s p h e n o m e n o n was m o r e o b v i o u s for the specimens u n d e r larger e c c e n t r i c i t y (e =  i/3).  T h e g r o u t d i d not prevent the d e b o n d i n g o f the m o r t a r j o i n t s o n the u n l o a d e d side, as i n d i c a t e d b y the s u b s t a n t i a l d e f o r m a t i o n m e a s u r e d across the j o i n t ( L V D T # 6 )  for the case of  e = i / 3 , a l t h o u g h i t appears t h a t the v e r t i c a l c o n t i n u i t y was i m p r o v e d b y the g r o u t i n g as the o p e n i n g of the j o i n t s was s m a l l e r c o m p a r e d w i t h t h e i r u n g r o u t e d c o u n t e r p a r t s . T h e face-shell o n the u n l o a d e d side d i d not transfer l o a d essentially for the whole l o a d i n g range, as s h o w n by the s t r a i n m e a s u r e d at l o c a t i o n 5, i n d i c a t i n g t h a t d e b o n d i n g took place as soon as the s p e c i m e n was loaded. Before f i n a l f a i l u r e , no p r e m a t u r e v e r t i c a l cracks were observed d u r i n g the tests (see also the d e f o r m a t i o n m e a s u r e m e n t s at l o c a t i o n s 3 a n d 4 as s h o w n i n F i g s . 10.1 a n d 10.2), w h i c h is i n s h a r p c o n t r a s t to w h a t was observed for the p r i s m s under u n i a x i a l l o a d i n g , suggesting t h a t the cross-sectional i n c o m p a t i b i l i t y is not a p r o b l e m for g r o u t e d m a s o n r y under eccentric l o a d i n g . T h i s is a n o t h e r s u p p o r t i n g i n d i c a t i o n t h a t the c o n t r i b u t i o n of the grout  to the c a p a c i t y is  F I G . 10.1 M e a s u r e d D e f o r m a t i o n s at C e r t a i n L o c a t i o n s of G r o u t e d P r i s m s under E c c e n t r i c C o m p r e s s i o n : a) N 2 6 - 1 , e = t / 6 ; b) N 2 6 - 2 , e = t / 6 ; c) M 2 6 - 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 AVERAGE STRAN ( 1/1000 N/N )  9  11  13  AVERAGE STRAN( 1/1000N/N) F I G . 10.2  Measured D e f o r m a t i o n s at C e r t a i n L o c a t i o n s of G r o u t e d P r i s m s under E c c e n t r i c C o m p r e s s i o n : a) M 2 6 - 3 , e = t / 3 ; b) S 2 5 - 1 , e = t / 3 ; c) S 2 5 - 1 , e = t / 3  161 r e l a t i v e l y m i n o r w h e n the m a s o n r y is under eccentric l o a d i n g . T h e s e observations are essentially the same as those for p l a i n concrete m a s o n r y .  This  encourages us to a p p r o a c h the p r o b l e m as we d i d for p l a i n concrete m a s o n r y under eccentric l o a d i n g . T h a t is, f a i l u r e is a s s u m e d to be governed b y the b l o c k shell a n d c a p a c i t y e s t i m a t i o n is based o n the u n i t s t r e n g t h r a t h e r t h a n o n the u n i a x i a l p r i s m s t r e n g t h . T h e force shared b y the g r o u t at f a i l u r e is e s t i m a t e d b y c o n s i d e r i n g the v e r t i c a l d e f o r m a t i o n c o m p a t i b i l i t y .  10.3 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 T h e c a p a c i t y o f reinforced concrete m a s o n r y under eccentric l o a d w i l l be expressed here i n t e r m s of the t r a d i t i o n a l f o r c e - m o m e n t  curve. S u c h a r e l a t i o n s h i p depends not o n l y on the  m a t e r i a l properties of the m a s o n r y c o n s t i t u e n t s , i n c l u d i n g b l o c k u n i t , g r o u t , r e i n f o r c i n g steel a n d m o r t a r , b u t also o n i t s geometry, w h i c h is further c o m p l i c a t e d b y 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 combinations. To  make  the  situation simpler,  properties, the geometry,  attempts  will  be  made  to  quantify  the  material  the 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 b y some p a r a m e t e r s , expressed  m a i n l y i n t e r m s of the m o d u l u s a n d d i m e n s i o n a l ratios. T h e usefulness of s u c h p a r a m e t e r s w i l l be i l l u s t r a t e d . F o r e x a m p l e , i f l i n e a r - e l a s t i c b e h a v i o u r is a s s u m e d , the forces shared b y the b l o c k shell, g r o u t , a n d the r e i n f o r c i n g steel c a n be c a l c u l a t e d based on d e f o r m a t i o n m o d u l u s ratios. A linear s t r e s s - s t r a i n r e l a t i o n s h i p m a y be a good a p p r o x i m a t i o n for concrete m a s o n r y as i n d i c a t e d b y v a r i o u s e x p e r i m e n t s , i n c l u d i n g those b y the a u t h o r , w h i c h show t h a t n o n l i n e a r i t y before f a i l u r e appears to be rather l i m i t e d . T h e analyses i n the preceding chapters based o n t h i s a s s u m p t i o n d o y i e l d reasonable e s t i m a t i o n s for the m a s o n r y c a p a c i t y . Further,  if linear strain along  the cross-section (plane sections r e m a i n i n g  plane) is  a s s u m e d , the i n t e r n a l force P a n d m o m e n t M c a n be r e a d i l y expressed i n t e r m s of the outer stresses a , x  <r  2  o r  *  n e  fibre  c r a c k d e p t h c (depending whether the cross-section is c r a c k e d or n o t ) , as  162 s h o w n i n F i g . 10.3. N o t e , i n the f o l l o w i n g expressions, the c o n t r i b u t i o n of the v e r t i c a l reinforcement,  which  p l a y s a n i m p o r t a n t role when the cross-section is c r a c k e d , is i n c l u d e d . T h i s has been neglected i n the a n a l y s i s for g r o u t e d m a s o n r y under u n i a x i a l c o m p r e s s i o n , since the c o n t r i b u t i o n is u n r e l i a b l e unless the steel is t i e d against b u c k l i n g . M o r e o v e r , for n o r m a l steel r a t i o s , the c o n t r i b u t i o n  in  s u s t a i n i n g compressive force is s m a l l c o m p a r e d to the concrete m a t e r i a l s , even it is i n c l u d e d . T h i s is especially true for the case of eccentric l o a d i n g . H o w e v e r , i n the f o l l o w i n g expressions, the force shared by the reinforcement  w i l l be i n c l u d e d for c o n t i n u i t y . T h e r e i n f o r c i n g steel is  a s s u m e d to be p l a c e d i n the m i d d l e of the cross-section as is the c o m m o n p r a c t i c e . W h e n the e c c e n t r i c i t y e is s m a l l , the cross-section r e m a i n s u n c r a c k e d , so the force a n d the m o m e n t c a n be expressed as (see F i g . 10.3)  10.1  10.2  where 6 is the h a l f thickness ( 6 = t / 2 ) of the m a s o n r y , b l o c k u n i t ; a n d / is the l e n g t h of the w a l l . a  x  maximum  a is the h a l f w i d t h of the inner core of  and a, 2  as have been m e n t i o n e d ,  a n d m i n i m u m e x t r e m e fiber stresses ( i n compression)  denote  the  respectively, p represents the  steel r a t i o w i t h respect to the gross cross-sectional area, a n d n stands for the m o d u l a r r a t i o ; i.e. the elastic m o d u l u s of steel to t h a t of the m a s o n r y block shell. T h e p a r a m e t e r A is i n t r o d u c e d here to characterize the g r o u t i n g , b e d d i n g c o n d i t i o n s a n d cross-sectional geometry i n the transverse d i r e c t i o n :  A  =  J_Wg m  g  i  10.3  163 where w a n d w  g  are the s u m of the m o r t a r e d web 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 along the w a l l  length, respectively (see F i g . 10.4). m  is the m o d u l a r r a t i o : the elastic m o d u l u s o f the b l o c k  g  shell to t h a t of the grout, a p p r o x i m a t e d as:  m, 1 + m  2  R e c a l l that m j a n d m  2  10.4  U h  0  are the m o d u l a r ratios o f u n i t t o g r o u t , a n d u n i t t o m o r t a r j o i n t ,  respectively. h is the height of the m a s o n r y unit a n d t is the thickness of the m o r t a r j o i n t . 0  0  T h u s whether the m a s o n r y is f u l l y bedded o r not, a n d whether i t i s p l a i n or f u l l y o r p a r t i a l l y grouted, c a n be expressed t h r o u g h the p a r a m e t e r A. F o r e x a m p l e , A = 0 corresponds to the case of a s o l i d section; w  g  = 0, A = ( 1 — w/l ) stands for the case of u n g r o u t e d m a s o n r y ;  s i m i l a r l y , w = 0, A = ( 1 — w /m l) is for the face-shell bedded m a s o n r y ; A = 1, when w = g  g  w  2a 2b Uncracked Section  2b P  =  Cracked Section  2b-c  2b a dx  .  M  =  xcr d x  P  =  | 0  2b-c cr d x  M =  |  xcr d x  0  F I G . 10.3 A s s u m e d Stress D i s t r i b u t i o n of an U n c r a c k e d S e c t i o n a n d a C r a c k e d Section  g  F I G . 10.5  Stress D i s t r i b u t i o n along a Section a n d Its C o m p o s i t i o n  165 = 0, o f course, represents the case of the face-shell bedded, u n g r o u t e d m a s o n r y . B y t h i s means, a l l the c o m b i n a t i o n s c a n be i n c l u d e d a n d the relations d e r i v e d here are generally useful; they have, i n c i d e n t a l l y , been a p p l i e d i n C h a p t e r V I for p l a i n concrete m a s o n r y . E q . 1 0 . 1 a n d E q . 1 0 . 2 are o b t a i n e d based o n the p r i n c i p l e of s u p e r p o s i t i o n . D u e to the difference i n d e f o r m a t i o n m o d u l u s of the m a s o n r y b l o c k shell, grout, a n d r e i n f o r c i n g steel, the stress d i s t r i b u t i o n a l o n g the cross-section m u s t  be d i s c o n t i n u o u s at the b o u n d a r i e s  of these  m a t e r i a l s as d e p i c t e d i n F i g . 10.5(a). T h i s stress d i s t r i b u t i o n , i n a n average sense a l o n g the w a l l l e n g t h , c a n be d e c o m p o s e d i n t o the stress d i s t r i b u t i o n s as s h o w n i n F i g . 10.5(b), (c) arid  (d),  where d i s t r i b u t i o n (b) corresponds to a s o l i d section a n d d i s t r i b u t i o n (c) represents the difference i n stress d i s t r i b u t i o n s between a s o l i d section a n d a g r o u t e d section. T h e p o i n t force d e p i c t e d i n F i g . 10.5(d), o f course, s t a n d s for the c o n t r i b u t i o n of the r e i n f o r c i n g steel. C l e a r l y , d i s t r i b u t i o n (c) is w e i g h t e d by p a r a m e t e r A a n d d i s t r i b u t i o n (d) b y np. the cross-section f a c t o r a/b  These p a r a m e t e r s are c o m b i n e d w i t h  i n E q . 10.1 a n d E q . 10.2.  T h e s a m e p r i n c i p l e is used i n the d e r i v a t i o n of the f o l l o w i n g equations. If the tensile resistance of the cross-section is neglected, the cross-section w i l l crack (by o b s e r v a t i o n , cracks a l w a y s occur at the m o r t a r j o i n t s , see F i g . 10.1 a n d F i g . 10.2) w h e n c o m p r e s s i o n ) . It c a n be s h o w n t h a t for 0 < c <  cr /o- < 2  1  0 (positive for  b — a  10.5  M :  (2 -  c/b)  {(i- + (x) + (x))- + ^)} 2 +  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  2(2 -  2  -  c/b) c/b)  +  2np  12-  c/b c/b  10.7  166  M  =  1+2  F i n a l l y , for 6 +  a <  t-x  10.8  c < 26  10.9  10.10  A g a i n , i f the m a s o n r y concrete m a s o n r y  u n i t strength is used to define the c r i t i c a l state, as for p l a i n  under eccentric l o a d i n g , we r e a d i l y o b t a i n the short w a l l c a p a c i t y curve by  l e t t i n g the e x t r e m e fibre stress <r  1  P—M  be equal to f , u  the u n i t compressive s t r e n g t h . T h a t is, the  curve c a n be developed b y v a r y i n g cy l<>'\ f r o m u n i t y to zero, w h e n <r 2  2  >  0; a n d  by  s t e p p i n g c f r o m 0 to 26, when the cross-section has c r a c k e d . N o t e t h a t i n the above expressions, the r e i n f o r c i n g steel is i m p l i c i t l y assumed not  to  reach its y i e l d s t r e n g t h . B y e x p e r i m e n t a l observations, we k n o w t h a t the c o m p r e s s i v e failure s t r a i n for concrete m a s o n r y is u s u a l l y s m a l l (less t h a n 0.002), so t h a t t h i s a s s u m p t i o n m a y be good as l o n g as t h e e c c e n t r i c i t y e is not  too s m a l l . F o r  c o n c e n t r i c loads, E q .  