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Strength of Glulam rivet connections under eccentric loading Karacabeyli, Erol 1986

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STRENGTH OF GLULAM RIVET CONNECTIONS UNDER ECCENTRIC LOADING by EROL KARACABEYLI M . S . , Y I L D I Z UNIVERSITY , ISTANBUL, TURKEY, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Depar tmen t o f C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e a u i r e d s t a n d a r d THE UNIVERSITY OF BRIT ISH COLUMBIA A p r i l , 1986 © EROL KARACABEYL I , 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t 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 f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s o r h e r r e p -r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa r tmen t of C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 M a i n M a l l V a n c o u v e r , Canada V6T 1Y3 Da te A p r i l , 1986 ABSTRACT Theo r e t i c a l and experimental studies of the strength of Glulam r i v e t connections under e c c e n t r i c loading are presented. Two f a i l u r e modes are studied: 1. Rivet y i e l d i n g i n bending with simultaneous bearing f a i l u r e of the wood under the r i v e t ' s shank. 2 . Wood f a i l u r e around the r i v e t c l u s t e r . A f i n i t e element stress a n a l y s i s of the connection i s c a r r i e d out, and the e f f e c t s of d i f f e r e n t parameters on strength and f a i l u r e modes are presented. Experimental r e s u l t s on f u l l - s i z e t e s t specimens are used to compare tests and t h e o r e t i c a l f a i l u r e loads. F i n a l l y , a design procedure i s proposed. - i i -TABLE OF CONTENTS Page Abstract i i L i s t of Figures v L i s t of Tables v i i Acknowledgements v i i i CHAPTER 1 - INTRODUCTION 1 1.1 General 1 1.2 Background and Previous Research 2 1.3 Objective and Scope of the Thesis 3 CHAPTER 2 - RIVET YIELDING FAILURE MODE 5 2.1 Kinematic Assumptions 5 2.2 Nonlinear Load-Slip C h a r a c t e r i s t i c s of Glulam Rivets 7 2.3 Determination of Plate Displacements u,v and Rotation 6 .. 10 2.3.1 Unconstrained Nonlinear Function Minimization 10 2.3.2 An Approximate Method to Estimate Load Carrying Capacity of the Connection Based on F i r s t Rivet F a i l u r e 12 2.4 Numerical Results 16 CHAPTER 3 - WOOD FAILURE MODE 27 3.1 S t r a i n Energy and Load P o t e n t i a l 27 3.2 Determination of Displacements i n the Wood Member 30 3.2.1 The Method 30 3.2.2 The Derivation of Element S t i f f n e s s Matrix and Consistent Load Vector 32 3.2.2.1 D e f i n i t i o n of Unknowns, Shape Functions and Their Derivatives 32 3.2.2.2 Construction of Element S t i f f n e s s Matrix 35 3.2.2.3 Construction of a Consistent Load Vector 37 3.2.2.4 The Global System of Equations and Solution 39 3.3 Determination of Stresses 41 3.4 Evaluation of Wood F a i l u r e Load Using Weibull's Weakest Link Theory 43 3.5 Numerical Results f o r D i f f e r e n t Connection Configurations 48 CHAPTER 4 - EXPERIMENT 51 4.1 Experimental Set-up and Preparation of Test Specimens .... 51 4.2 Test Results 55 4.2.1 Connection Type #1 55 4.2.2 Connection Type #2 55 4.2.3 Connection Type #3 61 4.2.4 Connection Type #4 61 4.3 Comparison 61 - i i i -TABLE OF CONTENTS (Continued) Page CHAPTER 5 - DESIGN CRITERION 68 5.1 General 68 5.2 Recommendations f o r Avoiding the Occurrence of Wood Fa i l u r e Mode 68 5.3 Steps of S i m p l i f i e d Design Procedure 68 CHAPTER 6 - CONCLUSIONS 72 6.1 Summary and Conclusions 72 6.2 Future Studies 73 REFERENCES . 75 APPENDIX I 76 APPENDIX II - CONVERGENCE OF THE FINITE ELEMENT PROGRAM 77 APPENDIX III - THE VALUES OF FACTOR 'C FOR 12 RANDOMLY SELECTED CONNECTIONS 80 - i v -LIST OF FIGURES Page Figure 1.1 Problem of Interest 3 2.1 Configuration of the Connection 5 2.2 Load-Slip Curve f o r i t h Rivet 7 2.3 The Displacements and S l i p of the i t h Rivet 9 2.4 Forces on i t n Rivet Due to Moment and Shear 12 2.5 Configuration of the i t h Rivet 14 2.6 The Results for the F i r s t Connection: NR=5; NC=10; y=0° 18 2.7 The Results f o r the Second Connection: NR=10; NC=5; y=0° .... 19 2.8 The Results for the F i r s t and Second Connections In Non-Dimensional Basis 20 2.9 The Results for the Third Connection: NR=5; NC=10; y=90° .... 21 2.10 The Results f o r the Fourth Connection: NR=10; NC=5; y=90° ... 22 2.11 The Results for the Third and Fourth Connections i n Non-Dimensional Basis 23 2.12 The Results for a l l Four Connections i n Non-Dimensional Basis 25 2.13 Representative 4th Degree Curve Versus Various Connection Results 26 3.1 An E c c e n t r i c a l l y Loaded Glulam Rivet Connection 28 3.2 The Load Configuration of the i t h Rivet 29 3.3 Quadratic Isoparametric Element (8 Nodes) 32 3.4 Element S t i f f n e s s Matrix 38 3.5 The Components of the Load p^(n) 39 3.6 Stress Concentrations and Observed Cracks 1" = 25.4 mm 47 3.7 The Dimensions and the Parameters of Connection (1 i n . = 25.4 mm) 50 4.1 Experimental Set-Up 52 4.2 Specimen Under E c c e n t r i c Loading 53 4.3 Specimen Under Tension Perpendicular-to-Grain Loading (Zero E c c e n t r i c i t y ) 54 4.4 Experimental and Theoretical Results for Connection Type #1 57 4.5 Wood F a i l u r e Under Ec c e n t r i c Loading (Type #1) 58 4.6 Experimental and Theoretical Results for Connection Type #2 59 4.7 Specimen F a i l e d i n Ri v e t - Y i e l d i n g F a i l u r e Mode (Type #2) .... 60 4.8 Experimental and Theoretical Results for Connection Type //3 62 4.9 Experimental and Th e o r e t i c a l Results for Connection Type #4 63 4.10 Wood F a i l u r e Mode Due to Pure Tension Perpendicular-to-Grain Loading (Type #4) 64 4.11 Comparison of Theoretical and Test Fa i l u r e Load (1 kip = 4,450 N) 65 4.12 Load D i s t r i b u t i o n Pattern for Connection Types #1 and #3 .... 66 4.13 Load D i s t r i b u t i o n Pattern for Connection Type #2 66 4.14 Load D i s t r i b u t i o n Pattern for Connection Type #4 67 - v -LIST OF FIGURES (Continued) Page Figure 5.1 Recommended Geometric Shape of the Rivet Clusters for Beam-Column Connection. Note: 1 inch = 25.4 mm 69 5.2 Recommended Geometric Shape of the Rivet Clusters f o r Beam Spl i c e s . Note: 1 inch = 25.4 mm 69 5.3 Load-Moment Interaction Graph for Design Purposes 71 I I . 1 Connection Subjected to Zero Eccentric Loading 79 - v i -LIST OF TABLES Page Table 3.1 The Results for Selected Two-Sided Connections 49 4.1 Experimental Results 56 4.2 Experimental and T h e o r e t i c a l Results 64 II.1 Stresses Around the Rivet Cluster f or Di f f e r e n t Pairs of N,M Values 78 - v i i -ACKNOWLEDGEMENTS The author takes t h i s opportunity to g r a t e f u l l y acknowledge the guidance of h i s advisor Dr. R.O. Foschi throughout the research and preparation of t h i s t h e s i s . His suggestions and advice during our discussions proved invaluable. The f i n a n c i a l support i n the form of a Graduate Research A s s i s t a n t s h i p from the National Science and Engineering Research Council of Canada, and from Forintek Canada Corp. f o r t e s t materials, i s g r a t e f u l l y acknowledged. F i n a l l y I wish to express my thanks to Professor B. Madsen, Mr. J.D. Dolan, the f a c u l t y and t e c h n i c a l s t a f f of the C i v i l Engineering Department at the University of B r i t i s h Columbia, fellow graduate students, my wife, and my parents f o r t h e i r encouragement, advice and assistance. - v i i i -1. CHAPTER 1  INTRODUCTION 1.1 General Glulam r i v e t connections under e c c e n t r i c loading are not covered e x p l i c i t l y i n the Code for Engineering Design i n Wood (CAN3-086-M84). Design procedures are given only f o r loadings with zero e c c e n t r i c i t y , and some assumptions would have to be made by the designer i f the loads are eccentric. Beam to column connections and »beam s p l i c e s are some types of e c c e n t r i c a l l y loaded connections f o r which a design procedure i s lackin g . Glulam r i v e t s have numerous advantages. They permit a greater load tran s f e r per unit contact area than other fasteners, r e s u l t i n g i n substan-t i a l savings i n s t e e l . Reductions i n glued laminated member s i z e s are possible i n some instances since member design can be based on the gross c r o s s - s e c t i o n a l area instead of the net area. F i e l d assembly i s simple and the chances of manufacturing error are reduced. We need to focus our att e n t i o n on the following aspects of the problem: • What are the f a i l u r e modes f o r an e c c e n t r i c a l l y loaded connection? • What i s the t h e o r e t i c a l f a i l u r e load f o r a connection where r i v e t y i e l d -ing i n bending i s the f a i l u r e mode? • What i s the t h e o r e t i c a l f a i l u r e load f o r a connection where wood f a i l u r e around the r i v e t c l u s t e r i s the f a i l u r e mode? • What i s the c o r r e l a t i o n between predicted f a i l u r e loads and experimental test r esults? • How do various connection parameters e f f e c t the ultimate load capacity? • How to e s t a b l i s h a design c r i t e r i o n ? 2. 