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Behaviour of headed stud connections for precast concrete panels under monotonic and cycled shear loading Neille, Donald Stewart 1977

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BEHAVIOUR OF HEADED STUD CONNECTIONS FOR PRECAST CONCRETE PANELS UNDER MONOTONIC  AND CYCLED SHEAR LOADING by DONALD STEWART NEILLE M.Sc.(Eng)(Witwatersrand) A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard The University of B r i t i s h Columbia May, 1977 ® Donald Stewart N e i l l e , 1977 In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publ i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Mall, Vancouver, B.C. Canada. V6T 1W5 i i R e s e a r c h S u p e r v i s o r : Dr. R i c h a r d A. S p e n c e r ABSTRACT The r e s e a r c h on h e a d e d s t u d c o n n e c t i o n s d e s c r i b e d i n t h i s d i s s e r t a t i o n f o r m s a p a r t o f an o v e r a l l o b j e c t i v e o f p r e d i c t i n g t h e b e h a v i o u r o f p r e c a s t c o n c r e t e p a n e l b u i l d i n g s u n d e r e a r t h q u a k e l o a d s . E x i s t i n g l a b o r a t o r y t e s t d a t a and c u r r e n t d e s i g n p r o c e d u r e s o f h e a d e d s t u d c o n n e c t i o n s a r e b r i e f l y r e v i e w e d . I t i s p o s t u l a t e d t h a t s h e a r l o a d s a r e t r a n s m i t t e d v i a a c o n n e c t i o n t o t h e s u r r o u n d i n g c o n c r e t e by t h r e e d i s -t i n c t m e chanisms: 1. f r i c t i o n b e tween f a c e p l a t e and c o n c r e t e 2. b e a r i n g o f end o f f a c e p l a t e on c o n c r e t e 3. i n t e r a c t i o n b etween s t u d s and c o n c r e t e T e s t s on l a b o r a t o r y m o d e l s d e s i g n e d t o i s o l a t e i n d i v i d u a l a s p e c t s o f t h e s e mechanisms c o n f i r m t h a t a l l t h r e e e x i s t . F r i c t i o n f o r c e s between f a c e p l a t e and c o n c r e t e a r e s m a l l i n c o m p a r i s o n w i t h t h e r e m a i n i n g f o r c e s a c t i n g i n a c o n n e c t i o n , p a r t i c u l a r l y u n d e r c y c l e d l o a d i n g . B e a r i n g o f t h e end o f t h e f a c e p l a t e on c o n c r e t e and i n t e r a c t i o n between s t u d s and surrounding concrete are shown to be the main c o n t r i b u t i o n s to the t o t a l load c a r r i e d by a connection. A simple analy-t i c a l model i s presented f o r the p r e d i c t i o n of the u l t i m a t e shear load c a p a c i t y of a connection and a computer a l g o r i t h m i s proposed f o r the p r e d i c t i o n of the load versus d e f l e c t i o n behaviour of a connection under both monotonic and c y c l i c c o n d i t i o n s . E x i s t e n c e of the three mechanisms whereby a connec-t i o n t r a n s f e r s a p p l i e d shear f o r c e s t o the surrounding con-c r e t e c o n t r a d i c t s the shear f r i c t i o n equation which i s c u r r e n t -l y used i n the design of connections. The a n a l y t i c a l equa-t i o n s developed i n the i n v e s t i g a t i o n i n d i c a t e t h a t the s t r e n g t h of a connection i s d i r e c t l y dependent upon the s t r e n g t h of the surrounding concrete, as opposed to the e x p r e s s i o n f o r shear f r i c t i o n , which does not c o n t a i n concrete s t r e n g t h as a v a r i a b l e . i v TABLE OF CONTENTS PAGE Rights of Publication, Copying and Loan i i Abstract i i i L i s t of Tables v i i i L i s t of Figures i x Dedication xvi Acknowledgement x v i i CHAPTER 1 INTRODUCTION 1 2 CURRENT INFORMATION ON HEADED STUD CON-NECTIONS 7 2.1 E x i s t i n g S t a t i c Load Test Informa-t i o n . 7 2.2 Current Design Procedure. 11 2.3 Recent C y c l i c Tests on Headed Stud Connections. 13 3 THE SHEAR LOAD-RESISTING COMPONENTS OF A CONNECTION 2 0 3.1 Probable Mechanisms. 20 3.2 Laboratory Models of Mechanisms. 21 3.3 Testing Apparatus. 29 v CHAPTER PAGE 4 LABORATORY TESTS AND RESULTS 35 4.1 Materials Tests. 35 4.2 Tests on F r i c t i o n Specimens. 37 4.3 Tests on End Bearing Specimens. 37 4.4 Tests on Single Studs i n Con-crete. 46 4.5 Tensile and Bending Tests on Studs. 60 5 ANALYTICAL MODELS 76 5.1 Approximations. 76 5.2 Non-Linear Plane Stress F i n i t e Element Program for P l a i n Con-crete. 77 5.3 Non-Linear Plane Frame Program. 83 6 COMPARISON OF COMPUTER ANALYSES WITH LABORATORY RESULTS 93 6.1 Properties of Steel Anchor Bars and Studs. 93 6.2 F r i c t i o n Specimen. 95 6.3 End-Bearing Specimens. 95 6.4 Single Studs i n Concrete. 105 6.5 Complete Headed Stud Connection. 112 7 SIMPLIFIED ANALYTICAL MODELS 126 7.1 Ultimate Shear Load of a Stud in Concrete. 126 7.2 Ultimate End-Bearing Capacity of Faceplate.. 129 7.3 Comparison of Proposed Theory and Current Shear F r i c t i o n Theory with Laboratory Measurements. 129 v i CHAPTER PAGE 8.2 BIBLIOGRAPHY APPENDIX A A. 1 A. 2 A. 3 A. 4 A. 5 APPENDIX B B. l B. 2 B. 3 7.4 F r i c t i o n Between F a c e p l a t e and Concrete. 7.5 L o a d - D e f l e c t i o n Model f o r a Connection. CONCLUSION 8.1 C o n f i r m a t i o n of I n i t i a l Assump-t i o n s . Future Research. APPENDIX C TRIANGULAR AND QUADRILATERAL PLANE STRESS/STRAIN FINITE ELEMENTS. Plane S t r e s s / S t r a i n T r i a n g l e . Plane S t r e s s / S t r a i n Q u a d r i l a t e r a l . Patch T e s t s on Elements. Element S t i f f n e s s and S t r a i n C a l -c u l a t i o n S u b r o u t i n e s . Performance of Elements i n S e l e c t e d Problems. LOAD-DEFLECTION CURVE GENERATOR FOR HEADED STUD CONNECTIONS L i s t of V a r i a b l e s i n Subroutine I n -put/Output L i s t . Notes. Subroutine STUDCO. CONTROL OF AXIAL LOAD AND MOMENT INTERACTION FOR STUDS IN ITERATIVE CALCULATIONS 132 132 142 142 144 146 150 150 163 173 177 194 205 205 206 207 220 v i i L I S T OF TABLES T I T L E Summary o f L o a d D a t a f o r C o n n e c t i o n s C o n c r e t e S t r e n g t h s o f L a b o r a t o r y Spe-c i m e n s T e n s i l e S t r e n g t h o f S t e e l Samples C u m u l a t i v e R o t a t i o n s o f S t u d B e n d i n g S p e c i m e n s S u b j e c t e d t o C y c l i c L o a d i n g C o m p a r i s o n Between A p p r o x i m a t e F o r m u l a and R i g o r o u s D e r i v a t i o n o f S t u d I n t e r -a c t i o n D i a g r a m T r i l i n e a r P r o p e r t i e s o f S t e e l A n c h o r B a r s and S t u d s C o m p a r i s o n o f C a l c u l a t e d and M e a s u r e d U l t i m a t e S t r e n g t h s o f C o n n e c t i o n s S t r a i n E n e r g i e s f r o m P a t c h T e s t s on E l e m e n t s C o m p a r i s o n o f S t r e s s e s and D e f l e c t i o n s f o r P a r a b o l i c a l l y L o a d e d S q u a r e P l a t e LIST OF FIGURES TITLE T y p i c a l p r e c a s t c o n c r e t e p a n e l b u i l d i n g . T y p i c a l headed stud c o n n e c t i o n showing two common stud c o n f i g u r a t i o n s and the ease w i t h which minor i m p e r f e c t i o n s and misalignments may be accommodated. T y p i c a l push-out specimen. Shear l o a d - d e f l e c t i o n curves f o r 3/4 i n . d i a . x 4 i n . s t u d s . L o a d - d e f l e c t i o n curves t o u l t i m a t e f o r 3/4 i n . d i a . x 4 i n . studs. D e t a i l s of connections t e s t e d . T e s t r i g f o r c y c l i c l o a d i n g of connec-t i o n s . L o a d - d e f l e c t i o n curve - c o n n e c t i o n A l . L o a d - d e f l e c t i o n loops - c o n n e c t i o n A3. Specimens f o r i n v e s t i g a t i o n of f r i c t i o n between f a c e p l a t e and c o n c r e t e . T e s t p i e c e s f o r i n v e s t i g a t i n g f a c e p l a t e end b e a r i n g . I s o l a t i o n of bars and back of f a c e p l a t e from conc r e t e w i t h p l a s t i c foam. Machined studs ready f o r c a s t i n g i n con-c r e t e . i x FIGURE TITLE PAGE 3.5 Models for examination of in t e r a c t i o n between stud and concrete. 27 3.6 Specimens for testing studs i n tension and bending. 2 8 3.7 Apparatus for tes t i n g single studs i n concrete, end bearing and f r i c t i o n spe-cimens. 30 3.8 Measurement of displacements of f r i c t i o n and end bearing specimens. 