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Slenderness of masonry block walls Man, Eric 1981

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SLENDERNESS OF MASONRY BLOCK WALLS By ERIC MAN B.A.Sc, University of B r i t i s h Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © E r i c Man, 1981 In present ing t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission fo r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my department or by h i s or her represen ta t i ves . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion . E r i c H.Y. Man Department of C i v i l Engineer ing The U n i v e r s i t y of B r i t i s h Columbia 2324 Main Mal l Vancouver, B . C . , Canada V6T 1W5 - i i -ABSTRACT The slenderness e f f e c t of masonry wal ls i s examined through the combination of e x i s t i n g a n a l y t i c a l methods and experimental r e s u l t s for masonry w a l l s . The current code design c a p a c i t i e s , when compared with the t h e o r e t i c a l va lues , are found to be i n c o n s i s t e n t and over -conserva t i ve . A L im i t State Design approach for p l a i n and r e i n f o r c e d masonry wal ls i s proposed based on the Moment Magni f ier Method, which i s the current ACI design format for slender concrete columns. - i i i -TABLE OF CONTENTS Page ABSTRACT - i -TABLE OF CONTENTS - i i i -LIST OF FIGURE -v-LIST OF TABLES - v i i -ACKNOWLEDGEMENTS - i x -I. INTRODUCTION 1 1.1 Background 1 1.2 L i t e r a t u r e Review 1 1.3 Scop and Purpose 4 9 I I . EQUIVALENT STRESS-STRAIN CURVES FOR PLAIN AND REINFORCED 11 MASONRY WALLS 2.1 General 11 2.2 The Computer Program for Simulation of the Wall's Behaviour 11 2.3 Equivalent Stress-Strain Curve 14 2.4 Experimental Data 15 2.5 Development of the Stress-Strain Curves 20 2.6 F i n a l S t r e s s - S t r a i n Curves f o r P l a i n and P a r t i a l l y Grouted 25 Masonry Walls 2.7 E f f e c t of Tensile Strength 26 I I I . VERIFICATION OF THE ANALYSIS 28 3.1 Theoretical Interaction Diagrams 28 3.2 Comparison of the Experimental and Theoretical Results f o r 28 Face-Shell Constructed P l a i n Masonry Walls 3.2.1 Short Wall Capacity 33 3.2.2 F u l l Size Walls 34 3.3 Comparison of the Experimental and the Theoretical Results 37 for P a r t i a l l y Grouted Reinforced Walls 3.3.1 Short Wall Capacity 37 3.3.2 F u l l Size Walls 39 - i v -3.4 Joint Reinforcement IV. EVALUATION OF THE CODE DESIGN METHOD 4.1 Introduction 4.2 Code Design Equations 4.3 Comparison of the Theoretical Capacities and the Code Design Values f or Masonry Walls 4.3.1 General 4.3.2 The Theoretical Walls 4.3.3 Ca l c u l a t i o n of the Code Design Values 4.3.4 Comparison of the Results 4.3.5 General Remarks V. MODIFICATION OF THE MOMENT MAGNIFIER METHOD FOR MASONRY  DESIGN 5.1 The Moment Magnifier Method 5.2 The Modulus of E l a s t i c i t y of Masonry Assemblages 5.3 General Function for the R i g i d i t y Reduction Factor (X) 5.4 The Function (K g) 5.5 The Expression f or the Load Influence Factor K (P/P ) p o 5.6 Improvement of the Established Functions 5.7 Discussions on the Implied Interaction Diagrams 5.8 Application for Unreinforced Masonry Walls 5.9 Design of Slender Masonry Walls VI. CONCLUSIONS AND RECOMMENDATIONS REFERENCES - v -Page LIST OF FIGURES 1.1 Interaction Curve for a Pinned Column With Equal End 3 E c c e n t r i c i t i e s 2.1 A family of Load-Moment Interaction Curves for Various Heights 13 of the Member 2.2 Typical Section of Masonry Walls ( P l a i n and P a r t i a l l y Grouted) 17 2.3a E f f e c t i v e Section for 8" P l a i n Wall Masonry and i t s Coordinates 19 2.3b E f f e c t i v e Section for 8" ( P a r t i a l l y Grouted) Reinforced Masonry 19 Wall and i t s Coordinates 2.4 Experimental and Computed Deflected Shapes of a Face-Shell 21 Constructed 8" P l a i n Masonry Wall 2.5 Experimental and Computed Deflected Shapes of a 8" Reinforced 23 Masonry Wall 2.6 Stress-Strain Curves for Masonry 24 3.1 Interaction Diagram of 8" Face-Shell Constructed P l a i n Masonry 29 Wall and the Experimental Results 3.2 Theoretical Interaction Diagram for 8" Reinforced Masonry Walls 30 (3-#3 @ <£) With the Associated Experimental Results 3.3 Theoretical Interaction Diagram for 8" Reinforced Masonry Walls 31 (3-#6 @ C_) with Associated Experimental Results 3.4 Theoretical Interaction Diagram for,8" Reinforced Masonry Walls 32 (3-#9 @ <£) With the Associated Experimental Results 4.1 Theoretical Interaction Diagram for 12" P l a i n Masonry Walls 48 5.1 Relationship Between Normalized R i g i d i t y Reduction Factor and 74 the Slenderness Ratios (L/r) 5.2 Function K and the Slenderness Ratios (L/r) 75 s 5.3 Relationship Between K p and the Load Ratios (P/P Q) for 80 Reinforced Masonry Walls 5.4 Comparison of Theoretical and Implied Interaction Diagram for 85 8" Reinforced Masonry Wall (3-#3 @ <£) 5.5 Relationship of the Cut-off Value for K p and the V e r t i c a l 88 Reinforcement Ratio (p) - v i -Page 5.6 Comparison of Theroetical and Implied Interaction Diagram for 90 8" Reinforced Masonry Walls (3-#3 @ C_) @ Low (P/P Q) Ratios 5.7 Comparison of Theoretical and Implied Interation Diagrams 91 for 8" Reinforced Masonry Walls (3-#3 @ CJ 5.8 Comparison of Theoretical and Implied Interaction Diagrams 92 for Reinforced Masonry Walls (3-#3 @ C_) @ Low (P/P Q) Ratios 5.9 Comparison of Theoretical and Implied Interaction Diagrams 93 for 8" Reinforced Masonry Walls (3-#6 @ G_) 5.10 Comparison of Theoretical and Implied Interaction Diagrams 94 for 8" Reinforced Masonry Walls (3-#9 @ <£) @ Low (P/P Q) Ratios 5.11 Comparison of Theoretical and Implied Interaction Diagrams 95 for 8" Reinforced Masonry Walls (3-#9 @ <£) 5.12 A Plot of Normalized R i g i d i t y Reduction Factors and 97 Slenderness Ratios (L/r) for P l a i n Masonry Walls 5.13 Relationship.of K p and (P/P Q) for 8" P l a i n Masonry Walls 98 5.14 Theoretical and Implied Interaction Diagrams for 8" P l a i n 102 Masonry Walls 5.15 Theoretical and Implied Interaction Diagrams for 12" P l a i n 103 Masonry Walls - v i i -Page LIST OF TABLES 3.1 Comparison of Experimental Results and Theoretical Values of 35 Short Wall Capacities of Face-Shell Constructed P l a i n Masonry Walls 3.2 Comparison of Experimental Results and Theoretical Values f o r , 3 6 F u l l Size, Face-Shell Constructed P l a i n Masonry Walls 3.3 Comparison of Test Results and Theoretical Capacities for 38 A x i a l l y Loaded 8" Reinforced Masonry Walls 3.4a Comparison of Test and Theoretical Capacities for F u l l Size 40 Reinforced Masonry Walls (3-#9 @ C_) 3.4b Comparison of Test and Theoretical Capacities for 137.0 Heights, 8 i n . Reinforced 8" Masonry Walls i n . 41 4.1 Theoretical Capacities for 8" P l a i n Masonry Walls 49 4.2 Theoretical Capacities for 12" P l a i n Masonry Walls 49 4.3 Theoretical Capacities @ £) for 8" Reinforced Masonry Walls (3-#3 50 4.4 Theoretical Capacities @ £) for 8" Reinforced Masonry Walls (3-#6 51 4.5 Theoretical Capacities @ 9 for 8" Reinforced Masonry Walls (3-#9 52 4.6 Allowable Loads for 8" P l a i n Masonry Walls 54 4.7 Allowable Loads for 12' ' P l a i n Masonry Walls 54 4.8 Allowable Loads for 8" Reinforced Masonry Walls (3-#3 @ 55 4.9 Allowable Loads for 8" Reinforced Masonry Walls (3-#6 @ 56 4.10 Allowable Capacities for Reinforced Masonry Walls (3-#9 @ <£) 57 4.11 Factors of Safety for 8" P l a i n Masonry Walls i n Current Design <£ 59 4.12 Factors of Safety for 12" P l a i n Masonry Walls i n Current Masonry Design Code 59 4.13 Factor of Safety for 8' Current Masonry Design ' Reinforced Masonry Walls (3-#3 Code @ <£) i n 60 4.14 Factor of Safety for 8' Current Masonry Design ' Reinforced Masonry Walls (3-#6 Code @ «2) i n 61 - v i i i -Page 4.15 Factor of Safety for 8" Reinforced Masonry Walls (3-#9 @ C_) i n 62 Current Masonry Design Code 5.1 R i g i d i t y Reduction Factors for 8" P l a i n Masonry Walls 68 5.2 R i g i d i t y Reduction Factors for 12" P l a i n Masonry Walls 69 5.3 R i g i d i t y Reduction Factors for 8" Reinforced Masonry Walls 70 (3-#3 @ <E) 5.4 R i g i d i t y Reduction Factors for 8" Reinforced Masonry Walls 71 (3-#6 @ <E) 5.5 R i g i d i t y Reduction Factors for 8" Reinforced Masonry Walls 72 (3-#9 @ G_) 5.6 Normalized R i g i d i t y Reduction Factors for 8" Reinforced 76 Masonry Walls (3-#3 @ (£) 5.7 Normalized R i g i d i t y Reduction Factors for 8" Reinforced 77 Masonry Walls (3-#6 @ <£) 5.8 Normalized R i g i d i t y Reduction Factors for 8" Reinforced 78 Masonry Walls (3-#9 @»<Q 5.9 Overall Average Values of Normalized R i g i d i t y Reduction 82 Factors f o r 8" Reinforced Masonry Walls 5.10 Average Values of K p for Reinforced Masonry Walls (3-#3 @ <£) 82 5.11 Average Values of K p for Reinforced Masonry Walls (3-#6 @ <£) 83 5.12 Average Values of K p for Reinforced Masonry Walls (3-#9 @ C_) 83 5.13 Values of K g for 8" P l a i n Masonry Walls 100 5.14 Values of K g for 12" P l a i n Masonry Walls 100 5.15 Average Values of Kp for 8" P l a i n Masonry Walls 101 5.16 Average Values of K for 12" P l a i n Masonry Walls 101 - i x -ACKNOWLEDGEMENT The author wishes to thank Drs- N.D. Nathan and D.L. Anderson for t h e i r advise and assistance i n the preparation of t h i s t h e s i s . F i n a n c i a l support was provided by the National Science and Engineering Research Council through grant number 67-7603. 1 I. INTRODUCTION 1.1 Background The idea of us ing masonry as a b u i l d i n g mater ia l has been around fo r c e n t u r i e s . In the 'pas t two decades, with economy on i t s s i d e , masonry mater ia l has been widely employed i n many i n d u s t r i a l l o w - r i s e s t r u c t u r e s , warehouses, and shopping m a l l s . Due to both the economy of cons t ruc t ion and the p l e a s i n g appearance presented by masonry s t r u c t u r e s , the number of h i g h - r i s e apartments and o f f i c e b u i l d i n g s , with masonry as the b a s i c const ruc t ion m a t e r i a l , i s ever i n c r e a s i n g . As the degree of complexity of masonry s t ructures i n c r e a s e s , the demand for a s o p h i s t i c a t e d design procedure for masonry i s inexorab le . When compared with design codes fo r other mater ia ls (eg: s t e e l and concre te ) , the present Canadian masonry design c o d e 1 i s not u p - t o - d a t e . The design of load bear ing members i s s t i l l based on an empi r i ca l al lowable s t ress design approach, and some important aspects i n d e s i g n , such as sec t ion geometry and support c o n d i t i o n s , have been neg lec ted . In order to keep pace with the other design codes, the next important step i n the evo lu t ion of"masonry design i s the in t roduct ion of l i m i t s ta tes des ign . The l i m i t s tates design approach has already been adopted i n Mexican and some European codes for masonry. The design method al lows the designer to e x p l o i t the p o t e n t i a l of the m a t e r i a l , and deal i n d i v i d u a l l y with each aspect of des ign , such as mater ia l s t rength , loading c o n d i t i o n s , end c o n d i t i o n s , and slenderness e f f e c t s . Th is gives the l i m i t s ta tes design approach o v e r a l l supremacy over the convent ional a l lowable s t r e s s des ign . When accounting for slenderness e f f e c t s i n masonry wal l d e s i g n , the l i m i t a t i o n of the slenderness r a t i o i n the current code may be over conservat ive . Recent ly , the development of concrete const ruc t ion technology has been very r a p i d , and an improvement i n the eva luat ion of slenderness e f f e c t s could govern the choice of masonry or concrete t i l t - u p wal l c o n s t r u c t i o n . With t h i s important economic aspect i n mind, t h i s study w i l l deal with the eva luat ion of the slenderness e f f e c t i n masonry w a l l s . When designing load bear ing members, i t i s important to bear i n mind that the capaci ty of the members depends on the c r i t e r i a of mater ia l f a i l -ure and i n s t a b i l i t y f a i l u r e . Th is p a r t i c u l a r aspect- i s c l e a r l y d iscussed by Nathan 2 . F i g . 1.1 shows the i n t e r a c t i o n curve for a pinned column with equal end moments tending to cause s i n g l e curvature . The l i n e 0-A def ines the r e l a t i o n s h i p between load and end moment. However the midspan moment i s magnif ied by the member d e f l e c t i o n , and the corresponding load-moment r e l a t i o n s h i p i s def ined by 0-B. Mater ia l f a i l u r e occurs at Point B, when the end condi t ions are as ind ica ted at Point C . At greater . e c c e n t r i c i t i e s , the midspan d e f l e c t i o n increases to a po in t such as D and the member becomes unstab le . D e f l e c t i o n then increases suddenly to f a i l u r e , and the end condi t ions at maximum load are given by E . Thus MOMENT F ig . 1.1 Interaction Curve for a Pinned Column with Equal End Eccent r i c i t i es i n t h i s i n s t a b i l i t y case , maximum load i s a c t u a l l y independent of the short-column i n t e r a c t i o n curve — independent of mater ia l f a i l u r e . I t i s c l e a r , the re fo re , that the moment magnif ier procedure, i n which the design moment i s magnif ied and compared with the short column moment, i s qu i te d ivorced from the r e a l i t y i n cases of i n s t a b i l i t y f a i l u r e . In most cases , wal ls with low a x i a l loads and high moments a r i s i n g from wind pressure , do f a i l i n the i n s t a b i l i t y mode. Use of the moment magni f ier method i s then an a r t i f i c i a l empi r ica l device with but two j u s t i f i c a t i o n s ; 1) i t i s f a m i l i a r to Engineers , 2) we do not as yet have any other way of handl ing the problem by a simple design o f f i c e procedure. 