10.1  may  o v e r e s t i m a t e the f a i l u r e l o a d , because the steel c o u l d y i e l d . H o w e v e r , t h i s p a r t of the c a p a c i t y curve is not o f interest here since the c o n c e n t r i c c a p a c i t y is treated separately, as i n C h a p t e r  VII  a n d VIII. M o r e o v e r , as m e n t i o n e d earlier, the c o n t r i b u t i o n to the compressive c a p a c i t y of the r e i n f o r c i n g steel is u s u a l l y s m a l l c o m p a r e d to t h a t of the s u r r o u n d i n g cross-section. H o w e v e r , i f the crack extends b e y o n d the h a l f d e p t h o f the w a l l , the r e i n f o r c i n g steel m a y y i e l d i n t e n s i o n . T h i s m a y h a p p e n w h e n the crack d e p t h at the b a l a n c e d l o a d  167  10.11 1 + nfu  is less t h a n t h a t c o r r e s p o n d i n g to the pure m o m e n t c a p a c i t y (the c w h i c h m a k e s E q . 10.7 or E q . 10.9 v a n i s h ) ; f  here is the y i e l d s t r e n g t h o f the steel. A l t h o u g h y i e l d i n g of the steel is not  y  desirable a n d is not a l l o w e d i n the c u r r e n t design code, for a n a l y s i s , we m a y replace the t e r m 2 n p ( l — c / 6 ) / ( 2 — c/b)  i n E q . 10.7 or E q . 10.9 b y 2 p / / < x to i n c l u d e t h i s s i t u a t i o n . !/  1  10.4 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 In s u m m a r y , parameters: f for  the c a p a c i t y curve for concrete m a s o n r y m , np,  , fy,  u  2  a, b, I, w,  p r a c t i c a l reasons, the m o d u l u s  w  g  a n d t /h ,  r a t i o s TOJ a n d  0  w h i c h c a n a l l be m e a s u r e d .  0  m  may  2  is d e t e r m i n e d b y the f o l l o w i n g  be related to the  However,  corresponding  s t r e n g t h r a t i o s . In the f o l l o w i n g c o m p a r i s o n , the same square root c o r r e l a t i o n is used as i n Chapter VII. T h e P—M  curves generated for the a u t h o r ' s specimens a n d those tested b y D r y s d a l e a n d  H a m i d (1983) are p l o t t e d i n F i g . 10.6 to F i g 10.15 w i t h the e x p e r i m e n t a l d a t a . T h e p l o t is nondimensionalized moments by M  k  by  = P i/6, 0  dividing  vertical load by  P =/ tl, 0  U  the m o m e n t c a p a c i t y w h e n P  0  the n o m i n a l  axial capacity;  and  is a p p l i e d at the k e r n e c c e n t r i c i t y of a  solid section. F o r m o s t cases, the agreement is reasonably good. F o r the e x p e r i m e n t s b y D r y s d a l e a n d H a m i d , the curves appear to u n d e r e s t i m a t e the b e n d i n g c a p a c i t y of the specimens tested at the biggest e c c e n t r i c i t y c o n s i s t e n t l y , a l t h o u g h b y a s m a l l a m o u n t . T h i s is p r o b a b l y caused b y the a s s u m p t i o n t h a t the cross-section does not resist a n y tensile force, w h i c h is closer to r e a l i t y for p l a i n m a s o n r y t h a n for g r o u t e d  masonry.  N o efforts are m a d e here to c o m p a r e the results n u m e r i c a l l y , since a n u m b e r o f the d a t a  168 p o i n t s are o b t a i n e d at large e c c e n t r i c i t y w h e n the l o a d i n g p a t h M=eP  is a l m o s t p a r a l l e l to the  lower b o u n d a r i e s o f the c a p a c i t y curves. In t h i s s i t u a t i o n s m a l l e x p e r i m e n t a l errors c a n lead to large n u m e r i c a l v a r i a t i o n s i n l o a d or m o m e n t . T h e m o d e l based o n linear s t r a i n a n d stress appears to give reasonable p r e d i c t i o n s . T h e c o m p a r i s o n a g a i n s u p p o r t s the a s s u m p t i o n t h a t the c a p a c i t y of e c c e n t r i c a l l y l o a d e d m a s o n r y is m o r e closely c o r r e l a t e d w i t h the u n i t  strength t h a n  w i t h the c o n c e n t r i c c a p a c i t y . T h u s  in  p r a c t i c a l design, i t m a y be a g a i n r e c o m m e n d e d t h a t the c o n c e n t r i c c a p a c i t y a n d the eccentric c a p a c i t y be treated separately, as i n the case of p l a i n m a s o n r y . T h e expressions developed here p r o v i d e convenient w a y s to e s t i m a t e the eccentric c a p a c i t y o f concrete m a s o n r y grouting and bedding conditions.  w i t h various  F I G . 10.7 C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s by Drysdale and H a m i d : N o r m a l Block, T y p e N G r o u t  170 0.7  0.6  0.5  -  0.4  -  0. CL  0.3  0.2  -  0.1  -  0.4  0.6  M/M<  F I G . 10.8 C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s by D r y s d a l e and H a m i d : N o r m a l B l o c k , T y p e W  Grout  0.9  a.  0.3  0.2  0.4 M/Mc  F I G . 10.9 C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s by D r y s d a l e 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  0.6  0  0.2  0.4  0.6  M/Mt  F I G . 10.10  C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s by D r y s d a l e a n d H a m i d : W e a k B l o c k , T y p e N  Grout  0 . 8 -|  0  F I G . 10.11  0.2  0.4  0.6  C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s by Drysdale a n d H a m i d : S t r o n g B l o c k , T y p e N  Grout  172 0.8  0  0.2  0.4  .  0.6  M/Mk  F I G . 10.12 C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s by D r y s d a l e a n d H a m i d : 7 5 % S o l i d B l o c k , T y p e N G r o u t  a.  0  0.2  0.4 M/M<  F I G . 10.13 C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s by D r y s d a l e a n d H a m i d : F u l l B l o c k  0.6  0  0.2  0.4  0.6  F I G . 10.14 C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s b y D r y s d a l e and H a m i d : 6 i n c h B l o c k , T y p e N G r o u t  0 . 8 -,  0  :  0.2  0.4  M/Mk F I G . 10.15 C o m p a r i s o n of P r e d i c t e d I n t e r a c t i o n C u r v e w i t h E x p e r i m e n t s b y D r y s d a l e a n d H a m i d : 10 i n c h B l o c k , T y p e N G r o u t  174 CHAPTER  XI  SLENDERNESS OF C O N C R E T E  MASONRY  11.1 I n t r o d u c t i o n M o d e r n m a s o n r y structures are b e c o m i n g t a l l e r , not o n l y i n t e r m s of the e l e v a t i o n of the b u i l d i n g , b u t also i n t e r m s o f storey heights. Besides advances i n engineering k n o w l e d g e , changes i n the m a s o n r y  c o n s t i t u e n t s have c o n t r i b u t e d  to t h i s d e v e l o p m e n t .  Structural behaviour  is  greatly i m p r o v e d b y h i g h strength concrete u n i t s w i t h steel reinforcement. T a l l , slender concrete m a s o n r y c a n be seen i n m a n y places, such as a p a r t m e n t highrises, department building  stores, warehouses, s u p e r m a r k e t s , g y m n a s i u m s  taller and  m o r e slender m a s o n r y  are o b v i o u s ;  and auditoriums.  The  benefits  of  besides space savings, m a t e r i a l  and  c o n s t r u c t i o n costs are reduced. A s the w a l l becomes lighter, s m a l l e r footings are required  and  lower s e i s m i c forces are i n d u c e d . These are i m p o r t a n t reasons w h y m o d e r n m a s o n r y structures f i n d a place i n t o d a y ' s c o m p e t i t i v e b u i l d i n g m a r k e t . H o w e v e r , the d e v e l o p m e n t of t a l l , slender m a s o n r y is s t i l l largely h a m p e r e d b y a l i m i t e d u n d e r s t a n d i n g of the m e c h a n i c a l b e h a v i o u r , a n d p r o b a b l y b y a n h i s t o r i c prejudice t h a t m a s o n r y is not s o u n d w h e n it is t a l l . T h i s is reflected i n the stringent slenderness requirements i n the c u r r e n t m a s o n r y design code ( C A N 3 - S 3 0 4 - M 8 4 ,  1984).  In the last t w o decades, reinforced slender w a l l s have been s t u d i e d e x t e n s i v e l y . S o m e e x p e r i m e n t s have s h o w n excellent f l e x u r a l performance; for e x a m p l e , the tests c o n d u c t e d in the early  80's b y  ACI  Southern  California Chapter  (Athey  1982), w h i c h lead to some  limited  r e l a x a t i o n s o f the slenderness requirements i n b u i l d i n g codes. H o w e v e r , since f u l l scale w a l l tests are very expensive a n d t i m e c o n s u m i n g , it is very  d i f f i c u l t to observe the b e h a v i o u r  under  v a r i o u s l o a d c o m b i n a t i o n s . T h e a n a l y s i s of slenderness effects have so far been largely l i m i t e d to the t r a d i t i o n a l a p p r o a c h , i.e. the m o m e n t m a g n i f i e r  method  has been a p p l i e d a n d thus  an  175 effective r i g i d i t y o f the m e m b e r has h a d to be a s s u m e d . In t h i s chapter, a m o r e r a t i o n a l a n a l y s i s w i l l be presented i n the c o n t e x t of these e x p e r i m e n t a l observations, a n d of the f i n d i n g s i n the preceding chapters, w h i c h have been focused o n the short w a l l or c o l u m n c a p a c i t y . T h i s w i l l f o l l o w a brief review of b a c k g r o u n d i n f o r m a t i o n .  11.2 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 T h e slenderness effects discussed here refer to m a s o n r y under eccentric l o a d . W a l l s under c o n c e n t r i c l o a d i n g are not o f p r a c t i c a l concern since a m i n i m u m e c c e n t r i c i t y has a l w a y s to be a s s u m e d (0.12 or 25 m m , specified b y the c u r r e n t design code ( C A N 3 - S 3 0 4 - M 8 4 ,  1984)) to t a k e  a c c o u n t o f m e m b e r i m p e r f e c t i o n s a n d a l i g n m e n t error. W h e n a slender m e m b e r carries a n eccentric l o a d , i t is i m p o r t a n t to bear i n m i n d t h a t it m a y suffer loss o f c a p a c i t y either due to m a t e r i a l f a i l u r e or b y i n s t a b i l i t y . T h i s p a r t i c u l a r p o i n t has been c l e a r l y e x p l a i n e d b y N a t h a n (1977). F i g . 11.1 shows the i n t e r a c t i o n curve for a c o l u m n subject  to  a  compressive  load  with  equal  end  eccentricities.  The  line  0 —A  defines  r e l a t i o n s h i p between l o a d a n d end m o m e n t . H o w e v e r , due to the slenderness, the m o m e n t is m a g n i f i e d b y the m e m b e r  the  midheight  deflection, a n d the c o r r e s p o n d i n g l o a d - m o m e n t  p a t h is  defined b y 0 — B . M a t e r i a l f a i l u r e occurres at p o i n t B , w h e n the end c o n d i t i o n s are as i n d i c a t e d at p o i n t C . T h e o r e t i c a l l y , i f the m o m e n t — c u r v a t u r e l i n e a r , m a t e r i a l f a i l u r e a l w a y s governs  r a l a t i o n s h i p of the b e a m c o l u m n  the b e h a v i o u r ,  since the m i d s p a n  remains  deflection w i l l  be  u n b o u n d e d w h e n the E u l e r l o a d is a p p r o a c h e d . T h e m o m e n t m a g n i f i e r m e t h o d perfectly predicts this failure mode.  W h e n the m e m b e r  develops some n o n l i n e a r i t y  i n its  moment—curvature  r e l a t i o n s h i p , the m e t h o d 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 p r o v i d e d a n a p p r o p r i a t e effective crosssectional  rigidity  is  used.  However,  when  the  cross-section  has  developed  substantial  n o n l i n e a r i t y , u s u a l l y at greater eccentricities, the m i d s p a n m o m e n t increases w i t h deflection to a p o i n t such as D i n F i g . 