31 3.9 Set of displacement transducers for mea-suring deflected shapes of studs. 31 3.10 Data a c q u i s i t i o n and recording i n s t r u -ments used i n tests on concrete. 33 3.11 Apparatus for bending tests on studs. 34 3.12 Studs firmly clamped to concrete block. 34 4.1 Bending t e s t on 1/2 i n . dia. bars. 39 4.2 Load-deflection curves for f r i c t i o n spe-cimen F l . 40 4.3 Load-deflection curves for f r i c t i o n spe-cimen F2. 41 4.4 Load-deflection curves for f r i c t i o n spe-cimen F3. 42 4.5 Load-deflection curves for end-bearing specimens Bl & B2. 4 3 4.6 Load-deflection curves for end-bearing specimens B3 & B4. 44 4.7 Load-deflection curves for end-bearing specimens B5 & B6. 45 4.8 Typical concrete f a i l u r e at end of face-plate. 47 4.9 Load-deflection curve for stud SI. 48 x FIGURE TITLE PAGE 4.10 Deflected shape of stud SI. 49 4.11 Load-deflection curve for stud S2. 50 4.12 Deflected shape of stud S2. 51 4.13 Load-deflection curve for stud S3. 52 4.14 Deflected shape of stud S3. 53 4.15 Load-deflection curve for stud S4. 54 4.16 Deflected shape of stud S4. 55 4.17 Load-deflection curve for stud S5. 56 4.18 Deflected shape of stud S5. 57 4.19 Load-deflection curve for stud S6. 58 4.20 Deflected shape of stud S6. 59 4.21 S t r e s s - s t r a i n curves from t e n s i l e tests on studs. 61 4.22 Monotonic bending te s t on stud specimen SB1. 65 4.23 C y c l i c bending test on stud specimen SB2. 6 6 4.24 C y c l i c bending te s t on stud specimen SB3. 67 4.25 Monotonic bending te s t on stud specimen SB4. 68 4.26 C y c l i c bending te s t on stud specimen SB5. ' 69 4.27 C y c l i c bending test on stud specimen SB6. 70 4.28 Monotonic bending test on stud specimen SB7. 71 4.29 C y c l i c bending test on stud specimen SB8. 72 4.30 C y c l i c bending test on stud specimen SB9. 73 x i FIGURE TITLE PAGE 4.31 C y c l i c loading caused t y p i c a l fracture of studs i n fusion welds. 74 4.32 Normalised load decrement versus norma-l i s e d d e f l e c t i o n from stud bending tests. Quadratic equation f i t t e d by least squares. 75 5.1 The approximation of a stud i n concrete by means of a beam on Winkler springs. 78 5.2 Triangular and q u a d r i l a t e r a l f i n i t e e l e -ments. 8 0 5.3 Concrete s t r e s s - s t r a i n model adapted from work by Karsan & J i r s a . 82 5.4 B i a x i a l strength envelope for concrete afte r Kupfer & Gerstle. 84 5.5 Five types of l i n e members b u i l t into plane frame model. 8 6 5.6 Load-deflection model for concrete springs. 86 5.7 T r i l i n e a r hysteresis loops used for both s t r e s s - s t r a i n and moment-curvature r e l a -tionships for studs. 88 5.8 Stud cross-section under ultimate a x i a l force and bending moment. 90 5.9 Stud i n t e r a c t i o n diagram. 90 6.1 Bending test on 1/2 i n . dia. bars compared with computer c a l c u l a t i o n s . 96 6.2 Computer model of f r i c t i o n specimen with 6 i n . long anchor bars. 97 6.3 Measured and computed load-deflection curves for f r i c t i o n specimen F2. 98 6.4 F i n i t e element gri d of one quadrant of con-crete test specimen used by Karsan & J i r s a . 100 6.5 Comparison of computed s t r e s s - s t r a i n curve with test by Karsan & J i r s a . 101 x n FIGURE TITLE PAGE 6.6 3/8 i n . end bearing specimen modelled by f i n i t e elements. 102 6.7 Load-deflection curve for 3/8 i n . end bearing specimens. 103 6.8 Comparison of calculated and measured maximum loads for end bearing specimens. 104 6.9 Computer model of end bearing specimen. 106 6.10 Measured and computed load-deflection curves for end-bearing specimens B3 & B4. 107 6.11 F i n i t e element g r i d of one half of con-crete s l i c e containing stud shank. 108 6.12 Load-deflection curve for concrete s l i c e containing stud shank. 109 6.13 Computer model of stud i n concrete. I l l 6.14 Measured and computed load-deflection curves for stud SI. 113 6.15 Deflected shape comparison and computed forces on stud SI well a f t e r f i r s t y i e l d . 114 6.16 Measured and computed load-deflection curves for stud S2. 115 6.17 Deflected shape comparison and computed forces on stud S2 well a f t e r f i r s t y i e l d . 116 6.18 Measured and computed load-deflection curves for stud S3. 117 6.19 Deflected shape comparison and computed forces on stud S3 well a f t e r f i r s t y i e l d . 118 6.2 0 Measured and computed load-deflection curves for studs S4, S5 & S6. 119 6.21 Deflected shape comparison and computed forces on studs S4, S5 & S6 well a f t e r f i r s t y i e l d . 120 6.22 Stud connection E2, due to Spencer, simu-lated by computer model. 122 x i i i FIGURE TITLE PAGE 6.23 Measured and computed load-deflection curves for connection E2. 123 6.24 Stud a x i a l s t r a i n s for connection E2. 124 7.1 Approximation of forces acting on a stud i n concrete at maximum load. 12 8 7.2 Approximate load-deflection model of a connection. 128 7.3 Calculated load-deflection curve for studs S4, S5 & S6. 136 7.4 Calculated load-deflection curve for connection E2. 137 7.5 Calculated load-deflection loops for stud bending specimen SB5. 138 7.6 Calculated load-deflection loops connection A3. 139 7.7 Load-deflection loops for connection E l measured by Spencer. 140 7.8 Calculated load-deflection loops for con-nection E l . 141 A. 1 Complete cubic t r i a n g l e . 152 A. 2 A t y p i c a l side of an element. 152 A. 3 Constrained cubic t r i a n g l e . 159 A.4 Numerical integration points for t r i a n g l e . 15 9 A.5 Complete cubic q u a d r i l a t e r a l . 164 A. 6 Constrained cubic q u a d r i l a t e r a l . 164 A.7 Cantilever with parabolic end load. 196 A.8 Stra i n energy convergence for cant i l e v e r . 197 A. 9 Stresses i n cant i l e v e r at cross-section 12 i n . from support for 4 x 16 g r i d . 198 A.10 One quadrant of p a r a b o l i c a l l y loaded square plate. 199 xiv FIGURE TITLE PAGE A.11 S t r a i n energy convergence f o r p a r a b o l i c a l -l y loaded square p l a t e . 2 00 A.12 One h a l f of s h o r t deep beam. 202 A.13 S t r a i n energy versus g r i d s i z e f o r s h o r t deep beam. 20 3 A.14 S t r e s s e s i n s h o r t deep beam at c r o s s - s e c t i o n 3 i n . from midspan f o r 8 x 8 g r i d . R e s u l t s are compared w i t h an approximate s e r i e s s o l u -t i o n . 204 C . l Stud i n t e r a c t i o n diagram 224 C.2 Moment-curvature diagram 224 xv DEDICATED TO MY PARENTS AND TO MY WIFE ACKNOWLEDGEMENT OF THEIR MANY SACRIFICES ON MY BEHALF xvi ACKNOWLEDGEMENT The w r i t e r i s i n d e b t e d t o Dr. R.A. S p e n c e r f o r h i s a d v i c e and g u i d a n c e and f o r h i s p e r m i s s i o n t o r e f e r t o some r e s u l t s f r o m h i s own l a b o r a t o r y t e s t s w h i c h a r e , as y e t , u n p u b l i s h e d . S p e c i a l t h a n k s go t o t h e w r i t e r ' s w i f e , Gaye, f o r h e r e n t h u s i a s t i c s u p p o r t and e n c o u r a g e m e n t , and f o r h e r p a t i e n t t y p i n g o f t h i s t h e s i s . The r e s e a r c h was f i n a n c e d by t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada. T h a n k s a r e due t o M e s s r s . B o r d i g n o n M a s o n r y L t d . who s u p p l i e d t h e c o n c r e t e p a n e l s w i t h c o n n e c t i o n s f o r t h e p r e l i m i n a r y t e s t s . x v i i 1 CHAPTER 1. INTRODUCTION I t h as b e e n c u s t o m a r y i n N o r t h A m e r i c a t o use t h e c o n v e n t i o n a l s k e l e t o n t y p e o f r e i n f o r c e d c o n c r e t e c o n s t r u c -t i o n o f m u l t i s t o r e y b u i l d i n g s f o r b o t h c o m m e r c i a l and d w e l l -i n g p u r p o s e s , i n w h i c h t h e f l o o r s and f r a m e s a r e c a s t i n p l a c e and t h e w a l l p a n e l s f i l l e d i n a t a l a t e r d a t e . S o c i a l and e c o n o m i c p r e s s u r e s have i n d i c a t e d t h a t g r e a t e r p r e f a b r i -c a t i o n i n t h e b u i l d i n g i n d u s t r y i s i n e v i t a b l e , and t h e t e n d -e n c y t o d e p a r t f r o m t h e c o n v e n t i o n a l t y p e o f c o n s t r u c t i o n i s i n c r e a s i n g . T h e r e i s a m a r k e d l y i n c r e a s e d use o f p r e c a s t c o n c r e t e i n b u i l d i n g s , e i t h e r a s p r e c a s t p a n e l c l a d d i n g t o p r e c a s t o r c o n v e n t i o n a l f r a m e s , o r as p r e c a s t l a r g e p a n e l b o x - t y p e b u i l d i n g s i n w h i c h t h e c o n v e n t i o n a l frame i s o m i t t e d a l t o g e t h e r and t h e l o a d s c a r r i e d by t h e p r e c a s t u n i t s them-s e l v e s . T y p i c a l p r e c a s t u n i t s i n a s i n g l e - s t o r e y l a r g e p a n e l b u i l d i n g a r e shown i n t h e e x p l o d e d p e r s p e c t i v e v i e w i n F i g . 