1.2 L i t e r a t u r e Reivew In the past decade, a s u b s t a n t i a l amount of masonry research has been performed i n Canada. The understanding of the mater ia l behaviour i n masonry has improved and much of the engineer ing knowledge has been put in to upgrading masonry design methods. A few design methods based on r a t i o n a l ana lys is have been put forward and a b r i e f review of work done i n t h i s area i s presented i n t h i s sec t ion . -Ojinaga and Turkst ra recommended a design method which was based on the convent ional d i r e c t P-A approach. It invo lved est imat ing the e f f e c t i v e moment of i n e r t i a of the sec t ion according to the load ing c o n d i t i o n . The maximum moment was found by combining the primary bending moments with the bending moments due to a x i a l forces ac t ing through the d e f l e c t i o n (the s o - c a l l e d P-A moment). The short column i n t e r a c t i o n equations der ived by Y o k e l , Mathey and D i k k e r 1 6 for p l a i n masonry wal ls with no t e n s i l e strength were used fo r c a l c u l a t i n g sec t ion c a p a c i t i e s . When no t ransverse load a c t s , the 5. upper l i m i t for the use of the short column capaci ty was assessed a s : L / r = 35 - 17.5 e-^/ e2 f ° r 0 * e l / / e 2 * ^ ( f ° r s i n g l e curvature) (1.1.1) and L / r = 35 - 35 e- j . / e 2 f ° r 0 * e l ^ e 2 * ^ o r double curvature) (1.1.2) The recommended minimum e c c e n t r i c i t y was 1/12 of the wal l th ickness and the maximum slenderness r a t i o (L / r ) allowed was 80 .0 . I f the slenderness e f f e c t had to be c a l c u l a t e d , the e f f e c t i v e moment of i n e r t i a was determined as: I = (I + i 0 /4 fo r 0 < e / e o < 0 (1.2.1) e f f end 1 end 2 1 2 and I ^ = min f - ( I , n + I) /4 (1.2.2) e f f - ( I e n d 1 + - ( I . + -(I  I ) /4 for -1 < e , / e „ * 0 (1.2.3) end 2 1 2 I , , and I , „ = uncracked moment of i n e r t i a , cracked i f appropr ia te , end 1 end 2 I = uncracked moment of i n e r t i a . The d e f l e c t i o n s were c a l c u l a t e d us ing an e l a s t i c d e f l e c t i o n equation and the primary bending moment. Super- imposi t ion of the moment diagram due to end moments, t ransverse loads and the v e r t i c a l load a c t i n g through the de f lec ted shape gave the maximum moment for des ign . Turning from p l a i n to r e i n f o r c e d masonry w a l l s , Ojinaga and Turkst ra recommended the same formulat ion and procedure i n "The Design of Re in for -D ced Masonry Walls and Columns." Since the a n a l y s i s on which the method depends considers only mater-i a l f a i l u r e , a l i m i t on slenderness had to be imposed to avoid p o t e n t i a l cases of i n s t a b i l i t y . For a s o l i d s e c t i o n , the imposed maximum s lender -ness r a t i o of 80 i s more conservat ive than the current code p r o v i s i o n , and appears to be unecessary at small e c c e n t r i c i t i e s and high loads . However, when a member i s loaded with a combination of high end-moment and low v e r t i c a l l o a d , i n s t a b i l i t y f a i l u r e can occur at a slenderness r a t i o much l e s s than 80. 6. Yokel and Dikker studied the possibility of adopting the Magnifier Method presently used in concrete design for masonry design i n Strength of Load Bearing Masonry Walls, 5. In the paper, a linear stress-strain relationship was assumed, flexural compressive strength (as defined by the Uniform Building Code) was replaced by the axial compressive strength (again as defined by the Uniform Building Code), and a strain gradient effect was considered at high eccentricity. Based on the experimental results of 76 concrete masonry walls, some hollow and some reinforced, Yokel and Dikker believed that at high eccentricity, the flexural compressive strength was 1.6 times the axial prism compression strength. When computing the short wall interaction diagram 1.6 fm' was used as the compressive strength of the material when the eccentricity exceeded 1/6 the wall thickness. For pure axial load, the prism strength (fm') was used; and for eccentricities between 0 and 1/6 the wall thickness a linear interpolation between these values was recommended. When calculating the slenderness effect, the buckling load had to be computed. The authors recommended fixed values for the rig i d i t y reduction factors, such that, F T ( E I ) e f f = - ( 1 ' 3 ) where EI = f u l l section r i g i d i t y A = 3 . 5 for plain masonry walls A = 2.5 for reinforced masonry walls These values were used in a formula similar to that used by the ACI code for concrete. 7 . Because of the use of constant r i g i d i t y reduction f a c t o r s , the slenderness e f f e c t cannot be modified to f i t a l l circumstances. It i s found that the r e s u l t i s very conservative at small e c c e n t r i c i t i e s , but becomes unconservative at large e c c e n t r i c i t i e s . Y o k e l 6 made a s t a b i l i t y and capacity analysis of compression members with no t e n s i l e strength. He assumed that the material s t r e s s - s t r a i n r e l a t i o n s h i p was l i n e a r i n the compression range, and the magnitude of d e f l e c t i o n was small, so that a small d e f l e c t i o n approximation could be applied. The section of the member was assumed to be s o l i d and rectangular, and the resultant c r i t i c a l load was computed as: P = 0.64 n% bu 3 / h 2 (1.4) cr 1 where b = width of wall h = height of wall u^ = distance between thrust l i n e and the compression edge of section, or, = (t/2 - e) E = i n i t i a l modulus of material Applying the above r e s u l t to a s o l i d masonry wall, the general equations f o r wall capacity as a f r a c t i o n of P give: At e = t/6: = min 0.8 u /u (related to material compr o 1 sion f a i l u r e : 1.4.1) 192/(h/t) (related to i n s t a b i l i t y f a i l u r e : 1.4.2) u Q = distance between l i n e of action of compressive load and compressive face of member at mid-height 8 . At e = t/3: p— = min 0.4 u Q/u^ (compression f a i l u r e : 1.4.3) L _ 2 4 / ( h / t ) 2 ( i n s t a b i l i t y f a i l u r e : 1.4.4) Hatzinikolas, Longworth, and Warwaruk 7' 8, adopted the moment magnifier method i n masonry design. When evaluating the slenderness e f f e c t , they performed a s t a b i l i t y analysis s i m i l a r to the work by Y o k e l 6 and ar r i v e d at the same r e s u l t . They further s i m p l i f i e d equation 1.4 and obtained the following r e s u l t . P = 8(1/2 - e / t ) 3 E n 2 I / h 2 (1.5) cr o b t 3 where I Q = -JTJ", moment i n e r t i a of the (rectangular) section. They believed that t e n s i l e strength was important for analysis of load bearing masonry members, and included t h i s i n the design formulation, which became: P = n 2 E I / h 2 cr 1 where I ± = 8(0.5 - e/t + - ^ ) 3 I Q « " 5 f t , / f m a x 5 = 2tP/A f max 2e. max t f 1 = t e n s i l e strength of masonry (1.6.1) (1.6.2) (1.6.3) (1.6.4) (1.6.5) If the member was loaded i n double curvature bending, the design method was more complicated; an "equivalent stepped column", with a jump i n the value of the moment of i n e r t i a was introduced. The buckling load was then: 9. P a E I / h 2 o (1.7) cr where ot = buckling c o e f f i c i e n t for the stepped column or wall ( i t i s a function of l o c a t i o n of i n f l e c t i o n point and r a t i o of e f f e c t i v e moment of i n e r t i a and gross-section i n e r t i a ) They also proposed a lower l i m i t on the f l e x u r a l s t i f f n e s s f o r a reinforced masonry member, namely, The proposed method was compared to a series of experiments they had performed, and the re s u l t was said to be s a t i s f a c t o r y . There are some draw-backs to t h i s design method. The analysis was o r i g i n a l l y based on the assumption that the section i s s o l i d and rectangular and i t does not seem j u s t i f i a b l e to apply the same analysis to p l a i n masonry walls with a hollow section, which have a completely d i f f e r e n t section geometry. At a combination of high moment and low a x i a l load, the design method does not account for the contribution of v e r t i c a l reinforcement. 1.3 Scope and Purposes The current ACI format f o r concrete design works well f o r load bearing members with high a x i a l load and low slenderness r a t i o . Since masonry walls usually serve as panel elements i n structures, the a x i a l applied load can be low with high applied bending moment, and the slenderness r a t i o i s usually higher than i n concrete load bearing members, so that i t i s not reasonable to apply the ACI design format for concrete d i r e c t l y to masonry design. Recently, a great number of tests have been performed on masonry walls with various combinations of a x i a l load and moment. With the E I (0.5 - e/t) > 0.10 E I (1.8) o v ' o ID. a v a i l a b i l i t y of these experimental r e s u l t s and some e s t a b l i s h e d theory , r a t i o n a l a n a l y s i s can be made to evaluate the slenderness e f f e c t i n masonry w a l l s . Since the ACI design method i s f a m i l i a r to a l l design engineers and there i s no new dependable a l t e r n a t i v e , t h i s study w i l l concentrate on modifying the current ACI design format, such that i t i s app l icab le for masonry des ign . 11. I I . DEVELOPMENT OF STRESS-STRAIN CURVES FOR PLAIN AND  REINFORCED MASONRY WALLS 2.1 General The c o n s t i t u t i v e p roper t i es of a mater ia l govern i t s behaviour under external i n f l u e n c e s . Knowing the c o n s t i t u t i v e proper t ies of the mater ia l and the state of s t r e s s , deformation and f a i l u r e c r i t e r i a for the element can be p red ic ted with the cor rec t mechanism or theory. The r e c i p r o c a l procedure can be a p p l i e d : we can work backwards and f i n d the mater ia l p roper t ies i f we know the cor rec t mechanism and the state of the member. For masonry w a l l s , i f we know the deformations of the wal ls under loads and the f a i l u r e c o n d i t i o n s , we can deduce the proper t ies of the masonry wal l u n i t qu i te a c c u r a t e l y . 2.2 The Computer Program fo r Eva luat ing the Capaci ty of Masonry Walls  Under Concentr ic or E c c e n t r i c Loads A computer program i s used to evaluate the behaviour of wal ls under a x i a l loads and bending. The program was o r i g i n a l l y developed fo r p r e s t r e s s e d , or r e i n f o r c e d concrete wal ls or c o l u m n s 2 ' 9 ' 1 0 ' 1 1 . By tak ing advantage of the capaci ty of present computer technology, numerical i n t e g r a t i o n and regress ion rout ines are used f requent ly i n t h i s program to compute s t r e s s d i s t r i b u t i o n , f o r c e s , moments, and d e f l e c t e d shapes of members with var ious c r o s s - s e c t i o n s and he igh ts . Mater ia l f a i l u r e i s p red ic ted by numerical ly i n t e g r a t i n g the s t resses over the c r o s s - s e c t i o n and i n s t a b i l i t y f a i l u r e i s monitored by const ruc t ing the column d e f l e c t i o n curves to locate the maximum end moment c a p a c i t y . A b r i e f d e s c r i p t i o n of the method i s as fo l lows: With the assumption that plane sections remain plane under bending action, the strain distribution, and hence the stresses (by referring to the stress-strain curve), are found at various combinations of the curva-ture and the neutral axis depth. The axial load and moment are evaluated for each curvature and neutral axis depth, by numerical integration of the stresses, and a contour of constant curvature may be plotted on the load-moment interaction diagram. By means of such contour lines, the moment-curvature relationship for any value of load may be constructed. Then the column deflection curves can be computed for each axial load, and they are used to find the maximum end moment for each of the number of chosen member lengths. Fig. 2.1 shows an example of a family of curves, which represents the maximum moment and load that the member can carry at various given lengths. Application of the above computer program to masonry wall analysis requires no significant alteration. The basic theory and method of compu-tation are the same. Due to the differences in material properties, the stress-strain curve of concrete w i l l be replaced by a stress-strain curve which represents the properties of the masonry wall unit. 13.' F i g . 2.1 A Family o f Load-Moment I n t e r a t i o n Curves f o r Var ious Heights of the Member 14. 2.3 S t r e s s - S t r a i n Curve The s t r e s s - s t r a i n curve of a mater ia l contains the fo l low ing informa-t i o n : the shape of the curve ( l i n e a r / n o n - l i n e a r ) which governs the p r o p o r t i o n a l i t y of s t r e s s and s t r a i n of the m a t e r i a l ; the peak s t ress and s t r a i n which l i m i t s the ul t imate capaci ty and corresponding deformation; and the gradient of the curve which descr ibes the e l a s t i c i t y of the mater ia l at any state of s t r e s s . A masonry wal l c o n s i s t s of b locks and mortar, and i n the case of a r e i n f o r c e d masonry w a l l , of grout and s t e e l as w e l l . I t i s very d i f f i c u l t to combine a l l the 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 for each mater ia l to obta in an e f f e c t i v e s t r e s s - s t r a i n curve for the masonry wal l u n i t , because i t i s hard to evaluate the c o n t r i b u t i o n of each mater ia l and t h e i r i n t e r a c t i o n . The a l t e r n a t i v e i s to const ruct a hypothe t ica l equivalent s t r e s s - s t r a i n curve, deduce the response of the w a l l , and compare t h i s with experimental r e s u l t s under s i m i l a r load ing c o n d i t i o n s . The method i s a " t r i a l and error" approach; i t invo lves us ing a sample s t r e s s - s t r a i n curve and computing the de f lec ted shape or u l t imate capaci ty of the wal l with an es tab l i shed computer program. The computed behaviour of the wal l un i t i s compared with the exper imental ly observed behaviour , and changes are made to the hypothet ica l s t r e s s - s t r a i n curve to improve the correspondence. The f i n a l s t r e s s - s t r a i n curve w i l l r e f l e c t the p roper t i es of the masonry wal l as an equiva lent "monol i th ic" m a t e r i a l . Since the bas ic mater ia l fo r masonry wal ls i s concre te , a non- l inear s t r e s s - s t r a i n curve as i n concrete was expected for the "equivalent" s t r e s s - s t r a i n curve of the masonry wa l l u n i t . The s t r e s s - s t r a i n r e l a t i o n -ship for prisms under a x i a l compression recorded by Drysdale and H a m i d 1 2 • 15. was used as the starting point. Improvements were made as described above. At this point, the author would like to emphasize that the f i n a l stress-strain curve for the masonry wall unit may be different from the curve recorded from an axially compressed prism test. Governed by the loading condition, the stress-strain relationship is affected by the strain gradient e f f e c t 1 0 . The stress-strain relationship may also be governed by the number of grouted cores, as section geometry may be an important factor. 