11.1, a n d the m e m b e r becomes u n s t a b l e i n the sense t h a t cannot  be  maintained  even  though  the  material  of  the  cross-section  is  equlibrium  still  sound.  0  F I G . 11.1  MOMENT  A L o a d — M o m e n t Interaction C u r v e a n d L o a d i n g P a t h s of a C o m p r e s s i o n M e m b e r  177 The member  w i l l f a i l at t h i s p o i n t , c o r r e s p o n d i n g to end c o n d i t i o n E , unless the l o a d c a n be  shed b y other m e a n s , to l e a d to m a t e r i a l f a i l u r e at p o i n t F. T h e m o m e n t m a g n i f i e r procedure, i n w h i c h the design m o m e n t  is c o m p a r e d  w i t h the short  column moment,  no  longer  applies  r i g o r o u s l y to t h i s s i t u a t i o n . T h e procedure a d o p t e d i n the current code ( C A N 3 - S 3 0 4 - M 8 4 ,  1984)  is, at best, a n a r t i f i c i a l e m p i r i c a l device for the m e m b e r governed b y i n s t a b i l i t y . It is clear t h a t for the m o m e n t m a g n i f i e r m e t h o d a p p l i e d succesfully to the design p r o b l e m , development  (albeit i n a n a r t i f i c i a l w a y ) to be  the key issue is h o w  to e s t i m a t e the  o f the cross-section. A s i n a concrete c o l u m n , the n o n l i n e a r  nonlinear  development  of a  m a s o n r y m e m b e r is due to m a t e r i a l n o n l i n e a r i t y as w e l l as to the c r a c k i n g of the cross-section. T o e s t i m a t e these effects a c c u r a t e l y is d i f f i c u l t since they are c o u p l e d w i t h the m a g n i t u d e as well as the e c c e n t r i c i t y of the l o a d . Therefore it is not s u r p r i s i n g t h a t the c u r r e n t design process is subject to m a n y l i m i t a t i o n s , since these effects c a n n o t be i n c l u d e d i n a single a s s u m e d "effective cross-sectional r i g i d i t y " . F u r t h e r , after the cross-section has c r a c k e d , the r i g i d i t y is a v a r i a b l e a l o n g the m e m b e r height rather t h a n a single c o n s t a n t represented b y the "effective r i g i d i t y " ; the p h y s i c a l p i c t u r e of the s i m p l i f i c a t i o n is not clear. T o i n c l u d e a l l the n o n l i n e a r effects, a r a t i o n a l a n a l y s i s w i t h some n u m e r i c a l procedures is often necessary. F o r  a n a l y s i s of concrete b e a m  columns, Nathan  (1985) has developed  c o m p u t e r p r o g r a m based on some w e l l established p r i n c i p l e s . B y n u m e r i c a l i n t e g r a t i o n , it finds  force-moment-curvature  r e l a t i o n s h i p s for a n y cross-sectional geometry,  a  first  a n d for m a t e r i a l s  w i t h a n y c o n s t i t u t i v e l a w . A n i t e r a t i o n scheme is then used to give the c o l u m n deflection curve w h i c h m a t c h e s to a n y b o u n d a r y c o n d i t i o n s . It is of course very general a n d useful, a n d m a y be a p p l i c a b l e to concrete m a s o n r y  w i t h a few m o d i f i c a t i o n s . O n  the other h a n d , S u w a l s k i a n d  D r y s d a l e (1986) have used a finite element m o d e l to d i r e c t l y a n a l y z e the slenderness influence of the c a p a c i t y of concrete m a s o n r y . T h e s e approaches are useful i n the sense t h a t they m a y i n c l u d e a l l the factors w h i c h affect the b e h a v i o u r . H o w e v e r , at the same t i m e , they require m o r e i n p u t p a r a m e t e r s , w h i c h , i n  178 design p r a c t i c e , often  must  be assumed rather  than  measured.  It  appears  that  with  these  approaches, the w a l l s m u s t be s t u d i e d i n d i v i d u a l l y , a n d i t is d i f f i c u l t to p e r f o r m a p a r a m e t r i c s t u d y w h i c h m a y y i e l d some s i m p l i f i e d relations governed b y some key factors. In  the f o l l o w i n g  a n a l y s i s , some  assumptions  will  be m a d e  based  on  the  observed  c h a r a c t e r i s t i c s of concrete m a s o n r y . W i t h these a s s u m p t i o n s , some a n a l y t i c a l r e l a t i o n s w i l l be developed to e x p l i c i t l y reveal some key factors representing the m a s o n r y slenderness effect. T h i s w i l l be s h o w n to lead to a r e l a t i v e l y s i m p l e b u t yet r a t i o n a l a p p r o a c h to the p r o b l e m .  This  a p p r o a c h w i l l be s h o w n to be easily a d a p t e d to design a n a l y s i s . T h e usefulness a n d l i m i t a t i o n s of t h i s a p p r o a c h w i l l t h e n be discussed.  11.3 M a s o n r y C h a r a c t e r i s t i c s a n d S o m e A s s u m p t i o n s Compared  w i t h concrete c o l u m n s , concrete m a s o n r y is m o r e prone to c r a c k w h e n the  cross-section is subjected to tension because of the m a t e r i a l d i s c o n t i n u i t y at the m o r t a r j o i n t . T h i s is c l e a r l y evident f r o m the d e f o r m a t i o n measurements across the j o i n t s , (see F i g s . 10.1, 10.2, also see F i g s . 6.1 a n d 6.2 for p l a i n m a s o n r y ) . S i m i l a r observations were also reported  by  F a t t a l et a l (1976) a n d b y H a t z i n i k o l a s et a l (1978). T h u s for a l l p r a c t i c a l purposes this tensile b o n d c a n be a s s u m e d to be zero. A n d since the bed j o i n t s are evenly spaced, i t is reasonable to treat the p r o b l e m i n a n average sense, w h i c h is necessary to lead to a c o n t i n u o u s a l o n g the m e m b e r Another nonlinearity  formulation  height. significant  is r a t h e r  observation,  mentioned  many  times  earlier,  is  that  material  limited up  to the f a i l u r e stress, a n d the l i n e a r s t r e s s - s t r a i n m a t e r i a l  r e l a t i o n 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 a n d D i k k e r s 1971, H a t z i n i k o l a s et a l 1978,  W a r w a r u k et a l 1986). T h e linear m a t e r i a l a n d the zero tensile b o n d a s s u m p t i o n s are e q u i v a l e n t to s u p p o s i n g t h a t the n o n l i n e a r i t y i n the m o m e n t — c u r v a t u r e r e l a t i o n s h i p of a concrete m a s o n r y member  is m a i n l y due to the c r a c k i n g of the cross-section. Indeed, the cross-sectional r i g i d i t y ,  w h i c h is p r o p o r t i o n a l to the t h i r d power of the section d e p t h , w i l l decrease d r a s t i c a l l y as the  179 d e p t h is reduced b y c r a c k extension. T h e t h i r d a s s u m p t i o n is t h a t of the plane section r e m a i n i n g plane, c o r r e s p o n d i n g to a l i n e a r s t r a i n d i s t r i b u t i o n across the section. T h i s is a c o m m o n l y accepted a s s u m p t i o n , however r i g o r o u s l y s p e a k i n g , it i m p l i e s , i n the c o n t e x t of the first a s s u m p t i o n , a n i n f i n i t e s i m a l c r a c k i n g s p a c i n g w h e n the side of a cross-section is subject to tension. Since the tension cracks occur o n l y at the m o r t a r j o i n t s , the m a t e r i a l s between t w o c r a c k e d j o i n t s m u s t transfer some tension force and  thus a l t e r the p l a n e sections. Therefore  the linear s t r a i n d i s t r i b u t i o n m a y  be a good  a p p r o x i m a t i o n o n l y w h e n the crack d e p t h is not b i g c o m p a r e d to the u n i t height. W i t h these m a i n a s s u m p t i o n s , i t is possible to e s t a b l i s h r e l a t i v e l y c o m p a c t r e l a t i o n s h i p s g o v e r n i n g the m e c h a n i c a l b e h a v i o u r of a m a s o n r y m e m b e r under v a r i o u s l o a d i n g c o n d i t i o n s , a n d thus i t is easier to p e r f o r m  some p a r a m e t r i c studies on slenderness effects. T h e  equations  g o v e r n i n g the cross-sectional b e h a v i o u r d e r i v e d i n the preceding c h a p t e r are s t i l l v a l i d a n d w i l l be q u o t e d w i t h o u t f u r t h e r c o m m e n t s .  11.4 D i f f e r e n t i a l E q u a t i o n s G o v e r n i n g C o n c r e t e M a s o n r y w i t h C r a c k e d S e c t i o n Equal  end  eccentricities w i l l  first be  investigated, and  the  approach  will  then  be  extended to m o r e general l o a d i n g c o n d i t i o n s . Different differential equations are used to describe the b e h a v i o u r ,  depending  o n whether  the cross-section has c r a c k e d a n d how deep the crack  extends. F i g . 11.2 d e p i c t s the m o s t general case: a concrete m a s o n r y m e m b e r under eccentric l o a d w i t h u n c r a c k e d sections at the t w o ends a n d , due to d e f l e c t i o n , a c r a c k e d s e c t i o n i n the m i d d l e range. N o t e , c represents the c r a c k l e n g t h or c r a c k e d sectional d e p t h . M , F , a n d C denote the cross-section at m i d s p a n , the cross-section at w h i c h the crack extends to the flange ( face-shell) d e p t h a n d the cross-section at w h i c h the crack begins to e x t e n d , respectively. T h e  variables  s u b s c r i p t e d w i t h these letters (in lower case) s t a n d for the c o r r e s p o n d i n g values at these crosssections.  Enlarged Diagram Stress  c  compression side  i  '  ls  boundary of cracked zone  — 2b-c—»-j  FIG. 11.2 A Cross-Section View of A  Reinforced Concrete Wall under Eccentric Compression  181 B y symmetry,  we need o n l y s t u d y the upper h a l f o f the m a s o n r y  w a l l . F o r t h e end  p o r t i o n o f the w a l l , c r a c k i n g does n o t take place. R e f e r r i n g t o the selected coordinates i n w h i c h the x a x i s coincides w i t h t h e t h r u s t line a n d y lies t h r o u g h t h e s y m m e t r i c s e c t i o n , we have, f o r the curve d e f i n i n g the c o m p r e s s i o n face  EI^Xdx  P(b-y)  = 0  h/2  < x < h/2  11.1  with boundary conditions:  y(h/2) =  b- t  11.2  0  at the end; a n d  y(h /2) = b c  e  11.3  c  with  dx\2)  ~  11.4  ^  at the C — c r o s s - s e c t i o n ; where e is the v i r t u a l l o a d i n g e c c e n t r i c i t y a n d <p is the r o t a t i o n a t this c  cross-section.  N o t e , f o r this l o a d i n g c o n d i t i o n , the e n d e c c e n t r i c i t y e  c  0  is s m a l l e r t h a n the  cracking eccentricity e . c  W h e n E q . 11.1 is i n t e g r a t e d a n d m a t c h e d t o the b o u n d a r y c o n d i t i o n s (see a p p e n d i x ) , we obtain  11.5  182 where P  is the E u l e r l o a d corresponding to the gross section:  cr  p  " ~ ~ir  Pcr  ~  7T Hill  11 c  w  1 L 6  a n d cj is the r a t i o o f the m o m e n t of i n t e r i a of the net cross-section to t h a t o f the gross-section  i n w h i c h A a n d a/b  are defined as i n the preceding chapter. F o r a g i v e n cross-section, e  c a n be  c  w r i t t e n as  e  in which M  c  and P  c  c  =  Mc = —, 3 (l - X-f +  T  np)  11.9  b  are expressed t h r o u g h E q s . 10.1 a n d 10.2 w i t h c  2  b e i n g set equal to zero b y  neglecting the tensile resistance of the cross-section. E q . 11.5 gives the r e l a t i o n between l o a d P a n d t w o u n k n o w n s , n a m e l y h  c  a n d <p , w h i c h c  w i l l be f o u n d b y the equations g o v e r n i n g the c r a c k e d section as s h o w n f u r t h e r below. T h e d i f f e r e n t i a l e q u a t i o n for the c r a c k e d section c a n be d e r i v e d b y first c o n s i d e r i n g the g e o m e t r i c r e l a t i o n . A s s h o w n b y the enlarged d i a g r a m i n F i g . 