1.2 Background and Previous Research Glulam r i v e t s are s p e c i a l l y designed f o r connections i n glued lamina-ted timber construction, McGowan and Madsen (1965). These r i v e t s are also known as Griplam n a i l s . They are made from a high-yield-point s t e e l and are driven into a wood member through a p r e - d r i l l e d s t e e l side plate. Load-slip c h a r a c t e r i s t i c s of Glulam r i v e t s have been studied by Foschi (1974) and ultimate loads have been derived f o r a f a i l u r e mode In which the r i v e t bends and y i e l d s while the wood under the shank f a i l s i n bearing. Glulam r i v e t connections can be designed under zero load e c c e n t r i c i t y using the e x i s t i n g code c r i t e r i o n , which i s based on work by Foschi and Longworth (197 5). This paper describes the analysis and design of Glulam r i v e t connections f o r loads p a r a l l e l and perpendicular to the grain with zero e c c e n t r i c i t y . Two modes of f a i l u r e were studied: 1) Rivet y i e l d i n g f a i l u r e mode ( y i e l d i n g of the r i v e t s i n bending with simultaneous bearing f a i l u r e of the wood under the r i v e t ' s shank); and 2) Wood f a i l u r e around the group of r i v e t s . To evaluate the three-dimensional state of stress around the r i v e t c l u s t e r , a f i n i t e element analysis was used, and Weibull's theory of b r i t t l e f racture was applied to study f a i l u r e s as c o n t r o l l e d by the l o n g i t u d i n a l shear strength of the wood. The l a t t e r was investigated by Foschi and Barrett (1976) for Douglas-fir. For loads p a r a l l e l to the grain, Foschi and Longworth (197 5) studied the influence of connection's geometrical parameters on the stress d i s t r i -bution around the r i v e t c l u s t e r : number of rows, number of r i v e t s per row, r i v e t penetration, r i v e t spacing, and distances between the r i v e t group and edges of the member. Ultimate loads were derived for both modes of f a i l u r e and allowable loads were thus determined. Very good agreement was shown between t h e o r e t i c a l f a i l u r e loads and f u l l - s i z e test r e s u l t s using 3. Douglas-fir, and design procedures were proposed and now form the basis of the current CAN3-086-M84 recommendations. Connections under loads perpendicular to the grain, were al s o studied by Foschi (1973) and Barrett, Foschi and Fox (1975). The f i r s t paper describes the s t r e s s analysis f o r a c l u s t e r of r i v e t s i n a connection loaded i n the d i r e c t i o n perpendicular to the grain. In the second paper, Barrett, Foschi and Fox (1975) used the r e s u l t s of the s t r e s s a n a l y s i s , by introducing Weibull's theory of b r i t t l e f r a c t u r e , determined the strength of Douglas-fir wood i n tension perpendicular to the grain, and studied the wood f a i l u r e mode. 1.3 Objective and Scope of the Thesis The aim of t h i s thesis i s to develop a method of analysis of Glulam r i v e t connections under eccentric loading. An attempt i s also made to e s t a b l i s h a corresponding design c r i t e r i o n . The s p e c i f i c problem on which we focus our attention i s shown i n F i g . (1.1), a Glulam r i v e t connection under load 'P' with an e c c e n t r i c i t y 'E'. glulam rivet F i g . 1.1 Problem of Interest 4. The i n v e s t i g a t i o n presented i n t h i s thesis i s divided i n t o three parts. Chapter 2 involves an a n a l y t i c a l study to determine the ultimate loads i n the r i v e t yielding/wood bearing f a i l u r e mode. Next, the three-dimensional state of stress around the r i v e t c l u s t e r i s i n v e s t i g a t e d i n order to determine the ultimate loads due to wood f a i l u r e around the group of r i v e t s . Wood f a i l u r e mode i s described i n Chapter 3 . The study then focusses on the experimental v e r i f i c a t i o n of the analysis and on the development of a design c r i t e r i o n . The comparison between t h e o r e t i c a l and experimental r e s u l t s i s discussed i n Chapter 4 and a design c r i t e r i o n i s presented i n Chapter 5. F i n a l l y Chapter 6 contains a summary and conclusions of the r e s u l t s . 5. CHAPTER 2  RIVET YIELDING FAILURE MODE 2.1 Kinematic Assumptions Consider the group of rivets i n the connection of Fig. (2.1). The distribution of the applied load, P, and moment P E among the rivets i s not uniform, as each rivet tends to displace with a different angle to the grain. As the load P is increased, each rivet eventually reaches i t s u l t i -mate capacity resulting i n the failure load P . The individual rivet load r capacity depends on the angle between the slip direction and the grain of the Glulam member. • Y Fig. 2.1 Configuration of the Connection The steel plate w i l l be assumed to be sufficiently s t i f f to displace as a rigid body with respect to Glulam member. Let u and v be, respect-ively, the plate displacements at point 0 i n the X and Y directions. Also, let 6 be the rotation of the rigid plate. 8 is assumed positive counter-clockwise, while u and v are positive i f in the positive direction of the coordinate axes. 6. P o i n t 0 i s t a k e n a t t h e g e o m e t r i c c e n t r e o f t h e G l u l a m r i v e t a r r a n g e -ment i n F i g . ( 2 . 1 ) . The d i s t a n c e r^ be tween t h e i r i v e t and t h e p o i n t 0 i s : r i = / X i + Y i ( 2 ' 1 ) where and a r e t h e c o o r d i n a t e s o f t he i*"^ r i v e t i n t he l o c a l c o -o r d i n a t e s y s t e m . When t h e s t e e l p l a t e r o t a t e s a n a n g l e 6 , and a s s u m i n g s m a l l d i s p l a c e m e n t s , p o i n t A i n F i g . ( 2 . 1 ) d i s p l a c e s an amount ( 6 r ) . The x and y d i s p l a c e m e n t s o f p o i n t A i n te rms of t he p l a t e d i s p l a c e m e n t s a r e t h u s : u A = u + 0 E ( 2 . 2 a ) A v A = v - 6 H ( 2 . 2 b ) where E i s t he a p p l i e d e c c e n t r i c i t y and H i s h a l f w i d t h o f t h e s t e e l p l a t e . F u r t h e r m o r e , t h e p o i n t a t t h e l o c a t i o n o f t h e i*"* 1 r i v e t w i l l d i s p l a c e ( u ^ ) and ( v ^ ) i n t h e X , Y d i r e c t i o n s r e s p e c t i v e l y : u = u + 6 Y . ( 2 . 3 a ) i l v± = v - 6 X ± ( 2 . 3 b ) and s l i p ( A ^ ) a t t h i s p o i n t i s : A i = / u i Z + V i Z ( 2 . 4 a ) o r A = / ( u + 9 Y ) * + (v - 0 X ± ) * ( 2 . 4 b ) Thus, I f the plate displacements u,v and r o t a t i o n 6 are known, then the displacements u^, and the s l i p at any r i v e t l o c a t i o n can be calc u l a t e d by using Eq. (2.3a,b) and Eq. (2.4a,b). 2.2 Nonlinear Load-Slip C h a r a c t e r i s t i c s of Glulam Rivets The nonlinear l o a d - s l i p c h a r a c t e r i s t i c s of Glulam r i v e t s have been studied by Foschi (1974), considering that the r i v e t y i e l d s i n bending an the wood under the r i v e t shank f a i l s i n bearing. For the i C ^ r i v e t , a t y p i c a l l o a d - s l i p curve i s shown i n F i g . (2.2). A mathematical expression for the l o a d - s l i p curve may be formulated follows: 0.25 in. (6mm) F i g . 2.2 Load-Slip Curve for i th Rivet P ± = (pj + ?\ A 1 ) ( l - e ) ( 2 . 5 ) 8. where p^ , i s the load at the i r i v e t A^ i s the s l i p of the i ^ r i v e t , and the other parameters represent the i n i t i a l s t i f f n e s s k^, the asymptotic s t i f f n e s s p* and the intercept of the asymptote pO. They depend on the angle 3^ between the grain and the d i r e c t i o n of the s l i p . Since tests have only been conducted to f i n d these parameters f o r 8 = 0° and 8 = 90°, intermediate values at 8^ may be found by i n t e r p o l a t i o n according to: P i = P / / P l 1 ( P / / S i n 2 f 3 i + P l c o s 2 V <2,6a> P I = p// p i 1 ( p / / s i n 2 e i + p i cos2si) (2-6b) k ± = k ^ k^ / ( k ^ sin2g i + 00828^ (2.6c) where P^y> a n ^ k ^ correspond to tests p a r a l l e l to grain ( 8 = n ° ) and p°, pj and k^ are the corresponding perpendicular to grain values (8=90°). In general, the i n i t i a l modulus (k) and the constants (p°, p 1 ) may be determined by, for example, nonlinear least-squares f i t t i n g of experimental data f o r any p a r t i c u l a r n a i l type and wood species, Foschi (1974). For loadings p a r a l l e l and perpendicular to the grain, these parameters are shown i n Appendix 1 f o r Douglas-fir specimens with a moisture content about 12%. Let 8^ be the angle between the s l i p (A^) of the i r i v e t and the grain of the wood member as shown i n F i g . (2.3). F i g . 2.3 The Displacements and S l i p of the i Rivet (3^  can be found i n terms of a^, the angle between the s l i p (A^ ) and x axis and y, the angle between grain of the wood member and x a x i s . Thus, J ± = a. - y (2.7) where the angle (ct^) i s : a i * S i n _ 1 (2.8a) or using Eq. (2.3b) and Eq. (2.4b) v - 6 X a - sin-1 r J j (2.8b) 1 •(u + 6 Y ) i + (v - 6 X )^ F i n a l l y , test r e s u l t s have shown that the expression of Eq. (2.5) i s only v a l i d up to a maximum value of the s l i p . Accordingly, i n the subsequent development, the l o a d - s l i p curve of F i g . (2.2) was modified to l i m i t the load p to that corresponding to a maximum s l i p of 0.25 i n (6 mm). 1 0 . 2 . 3 D e t e r m i n a t i o n o f P l a t e D i s p l a c e m e n t s u , v and R o t a t i o n 6 2 . 3 . 1 U n c o n s t r a i n e d N o n l i n e a r F u n c t i o n M i n i m i z a t i o n A t any g i v e n l o a d P , t h e d i s p l a c e m e n t s u , v and r o t a t i o n 6 w h i c h d e s c r i b e t he r i g i d body m o t i o n o f t h e s t e e l p l a t e were d e t e r m i n e d by u n c o n s t r a i n e d m i n i m i z a t i o n o f t h e f u n c t i o n Y = U - W ( 2 . 9 ) where 4" i s t h e t o t a l p o t e n t i a l e n e r g y U i s the e n e r g y s t o r e d i n g l u l a m r i v e t s , a n d , W i s t h e l o a d p o t e n t i a l . The e n e r g y s t o r e d i n g l u l a m r i v e t s i s : N A i U = Z J p . dA ( 2 . 1 0 ) 1=1 o 1 where p^ i s g i v e n i n E q . ( 2 . 5 ) , and N i s t h e number o f r i v e t s i n t h e c o n n e c t i o n . P e r f o r m i n g t h e i n t e g r a l i n E q . ( 2 . 1 0 ) l e a d s t o : k i A i U = N A.2 ( P 0 ) 2 I { p ° A + p l 2 ^ + ^ — (« i - 1 1 Z i k i A i i i P l P 0 A i 6 ( P ° ) 2 ?l K i k i A i (p0)2 p l -} ( 2 . 