2.4 Experimental Data When searching for the stress-strain relationship of masonry mater-i a l , experimental results were required for comparison. For simplicity, the experimental results for concrete masonry walls recorded in Ref. (7) were used throughout this study. A l l walls were loaded vertically with top and bottom pinned tb a r i g i d support to ensure no lateral movement at either end. The basic units used in constructing a l l test specimens were 8" x 8" x 16" stretcher blocks, 8" x 8" x 16" end blocks, and 8" x 8" x 8" half blocks with an average strength of 2350 psi. Type S mortar was used throughout, and a l l walls were built to 2 1/2 blocks wide (39.625") in running bond. Face-shell construction was used on a l l plain walls. For the reinforced walls, three bars with ultimate strength of 60 ksi were used as vertical reinforcement at the center line of the wall, one in each alternate hole and these holes with vertical reinforcement were f i l l e d with grout of mean compressive strength 2380 psi. The bar sizes used were 16. #3, #6, and #9. T y p i c a l sect ions of p l a i n and r e i n f o r c e d wal ls are shown on F i g . 2 .2 . Data extracted were: Def lec ted shapes of a p l a i n w a l l , 105" i n he ight , with var ious l o a d -ings (20 k i p s , 40 k i p s , 60 k i p s , 80 k i p s , 100 k i p s , 120 k i p s , and 140 kips) at a constant end e c c e n t r i c i t y of 1.27"; Def lected shapes of a r e i n f o r c e d wal l 185" i n he ight , with var ious loadings of 20 k i p s , 40 k i p s , 60 k ips and 200 k ips at a constant end e c c e n t r i c i t y of 2.54"; 17. .T m|oo I S 8"X8"X 16" SINGLE CORNER , / , MORTARED / / / , AREA 5|" MORTAR JOINT ^ 8 " X 8 " X 16" STRETCHER V?777r//7777?/A\?77777//77777/, J n 8 X 8 X 8 HALF BLOCK 39 | -TYPICAL PLAIN WALL S E C T I O N ( F A C E - S H E L L CONSTRUCTED) VERTICAL REINFORCEMENT GROUT A • A • •A MA. A }77777y^7777rXS77777'A AV 39 | TYPICAL REINFORCED WALL SECTION ( PARTI ALLY GROUTED) F ig . 2.2 Typical Section of Masonry Walls (P la in and P a r t i a l l y Grouted) 18. F a i l u r e loads fo r wa l ls of var ious heights loaded at var ious end e c c e n t r i c i t i e s . Since the f a c e - s h e l l area i s the e f f e c t i v e area i n r e s i s t i n g v e r t i c a l load for a p l a i n w a l l , when enter ing the c r o s s - s e c t i o n of the p l a i n wal l in to the computer program, the c r o s s - s e c t i o n can be modelled as a s e r i e s of box s e c t i o n s . Since the assumption i s made that plane sect ions remain p lane , the web elements are a l l s i m i l a r l y s t r e s s e d , and the computation i s incapable of d i s t i n g u i s h i n g between box and I - s e c t i o n s . Thus, for convenience, an I - s e c t i o n as shown i n F i g . 2.3a was a c t u a l l y used i n the present c a l c u l a t i o n s . The net c r o s s - s e c t i o n s fo r r e i n f o r c e d wal ls were not def ined c l e a r l y i n Ref. (7) . When model l ing the r e i n f o r c e d w a l l s , the c r o s s - s e c t i o n was def ined as f o l l o w s : The f lange i s the same as the f a c e - s h e l l area i n a p l a i n w a l l , and the web area i s c a l c u l a t e d by subt rac t ing the f lange area from the t o t a l net area der ived by d i v i d i n g the f a i l u r e load by the average s t resses noted on Page 148 of Ref. (7) . Width of the web i s found by d i v i d i n g the web area by the known web depth. The modelled net c r o s s -sec t ion and the corresponding coordinates are shown in F i g . 2.3b. 19. Y A (0. , 7.63) t (0., 6.13) (0., 1.5) t ( 0 . , 0.) ( 19.73,6.13) (19.73, 1.5) (19.91,6.13) (19.91, 1.5) (39.63 ,7.63) (39.63,6.13) (39.63, 1.5) (39.63 , 0. ) X Fig. 2.3a Effective Section fer 8" Plain Masonry Wall and its Coordinates Y A (0 . , 7.63) ( 0. , 6.13)] (0. , 1.5 ) I (3.6, 6.13 ) (3 .6 , 1.5) (36.02,6.13) (39.63,7. 63 ) 3 (39 .63 , 6.13) (0 . , 0. ) ( 36 .02 , 1.5 )1 ( 3 | . 6 3 , 1.5) ( 39 .63 , 0 . ^ X Fig. 2.3b Effective Section for 8" (Partially Grouted) Reinforced Masonry Wall and its Coordinates 2 0 . 2.5 Development of the S t r e s s - S t r a i n Curves The procedure for finding the equivalent s t r e s s - s t r a i n curve for p l a i n masonry walls was as follows: A sample s t r e s s - s t r a i n curve, along with section coordinates (shown i n F i g . 2.3a or 2.3b), wall height and the loads of i n t e r e s t were entered into the computer program. The deflected shapes of the wall under the defined loads were computed and plotted against the deflected shapes reported i n Ref. (7). The s t i f f n e s s of the modelled wall i s governed by the shape of the input s t r e s s - s t r a i n curve. By looking at the differences i n d e f l e c t i o n , one can t e l l i f the modelled wall i s too soft or too s t i f f , and the slope of the s t r e s s - s t r a i n curve can be adjusted accordingly u n t i l the computed d e f l e c t i o n matches the experimental d e f l e c t i o n . The f i n a l comparison plot between the computed de f l e c t i o n s and the experimental d e f l e c t i o n s i s shown i n F i g . 2.4. Once the deflected shapes matched, i t was assumed that the shape of the equivalent s t r e s s - s t r a i n curve was c o r r e c t , as the modelled wall and the r e a l wall had the same s t i f f n e s s . F i n a l l y the peak s t r a i n and the ultimate s t r a i n of the curve were varied u n t i l the f a i l u r e load equalled the experimental f a i l u r e load. Due to the difference i n composition of cross-section between a p l a i n masonry wall and a reinforced wall - the presence of grout i n alternate cores — another s t r e s s - s t r a i n curve i s needed for the l a t t e r . The same procedure as described i n the previous paragraph was used, and the f i n a l plot of the superimposed deflected shapes are shown i n F i g . 2.5. The f i n a l s t r e s s - s t r a i n curves f o r p l a i n and reinforced walls, under a x i a l load and bending, are shown on F i g . 2.6. The developed equivalent 21. -0.15 0 0.15 0.30 0.45 0.60 DEFLECTION, in. F ig . 2.4 Experimental and Computed Deflected Shapes of a Face-Shell Constructed 8" P la in Masonry Wal1 22. s t r e s s - s t r a i n curves may be a p p l i c a b l e to wal ls with the same nature c r o s s - s e c t i o n . 23. -0.5 0 0.5 1.5 2.5 3.5 4.5 DEFLECTION, in. £ig.. 2.5 Experimental and Computed Deflected Shapes of a 8" Reinforced Masonry Wall 24. 3.0 STRAIN F ig . 2.6 Stress-Strain Curves for Masonry 25. 2.6 S t r e s s - S t r a i n Curves fo r P l a i n and P a r t i a l l y Grouted Masonry Walls F i g . 2.6 shows the f i n a l p l o t fo r the s t r e s s - s t r a i n curves fo r both p l a i n and p a r t i a l l y grouted masonry w a l l s . The s t r e s s - s t r a i n curves recorded by Drysdale and H a m i d 1 2 , and H a t z i n i k o l a s 7 fo r ungrouted prisms under a x i a l compression t e s t are a l s o p l o t t e d . Both der ived s t r e s s - s t r a i n curves are non- l inear as expected. For p l a i n w a l l s , the peak s t ress i s 2.1 k s i with the corresponding peak s t r a i n of 0.0022, and the i n i t i a l modulus of e l a s t i c i t y and u l t imate s t r a i n are 1389 k s i and 0.0025 r e s p e c t i v e l y . For p a r t i a l l y grouted w a l l s , the peak s t ress and s t r a i n are 2.35 k s i and 0.0022 r e s p e c t i v e l y . When comparing the der ived s t r e s s - s t r a i n curves , the one for p a r t i a l -l y grouted wal ls has higher modulus of e l a s t i c i t y and peak s t r e s s , with the same peak s t r a i n and u l t imate s t r a i n . I t g ives the p a r t i a l l y grouted wal l a greater capac i ty than the p l a i n wal l under e c c e n t r i c load ing c o n d i -t i o n s . C u r r e n t l y , the e f f e c t of grout on masonry wal ls i s not we l l estab-l i s h e d . Drysdale and Hamid report that i n a x i a l compression t e s t s , grouted prisms exh ib i t lower compressive strength than ungrouted pr isms. On the cont ra ry , B o u l t ^ reports that i f the modulus and l i m i t i n g s t r a i n of both block and grout are s i m i l a r , the r e s u l t i n g prism has u l t imate proper t ies i n excess of the i n d i v i d u a l elements. It was a lso reported that masonry assemblages have d i f f e r e n t modes of f a i l u r e under a x i a l l y loaded and e c c e n t r i c a l l y loaded c o n d i t i o n s ; the former f a i l by t e n s i l e s p l i t t i n g of the un i t and the l a t t e r f a i l when the s t ress of the compres-s ion side reaches the strength of the m a t e r i a l s . Th is comment together with the remarks made by Boult i n d i c a t e that p a r t i a l l y grouted wal ls may be expected to behave as though stronger than ungrouted wal ls i n f l e x u r e , but more experimental s tud ies are needed to confirm t h i s . 26. The s t r e s s - s t r a i n r e l a t i o n s h i p recorded by H a t z i n i k o l a s 7 was der ived from tes ts i n a x i a l compression one and one-hal f b locks wide, f i v e b locks h i g h , of ten ungrouted specimens. With the assumption of a l i n e a r s t r e s s -s t r a i n r e l a t i o n s h i p , the modulus of e l a s t i c i t y i s 1120 k s i , and the f a i l -ure s t ress i s 2.056 k s i . Th is s t r e s s - s t r a i n curve i s compared with the der ived s t r e s s - s t r a i n curves , which are based on experimental data ( d e f l e c t i o n shapes and ul t imate loads) of e c c e n t r i c a l l y loaded w a l l s , i n F i g . 2 .6 . In s p i t e of the d i f f e rence i n loading c o n d i t i o n s , the der ived s t r e s s - s t r a i n curve for p l a i n wa l ls does resemble the prism t e s t r e s u l t . The moduli of e l a s t i c i t y are almost the same, and the peak s t ress for both curves are the same at 2.1 k s i . The s i m i l a r i t i e s suggest that load ing condi t ions have no s i g n i f i c a n t e f f e c t on the strength of a f a c e - s h e l l p l a i n masonry assemblage. 2.7 E f f e c t of T e n s i l e Strength The al lowable t e n s i l e strength permitted i n the current c o d e 1 fo r design of masonry wal ls i s 23 p s i and 36 p s i for f a c e - s h e l l constructed and f u l l y bedded wal ls r e s p e c t i v e l y , Ref . (7) recorded a mean t e n s i l e strength of 350 p s i and i t was inc luded i n the formulat ion for computation of the wal l c a p a c i t y . But i n most s t u d i e s , s ince the magnitude of t e n s i l e strength i s small compared to compressive s t rength , the t e n s i l e strength fo r masonry mater ia l i s assumed to be zero . In order to study the cont r ibu t ions of t e n s i l e strength to the wa l l c a p a c i t y , two sets of computations were made on a f a c e - s h e l l constructed p l a i n masonry w a l l , 105" i n height and 2 1/2 b locks wide, with a load of 140 k ips at an e c c e n t r i c i t y of 1.27". One set was computed with no ten-s i l e strength and the other had an assumed t e n s i l e strength of 360 p s i 27. with a corresponding t e n s i l e s t r a i n of 0.00017. The value of 360 p s i was based on the recomendation of the concrete design c o d e 1 5 . ( i e : f = t 7 .5 / f ') and t h i s value agreed with the one reported i n Ref. (7 ) . c The computed d e f l e c t e d shapes and short wal l load-moment i n t e r a c t i o n c a p a c i t i e s were compared. I t was found that there was no not iceab le d i f fe rence between the two sets of r e s u l t s . Th is impl ied that the e f f e c t of t e n s i l e strength i s i n s i g n i f i c a n t i n the computation of the wal l c a p a c i t y . 28. I I I . VERIFICATION OF THE ANALYSIS 3.1 Theoretical Interaction Diagrams As the f i n a l s t r e s s - s t r a i n r e l a t i o n ships were developed i n the pre-vious chapter for p l a i n and p a r t i a l l y grouted masonry walls, d i f f e r e n t i n t e r a c t i o n diagrams can now be produced for the walls of those types. It was found that (as suggested i n Section 2.5) the s t r e s s - s t r a i n r e l a -tionship depends upon the cross-section geometry of the wall, since two d i f f e r e n t s t r e s s - s t r a i n curves were obtained separately for f a c e - s h e l l constructed and p a r t i a l l y grouted walls. With t h i s l i m i t a t i o n , the i n t e r -action diagram i s governed by the composition of the cross-section and the strength of the components. Since the purpose of t h i s study i s to model the general behaviour of masonry walls and the slenderness e f f e c t , the a v a i l a b l e information i s adequate to meet the task. The t h e o r e t i c a l i n t e r a c t i o n diagrams for p l a i n and reinforced walls were produced with the computer program described i n Section 2.2 The slenderness e f f e c t was included i n these i n t e r a c t i o n diagrams with the wall heights varying from 95.6" to 375.6", which was a reasonable range for consideration. The composition of the walls modelled was s i m i l a r to those s p e c i f i e d i n Section 2.4. The i n t e r a c t i o n diagrams are shown on F i g . 