11.2, the cross-section w i l l rotate due t o the uneven c o m p r e s s i o n w h i c h produces the outer fiber s t r a i n e  x  at the c o m p r e s s i o n face  b u t zero at the b o u n d a r y between c r a c k e d a n d u n c r a c k e d zones. T h e change of the r o t a t i o n of a s m a l l s e c t i o n , therefore, c a n be a p p r o x i m a t e d as  e ds 1  e dx 1  = -2b-=T * -W=T  B y recognizing  1L1  °  183 11.11  it follows t h a t  T h e assumed linear stress-strain relation allows us t o write  fl =  Finally, by  11.13  -j  cr 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 1  theequilibrium  condition, either t h r o u g h  E q . 10.5  crack has extended b e y o n d the flange. T h u s for 0 <  2  dy dx z  depending  2Eb  o nwhether the  b— a, w e h a v e  P  -  1  c<  o r E q . 10.7,  11.14  ''(('-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 y r e c o g n i z i n g e=M/P S i m i l a r l y , for  b  V  2b_J_\  b_J  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  \JJ  b  u.i5  10.6.  b—a< c < 6 + a w e o b t a i n  2  dy  11.16  dx" 2  2Eb l  184 and  ('- &)'(•+1) -  {(f)'- i O - f -  f  )) 11.17  y =  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  dy dx  _ ~  ["FT \Ebl  11.18  \  for 0 < c < 6—a, in which C± is a constant of integration and  4  1  ( ~ A)' 6  ( (  i  A  - 2 i )  3  3  A  is a function of c expressed as  1  ( t ) - ( t - "'X - t)' 2  -  ( n - " X  1  - 1 ) )  11.19  2  And similarly for 6—a< c < 6+a  dy dx  _ ~  \P_ iEbl  ( \  C - £l (c)) 2  2  11.20  with  11.21  185  T h e c o n s t a n t s of i n t e g r a t i o n C i a n d C  2  c a n be d e t e r m i n e d b y m a t c h i n g to the k n o w n  c o n d i t i o n s o n the r o t a t i o n . B y s y m m e t r y , we have  11.22  w h i c h leads to  11.23  c  C — Q ( ">) 2  where c  m  2  denotes the m i d s p a n crack l e n g t h . T h u s the r o t a t i o n at section F is  11.24  w h i c h also leads, by c o n t i n u i t y of the r o t a t i o n , to a n expression for C  Ci =  where Cj =  b—a,  fi (c ) 2  m  -  fi ( ) 2 C/  +  il^cj)  x  11.25  the c r a c k l e n g t h at section F.  E q s . 11.18 a n d 11.20 c a n t h e n be rearranged a n d i n t e g r a t e d a l o n g the w a l l height to give  Cc  p hc/h—hj/h 2  \|Pc7  for  0 <  c <  b—a,  where c = c  7T  =  dy  0, the c r a c k l e n g t h at section C ; a n d  11.26  186  p_  hf/h -7T  _3_ N 2  =  dy  c < 6+a.  11.27  2  2  Cm  for b—a<  fi (c)  b\C -  <fy c a n be expressed i n t e r m s o f dc b y d i f f e r e n t i a t i o n o f E q . 11.15 o r E q .  11.17 w i t h i n their specified d o m a i n s . T h u s , for g i v e n a c  m  the r i g h t h a n d sides o f E q s . 11.26 a n d  11.27 c a n be i n t e g r a t e d n u m e r i c a l l y . Further,  the r o t a t i o n a t section C , w h i c h i s c o n t a i n e d i n E q .  11.5, c a n be r e a d i l y  o b t a i n e d i n v i e w of E q . 11.18  fc  = ^  ( fi (c ) 2  m  Finally, by summing  -  J2 ( ) 2  Eqs.  +n ( )  C /  i  11.26,  -  Cf  11.27  a p p l i e d l o a d P a n d the m i d s p a n crack d e p t h c  fi^Ce))  and  11.28  11.5, a d e f i n i t i v e r e l a t i o n between t h e  is reached  m  c./b 3_  P Per  N 2  dy/ dc Cm/b  •\ ^ ( c ) - ^ ( ) 2  m  2  c  <+)  cc/b  +  dy/dc  c./b  + H ^ sin"  e /6 c  1  e /b 0  — sin  11.29  T h i s e q u a t i o n i m p l i c i t l y defines the f o r c e — d e f l e c t i o n r e l a t i o n of a concrete m a s o n r y w a l l  187 t h r o u g h the p a r a m e t r i c v a r i a b l e c , w h i c h c a n be used to s t u d y the slenderness effects a n d the m  s t a b i l i t y o f the w a l l . B y e x a m i n i n g E q . 11.29, one finds t h a t the three t e r m s o n the r i g h t h a n d side a c t u a l l y represent the c a p a c i t y c o n t r i b u t i o n s o f three sections o f the m a s o n r y w a l l , n a m e l y , the c r a c k e d section i n w h i c h the crack has extended i n t o the g r o u t core, the c r a c k e d section i n w h i c h the crack  extends  within  the  face-shell, and  the  u n c r a c k e d section. Therefore,  by  adding  or  s u b t r a c t i n g the c o n t r i b u t i o n s , the results c a n be extended t o m o r e general l o a d i n g cases. F o r (equal) e n d eccentricities e  0  larger t h a n c r a c k i n g e c c e n t r i c i t y e , the c r a c k e d zone c  w i l l e x t e n d over the entire height, a n d E q . 11.29 reduces t o Cj/b  11.30  where c  0  is the e n d c r a c k i n g c o r r e s p o n d i n g t o e , f o u n d t h r o u g h E q s . 10.5 a n d 10.6. 0  If t  0  is greater t h a n the flange c r a c k i n g e c c e n t r i c i t y t.,  E q . 11.30  f u r t h e r reduces to  11.31  where c  0  is d e t e r m i n e d t h r o u g h E q s . 10.7 a n d 10.8. S i m i l a r l y , w h e n the m i d s p a n c r a c k i n g c  m  happen when e  0  is less t h a n t,,  cc/b  is less t h a n the flange c r a c k i n g Cy, w h i c h m a y  E q . 11.29 a n d E q . 11.30 become  188  + ft  sin"  1  e  ^ l L .  -  sin"  1  'olh  ,  )•  11.32  and  c /b 0  P P  dy/dc  _ 6 —  11.33  <+)  2 Cm/O  respectively. T h u s , a l l the possible c o m b i n a t i o n s f o r equal e c c e n t r i c i t y l o a d i n g are i n c l u d e d . By  t h e same  p r i n c i p l e , t h e results c a n also be extended  t o the case o f u n e q u a l  e c c e n t r i c i t y l o a d i n g . A c c o r d i n g t o N a t h a n (1972), t h e c o n f i g u r a t i o n o f a c o l u m n loaded w i t h a r b i t r a r y eccentricities c a n be represented b y a p o r t i o n o f a wave o f a n i m a g i n a r y , i n f i n i t e l y l o n g c o l u m n under t h e a c t i o n o f the a x i a l l o a d , as s h o w n i n F i g . 11.3. W i t h o u t loss o f g e n e r a l i t y , we assume t h a t t h e m a g n i t u d e o f the b o t t o m e c c e n t r i c i t y is n o t less t h a n t h a t o f the t o p one. T h u s the m a x i m u m deflection f r o m t h e t h r u s t line a l w a y s lies i n t h e lower p o r t i o n o f t h e c o l u m n . T h i s p o i n t , a t w h i c h dy/dx=0, corresponds t o t h e m i d s p a n o f t h e case o f equal e c c e n t r i c i t y loading  studied  above.  T h e column  loaded  with  arbitrary  eccentricities then  is a c t u a l l y  c o m p o s e d o f a p o r t i o n s y m m e t r i c a l a b o u t t h e m a x i m u m deflection p o i n t , w i t h a n extension at the  t o p e n d a s f a r as t h e a p p r o p r i a t e  corresponding  capacity  contributions  value  c a n easily  o f eccentricity. F r o m be e v a l u a t e d  this  viewpoint, the  and summed  t o give the  force—cracking relation. It s h o u l d be i n d i c a t e d t h a t f o r t h e case o f d o u b l e c u r v a t u r e l o a d i n g (e /e t  b  negative),  there are a p p a r e n t l y t w o possible e q u i l i b r i u m c o n f i g u r a t i o n s d e p i c t e d b y sections A B a n d A C i n F i g . 11.3. H o w e v e r , as f a r as t h e lowest b u c k l i n g l o a d is concerned, t h e c o n f i g u r a t i o n A B w i l l be under c o n s i d e r a t i o n . T h i s c o n f i g u r a t i o n s h o u l d also be realized f o r t h e case o f a n t i - s y m m e t r i c l o a d i n g (e /e t  b  = — l). T h i s h a s been s h o w n b y t h e e x p e r i m e n t s ( H a t z i n i k o l a s et a l , 1978; F a t t a l  F I G . 11.3 C o l u m n D e f l e c t i o n C u r v e  190 et al 1976), and a theoretical explanation will be presented in appendix. Consider two most general cases. First 0 < e  < e  t  < e ; the corresponding relation is  b  c  Cj/b  _P_ _  4  Per  r  dy/ dc  T  1 >  2  2  \ V (c )  J  2  Jl(c) if)  -  m  2  Cm/b  cc/b  + _E  dy/ dc Cj/b  e /b c  1  + ft  if)  ^  I sin- -^  ((  1  -l  sin  N(T)  l H ("  +  2(Cm)  ~  a  n  + i( /) - i(<=«)) fi  M  c  1  + - ± - ^ ^ sin" -  e /b  -l  t  sin  ( l )  The second case is when  2  +  e  b  iN (  > tj  fi  2(Cm)  > 0, e  2  ^  Cm/b  +« (C )  2  1  /  A| ^ ( c ) 2  m  -  ^ ( C c ) )  > |e | > e^, the relation becomes  6  t  dy/ dc  4 r  fi (C/)  < 0, but e  t  c„/b  P_ _ Per  -  11.34  c  ft ( ) 2  <+)  191  c./b  cj/b dy/ dc  +  ,Jfi (c ) 2  m  <f)  fi (c) 2  dy/ dc  + +  ]tt (c ) 2  m  -  c  0 ( ) 2  <+)  cc/b  +  dy/dc  <-fr)  Cj/b  sin  1  e /6  ,  c  0 (c ) 2  where c a n d c 6  t  m  11.35  fi ( ) 2  C/  -rfiiCc/) -  are the crack depths c o r r e s p o n d i n g to e  b  l o a d i n g c o n d i t i o n s , the c r a c k i n g c  m  fi^cc))  a n d e , respectively. I n these t w o (  is a s s u m e d to be greater t h a n Cj.  T h e s e results are r e a d i l y generalized to a n y other l o a d c o m b i n a t i o n s for u n e q u a l ^  end  eccentricity loading. A computer program written in F O R T R A N - G  was developed based on the equations  d e r i v e d above. A l i s t i n g of the p r o g r a m is g i v e n i n a n a p p e n d i x .  11.5 R e s u l t s a n d A p p l i c a t i o n s The algorithm  developed above w i l l be used to s t u d y t w o m a i n aspects o f concrete  m a s o n r y w a l l b e h a v i o u r , n a m e l y the s t a b i l i t y , a n d the f o r c e — d e f l e c t i o n r e l a t i o n ; the l a t t e r also affects the w a l l c a p a c i t y . T h e focus w i l l be o n the case of equal e c c e n t r i c i t y l o a d i n g . A c c o r d i n g to the m o d e l , the b u c k l i n g l o a d of concrete m a s o n r y c a n be f o u n d b y s t e p p i n g c  m  by  f r o m zero to some c r i t i c a l d e p t h , at w h i c h the l o a d P reaches a m a x i m u m . T h i s is i l l u s t r a t e d the  c — P r e l a t i o n s h i p for m  eccentricity represents  (e = t/6), 0  as s h o w n  a n undeflected  a p l a i n , s o l i d section (X = a/b=np = 0) i n Fig.  member.  Thus  11.4.  O n the h o r i z o n t a l  t h i s corresponds  loaded  a x i s , the  t o no axial load  a t the  kern  value  c /t=0  or  P/P =0.  O b v i o u s l y , no l o a d c o u l d be m a i n t a i n e d if the whole cross-section were c r a c k e d , c /i=l. m  m  cr  Thus  192  0.3 0.28 .0.26 0.24 0.22 0.2 u o fx, \  0.18 _ 0.16 0.14  cu 0.12 0.1 0.08 0.06 0.04 0.02 0  Cm/t FIG. 11.4 Critical Load versus Crack Depth at Middle Section of a Plain, Solid Member Loaded at e=t/6  l  0.9 PRESENT  RESULT  0.8 0.7 -  YOKEL  0.6 o  a,  0.5 0.4 0.3 0.2 0.1 0 0  0.1  FIG. 11.5 Critical Load versus Loading Eccentricity for a Solid Section  0.5  193 this p o i n t also corresponds t o the value  P/P = Q. W h e n P is a p p l i e d a n d increased, deflection cr  w i l l increase together w i t h the c r a c k d e p t h . A s s u m i n g there is no compression f a i l u r e d u r i n g the l o a d i n g stage, the l o a d P w i l l reach a m a x i m u m corresponding t o some c r a c k d e p t h and  PjPer~0.28  member  (c /zss0.4 m  for this case). A n y further increase o f P b e y o n d this p o i n t w i l l cause the  t o collapse. It i s clear t h a t  the r e l a t i o n before  represents stable e q u i l i b r i u m . B e y o n d this p o i n t , dP/dc A t the p o i n t , dP/dc  m  = 0; P=P ax m  m  this c r i t i c a l p o i n t ,  dP/dc  m  > 0,  < 0 represents unstable e q u i l i b r i u m .  of course, stands for the b u c k l i n g l o a d .  O b v i o u s l y , t h e cross-sectional c r a c k i n g of a m e m b e r w i l l depend h e a v i l y o n the l o a d i n g eccentricity a n d so, therefore, w i l l the b u c k l i n g l o a d . F o r a p l a i n , s o l i d section, the b u c k l i n g l o a d is p l o t t e d against the e c c e n t r i c i t y i n discrete f o r m i n F i g . 