1 1 ) 1 1 . The energy U i n Eq. (2.11) i s a function of p l a t e displacements u,v and r o t a t i o n 6 since, according to Eqs. (2.4b), (2.6a,b,c), (2.7) and (2.8b) A^, k^, pO and p| are dependent on u,v and 9. The load p o t e n t i a l i s W = P(u + E 6) (2.12) i n which P and E are the applied load and the e c c e n t r i c i t y on the connection. By a p p l i c a t i o n of p r i n c i p l e of v i r t u a l work, the problem i s reduced to the minimization of the t o t a l energy function ¥ with respect to the plate displacements u,v and the r o t a t i o n 0. That i s , one has to solve the following system of equations: ¥ = 0 (2.13a) 9u v ' | ^ - 0 (2.13b) 9v v ' f - 0 (2.13c) Eqs. (2.13a,b,c) are highly nonlinear, and a technique of d i r e c t , unconstrained nonlinear function minimization (Fletcher, Powell, 1963) was used to determine the plate displacements u,v and the r o t a t i o n 9. To determine the f a i l u r e load P^_ f o r the connection, the load P was increased from about 80% of an estimated f a i l u r e load P* determined by the approximate a n a l y t i c a l method explained i n Section (2.3). The load increment was taken as 10% of P*. r A r i v e t was assumed to reach i t s ultimate load c a r r y i n g capacity ( p ^ ^ when i t s s l i p reached 0.25 i n . (6 mm), as shown i n F i g . (2.2). 12. For each load step, the norm of the vector of plate displacements u,v and r o t a t i o n 6 was computed and compared to the norm for the previous load step, and I f (/(u + E e>* + (v - e H ) * ) - > 4 (2.14) /Iu + E 6 ) z + (v - 6 H)^] L ' V ' JPREVIOUS the connection was assumed to be reaching i t s load carrying capacity P . The maximum load carrying capacity of the connection i n t h i s f a i l u r e mode was f i n a l l y estimated by: P r • + 2 ? r p R E V I 0 U S (2.15) 2.3.2 An Approximate Method to Estimate Load Carrying Capacity of the Connection Based on F i r s t Rivet F a i l u r e F i g . 2.4 Forces on i Rivet Due to Moment and Shear 13 . C o n s i d e r t h e i t h r i v e t i n F i g . ( 2 . 4 ) . A s s u m i n g t h a t , due t o t h e moment M, t he r i v e t f o r c e s a r e p r o p o r t i o n a l t o t h e d i s t a n c e r^ f r om t h e M g e o m e t r i c c e n t e r o f t he c o n n e c t i o n , t h e f o r c e F ^ due t o moment i s : F*J = X r . ( 2 . 1 6 ) M I t i s a l s o assumed t h a t the d i r e c t i o n o f F ^ i s p e r p e n d i c u l a r t o t h e r a d i u s r^ . The t o t a l moment on t he c o n n e c t i o n i s : N M M = Z F r ( 2 . 1 7 a ) i = l and a l s o , M = P E ( 2 . 1 7 b ) where N i s t he number of r i v e t s . S u b s t i t u t i n g E q . ( 2 . 1 6 ) i n t o E q . ( 2 . 1 7 ) l e a d s t o : N Z i = l M = X  r 2 ( 2 . 1 8 ) and I = -N M \ ' S ( 2 - 1 9 ) E r 2 i - 1 1 E q . ( 2 . 1 6 ) , now c a n be w r i t t e n a s : M r i M " i P E F i = ^  =-15 <2'20> Z r 2 I r 2 i= 1 1 i= 1 i The r i v e t f o r c e s due t o -shear may be assumed t o be u n i f o r m l y d i s t r i b u t e d . T h u s , 14. F ^ = | (2.21) The resultant force on the i * " * 1 r i v e t i s , therefore, F i - p / £ + -Vi> 2+ (-ry ±) 2 ( 2' 2 2 ) T The r e s u l t ant force (F^) cannot exceed a maximum which can be expressed as follows: u l t u l t T p// p i -"<Fi> = u l t , " u l t — < 2' 2 3> p ^ s i n ^ + p^ c o s 2 $ ± u l t u l t where pjj and p^ are, r e s p e c t i v e l y , the ultimate load c a r r y i n g capacity f o r one Glulam r i v e t f or loadings p a r a l l e l and perpendicular to the grain. These values have been reported by Foschi (1974) and are presented i n Appendix 1. i s the angle between s l i p and the grain as shown i n F i g . (2.5). F i g . 2.5 Configuration of the i t n Rivet 15. The angle (B^) can be calculated by: 6 ± - o ± - y (2.24) where y i s the angle between x axis and the gr a i n of the wood member i s the angle between s l i p d i r e c t i o n and x axis which can be determined by: 1 E N + — x i i r ? a± = sin-1 ( ) (2.25) . /(£ + -£-, )* + ( J L y ) 2 Zr? £r? x x The value of P* which w i l l cause the f i r s t r i v e t to reach i t s ultimate r load can now be computed. For the i t h r i v e t , equating Eq. (2.22) to Eq. (2.23) leads to a load value ( p^) which causes the y i e l d i n g of th i s r i v e t . Thus, a f t e r determining the values of P for each r i v e t by using Eq. (2.26), the minimum of these P values can be taken as the load P*. i r u l t u l t P / / P l p . " x ; (2.26) j, 1 , E ~Z ' , E 77 , u l t , ~ . u l t . / ( - + x )2 + ( y )2 ( p sin2g + p c o s 2 6 ) *2 T*r2 '' i r | z r^ P* = min ( P J i = 1, .. .N (2.27) r i where, the angles 8^ and the corresponding are obtained from Eqs. (2.24 and 2.25). 80% of P* was taken for the i n i t i a l load as a guess f or the nonlinear function minimization, and 10% of P* was taken as the r corresponding load increment. As w i l l be discussed i n Chapter 5, the connection f a i l u r e load P* r 16. b a s e d on t h i s a p p r o x i m a t e a n a l y s i s c a n a l s o be used f o r t h e deve lopmen t of a d e s i g n c r i t e r i o n f o r e c c e n t r i c a l l y l o a d e d G l u l a m r i v e t c o n n e c t i o n s . 2 . 4 N u m e r i c a l R e s u l t s The r i v e t y i e l d i n g f a i l u r e mode was exam ined f o r f o u r G l u l a m r i v e t c o n n e c t i o n s unde r d i f f e r e n t e c c e n t r i c l o a d i n g s : 1. NR = 5 ; NC = 1 0 ; Y - 0 ° ; h = 2 " ( 5 0 . 8 mm) 2 . NR = 1 0 ; NC = 5 ; Y = 0 ° ; h = 2 " ( 5 0 . 8 mm) 3 . NR = 5 ; NC = 1 0 ; Y = 9 0 ° ; h = 2 " ( 5 0 . 8 mm) 4 . NR = 1 0 ; NC = 5 ; Y = 9 0 ° ; h = 2 " ( 5 0 . 8 mm) where NR = number o f r i v e t rows i n t he d i r e c t i o n p a r a l l e l t o t he x - a x i s NC = number o f r i v e t s p e r row y = t h e a n g l e be tween t h e g r a i n and t h e x - a x i s h = t h e p e n e t r a t i o n l e n g t h o f t h e g l u l a m r i v e t s . The t h e o r e t i c a l load-moment (P v s . M) i n t e r a c t i o n a t f a i l u r e f o r t h e s e f o u r c o n n e c t i o n s a r e shown i n F i g s . ( 2 . 6 - 2 . 1 2 ) i n d i m e n s i o n a l and n o n -d i m e n s i o n a l b a s i s f o r d i f f e r e n t e c c e n t r i c i t i e s . P i s t h e t h e o r e t i c a l o f a i l u r e l o a d when e c c e n t r i c i t y i s z e r o i n r i v e t - y i e l d i n g f a i l u r e mode, and i s t h e t h e o r e t i c a l moment c a r r y i n g c a p a c i t y under a b e n d i n g moment a c t i n g a l o n e . I n F i g s . ( 2 . 6 - 2 . 7 ) , r e s p e c t i v e l y , t he r e s u l t s o f t h e f i r s t (NR=5; NC=10; Y =0° ) and t h e s e c o n d (NR=10; NC=5; 7=0° ) c o n n e c t i o n s a r e shown. N o t e t h a t f o r z e r o e c c e n t r i c i t y , t h e t h e o r e t i c a l f a i l u r e l o a d s f o r t h e s e c o n n e c t i o n s a r e t h e same 69750 l b . and t h e y can a l s o be computed by u s i n g E q . ( 2 . 2 8 ) , ( F o s c h i , L o n g w o r t h , 1 9 7 5 ) : 17. P q = NR NC p U X L (2.28) where p U ^ t i s the maximum load-carrying capacity f o r a s i n g l e r i v e t and given i n Appendix 1 for loads p a r a l l e l and perpendicular to the grain. M o was extrapolated to P=0 from the computer runs with the largest e c c e n t r i c i t i e s . In F i g . (2.8), the t h e o r e t i c a l f a i l u r e load-moment i n t e r a c t i o n i s shown for the f i r s t (NR=5; NC=10; y=0°) and the second (NR=10; NC=5; Y=0°) connections together i n a non-dimensional basis ( P / P vs. M/M ), o o A fourth order polynomial was f i t t e d to these data. The r e s u l t i s shown i n Eq. (2.29) and al s o i n F i g . (2.8). | - - 1 - 1-087 (|-)2 - 1.170 (|-) 3 + 1.257 (|-)- (2.29) o o o o S i m i l a r l y , F i g s . (2.9-2.10) show, res p e c t i v e l y , the re s u l t s for the t h i r d (NR=5; NO10; Y=90°) and the fourth (NR=10; NC=5; T=90°) connections. Note that for zero e c c e n t r i c i t y , t h e o r e t i c a l f a i l u r e loads for these connections are same (44000 l b . ) and they can al s o be computed by using Eq. (2.28), Foschi, Longworth (1975). In F i g . (2.11), the t h e o r e t i c a l f a i l u r e load-moment i n t e r a c t i o n i s shown f o r these connections together i n non-dimensional basis ( P / P vs. M/M ). A fourth order polynomial was also o o f i t t e d to these data. The res u l t i s shown i n Eq. (2.30) and a l s o i n F i g . (2.30). | - = 1 - 2.424 (f-)2 + 0.849 ( f - ) 3 + 0.576 ( f - ) -o o o o (2.30) 18. F i g . 2.6 The Results f o r the F i r s t Connection: NR=5; NC=10; Y=0°. Note: 1 l b = 4.45 Newton, 1 inch = 25.4 mm. 19. J + + Computer Data ^ 168000. g r a i n + + g = 69750. T ' I I I 1 1 1 1 1 1 1 1 1 1 1 1—•—| j — 80.0 160.0 240.0 320.0 400.0 480.0 560.0 640.0 720.0 800.0 OX) P = Load (Lb.) (X102 ) F i g . 2.7 The Results f o r the Second Connection: NR=10; NC=5; Y=0°. Note: 1 l b = 4.45 Newton, 1 inch = 25.4 mm. F i g . 2.8 The Results for the F i r s t and Second Connections i n Non-Dimensional Basis. o o Computer Data ^ M = 168000. o 0 o 0 o o groin F£= 44 000. _ ~ i 1 1 1 1 — 1 — 1 — 1 — 1 — 1 — ? — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — . OX) 80.0 160X) 240.0 S20X) 400.0 480X) 560X> 640X) 720X) 800 P = Load (Lb.) ( X 1 0 2 ) F i g . 2.9 The Results f o r the Third Connection: NR=5; NC=10; ^=90°. Note: 1 l b = 4.45 Newton, 1 inch = 25.4 mm. 22. O q X 8 c c i f O 8. II - J © © Computer Data MQ= 108 000. o o o o o o groin £ = A A 000. °-° •O- 0 «<«> ' 240.0 ' 320JO ' 400.0 ' 4M.0' 560.0 ' 640.0 ' 720.0 ' 800.0 P = Lood (Lb.) (X10 2 ) F i g . 2 . 1 0 The Results f o r the Fourth Connection: NR=10; NC=5; Y = 9 0 ° . Note: 1 l b = 4 . 4 5 Newton, 1 inch = 2 5 . 4 mm. 23. F i g . 2 .11 The R e s u l t s f o r t h e T h i r d and F o u r t h C o n n e c t i o n s i n N o n - D i m e n s i o n a l B a s i s . 24. A l l the data f o r these four connections may be combined to f i t a s i n g l e fourth-order polynomial. The r e s u l t i s shown i n Eq. (2.31) and also i n F i g . (2.12) . | - = 1 - 1.593 ( f - ) 2 - 0.576 (|-)3 + 1.169 ( f - ) 1 * (2.31) o o o o Also F i g . 