3.1, 3.2, 3.3 and 3.4, and the corresponding experimental r e s u l t s recorded i n Ref. (7) are also plotted for comparison. Each datum i s represented by a s o l i d dot with an arrow leading to the related theore-t i c a l l o c a t i o n . 3.2 Comparison of the Experimental and the Theoretical Values for Face- Shel l Constructed P l a i n Masonry Wall F i g . 3.1 represents the t h e o r e t i c a l load-moment i n t e r a c t i o n capaci-29. 0 5 10 15 20 25 30 35 MOMENT, kip-ft. F ig . 3.1 Theoretical Interact ion Diagram for 8" Face-Shell Constructed Pla in Masonry Walls and the Associated Experimental Results 30. F ig . 3.2 Theoretical Interact ion Diagram for 8" Reinforced Masonry Walls (3-#3 @ ([} with the Associated Experimental Results 31. 900i 800 • Experimental r e s u l t 1 10 20 30 40 50 MOMENT, kip-ft. 60 F i g . 3.3 Theoret ica l In teract ion Diagram fo r 8" Reinforced Masonry Walls (3-#6 @ with the Associa ted Experimental Results 32. 900 F i g . 3.4 Theoret ica l In teract ion Diagram f o r 8" Reinforced Masonry Walls (3-#9 @ C) with the Associated Experimental Results 33. t i e s of the 2 1/2 blocks' wide f a c e - s h e l l constructed masonry walls with various lengths. A l l p l a i n walls used for t h i s comparison purpose are f a c e - s h e l l constructed with no j o i n t reinforcement. 3.2.1 Short Wall Capacity Unreinforced short walls, 39.625" i n height, were loaded a x i a l l y and e c c e n t r i c a l l y to r e s u l t i n f a i l u r e . The r e s u l t s were recorded on P. 117 and P. 118 of Ref. (7) respectively and were also plotted i n F i g . 3.1 for comparison. For c l a r i t y , Table 3.1 shows the comparison of the corres-ponding experimental f a i l u r e loads and the t h e o r e t i c a l l y predicted values. The spread between repeated experimental r e s u l t s was as high as 33% which was registered with the specimens loaded at an e c c e n t r i c i t y of t/3 (e = 2.54"). The comparison indicates that the t h e o r e t i c a l short wall i n t e r a c t i o n curve does predict the capacity of the wall accurately with the highest percentage error of 8.9% which was recorded under a x i a l loading condi-tions. It i s i n t e r e s t i n g to note that the t h e o r e t i c a l value over-estimates the experimental f a i l u r e loads under concentric loading condi-tions, but the differences are quite small (8.9% difference between the t h e o r e t i c a l value and the average of the test r e s u l t s ) compared to the spread i n the experimental data. At t h i s point, i t i s noted that the differences mentioned may be caused by the imperfections of the experi-mental specimens — eg: i n i t i a l d e f l e c t i o n s . The e f f e c t of s t r a i n gradient reported i n most studies (eg: Refs. 5, 7, and 13) i s not apparent. It suggests that the e f f e c t s of s t r a i n gradient are i n s i g n i f i -cant for f a c e - s h e l l constructed unreinforced masonry walls. 34. 3.2.2 F u l l Size Walls Walls, with heights of 105", 121", 137", and 187" were loaded a x i a l l y and e c c e n t r i c a l l y u n t i l f a i l u r e and loading conditions varied from concentric to an e c c e n t r i c i t y of 3.0". The r e s u l t s were recorded on P. 148 and P. 149 of Ref. (7) and are plotted i n F i g . 3.1 accordingly. Table 3.2 shows the comparison between the t h e o r e t i c a l l y predicted values and the experimental r e s u l t s . There i s an error of 39.5% for the 105" high wall with 2.54" e c c e n t r i c i t y , but otherwise errors are 17% or les s -generally l e s s than the spread i n the duplicated experimental r e s u l t s for the short-wall. It indicates that the present a n a l y t i c a l method i s capable of modelling the slenderness e f f e c t and predicting the f a i l u r e combinations of loads and moments for p l a i n masonry walls. 35. Load Eccentricity (in.) Experimental Capacity . . . . (kip) Experimental Discrepancy (%) Theoretical Capacity (kip) Error 0.00 0.00 215.5 249.1 232.3 (Ave.) .. 15.6 253. 8.9 1.27 1.27 196.9 150.1 173.5 (Ave.) 31.2 178. 2.6 2.54 119.3 158.7 139.0 (Ave.) 33.0 137. -1.4 Table 3.1 Comparison of Experimental Results and Theoretical Values for Short Wall Capacities of Face-Shell Constructed Plain Masonry Walls. 36 . Height of Wall (in.) (in.) Load Eccentricity (in.) Experimental Capacity (kips) Theoretical Capacity (kips) Error 105. 0 262.5 238.6 227. - 9.4 105. 1.27 159.2 154. - 3.3 105. 2.54 80.3 112. 39.5 105. 3.00 26.1 30. 15. 121. 0.00 190.0 223. 17.4 137. 0.00 218.3 218. - 0.1 185. 0.00 207.8 200. - 3.8 185. 1.27 120. 135. 12.5 Table 3.2 Comparison of Experimental Results and Theoretical Values for Full Size, Face-Shell Constructed Plain Masonry Walls. 37. 3•3 Comparison of the Experimental and the Theoretical Results for Pa r i t i a l l y Grouted Reinforced Walls As described earlier, a l l reinforced walls tested were 2 1/2 blocks wide with 3 alternate cores f i l l e d with grout. Reinforcing bars were placed at the centre of each grouted core. There were three types of reinforced walls examined, and their differences were in the size of the reinforcing bars - 3-#3. 3-#6, 3-#9. The corresponding theoretical interaction diagrams are shown on Figs. 3.2, 3.3, and 3.4. 3.3.1 Short Wall Capacity Unfortunately, no reinforced short wall was loaded eccentrically to failure in Ref. (7). There are thus no experimental results to study the accuracy of the theoretical short wall interaction curves. Data for axially loaded reinforced short walls (39.625") are available and the failures were recorded on P. 120, Ref. (7). Table 3.3 shows the comparison between the predicted axial capacities and the experimentally recorded results, for the three types of reinforced walls. It was found that the experimental results were a lot less than the predicted capacities. They are 48%, 42.2%, and 40% of the capacities calculated from the flexural compressive strength of the masonry unit. It i s very possible that the loss in capacity i s related to the tensile stress induced on vertical planes by lateral deformation of the mortar. V e r t i c a l Reinforcement (1) Expe r imenta1 Capaci ty (2) T h e o r e t i c a l Capacity (kips) d ) / ( 2 ) 3 - #3 @ <£ 348.3 280.3 314.3 (Ave.) 651.6 0.48 3 - #6 @ <£ 334.1 265.3 299.7 (Ave.) 711.0 0.42 3 - #9 @ <£ 386.6 275.3 330 .9 . (Ave . ) . 811.8 0.40 Table 3.3 Comparison of Test and Theore t i ca l Capac i t i es for A x i a l l y Loaded 8" Reinforced Masonry Walls 39. The effect does not appear to be present in plain walls of face-shell construction, so i t could be due to the interaction of blocks, mortar, and grout as suggested by Turkstra and Thomas1 . The lateral displacement of the grout and the mortar may increase the lateral tensile stress in the blocks near the joint and i t may cause the change of failure mode and the significant loss in capacity. A detailed study on this matter is recommended in order to have a better understanding of the failure mechanism. 3.3.2 Full Size Walls F u l l size walls reinforced with 3-#9 bars, of 105", 127", 137", and 185" heights were loaded at end eccentricities of 1.27", 2.54", 3.00", and 3.50"; while the walls reinforced with 3-#3, and 3-#6 bars were tested under eccentric loads at one wall height of 137". The values of failure loads were recorded on P. 148 and P. 172 of Ref. (7). Table 3.4(a and b) shows the comparison of the experimental results and the predicted values. As in the case of short walls, the theoretical values tend to over estimate the capacities by a big margin under pure axial compression. When ignoring the results for the axial loading condition, the correlation between the test results and the predicted values is good, and most of the errors are within 10%. The results shown on Table 3.4b for walls reinforced with 3-#6 and 3-#3 bars 40. Height of Wall ( in . ) Load E c c . ( in . ) Experimental Capaci ty (kips) T h e o r e t i c a l Capaci ty (kips) Error 105.0 1.27 320.0 314.0 - 1.9 2.54 140.0 146.0 4.3 3.00 155.0 124.0 -20.0 3.50 114.9 106.5 - 7.3 121.0 0.00 315.0 620.0 97.0 1.27 249.6 278.0 . 11.4 2.54 125.0 131.5 5.2 3.00 122.5 112.0 - 8.6 3.50 90.0 97.5 8.3 137.0 0.00 400.0 580.0 45.0 1 .27 200.0 246.5 23.0 2.54 108.8 117.5 8.0 3.00 94.5 101.5 7.4 3.50 83.0 89.0 7.2 185.0 0.00 383.5 445.0 16.0 1.27 150.0 176.0 17.0 2.54 90.0 84.0 - 6.7 3.00 80.0 74.0 - 7.5 3.50 73.3 66.7 - 9.0 Table 3.4a Comparison of Test and T h e o r e t i c a l Capac i t i es fo r F u l l Size Reinforced Masonry Walls (3 - #9 g ) 41. V e r t i c a l Re in f . Load E c c . ( in . ) Experimental Capaci ty (kips) T h e o r e t i c a l Capacity (kips) Er ror (%) 3-#6 @ <£ 0.00 375.2 540.0 44.0 1.27 259.5 254.0 - 2.1 2.54 86.3 98.0 13.6 3.00 65.1 82.0 26.0 3.50 56.0 69.5 24.0 3-#3 @ <£ 0.00 305.0 515.0 68.9 1.27 217.0 259.0 19.4 2.54 54.0 68.5 26.9 3.00 32.9 44.0 34.0 3.50 24.3 32.5 34.0 Table 3.4b Comparison of Test and Theore t i ca l Capac i t i es fo r 137.0 i n Height , 8 i n . Reinforced Masonry Walls are misleading. Due to the small absolute magnitude, with the failure load small compared to the accuracy of the experimental set-up, a d i f f e r -ence of a few kips caused by inaccurate reading w i l l be shown as a big percentage error, when comparing i t with the actual failure load i t s e l f . This explains the high percentage error (as much as 34%) recorded for walls reinforced with 3-#6 and 3-#3 bars. Figs. 3.3 and 3.4 give a better picture of the comparison and the results appear to give good correla-tions. 3.4 Joint Reinforcement The normal function of the joint (horizontal) reinforcement is to provide a two-way action for resisting lateral forces. The presence of joint reinforcement can also restr i c t lateral expansion of mortar joints under in-plane compressive forces and hence increase the compression capa-city of the wall. The effect of the joint reinforcement has been investi-gated in both Refs. (7) and (8). 1 9 Supported by the results of the prism tests, Dyrsdale and Hamid1^ reported that joint reinforcement of normal gauge wires did not provide much beneficial confining effect or any- deterimental effect due to stress concentration. In Ref. (7) Hatzinikolas, Longworth, and Warwaruk, reported that there were reductions in strength for walls with normal joint-reinforcement compared to the walls without joint reinforcement. The reduction of capacities, for walls without vertical reinforcement, varied from 18% to 22% and the reduction of capacities for reinforced short walls was 6%. Both papers asserted that flattened joint reinforce-ment was more beneficial than normal gauge wire joint reinforcement. It i s d i f f i c u l t to introduce the effect of joint reinforcement in the present a n a l y s i s , and, s ince both papers are i n favour of the wal ls with no j o i n t re inforcement , a l l experimental r e s u l t s chosen for comparison purposes, and the a n a l y t i c a l r e s u l t s , are based on wal ls without j o i n t re inforcement. 44. IV. EVALUATION OF THE CODE DESIGN METHOD 4.1 Introduct ion The current Canadian Design Code fo r Masonry Walls and Columns has adopted al lowable s t ress des ign . The p r o v i s i o n s are c l e a r l y s ta ted i n "Masonry Design and Construct ion fo r B u i l d i n g s . " (CAN3-S304-M78). The code i s mainly based on the work done by the Br ick I n s t i t u t e of America, and gives p r o v i s i o n s fo r many aspects of design and const ruc t ion d e t a i l . Mater ia l p r o p e r t i e s , const ruc t ion requirements, and design procedures are covered. The top ic to be focused on here i s the design equations used for c a l c u l a t i n g the capaci ty of load bear ing members or w a l l s . 4.2 Code Design Prov is ions In the c o d e 1 , the design sec t ion capaci ty of a masonry load bear ing member i s dependent on the al lowable s t ress of the masonry un i t and the net sec t ion a rea , where the al lowable s t r e s s for design i s based on the type of s t ress and the prism a x i a l strength of the masonry u n i t . The al lowable design load i s obtained by m u l t i p l y i n g the capaci ty by two reduct ion f a c t i o n fac to rs C and C , accounting for the e f f e c t of the e s magnitude and pat tern of the app l ied end moment, and the slenderness e f f e c t r e s p e c t i v e l y . A b r i e f d e s c r i p t i o n of the design equations f o l l o w s : When the maximum v i r t u a l e c c e n t r i c i t y exceeds t / 3 , P = C C f A fo r p l a i n masonry wal ls 4.1 e s m n P = C C (f +0.8 p f )A fo r r e i n f o r c e d masonry e s m s n w a l l s . 