11.5. A t the p o i n t where e /l = 0, when o  the m e m b e r is loaded c o n c e n t r i c a l l y , P / P  c r  = l , a n d the b u c k l i n g l o a d coincides w i t h the E u l e r  l o a d . T h e b u c k l i n g l o a d decreases d r a s t i c a l l y w i t h increase i n e c c e n t r c i t y . W h e n  e /t = 0.5, o  P/P r = 0, i.e., no l o a d c a n be sustained i f the l o a d is a p p l i e d at the edge of a m e m b e r w i t h no C  tension resistance. T h e classic p r o b l e m of the b u c k l i n g of a p l a i n , s o l i d m e m b e r w i t h no tension resistance was first i n v e s t i g a t e d b y R o y e n (1937). T h e p r o b l e m a n d i t s a p p l i c a t i o n t o b r i c k w o r k have been s u b s q u e n t l y s t u d i e d b y C h a p m a n a n d S l a t f o r d (1957), S a h l i n (1971), Y o k e l (1971), H a t z i n i k o l a s (1978). I n F i g . 11.5, t h e results o b t a i n e d b y the a l g o r i t h m are c o m p a r e d w i t h a closed f o r m s o l u t i o n f o r l o a d i n g e c c e n t r i c i t y larger t h a n t / 6 given b y Y o k e l . F o r the range c o m p a r e d , the results are essentially i d e n t i c a l . T h e f o l l o w i n g are some of the interesting p r e d i c t i o n s given b y the a l g o r i t h m . A s shown by F i g . 11.6 for the s o l i d section ( A = a / 6 = 0 , np varies), w h i l e the b u c k l i n g loads corresponding to s m a l l eccentricities are essentially unaffected, the s t a b i l i t y o f the w a l l i s g r e a t l y w i t h increase o f the reinforcement  improved  r a t i o a t large eccentricities. T h e rather flat t a i l s a t large  eccentricities i m p l y t h a t the c a p a c i t y o f reinforced w a l l s i s largely governed b y the b e n d i n g rigidity.  1 0.9  H  "X =0.0 0.8  a/b=0.0 0.7 0.6 0.5 -  np = 0.05 0.025 0.005 0.0  0.1 0.3 0.2 0.1 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  a/b=0  \\V^-  1-0.5  a/b=0.65  np =0.05  0.7  0.4 H a/b=0.75  \  \  0.3 H 0.2 0.1 0 0.4  0.2  o.e  Eo/T  FIG. 11.7 Critical Load versus Loading Eccentricity: A = 0.5, np = 0.05, a/b Varies  195  0.2  0  0.4 Eo/T  F I G . 11.8 C r i t i c a l L o a d versus L o a d i n g E c c e n t r i c i t y : a / 6 = 0.65, 71/9 = 0,  In contrast, the v a r i a t i o n of a/b  A Varies  o n l y affects the s t a b i l i t y at s m a l l eccentricities as  s h o w n b y F i g . 11.7 (A = 0.5, corresponding t o a p a r t i a l l y grouted w a l l ; 71/9 = 0.05; a / 6 = 0 , 0.65, 0.75). T h e effect of changes i n A is i l l u s t r a t e d t h r o u g h a n e x a m p l e c o m p a r i n g different  bedding  conditions. F i g . 11.8 shows the b u c k l i n g loads for a t y p i c a l 8 i n c h p l a i n section ( a / 6 = 0 . 6 5 ; n p = 0 . A = l for face-shell b e d d i n g a n d A = 0.75 for f u l l b e d d i n g ; A = 0 represents a s o l i d , or f u l l y grouted section, i n c l u d e d here as a reference). A l t h o u g h the b u c k l i n g l o a d of the s o l i d section is higher  at very  s m a l l eccentricities, i t drops  rapidly  as the e c c e n t r i c i t y increases a n d soon  becomes the lowest. F a c e - s h e l l bedded m a s o n r y , on the c o n t r a r y , has lower b u c k l i n g loads a t s m a l l eccentricities b u t remains r e l a t i v e l y higher at larger eccentricities. A f u l l y bedded section falls in between. Since b u c k l i n g u s u a l l y only governs failure at larger eccentricities, one m a y conclude that i n terms of s t a b i l i t y , face-shell bedded m a s o n r y  is better t h a n its fully  bedded  counterpart w h i c h is in t u r n better t h a n a s o l i d section. T h i s is not s u r p r i s i n g considering t h a t  196 face-shell bedded m a s o n r y is least prone to crack under eccentric l o a d i n g . W e m a y infer, i n the c o n t e x t o f the s t r e n g t h studies presented i n the preceding chapters, t h a t f a c e - s h e l l  bedded  m a s o n r y is m o r e l i k e l y to be governed b y s t r e n g t h t h a n b y s t a b i l i t y . T h e present a p p r o a c h is c o m p a r e d w i t h some of the e x i s t i n g d a t a o b t a i n e d f r o m f u l l scale concrete w a l l tests. These i n c l u d e eleven 137 i n c h a n d 105 i n c h h i g h w a l l s ( 8 x 4 0 x 1 2 8 i n c h a n d 8 x 4 0 x 9 6 i n c h n o m i n a l ) w i t h different reinforcement tested under e q u a l e n d eccentricities b y H a t z i n i k o l a s et a l (1978): A s discussed at the b e g i n i n g o f the chapter, t a l l m a s o n r y w a l l s m a y lose s t r e n g t h either b y m a t e r i a l f a i l u r e or b y i n s t a b i l i t y . T h e e x a m i n a t i o n of the 137 i n c h h i g h w a l l w i t h  3#3  r e i n f o r c i n g steel (np=0.027) provides a n excellent i l l u s t r a t i o n of t h i s p o i n t . In F i g . 11.9, the l o a d — m o m e n t i n t e r a c t i o n curve is developed b y the e q u a t i o n s g i v e n in C h a p t e r X . T h e s t r a i g h t lines r a d i a t i n g f r o m the o r i g i n define the end c o n d i t i o n s for  different  l o a d i n g eccentricities. These e x p e r i m e n t a l lines are t e r m i n a t e d b y the d a t a p o i n t s a n d w i t h the p r e d i c t e d curves representing the l o a d — m o m e n t  paired  r e l a t i o n s h i p s at m i d h e i g h t . It is clear  t h a t the m o m e n t is m a g n i f i e d due to the slenderness. W h e n the cross-section r e m a i n s u n c r a c k e d , u s u a l l y under s m a l l eccentricities w i t h l o w l o a d m a g n i t u d e , the m a g n i f i e r is g i v e n b y the linear solution:  11.36  Recall that P  cr  represents the E u l e r l o a d for the gross section w h i c h m u s t be a d j u s t e d by £ for  p a r t i c u l a r c o n d i t i o n s . W h e n the cross-section is c r a c k e d , the m a g n i f i e r 6 = b y the a l g o r i t h m , for every m i d s p a n crack d e p t h c . m  the s m a l l e s t e c c e n t r i c i t y e  0  =  e /e m  0  is c a l c u l a t e d ,  It is i n t e r e s t i n g to note t h a t for the case of  i / 6 , as d e p i c t e d b y the lines w i t h the steepest i n i t i a l slope, the  p o i n t d e f i n i n g the e n d c o n d i t i o n s is w i t h i n the P—M  c a p a c i t y curve w h i l e the  p o i n t representing the m i d s p a n c o n d i t i o n s ( m o m e n t has been m a g n i f i e d )  corresponding  is o u t s i d e b u t f a i r l y  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 0.0  1  :  -I -  — CL  0  0.2  0.4 Eo/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 -  0.4  -  0.3  -  0.2  -  0.1  -  \  \  STABLITY GOVERNS  MATERIAL GOVERNS  1  0 0  —  1  r—  1  0.2  0.4  Eo / T F I G . 11.11 T h e o r e t i c a 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 E x p e r i m e n t s by H a t z i n i k o l a s et a l : 137 inch H i g h W a l l w i t h R e i n f o r c e m e n t 3 # 6  \ >v  N.  I  STABUTY FALURE  O  MATERIAL FAILURE  STABLITY GOVERNS  MATERIAL GOVERNS  0  •  >s  i  1  0.2  1 0.4  Eo/T F I G . 11.12 T h e o r e t i c a 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 E x p e r i m e n t s by H a t z i n i k o l a s et a l : 105 i n c h H i g h P l a i n W a l l  199 close to the c u r v e . T h i s i n d i c a t e s m a t e r i a l failure since the cross-sectional c a p a c i t y is reached at m i d s p a n . H o w e v e r , for the cases of larger eccentricities ( e = </3, 0  e = 3 i n a n d e = 3.5 i n ) , a l l 0  0  the end p o i n t s are w e l l w i t h i n the c a p a c i t y 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 w h i l e the cross-sectional c a p a c i t y is not exceeded as s h o w n b y the curves d e f i n i n g the m i d s p a n c o n d i t i o n s . It is clear t h a t these are the cases of i n s t a b i l i t y failure. ( A t i n s t a b i l i t y , the m i d h e i g h t l o a d p a t h s h o u l d reach a h o r i z o n t a l tangent. It is seen t h a t t h i s is a p p r o x i m a t e l y true of the p r e d i c t e d curves.) T h e c o m p a r i s o n i n terms of the failure loads m a y be better i l l u s t r a t e d b y p l o t t i n g against  e /t 0  P/P  cr  as s h o w n i n F i g . 11.10. T h e p l o t includes t w o curves, one of w h i c h represents  i n s t a b i l i t y f a i l u r e generated b y the a l g o r i t h m s i m i l a r to the curve i n F i g . 11.5. T h e other defines m a t e r i a l f a i l u r e , w h i c h is converted a n d s h r u n k  (by  the slenderness effect) f r o m the  P—M  c a p a c i t y curve g i v e n i n F i g . 11.9. It is clear t h a t when l o a d i n g e c c e n t r i c i t y is s m a l l , the w a l l is governed b y m a t e r i a l f a i l u r e . W h e n the e c c e n t r i c i t y is great, the w a l l w i l l f a i l b y i n s t a b i l i t y . T h e agreement w i t h the e x p e r i m e n t s i n t e r m s o f the f a i l u r e loads is very good. A s expected, a n increase of the reinforcement w i l l overcome the brittleness of the w a l l a n d prevent i n s t a b i l i t y f a i l u r e . F i g . 11.11 shows the P—e configuration  as the ones s t u d i e d above  but  r e l a t i o n s h i p for w a l l s of the same  w i t h heavier reinforcement  (3#6,  np = 0.108).  M a t e r i a l f a i l u r e governs for the whole e c c e n t r i c i t y range as i l l u s t r a t e d i n the p l o t . W h e n w a l l s are lower, m a t e r i a l failure w i l l a g a i n g o v e r n the b e h a v i o u r , as s h o w n i n F i g . 11.12 for the case of the 105 i n c h h i g h p l a i n concrete w a l l w i t h s m a l l e r eccentricities. W e see a g a i n , b y c o m p a r i n g F i g s . 11.10, 11.11 a n d 11.12, t h a t b e h a v i o u r under large eccentricities is s i g n i f i c a n t l y enhanced b y a n increase i n the reinforcement.  11.6 Usefulness a n d L i m i t a t i o n s T h e a n a l y s i s presented leads to a very a t t r a c t i v e a p p r o a c h to the slenderness of concrete m a s o n r y . F o r a g i v e n w a l l , i.e. w h e n the d i m e n s i o n s a n d the p a r a m e t e r s f  u  , E,  A, a/b  a n d np  of  200 the  w a l l are  known,  the  cross-sectional c a p a c i t y curve  P—M  and  the curve d e f i n i n g  r e l a t i o n s h i p between b u c k l i n g l o a d a n d e c c e n t r i c i t y (such as the one i n F i g . 