2.12 shows a simpler st r a i g h t l i n e r e l a t i o n s h i p M P M" = 1 ~ P" ( 2 - 3 2 ) o o which may be used conservatively i n most cases, except at low e c c e n t r i c i t i e s . Since Eq. (2.31) i s able to represent a l l data f o r these four connections reasonably well, i t was decided to study i t s performance f o r other d i f f e r e n t connections with d i f f e r e n t e c c e n t r i c i t i e s . As shown i n F i g . (2.13), Eq. (2.31) may be used to represent the t h e o r e t i c a l f a i l u r e load-moment i n t e r a c t i o n f o r r i v e t - y i e l d i n g f a i l u r e mode with reasonable accuracy. As w i l l be discussed i n Chapter 5, a design c r i t e r i o n w i l l be proposed based on Eq. (2.31). 2 5 . Fig. 2.12 The Results for a l l Four Connections in Non-Dimensional Basis. 26. F i g . 2.13 Representative 4th Degree Curve Versus Various Connection Results. 27, CHAPTER 3 WOOD FAILURE MODE 3.1 St r a i n Energy and Load P o t e n t i a l Consider F i g . (3.1), where the d i s t r i b u t e d load, p^(z) represents the t tl load transferred to the wood member through the shank of i r i v e t . Let u,v and w, be, res p e c t i v e l y , the displacements i n the wood member i n x,y and z d i r e c t i o n s . The three dimensional state of stress around the r i v e r c l u s t e r w i l l be studied by assuming that w=0 while u=u(x,y,z) and v=v(x,y,z). Assuming that the wood member behaves as a l i n e a r orthotropic s o l i d , the s t r a i n energy U i n the member i s given by: + ! i £ + % + ^ KX (3.1, i n which E^ = the modulus of e l a s t i c i t y of the wood along the x ax i s , E^ = the modulus of e l a s t i c i t y of the wood along the y axis, G ,G and G = shear modulus f o r xy, xz and yz planes, xy xz yz u = Poisson's r a t i o f o r xy plane. Hxy The external load p o t e n t i a l , W can be calculated as follows. The s t e e l plate applies the resultant force F^ to the i ^ r i v e t , which trans-fers i t to the wood member as a dis t r i b u t e d load p^(z) over half of the r i v e t penetration h, as shown i n F i g . (3.2): 28. ( 3 . 2 ) where i t has been assumed that p^(z) i s l i n e a r l y d i s t r i b u t e d with z and that a l l the force i s transferred over the half of shank length c l o s e r to the s t e e l plate. This l a t t e r assumption cl o s e l y r e f l e c t s test r e s u l t s (Foschi, 1974). F. i s taken from the l o a d - d i s t r i b u t i o n (0 < z < Tr) obtained i n the r i v e t - y i e l d i n g f a i l u r e mode an a l y s i s . It must be noted that the analysis w i l l be approximate i n the sense that v e r t i c a l forces tending to withdraw the r i v e t w i l l not be taken into account. However, for the standard length 29. F i g . 3.2. The load configuration of the i * " * 1 r i v e t . of r i v e t s , the e f f e c t of withdrawal i s secondary. Since the d i r e c t i o n of the load p.j.(z) makes an angle (a^) with the x axis, the load p o t e n t i a l W can be computed as follows: NP h/2 W = I {/ p.(z) u (x ,y ,z) cos a dzl . , l J 1 i i i i ' 1=1 o NP h/2 + I {/ P i(z) v i ( x i , y i , z ) s i n a± dz} (3.3) i = l o The angle (the d i r e c t i o n of the s l i p ) i s also taken from r i v e t - y i e l d i n g f a i l u r e mode ana l y s i s , and NP i s the number of the r i v e t s the r i v e t c l u s t e r . 30. The displacement functions u(x,y,z) and v(x,y,z) w i l l be determined by minimizing the t o t a l p o t e n t i a l energy = U-W). 3.2 Determination of Displacements i n the Wood Member 3.2.1 The Method Although the displacement functions, u(x,y,z) and v(x,y,z), could be obtained by a standard three-dimensional f i n i t e element a n a l y s i s , the problem may be s i m p l i f i e d by adopting a semi-analytical f i n i t e element approach, Zienkiewicz (1971). Considering that at the boundaries x=0, x=a, the displacements u(x,y,z) and v(x,y,z) are zero, the functions u(x,y,z) and v(x,y,z) can be taken i n the following se r i e s form: N u(x,y,z) = Z F (y,z) s i n 21* ( 3.4a) n=l M v(x,y,z) = z G m(y,z) s i n ^  (3.4b) m=l i n which the functions F (y,z) and G (y,z) are unknown, but may be n m determined by a two-dimensional f i n i t e element a n a l y s i s . Introducting Eqs. (3.4a,b) i n t o Eq. (3.1), and performing the s t r a i n energy i n t e g r a l i n the x d i r e c t i o n , U becomes, 31. b c E„ N o 2 E„ M 3G m(y,z) y=0 z=0 n=l m=l G N 8F (y,z) G M , , n=l m=l N M 3F (y,z) + G Z Z — 2 G (y,z) 2 n m n=l m=l y n^ -m^  G N 3F (y,z) G M 3G (y,z) 2 , 1 3z J 2 2 \ >• 3z > 2 n=l m=l N M 3G (y,z) -+ ^xv E x 1 1 F n ( y ' z ) — | ; — ] d y d z (3.5) X n - l m-1 n 9 7 m2-n2 Introducing Eqs. (3.4a,b) into Eq. (3.3), the external load p o t e n t i a l i s : N NP h/2 mrx W - Z Z { / p ± ( z ) F n ( y i , z ) s i n — — cos o^dz} n=l 1=1 o M NP h/2 mux + Z Z { / p ± ( z ) G m( y ; L,z) s i n - - — s i n a ± dz} (3.6) m=l i = l o The three-dimensional v a r i a t i o n a l problem: 6(U - W) = 0 (3.7) i s thus reduced to a set of two-dimensional v a r i a t i o n a l problems i n y-z plane, one f o r each combination of n and m. For each p a i r n and m, the problem of Eq. (3.7) i s solved by f i n i t e element a n a l y s i s . The y-z plane of the wood member i s divided i n t o f i n i t e elements. A quadratic isopara-metric element (8 nodes) was chosen. Within each element, the functions F (y,z) and G (y,z) were approximated by an incomplete cubic polynomial n m uniquely determined by the values of functions at the nodes. 32. Since the energy i n t e g r a l has de r i v a t i v e s of the f i r s t order, continu-i t y c r i t e r i o n requires the functions F (y,z) and G (y,z) themselves should n m be continuous. Therefore, quadratic isoparametric element (8 nodes) shown i n F i g . ( 3 . 3 ) may be used f o r s o l v i n g the problem. F i g . 3 . 3 . Quadratic isoparametric element (8 nodes) 3 . 2 . 2 The Derivation of Element S t i f f n e s s Matrix and Consistent Load Vector 3 . 2 . 2 . 1 D e f i n i t i o n of Unknowns, Shape Functions and Their Derivatives Since the element has 8 nodes, for each n (n=l,...N) and m (m=l,...M), the eight node values of F (y,z) and the eight G (y,z) are taken as the n m unknowns or degrees of freedom. These are arranged i n two vectors, {f } T = {Fl F2 F3 F4* F5 F6 F? F8} ( 3 . 8 a ) 1 n l n n n n n n n n J {g } T = {G1 G2 G3 G1* G5 G6 G? G8} ( 3 . 8 b ) lomJ l m m m m m m m m J v ' 33. and thus, the vector of unknowns f o r one element can be written as: { 6} T = {f*[ . . . f N g* gT2 ...gM> (3.9) Introducing the natural coordinates £ and n as l o c a l coordinates, the gl o b a l coordinates y,z and the functions F (y,z), G (y,z) can be expressed n m as: 8 y = Z N U,n) y (3.10a) i = l 1 1 8 z = Z N (£,n) z (3.10b) i = l 8 F (y,z) = Z N ( C n ) F (3.11a) n . , i n. 1=1 i 8 G (y,z) = z N (£,n) 6 (3.11b) m i = 1 i m± where N^(£,n) are the shape functions of the element: N ^ . n ) - -(l-£)(l-Ti)(l+5+n)/* N 2(c,r,) = ( l - ? 2 ) ( l - n ) / 2 N3(5,n) = -(l+5)d+n)(l-5-n)/4 N^.n) = (l-n 2)d+5)/2 (3.12) N5(5,n) = -(l+S)(l + n)(l-5 - n ) M N 6 ( C,n) = (l-5 2)d + n)/2 N7(?,n) = -(l - 0(l + n)(l + 5 - n)/4 N8(5,n) = ( l - n 2 ) d - 0 / 2 Shape functions and t h e i r d e r i v a t i v e s are given i n Eqs. (3.13a,b,c) i n vector form: /N\"^=/N N N N N N N N T { Hi> l w i "2 3 *• 5 6 7 8t A 1 l3y ' - { 3 N X 3N 2 3N 3 3N1, 3N 5 3N 6 3N 7 3N8 3y 3y 3y 3y 3y 3y 3y 3y 3 ^ 3N 2 3N 3 SN^ 3N 5 3N6 3N 7 3N8 3z 3z dz dz 3z 3z dz dz } 34. (3.13a) (3.13b) (3.13c) The strain-displacement r e l a t i o n s h i p s are written i n terms of the l o c a l coordinates £ and n. This requires that the following coordinate transformation of derivatives be invoked, Cook (1981). Let <j> be some function of y and z, then the chain r u l e y i e l d s : or 94> d(j> dy t 3j> dz d^ 3y 35 3z 35 3$ 3<t> 3y | 3<)> 3z 3n ~ 3y 3n 3z 3n f 9 ^ " i i i • = [J] - > i i i i L 3zJ (3.14a) (3.14b) (3.14c) where [J] i s the Jacobian matrix: [J] = 3y i £ 35 35 J £ _3n 3n_ J J 11 12 J J 21 22 (3.15) From Eq. (3.14c) the inverse r e l a t i o n s are: 37 3z "i£1 35 > J L 3nJ (3.16) Eqs. (3.14a,b,c), (3.15) and (3.16) are general. For our s p e c i f i c problem 35. f u n c t i o n <(> becomes e i t h e r F n ( y , z ) o r G m ( y , z ) antf t h e n u m e r i c a l v a l u e s o f c o n s t a n t s i n [J ] c a n be d e t e r m i n e d by u s i n g E q s . ( 3 . 1 0 a , b ) : J l l ! _ 3y _ " 35 8 E i = l 9 N i 35 J 1 2 = 3z 35 8 E i = l z i 3 N ± 35 J2 1 = 3y _ 3n 8 E i = l y i 3 N ± 3n J2 1 = 3z an 8 E i = l z i 3 N ± 3n ( 3 . 1 7 ) 3 . 2 . 2 . 2 C o n s t r u c t i o n o f E l e m e n t S t i f f n e s s M a t r i x F o r one e l e m e n t , t h e s t r a i n ene rgy i n t e g r a l f o r t h e wood member may be w r i t t e n i n m a t r i x f o r m a s : x „ , ,T n z i r2 U = T \ { f n > t A l H f n > -TT n=l E M + / S {Sm> [ A 2 ] { g m } f m=l + % ^ { f n } T t A 2 H V f n= l G M 9 9 m=l 2a + G x y * " { f n } T t A 3 H g m } ( 3 . 1 8 ) n= l m=l n z - m z + % * t fn) TtM{f n} J n= l G M j _ T r A 1 r i 3 + — Z {Sm> I \ H g m } 2 m=l N M „ 2nm Pxy E x £, I { gm} l A 5 H f n ) n= l m=l m 2 - n 2 36. where { f n ) and {g^} are the unknown vectors defined i n Eqs. (3.8a,b), and [AjJ through [A,.] are 8x8 matrices obtained as follows: 1 ' / 1 / -1 -1 1 1 / / -1 -1 1 1 / / -1 -1 1 1 : J / -1 -1 1 1 ' / -1 / -1 •ay (3.19) ' 3z 3z 3N i,T 3y These matrices do not depend on n or m, and are thus computed only once during the s o l u t i o n process. The integrations i n Eq. (3.19) were numeric-a l l y performed using a 3x3 Gaussian procedure, which gives an exact answer t t i f o r polynomials of up to 5 degree. The energy i n t e g r a l requires the th numerical i n t e g r a t i o n of polynomials of up to the A degree. Using s t r a i n energy expression i n Eq. (3.18), the element s t i f f n e s s matrix can now be constructed. Let: n 2 i r 2 f k l J n = E x t A l l -27" IVm = E y ^ f - Gxy l A 2 > f m 2ir 2 [Mm = Gxy t A l l ~2a~ (3.20) 5Jnm G xy ^ 2 n^ 1 6 Jn G xz [A,] f ^Vm = G yz [A,] f ^Vnm = E x 2nm m 2-n 2 37 . Using Eq. (3.20) the form of the element s t i f f n e s s matrix [K] f o r values of n=l,... ,N and m=l,...,M i s shown i n F i g . (3.4). 