1.1/ and a number o f s t r i k i n g e x a m p l e s o f p r e c a s t c o n -s t r u c t i o n may be f o u n d i n R e f . 1. B e c a u s e o f t h e r e l a t i v e l y e a r l y s t a g e o f d e v e l o p m e n t o f p r e c a s t c o n c r e t e c o n s t r u c t i o n , k n owledge o f t h e a n a l y s i s and d e s i g n o f p r e c a s t c o n c r e t e b u i l d i n g s i s l i m i t e d i n some a r e a s . W h i l e q u a n t i t a t i v e i n f o r m a t i o n e x i s t s and r e s e a r c h c o n t i n u e s on t h e e l a s t i c and i n e l a s t i c b e h a v i o u r o f p r e c a s t c o n c r e t e s t r u c t u r a l e l e m e n t s u n d e r b o t h m o n o t o n i c and c y c l i c l o a d i n g , l i t t l e i s known a b o u t t h e b e h a v i o u r , u n d e r s i m i l a r c o n d i t i o n s , o f t h e c o n n e c t i o n s between p r e c a s t e l e m e n t s . M o r e o v e r , t h e l a r g e v a r i e t y o f c o n n e c t i o n s i n w o r l d w i d e us e t o d a y 2 i s an added c o m p l i c a t i o n . The i n f o r m a t i o n t h a t i s a v a i l a b l e c o n c e r n s t h e b e h a v i o u r o f a few t y p e s o f c o n n e c -t i o n s m a i n l y u n d e r m o n o t o n i c l o a d i n g w i t h l i m i t e d i n f o r m a -t i o n f r o m f a t i g u e t e s t s . 3 - 1 5 Some work on p r e c a s t c o n c r e t e p a n e l s t r u c t u r e s 1 6 ~ 2 3 has b e en done, p r i n c i p a l l y i n R u s s i a and J a p a n , and i n c l u d e s t e s t s on f u l l - s c a l e s t r u c t u r e s , u n d e r l a t e r a l l o a d o r u n d e r b l a s t l o a d i n g . T h e s e t e s t s were done on d w e l l i n g s t r u c t u r e s i n t e n d e d f o r mass p r o d u c t i o n , and t h e r e s u l t s c a n o n l y be r e g a r d e d as q u a l i t a t i v e w i t h r e s p e c t t o p r e c a s t p a n e l s t r u c t u r e s i n g e n e r a l . Of p a r t i c u l a r c o n c e r n t o t h e s t r u c t u r a l e n g i n e e r i s t h e e a r t h q u a k e d e s i g n o f p r e c a s t c o n c r e t e b u i l d i n g s , i n t h a t t h e c y c l i c l o a d s w h i c h may be e x p e c t e d d u r i n g t h e l i f e s p a n o f a s t r u c t u r e a r e o f t e n l a r g e r t h a n any o f t h e o t h e r d e s i g n l o a d s . I t i s t o be e x p e c t e d t h a t a s t r u c t u r e w i l l s u f f e r damage u n d e r t h e s e o v e r l o a d s d u r i n g a s e v e r e e a r t h q u a k e . L i m i t a t i o n o f s u c h damage i s a t t e m p t e d by d e s i g n i n g members and c o n n e c t i o n s w h i c h c a n d e f o r m i n e l a s t i c a l l y w i t h o u t f r a c -t u r e , e v e n u n d e r s e v e r a l c y c l e s o f l a r g e d i s p l a c e m e n t r e v e r -s a l s , and s t i l l m a i n t a i n t h e i r u l t i m a t e c a p a c i t i e s . W h i l e f o r some t i m e d e s i g n e r s have been e n s u r i n g t h e d u c t i l i t y and i n t e g r i t y o f p r e c a s t s t r u c t u r a l e l e m e n t s , t h e d e s i g n o f c o n n e c t i o n s f o r s i m i l a r r e q u i r e m e n t s u n d e r e a r t h q u a k e l o a d -i n g i s s t i l l i n i t s i n f a n c y . T h e r e i s no i n f o r m a t i o n a v a i l -a b l e as t o how much d u c t i l i t y i s r e q u i r e d f r o m c o n n e c t i o n s , n o r , on t h e o t h e r hand, w h e t h e r t o e n s u r e t h a t t h e c o n n e c t e d s t r u c t u r a l e l e m e n t s y i e l d i n s t e a d . The l a t e r a l l o a d s i n a p r e c a s t c o n c r e t e s t r u c t u r e a r e p r i n c i p a l l y f u n n e l l e d t h r o u g h t h e c o n n e c t i o n s , and m a i n t e n a n c e o f t h e i r i n t e g r i t y d u r i n g an e a r t h q u a k e i s p a r a m o u n t . As f a r as t h e a n a l y s i s o f s u c h s t r u c t u r e s i s c o n c e r -n e d , t h e r e a r e a number o f co m p u t e r p r o g r a m s , e m p l o y i n g t h e d i s p l a c e m e n t method, w h i c h have been u s e d f o r l i n e a r and non-l i n e a r a n a l y s e s o f r e i n f o r c e d and p r e s t r e s s e d c o n c r e t e frame s t r u c t u r e s u n d e r v a r i o u s l o a d c o n d i t i o n s i n c l u d i n g e a r t h q u a k e l o a d i n g . F u r t h e r e x t e n s i o n s t o i n c l u d e p r e c a s t c o n c r e t e e l e -m e n t s, p a n e l s , c o n n e c t i o n s and j o i n t s i n s u c h p r o g r a m s have been u n d e r t a k e n . 2 " * - 2 6 Beams and col u m n s a r e m o d e l l e d as u s u a l , by l i n e members. P a n e l s a r e m o d e l l e d by f i n i t e e l e m e n t s w i t h a l a r g e number o f d e g r e e s o f f r e e d o m c o n d e n s e d o u t so t h a t o n l y t h e d e g r e e s o f f r e e d o m a t c o n n e c t i o n s and j o i n t s r e m a i n . The c o n n e c t i o n s t h e m s e l v e s a r e m o d e l l e d by n o n - l i n e a r s p r i n g s , t h e p r o p e r t i e s o f w h i c h a r e , as y e t , l a r g e l y u n d e t e r m i n e d . T h u s , one o f t h e p r e r e q u i s i t e s f o r i n c r e a s i n g t h e 5 q u a n t i t a t i v e knowledge o f t h e a n a l y s i s and d e s i g n o f p r e c a s t c o n c r e t e b u i l d i n g s i s a d e t a i l e d f a m i l i a r i t y w i t h t h e b e h a -v i o u r o f c o n n e c t i o n s u n d e r d i f f e r e n t l o a d i n g c o n d i t i o n s . I t i s p r o b a b l e t h a t w i t h i n c r e a s e d p r e f a b r i c a t i o n i n t h e b u i l d -i n g i n d u s t r y , s t a n d a r d c o n n e c t i o n t y p e s w i l l be more u n i v e r -s a l l y a d o p t e d and w i l l t h u s r e d u c e t h e l a r g e v a r i e t y o f c o n -n e c t i o n s t h a t w i l l p o s s i b l y y e t be i n v e s t i g a t e d . L o c a l l y , many c o n n e c t i o n s e m p l o y i n g one common d e n o m i n a t o r - t h e f u s i o n w e l d e d h e a d e d s t u d - have e n j o y e d p r o m i n e n t f a v o u r i n r e c e n t y e a r s . A t y p i c a l h e a d e d s t u d c o n n e c t i o n i s shown i n F i g . 1 .2 . T h i s i n v e s t i g a t i o n d e s c r i b e s b r i e f l y t h e l a b o r a t o r y t e s t i n g o f h e a d e d s t u d c o n n e c t i o n s f o r p r e c a s t c o n c r e t e p a n e l s u n d e r m o n o t o n i c and c y c l e d s h e a r l o a d i n g , and summarizes t h e l o a d d e f l e c t i o n b e h a v i o u r o f t h e c o n n e c t i o n s . T h r e e d i f f e r -e n t mechanisms whereby a c o n n e c t i o n may c a r r y s h e a r l o a d s a r e i n v e s t i g a t e d e x p e r i m e n t a l l y and m o d e l l e d m a t h e m a t i c a l l y . The m o d e l s o f t h e s e mechanisms a r e s y n t h e s i z e d i n t o a c o m p u t e r m o d e l w h i c h g i v e s an i n s i g h t i n t o t h e b e h a v i o u r o f h e a d e d s t u d c o n n e c t i o n s u n d e r l o a d . From t h e d a t a t h u s a s s e m b l e d , s i m p l i f i e d m a t h e m a t i c a l p r o c e d u r e s a r e p r e s e n t e d as a i d s i n t h e a n a l y s i s , and a d d i t i o n a l a i d s i n t h e d e s i g n o f s u c h c o n n e c -t i o n s . F ig . 1.2 Typica l headed stud connect ion showing two common stud c o n f i g u r a t i o n s and the ease with which minor imperfect ions and misal ignments may be a c c o m m o d a t e d . CHAPTER 2. CURRENT INFORMATION ON HEADED STUD CONNECTIONS 2.1 E x i s t i n g s t a t i c l o a d t e s t i n f o r m a t i o n W elded headed s t u d c o n n e c t o r s were d e v e l o p e d f o r u s e i n c o m p o s i t e c o n s t r u c t i o n and a r e w i d e l y a c c e p t e d i n t h i s a p p l i c a t i o n . A summary o f t h e r e s e a r c h i n t h i s f i e l d and r e f e r e n c e s t o most o f t h e i n v e s t i g a t i o n s may be f o u n d i n R e f . 