4.2 45. where e = maximum v i r t u a l e c c e n t r i c i t y which i s def ined as the maximum value of the primary moment along the member d iv ided by the app l ied v e r t i c a l l o a d . t = e f f e c t i v e th ickness of the member P = al lowable working load f = al lowable compressive s t ress fo r masonry m A = net c r o s s - s e c t i o n area n p = reinforcement r a t i o f = a l lowable s t e e l s t r e s s s C e = e c c e n t r i c i t y c o e f f i c i e n t C = slenderness c o e f f i c i e n t s The slenderness c o e f f i c i e n t , C , i s a funct ion of the h / t r a t i o and s the end e c c e n t r i c i t y r a t i o , e-^/e2' w n e r e e ^ a n ( i e2 a r e the e c c e n t r i c i t i e s at both ends with > e^. The empi r ica l equation for C g i s : e C = 1.20 - ^ | (5.75 + (1.5+-i)2) < 1.0 4.3 s 300 e 2 The e c c e n t r i c i t y c o e f f i c i e n t , Ce , i s a funct ion of the maximum v i r t u a l e c c e n t r i c i t y , e , the e f f e c t i v e t h i c k n e s s , t , and the end e c c e n t r i c i t y r a t i o , e^/ e2 • T l i e func t ion i s descr ibed by two equations governed by the magnitude of the maximum e c c e n t r i c i t y . The equations are: C e = I T T T i A ) + H ~ z 0 ) ( 1 " I T - * « t/20 < e < t/6 4.4 e C = 1.95(i - §) + i ( f - ^ ) ( 1 — - ) fo r t/6 < e < t/3 4.5 e 2. t z t 20 e^ 46. When the maximum v i r t i a l e c c e n t r i c i t y exceeds t / 3 , the a l lowable sec t ion capaci ty of the (p la in or re in forced) wal l i s based on the t r a n s -formed sec t ion and the assumption of l i n e a r s t ress d i s t r i b u t i o n . Then the sec t ion capaci ty i s modi f ied by the slenderness c o e f f i c i e n t , C , to obta in s the al lowable working l o a d . When designing the load bear ing member, a l i m i t a t i o n on the s lender -ness of the member i s imposed such that the r a t i o of h / t does not exceed the value of 10(3-e^/e^). And a minimum e c c e n t r i c i t y of t /20 i s recom-mended i n the code to account for the imperfect ion of the member. 4.3 Comparison of the T h e o r e t i c a l Capac i t i es and the Code Design Values  for Masonry Walls 4.3.1 General In order to examine the e f fec t i veness of the current code design method, a comparison t e s t was made between the code design values and the t h e o r e t i c a l l y "exact" values evaluated with the computer programme descr ibed i n Sect ion 2 .2 . 4.3.2 The T h e o r e t i c a l Values The t h e o r e t i c a l i n t e r a c t i o n diagrams for 8" p l a i n and r e i n f o r c e d masonry wal ls shown on F i g s . 3.1, 3 .2 , 3 .3 , and 3.4 were used for de te r -mining the t h e o r e t i c a l capac i ty of the w a l l s . As suggested i n sec t ion 2 .6 , the wal ls with s i m i l a r composit ion of the c r o s s - s e c t i o n can have s i m i l a r c r o s s - s e c t i o n p roper t i es and therefore can be governed by the same 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 . For f a c e - s h e l l constructed p l a i n w a l l s , the c r o s s - s e c t i o n geometries fo r wa l ls constructed with var ious block s i z e s are s i m i l a r and t h e i r e f f e c t i v e sect ions are represented by two s t r i p s of concrete with d i f f e r e n t spacings between them. If the mater ia ls of construction are the same, the i n t e r a c t i o n diagram for the 12" f a c e - s h e l l constructed masonry wall can be constructed by using the s t r e s s - s t r a i n curve developed f or the 8" f a c e - s h e l l constructed p l a i n w a l l . Based on thi s argument, the t h e o r e t i c a l load-moment i n t e r a c t i o n diagram for the 12" p l a i n masonry wall i s produced and shown on F i g . 4.1. This diagram w i l l be used along with F i g s . 3.1, 3.2, 3.3, and 3.4 i n the l a t t e r course of th i s study, and i t w i l l provide more information when studying the slen-derness e f f e c t on masonry walls. Straight l i n e s representing constant end e c c e n t r i c i t i e s were drawn on the t h e o r e t i c a l load-moment i n t e r a c t i o n diagrams. Each i n t e r s e c t i o n of the st r a i g h t l i n e and the i n t e r a c t i o n curve, representing a p a r t i c u l a r wall height, gave the f a i l u r e condition as a combination of load and moment. A l l of the t h e o r e t i c a l capacities f o r walls with various wall heights were extracted i n t h i s manner and were recorded i n Tables 4.1, 4.2, 4.3, 4.4, and 4.5. 48. 0 10 20 30 40 50 60 MOMENT, kip-ft . F i g . 4.1 Theoret ica l In teract ion Diagram fo r 12" P la in Masonry Walls 49. Height (in.) 95.63 135.63 175.63 215.63 255.63 295.63 335.63 375.63 Ecc. 0 232.0 218.0 205.0 185.0 160.0 133.0 110.0 92.0 t/20 207.0 194.0 181.0 162.0 139.0 117.0 97.0 82.0 t/15 199.0 187.0 175.0 154.0 133.0 111.0 93.0 77.0 t/10 187.0 175.0 162.0 142.0 122.0 100.0 83.0 69.0 t/6 165.0 154.0 141.0 121.0 98.0 80.0 66.0 54.0 t/5 156.0 145.0 132.0 110.0 87.0 70.0 57.0 48.0 t/4 143.0 134.0 113.0 88.0 68.0 53.0 43.0 32.0 t/3 121.0 88.0 59.0 45.0 25.0 20.0 .15.0 7.0 Table 4.1 Theoretical Capacities (in kips) for 8" Plain Masonry Walls Height (in.) 95.63 135.63 175.63 215.63 255.63 295.63 335.63 375.63 Ecc. 0 251.0 250.0 239.0 232.0 222.0 212.0 200.0 175.0 t/20 224.0 . 217.0 208.0 202.0 192.0 182.0 166.0 155.0 t/15 214.0 207.0 200.0 192.0 184.0 173.0 160.0 150.0 t/10 197.0 192.0 185.0 176.0 168.0 161.0 150.0 140.0 t/6 175.0 170.0 162.0 156.0 151.0 143.0 134.0 123.0 t/5 167.0 161 .0 155.0 150.0 144.0 136.0 125.0 111.0 t/4 155.0 150.0 145.0 140.0 133.0 122.0 104.0 87.0 t/3 138.0 134.0 127.0 112.0 80.0 68.0 58.0 41.0 Table 4.2 Theoretical Capacities (in kips) for 12" Plain Masonry Walls 50.... T h e o r e t i c a l Capac i t i es (kips) Height ( in . ) 95.63 135.63 175.63 215.63 255.63 295.63 335.63 375.63 E c c e n t r i c i t y 0 570.0 520.0 439.0 362.0 302.0 252.0 210.0 163.0 t /20 484.0 434.0 361.0 296.0 240.0 194.0 160.0 137.0 t /15 462.0 405.0 339.0 271.0 220.0 178.0 147.0 127.0 t /10 419.0 359.0 291.0 229.0 184.0 150.0 126.0 103.0 t /6 332.0 265.0 203.0 159.0 124.0 102.0 80.0 57.0 t / 5 286.0 220.0 163.0 128.0 100.0 81.0 52.0 40.0 t /4 215.0 152.0 110.0 86.0 64.0 36.0 29.0 20.0 t / 3 110.0 70.0 44.0 . 30.0 21.0 16.0 13.0 11.0 51/12 60.0 . 38.0 26.0 20.0 16.0 13.0 11.0 8.0 Table 4.3 T h e o r e t i c a l Capac i t i es for 8" Reinforced Masonry Walls ( 3 - #3 @ £ ) 51... T h e o r e t i c a l Capac i t ies (kips) Height ( in . ) 95.63 135.63 175.63 215.63 255.63 295.63 335.63 375.63 E c c e n t r i c i t y 0 610.0 560.0 460.0 377.0 308.0 250.0 204.0 173.0 t /20 512.0 442.0 360.0 297.0 239.0 191.0 159.0 137.0 t /15 484.0 411.0 332.0 274.0 218.0 172.0 146.0 126.0 t /10 433.0 356.0 287.0 228.0 181.0 145.0 123.0 103.0 t /6 334.0 256.0 197.0 150.0 120.0 100.0 80.0 50.0 t / 5 282.0 210.0 156.0 121.0 97.0 73.0 60.0 32.0 t / 4 212.0 150.0 108.0 79.0 61.0 50.0 30.0 24.0 t /3 143.0 98.0 70.0 52.0 42.0 31.0 24.0 19.0 5t/12 107.0 75.0 54.0 42.0 33.0 25.0 20.0 17.0 • t /2 87.0 62.0 45.0 36.0 27.0 23.0 18.0 14.0 Table 4.4 T h e o r e t i c a l Capac i t i es fo r 8" Reinforced Masonry Walls ( 3 - #6 @ £ ) 52. T h e o r e t i c a l Capac i t i es (kips) Height ( in . ) 95.63 135.63 175.63 215.63 255.63 295.63 335.63 375.63 E c c e n t r i c i t y 0 690.0 600.0 463.0 384.0 311.0 255.0 207.0 182.0 t /20 557.0 465.0 372.0 300.0 243.0 196.0 160.0 135.0 t /15 520.0 426.0 338.0 273.0 220.0 175.0 144.0 123.0 t/10 454.0 354.0 288.0 227.0 175.0 145.0 119.0 103.0 t /6 343.0 261.0 200.0 153.0 123.0 101.0 80.0 68.0 t / 5 276.0 204.0 151.0 117.0 93.0 74.0 60.0 50.0 t /4 211 .0 156.0 115.0 87.0 68.0 58.0 43.0 38.0 t /3 154.0 118.0 90.0 67.0 55.0 46.0 33.0 31.0 5t/12 124.0 99.0 77.0 58.0 50.0 39.0 31.0 28.0 Table 4.5 T h e o r e t i c a l Capac i t i es for 8" Reinforced Masonry Walls ( 3 - #9 @ ) 53. 4.3.3 C a l c u l a t i o n of the Code Design Values In coord inat ion with the t h e o r e t i c a l c a p a c i t i e s , the al lowable loads f o r p l a i n and r e i n f o r c e d wal ls with heights vary ing from 95.625" to 375.625" — with i n t e r v a l s of 40" (5 courses) — were c a l c u l a t e d according to the present code p r o v i s i o n s . The equal end e c c e n t r i c i t i e s were v a r i e d from 0 to t /3 for p l a i n wal ls and from 0 to 5t/12 r e i n f o r c e d w a l l s . The corresponding al lowable c a p a c i t i e s are shown i n tab les 4 .6 , 4 .7 , 4 .8 , 4.9 and 4.10. 54. Allowable Loads (kips) Height ( in . ) 95.63 135.63. 175.63* 215.63* E c c e n t r i c i t y 0 - t /20 26.18 18.3 10.45 2.58 t /15 24.31 17.0 9.70 2.40 t/10 21.27 14.88 8.49 2.10 t /6 17.02 11.90 6.79 1.68 t / 5 15.32 10.71 6.11 1.51 t /4 12.76 8.93 5.09 1.26 t /3 8.51 5.95 3.40 0.84 Table 4.6 Al lowable Loads fo r 8" P l a i n Masonry Walls Height ( in . ) 95.63 135.63 175.63 215.63 255.63* 295.63* 335.63* E c c e n t r i c i t y 0 - t /20 32.64 27.49 22.44 17.17 12.01 6.85 1.69 t /15 30.31 25.52 20.73 15.94 11.15 6.36 1.57 t /10 26.52 22.33 18.14 13.95 9.76 5.57 1.38 t /6 21.22 17.87 14.51 11.16 7.81 4.45 1.10 t / 5 19.10 16.08 13.06 10.04 7.03 4.01 0.99 t /4 15.91 13.40 10.88 8.37 5.85 3.34 0.83 t /3 10.61 8.93 7.26 5.58 3.90 2.23 0.55 Table 4.7 Allowable Loads for 12" P l a i n Masonry Walls * Values not permitted i n code 55. Allowable Loads (kips) Height ( in . ) 95.63 135.63 175.63* 215.63* E c c e n t r i c i t y 0 - t /20 70.28 49.16 28.05 6.93 t /15 65.26 45.65 26.04 7.47 t/10 57.10 39.94 22.79 5.63 t /6 45.68 31.95 18.23 4.51 t / 5 41.12 28.76 16.41 4.06 t / 4 34.26 23.97 13.67 3.38 t /3 22.84 15.98 9.11 2.25 51/12 11.77 8.23 4.70 1.16 Table 4.8 Al lowable Loads for 8" Reinforced Masonry Walls ( 3 - #3 <§ g ) Allowable Loads (kips) Height (in.) 95.63 135.63 175.63* 215.63* Eccentricity 0 - t/20 83.52 58.42 33.33 8.24 t/15 77.55 54.25 30.95 7.65 t/10 67.86 47.47 27.08 6.69 t/6 54.29 37.97 21.66 5.36 t/5 48.86 34.17 19.56 4.82 t/4 40.72 28.48 16.25 4.02 t/3 27.14 18.99 10.83 2.68 5t/12 14.63 14.63 5.84 1.44 Table 4.9 Allowable Loads for 8" Reinforced Masonry Walls ( 3 - #6 @ g ) Allowable Loads (kips) Height ( in . ) 95.63 135.63 175.63* 215.63* E c c e n t r i c i t y 0 - t /20 106.23 74.32 42.39 10.48 t /15 98.65 69.00 39.36 9.7 t /10 86.32 60.37 34.44 8.52 t /6 69.05 48.30 27.55 6.81 t /5 62.15 43.47 24.80 6.13 t / 4 51.79 36.22 20.67 5.11 t / 3 34.53 24.15 13.78 3.41 5t/12 15.64 . 10.94 6.24 1.54 Table 4.10 Al lowable Loads for 8" Reinforced Masonry Walls ( 3 - #9 @ £ ) 5 8 . 4.3.4 Comparison of the Resul ts Tables 4 .11, 4.12, 4 .13, 4.14, and 4.15 record the safe ty f a c t o r s involved i n the current code design method. The safety fac tor i s def ined as the r a t i o of the t h e o r e t i c a l capac i ty to the corresponding al lowable design l o a d . Since the slenderness r a t i o of each wal l i s l i m i t e d to not exceeding 10(3 - e^ /e^) , with equal end e c c e n t r i c i t i e s (e^=e^), the height l i m i t a t i o n for 8" wal ls i s 160", and for 12" wal ls i s 240", regard less of the number of cores grouted. However, the equation for C quoted from the s code does al low one to c a l c u l a t e loads for heights up to 229" for 8" w a l l s , and 349" for 12" w a l l s . Tables 4.11 and 4.12 show the safe ty f a c t o r s for the 8" and 12" p l a i n w a l l s . Within the range permitted by the code, the safety fac tor v a r i e s from 7.91 to 14.79. Beyond the slenderness r a t i o l i m i t recommended i n the code, the safety fac tor v a r i e s from 17.92 to 72.85. The values are i n c o n s i s t a n t and they increase with height and e c c e n t r i c i t y . The same pat tern i s found on 12" p l a i n w a l l s . The safety fac to rs vary from 6.83 to 20.07 wi th in the range permit ted by the code, and when the slenderness r a t i o exceeds the l i m i t a t i o n , the values range from 15.99 to 126.26. Tables 4.13, 4 .14, and 4.15 show the comparison of r e s u l t s fo r 8" p a r t i a l l y grouted wal ls r e i n f o r c e d with 3-#3, 3-#6, and 3-#9 bars r e s p e c t i v e l y . Within the slenderness r a t i o permitted by the code, the values vary from 4.38 to 10.58 Factors of Safety Height ( in.) 95.63 135.63 175.63* 215.63* E c c e n t r i c i t y 0 8.86 11.91 19.62 71.71 t/20 7.91 10.60 17.32 62.79 t/15 8.19 11.00 i 18.04 64.17 t/10 8.79 11.76 19.09 67.68 t/6 9.70 12.94 20.77 72.02 t/5 10.19 13.54 21.60 72.85 t/4 11.13 15.01 22.19 69.70 t/3 14.22 14.79 17.38 53.62 Table 4.11 Factors of Safety for 8" Pl a i n Masonry Walls i n Current Design Code Factors of Safety Height ( in.) 96.63 135.63 175.63 215.63 255.63* 295.63* 335.63* E c c e n t r i c i t y 0 7.69 9.09 10.70 13.51 18.48 30.95 118.34 t/20 6.86 7.89 9.31 11.76 15.99 26.57 98.22 t/15 6.83 8.11 9.65 12.05 16.50 27.20 101.91 t/10 7.43 8.60 10.20 12.62 17.21 28.90 108.70 t/6 8.25 9.51 11.16 13.98 19.33 32.13 121.82 t/5 8.74 10.01 11.87 14.94 20.48 33.92' 126.26 t/4 9.74 11.19 13.33 16.73 22.74 36.53 125.30 t/3 13.01 15.01 17.49 20.07 20.51 30.49 105.45 Table 4.12 Factors of Safety f o r 12" P l a i n Masonry Walls i n Current Masonry Design Code * Values not permitted i n code 60.. Factors of Safety Height ( in . ) 95.63 135.63 175.63* 215.63* E c c e n t r i c i t y 0 8.11 10.58 15.65 52.24 t/20 6.89 8.83 12.87 42.71 t /15 7.08 8.87 13.02 36.28 t /10 7.34 8.99 12.77 40.67 t /6 7.27 8.29 11.14 35.28 t / 5 6.96 7.65 9.93 31.53 t /4 6.28 6.34 8.05 25.44 t /3 4.82 4.38 4.83 13.33 5t/12. . 5.10 4.62 5.53 17.24 Table 4.13 Factors of Safety for 8" Reinforced Masonry Walls ( 3 - #3 § ^ ) i n Current Masonry Design Code Factors of Safety Height (in.) 95.63 135.63 175.63* 215.63* E c c e n t r i c i t y 0 7.30 9.59 13.80 45.75 t/20 6.13 7.57 10.80 36.04 t/15 6.54 7.58 10.73 35.82 t/10 6.38 7.50 10.60 34.08 t/6 6.15 6.74 9.10 27.99 t/5 5.77 6.15 8.00 25.10 t/4 5.21 5.27 6.65 19.65 t/3 5.27 5.16 6.46 19.40 5t/12 7.31 5.13 9.23 29.17 Table 4.14 Factors of Safety f o r 8" Reinforced Masonry Walls ( 3 - #6 @ £ ) i n Current Masonry Design Code Factors of Safety Height ( in . ) 95.63 135.63 175.63* 215.63* E c c e n t r i c i t y 0 6.6,0 8.08 10.92 36.64 t /20 5.24 6.26 18.78 28.63 t /15 5.27 6.17 18.59 28.05 t/10 5.26 5.86 18.36 26.66 t /6 4.97 5.40 7.26 22.46 t / 5 4.44 4.69 6.09 19.08 t /4 - 4.07 4.31 5.56 17.03 t /3 4.46 4.89 6.53 19.67 5t/12 9.55 . 