3.2.2.3 Construction of a Consistent Load Vector The load p o t e n t i a l expression may be written i n matrix form as: W = Z { Z {f } T[ / P,(n) {N.(£.,n)} J dn cosa, s i n i = l n=l -1 + I {gm} T[ / P^n) { N i(e i i n ) } J dn sino^ s i n - ^ j ) (3.21) -1 th i n which, p^(n) i s the tr i a n g u l a r d i s t r i b u t e d load assuming that the i r i v e t t r a n s f e r s the t o t a l load F^ to the wood member over h a l f i t s penetration h: J i s Jacobian f o r the coordinate transformation from z to n: (3.23) The f i n i t e element mesh was constructed with rectangular elements, and the height of the loaded element was chosen equal to hal f the penetration length of the r i v e t s , as shown i n F i g . (3.5). th The shape functions are evaluated at the location of i r i v e t (£ i n F i g . (3.5)) and the in t e g r a t i o n i n Eq. (3.21) was numerically performed by using a 3-point Gauss' procedure i n the z d i r e c t i o n . Since the external load p o t e n t i a l i s : P ±(n) = (3.22) [kj! + [k3ll + [Ml [k5]M + |k8]M [k5]12+ [ke]I2 t k 5 l l M + l k 8 l l M + [k6]2 |k5]21 + [k,l21 |k 5l 2 2 + [k,l2 [k5]2M + [k8l2M ===============: ============rr= • • • * • • ==r=r========r==== • • • • • • • • • • • • • • • Iklln + l k 3 l n + lk6ln Ik5]M1 + |k.lH1 I ksJH2 + t k8 JN2 l k s ' N M + l k s ' N M [k 5] u + [k e] n [k5l2l + [k,l21 t k 5 l N l + [k8lHl Ik2ll + Ik,,], * [k 7l, |k5lia+ [k al 1 2 [k5l2 2 + tk,]2 [k5]N2 + [ k a l N 2 [k2]2 + lk„J2 + Ik7l2 • • • • • • • • • • • • • • • • • • • • • • • * [k5l2M + l k e J 2 M l k5^NM + t ke'NM [k2]„ + [\}H + [k7l„ Figure 3 . 4 . Element s t i f f n e s s matrix. 39. stee plate P.(T?) Sina. p.iv) Cosa; F i g . 3.5. The components of the load P . ( T I ) . W = {6} {C} (3.24) the c o n t r i b u t i o n of the r i v e t s at r i v e t row ( p a r a l l e l to x axis) to the consistent load vector may be computed using Eq. (3.25): N L 1 {C} = I ( I [ / p (n) {N (f n)} £ d n cosa n - l i = l -1 J J J 3 n-r ix . — j M L 1 mirx. + E ( l [ | p . . W {N..(L . , l ) ) N n sin a s i n - i i ] ) (3.25) m=l i = l -1 J J 3 J where L i s the number of r i v e t s on j ' ^ row 3.2.2.4 The Global System of Equations and Solution The s o l u t i o n to the v a r i a t i o n a l problem 6(U-W) = 0 r e s u l t s f o r each element i n a system of l i n e a r equations [K]{6} = {C} 40. (3.26) i n which [K] i s the s t i f f n e s s matrix previously discussed, a 8(N+M)x8(N+M) symmetric matrix, and {6} and {C} are 8(N+M) vectors corresponding, r e s p e c t i v e l y , to the unknowns and to the set of consistent loads. These element matrices and vectors were used to construct the g l o b a l system of equations following standard procedures. However, to obtain a global matrix with a band as narrow as possible, the element s t i f f n e s s matrices and consistent load vectors were re-organized to correspond to a rearrange-ment of the unknown vector as shown i n Eq. (3.27). { 6new } T = { F1 F2'-- FN G l • • - ] ? f Fg...FJ} Gf G8...G8} (3.27) In t h i s new form for the vector, the superscripts designate l o c a l node numbers and subscripts r e f e r to the order i n s e r i e s . There are (N+M) degrees of freedom at each node, and, therefore, the number of equations i n the system i s : (N+M) x (Number of Nodes). The h a l f band width of the g l o b a l s t i f f n e s s matrix i s : HALF BAND WIDTH = (N+M) MPD (3.28) where MPD i s the maximum difference between the node numbers for an element. The values of the functions F (y,z) and G (y,z) are determined by n m solving the global system of equations and using the shape functions for the element. The displacements u(x,y,z) and v(x,y,z) are then c a l c u l a t e d by using Eqs. (3.4a,b). 3.3 Determination of Stresses The displacement functions u(x,y,z) and v(x,y,z) are used to compute the stresses i n the wood member. a) Normal stresses i n x d i r e c t i o n : a = E + ]i E — - (3.29) x x 3x Mxy x 3y v ' b) Normal stresses i n y d i r e c t i o n : o = E - ^ + y E - ^ - (3.30) y y 3y yx y 3x where the product u u was neglected i n comparison to unity. xy yx c) L a t e r a l shear stresses: T = (-|^- + —-) G (3.31) xy v3y 3x' xy d) Bottom shear stresses: T =|^G (3.32) xz 3z xz e) Transfer shear stresses: T - p- G (3.33) yz 3z yz In matrix form, l e t us define the stress vector, strain vector T {a} = {a a T x T } x y xy xz yz {e} = {e e Y Y Y } x y xy xz 1 yz 42. (3.34) (3.35) and e l a s t i c i t y matrix [D] * Ji E yx y o 0 o U E xy x 0 0 0 0 0 'xy 0 0 0 0 0 'xz 0 0 0 0 0 Jyz (3.36) Since numerical Gauss integration was used for the formulation of the element stiffness matrix and, the displacements u(x,y,z) and v(x,y,z) were obtained at the 3x3 sampling points, the stress values may be computed at these integration points by using Eq. (3.37): {a} = [D] {e} (3.37) where the components of strain vector are: 'xy •xz 'yz 3u 9x 3v 3y 3u [ av 3y 3x 3u 3z 3y_ 3z (3.38) where u(x,y,z) and v(x,y,z) are given i n Eqs. (3.4a,b) in terms of F (y,z) n and G (y,z). To be able to obtain reasonable stress values, at least five m 43. terms have to be taken i n each of the series i n Eqs. (3.4.a,b). This was concluded by determining the equilibrium between the external forces and the stresses around the r i v e t c l u s t e r , as shown i n Appendix I I . It i s also important to point out that N and M may be odd or even numbers depending upon loading conditions. For example for an e c c e n t r i c load p a r a l l e l to x-axis, N should be odd and M should be even numbers. 3.4 Evaluation of Wood F a i l u r e Load Using Weibuill's Weakest Link Theory Weibull showed that the strength of a b r i t t l e m a t erial decreases with increasing specimen volume. Barrett, Foschi and Fox (197 5) and Foschi, Barrett (1976) applied, r e s p e c t i v e l y , Weibull's theory of b r i t t l e f r a c t u r e to the determination of strength of Douglas-fir wood i n tension perpendicu-l a r to the g r a i n and i n l o n g i t u d i n a l shear. B r i e f l y , i f V i s a volume with a d i s t r i b u t i o n of stresses, a(x,y,z), the p r o b a b i l i t y of f a i l u r e F^ of V i s given, according to Weibull's b r i t t l e f r acture theory, F y = 1.0 - exp[- ^ / (^V dV] (3.39) v where m>k and CTq are material constants, V* i s a reference volume, and 0 q corresponds to the minimum strength of the material. Since three material constants are involved, Eq. (3.39) i s referred to as a 'three parameter 1 Weibull model. A simpler, 'two parameter' model i s used here by assuming a Q = 0. The assumption of zero minimum strength may appear to be u n r e a l i s t i c , but Foschi and Barrett (1976) have shown that, f o r p r a c t i c a l purposes, both models gave approximately the same res u t l s at p r o b a b i l i t i e s of f a i l u r e l a r g e r than or equal to 0.05. 44. Assume then a two-parameter model and, f o r s i m p l i c i t y , consider the reference volume V* as a unit volume under a uniform stress a*. V* and the volume V w i l l have the same p r o b a b i l i t y of f a i l u r e i f the following r e l a t i o n s h i p holds between the stress o* and the stresses o: S # k dV . V or / a k dV = a * k v Barrett, Foschi and Fox (1975) and Foschi and Barrett (1976) have focussed on the determination of k, m and a* for tension perpendicular to the gr a i n and l o n g i t u d i n a l shear: a* = m [-*n ( l - p ) ] 1 / k (3.42) where: p = p r o b a b i l i t y l e v e l (0 < p < 1) m = scale parameter which i s : m^  = 508 p s i (3502 kN/m2) for tension perpendicular to grain mg = 2700 p s i (18616 kN/m2) for l o n g i t u d i n a l shear k = shape parameter which i s : k^ = 4.63 for tension perpendicular to grain k g = 5.53 for l o n g i t u d i n a l shear Consider a connection i n which, when the load P Q i s applied, o Q i s the stress induced i n a d i r e c t i o n of i n t e r e s t ( f o r example tension perpendicular to the gr a i n ) . I f , P i s the wood f a i l u r e load corresponding w (3.40) (3.41) 45. to the mode f o r the d i r e c t i o n of a„, and a are the stresses induced by the 0 w load P , then assuming l i n e a r behaviour, Eq. (3.41) can now be written as P k - | / a Q k dV - a * k (3.44) P v Le t t i n g I Q = J a Q dV, the wood f a i l u r e loads P may be computed at any v p r o b a b i l i t y l e v e l by where a* i s the strength of a uniformly stressed unit volume and i s given by Eq. (3.42). Since we are int e r e s t e d i n f a i l u r e s c o n t r o l l e d by shear and tension perpendicular to the grain, l e t k k shear = / (T S + T S ) d V (3.46) 0 s J_ xy Q x z o v j t e n s i o n l = j J^1 dV ] (3.47) 0 1 „ y n where k,, k are defined i n Eq. (3.42). I s n The wood f a i l u r e load P w for t h i s combined case can be computed using a g e n e r a l i z a t i o n of Eq. (3.40): 46. P k I P k I , _ W v 1 1 , wN s _s K ? > k + t p n ' k 0 _ 1 0 m s (3.48) m m m Thus, the wood f a i l u r e load P was computed for three s i t u a t i o n s : w 1) considering l o n g i t u d i n a l shear stresses only (T and i ^ ) by using Eq. (3.45) with 1 = 1 . u s 2) considering tension perpendicular to the g r a i n stresses only (o v) D V using Eq. (3.45) with I Q = 1^. 