46. One t y p e o f t e s t , p e r t i n e n t t o t h i s i n v e s t i g a - . t i o n , w h i c h has been u s e d by many i n v e s t i g a t o r s , i s t h e d i r e c t s h e a r t e s t on p u s h - o u t s p e c i m e n s ' * ' 6 ' 1 1 o f t h e t y p e shown i n F i g . 2.1. M o s t o f t h e s e t e s t s were s t a t i c , w i t h a few u n l o a d i n g and r e l o a d i n g b r a n c h e s i n t h e l o a d - d e f l e c t i o n c u r v e s i n some c a s e s . E a r l i e r t e s t s 7 were p r i m a r i l y c o n -c e r n e d w i t h t h e l i m i t a t i o n o f s l i p between s t e e l and c o n c r e t e w i t h t h e r e s u l t t h a t d a t a r e c o r d i n g was t e r m i n a t e d a t a low d e f l e c t i o n , as i n F i g . 2.2 f o r i n s t a n c e . I n v e s t i g a t o r s l a t e r became more c o n c e r n e d w i t h u l t i m a t e c a p a c i t i e s o f t h e c o n n e c -t o r s and r e s u l t s were r e c o r d e d t o f a i l u r e as t y p i c a l l y i n d i -c a t e d f o r r e s u l t s by O l l g a a r d e t a l . 1 1 i n F i g . 2.3. Mc M a c k i n e t a l . 1 2 c o n d u c t e d s i x t y c o m b i n e d s h e a r and t e n s i o n t e s t s on i n d i v i d u a l s t u d s o f v a r i o u s s i z e s , and d e -8 Deflections measured b e t w e e n c o n c r e t e a n d s t e e l b e a m . o CM .„?|D.'Q| 8 n - - i i Sec t i on A - A Fig.2.1 Typica l push-out spec imen. D. a Z) -1—' Q. O c 25 20 15 10 + 0 9/ *j y °/ f _ Thiir l imann Viest BGnjamin Spec. Quit No. (p.s.i.) (kip) 1 5080 ® 24.2 6A2 3870 0 32.0 6B2 A 32.5 6A/4 3360 0 21.2 6B/* 3260 A 22.5 2B 3650 0 8A /4200 1 B I 3650 • + + 0 0.02 0.06 0.08 0.04 D e f l e c t i o n (in.) Fig. 2.2 Shear l o a d - d e f l e c t i o n curves for 3/4 in. dia. x 4 in. studs 7 . 30 10 0 o' f 23 Norma l we ight c o n c r e t e L i gh t w e i g h t c o n c r e t e 0 0.1 0.2 D e f l e c t i o n (in.) 0.3 L o a d - d e f l e c t i o n curves to u l t imate for '4 in. dia. x 4 in. studs 3// C U J1 r i v e d an i n t e r a c t i o n c u r v e u P' uc where P u V u P' uc V uc + v u v uc £ 1 (2.1) A p p l i e d u l t i m a t e t e n s i o n . A p p l i e d u l t i m a t e s h e a r . U l t i m a t e p u l l o u t s t r e n g t h o f s t u d c o n t r o l l e d by c o n c r e t e . U l t i m a t e c o n c r e t e s h e a r c a p a c i t y o f s t u d . Chadha r e c e n t l y c o m p l e t e d a s e r i e s o f t e s t s on s t u d c o n n e c t i o n s u n d e r c o m b i n e d s h e a r and moment. 2.2 C u r r e n t D e s i g n P r o c e d u r e The d e s i g n p r o c e d u r e . c u r r e n t l y recommended by t h e "PCI D e s i g n Handbook" 1 and t h e "PCI M a n u a l on D e s i g n o f Con-n e c t i o n s " 2 7 i s as f o l l o w s : The u l t i m a t e p u l l o u t s t r e n g t h o f a s t u d c o n t r o l l e d by e n c a s i n g c o n c r e t e i s d e t e r m i n e d f r o m (2.2) P' uc where <J> 4 (j) A f o c 0.85 A o f ' c A r e a o f s h e a r c o n e c o n c r e t e c y l i n d e r s t r e n g t h The u l t i m a t e c o n c r e t e s h e a r s t r e n g t h o f a s t u d a t t a c h e d t o a f a c e p l a t e c a n be d e t e r m i n e d by t h e s h e a r f r i c -t i o n c o n c e p t V uc * \ » f s u (2.3) where <j> su 0. 85 Cross-sectional area of stud shank F r i c t i o n c o e f f i c i e n t = 0.9 Ultimate t e n s i l e strength of stud The ultimate t e n s i l e capacity of a stud exclusive of concrete can be calculated from P' us 0.9 b^ ^su (2.4) The ultimate shear capacity of a stud exclusive of concrete can be calculated from V us 0.75 A, f D S U (2.5) The ultimate concrete capacity for combined tension and shear loading of headed studs welded to a faceplate can be determined from the i n t e r a c t i o n equation u P' uc 7, + v u V uc 7, £ 1 (2.6) The ultimate stud capacity exclusive of concrete for combined tension and shear loading should s a t i s f y 2 2 u P' us + V u V us £ 1 (2.7) It should be noted that PCI recommends an additional load factor of 4/3 for the design of t h i s type of connection 2 7 This e f f e c t i v e l y reduces the calculated design loads by 25%. 2. 3 Recent C y c l i c tests on Headed Stud Connections As preliminary exploratory research for t h i s i n v e s t i -gation, s ix headed stud connections of the type shown i n F i g . 1.2 were tested under q u a s i - s t a t i c monotonic and c y c l i c shear loading. A comprehensive report of t h i s research has been published by Spencer and N e i l l e ; 2 8 consequently, only s a l i e n t points w i l l be included here. Details of the six connections tested are shown i n F i g . 2.4. Each connection was cast i n a 2 x 4 f t . panel x 8 i n . thick and tested i n the r i g shown i n F i g . 2.5. The loading yoke was bolted with high-strength f r i c t i o n grip bolts to a 5 x 12 x 5/8 i n . faceplate which had been welded to the connec-t i o n angle. A displacement-controlled hydraulic jack applied a v e r t i c a l c y c l i c force to the s t e e l plate v i a the loading yoke which prevented neither pullout of the studs nor rotation of the connection about a horizontal axis. Deflections of the connection were measured between the top of the 5/8 i n . plate and the top of the concrete panel. Connection A.1 was loaded monotonically to f a i l u r e . Its load-deflection curve i s shown i n F i g . 2.6. The remaining connections were subjected to c y c l i c loading at frequencies 14 Connec t i on D e t a i l s of s t u d s , c o n n e c t i o n angles and pane l re in fo rcement . A'1 A2, A 3 B 1 B 2 B 3 CM / X3WLx12 #Long 3 ^ 3 # x 3 t f L x 1 2 ' L o n g 3 ' x 2 ' x 3 # L X 1 0 # L o n g MHO.* S ^ W L x l o ' L o n g J ^ L 5' m 5-f s u = 60 000p.s.i. fcy = A 600p.s.i. • Denotes no. U grade 60 bar. Fig.2.4 De ta i l s of c onnec t i on s t e s ted . 15 30 20 10 -H 0 0 Fig. 2.6 0.1 0.2 . 0 . 3 0.4 0 Def l e c t i on (i n.) L o a d - d e f l e c t i o n curve - c o n n e c t ion A t ranging from 0.01 to 0.02 Hz., with a few cycles i n the e l a s -t i c range followed by cycles of increasing amplitude up to f a i l u r e . Typical load-deflection loops for connection A.3 are shown in F i g . 2.7. A summary of the ultimate loads and design loads for the connections i s given i n Table I. The design ultimate loads for these connections, c a l -culated by PCI design procedures but i n which the 4/3 load factor was omitted, were found to be conservative. Under mono-tonic loading, connection A.1 sustained a high load after y i e l d i n g , equal almost to the maximum load at f i r s t y i e l d , for a d e f l e c t i o n up to 14 times that at f i r s t y i e l d . The remain-ing connections tended to some sort of r a p i d l y degrading y i e l d strength envelope under c y c l i c loading with gradually increased amplitude. The c y c l i c load-deflection loops exhibited a s t a b i l i t y l i m i t . For loads within t h i s l i m i t , the loops remained stable, and for loads outside the l i m i t they continued to degrade. Both the y i e l d strength envelopes and the s t a b i -l i t y l i m i t s were a r b i t r a r i l y chosen i n F i g . 2.7. P r i o r to t h i s research, Al-Yousef 4 8 performed reversed c y c l i c loading tests on a number of push-out type specimens. Many of the l o a d - s l i p curves from his r e s u l t s bear s t r i k i n g resemblances to the load-deflection curves of the above i n -vestiga t i o n . 1 1 + 30 Fig.2.7 Load -de f l ec t i on l oop s - connect ion A3 19 TABLE I SUMMARY OF LOAD DATA FOR CONNECTIONS Connection Design Ultimate Strength Maximum Load Mode of Fai l u r e Concrete (kip) Steel (kip) Up (kip) Down (kip) A l 27. 2 27.2 35. 4 - Concrete A2 27.2 27.2 31.5 30.2 Stud A3 27.2 27.2 29.3 27.0 Stud Bl 26.4 27.2 31. 8 28.0 Concrete B2 25.5 26.9 30.0 32. 0 Stud B3 25.5 26. 9 23.4* 24.0* Stud * T e s t r e s u l t s u n r e l i a b l e b e c a u s e o f c o n n e c t i o n f a b r i c a t i o n e r r o r . 