10.90 14.85 45.26 Table 4.15 Factors of Safety for 8" Reinforced Masonry Walls ( 3 - #9 @ £ ) i n Current Masonry Design Code 63. fo r wa l ls r e i n f o r c e d with 3 - #3 b a r s , from 5.13 to 9.59 for wal ls r e i n f o r c e d with 3 - #6 bars and from 4.31 to 10.90 for wal ls r e i n f o r c e d with 3 - #9 b a r s . At wa l l heights exceeding the code slenderness l i m i t a t i o n , the values of the safety f a c t o r s vary from 4.83 to 52.24, from 6.46 to 45.75 and from 5.56 to 45.26 for wal ls r e i n f o r c e d with 3 - #3, 3 -#6, and 3 - #9 bars r e s p e c t i v e l y . The ranges of discrepancy are smal ler than those exh ib i ted i n p l a i n masonry w a l l s . The safety fac to rs increase with height of the wal l or slenderness r a t i o , but decrease with the magnitude of the end e c c e n t r i c i t y . 4 .3 .5 Remarks In genera l , the r e s u l t s above show that the current code design method gives i n c o n s i s t a n t r e s u l t s when compared with t h e o r e t i c a l v a l u e s , and the r e s u l t s for the p l a i n wal ls are more conservat ive than the r e i n f o r c e d w a l l s . They a lso show that the code design c a p a c i t y , modi f ied by the recommended slenderness c o e f f i c i e n t , gives over -conservat ive r e s u l t s for t a l l masonry w a l l s , as the safety fac tor increases with the slenderness r a t i o . The l i m i t a t i o n on slenderness r a t i o recommended by the code i s found to be a very conservat ive measure, s ince there appears to be a u s e f u l load capaci ty at heights greater than impl ied thereby. Furthermore, the use of the h / t r a t i o as the slenderness parameter i s a bad c h o i c e , s ince i t does not r e f l e c t the sec t ion geometry of the members, which i s an important f a c t o r i n eva luat ing the slenderness e f f e c t . Compared with a p a r t i a l l y grouted w a l l , a f a c e - s h e l l constructed masonry wal l may have d i f f e r e n t c h a r a c t e r i s t i c s when cons ider ing the slenderness of the member. The use of the r a t i o of the e f f e c t i v e height to the radius of gyrat ion of the member, h / r , (or L / r ) as the slenderness parameter should be an improvement. 64. V. MODIFICATION OF THE MOMENT MAGNIFIER METHOD  FOR MASONRY WALL DESIGN 5.1 The Moment Magnifier Method The Moment Magnifier Method was o r i g i n a l l y developed for s t r u c t u r a l s t e e l . It has been well adapted by the ACI code for the design of concrete structures, accounting for the e f f e c t s of material properties, the presence of cracks and reinforcement, and the slenderness of the load bearing member. This method considers e c c e n t r i c i t y and the slenderness e f f e c t s by magnifying end moments. The increased moment and corresponding load are then checked against the l i m i t imposed by the short column (or wall) capacity. The magnifier has the form: C 0 " 1.0 - P/cJP 3 , J -cr where C = 0.6 +'0.4 e,/e 0 5.1.1 m 1 2 TT2 (EI) P = ,...2 y& 5.1.2 cr ( k L ) A el>e2 = end e c c e n t r i c i t i e s k = e f f e c t i v e length factor (EI) = e f f e c t i v e r i g i d i t y of section, e The capacity reduction f a c t o r , <|>, w i l l be omitted at t h i s stage of the discussion. (EI) i n l i g h t l y reinforced sections such as are t y p i c a l of masonry e may be expressed as: (EI) = EI/X 5.1.3 e where E = i n i t i a l modulus of masonry assemblage. I = net section uncracked moment of i n e r t i a . X = a r i g i d i t y reduction f a c t o r . 65. 5.2 The Modulus of E l a s t i c i t y of Masonry Assemblages In ultimate state masonry design, the strength and d e f l e c t i o n are governed by the modulus of e l a s t i c i t y of the material. At t h i s stage of time, l i t t l e attention has been paid to the e l a s t i c modulus of masonry and experimenal information i s lacking. The equivalent modulus of e l a s t i c i t y of masonry assemblages, besides being influenced by the moduli of masonry constituents (namely, block, mortar, and grout — i f a p p l i c a b l e ) , also depends on the geometry of the section, the thickness of the mortar j o i n t s and the loading condition. In-depth i n v e s t i g a t i o n of t h i s parameter i s out of the scope of t h i s paper, but i t w i l l be an i n t e r e s t i n g topic for future study. In the preceding c a l c u l a t i o n s , the i n i t i a l gradient of the s t r e s s -s t r a i n curve shown i n F i g . 2.6 was used as the modulus of e l a s t i c i t y of masonry. The values used were 1389 k s i and 2083 k s i for p l a i n and p a r t i a l l y grouted walls r e s p e c t i v e l y . Thus under f l e x u r a l type loading, a p a r t i a l l y grouted wall i s s t i f f e r than a p l a i n masonry wa l l . Since the ava i l a b l e experimental data are l i m i t e d , i t i s impossible to r e l a t e the e f f e c t i v e e l a s t i c modulus of the masonry assemblage to i t s section geometry. Under the provisions of the current Canadian Masonry Code S-304-M78, the modulus of masonry depends s o l e l y on the compressive strength of masonry, which i s governed by the strength of the unit and mortar type used (E = lOOOf'm). With the average unit strength of 2350 p s i and S m type mortar being used, the nominal compressive strength of the masonry (fm') i s 1520 p s i and the recommended modulus i s 1520 k s i . ' Eskenazi, Ojinaga, and T u r k s t r a 1 7 suggest that E m = 440 x fm' + 731,000 p s i , and the modulus of e l a s t i c i t y would then be 1400 k s i . 66,. H a t z i n i k o l a s 7 recommends a lower m u l t i p l i e r than the code, ( i e : E = m 750 f m ) and h is recommended value of e l a s t i c modulus would be 1140 k s i . The empi r ica l expressions above fo r eva luat ing the modulus of e l a s t i c i t y of masonry are a l l based on compression tes ts of a x i a l l y loaded ungrouted pr isms. These values are used fo r a l l masonry assemblages with the same un i t strength and mortar type regard less of the d i f f e rences i n c r o s s - s e c t i o n composition ( p l a i n , p a r t i a l l y grouted, or f u l l y grouted) . It i s i n t e r e s t i n g to note that the modulus for f a c e - s h e l l constructed p l a i n wal ls obtained from the s t r e s s - s t r a i n diagram der ived i n Chapter II , does agree with the values p red ic ted with the expression suggested by E s k e n a z i , O j inaga , and T u r k s t r a 9 , while the e l a s t i c modulus of p a r t i a l l y grouted wal ls i s about 1.5 times the modulus for p l a i n w a l l s . In the absence of s u f f i c i e n t knowledge on the i n t e r a c t i o n of masonry components, i t i s bet ter to use d i f f e r e n t moduli for p l a i n and p a r t i a l l y grouted masonry w a l l s . 5.3 General Funct ion fo r the R i g i d i t y Reduction Factor (X) The fo l low ing a n a l y s i s concentrates on developing a l i m i t s tate design method based on the moment-magnifier procedure. For s i m p l i c i t y , a l l wal ls were assumed to be loaded with equal end e c c e n t r i c i t i e s : e, = e„ and C = 1.0 1 2 m 1 Then equation 5.1 becomes: o = _ — 5.2 c r By s u b s t i t u t i n g 5.1.2 and 5.1.3 in to 5.2, we have: 67 . 6= i 5.3 1 - [ P k 2 L 2 A / ( T f 2 E I ) ] If the top and bottom are assumed to be pinned, k i s equal to 1, and the above equation becomes 6 1 1 - (PL 2X)/(n 2EI) leading to 1/6 = 1.0 - (PL 2A)/( TT 2EI) or X = ( TT 2 EI)(1-1/6)/(PL 2) 5.4 By f i n d i n g the r i g h t r i g i d i t y reduction factor the magnifier can be evaluated and the reduced capacity of wall due to slenderness e f f e c t can be obtained. Since the r i g i d i t y E I and length L, are constant for a p a r t i c u l a r wall, the values of f a i l u r e load and moment give the r e l a t i o n -ship between X and 1/6 at any assigned load value, P. Values for X were calculated by deducing the magnification factor <$ d i r e c t l y from the theo-r e t i c a l i n t e r a c t i o n curves obtained from the previous chapter. The calcu-l a t i o n s for X for each section are shown i n Table 5.1, 5.2, 5.3, 5.4 and 5.5. In the concrete codes, the r i g i d i t y has been reduced by factors depending on the s t e e l r a t i o , but X may be related to the r a t i o s of applied load l e v e l to the a x i a l capacity of section, (P/P Q) which i s a means of monitoring the degree of cracking of the section, and the s l e n -derness r a t i o , ( L / r ) , where r i s the radius of gyration of the section. If the r i g i d i t y reduction factor can be assumed to be a separable function of slenderness r a t i o (L/r) and the load r a t i o (P/Po), the expression for X can be written as a product of two functions, such that: X = K (L/r) * K (P/P ) s p o With the above assumptions, the slenderness e f f e c t and the load e f f e c t can be dealt with i n d i v i d u a l l y . R i g i d i t y Reduction Factors ( A) Height ( in . ) 95.63 135.75 175.63 215.63 255.63 295.63 335.63 375.63 (L / r ) 30.92 43.85 56.78 69.71 82.65 95.58 108.51 121.44 (P/P Q ) 0.080 9.78 6.25 4.56 4.12 3.32 2.92 2.50 2.35 0.099 8.24 5.23 3.80 3.42 2.82 2.35 2.12 1.90 0.119 5.52 4.04 2.98 2.62 2.32 2.01 1.77 1.66 0.159 5.23 3.35 2.44 2.06 1 .78 1.64 1.49 1.38 0.199 4.31 3.03 2.34 1.91 1.67 1.53 1.41 1.29 0.239 3.57 2.54 2.07 1.74 1.55 1.42 1.33 1.24 0.318 2.48 2.02 1.73 1.58 1.44 1.36 1.27 1.20 0.398 2.16 1.89 1.61 1.55 1.44 1.36 1.25 0.477 2.36 2.00 1.77 1.62 1.42 1.33 0.557 1.89 1.79 1.70 1.58 1.45 0.636 2.15 1.98 1.72 1.60 Table 5.1 R i g i d i t y Reduction Factors for 8" P l a i n Masonry Walls R i g i d i t y Reduction Factors (X) Height ( in . ) 95.63 135.75 175.63 215.63 255.63 295.63 335.63 375.63 (L / r ) 18.82 26.69 34.57 42.44 50.31 58.18 66.06 73.93 (P /P Q ) 0.080 14.68 8.51 6.53 5.29 4.45 3.84 3.38 3.01 0.119 14.84 8.47 6.03 4.65 3.77 3.16 2.72 2.28 0.159 8.21 5.34 4.12 3.23 2.74 2.38 2.10 1.88 0.199 6.10 4.25 3.26 2.72 2.34 • 2.05 1.82 1.66 0.239 5.66 3.80 3.10 2.45 2.06 1.87 1.70 1.60 0.318 3.44 2.69 2.53 2.35 2.06 1.85 1.72 1.56 0.398 2.79 2.40 2.13 1.88 1.76 1.67 1.60 1.52 0.477 3.16 2.69 2.21 1.98 1.80 1.69 1.59 1.50 0.596 2.35 2.34 2.24 1.95 1.81 1.69 1.64 1.54 Table 5.2 R i g i d i t y Reduction Factors for 12" P l a i n Masonry Walls Rigidity Reduction Factors (A) Height (in.) 9 5 . 6 3 1 3 5 . 7 5 1 7 5 . 6 3 2 1 5 . 6 3 2 5 5 . 6 3 2 9 5 . 6 3 3 3 5 . 6 3 3 7 5 . 6 3 (L/r) 4 1 . 8 3 5 9 . 3 4 7 6 . 8 4 9 4 . 3 4 1 1 1 . 8 4 1 2 9 . 3 4 1 4 6 . 8 4 1 6 4 . 3 4 ( P / P Q ) 0 . 0 3 1 4 3 . 3 5 2 8 . 3 1 2 2 . 0 6 1 7 . 3 1 1 3 . 8 6 1 0 . 9 0 9 . 0 4 7 . 6 3 0 . 0 4 6 2 8 . 6 4 1 9 . 7 6 1 5 . 1 8 1 1 . 2 7 8 . 9 7 6 . 9 3 5 . 6 7 4 . 8 0 0 . 0 6 1 2 1 . 7 4 1 5 . 3 6 1 1 . 2 6 8 . 3 3 6 . 5 6 5 . 1 7 4 . 2 4 3 . 6 2 0 . 0 9 2 1 4 . 3 4 1 0 . 1 1 7 . 1 7 5 . 3 3 4 . 1 0 3 . 3 6 2 . 8 4 2 . 4 7 0 . 1 2 3 1 1 . 3 6 7 . 4 0 5 . 3 4 3 . 9 3 3 . 1 0 2 . 5 3 2 . 1 9 1 . 9 0 0 . 1 5 4 1 0 . 8 9 6 . 8 4 4 . 7 9 3 . 5 0 2 . 7 4 2 . 2 4 1 . 8 9 1 . 6 4 0 . 2 0 0 6 . 5 1 4 . 5 5 3 . 3 5 2 . 6 1 2 . 1 5 1 . 8 1 1 . 5 5 1 . 3 6 0 . 2 4 6 4 . 9 6 3 . 6 9 2 . 7 9 2 . 2 4 1 . 8 7 1 . 6 0 1 . 4 0 1 . 2 5 0 . 3 0 7 3 . 8 6 2 . 9 9 2 . 3 9 1 . 9 7 1 . 6 7 1 . 4 3 1 . 2 5 0 . 3 8 4 3 . 1 5 2 . 5 2 2 . 1 0 1 . 8 0 1 . 5 3 1 . 3 1 0 . 4 6 0 2 . 6 5 2 . 3 3 1 . 9 6 1 . 6 8 1 . 4 5 Table 5 . 3 Rigidity Reduction Factors for 8 " Reinforced Masonry Walls ( 3 - #3 @ g ) R i g i d i t y Reduction Factors ( X) Height ( in . ) 95.63 135.75 175.63 215.63 255.63 295.63 335.63 375.63 (L / r ) 41.83 59.34 76.84 94.34 111.84 129.34 146.84 164.34 (P /P Q ) 0.028 15.67 17.26 16.85 16.19 13.35 11.38 10.02 8.70 0.042 13.54 14.62 14.11 11.99 9.96 8.42 6.99 5.83 0.056 12.54 13.01 11.80 9.60 7.65 6.25 5.20 4.34 0.084 10.72 10.33 8.61 6.84 5.10 4.04 3.26 2.75 0.113 8.79 8.34 6.58 4.92 3.72 2.98 2.44 2.04 0.141 7.53 6.91 5.13 3.81 2.94 2.37 1.98 1.67 0.183 5.87 5.03 3.78 2.88 2.30 1.89 1.60 1.38 0.225 4.84 4.46 3.03 2.38 1 .96 1.65 1.42 1.22 0.281 3.96 3.19 2.52 2.02 1.69 1.46 1.26 0.422 2.71 2.40 2.02 1.69 1.44 0.492 . .2 .52 . . . 2.32 . 1.98 1.62 Table 5.4 R i g i d i t y Reduction Factors for 8" Reinforced Masonry Walls ( 3 - #6 @ ) Rigidity Reduction Factors (X) Height (in.) 95.63 135.75 175.63 215.63 255.63 295.63 335.63 375.63 (L/r) 41.83 59.34 76.84 94.34 111.84 129.34 146.84 164.34 (P/PQ) 0.025 9.21 9.38 8.84 9.06 9.02 8.63 8.37 7.65 0.049 8.49 7.81 7.58 7.49 6.76 6.04 5.28 4.30 0.099 6.95 6.48 5.97 5.08 3.99 3.15 2.55 2.11 0.160 5.59 5.03 4.08 3.08 2.43 1.96 1.64 1.40 0.197 4.97 4.21 3.28 2.53 2.04 1.67 1.42 1.22 0.246 4.16 3.38 2.61 2.07 1.71 1.44 1.25 0.308 3.41 2.79 2.20 1.78 1.50 1.29 0.369 2.91 2.53 2.00 1.67 1.43 0.431 2.73 2.41 1.94 1.61 0.493 2.51 2.28 1.91 Table 5.5 Rigidity Reduction Factors for 8" Reinforced Masonry Walls ( 3 - #9 @ £ ) 5.4 The Function (K ) s In order to f i n d a general function for K g, the t h e o r e t i c a l values of X had f i r s t to be normalized for each value of ^l^Q- Tables 5.6, 5.7, and 5.8 contain a l l values of X normalized with respect to the value where the slenderness r a t i o (L/r) i s 76.86. The average values of the normalized X for a l l (P/P ) r a t i o s are shown on Table 5.4 along with the corresponding o (L/r) values. The above r e s u l t s were plotted on F i g . 5.1, and, by using a U.B.C. non-linear f i t t i n g routine "LQF" with 3 parameters requested, a quadratic equation was f i t t e d through the average values of Tables 5.4. The approximate function was found to be: K (L/r) = -0.294 + 130(L/r)~ 1 - 2325(L/r)" 2 5.5 s The complete plot of the K g function i s shown on F i g . 5.2. F i g . 5.1 shows the plot of normalized X vs (L/r) r a t i o along with the f i t t e d function of K . There i s a l o t more scatter at low (L/r) r a t i o s . s The data for walls with reinforcement r a t i o of 0.00123 and 0.00491 bunch together well at (L/r) r a t i o more than 76.92, while walls with reinforcement r a t i o of 0.01116 tend to scatter more at that range. At the low range of (L/r) values, the scattering c h a r a c t e r i s t i c i s i n d i f f e r e n t for a l l walls. By observation, the derived function does describe the influence of slenderness r a t i o on the reduction factor reasonably w e l l . The complete plot of the function K on F i g . 