3) considering the combination of shear and tension perpendicular stresses by using Eq. (3.48). The minimum of these r e s u l t s i s chosen as P . w The stresses i n the small volume which contains the r i v e t c l u s t e r were excluded from the integrations since i t i s not l i k e l y that f a i l u r e occurs within that volume due to the e f f e c t of compression stresses under the r i v e t shanks. This has been confirmed by the experiments performed. Also, the analysis assumed that the r i v e t loads p (n) were l i n e loads which, i n l theory, would produce compression ahead of the shank and tension behind the r i v e t . The l a t t e r are highly concentrated stresses e s p e c i a l l y i n e c c e n t r i c loading case and i n r e a l i t y these stresses cannot occur since the shank separates from the wood. It i s d i f f i c u l t to obtain an accurate representa-t i o n of the state of stress within and immediately around the c l u s t e r of r i v e t s . The model was then c a l i b r a t e d to one of the experimental r e s u l t s by excluding from the i n t e g r a t i o n domain the c l u s t e r i t s e l f plus a small area around i t , as shown i n F i g . 3.6. The width of t h i s small area was found to be approximately 3 inches i n the x d i r e c t i o n i n order to f i t the t h e o r e t i c a l median f a i l u r e load P to the mean-experimental f a i l u r e load. w The width was 1/2" i n the y d i r e c t i o n which, from F i g . 3.5, implies that the element containing a r i v e t was not included i n the i n t e g r a t i o n domain. 47. stress observed crack in the exper irnent observed crack in the experiment stress concentration rivet cluster boundary of integration domain g. 3.6. Stress concentrations and observed cracks 1" = 25.4 mm. 48. 3 . 5 N u m e r i c a l R e s u l t s f o r D i f f e r e n t C o n n e c t i o n C o n f i g u r a t i o n s To o b t a i n t he t h e o r e t i c a l f a i l u r e l o a d f o r a c o n n e c t i o n , i t i s n e c e s -s a r y t o compute t h e minimum be tween t h e l o a d s f o r two f a i l u r e modes: P f o r r t h e r i v e t y i e l d i n g f a i l u r e mode and P w f o r the wood f a i l u r e mode. F o r a 5 " x l 8 " ( 1 2 . 7 mm x 4 5 7 . 2 mm) G l u l a m beam w i t h a l e n g t h of a=6 ' ( 1 8 2 8 . 8 mm) and 2 . 5 " ( 6 3 . 5 mm) l o n g G l u l a m r i v e t s , t h e s e l o a d s were com-p u t e d f o r t he f o l l o w i n g t w o - s i d e d c o n n e c t i o n s under d i f f e r e n t e c c e n -t r i c i t i e s : 1 . NR=5; NC=10; y = 0 ° ; d = l " ( 2 5 . 4 mm) 2 . NR=5; NC=10; y=0°; d=2" ( 5 0 . 8 mm) 3 . NR=10; NC=5; y = 0 ° ; d=2" ( 5 0 . 8 mm) 4 . NR=10; NC=7; y=0° ; d=2" ( 5 0 . 8 mm) 5 . NR=10; NC=10; y = 0 ° ; d=2" ( 5 0 . 8 mm) where NR = Number o f rows o f r i v e t s i n t he d i r e c t i o n p a r a l l e l t o x a x i s NC = Number o f r i v e t s p e r row Y = A n g l e be tween t h e g r a i n and t he x a x i s d = Minimum d i s t a n c e be tween t h e l o a d e d edge o f t h e G l u l a m member and t h e r i v e t c l u s t e r as shown i n F i g . ( 3 . 7 ) . R i v e t s were s p a c e d i n a l " x l " g r i d ( 2 5 . 4 mm x 2 5 . 4 mm). Two t h e o r e t i c a l f a i l u r e l o a d s f o r t h e s e c o n n e c t i o n s a r e shown i n T a b l e ( 3 . 1 ) . No te t h a t b e c a u s e of t h e g e o m e t r i c a l c o n f i g u r a t i o n of t h e r i v e t c l u s t e r , t h e f a i l u r e mode c o u l d change f r o m r i v e t y i e l d i n g t o wood f a i l u r e . 4 9 . Table 3.1. The r e s u l t s f o r selected two-sided connections. Connection NR NC d E y P r P r o b a b i l i t y Number Pre-Number ( r i v e t - level=0.50 of Theory di c t e d y i e l d i n g ) Pw Rivets F a i l u r e (Wood at Two Mode Fa i l u r e ) Sides - - - i n . i n . deg. kips kips - kips -0. 139.5 162.0 139.5 r i v e t 0.5 106.7 70.4 70.4 wood 1.0 92.2 58.0 58.0 wood 1 5 10 1. 2.0 0. 70.9 44.6 100 44.6 wood 9.0 22.3 14.0 14.0 wood 10.0 20.2 12.8 12.8 wood 12.5 16.3 10.2 10.2 wood 100. 2.1 1.4 1.4 wood 0. 139.5 292.8 139.5 r i v e t 0.5 106.7 93.8 93.8 wood 1.0 92.2 75.2 75.2 wood 2 5 10 2. 2.0 0. 70.9 57.0 100 57.0 wood 9.0 22.3 17.6 17.6 wood 10.0 20.2 16.0 16.0 wood 12.5 16.3 13.0 13.0 wood 100. 2.1 1.6 1.6 wood 0. 139.5 720.0 139.5 r i v e t 0.5 119.3 380.0 119.3 r i v e t 1.0 106.9 400.0 106.9 r i v e t 3 10 5 2. 2.0 0. 88.6 262.0 100 88.6 r i v e t 9.0 33.0 104.0 33.0 r i v e t 10.0 29.7 99.0 29.7 r i v e t 12.5 24.9 76.0 24.9 r i v e t 100. 3.2 10.2 3.2 r i v e t 0. 195.3 738.0 195.3 r i v e t 0.5 163.1 270.6 163.1 r i v e t 1.0 148.4 261.6 148.4 r i v e t 4 10 7 2. 2.0 0. 122.0 203.6 140 122.0 r i v e t 9.0 45.9 71.6 45.9 r i v e t 10.0 41.9 65.1 41.9 r i v e t 12.5 34.4 52.9 34.4 r i v e t 100. 4.4 6.7 4.4 r i v e t 0. 279.0 769.6 279.0 r i v e t 0.5 235.7 203.1 203.1 wood 1.0 209.0 177.3 177.3 wood 5 10 10 2. 2.0 0. 175.9 136.8 200 136.8 wood 9.0 66.4 54.8 54.8 wood 10.0 61.5 50.8 50.8 wood 12.5 50.3 41.4 41.4 wood L 100. 6.5 5.4 5.4 wood Note: 1 i n . = 25.4 mm 1 kip = 4,450 N 50. X il o I 1-6' Fig. 3 . 7 . The dimensions and the parameters of connection (1 in.= 25 .4 mm) The geometric configuration of the rivet cluster is found to be the most important factor on the load carrying capacity of eccentrically loaded connections. As shown in Table 3 . 1 , the third and fourth connections (10x5 and 10x7) always failed i n rivet-yielding mode and they carried more load than the f i r s t and second connections ( 5 x 1 0 ) . This w i l l be discussed in more details in Chapter 4 . CHAPTER 4 5 1 . EXPERIMENT 4.1 Experimental Set-up and Preparation of Test Specimens An experimental set-up was designed to test a series of f u l l - s i z e connections to v e r i f y the t h e o r e t i c a l predictions f o r ultimate loads as shown i n F i g s . (4.1-4.3). Fourteen specimens which were tested under various e c c e n t r i c loadings were held by a r i g i d s t e e l frame as shown i n Fi g s . (4.1-4.2). One specimen under pure tension perpendicular to grain load was tested as a simply-supported beam as shown i n F i g . (4.3). The glued-laminated members were fabricated using ll/2 i n . (38 mm) Douglas-fir laminations and the moisture content at time of t e s t i n g was approximately 13%. The size of the Glulam members were same: 5"xl8" (12.7 mm x 457.2 mm) with a length of 6' (1828.8 mm). A l l connection plates were f a b r i c a t e d of A36 s t e e l and p r e d r i l l e d for n a i l i n g the r i v e t s . They were not reused and applied to both sides of the wood member. Glulam r i v e t s 2 1 / 2 ± n, (63.5 mm) long were used for a l l connections, giving a penetration length of approxi-mately 2 i n . (50.8 mm). The applied load was c o n t r o l l e d by the d i s p l a c e -ment at the t i p of the load c e l l at a constant rate of 0.2 i n . (5.08 mm)/ minutes, and t e s t s were continued u n t i l wood f a i l u r e occurred or excessive n a i l bending was achieved. A t o t a l of four connection types were tested. The spacing of the r i v e t s : e = 1 i n . (25.4 mm) and e = 1 i n . (25.4 mm) was the same for a l l x y cases and a l l the connections were applied to both sides of the wood member. Below i s a l i s t which describes the c h a r a c t e r i s t i c s of these four connection types: TYPE #1: NR = 5; NC = 10; E = 9.0" (228.6 mm); d = 1" (25.4 mm) Number of r e p l i c a t i o n s = 5; Expected f a i l u r e mode = Wood F i g . 4 . 1 . Experimental set-up. F i g . 4.2. Specimen under eccentric loading. 5 4 . F i g . 4 . 3 . Spec imen unde r t e n s i o n p e r p e n d i c u l a r - t o - g r a i n l o a d i n g ( z e r o e c c e n t r i c i t y ) . 55. TYPE #2: NR = 10; NC = 5; E = 12.5" (317.5 mm); d = 2" (50.8 mm) Number of r e p l i c a t i o n s = 5; Expected f a i l u r e mode = Rivet Y i e l d i n g TYPE #3: NR = 5; NC = 10; E = 10" (254 mm); d = 2" (50.8 mm) Number of r e p l i c a t i o n s = 4; Expected f a i l u r e mode = Wood TYPE #4: NR = 5; NC = 10; E = 0 (Tension!); d = 2" (50.8 mm) Number of r e p l i c a t i o n s = 1; Expected f a i l u r e mode = Wood Note that i n t h i s case the load P i s applied i n the y - d i r e c t i o n i n order to have tension perpendicular loading with zero e c c e n t r i -c i t y . The parameters NR, NC, E and d are given i n Chapter (3.5) and shown i n F i g . (3.9). 4.2 Test Results The ultimate t e s t loads, displacements, t e s t mean and c o e f f i c i e n t of v a r i a t i o n f or each connection type i s given i n Table (4.1). 4.2.1 Connection Type ffl; NR=5; NC=10; E=9" (228.6 mm); d=l" (25.4 mm) Five specimens were tested. Experimental and t h e o r e t i c a l r e s u l t s are presented i n F i g . (4.4). A l l the specimens f a i l e d i n wood f a i l u r e mode due to tension perpendicular to gr a i n stresses. A t y p i c a l wood f a i l u r e i s shown i n F i g . (4.5). 4.2.2 Connection Type ff2: NR=10; NC=5; E=12.5" (317.5 mm); d=2"(50.8 mm) Five specimens were tested. Experimental and t h e o r e t i c a l r e s u l t s are presented i n F i g . (4.6). A l l the specimens f a i l e d i n r i v e t - y i e l d i n g f a i l u r e mode due to r i v e t y i e l d i n g i n bending with simultaneous bearing f a i l u r e of the wood under the r i v e t ' s shank. This type of f a i l u r e i s shown i n F i g . (4.7) . 56. Table 4.1. Experimental Results Replicate No. P u l t (kips) 6 u l t (in) T.mean P u l t (kips) cov (%) Test Type #1 1 16.575 0.290 15.85 6 2 17.063 0.365 3 14.325 0.418 4 15.613 0.374 5 15.650 0.214 Test Type #2 1 23.838 1.187 24.93 4 2 25.775 1.279 3 24.075 1.465 4 24.763 1.067 5 26.175 1.821 Test Type #3 1 16.950 0.283 16.29 3 2 15.825 0.421 3 16.853 0.433 4 15.538 0.411 Test Type #4 1 13.505 0.169 13.51 0 Note: <5u2t i s the displacement at the load point 1 kip = 4.45 kN and 1 inch = 25.4 mm. 5 7 . o TYPE 1 ; 5x10 d=1.!n. E=9.in. 9 K . CN O CX Predicted Rivet -yielding Failure Mode OX) t -riv»i dust*f 1-f Predicted Wood Failure (pr0.50) ~~r~ OA 0.2 0.6 I 0.8 - r ~ to 1-2 14 16 18 2.0 Displacement at load point (In.) F i g . 