20 CHAPTER 3. THE SHEAR LOAD-RESISTING COMPONENTS OF A CONNECTION 3 .1 P r o b a b l e Mechanisms From t h e e x p l o r a t o r y c y c l i c t e s t s , d e s c r i b e d b r i e f l y i n t h e p r e v i o u s c h a p t e r , i t was e v i d e n t t h a t b e h a v i o u r o f t h e h e a d e d s t u d c o n n e c t i o n s was complex and t h a t a much more d e t a i l e d i n v e s t i g a t i o n was w a r r a n t e d . A f t e r f u r t h e r s c r u -t i n y , two r o u t e s o f r e s e a r c h emerged: t h e f i r s t was an exam-i n a t i o n o f t h e v a r i o u s mechanisms by w h i c h a t y p i c a l h e a d e d s t u d c o n n e c t i o n t r a n s m i t t e d e x t e r n a l l y - a p p l i e d s h e a r l o a d s t o t h e s u r r o u n d i n g c o n c r e t e . The s e c o n d was a f u r t h e r , more e x t e n s i v e s e r i e s o f m o n o t o n i c and c y c l i c s h e a r l o a d t e s t s on c o n n e c t i o n s i n c o n c r e t e p a n e l s , w i t h e x a m i n a t i o n o f t h e e f f e c t s o f v a r i a t i o n s i n p a n e l r e i n f o r c e m e n t , s t u d and f a c e -p l a t e a r r a n g e m e n t s . The l a t t e r r e s e a r c h i s t h e s u b j e c t o f a s e p a r a t e i n v e s t i g a t i o n and i s n o t i n c l u d e d i n t h i s d i s c o u r s e . O b s e r v a t i o n s o f t h e p r e l i m i n a r y c y c l i c t e s t s i n d i -c a t e d t h a t t h e r e were s e v e r a l p e r t i n e n t p h y s i c a l f e a t u r e s common t o a l l s p e c i m e n s . F i r s t l y , t h e c o n c r e t e a t t h e end o f e a c h c o n n e c t i o n a n g l e f a i l e d i n c o m p r e s s i o n and s p a l l e d o f f as t h e l o a d r e a c h e d a maximum and y i e l d i n g commenced. I t was c o n c l u d e d t h a t t h e end b e a r i n g o f t h e f a c e p l a t e on t h e a d j a c e n t c o n c r e t e p r o b a b l y c o n t r i b u t e d s i g n i f i c a n t l y t o t h e i n i t i a l s t i f f n e s s o f e a c h c o n n e c t i o n . I n t h e t e s t s , e a c h f a c e p l a t e f i n a l l y b r o k e away f r o m t h e s t u d s , o r t h e e n t i r e c o n n e c t i o n p u l l e d o u t o f t h e c o n c r e t e and i t was p o s s i b l e t o see what had h a p p e n e d b e h i n d t h e f a c e p l a t e . A p a r t f r o m an ob-v i o u s i n t e r a c t i o n b etween s t u d s and c o n c r e t e , s t r i a t i o n s and d e p o s i t e d m i l l s c a l e on t h e c o n c r e t e s u r f a c e s b e h i n d t h e f a c e -p l a t e s , a l i g n e d i n t h e d i r e c t i o n o f s h e a r , i n d i c a t e d f r i c t i o n t r a n s f e r b etween t h e f a c e p l a t e s and t h e c o n c r e t e . As a r e s u l t o f t h e s e o b s e r v a t i o n s i t was p o s t u l a t e d t h a t e x t e r n a l s h e a r f o r c e s were t r a n s m i t t e d v i a a c o n n e c t i o n t o t h e s u r r o u n d i n g c o n c r e t e by t h r e e m echanisms: 3.1.1 F r i c t i o n b etween i n s i d e o f f a c e p l a t e and s u r r o u n d i n g c o n c r e t e . 3.1.2 B e a r i n g o f end o f f a c e p l a t e on c o n c r e t e . 3.1.3 I n t e r a c t i o n between s t u d s , f a c e p l a t e and c o n c r e t e by b e a r i n g o f t h e s t u d s on c o n c r e t e and b e n d i n g o f t h e s t u d s . 3.2 L a b o r a t o r y M o d e l s o f Mechanisms F o u r s e t s o f l a b o r a t o r y t e s t s p e c i m e n s were d e v i s e d . E a c h s e t was d e s i g n e d t o i s o l a t e some a s p e c t o f t h e t h r e e s h e a r f o r c e - r e s i s t i n g components t a b u l a t e d a b o v e . A l o n g w i t h e a c h c o n c r e t e s p e c i m e n t h a t was c a s t , s i x 4 i n . d i a m e t e r by 8 i n . c o n c r e t e c y l i n d e r s were c a s t f o r measurement o f c y l i n d e r s t r e n g t h s . A l l s p e c i m e n s and c y l i n d e r s were p r o p e r l y c u r e d f o r a t l e a s t 28 d a y s i n a c u r i n g room. 3.2.1 F r i c t i o n Specimens: These specimens were designed for the invest i g a t i o n of f r i c t i o n between the inside of a faceplate and the adjacent concrete. Details of the three specimens made are given i n F i g . 3.1. Each faceplate was held up against the concrete by two 1/2 i n . diameter s t e e l bars which were welded to the faceplate at one end and to an angle section embedded i n the concrete at the other. These s t e e l bars had a d i f f e r e n t length i n each specimen and were i s o l a t e d from the surrounding concrete with r i g i d p l a s t i c foam possessing l i t t l e bearing strength. The t e n s i l e and bending properties of the bars were measured from a set of t e n s i l e and bending t e s t s , the res u l t s of which appear i n the following chapter. 3.2.2 End bearing Specimens: Six tes t pieces were construc-ted of 1/2 i n . thick faceplates with thicknesses of 1/4, 3/8 and 1/2 i n . bearing on the concrete, as shown i n F i g . 3.2. Again the faceplates were held i n place by two 1/2 i n . d i a -meter s t e e l bars possessing the same properties as the bars used for the f r i c t i o n t e s t s . The bars and the back of each faceplate were i s o l a t e d from the concrete with r i g i d p l a s t i c foam, except for approximately 1/8 i n . of the lower edge and the entire end of the faceplate which were i n contact with the concrete. A t y p i c a l specimen, ready for casting i n concrete, i s shown i n Fi g . 3.3. 3.2.3 Single Studs i n Concrete: The studs for these labo-ratory models were machined from bright bar s t e e l , as opposed 1 7 / '/? d ia. bar 1// Z x 2 Y / 4 L s/g f a c e p l a t e P l a s t i c f o a m 4,6 or 8 anchor bar l eng th r 4;in. Sect ion A-A Fig.3.1 Spec imens for i n v e s t i g a t i o n of f r i c t i o n between f a c e p l a t e and c o n c r e t e . CM A CO F ig . 3.2 Test p ieces for i n ve s t i g a t i n g f a c e p l a t e end bea r i ng . to commercial studs which have a cold-formed head at one end of the shank. The t e n s i l e properties of the s t e e l were the same as those for the bars used in the f r i c t i o n specimens. Three specimens were cast with 6 i n . studs of 1/2, 5/8 and 3/4 i n . diameter (Fig. 3.4) and a further three were cast, a l l with studs of 5/8 i n . diameter and a s p e c i a l s t i f f e n e r at the faceplate, as shown i n F i g . 3.5. The faceplates were a l l of 5/8 i n . plate and were i s o l a t e d from the concrete by a 1/2 i n . gap. During t e s t i n g , the lower ends of the face-plates were supported by load c e l l s instead of the usual second stud. A 1/8 i n . wide s l i t extended from the top of the stud to the top of the concrete over the whole length of the stud. This was provided to allow for l a t e r access to the stud by means of displacement transducer probes so that the deformed shape of the stud could be measured during t e s t i n g . The studs had small dimples d r i l l e d at 5/8 i n . between centres along t h e i r top surfaces to f a c i l i t a t e accurate location of the probes as shown in F i g . 3.4. 3.2.4 Studs i n Tension and Bending. Three t e n s i l e t e s t specimens were assembled by fusion welding three 6 i n . by 5/8 i n . diameter commercial studs, head to end, as shown in Fi g . 3.6. Nine bending specimens were constructed, each with two of the same studs fusion-welded to 5/8 i n . face-plates. The method of t e s t i n g these specimens i s discussed below. 26 Fig. 3.3 I s o l a t i o n of bar s and back of f a cep l a te from c o n c r e t e w i t h p l a s t i c foam. Fig. 3.4 M a c h i n e d studs ready for ca s t i ng in conc re te . 15 c CD •••00 1 7 / '/8 s l i t f o r -probes R \ \ \ \ V 5/g f a cep l a t e S e c t i o n A - A of probes D e t a i l s of s t ud s Fig.3.5 M o d e l s for examina t i on of i n t e r a c t i o n be tween s tud and c o n c r e t e . 19 7 2 i 1 ' 1 I u T e n s i l e s p e c i m e n s P Fig.3.6 S p e c i m e n s f o r te s t ing studs in t e n s i o n and bending. 3. 