5.2, becomes negative at s slenderness r a t i o s above 500 and below 20. 75. Normalized R i g i d i t y Reduction Factor (L / r ) 41.84 59.34 76.84 94.34 111 .84 129.34 146.84 164.34 (P /P Q ) 0.031 1.97 1.28 1.00 0.78 0.63 0.49 0.41 0.35 0.046 1.89 1.30 1.00 0.74 0.59 0.46 0.37 0.32 0.061 1.93 1.36 1.00 0.74 0.58 0.46 0.38 0.32 0.092 2.00 1.42 1.00 0.74 0.57 0.47 0.40 0.34 0.123 2.13 1.38 1.00 0.74 0.58 0.47 0.41 0.36 0.154 2.27 1.43 1.00 0.73 0.57 0.47 0.40 0.34 0.200 1.94 1.36 1.00 0.78 0.64 0.54 0.46 0.41 0.246 1.78 1.32 1.00 0.80 0.67 0.57 0.50 0.45 0.307 1.61 1.25 1.00 0.82 0.79 0.60 0.49 0.384 1.50 1.20 1.00 0.85 0.73 0.62 0.460 1.35 1.19 1.00 0.86 0.74 Table 5.6 Normalized R i g i d i t y Reduction Factors for 8" Reinforced Masonry Walls ( 3 - #3 @ £ ) Normalized R i g i d i t y Reduction Factor (L / r ) 41.84 59.34 76.84 94.34 111.84 129.34 146.84 164.34 (P/P Q ) 0.028 0.93 1.02 1.00 0.96 0.80 0.68 0.60 0;52 0.042 0.96 1.04 1.00 0.85 0.71 0.60 0.50 0.41 0.056 1.06 1.10 1.00 0.81 0.65 0.53 0.44 0.37 0.084 1.25 1.20 1.00 0.80 0.60 0.47 0.38 0.32 0.113 1.34 1.27 1.00 0.75 0.57 0.45 0.37 0.31 0.141 1.47 1.35 1.00 0.74 0.57 0.46 0.39 0.33 0.183 1.55 1.33 1.00 0.76 0.61 0.50 0.42 0.37 0.225 1.60 1.47 1.00 0.79 0.65 0.54 0.47 0.41 0.281 1.57 1.27 1.00 0.80 0.67 Q.58 0.50 0.422 1.34 1.19 1.00 0.84 0.71 0.492 1.27 1.19 1.00 0.82 —— Table 5.7 Normalized R i g i d i t y Reduction Factors for 8" Reinforced Masonry Walls ( 3 - #6 @ %) Normalized R i g i d i t y Reduction Factor (L / r ) 41.84 59.34 76.84 94.34 111 .84 129.34 146.84 164.34 (P/P Q ) 0.025 1.04 1.06 1.00 1.02 1.02 0.98 0.95 0.87 0.049 1.12 1.03 1.00 0.99 0.89 0.80 0.70 0.57 0.099 1.16 1.09 1.00 0.85 0.67 0.53 0.43 0.35 0.160 1.37 1.23 1.00 0.76 0.60 0.48 0.40 0.34 0.197 1.51 1.28 1.00 0.77 0.62 0.51 0.43 0.37 0.246 1.60 1.30 1.00 0.80 0.66 0.55 0.48 0.308 1.55 1.27 1.00 0.81 0.68 0.58 0.370 1.45 1.27 1.00 0.84 0.72 0.431 1.41 1.24 1.00 0.83 0.493 1.31 1.20 1.00 . — ~ . . -.-—- . Table 5.8 Normalized R i g i d i t y Reduction Factors for 8" Reinforced Masonry Walls ( 3 - #9 @ £ ). This e f f e c t can be ignored as at slenderness r a t i o more than 500, the wall i s way over the p r a c t i c a l l i m i t a t i o n on slenderness, and at a slenderness r a t i o of l e s s than 20 the wall i s generally treated as a short w a l l . In order to avoid having negative numbers for K , i t i s h e l p f u l to set a s sensible l i m i t for K g which i s picked to be 40 < L/r < 400. The function of Ks does correlate reasonably well with the somewhat scattered data and i t does r e f l e c t the influence of slenderness on the r i g i d i t y reduction factor i n the simplest form. 5.5 The Expression for Load Influence Factor, K (P/P ) P o Accepting the above function for Kg, the corresponding function for K (P/P ) was solved e a s i l y with s i m i l a r a n a l y t i c a l techniques. As P o defined previously, the r i g i d i t y reduction f a c t o r , X, i s the product of two functions, K ( L / r ) , and K (P/P ). The values of K were obtained by s p o p d i v i d i n g each t h e o r e t i c a l X by the corresponding value of K g calculated from the expression i n the previous section. The computed average values of K for each (P/P ) r a t i o are shown i n Tables 5.10, 5.11, and 5.12. p o When studying the plot of vs (P/P q) i n F i g . 5.3, the best expression f o r ^ (P/P q) i s seen to be not a quadratic function, but an exponential or inverse function. For s i m p l i c i t y , an inverse function was chosen and the expression had the form: K p (P/P q) = a + b ( p / p o ) - 1 = a + b (p Q/p) The expression for K^ was found by using the same least square f i t -t i n g routine "LQF" as i n the previous secction. Instead of using the value of (P/P ), the inverse of the values were entered as the variables o 30 0 0.1 0.2 0.3 0.4 0.5 P/P0 F i g . 5.3 Relat ionship Between Kp and the Load Ratios (P/PQ) f o r Reinforced Masonry Walls 81.. along with the average values of as the dependent v a r i a b l e s . By request ing 2 parameters, the r e s u l t i n g expression for i s shown below: K = 1.207 + 0.457 (p /p) 5.7 p r o r (L/r) 41.84 59.34 76.84 94.34 111.84 129.34 146.84 164.34 Average Normalized R i g i d i t y Reduction Factors 1.48 1.23 1.00 0.82 0.68 0.57 0.48 0.41 Table 5.9 Over-all Average Values of Ri g i d i t y Reduction Factors f o r 8" Reinforced Masonry Walls (P/P 0) 0.031 . 0.046 0.061 0.092 0.123 0.154 0.200 0.246 0.307 0.384 0.461 Average Value 21.56 14.15 10.62 7.10 5.25 4.74 3.44 2.93 2.48 2.17 1.97 Table 5.10 Average Values of for Reinforced Masonry Walls ( 3 - #3 @ £ ) (P/PcJ 0.028 0.042 0.056 0.084 0.113 0.141 0.183 0.225 0.281 0.422 0.492 Average Value of K p 17.88 13.50 10.78 7.59 5.76 4.65 3.63 3.07 2.56 2.00 1.89 Table 5.11 Average Values of K p for Reinforced Masonry Walls ( 3 - #6 @ ) (P/P 0> 0.025 0.049 0.099 0.160 0.197 0.246 0.308 0.370 0.431 0.493 0.554 Average Value of K p 14.17 8.85 5.50 3.69 3.12 2.62 2.24 2.03 1.93 1.82 1.83 Table 5.12 Average Values of K,* for Reinforced Masonry Walls ( 3 - #9 @ ) 84. F i g . 5.3 shows that at (P/P Q) > 0.2, values of K p a l l bunch together and can be f i t t e d well by the expression found above. At ( P / P Q ) < 0 .2 , scattering of data becomes severe as (P/P q) decreases. The expression developed above by using the average values of for each type of r e i n -forced wall can only represent a function describing the general behavior of the data. The use of t h i s expression for design purposes w i l l lead to unconservative r e s u l t s for most walls, e s p e c i a l l y at low (P/P ) r a t i o s . o The above problem can be i l l u s t r a t e d c l e a r l y by comparing the theore-t i c a l i n t e r a c t i o n curves with the i n t e r a c t i o n curves reproduced by apply-ing the derived functions of K (L/r) and K (P/P ) to the moment magni-s p o f i e r method. F i g . 5.4 shows the comparison for walls with 3 - #3 r e i n f o r c i n g bars. It shows good c o r r e l a t i o n but i s unconservative i n the region where the (P/P ) r a t i o i s low. As mentioned i n the introduction of o t h i s Chapter, masonry walls often serve as panel elements i n structures, and are required to carry higher r a t i o s of moment to v e r t i c a l load than generally occur i n column design. For that p a r t i c u l a r reason, the portion of i n t e r a c t i o n curves with low (P/P ) r a t i o i s important for masonry wall o design and the presently derived functions are not s a t i s f a c t o r y . Another problem i n using the present function i s that when the r a t i o of (P/P ) approaches zero, the value of K o p 85. OL < O 800 700 600 500 400 300 200 100 THEORETICAL CURVE IMPLIED CURVE 50 60 20 30 40 MOMENT , kip-ft. F i g . 5.4 Comparison of Theoret ica l and Implied Interact ion'Diagram f o r 8" Reinforced Masonry Walls ( 3-#3 @ Q_) 86. becomes i n f i n i t e . In applying t h i s to the moment magnifier method, the r e s u l t i s very unstable as P becomes zero and the resultant magnifier 6 approaches i n f i n i t y , while the t h e o r e t i c a l i n t e r a c t i o n curves show that the magnifier becomes 1. as (P/P ) r a t i o approaches zero. This w i l l pro-o vide very inaccurate r e s u l t s at very low (P/P 0) r a t i o s . 5.6 Improvement of the Established Functions For the two functions derived currently, the K (L/r) function i s a s general function which only ref e r s to the e f f e c t of slenderness on r i g i d -i t y . Since the two problems encountered i n the previous section mainly concern the load r a t i o e f f e c t , there i s no obvious reason to make any change i n the K function. The K (P/P ) expression i s the one which has s p o to be improved. In attacking the f i r s t problem, F i g . 5.3 shows the complete picture which has to be contended with, as i t i s o l a t e s the e f f e c t of load r a t i o on the reduction f a c t o r . By modifying the present expression for K to make i t an upper bound for a l l data i n F i g . 5.3, the predicted values of are always higher than the t h e o r e t i c a l values, and the resultant values of the magnification factors are conservative. The above objective i s f u l f i l l e d by s h i f t i n g the present curves to the ri g h t i n order to cover most of the data points at low P/P r a t i o , and the improved expression i s : o K (P/P ) = 0.7 + 0.75 (P/P ) - 1 5.8 p o o The plot i s superimposed on F i g . 5.3. The second problem which i s about a mismatch of the o r i g i n was solved by applying a cut-off point to the function, so that at values of (P/P ) from the cut-off point to zero, K i s a constant instead of a value o' p approaching i n f i n i t y . 87. i On the load-moment i n t e r a c t i o n diagram, the introduction of a cut-off value i n produces an approximately straight l i n e j o i n i n g the point at the cut-off value of (P/P q) to the pure moment capacity of the short wall. To select a proper cut-off point, i t i s important to refer to the reproduced i n t e r a c t i o n diagram. I f the cut-off value of K i s too low, P the resultant i n t e r a c t i o n curves w i l l be unconservative; the opposite occurs i f the cut-off value of K i s too high. It i s found that d i f f e r e n t P s t e e l r a t i o s require cut-off points at d i f f e r e n t (P/P q) r a t i o s . For walls with a high s t e e l r a t i o , the cut-off point should be at a high (P/P Q) r a t i o . Through observation, an empirical l i n e a r function r e l a t i n g the cut-off value of ( P / P ) to the s t e e l r a t i o i s found as follows: (P/P ) = 0.0045 + 3.57 (p) O CO where (P/P ) = value of (P/P ) r a t i o at the cut-off i n the O CO o Kp function p = v e r t i c a l reinforcement r a t i o The corresponding plot i s shown on F i g . 5.5. A 0 0.005 0.010 0.015 0.020 S T E E L RATIO , • p F i g . 5.5 Relat ionship of the Cut -o f f Value for and V e r t i c a l Reinforcement Ratio (p) 8 9 . 5.7 Discussion of the Implied Interaction Diagrams Figs 5.6 and 5.7 show the blown up section of i n t e r a c t i o n diagrams for small (P/P ) r a t i o s and the complete i n t e r a c t i o n curves f o r a 40" wide o wall with alternate cores grouted and with 3-#3 v e r t i c a l reinforcement. The s o l i d dark l i n e s are the t h e o r e t i c a l i n t e r a c t i o n curves for various lengths, while the dotted l i n e s are the corresponding ones developed using the current equations. With a s t e e l r a t i o of 0.00123, the cut-off point for the K (P/P ) function i s 0.009, with the associated K value of 87.9. p o p It i s seen that the implied i n t e r a c t i o n curves are conservative for wall heights above 135.6". For walls which are 95.63" or shorter the r e s u l t s are s l i g h t l y unconservative, but the deviation i s small and can be neglec-ted. At high v e r t i c a l loads, the reproduced curves become unconserva-t i v e , but the problem vanishes when the minimum e c c e n t r i c i t y of t/20, recommended i n the current code, i s introduced. Fi g s . 5.8, 5.9, 5.10, and 5.11 show sim i l a r plots f o r walls r e i n -forced with 3 - #6 or 3 - #9. The e f f e c t of the cut-off value for K i s P more pronounced at these reinforcement r a t i o s . Due to the imposed cut-off values of the K function, the implied curves are angular at the cut-off P (P/P Q) r a t i o , but they do match the pronounced double curvature character-i s t i c of the t h e o r e t i c a l i n t e r a c t i o n curves due at the high reinforcement 90. 0 10 20 30 MOMENT, kip-ft. F i g . 5 . 6 Comparison of Theoret ica l and Implied In teract ion Diagram fo r 8" Reinforced Masonry Walls (3-#3 @ Cj @ Low (P /P Q ) Ratio F i g . 5.7 Comparison of Theoret ica l and Implied In teract ion Diagrams fo r 8" Reinforced Masonry Walls (3-#3 @ (Q MOMENT, kip-ft. F i g . 5.8 Comparison- of Theoret ical and Implied In teract ion Diagrams f o r 8" Reinforced Masonry Walls (3-#6 @ CJ at Low ( P / P ) Ratios 93, 900 800 THEORETICAL CURVE IMPLIED CURVE 20 30 40 MOMENT, kip - ft. 50 60 F i g . 5.9 Comparison of Theoret ica l and Implied In teract ion Diagrams fo r 8" Reinforced Masonry Walls (3-#6 @ Gj 0 10 20 30 40 50 M O M E N T , k ip - f t . F i g . 5.10 Comparison of Theoret ical and Implied Interact ion Diagrams f o r 8" Reinforced Masonry Walls (3-#9 @ Gj at Low (P/P Q ) Ratios 95 . 900 0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 MOMENT , kip-ft. F i g . 5.11 Comparison of Theoret ica l and Implied In teract ion Diagrams f o r 8" Reinforced Masonry Walls (3-#9 @ Gj 96. r a t i o . The implied values are a l l conservative for these two types of walls. One drawback to the above expression i s that the degree of conserva-tism varies with the amount of reinforcement. The implied i n t e r a c t i o n diagrams for walls reinforced with 3 - #9 bars are more conservative than those f o r other quantities of reinforcement. This must be accepted i f only one expression of K i s to be used for a l l walls i n a trade o f f of P accuracy for s i m p l i c i t y . 5.8 A p p l i c a t i o n to P l a i n Masonry Walls In the following section, the a p p l i c a t i o n of these formulas to unreinforced walls i s investigated. If possible, the same formulation w i l l be used to compute the r i g i d i t y reduction f a c t o r , so that the pro-posed moment magnifier method for design of concrete masonry walls can be simple, straight-forward and complete. The same a n a l y t i c a l approach as i n the previous sections i s employed. The r i g i d i t y reduction factor X i s assumed to be the product of two func-t i o n s , K (P/P ) and K ( L / r ) , such that: p o s X = K (P/P ) * K (L/r) p v o' s v ' Since the p l a i n walls have d i f f e r e n t section geometry from reinforced walls, the radius of gyration i s changed. In order to study the charac-t e r i s t i c s of the section, F i g . 