11.5)  the  can be  d e v e l o p e d . T h e designer m u s t first ensure t h a t the design l o a d at the design e c c e n t r i c i t y does not exceed the b u c k l i n g v a l u e . H e is t h e n r e q u i r e d to m a k e sure t h a t the design l o a d a n d the design m o m e n t at m i d s p a n (or the p o i n t of m a x i m u m deflection for u n e q u a l eccentricities) lie inside the P—M  c a p a c i t y curve so t h a t m a t e r i a l f a i l u r e w i l l not h a p p e n . T h e end m o m e n t is m a g n i f i e d  t o give the m i d s p a n m o m e n t . T h e m a g n i f i e r , w h i c h varies w i t h P/P , cr  is a b y p r o d u c t of the  d e r i v a t i o n o f the b u c k l i n g curve. T h e a t t r a c t i v e n e s s o f the a p p r o a c h lies i n the fact t h a t the b u c k l i n g c u r v e as a f u n c t i o n of the l o a d i n g e c c e n t r i c i t y is u n c o u p l e d f r o m the specific m a t e r i a l properties a n d d i m e n s i o n s o f a w a l l . T h e curve is dependent o n l y o n the three c r o s s - s e c t i o n a l p a r a m e t e r s : n a m e l y , A, the extent o f the g r o u t a n d the b e d d i n g ; a/b,  the hollowness of the block u n i t ; a n d r i p , the reinforcement  p a r a m e t e r s . T h u s , for a n y c o m b i n a t i o n s of these p a r a m e t e r s , the curve m a y be p r e - p r e p a r e d . A designer is t h e n o n l y r e q u i r e d to w o r k w i t h these prepared curves a n d the P—M  cross-sectional  c a p a c i t y b o u n d , w h i c h c a n be developed for a specific w a l l b y equations g i v e n i n C h a p t e r X or b y a n y other s i m p l i f i e d means, t o d e t e r m i n e i f the w a l l is adequate. W i t h o u t p e r f o r m i n g  a  s p e c i a l , c o s t l y a n a l y s i s for a n i n d i v i d u a l w a l l , the designer is able t o a p p r o a c h the p r o b l e m w i t h assured a c c u r a c y . T h i s a p p r o a c h , the a u t h o r believes, is m u c h m o r e r a t i o n a l t h a n the current design a n a l y s i s at the cost of very l i m i t e d a d d i t i o n a l effort. The  independence  of  the  buckling load  from  the  specific m a t e r i a l  properties  and  d i m e n s i o n s o f a w a l l arises f r o m the a s s u m p t i o n of l i n e a r m a t e r i a l r e l a t i o n s h i p s . F u r t h e r , the v a l i d i t y o f the a p p r o a c h is also based o n the a s s u m p t i o n t h a t plane sections r e m a i n p l a n e . T h e a p p r o a c h is g o o d as l o n g as these a s s u m p t i o n s are s t i l l close t o r e a l i t y ; otherwise it is subject to limitations. Substantial nonlinearity  may  be caused by  the y i e l d i n g of the r e i n f o r c i n g steel i n  t e n s i o n . T h i s m a y h a p p e n w h e n the necessary c o n d i t i o n specified i n C h a p t e r X (see the c o n t e x t  201 of E q . 10.11) is s a t i s f i e d , w h i c h u s u a l l y corresponds to a l o w steel r a t i o . A l t h o u g h the steel y i e l d i n g c a n be i n c o r p o r a t e d i n the a l g o r i t h m w i t h o u t m u c h d i f f i c u l t y , b y c h a n g i n g the np  value  for a p p r o p r i a t e sections at w h i c h the y i e l d s t r a i n is exceeded, the a d v a n t a g e of s i m p l i c i t y is lost. If t h i s happens i t appears t h a t the w a l l m u s t be s t u d i e d i n d i v i d u a l l y . H o w e v e r , f u r t h e r i n v e s t i g a t i o n i n d i c a t e s t h a t w h e n the steel r a t i o is l o w , the b e h a v i o u r of the w a l l w i l l be governed m a i n l y b y the s u r r o u n d i n g concrete. B u c k l i n g u s u a l l y takes place before the y i e l d s t r a i n is reached, as i n the 3 # 3 reinforced w a l l s s t u d i e d above. Indeed, y i e l d i n g of the steel was never observed i n the e x p e r i m e n t s ( H a t z i n i k o l a s et a l 1978), a n d the proposed procedures do give very g o o d p r e d i c t i o n s , as s h o w n above. F u r t h e r , for very l o w steel ratios, the c h a n g i n g of np  i n the a l g o r i t h m m a k e s very l i t t l e  difference i f the l o a d i n g e c c e n t r i c i t y is not too large. A n y h o w , corresponding  to  large  deflections  is  unfavorable  and  may  the steel y i e l d i n g i n t e n s i o n be  prevented  through  design  requirements. The  presented procedure  tends to overestimate the deflections for  reinforcement, as is seen w i t h the 3 # 9  walls w i t h  heavy  ( n p = 0 . 2 4 5 ) reinforced w a l l s tested b y H a t z i n i k o l a s et a l  (1978). T h i s is believed to be m a i n l y caused b y the v i o l a t i o n of the p l a n e section a s s u m p t i o n . W i t h h e a v y reinforcement, a w a l l tends to a l l o w d e v e l o p m e n t of deeper c r a c k i n g s i n i t s m i d s p a n region. Since the cracks occur u s u a l l y o n l y at the bed j o i n t s , the c o m p r e s s i v e s t r a i n s between t w o j o i n t s , i.e., w i t h i n a b l o c k u n i t , w i l l depart c o r r e s p o n d i n g l y f r o m the l i n e a r d i s t r i b u t i o n as the  crack  depths  increase.  As  indicated,  the  model  assumes  a  linear  strain  distribution  c o r r e s p o n d i n g t o a n i n f i n i t e s i m a l c r a c k i n g s p a c i n g , w h i c h , of course, u n d e r e s t i m a t e s the r i g i d i t y of  the  cross-section. T h e  underestimation  may  be  substantial when  crack d e p t h  is  large  c o m p a r e d to c r a c k s p a c i n g ( u n i t h e i g h t ) , l e a d i n g to erroneous results. v. .  T h e n o n l i n e a r i t y o f concrete m a y also affect the a c c u r a c y of the a p p r o a c h . H o w e v e r , the a s s u m p t i o n of linear m a t e r i a l tends to o v e r e s t i m a t e the r i g i d i t y of the cross-section. For  the cases c o m p a r e d ,  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 crosssectional 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 (e = l/9t), the 0  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 (e = k  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 P . k  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, P , and the k  203  0.6  Linear, Exact 0.5  Linear, Approxiiiated 0.4  Eo/T=1/9  0.3  /  Eo/T-1/6  0.2  Eo/T=2/9 0.1  ^  —  Eo/T-5/18  1.2  1.4  1.6  1.8  2.2  2.4  MAGNRER  F I G . 11.13  M o m e n t M a g n i f i e r versus L o a d f o r a P l a i n S e c t i o n  0.6  Linear Solution 0.5  -  np =0.05  0.4  0.3  0.2  0.1  2.2 MAGMFER  F I G . 11.14 M o m e n t M a g n i f i e r versus L o a d for a R e i n f o r c e d S e c t i o n  2.4  204  CL  1  1.2  1.4  1.6  1.8  2  2.2  ,  2.4  MAGNFER  F I G . 11.15  F I G . 11.16  M o m e n t M a g n i f i e r : E x a c t versus A p p r o x i m a t i o n  C r i t i c a l Load and C r i t i c a l M o m e n t Magnifier Versus Eccentricity: for Purpose of Design A n a l y s i s  205 corresponding magnifier  If the curve c a n be c o n s t r u c t e d b y some m e a n s to end w i t h a  S. k  h o r i z o n t a l tangent at t h i s p o i n t , i t m a y be accurate enough for design. F o r t h i s purpose we i n t r o d u c e the f o r m  11.37  which  passes t h r o u g h  the  origin  and  reaches  the  end  point  with  zero  slope.  This  gives  s a t i s f a c t o r y f i t t i n g for the p l a i n section case, as s h o w n i n F i g . 11.15. F o r a reinforced crosssection at large e c c e n t r i c i t y , the curve m a y be t r u n c a t e d at the b e g i n n i n g of the f l a t p l a t e a u to y i e l d a good fit (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 u s u a l l y values w e l l below the b u c k l i n g l o a d are o f interest. T h r o u g h t h i s s i m p l i f i c a t i o n , we o n l y corresponding magnifier 8  need to k n o w  the b u c k l i n g l o a d  P  k  and  the  for a g i v e n cross-section at g i v e n e c c e n t r i c i t y . These t w o p a r a m e t e r s  k  can be p r e - d e t e r m i n e d a n d e x h i b i t e d i n the f o r m of tables, or graphs such as the one s h o w n i n F i g . 11.16. T h e figure is for a p l a i n section w i t h different cross-sectional factors A. N o t e t h a t t w o scales are used for the o r d i n a t e so t h a t P  k  and 6  k  c a n be p l o t t e d i n the same g r a p h .  F o r c o m m e r c i a l l y a v a i l a b l e b l o c k u n i t s , the range of v a r i a t i o n i n a/b p a r a m e t e r A varies f r o m a b o u t 0 u p to 1; np  is s m a l l . T h e  also has a n upper l i m i t (of 0.1 for the t i m e being).  T h u s the c o m b i n a t i o n s of these three p a r a m e t e r s are l i m i t e d a n d it is not i m p r a c t i c a l to prepare tables or g r a p h s of P  k  and 6  k  for design purposes.  F i n a l l y , it m a y be w o r t h repeating the a p p r o a c h w h i c h has been developed a n d w h i c h is s t r o n g l y r e c o m m e n d e d for design purposes: 1) Select the w a l l cross-section, a n d , u s i n g the m a t e r i a l properties a n d d i m e n s i o n s c o n s t r u c t the P—M  cross-sectional c a p a c i t y curve (or choose a p r e - p r e p a r e d one). T h i s is a w e l l developed  procedure except t h a t i t is r e c o m m e n d e d e q u a t i o n s i n C h a p t e r X m a y be used.)  t h a t the curve be based o n the u n i t s t r e n g t h .  (The  206 2) C a l c u l a t e the E u l e r l o a d for the gross section 3) D e t e r m i n e p a r a m e t e r s A, a/b  a n d np.  P rC  (these m a y have been d e t e r m i n e d i n the first step)  A c c o r d i n g t o these three p a r a m e t e r s , choose a n a p p r o p r i a t e p r e - p r e p a r e d b u c k l i n g l o a d g r a p h (such as the one s h o w n i n F i g . 11.16) or t a b l e . E x a m i n e the s t a b i l i t y b y c h e c k i n g whether the design l o a d  (P/P r) C  is below the b u c k l i n g l o a d  (P^/Pcr)  at the design e c c e n t r i c i t y (e /<). If not, 0  repeat f r o m step 1. 4) T o check m a t e r i a l f a i l u r e , read P  k  and 6  k  at the design e c c e n t r i c i t y ( i n t e r p o l a t i o n  often  necessary), a n d c a l c u l a t e the m o m e n t m a g n i f i e r 6 b y u s i n g E q . 11.37. 5) M a g n i f y  the design end m o m e n t  checking if this moment  b y S a n d ensure t h a t m a t e r i a l f a i l u r e w i l l not occur by  c o m b i n e d w i t h the design l o a d falls w i t h i n the P—M  cross-sectional  c u r v e . If not, repeat f r o m step 1. T h e r e c o m m e n d e d design a p p r o a c h , the a u t h o r believes, can be extended to the case of u n e q u a l eccentricities ( w h i c h is i n c l u d e d i n the a l g o r i t h m ) w i t h o u t m u c h d i f f i c u l t y .  207 CHAPTER SUMMARY AND  1)  XII  CONCLUSIONS  T h e m e c h a n i c a l properties of concrete m a s o n r y subject to a x i a l c o m p r e s s i o n a n d out  b e n d i n g have been i n v e s t i g a t e d e x p e r i m e n t a l l y , b y testing block p r i s m s w i t h v a r i o u s  plane  bedding  a n d g r o u t i n g c o n d i t i o n s under v a r i o u s eccentricities. 2)  S p l i t t i n g f a i l u r e has been e x a m i n e d a n d H i l s d o r f s m o d e l has been revised i n the l i g h t o f b o t h  e x p e r i m e n t a l a n d a n a l y t i c a l w o r k . It is c o n c l u d e d t h a t the s p l i t t i n g f a i l u r e m o d e of concrete masonry  under  a x i a l c o m p r e s s i o n c a n n o t s i m p l y be a t t r i b u t e d to the lower stiffness of the  mortar joints. 3)  B r i t t l e f a i l u r e under u n i a x i a l c o m p r e s s i o n has been i n v e s t i g a t e d at the f u n d a m e n t a l level. A  qualitative model  was proposed  to e x p l a i n the s p l i t t i n g failure, a n d to reveal some o f  the  c h a r a c t e r i s t i c s o f concrete a n d other b r i t t l e m a t e r i a l s under a x i a l c o m p r e s s i o n . 4)  T h e j o i n t effect o n m a s o n r y  s t r e n g t h c a n be a t t r i b u t e d to the d i s t o r t i o n of the  uniform  compressive stress i n the v i c i n i t y of the j o i n t . 5) T h e deep b e a m b e n d i n g m o d e l proposed by S h r i v e for f a i l u r e of face-shell bedded  masonry  under a x i a l c o m p r e s s i o n has been reviewed a n d verified e x p e r i m e n t a l l y . 6)  B a s e d o n the f a i l u r e m e c h a n i s m a n d j o i n t effect s t u d y , i t is c o n c l u d e d t h a t c o n c e n t r i c a n d  eccentric c a p a c i t i e s s h o u l d be treated differently. It is s h o w n t h a t eccentric c a p a c i t y c a n be s a t i s f a c t o r i l y p r e d i c t e d o n the basis of m a s o n r y u n i t compressive s t r e n g t h . 