3 T e s t i n g A p p a r a t u s A t e s t i n g r i g d e s i g n e d t o accommodate t h e t h r e e d i f f e r e n t t y p e s o f c o n c r e t e s p e c i m e n s was c o n s t r u c t e d as shown i n F i g . 3.7. E a c h c o n c r e t e s p e c i m e n was f i r m l y c l a m p e d i n p o s i t i o n on t h e t e s t i n g b e d and was t e s t e d by t h e a p p l i c a t i o n o f a v e r t i c a l downward l o a d t o t h e f a c e -p l a t e . B e c a u s e o f t h e s i z e o f t h e l o a d i n g h e a d o f t h e t e s t i n g m a c h i n e , i t p r o v e d t o be t o o d i f f i c u l t t o a p p l y l o a d s d i r e c t l y t o e a c h f a c e p l a t e . An i n t e r m e d i a t e l o a d i n g beam was u s e d i n s t e a d , one end o f w h i c h r e s t e d on a b e a r i n g p a d b o l t e d t o t h e f a c e p l a t e . The beam was l o a d e d a t one-t h i r d o f i t s s p a n , p r o d u c i n g a r e a c t i o n on t h e f a c e p l a t e e q u a l t o t w o - t h i r d s o f t h e l o a d a p p l i e d by t h e t e s t i n g m a c h i n e . D e f l e c t i o n s o f t h e f r i c t i o n and end b e a r i n g s p e c i -mens were m e a s u r e d by means o f a d i s p l a c e m e n t t r a n s d u c e r b e t w e e n a clamp n e a r t h e b o t t o m o f t h e c o n c r e t e a n d t h e f a c e p l a t e , a s shown i n F i g . 3.8. L o a d - d e f l e c t i o n c u r v e s f o r t h e s e s p e c i m e n s were p l o t t e d d i r e c t l y on an X-Y p l o t t e r . The d e f l e c t e d s hape o f e a c h s i n g l e s t u d i n c o n c r e t e was m e a s u r e d by means o f a b a t t e r y o f d i s p l a c e m e n t t r a n s -d u c e r s mounted i n two t i e r s i n t h e s p e c i a l frame shown i n F i g . 3.9. One o f t h e l o a d c e l l s p r e v i o u s l y m e n t i o n e d c a n a l s o be s e e n i n t h i s f i g u r e . D u r i n g t e s t i n g , a l o a d - d e f l e c -t i o n c u r v e f o r e a c h s t u d was p l o t t e d c o n t i n u o u s l y on an X-Y p l o t t e r . The d e f l e c t i o n f o r t h i s p l o t was r e a d f r o m t h e Fig. 3.7 Apparatus for test ing s ingle s tuds in conc re te , end bearing and f r i c t ion specimens. o Fig. 3.8 Measurement of d i s p l a c e m e n t s of f r i c t i o n and end b e a r i n g s p e c i m e n s . F ig. 3.9 Set of d i s p l a c e m e n t t r a n s d u c e r s f o r m e a s u r i n g d e f l e c t e d shapes of s t u d s . t r a n s d u c e r c l o s e s t t o t h e f a c e p l a t e . A t r e g u l a r i n t e r v a l s a l l l o a d and d i s p l a c e m e n t v o l t a g e s were r e a d by means o f an e l e c t r o n i c d a t a a c q u i s i t i o n s y s t e m . T h e s e r e a d i n g s were t r a n s m i t t e d t o a m i n i - c o m p u t e r w h i c h p r o c e s s e d them and a l -most i m m e d i a t e l y r e t u r n e d a l l m e a s u r e d l o a d s and d e f l e c t i o n s v i a t h e t e r m i n a l t y p e w r i t e r shown i n t h e f o r e g r o u n d i n F i g . 3.10. A p l o t o f t h e d e f l e c t e d shape o f t h e s t u d c o u l d a l s o be p r i n t e d a t w i l l on t h e t y p e w r i t e r a t any s t a g e o f t h e t e s t . i The a p p a r a t u s , i n F i g . 3.11, u s e d f o r t h e b e n d i n g t e s t s on c o m m e r c i a l s t u d s , was an a d a p t a t i o n o f t h e r i g shown i n F i g . 2.5. E a c h s e t o f s t u d s and f a c e p l a t e had a s t e e l s p a c e r p l a c e d between t h e s t u d s and was f i r m l y c l a m p e d t o a b l o c k o f c o n c r e t e as shown i n F i g . 3.12. The s p a n o v e r w h i c h t h e s t u d s were f r e e t o b end was v a r i e d between 1/2, 1 and 1-1/2 i n . D e f l e c t i o n s were m e a s u r e d between t h e t o p o f t h e c l a m p i n g d e v i c e and t h e t o p o f e a c h f a c e p l a t e . M o n o t o n i c o r q u a s i - s t a t i c c y c l i c l o a d s were a p p l i e d by a d i s p l a c e m e n t -c o n t r o l l e d h y d r a u l i c j a c k t h r o u g h a l o a d i n g y o k e b o l t e d t o t h e f a c e p l a t e . L o a d - d e f l e c t i o n c u r v e s were a g a i n p l o t t e d c o n t i n u o u s l y on an X-Y p l o t t e r . Fig. 3.10 Data a c q u i s i t i o n and r e c o r d i n g i n s t r u m e n t s used in t e s t s on s tuds in c o n c r e t e . F i g . 3.11 A p p a r a t u s F i g . 3.12 Studs f i rm l y for bending t e s t s on studs. c l a m p e d to c o n c r e t e block. 35 CHAPTER 4. f LABORATORY TESTS AND RESULTS 4 .1 M a t e r i a l s T e s t s T e s t s on t h e m a t e r i a l s c o n s t i t u t i n g t h e l a b o r a t o r y s p e c i m e n s a r e c o l l e c t i v e l y d e s c r i b e d b e l o w p r i o r t o p r e s e n -t a t i o n o f t e s t s on t h e s p e c i m e n s . 4.1.1 C o n c r e t e : E a c h c o n c r e t e c y l i n d e r s t r e n g t h m e a s u r e -ment was t h e a v e r a g e o f c o m p r e s s i v e s t r e n g t h s o f s i x 4 i n . d i a m e t e r by 8 i n . c y l i n d e r s . M easurements on t h e s e c y l i n -d e r s were made w i t h a s t a n d a r d l a b o r a t o r y t e s t i n g m a c h i n e w h i c h h a d a f i x e d l o w e r p l a t e n and a s p h e r i c a l s e a t on t h e u p p e r p l a t e n . The r a t e o f l o a d i n g f o r e a c h s t r e n g t h t e s t was a p p r o x i m a t e l y 1800 p . s . i . p e r m i n u t e . R e s u l t s o f t h e s e t e s t s a r e p r e s e n t e d i n T a b l e I I . 4.1.2 S t e e l : T h r e e s t a n d a r d t e n s i l e t e s t s p e c i m e n s were m a c h i n e d f r o m s a m p l e s o f t h e s t e e l u s e d f o r t h e a n c h o r b a r s and m a c h i n e d s t u d s . S t r a i n s were m e a s u r e d by means o f an a v e r -a g i n g e x t e n s o m e t e r - o v e r a gauge l e n g t h o f 2 i n . and s t r e s s -s t r a i n c u r v e s were d i r e c t l y p l o t t e d on an X-Y r e c o r d e r . TABLE II CONCRETE STRENGTHS OF LABORATORY SPECIMENS Specimen Description Specimen Number Pr i n c i p a l Dimension Cylinder Strength (p.s.i.) S p e c i f i c Mean Standard Deviation Weight (p.c.f.) F r i c t i o n between F l 4 i n . span 6192 210 153. 3 back of faceplate F2 6 i n . span 6165 502 155. 0 and concrete F3 8 i n . span 5663 239 154.1 Bl 1/4 i n . faceplate 4593 121 151.1 End bearing of B2 1/4 i n . faceplate 4810 305 153. 3 faceplate on B3 3/8 i n . faceplate 4926 146 153. 3 concrete B4 3/8 i n . faceplate 4933 86 152.8 B5 1/2 i n . faceplate 4812 96 151. 8 B6 1/2 i n . faceplate 4955 216 153.7 SI 1/2 i n . dia. stud 7704 165 154. 3 Interaction between S2 5/8 i n . dia. stud 6775 243 153.3 stud and S3 3/4 i n . dia. stud 6910 . 223 153. 9 surrounding S4 5/8 i n . dia. stud with s t i f f e n e r 5800 349 154.0 concrete S5 II II 5211 427 154. 7 S6 II II 4674 257 154. 8 A l l specimens exhibited a well-defined y i e l d point and f r a c -ture occurred at strains of about 20%. Relevant properties measured i n these tests are presented i n Table I I I . Bending tests were conducted on two 1/2 i n . diameter bars for l a t e r comparison with a computer model. Load versus d e f l e c t i o n curves from these tests appear i n F i g . 4.1. 4. 2 Tests on F r i c t i o n Specimens Each f r i c t i o n specimen was put through several load-ing and unloading loops as shown i n the load-deflection curves i n Figs. 4.2 to 4.4. The te s t i n g apparatus permitted loading i n one d i r e c t i o n only. Consequently, each loop was completed by removing the load e n t i r e l y and moving the face-plate back to zero displacement by means of a small hydraulic jack applied between the bottom of the faceplate and the bed of the t e s t i n g machine. The jack was then removed and re-loading commenced v i a the t e s t i n g apparatus. Reverse loads applied by the jack were not measured. Each te s t was con-ducted slowly and took approximately half-an-hour to complete. 4.3 Tests on End Bearing Specimens Deflections of these specimens were monotonically increased to a maximum of about 0.4 i n . , a f t e r which the load was completely removed. A l l load-deflection curves showed a marked drop i n load at a d e f l e c t i o n of between 0.05 and 0.12 i n . as shown i n Figs. 4.5 to 4.7. Simultaneously with t h i s drop i n load the concrete at the end of the faceplate TABLE III TENSILE STRENGTH OF STEEL SAMPLES Specimen Yi e l d Stress Ultimate Stress E l a s t i c Modulus Number (p. s . i . ) (p.s. i . ) (p. s .i. ) 1 67 800 74 000 29. 