4.4 Experimental and t h e o r e t i c a l r e s u l t s for connection type #1. Note: 1 k i p = 4.45 kN, 1 inch = 25.4 mm. F i g . 4.5 Wood f a i l u r e under eccentric loading (Type #1) 59. TYPE 2 ; 10x5 d=2.in. E=12.5 in. $ _ _ Predicted g_ Rivet - yielding Failure Mode Displacement at load point (In.) F i g . 4.6 Experimental and t h e o r e t i c a l r e s u l t s for connection type #2. Note: 1 k i p «= 4.45 kN, 1 inch = 25.4 mm. 4.2.3 Connection Type #3; NR=5; NC=10; E=10" (254.0 mm); d=2" (50.8 mm) Four specimens were tested. Experimental and t h e o r e t i c a l r e s u l t s are presented i n F i g . (4.8). A l l the specimens f a i l e d i n wood f a i l u r e mode due to tension perpendicular to grain stresses. 4.2.4 Connection Type U: NR=5; NC=10; E=0" (11 mm); d=l" (25.4 mm) This connection type was selected to v e r i f y the t h e o r e t i c a l f a i l u r e loads for zero ecc e n t r i c tension loading perpendicular to the grain. Only one specimen was tested and, as expected, f a i l e d i n wood due to tension perpendicular to grain stresses. However, because of the minimal sample size, only a rough v e r i f i c a t i o n was possible f o r the t h e o r e t i c a l f a i l u r e load. Experimental and t h e o r e t i c a l r e s u l t s are presented i n F i g . (4.9) and the wood f a i l u r e mode i s shown i n F i g . (4.10). 4.3 Comparison Table (4.2) shows the t h e o r e t i c a l loads, P and P , and the predicted r w f a i l u r e load, P^, for the connection. The mean of ultimate load recorded i n the t e s t s and the observed f a i l u r e mode are a l s o shown. The load, P , r was computed using the theory for r i v e t - y i e l d i n g f a i l u r e mode explained i n Chapter 2. The load, P , was computed using the theory f o r wood f a i l u r e w mode explained i n Chapter 3. The ultimate loads thus predicted correspond to a p r o b a b i l i t y of f a i l u r e of 0.5. As such they may be compared with the mean f a i l u r e load obtained i n t e s t s f o r a l l the r e p l i c a t i o n s of the same connection type. This comparison i s shown i n F i g . (4.11), where, for per-f e c t agreement between test r e s u l t s and t h e o r e t i c a l p r e d i c t i o n s , a l l points must l i e on the 45° l i n e through the o r i g i n . The agreement achieved i s s a t i s f a c t o r y , despite the small number of r e p l i c a t i o n s f o r each type of connection, allowing only a f i r s t approximation to the mean strength f o r that p a r t i c u l a r type, f o r the e c c e n t r i c a l l y loaded connections, i t i s 62. TYPE 3 ; 5x10 d=2.in. E=10.in. Predicted Rivet - yielding Failure Mode t *r,0" r — < i I 16 Displacement at load point (In.) 18 2.0 F i g . 4.8 Experimental and t h e o r e t i c a l r e s u l t s for connection type #3. Note: 1 k i p = 4.45 kN, 1 inch = 25.4 mm. 63. TYPE 4 ; TENSION ± ; E=0.0 Predicted Rivet-yielding Failure = 88. kips r?Z"eweoo^iw:d-* u . D Predicted Wood Failure ( p =0.50 ) 0.0 0.2 OA 0.6 0.8 10 12 14 16 16 2.0 Displacement at load point (In.) F i g . 4.9 Experimental and t h e o r e t i c a l r e s u l t s for connection type #4. Note: 1 k i p = 4.45 kN, 1 inch = 25.4 mm. 64 . F i g . 4.10 Wood f a i l u r e mode due to pure tension perpendicular to grain loading (Type #4). TABLE 4.2 Experimental and The o r e t i c a l Results Samp l e Size Type NR NC d E Th( >oretical F« i i l u r e I -oad Tei its Foschi and Longworth (197 5) Pw (Wood Fa i l u r e ) p r (Rivet-Yielding) P £ Theory Predicted F a i l u r e Mode Mean Failure Mode - - - - i n . i n . kips kips kips - kips - kips 5 1 5 10 1. 9. 14.0 22.3 14.0 wood 15.9 wood -5 2 10 5 2. 12.5 76.0 24.9 24.9 r i v e t 24.9 r i v e t -4 3 5 10 2. 10.0 16.0 20.2 16.0 wood 16.3 wood -1 4 5 10 2. 0. 16.2 88. 16.2 wood 13.5 wood 14.3 Note: 1 i n - 25.4 mm; 1 kip = 4.45 kN 65. q Theory (p=0.50) (Kips) F i g . 4.11 Comparison of t h e o r e t i c a l and test f a i l u r e loads (1 k i p = 4,450 N). apparent that the geometric arrangement of the r i v e t c l u s t e r i s the most important e f f e c t on ultimate load capacity. Connection types #1 and #3 (5x10) f a i l e d i n wood f a i l u r e mode because the forces due to moment caused high tension perpendicular to grain stresses as shown i n Figs. (4.5 and 4.12). Connection type #2 (10x5) f a i l e d i n r i v e t - y i e l d i n g f a i l u r e mode because the coupled forces due to moment caused mostly shear stresses which d i d not exceed the shear strength of the wood as shown i n Figs. (4.7 and 4.13). The behaviour of th i s connection was d u c t i l e and the ultimate load capacity of the connection was higher than the connection types #1 and #3 despite having a bigger e c c e n t r i c i t y . The ultimate load capacity for the connection #4 such as the hanger 66. 11 _• L d=1"for c o n n e c t i o n t y p e =1 d=2" f o r connection t y p e = 3 o b s e r v e d w o o d failure d , JH 1 - r i v e t c l u s t e r grain V (z) I o to + r-6" F i g . 4.12 Load d i s t r i b u t i o n pattern for connection types #1 and #3. x d=2" 0 - 9 " -rivet cluster M o grain r-6" F i g . 4.13 Load d i s t r i b u t i o n pattern for connection type #2 67. shewn i n F i g s . (A.3, 4.10, and 4.14), i s usually c o n t r o l l e d by the strength of the wood i n tension perpendicular to the grain. The f a i l u r e mode corresponds to an opening of the f i b e r s along l i n e A-A of F i g . (4.14), just above the top row of r i v e t s . The load capacity of such connections increases i f the r i v e t c l u s t e r i s moved towards the unloaded edge of the wood member, Foschi, Longworth (1975). F i g . 4.14 Load d i s t r i b u t i o n pattern f o r connection type #4. In F i g . 4.9, a f t e r complete s p l i t t i n g along the l i n e A-A of F i g . (4.14), the lower part of the beam was loaded i n bending and that i s why the load P started to increase again. 6 8 . CHAPTER 5  DESIGN CRITERION 5 .1 G e n e r a l In t h i s c h a p t e r , f o r e c c e n t r i c a l l y l o a d e d G l u l a m r i v e t c o n n e c t i o n s , a d e s i g n p r o c e d u r e w i l l be g i v e n f o r d e t e r m i n i n g t he u l t i m a t e l oad c a r r y i n g c a p a c i t y i n r i v e t - y i e l d i n g f a i l u r e mode. To a v o i d t h e o c c u r r e n c e o f wood f a i l u r e mode, some recommenda t i ons w i l l be made so t h a t t h e u l t i m a t e l o a d d e t e r m i n e d f o r r i v e t - y i e l d i n g f a i l u r e mode may be u s e d as t h e l o a d c a r r y i n g c a p a c i t y f o r t he c o n n e c t i o n . 5 . 2 Recommendat ions f o r A v o i d i n g t h e O c c u r r e n c e o f Wood F a i l a r e Mode I t i s f o u n d f r o m e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s t ha t t h e f a i l u r e mode o f t h e c o n n e c t i o n i s u s u a l l y dependen t on t he g e o m e t r i c c o n f i g u r a t i o n o f t h e r i v e t c l u s t e r . F o r beam-co lumn c o n n e c t i o n s and beam s p l i c e s , t o a v o i d t he o c c u r r e n c e o f wood f a i l u r e , t he g e o m e t r i c shape o f the r i v e t c l u s t e r s s h o u l d be a r r a n g e d a s shown i n F i g s . ( 5 . 1 and 5 . 2 ) . The p r o p o r -t i o n o f 6 / 1 0 be tween number o f rows and co lumns i n t he r i v e t c l u s t e r was f o u n d t o be a n i d e a l a r r a n g e m e n t t o a v o i d t h e o c c u r r e n c e o f wood f a i l u r e mode. The recommended edge and end d i s t a n c e s a r e a l s o shown i n F i g s . ( 5 . 1 and 5 . 2 ) f o r c o n n e c t i o n s w h i c h have up t o 10 r i v e t s i n a row o r c o l u m n . 5 . 3 S t e p s o f S i m p l i f i e d D e s i g n P r o c e d u r e 1 . S t e p : U s i n g t he a p p r o x i m a t e a n a l y t i c a l method w h i c h i s e x p l a i n e d i n C h a p t e r ( 2 . 3 . 2 ) , e s t i m a t e t h e p u r e moment c a r r y i n g c a p a c i t y o f t h e c o n n e c -t i o n , M * , b a s e d on f i r s t r i v e t f a i l u r e . T h i s can be a c h i e v e d &y u s i n g E q s . ( 2 . 2 6 and 2 . 2 7 ) t o compute t h e a p p l i e d moment M^ = P^ E , and then l e t t i n g E a p p r o a c h i n f i n i t y . T h u s , 69. F i g . 5.1 Recommended geometric shape of the r i v e t c l u s t e r s for beam-column connection. Note: 1 inch = 25.4 mm. F i g . 5.2 Recommended geometric shape of the r i v e t c l u s t e r s for beam s p l i c e s . Note: 1 inch = 25.4 mm. 70. M. = u l t u l t P / / P l 1 r i , u l t , o „ . u l t , „ . — (p j j s i n ^ fi^ + p ± cos^ 8J Z r i M* = min (M.) i = 1, ... N o v l ' (5.1) 2. Step: Eq. (5.2) can be used to compute the moment carrying capacity, MQ, for the r i v e t y i e l d i n g f a i l u r e mode. MQ = C M* (5.2) where the factor, C, was investigated f o r d i f f e r e n t type of connections and found that i t varied from '1.03' to '1.30'. The values of factor 'C for 12 randomly selected connections are l i s t e d i n Appendix I I I . The average value of C = 1.15 may be considered to approximate the moment carrying capacity of the connection. 3. Step: Using Eq. (5.3), the ultimate load carrying capacity f o r zero e c c e n t r i c i t y , P Q, may be computed, Foschi, Longworth (1975): P Q = NR NC p u l t (5.3) where NR = number of rows of r i v e t s NC = number of r i v e t s per row pU^ = load c a r r y i n g capacity f o r one Glulam r i v e t . In Appendix (I) the values of p11^*" f o r loadings p a r a l l e l and perpendicular to grain are given, Foschi (1974). 4. Step: Since P Q and MQ values are known, the fourth degree curve i n Figs. (2.12 and 2.13) and Eq. (2.31) which i s representing r i v e t y i e l d i n g 71. f a i l u r e mode, may be used to estimate moment carry i n g capacity of a connection f o r a given load, P, or vise-versa as shown i n F i g . (5.1). 