7)  T h e b e h a v i o u r o f g r o u t e d m a s o n r y is h i g h l y governed b y the d e f o r m a t i o n properties of the  m a s o n r y c o n s t i t u e n t s . P r e m a t u r e c r a c k i n g is caused b y the i n c o m p a t i b i l i t y between b l o c k shell a n d g r o u t . T h e u l t i m a t e c a p a c i t y is m o r e s t r o n g l y governed b y the s t r e n g t h of the b l o c k shell. 8)  B a s e d o n the above observations, a n a n a l y t i c a l m o d e l c o n s i d e r i n g v e r t i c a l as w e l l as cross-  sectional deformation  i n t e r a c t i o n has been presented w h i c h gives satisfactory p r e d i c t i o n s for  u l t i m a t e c a p a c i t y a n d c r a c k i n g loads.  208 9)  B a s e d o n the observations a n d studies o n the m a s o n r y p r i s m c h a r a c t e r i s t i c s , a t h e o r e t i c a l  m o d e l has been developed t o s t u d y the slenderness a n d the s t a b i l i t y of concrete m a s o n r y w a l l s . Compared  w i t h e x p e r i m e n t s , the m o d e l  gives very  good p r e d i c t i o n s for l o w a n d  moderate  reinforcement r a t i o s . 10)  T h e g e o m e t r y , g r o u t i n g , a n d b e d d i n g c o n d i t i o n s a n d the reinforcement are q u a n t i f i e d b y a  few p a r a m e t e r s , a n d the m o d e l is presented i n a r e l a t i v e l y s i m p l e f o r m . It is d e m o n s t r a t e d t h a t t h i s s i m p l e , r a t i o n a l a p p r o a c h can easily be a d a p t e d to the design a n d a n a l y s i s of slender w a l l s .  209 REFERENCES  A k i o B a b a a n d O s a m u S e n b u (1986). Influencing factors o n p r i s m s t r e n g t h of g r o u t e d and  fracture  Symposium. Ashby,  mechanism  under  uniaxial  loading.  of 4th  Canadian  Masonry  1078-1092.  M . F. and  Cooksley,  S . D . (1986).  "The  failure  cracks under compressive stress states." Acta metall, Athey,  Proc.  masonry  J . W . (1982,  Edt.)  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Fracture  Mechanics of  251-299.  aspects of the fracture of concrete" Fracture Mechanics E l s e v i e r , 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-  <r  ij(ij  Al  dr  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 + \ Q Vi t  ds  a  dr  ij£ij  r = 2U  ( in view of Eq. A l )  A2  as a result of the application of the divergence theorem,  equilibrium  and  compatibility  conditions; where T , u,- and Q , v denote the {  t  t  tractions and associated displacements on the external boundary T  x  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 FIG. A l An Elastic Body Containing a Single Crack  Q = 0. t  If the opposite surfaces of the crack  217 slide against each other, as is the case in the model, then =  0, so that Q v i  i  Q  ^  n  0 and v  t  ^ 0, but Q  t  =  v  n  = 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. O f course, for our model this is  U = - - i - FA  If  friction  between  the  crack  surfaces  is  included, the  complicated. Restricting attention to our model, we have Q  t  =  situation  .  A4  becomes  more  / , 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). v may be approximated by the t  geometric relation between the crack opening and the sliding displacement, as shown in Fig. A2  v « 6/s'mct  A6  t  In view of E q . A 2 , 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  21  (5/s  FIG. A2  ma  G e o m e t r i c R e l a t i o n between C r a c k . Opening and Sliding Displacement  FIG. A3  A Crack Extended by a P a i r of S p l i t t i n g Forces  d i r e c t i o n to t h a t of the s l i d i n g d i s p l a c e m e n t . A n expression for the crack opening 6 is s t i l l needed. C o n s i d e r a crack w i t h i n i t i a l l e n g t h 2l  0  being extended t o 21 under the a c t i o n of a pair of forces P, as s h o w n i n F i g . A 3 . A c c o r d i n g  to the energy theorems concerning the f o r m a t i o n a n d extension of cracks i n the elastic s o l i d ( G o o d i e r 1968), we have  I Gj  dl  R e a r r a n g i n g the equation a n d n o t i n g E q . 3.5, we o b t a i n  A8  219 rl 8 =  2  4P(1-^ )  AP(l-v )  _  2  Eb sin (rrl/b)  TTE  ( tan(7r//26)  \  A9  \ tan(7r/ /26) J 0  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  dR = 2a M f dv = 2a M fia sina d8 t  A10  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  All  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  ( recall F= aw)  A12  220 i n w h i c h r e l a t i o n s g i v e n b y E q . 3.2 a n d E q . 3.4 are a p p l i e d . E q . A 1 2 i s t h e n i n t e g r a t e d a n d m a t c h e d t o t h e i n i t i a l c o n d i t i o n g i v e n b y E q . 3.18. T h e s o l u t i o n takes t h e f o r m as g i v e n b y E q . 3.22 w h e n t h e r e l a t i o n s defined b y E q s . 3.6, 3.19, 3.20 a n d 3.21 are used. T h e s o l u t i o n i s v a l i d f o r t h e w h o l e range except t h e f r i c t i o n t r a n s i t i o n a l i n t e r v a l , i n w h i c h t h e f r i c t i o n force is c h a n g i n g m a g n i t u d e as w e l l as d i r e c t i o n ; t h e r e l a t i o n g i v e n b y E q . A 5 does n o t t h e n h o l d . C e r t a i n l y , after the t r a n s i t i o n a l i n t e r v a l , t h e sign p r e c e d i n g u s h o u l d be changed. T h e s t a r t i n g p o i n t o f t h i s t r a n s i t i o n a l i n t e r v a l m a y be f o u n d b y s e t t i n g dS e q u a l t o zero. T h i s condition follows b y differentiating E q . A 9 :  (recall 0 = a r c sm(<r/f ) , 2  c  0  define t h i s s t a r t i n g p o i n t b y 0  = a r c sin(<r /'fc) , 2  O  X  0  or a  x  a n d recognize also a r c sm(a/f ) = 2  c  irl/b ) W e  ( q u a n t i t i e s a t t h i s p o i n t denoted w i t h s u b s p r i p t 1). D u r i n g  the t r a n s i t i o n a l i n t e r v a l , t h e expression f o r t h e tensile s p l i t t i n g force becomes  P = 2a ( i r s i n a c o s a — / ) s i n a  A14  T h e c r a c k e x t e n s i o n c o n d i t i o n is s t i l l governed b y E q . 3.4. R e c o g n i z i n g t h a t d u r i n g t h i s i n t e r v a l t h e c r a c k o p e n i n g r e m a i n s c o n s t a n t , we have a n e x t r a c o n d i t i o n  2  *W-" )t laong( 0ptglM = 8, TTE /2) 6  = constant  A15  o  T w o cases need t o be discussed. F i r s t w e assume / increases d u r i n g t h e i n t e r v a l . I t i s o b v i o u s t h e n t h a t E q . 3.4 a n d E q . A 1 5 c a n n o t be satisfied s i m u l t a n e o u s l y . F u r t h e r i n s p e c t i o n i n d i c a t e s t h a t t h e force defined b y E q . A 1 5 is a l w a y s higher t h a n t h a t d e f i n e d b y E q . 3.4. T h i s  221 a c t u a l l y i m p l i e s t h a t the crack w i l l e x t e n d i m m e d i a t e l y a n d the m a t e r i a l w i l l f a i l a l m o s t at the i n s t a n t the s t a r t i n g p o i n t is reached. In the second case, i f we assume t h a t the a p p l i e d l o a d retreats so fast t h a t  / or  P  r e m a i n s u n c h a n g e d d u r i n g the i n t e r v a l , t h e n b o t h E q . 3.4 a n d E q . A 1 5 are s a t i s f i e d , a n d the f r i c t i o n force / c a n be f o u n d b y u s i n g the r e l a t i o n g i v e n b y E q . A 1 4 . In view of E q . A 7 , the s t r a i n energy is t h e n expressed as  FA  Further  since  P  =  M [ crsinacosa  — 2a  —— ' 2a s i n a / s i n a  r e m a i n s c o n s t a n t , the f i n i s h p o i n t ,  P  x  A16  defined as <7 , c a n be f o u n d b y 2  e q u a t i n g the expression for P at the b e g i n n i n g p o i n t to t h a t at the f i n i s h p o i n t , as  r, = '  Since db =  dl =  c o s a — zzsina — — cosa + ^ s i n a  . , _ A17  1  0, the d i f f e r e n t i a l e q u a t i o n of the energy r e l a t i o n reduces to  dV  -  =  dU  0  A f t e r s u b s t i t u t i n g E q . 3.11 a n d E q . A 1 6  a  l  <  a <  cr  2  A18  , the s o l u t i o n of E q . ' A 1 8 t u r n s out to be a linear  r e l a t i o n between stress a n d s t r a i n :  e -  a 6, c o s a • _ L_5 (- C7cr 2b  A19  2  where C is a n i n t e g r a l constant f o u n d by the c o n d i t i o n s at the s t a r t i n g p o i n t . T h i s r e l a t i o n is used i n p l o t t i n g F i g . 3.9.  222 A P P E N D I X C . S o l u t i o n o f e q u a t i o n 4.1 A series s o l u t i o n c a n be f o r m e d b y the eigen f u n c t i o n s o f the p r o b l e m :  oo u(x,y) = ^2  X (x)  A20  sm(a y)  n  n  n=l  w h i c h satisfies b o u n d a r y c o n d i t i o n s specified b y E q s . 4.3 a n d 4.4; where X (x) is a f u n c t i o n o f x n  a n d a = mr/t . W h e n A 2 0 is s u b s t i t u t e d i n t o E q . 4.1, i t follows t h a t n  0  X (x) n  Recall that  -  2  k al  X {x) n  - 0  n=l,2  A21  oo  K — ^{\-v)/2. A 2 1 i s i n t e g r a t e d a n d s u b s t i t u t e d b a c k i n t o A 2 0 , w h i c h ,  when  b o u n d a r y c o n d i t i o n E q . 4.2 is a p p l i e d , reduces t o  oo u(x,y) = ^2  An sinh(/ca £) sin(a„j/)  A22  n  71=1  where A  n  i s a c o n s t a n t , w h i c h i s then f o u n d b y m a t c h i n g A 2 3 t o the b o u n d a r y c o n d i t i o n E q .  4.5:  oo ^^AnKctn  cosh^^,""  j sin(a„2/) =  1  "  q  71=1 to  t KCX„ 0  COSh(^)  J  4(l-V )qto 2  ^ T T ^ C O S ^ ^ )  b  .  .  . .  when n i s odd A23 w h e n n is even  223  E q . 4.6 f o l l o w s w h e n A 2 3 is s u b s t u t e d i n t o A 2 2 .  A P P E N D I X D . Coefficients A , m  Eq.  4.11 is actually  i n stress f u n c t i o n 4> specified b y e q u a t i o n 4.11  B  m  a summation  of Levy's  type  solutions  for plate  bending  ( T i m o s h e n k o a n d K r i e g e r , 1959), so i t is clear t h a t  V  4  $  =  A24  0  A n d i t i s also o b v i o u s t h a t E q . 4.11 i s c o n s t r u c t e d a c o o r d i n g t o the d i a m e t r i c a l l y s y m m e t r i c properties o f t h e p r o b l e m ,  so t h a t o n l y b o u n d a r y  c o n d i t i o n s a t x=0  a n d ?/=0 need t o be  considered. R e f e r r i n g t o F i g . 4.7, i f we integrate the b o u n d a r y d a t a we h a v e  <3> = c y + c  A25  2  x  with ®  x  = c  A26  3  at t h e b o u n d a r y x = 0; a n d  <& = c x + c 4  A27  5  with •*v(*)  T y(xfi)  =  4qt ^ 0  I"  2  dx + C  X  ^  6  cosh[(2n-l)/c7r(:E-a/2)/< ]  ^  (2n-l) cosh[(2w-l)K7ra/2* ]  °  0  2  0  6  ^  224 at y = 0; where c t o c x  are constants of i n t e g r a t i o n . N o t e i n A 2 8 , r  6  b u t w i t h the shifted o r i g i n , q a n d c m u s t v a n i s h for s y m m e t r y 4  xy  is specified by E q . 4.7  o f the p r o b l e m . T h u s 4> is  c o n s t a n t a l o n g the boundaries, a n d E q . 4.11 assumes c = c = 0 by the fact t h a t it is i m m a t e r i a l 2  5  to a d d a c o n s t a n t t o <£. F u r t h e r , the slopes defined by A 2 6 a n d A 2 8 m u s t also v a n i s h at the o r i g i n (r=2/=0) because of the c o n s t a n c y . T h i s leads to c = 0 a n d 3  c  fi  = -  2*2  A29  W h e n E q . 4.11 is differentiated a n d set equal to the b o u n d a r y c o n d i t i o n s , we o b t a i n t w o equations:  oo  i —  _  \A <V (y) TO=1  and  + -fBmbm  m  m  [ -TT ma  =  s i n ( ^ z ) + 5 SG (x)~| =  A  53 m=l  s i n ^ y j  m  m  m  0  A30  A31  $ (i) 9  where  S («) m  =  =  !p  sinh  ^  !22E(W2)  sinh  a  m  = ^  6  m  = nm. ( ( / ? t a n h / ? - l ) s i n h / ?  -  ((<*mtanha: — l ) s i n h a m  m  m  m  m  ^ t a n h ^ c o s h ^ ^ j  -  a  t a n h a  m  —  a cosha j  -  /? cosh/? )  m  m  c o s h ^ % ^  m  m  m  a n d <&y(x) is given b y A 2 8 . B y o r t h o g o n a l i t y , A 3 0 a n d A 3 1 become t w o linear s y s t e m equations,  225 w h i c h c a n be w r i t t e n s y m b o l i c a l l y  A32  Am "f" kmnBn — On  m, n = 1, 2  oo  A33  where ni  kmn  b  k  - -L  / \  SG (i) n  • miry .  sin^P  (fe  fc (x) sin^P <fe y  T h e r e f o r e , for f i n i t e size iV, i t is a l w a y s possible to solve for A  {A} =[[!}-  a  [k )[k ]j\c} b  {5} = -[ *»] {^}  m  a n d B , m = l , 2, m  N:  A34  A35  i n view o f A 3 2 a n d A 3 3 ; where [ I ] is a n i d e n t i t y m a t r i x of size N, a n d hence to o b t a i n a n a p p r o x i m a t i o n for  I n C h a p t e r I V , N=8 was used.  226 APPENDIX  E . D e r i v a t i o n of e q u a t i o n 11.5  T h e general s o l u t i o n for E q . 11.1 is  y =  B=  — t  0  A sin  +  P_ EI  cos  B  A36  for the b o u n d a r y c o n d i t i o n at the t o p . W h e n the b o u n d a r y c o n d i t i o n s at section C are  a p p l i e d , we o b t a i n f o l l o w i n g t w o relations  A sin  P_  + t  A cos  P_ EI  COS  0  (^))  (h—h  A37  (^))  -  A38  w h i c h lead to A  2  e  2  +  e  (  o  o  yl  p  sin  /l-h /h\  .  c  w  '( — 2 — — )* + U P -— ,  +  c  +  e  fc  \—1/2 1  0  A39  -p  , it c a n be r e w r i t t e n  _!  sin  A40 ^  -l  e ^  2  +  e  +  2  J.4  2  -l  c  e  2  e  +  2  e„  -JA  2  E q u a t i o n 11.5 f o l l o w s w h e n the r e l a t i o n g i v e n by A 3 9 is used.  +  2  A41 e  2  227  A P P E N D I X F . Integration of equations 11.14 and 11.16 By letting  jC - ft)'( 1  1  + + )•- *(+)*]  A42  and  (1-  <h(c) =  i- f b  (A f-  n )(l P  -  f)  A43  Eqs. 11.14 and 11.15 can be written as  2  dy  P  =  2  A44 G (c)  2  dx  2Eb l  x  and y _ b ~  i  ^i(c) Gl  A45  (c)  A45 defines the relationship between y and c, which must be used in integration of A44. B y recognizing  2  4-(Q\ = 2 dx\dx)  d2  i  1  dx  A46  A44 becomes  \dx)  Ebl J G i ( c ) V 6 J  A45 is then substituted into A47, which becomes, after integration by parts  A47  228 2  'dy\ <dx)  EM 2Ebl  G\{c)  +  Gl(c)  dc  A48  where  *M = ^  = A[(i) S - 2 (1-)]  A49  i n v i e w o f A 4 2 . N o w the p r o b l e m becomes t o integrate  J G{{c)  A50  It is seen t h a t the i n t e g r a n d is a r a t i o n a l f u n c t i o n o f c. T h e d e n o m i n a t o r is f o r m e d by the square of the G  x  f u n c t i o n , w h i c h , i n view o f A 4 3 , is q u a d r a t i c i n c. T h e n u m e r a t o r can be b r o k e n i n t o  t w o t e r m s . T h u s the i n t e g r a l c a n be c a r r i e d out b y a n y s t a n d a r d a p p r o a c h , for e x a m p l e , see C R C S t a n d a r d M a t h e m a t i c a l T a b l e s 2 7 t h E d . p245 (Beyer 1986). A f t e r a p p r o p r i a t e c a l c u l a t i o n , the result t u r n s out t o be rather s i m p l e :  _ 1  _  W 6 Gi(c)  A51  W h e n A 5 1 is s u b s t i t u t e d back i n t o A 4 8 w i t h a c o n s t a n t o f i n t e g r a t i o n , we o b t a i n  (2)' = A (»'«>)  A52  where fi^c) is defined b y E q . 11.19. E q . 11.18 takes the p o s i t i v e square root of A 5 2 referring t o F i g . 11.2. A s i m i l a r a p p r o a c h i s used t o integrate E q . 11.16. H o w e v e r , the e q u i v a l e n t F a n d G f u n c t i o n s become  229  T  2l) (  1  i-  2  and  G (c) = a  ( 1 -  i  h  1  f  +  ) ~  ) - i( 2  1  A  ( ( f )  3  -  -  c 2 6  f )( 2  1  +  2  f - f ) ) ]  n  y) + K  a 6  A  5  3  A54  T h e i n t e g r a n d o f the e q u i v a l e n t i n t e g r a l  2  _ f  ^2  " J  G (c)  (C)  A55  2  is also a r a t i o n a l f u n c t i o n . H o w e v e r , F (c) contains three terms b y d i f f e r e n t i a t i o n of A 5 3 :  H (c) = i j ( 1 - A ) ( - | - )  2  -  2(1-A)( -) i  A( 1-  A56  2  (t) )  A f t e r a l e n g t h y b u t c o n t r o l l a b l e c a l c u l a t i o n , i t t u r n s out a g a i n i n s i m i l a r f o r m to A 5 1  1-c/b  A57  G (c) 2  E q . 11.20 f o l l o w s after a p p r o p r i a t e s u b s t i t u t i o n s .  A P P E N D I X G . C o n f i g u r a t i o n o f a c o l u m n loaded w i t h double c u r v a t u r e b e n d i n g W e t r y t o shed some l i g h t o n the p r o b l e m b y i n v e s t i g a t i n g a n 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 t o p e c c e n t r i c i t y e according t o  t  and b o t t o m eccentricity e , h  w h i c h w i l l deflect  230 y=  e, — e coskl . sinfc/ smkx + e coskx  A58  h  b  where k =  \P/EI.  A 5 8 m u s t v a n i s h a t the inflection p o i n t x , w h i c h leads t o 0  tan fcr 0  It  sinfc/ coskl —  is clear  column  that  e /e t  A59 b  when  e /e = — l, t  b  i.e.  the  i s loaded a n t i - s y m m e t r i c a l l y , x = l/2. 0  However, we w i l l show t h a t this c o n f i g u r a t i o n is  n o t stable  approached.  when  the Euler  load i s  F o r this purpose we i n t r o d u c e a  s m a l l p e r t u r b a t i o n e t o the l o a d i n g c o n d i t i o n s F I G . A 4 A C o l u m n Loaded with Double Curvature Bending  A60  (*>0)  e«/e =-(l-0 t  a n d examine the s e n s i t i v i t y o f the deflected c o n f i g u r a t i o n . W h e n A 6 0 is s u b s t i t u t e d i n t o A 5 9 , we obtain  F = (coskl + 1 — e)tankx  0  — sinkl = 0  A61  T h e sensitivity o f the c o n f i g u r a t i o n t o t h e p e r t u r b a t i o n is reflected i n the d e r i v a t i v e o f x w i t h 0  respect to e  231 dxo de  Fe  _  Fx _ 0e  Q  sin2fcE  0  2k(l+coskl)  sinfc/  2k(l + coskl)  A62  It is seen t h a t w h e n the l o a d P is r e l a t i v e l y l o w , the d e r i v a t i v e w i l l be s m a l l . H o w e v e r , w h e n P approaches the E u l e r l o a d , kl -» IT, i t becomes u n b o u n d e d (note A 6 2 is i n a n i n d e t e r m i n a t e f o r m , L ' H o s p i t a l ' s rule has been a p p l i e d once). T h e h i g h s e n s i t i v i t y is o b v i o u s . T h a t is, w h e n the E u l e r l o a d is a p p r o a c h e d , the c o l u m n w i l l have a very h i g h tendency t o d e p a r t f r o m its o r i g i n a l a n t i symmetric configuration. In r e a l i t y , i t is a l w a y s reasonable t o assume some i m p e r f e c t i o n 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 c a n be r e w r i t t e n as  A63  It is seen b y the second e q u a t i o n of A 6 3 t h a t  A64  T h a t is, the c o l u m n w i l l assume its lowest b u c k l i n g c o n f i g u r a t i o n , for a n y s m a l l i m p e r f e c t i o n e, w h e n the E u l e r l o a d is a p p r o a c h e d . F o r a n o n l i n e a r c o l u m n , the s i t u a t i o n becomes m u c h m o r e c o m p l i c a t e d . It appears t h a t a s i m i l a r tendency w o u l d c o n t r o l the b e h a v i o u r . F o r design purposes, i t is reasonable t o assume, conservatively, that this would happen.  232  A P P E N D I X H . E l e c t r o n i c C i r c u i t Used i n D e t e c t i n g M a c r o s c o p i c S p l i t t i n g  4017 1234 56  Vcc DATA 74LS373  74LS74  74LS04 74LS30  V.  O  LEDs  DQ TJ  T v v  n  v  •74LS32 ct  LEOE  Vcc  74LS02  100K NO  Vcc  Vcc 74LS373  DATA 74LS74  0  : LEDs  D Q >  TJ  V  LE OE  74LS32  V  (Part)  V  100K Vpf  v  >— 74LS02  /  233 A P P E N D I X J. Computer Program Calculating Buckling Load  and Moment Magnifier of Concrete Masonry C C C  PROGRAM TO EVALUATE MAXIMUM BUCKLING LOAD OF REINFORCED MASONRY EXTERNAL F 1 , F 2 , F 3 , F 4 COMMON R 0 , R C , R S , E 0 , E C , E F , E E , D 1 , D 2 DIMENSION TITLE(20)  C C  *  c  C C C C C C C C C C C C C C C C C C C  *  NOTATION OF VARIABLES RO = RC = RS = EO = EA = EB = EC = EF = D1.D2  CROSS-SECTIONAL FACTOR A/B CORE RATIO STEEL RATIO (EQUAL) END ECCENTRICITY SMALLER END ECCENTRICITY LARGER END ECCENTRICITY (EB.GE.ABS(EA)) CRACKING ( KERN ) ECCENTRICITY ECCENTRICITY CORRESPONDING TO FLANGE CRACKING = INTERGAL CONSTANTS  NOTE: ALL ECCENTRICITIES ARE TAKEN AS RATIOS TO HALF DEPTH OF CROSS - SECTION ****  *  *  *  ***  ' DEFINE PARAMETERS READ(5,1)(TITLE(I),1=1,20) WRITE(6,2)(TITLE(I),1=1,20) FORMAT(20A4) FORMAT(//,1H1,/,24X,20A4,/) READ(5,3)RO,RC,RS,EA,EB FORMAT(5F10.5) IF(RC.LT.O.O.OR.RS.LT.0.O.OR.EB.LT.ABS(EA))STOP  1 2 3  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 4 '  WRITE(6,4)RO,RC,RS,EC,EF.EU FORMAT!/,T25,'CROSS-SECTIONAL P R O P E R T I E S ' , / / , ' S E C . FACTOR = . F8 . 3,5X,'CORE RATIO = ' , F 8 . 3 , 5 X , ' S T E E L RATIO = ' , F 8 . 3 , / / / , .'EC/B = ' ,F8.3,5X,'EF/B = ' ,F8.3.5X,'EU/B = \ F 8 . 3 )  ',  C 5  WRITE(6,5)EA,EB F0RMAT(/,T25,'LOADING  CONDITIONS',//,'EA/B  = ',F8.3,5X,'EB/B = '  . , F 8 . 3 , / / / , T 2 5 , 'CM/B' ,11X, 'P/PCR' , 10X , 'EM/EO' . 10X, 'ERROR') EO=EB C C C  EVALUATE BUCKLING LOAD PM=0.0 DM=0.0 IF(EO.GT.EC)GO  C C C  END ECCENTRICITY EB LESS THAN CRACKING ECCENTRICITY CD=CMU/30. CM=6.0 DO 500 1=1,29 CM=CM+CD IF(CM.GT.CF)GO D1=01(CM) TH=D1-01(0.) CML=CM-5.E-6  C C  TO 200  SUB-FUNCTION  TO 110  CADRE PERFORMS NUMERICAL  INTEGRATION  C  110  505 500  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 . / ( P I ' P I ) * (DRT'SUSQ) * *2 IF(P.LE.PM)GO TO 1000 PM=P IF(EO.NE.0.0)DM=E1(CM)/EO GO TO 505 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 WRITE(6,470)CM,P,DM,ERROR CONTINUE WRITE(6,555) GO TO 1000 IF(EO.GE.EF)GO TO 300  200 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 C  210  515 510  SUBROUTINE ROOT FINDS CO FOR GIVEN EO 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) I F t E A . L E . E C . A N D . E A . G E . - 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 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 WRITE(6,470)CM,P,DM,ERROR CONTINUE WRITE(6,555) GO TO 1000 IF(E0.GE.EU)G0 TO 900  300 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  IF(EA.NE.E0.AND.EE.GT.EF)CALL ROOT(C1,CL,F4,ER) I F ( E E . L E . E F . A N D . E E . G T . E C ) C A L L 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) +0.5*CADRE(F1,CF,C2,0. 0001,0.,ERR) I F ( E A . L T . E G . A N D . E A . G E . -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 . 0 0 0 1 , 0 . , E R R ) + 0 . 5 / D R T * Q ( T H , E C , - 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 R 0 , R C , R S , E 0 , E C , E F , E E , D 1 , D 2 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 R 0 , R C , R S , E 0 , E C , E F , E E , D 1 , D 2  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 C  FUNCTION DEFINING CRACKING ECCENTRICTY 1 FUNCTION F3(C) COMMON RO,RC,RS,EO,EC,EF,EE,D1,D2 F3=EE-E1(C) RETURN END  C  C C  C C C  C C C  C C C  FUNCTION DEFINING CRACKING ECCENTRICTY 2 FUNCTION F4(C) COMMON R0,RC,RS,E0.EC,EF,EE,D1,D2 F4=EE-E2(C) RETURN END FUNCTION DEFINING THE TERM IN INTERGRAND 1 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 FUNCTION DEFINING THE TERM IN INTERGRAND 2 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./6./(T1*T1-0.25'RO*T4*T4 + RS'T2)"2 RETURN END INVERSING SIN FUNCTIONS FUNCTION Q(TH,EV,EU) COMMON RO,RC,RS,EO.EC,EF.EE,D1,D2  3.«T5*T2*T2)  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 C  FUNCTION DEFINING CRACKING ECCENTRICITY 1 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 C  FUNCTION DEFINING CRACKING ECCENTRICITY 2 FUNCTION E2(C) COMMON R O , R C , R S , E O , E C , E F , E E . D 1 , D 2 . T1=(1.-0.5*C)*(1.-0.5'C) T2=( 1 . - R C - C ) * ( 1 . - R C - 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 C  20  SUBROUTINE FINDING ZERO OF FUNCTION F 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 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) . G T . T D G O TO 20 A=X RETURN END  2  

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