71 x 10 6 2 67 900 73 500 30.08 x 10 6 3 68 300 74 000 30. 03 x 10 6 z A = = = = z A A >© — AW 4? I! p 1 \ Jl // if 5/2 in. - * l n> 3=— T " // J - II ! 1 1 0 0.1 0.2 0.3 0.4 Q5 0.6 0.7 D e f l e c t i o n A (in.) Fig. 4.1 Bend ing test on/2 in. dia. bars. 0 0.1 0.2 Def lee t ion (in.) 0.3 O.U 0.5 Fig.4.4 L o a d - d e f I e c t i o n curves for f r i c t i o n s p e c i m e n F3 . 10 8 6 2 0 Bea - ing t h i c k n e s s = 3/8in. ^ B4 / 1 II ^  Bh I B3 ^ 1 f r-—ZTZ.- t^i J 0 01 0.2 0.3 0M 0.5 Def I ect ion (in.) g. 4.6 Load - def I ect ion curves for e n d - b e a r i n g s p e c i m e n s B 3 & B4. 0 0.1 0.2 0.3 0.4 0.5 Def l e c t i on (in.) 4.7 Load - def l e c t ion cu rves for end -bea r i n g s pec imen s B 5 & B 6 . broke away, as shown for a t y p i c a l specimen i n F i g . 4.8. After t h i s concrete f a i l u r e , the load gradually b u i l t up again, probably because of f r i c t i o n between the lower end of the faceplate and the remaining concrete. Each test took approximately twenty minutes to complete. 4. 4 Tests on Single Studs i n Concrete Results of these tests are presented i n Figs. 4.9 to 4.20. The load on each stud was reduced to zero sev-e r a l times to produce loading and unloading branches i n the load-deflection curve which was recorded d i r e c t l y on an X-Y p l o t t e r . The load transducers, previously shown in F i g . 3.7, gave an i n d i r e c t measurement of the tension i n the stud. Measurements of stud tension were made at in t e r v a l s throughout each test and are included i n the load-d e f l e c t i o n curves. Tension measurements for stud S2 are omitted because i t was discovered aft e r the test that mea-surements from one of the load c e l l s were e r r a t i c , probably due to a poor e l e c t r i c a l connection. Included with each load-deflection curve are four deflected shapes of the stud which were measured at d i f f e r e n t stages of the test. I t should be noted that the scales on the d e f l e c t i o n axes of these graphs become progressively coarser with increased d e f l e c t i o n . Tests on studs SI to S3 indicated that the 1/2 i n . i s o l a t i o n gap between faceplate and concrete was probably 47 F ig . 4.8 Typ i ca l c o n c r e t e f a i l u r e at end of f a c e p l a t e . J 1 r "* — ~ F / - ^ . — ' ^ — — ^ Si l.b / / 1 i < i * • • CL 1 n • I ' 1 • / l • l l 11 / £ — S t u d t e i / I; n s i o n f<?' IF' I. u c o V) c / * < i ' / *' 1 \\ \\ 0 ;/ /< V? in. d ia . s tud 1' /" /'/ axial t( n 1 • F ]" 1 a 1 . il i I a i 1 I 1 /I i i /I /'/ \ I ( ^ ... -i—» o-0 0.05 0.1 0.15 0.2 0.25 Def I ec t ion (in.) Fig. 4.9 L o a d - d e f l e c t i o n c u r v e for stud S1. D i s t a n c e f r o m c o n c r e t e f a c e (in.) 0 . —I °*-o=fco=»o—O-KD-2 4 6 L o a d = 2.0 kip. 0-04-0-0-040-4 6 0 ^ 2 L o a d = 4.0 k i p . 7 ^ — 1 Q - o — Q j - e - o — Q 4 0 . lO? 2 4 6 L oad = 5.06 k ip . 2 4 L o a d = 5.32 k ip Fig.4.10 D e f l e c t e d shape of s tud S1 51 D i s t a n c e f r o m c o n c r e t e f a c e (in.) c o (_> CD Q 4 6 L o a d = 2.0 k ip . CT-10 / 2 L o a d = 6.04 k i p . 4 6 0 0 / 2 4 6 L oad = 7.5 k ip. L o a d = 8.0 kip. •08-O-I Fig. 4.12 D e f l e c t e d shape of s tud S2. D i s t a n c e f r o m c o n c r e t e f a c e (in.) 4 6 4 6 L o a d = 12.65 k ip . Q H 3 — 0 - „ Q 4 = Q ~ O 2 4 L o a d = 13.78 ki F i g . 4.14 D e f l e c t e d shape of s tud S D i s t a n c e f r o m c o n c r e t e f a c e (in.) 0 410=0-0^.©. 4 6 0014-} : © = . © - © 4 0 _ 4 ' 6 0 0 2 4 =0—©4©— ^2 4 6 ^{<-0—Q—,Q^Q_ © 0 / 2 4 L o a d =13.31 k i Fig.4.16 D e f l e c t e d shape of s tud S4 D i s t a n c e f r o m c o n c r e t e f a c e (in.) 57 c o - i — ' Q _^0=0—Of-O-O—O-fO-'2 4 6 p L o a d =2.0 k i p . 0}-0=0-Of-0-O—040— / 2 4 6 P L o a d = 6.97kip. d^°^°^04-o-o-o4 o— 0 / 2 4 6 L o a d =11.0kip. 0 p 2 4 6 Load= 13.13 kip. F i g . 4.18 D e f l e c t e d shape of s tud S5. F ig . 4.19 L o a d - d e f l e c t i o n cu r ve for s tud S6. D i s t a n c e f r o m c o n c r e t e f a c e (in.) •A 10 p 2 U Load = 1.94 kip. 0-fO— 6 003+ o -o -o L o a d = 6.78 kip. .0- 1 u ^ Q - o - i t t — •o-fo-4 6 Load = 11.18 k ip . 2 U L oad =12 .21 k i p Fig. 4.20 D e f l e c t e d shape of s tud S6. u n r e a l i s t i c . Studs S4 to S6 were thus manufactured with a s t i f f e n e r i n the 1/2 i n . gap, F i g . 3.5. The s t i f f e n e r probably formed a s t r e s s - r a i s e r because studs S4 to S6 fractured, whereas studs SI to S3 remained i n t a c t for l a r -ger d e f l e c t i o n s . Stud S5 was ac c i d e n t a l l y preloaded with approximately 8 kip. , during which no measurements were re-corded. The preload was removed and t e s t i n g commenced i n the usual manner. The e f f e c t s of the preload are r e f l e c t e d i n the deflected shapes, which are d i f f e r e n t from those of studs S4 and S6, and i n the load-deflection curve which exhibits a s l i g h t s t i f f e n i n g i n the f i r s t load branch. 4.5 Tensile and Bending Tests on Studs 4.5.1 Tensile Tests: Stress versus s t r a i n curves from these tests appear i n F i g . 4.21. Strains were measured over a gauge length of 2 i n . situated c e n t r a l l y on the shank of the middle stud. The welds i n a l l three specimens re-mained i n t a c t while fracture occurred i n one of the stud shanks. In each case necking of the stud shank occurred outside of the extensometer gauge length and, as a r e s u l t , the unloading portion of the s t r e s s - s t r a i n curve could not be measured. The r e s u l t s for the three specimens show greater v a r i a b i l i t y than i s normally expected for t e n s i l e tests on s t e e l . This i s a t t r i b u t a b l e to the probable lack of alignment and c o n c e n t r i c i t y of the three studs comprising each t e s t specimen. 70 60 50 40 30 >— o — © — o _ _ 0 — 0 — -. _ - — — 0-^A^Q' 1 1/ 1 o 1 7 o Spec A :imen ST 1 ST 2 •" G / o / ST3 / 0 - ! 1 I 1 —1 1-0 0.25 0.5 0.75 S t r a i n (%) 1.0 1.25 1.5 4.21 S t r e s s - s t r a i n curves f rom tens i le t e s t s on studs. 4.5.2 Bending T e s t s : Groups of t h r e e specimens were t e s -ted w i t h spans of 1/2, 1 and 1-1/2 i n . , F i g . 3.6. D e f l e c -t i o n s of the f i r s t specimen of each group were i n c r e a s e d m o n o t o n i c a l l y u n t i l a l a r g e d e f l e c t i o n had been reached, when the l o a d was reduced t o zero, as shown i n F i g s . 4.22, 4.25 and 4.28. No f r a c t u r e s o c c u r r e d i n the m o n o t o n i c a l l y -loaded specimens. The second specimen of each group was s u b j e c t e d t o q u a s i - s t a t i c c y c l i c l o a d i n g a t f r e q u e n c i e s of the order of .01 Hz., w i t h an i n c r e a s e i n amplitude i n each c y c l e u n t i l f r a c t u r e o c c u r r e d , F i g s . 4.23, 4.26 and 4.29. The f i n a l member of each group was s i m i l a r l y s u b j e c t e d to c y c l i c l o a d -i n g , but the amplitude was kept c o n s t a n t f o r t h r e e t o f i v e c y c l e s b e f o r e the next i n c r e a s e , F i g s . 4.24, 4.27 and 4.30. Without e x c e p t i o n , f r a c t u r e s i n the c y c l i c a l l y - l o a d e d s p e c i -mens o c c u r r e d i n the f u s i o n welds between stud and f a c e p l a t e , as shown f o r a t y p i c a l specimen i n F i g . 4.31. A d e g r a d a t i o n of l o a d - c a r r y i n g c a p a c i t y may be n o t i c e d i n a l l of the c y c l i c t e s t s , which i n c r e a s e s w i t h the amplitude of the c y c l e . Each monotonic l o a d - d e f l e c t i o n curve possesses a d i s t i n c t i v e t r a n s i t i o n p o i n t (D,P) i n d i -c d o c a t e d i n . F i g s . 4.22, 4.25 and 4.28. The l o a d decrement versus a s s o c i a t e d maximum d e f l e c t i o n i n any c y c l e was nor-m a l i z e d w i t h r e s p e c t t o P and D f o r the same span, and o o p l o t t e d on the graph shown i n F i g . 