5.12 shows a plot of X vs (L/r) normalized to a (L/r) r a t i o of 56.8. The r e s u l t s shows that i t has the same trend as the reinforced walls. For the sake of convenience and s i m p l i c i t y , the same function for K that was derived for reinforced walls i s used. s With equation 5.5, the values of K g were calculated f or p a r t i c u l a r (L/r) r a t i o s are given 9 7 . 2.0 I .5 ,< I . 0 I— 0.51— 0.04 F i g . 5.12 |;A P lot of Normalized R i g i d i t y Reduction Factors and Slenderness Ratios (L / r ) f o r P la in Masonry Walls 9 9 . i n Tables 5.13 and 5.14. The values of K were evaluated from K = X/K . P P s The average of for 8" and 12" p l a i n masonry walls are shown on Tables 5.15 and 5.16 re s p e c t i v e l y . F i g . 5.13 shows the corresponding plot of average values of K vs (P/P ) r a t i o . The expression for K used for p o p reinforced walls was also plotted and superimposed on the same diagram for comparison. It i s found that the t h e o r e t i c a l values are smaller than the values obtained from the previous expression for K , and the equation gives an P upper bound of a l l the data. Thus the same expression used for reinforced walls w i l l produce conservative r e s u l t s for p l a i n walls. Thus i t i s tempting to use the same expression for p l a i n walls, but i n the l a s t r e s ort, the choice must depend upon the accuracy of the implied i n t e r a c t i o n curves. The i n t e r a c t i o n diagrams for 8" and 12" p l a i n masonry walls with various slenderness r a t i o s as implied by the present moment magnifier method were superimposed on the t h e o r e t i c a l i n t e r a c t i o n diagrams as shown on Figs. 5.14 and 5.15. F a i r l y good agreement i s found between the implied diagrams and the t h e o r e t i c a l diagrams. Most values computed by the moment magnifier method are conservative, except for the 12" walls with wall heights of less than 100" i n which the computed value exceed the t h e o r e t i c a l l y predicted value. In the current code, walls with (h/t) r a t i o of 8 or l e s s , such as a wall of 100" height and 12" thickness are designed as showt walls, and the unconservative r e s u l t i n t h i s range i s of no concern. The o v e r a l l r e s u l t i s reasonably conservative and s a t i s f a c t o r y . Height ( in . ) 95.63 135.63 175.63 215.63 255.63 295.63 335.63 375.63 (L / r ) 30.92 43.85 56.78 69.71 82.65 95.58 108.51 121.44 K s 1.47 1.46 1.27 1.09 0.94 0.81 0.70 0.61 Table 5.13 Values of K s fo r 8" P l a i n Masonry Walls H e i g h t . ( i n . ) 95.63 135.63 175.63 215.63 255.63 295.63 335.63 375.63 (L / r ) . . 18.82 26.69 34.57 42.44 50.31 58.18 66.06 73.93 K s 0.04 1.31 1.52 1.48 1 .37 1.25 1.14 1.04 Table 5.14 Values of K s fo r 12" P l a i n Masonry Walls (P /P Q ) 0.080 0.100 0.119 0.159 0.199 0.239 0.318 0.398 0.477 0.557 0.636 Average Value 4.11 3.42 2.68 2.25 2.05 1.84 1.61 1.50 1.51 1.37 1.45 Table 5.15 Average Values of K for 8" P l a i n Masonry Walls i r ( P / P 0 ) 0.080 0.119 0.159 0.199 0.239 0.318 0.398 0.477 0.596 Average Value Of Kp 4.08 3.35 2.36 1.97 1.80 1.61 1.43 1.48 1.46 Table 5.16 Average Values of K for 12" P l a i n Masonry Walls 102. MOMENT, kip-ft. F i g . 5.14 Theoret ica l and Implied Interact ion Diagrams f o r 8" P la in Masonry Wal 1 s 103. ' 0 10 20 30 40 50 60 MOMENT, kip-ft. F i g . 5.15 Theoret ica l and Implied Interact ion Diagrams f o r 12" P la in Masonry Walls 104.. 5.9 Design of Slender Masonry Walls As discussed in the previous sections, the capacity of a masonry wall does not only depend on the strength of the material i t s e l f , but also on the effects of slenderness. Although i t i s of questionable applicability in the case of lightly loaded wall panels, the ACI moment magnifier approach has been used here, since i t i s familiar to most practicing engineers. For walls in single curvature bending, the maximum mid-height moment capacity i s obtained by multiplying the maximum applied moment by the moment magnifier 6, as shown in Eq. 5 . 1 . The magnified moment is then compared with the short column interaction curve. In double curvature bending, 6 is modified by a factor C depending on the ratio of end m moments. The tensile strength of the masonry material was found to have no significant effect on wall behaviour. It is important to point out that the production of the correct short wall interaction diagram i s very c r i t i c a l for the design of masonry walls. Unfortunately, due to the "strain gradient effect", the construction of a correct interaction diagram i s a very d i f f i c u l t task, as the compressive strength of the material appears to increase as the loading condition varies from axial to eccentric. The effect is very severe for partially or ful l y grouted walls. The suggested procedure for the design of slender masonry walls i s given in the step-by-step outline, where most of the equations are repeated for cla r i t y . 1. Compute the required design load and moment. 2. Compute the effective r i g i d i t y of the section: 105. ( E I ) e = EI/X where E = 1389 ( f o r p l a i n masonry wall) = 2083 (for p a r t i a l l y grouted wall) I = moment of i n e r t i a of net section, including grout X = (K )(K ) P s K g = -0.294 + 130/(L/r) - 2325/(L/r) 2 where 40 < L/r < 400 K = 0.7 + 0.75/p P P = P/Po, but not les s than 0.0045 + 3.57p p = s t e e l r a t i o , based on net section n2 (*De 3. Compute P = r — cr ( k L ) 2 Cm  4. Compute 6 = i _ p/ $p cr see ACI code for values of k, Cm and cj. 5. Check whether P and SM f a l l within the i n t e r a c t i o n curve for short walls. An i l l u s t r a t i o n of the design procedure i s shown below. The example i s based on a 40" wide and 7.265" thick f a c e - s h e l l constructed p l a i n w a l l . The wall i s 150" high and the block strength i s 2350 with S type mortar. The wall i s pinned and c a r r i e s a load of 100 kips at equal end e c c e n t r i c i -t i e s of 1.5". The e l a s t i c modulus of the wall unit i s 1389 k s i , and the short wall i n t e r a c t i o n diagram shown on F i g . 5.13 i s used for the follow-ing c a l c u l a t i o n with the a x i a l capacity (P ) of 251.5 k i p s . 106. Step 1: P = 100 kips Mn = 12.5 k i p s - f t . r 3.09 i n 2 I = 1137. ±nk P = 251.5 kips o Step 2: K g = -.294 + 130(3.09/150) - 2325(3.09/150) 2 = 1.397 P/P = 100/251.5 = 0.389 > .0045 + 3.57 p o K = .7 + .75/.398 P = 2.587 X = (1.39)(2.587) = 3.61 Step 3: P c r TT2( 1388) (1137) (150) 2(3.61) = 192 kips Step 4: 6 = !_ 1 0o/(192)(0.7) = 3.91 Step 5: <SM = (3.91)(12.5) = 48.8 k i p - f t . Since Mu from short wall i n t e r a c t i o n curve i s 27.4 k i p - f t , and i s less than the applied moment (48.8 k i p - f t ) , the section has to be redesigned. In the example above, since the capacity reduction factor (<(>) used i n the ACI design format for concrete has yet to be determined for masonry material, the value recommended for concrete was used to complete the i l l u s t r a t i o n . For walls with v e r t i c a l reinforcement, the procedure for design i s s i m i l a r . A d i f f e r e n t short wall i n t e r a c t i o n curve has to be used, and at low P/P 0 r a t i o , the cut-off point i s imposed on the function f o r the load influence f a c t o r , K^, according to the amount of r e i n f o r c e -ment i n the section. 107. VI CONCLUSIONS AND RECOMMENDATIONS Using numerical in t e g r a t i o n techniques to determine the section capa-c i t y and the column d e f l e c t i o n curves, and therefore the i n s t a b i l i t y f a i l u r e conditions, a t h e o r e t i c a l analysis was performed on masonry walls. A moment magnifier method was developed for masonry wall design based on the above an a l y s i s . In the moment magnifier method, the slenderness e f f e c t i s simulated by introducing the r i g i d i t y reduction factor which i s i n turn a function of the slenderness r a t i o (L/r) and the applied load r a t i o (P/P ). The design method was found to be s a t i s f a c t o r y and conser-o v a t i v e . For s i m p l i c i t y , there i s only one general expression used accounting for the slenderness e f f e c t for a l l types of masonry wall design ( p l a i n or r e i n f o r c e d ) . In general, the design method i s simple, st r a i g h t forward and adequate for design purposes, although i t may be considered over-conservative i n some cases. The current code, based on allowable stresses, was found to be incon-s i s t e n t and over-conservative i n most cases. The design method i s not capable of dealing i n d i v i d u a l l y with d i f f e r e n t design aspects. The use of the h/t r a t i o as the slenderness parameter does not d i s t i n g u i s h the difference between a f a c e - s h e l l constructed p l a i n wall and a f u l l y grouted w a l l . The l i m i t a t i o n on the slenderness r a t i o stated i n the present code apears to be an overconservative measure. Since the magnitude of the t e n s i l e strength of masonry material i s small, i t has no s i g n i f i c a n t e f f e c t on masonry wall design. Based on the deflected shapes and the f a i l u r e loads of some e x i s t i n g experimental r e s u l t s , the walls f i l l e d with grout of strength s i m i l a r to that of the block tend to have higher f l e x u r a l compressive strength and o v e r a l l : .103.: e l a s t i c modulus than the f a c e - s h e l l constructed p l a i n wal ls with the same block s t rength . Comparing a n a l y t i c a l to experimental r e s u l t s , the ' s t r a i n gradient e f f e c t ' appears to be more pronounced i n p a r t i a l l y grouted wal ls than i n f a c e - s h e l l constructed p l a i n w a l l s . In f a c t , there was no s i g n i f i c a n t -d i f f e r e n c e when comparing the experimental pure a x i a l load capaci ty of the f a c e - s h e l l constructed wal ls with the t h e o r e t i c a l a x i a l capac i ty based on the f l e x u r a l compressive s t rength . But f o r p a r t i a l l y grouted w a l l s , the experimental pure a x i a l capac i ty was approximately one-hal f of the theore-t i c a l capaci ty evaluated from f l e x u r a l compressive strength of the wal l assemblages. Due to the tremendous d i f f e r e n c e i n capac i ty caused by the " s t r a i n gradient e f f e c t " , the const ruc t ion of an accurate short wa l l i n t e r a c t i o n diagram i s d i f f i c u l t e s p e c i a l l y for grouted w a l l s . Further study should be d i r e c t e d towards r e l a t i n g the " s t r a i n gradient e f f e c t " with the s e c t i o n geometry of masonry wal l assemblages. 109'. REFERENCES 1. Canadian Standard . A s s o c i a t i o n , 1978, "Masonry Design and Const ruct ion  for B u i l d i n g s . " Nat ional Standard of Canada. Can 3 - 5304 - M 18, Rexdale, Ontar io . 2. Nathan, N.D. "Slenderness of Press t ressed Concrete Beam-Columns." PCI J o u r n a l , V o l . 17 - #6. Nov. -Dec. 1972. pp. 45-57. 3. Oj inaga, J . and T u r k s t r a , C . "The Design of P l a i n Masonry W a l l s . " Dept. of C i v i l Eng ineer ing . M c G i l l U n i v e r s i t y , Montreal . 4. Oj inaga, J . and Turkstra C . "The Design of Reinforced Masonry Walls  and Columns. I - concent r ic Loading and Minor Axis Bending." Dept. of C i v i l Engineer ing , M c G i l l U n i v e r s i t y , Sept. 1979. 5. Yoke l , D ikker . "Strength of Load Bearing Masonry Wa l l s . " S t r u c t u r a l D i v i s i o n , ASCE J o u r n a l , May 1971, pp. 1593-1609. 6. Yoke l , Dikker . " S t a b i l i t y and Load Capaci ty of Members with No Ten-s i l e S t rength ." S t r u c t u r a l D i v i s i o n , ASCE J o u r n a l , J u l y , 1971, pp. 1913-1925. 7. H a t z i n i k o l a s , Longworth, and Warwaruk, "Concrete Masonry W a l l s . " Ph.D. T h e s i s , S t r u c t u r a l Engineer ing repor t No. 70, Dept. of C i v i l Eng ineer ing , U n i v e r s i t y of A l b e r t a , Edmonton, Sept. 1978. 8. H a t z i n i k o l a s , M. , Longworth, J . and Warwaruk, J . "The A n a l y s i s of  E c c e n t r i c a l l y Loaded Masonry Walls by the Moment Magni f ier Method." Pro . 2nd Canadian Masonry Conference, 1980. pp. 245-252. 9. Nathan, N.D. and Chaudwani, R., "Precast Prest ressed Sect ions Under  A x i a l Load and Bending." PCI Journa l V o l . 16-#3. May-June 1971, pp. 10-19. 10. Nathan, N.D. " A p p l i c a t i o n of ACI Slenderness Computatins to Pre -s t ressed Concrete S e c t i o n s . " PCI J o u r n a l , V o l . 20-#3, May-June, 1975, pp. 68-75. 11. A lcock , W . J . , and Nathan, N.D. "Moment Magni f ica t ion Tests of Pre -s t ressed Concrete Columns." PCI J o u r n a l , V o l . 22-#4, Ju ly -Augus t , 1977. 12. Drysdale , R.G. and Hamid, A . A . "Behaviour of Concrete Block Masonry  Under A x i a l Compression." Techn ica l Paper. ACI J o u r n a l , June 1979. pp. 707-721. 13. T u r k s t r a , C , and Thomas, G. S t r a i n Gradient E f f e c t s i n Masonry. S t r u c t u r a l Masonry s e r i e s No. 78-1 , Dept. of C i v i l Engineer ing and Appl ied Mech. , M c G i l l U n i v e r s t i y , Montreal , A p r i l 1978. 110. References Cont'd... 14. Boult, B.F. Concrete Masonry Prism Testing, ACI Journal, A p r i l 1979, pp. 513-535. 15. Canadian Standard Association. Code for the Design of Concrete  Structures for Buildings. National Standard of Canada, CAN 3 - A 23.3 - M 77. 16. Yokel, F.Y., Mathey, R.G. and Dikkers, R.D., "Strength of Masonry Walls Under Compressive and Transverse Loads", National Bureau of Standards, Building Science Series 34, March 1981. 17. Eskenazi, A., Ojinaga, J . and Turkstra, C.J., "Some Mechanical Properties of Brick and Block Masonry". Interim Report, Dept. of C i v i l Engineering and Applied Mechanics, McGill University, Montreal, St r u c t u r a l Masonry Series 75-2, pp. 75, 1975. 

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