3-, — — P/Po F i g . 5 .3 Load-moment i n t e r a c t i o n graph for design purposes. As a simpler a l t e r n a t i v e , the l i n e a r r e l a t i o n s h i p (5.A) may be used. CHAPTER 6 72. CONCLUSIONS 6.1 Summary and Conclusions T h e o r e t i c a l and experimental studies of the strength of Glulam r i v e t connections under eccentric loading have been investigated i n four stages as follows: 1. An a n a l y t i c a l study was conducted to determine the ultimate load capa-c i t y of the connection due to r i v e t yielding/wood bearing f a i l u r e mode. The s t e e l plate was assumed to be s u f f i c i e n t l y s t i f f to displace as a r i g i d body with respect to the Glulam member. The plate displacements and r o t a -t i o n which describe the r i g i d body motion of the s t e e l plate were deter-mined by unconstrained non-linear minimization of p o t e n t i a l energy using non-linear l o a d - s l i p c h a r a c t e r i s t i c s of Glulam r i v e t s . An approximate a n a l y t i c a l method was a l s o developed to estimate load c a r r y i n g capacity of the connection based on f i r s t r i v e t f a i l u r e . This method was used f o r two purposes: f i r s t , i n the determination of i n i t i a l load and load increments for the nonlinear function minimization, and, secondly, i n the development of design procedure. 2. Three-dimensional state of stress around the r i v e t c l u s t e r was evalu-ated i n order to determine the ultimate loads due to wood f a i l u r e mode. F i n i t e element analysis was used and Weibull's theory of b r i t t l e f r a c t u r e was ap p l i e d to study f a i l u r e s c o n t r o l l e d by the l o n g i t u d i n a l shear and tension perpendicular-to-grain strength of the wood. 73. 3. A serie s of f u l l - s i z e connection tests were conducted to v e r i f y the t h e o r e t i c a l predictions f o r ultimate loads. Very good agreement was observed f o r r i v e t y i e l d i n g f a i l u r e mode and s a t i s f a c t o r y agreement was observed f o r wood f a i l u r e mode. 4. A design c r i t e r i o n has been developed to estimate the ultimate loads f o r the e c c e n t r i c a l l y loaded Glulam r i v e t connections. For t h i s purpose, the design procedure was given to compute the ultimate loads f o r r i v e t yielding/wood bearing mode and some recommendations were made to avoid the occurrence of wood f a i l u r e mode. From the r e s u l t s of t h i s study, the following conclusions may be offered: . Usually the geometric shape of the r i v e t c l u s t e r controls the mode of f a i l u r e of the connection. The r i v e t cluster should be arranged i n a manner such that no high tension perpendicular-to-grain stresses occur. . E c c e n t r i c i t y controls the magnitude of the load capacity, with a bigger e c c e n t r i c i t y producing lower load capacity. . Wider r i v e t spacings increase the load carrying capacity and decrease the chance of having wood f a i l u r e mode, thus optimizing r i v e t u t i l i z a t i o n . This should be balanced, however, against increased metal plate cost. 6.2 Future Studies The a n a l y s i s presented i n t h i s thesis i s a r a t i o n a l approach to timber connection design, i t i s general i n nature, and could be extended to other types of connectors such as bolts and truss plate j o i n t s . For the development of a de t a i l e d design procedure for the Code, the v e r i f i e d model developed i n t h i s t h e s i s may be used to investigate the 74. influence of other geometrical parameters of the connection: number of rows of r i v e t s ; number of r i v e t s per row; r i v e t penetration length; spacing of r i v e t s ; edge and end distances to develop comprehensive design tables or a i d s . There i s also a need to study on the long-term behaviour of the Glulam r i v e t connections under d i f f e r e n t types of loadings. 75. REFERENCES 1. BARRETT, J . D . , FOSCHI , R . O . and FOX, S . P . 1 9 7 5 . P e r p e n d i c u l a r - t o -G r a i n S t r e n g t h o f D o u g l a s - F i r . C a n . J . C i v . E n g . 2 ( 1 ) , p p . 5 0 - 5 7 . 2 . COOK, R . D . 1 9 8 1 . C o n c e p t s and A p p l i c a t i o n s o f F i n i t e E l e m e n t A n a l y s i s . J o h n W i l e y & S o n s , I n c . , pp . 1 1 3 - 1 2 7 . 3 . FLETCHER, R. and POWELL, M. 1 9 6 3 . A R a p i d l y C o n v e r g e n t D e s c e n t Method f o r M i n i m i z a t i o n . Comput . J . 6 ( 1 9 6 3 ) : 1 6 3 - 1 6 8 . 4 . FOSCHI , R . O . 1 9 7 3 . S t r e s s A n a l y s i s and D e s i g n o f G l u l a m R i v e t C o n n e c t i o n s f o r P e r p e n d i c u l a r t o G r a i n L o a d i n g s . C a n . F o r . S e r v . , W e s t . F o r . P r o d . L a b . , I n f . R e p . V P - X - 1 1 7 , V a n c o u v e r , B . C . 5 . F O S C H I , R . O . 1 9 7 4 . L o a d - S l i p C h a r a c t e r i s t i c s o f N a i l s . Wood S c i e n c e , V o l . 7 , N o . 1, p p . 6 9 - 7 6 . 6 . F O S C H I , R . O . and L0NGW0RTH, J . 1 9 7 5 . A n a l y s i s and D e s i g n o f G r i p l a m N a i l e d C o n n e c t i o n s . A S C E , J . S t r u c t . D i v . 101 ( S T 1 2 ) , p p . 2 5 3 7 - 2 5 5 5 . 7 . F O S C H I , R . O . and BARRETT, J . D . 1 9 7 6 . L o n g i t u d i n a l S h e a r S t r e n g t h o f D o u g l a s - F i r . C a n . J . C i v . E n g . 2 ( 1 ) , p p . 9 8 - 1 0 8 . 8 . McGowan, W.M. and M a d s e n , B . 1 9 6 5 . A R i g i d F i e l d J o i n t f o r G l u e d -L a m i n a t e d C o n s t r u c t i o n . P r e p u b l i c a t i o n r e p o r t i s s u e d f o r c o m m i t t e e members o f t h e C a n a d i a n S t a n d a r d s A s s o c i a t i o n . 9 . TDM 1 9 8 0 . T i m b e r D e s i g n M a n u a l , L a m i n a t e d T i m b e r I n s t i t u t e o f C a n a d a . 1 0 . Z IENKIEWICZ , O . C . 1 9 7 1 . The F i n i t e E l e m e n t Me thod i n E n g i n e e r i n g S c i e n c e . M c G r a w - H i l l P u b l i s h i n g C o . L t d . , L o n d o n , E n g l a n d , p p . 1 0 3 - 1 1 0 . 76. APPENDIX I The values of the i n i t i a l modulus (k) and the constants (p Q and Pj) for loadings p a r a l l e l and perpendicular to the gr a i n f o r various r i v e t penetrations, Foschi (1974). The ultimate loads are computed using the following formula f o r the s l i p value of A = 0.25 i n . (6 mm). P = (p Q + P L A) (1 - e k A / p 0 ) Rivet Penetration Loading P a r a l l e l to Grain Loading Perpendicular to Grain P° II P 1 II k II pult II p i k l p ? " i n . l b l b / i n l b / i n l b l b l b / i n l b / i n l b 3 1538 0. 66014 1538 551 2027 20580 1058 2 1395 0. 66895 1395 530 1400 20200 880 1 1116 0. 69937 1116 512 835 20012 721 Note: 1 l b = 4.45 N; 1 i n = 25.4 mm APPENDIX I I 77. CONVERGENCE OF THE F I N I T E ELEMENT PROGRAM To be a b l e t o o b t a i n r e a s o n a b l e s t r e s s v a l u e s , a t l e a s t f i v e t e rms h a v e t o be t a k e n i n e a c h o f t h e f o l l o w i n g s e r i e s w h i c h r e p r e s e n t d i s p l a c e m e n t s i n t h e wood member: N u ( x , y , z ) = I F ( y , z ) s i n • n ^ x -. n a n= l ( I I . l ) N v ( x , y , z ) = Z G m ( y » z ) s i n m=l T h i s was c o n c l u d e d by d e t e r m i n i n g t h e e q u i l i b r i u m be tween t h e e x t e r n a l f o r c e s and t h e s t r e s s e s a r o u n d t h e r i v e t c l u s t e r shown i n F i g . ( I I . l ) . F o r a z e r o e c c e n t r i c l o a d i n g i n x d i r e c t i o n , t h e c o n n e c t i o n has c a p a c i t y ( 6 9 , 7 5 0 l b o r 310 kN) a c c o r d i n g t o E q . ( I I . 2 ) : P = p U l t NR NC ( I I - 2 ) where p " ^ i s g i v e n i n A p p e n d i x I (1395 l b o r 6 . 2 1 k N , f o r t h i s examp le ) NR i s number o f rows (5 f o r t h i s e x a m p l e ) NC i s number o f r i v e t s p e r row (10 f o r t h i s e x a m p l e ) . The l o a d P was a p p l i e d t o t h e wood member by means o f l i n e l o a d s and t h e n o r m a l ( a ) , b o t t o m s h e a r (Z ) and l a t e r a l s h e a r (Z ) s t r e s s e s were x x z xy i n t e g r a t e d a r o u n d t h e r i v e t c l u s t e r f o r f i v e p a i r s o f N and M a s shown i n T a b l e I I . l . 7 8 . TABLE II.1 Stresses Around the Rivet Cluster f o r D i f f e r e n t Pairs of N,M Values N,M APPLIED LOAD (lb) RESULTANT FORCES DUE TO STRESSES °x (lb) Zxz (lb) Zxy (lb) TOTAL (lb) % OF APPLIED LOAD 1, 2 69750 4386 11472 6275 22133 32 3, 4 69750 7180 6861 4998 19039 27 5, 6 69750 7542 3232 3328 14102 20 7, 8 69750 5772 1289 1841 8902 13 9, 10 69750 3500 562 883 4845 7 TOTAL 69750 28280 23416 17325 69021 99 Note: 1 l b = 4.45 N o x = normal stresses i n x d i r e c t i o n (tension p a r a l l e l + compression p a r a l l e l ) Z X 2 = bottom shear stresses Z^„ = l a t e r a l shear stresses 79. Fig. II.1 Connection subjected to zero eccentric loading. 80. APPENDIX I I I THE VALUES OF FACTOR 'C FOR 12 RANDOMLY SELECTED CONNECTIONS NR NC Y (degree) E (inch) M o ( l b . i n ) M* o ( l b . i n ) M„ C = ° M o 10 20 90 100 1252477 963443 1.30 20 10 0 5 10 90 100 159650 130350 1.22 10 5 0 10 5 90 100 104500 97232 1.07 5 10 0 20 10 90 100 768780 745250 1.03 10 20 0 10 7 0 100 221984 184987 1.20 10 10 0 100 325400 276984 1.17 5 5 0 100 40792 37946 1.08 5 20 0 100 356247 323861 1.10 NR i s number of rows of r i v e t s i n the d i r e c t i o n p a r a l l e l to x axis, NC i s number of r i v e t s per row, y i s the angle between the grain of the wood member and x axis, M i s moment c a r r y i n g capacity of the connection i n r i v e t y i e l d i n g f a i l u r e mode, M i s approximate moment ca r r y i n g capacity, C i s the c o r r e c t i o n f a c t o r . Note: 1 lb = 4.45 N, 1 inch = 25.4 mm. 

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