4.32. A p a r a b o l a was f i t t e d to the data points by the method of least squares 2* AP 2.43 + 8.22 D D. + 93.5 D x 10 -3 (4.1) where AP D = Load decrement i n a cycle. = Maximum d e f l e c t i o n i n the same cycle. = Coordinates of t r a n s i t i o n point on monotonic load-deflection curve of specimen with same span. Equation 4.1 i s incorporated i n the load-deflection model of a connection i n Chapter 7. The cumulative ro t a t i o n of a stud bending specimen under c y c l i c loading i s an approximate i n d i r e c t measure of i t t o t a l energy d i s s i p a t i o n before frac t u r e : Sd 0 = up to fracture (4.2) where 0 = Cumulative r o t a t i o n to fracture. 6d = Faceplate d e f l e c t i o n increment. L = Bending span as defined i n F i g . 3.6 This may be used as an indicator of stud fracture in s t r u c t u r a l analysis programs. The cumulative rotations of the laboratory specimens are l i s t e d i n Table IV. TABLE IV CUMULATIVE ROTATIONS OF STUD BENDING SPECIMENS SUBJECTED TO CYCLIC LOADING Stud Bending Cumulative Rotation Specimen Number to Fracture SB2 3.136 SB3 2. 984 SB5 3.404 SB6 5.338 SB8 3 . 066 SB9 4.814 Mean 3.790 CTi F i g . 4.23 C y c l i c bend ing te s t on s tud s p e c i m e n SB2 . 20 15 10 (D 0 , £ ) Span 1 in. / 0 0.05 0.1 0.15 0.2 0.2 5 0.3 0.3 5 D e f l e c t i o n (in.) Fig. 4.25 Monoton ic bending t e s t on s tud s p e c i m e n SB4 . 00 F i g . 4.26 C y c l i c bend ing t e s t on s tud s pec imen SB5 . 4.27 C y c l i c bending test on stud s p e c i m e n S B 6 2 Span 11/2 in. ' I / 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 D e f l e c t i o n (in.) Fig.4.28 Monotonic bending tes t on s tud s p e c i m e n S B 7 . 12 12 H 1 1 1 1 1 1 1 1 1 H 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 D e f l e c t i o n (in.) Fig.4.29 C y c l i c bending tes t on s tud s p e c i m e n SB8. F ig . 4.30 C y c l i c bend ing t e s t on s tud s p e c i m e n S B 9 . 4.31 Cyc l i c loading cau sed typical f r a c t u r e of studs in fu s ion welds. < 0_c c CD £ i _ u CD "O T 3 CO O •D CD LO ro . , 0 . 5 ^ 1.0 N o r m a l i s e d d e f l e c t i o n F i g . 4.32 N o r m a l i s e d load d e c r e m e n t ver sus n o r m a l i s e d d e f l e c t i o n f rom s tud bending tes t s . Quadra t i c e q u a t i o n f i t t e d by l e a s t squares. CHAPTER 5. ANALYTICAL MODELS Two c o m p u t e r p r o g r a m s were w r i t t e n f o r a n a l y s i s o f t h e l a b o r a t o r y m o d e l s and t o s u b s e q u e n t l y p r o v i d e an a n a l y -t i c a l m o d e l o f a c o m p l e t e c o n n e c t i o n . A d e s c r i p t i o n o f t h e s e p r o g r a m s and t h e a p p r o x i m a t i o n s i n v o l v e d i s p r e s e n t e d b e l o w . 5 .1 A p p r o x i m a t i o n s I d e a l l y , a s t u d i n c o n c r e t e c a n be r e p r e s e n t e d by a l a r g e a s s e m b l a g e o f t h r e e - d i m e n s i o n a l f i n i t e e l e m e n t s . P r e -s e n t - d a y l a r g e c o m p u t i n g f a c i l i t i e s c a n p r o b a b l y p r o v i d e s u f f i c i e n t c o m p u t e r memory t o accommodate t h e l a r g e number o f d e g r e e s o f f r e e d o m i n v o l v e d . I n t h i s i n v e s t i g a t i o n t h e n o n - l i n e a r m a t e r i a l b e h a v i o u r o f a c o n n e c t i o n i s most i m p o r -t a n t . U n f o r t u n a t e l y t h e i t e r a t i v e s o l u t i o n s and t h r e e - d i m e n -s i o n a l c o n s t i t u t i v e r e l a t i o n s h i p s f o r c o n c r e t e and s t e e l , w h i c h a r e n e c e s s a r y i n n o n - l i n e a r a n a l y s e s , a r e as y e t e i t h e r t o o e x p e n s i v e , o r b eyond t h e s c o p e o f t h r e e - d i m e n s i o n a l f i n i t e e l e m e n t s . W i t h t h i s i n m i n d , a s i m p l e r i d e a l i z a t i o n i s 77 considered which i s modelled along the l i n e s of a beam on Winkler springs, 2 9 as depicted i n Fig . 5.1. A li n e a r e l a s -t i c form of t h i s analysis has previously been used for dowel j o i n t s i n concrete pavements 4 9 and channel shear connectors. 5 0 The springs have non-linear load-deflection properties and simulate concrete around a stud while the beam - which repre-sents a stud - incorporates the e l a s t o - p l a s t i c behaviour of s t e e l . The properties of each concrete spring are determined from a non-linear plane stress f i n i t e element analysis of a s l i c e of concrete which i s at r i g h t angles to the stud. This approximation ignores any d i r e c t or shear stress transfer from one concrete s l i c e to another. The t r i a x i a l state of stress i n concrete i s ignored and only a b i a x i a l state i n the plane of each s l i c e i s considered. This may be p a r t i c u l a r l y conservative i n the case of t r i a x i a l compression. Richart et a l . 3 0 and Balmer 3 1 have shown by means of t r i a x i a l tests on concrete under confining f l u i d pressure that con-crete strengths may be increased between 4.1 and 7.0 times the unconfined strength. On the other hand, Kupfer et a l . 3 2 recorded increases of up to only 27% above unconfined concrete strengths i n b i a x i a l compressive tests using s t e e l brush platens to avoid any p o s s i b i l i t y of a confining stress i n the t h i r d d i r e c t i o n . v 5.2 Non-Linear Plane Stress F i n i t e Element Program for Pla i n Concrete The concrete c o n s t i t u t i v e r e l a t i o n s h i p s included i n 78 S t u d e n c a s e d in c o n c r e t e . T y p i c a l s l i c e m o d e l l e d by f i n i t e e l e m e n t s gives p r o p e r t i e s of sp r i n g s . •/////////> C o m p u t e r m o d e l of s t u d in c o n c r e t e F i g . 5.1 The a p p r o x i m a t i o n of a stud in c o n c r e t e by means of a beam on Wink le r spr ings. the program are e s s e n t i a l l y the same as those used by Darwin and Pecknold; 3 3 thus only a summary w i l l be given here. Work i n t h i s f i e l d has also been described by Cervenka and Gerstle 3 1*' 3 5 and P h i l l i p s and Zienkiewicz ,36 Numerical solution i s achieved by means of the d i s -placement method using an incremental i t e r a t i v e solution i n which the e f f e c t s of non-linear material behaviour are i n c l u -ded as residual load terms i n a Newton-Raphson process. 3 7 The f i n i t e elements employed are triangular or q u a d r i l a t e r a l with corner nodes only and three degrees of freedom per node, as shown i n F i g . 5.2. Displacements along an element edge are compelled to be l i n e a r i n a d i r e c t i o n p a r a l l e l to the edge, and cubic at r i g h t angles to the edge. This allows the plane stress element to be compatibly combined with a plate bending element, with cubic displacements along i t s edges, i n folded plate and s h e l l applications. For readers who are interested a more complete description of these elements i s provided i n Appendix A. Increments of eith e r nodal displacements or loads are applied, allowing for the simulation of either load- or d i s -placement-controlled laboratory experiments. During the i n -cremental solution, a record of strains and associated p r i n -c i p a l stresses i s kept for the centroid only of each element. Cumulative b i a x i a l s t r a i n s are stored i n the form of "equiva-X r i gh t a n g l e s to edge. 5.2 T r i angu l a r and q u a d r i l a t e r a l f i n i t e e l ement s . CO o lent u n i a x i a l s t r a i n s " 3 3 defined by: e. = } Aa. (5.1) I U / 1 V ' E . l A l l load increments i n which Aa. = incremental change i n stress i n i d i r e c t i o n . t h E. = tangent e l a s t i c modulus i n i d i r e c t i o n . i = 1,2 refers to p r i n c i p a l stress d i r e c t i o n s 1 and 2. In each increment the change i n equivalent u n i a x i a l s t r a i n i s calculated from: Ae . = a. - a. (5.2) xu 1 1 . , new old E. l Stress increments are related to s t r a i n increments by: da-da, d T 12 1-v / E ^ Symmetrical 0 0 h(E1+E2-2/E1E2; de. de, dY 12 (5.3) where t h E^ = tangent e l a s t i c modulus in i d i r e c t i o n . Each d i r e c t i o n has a s t r e s s - s t r a i n model as shown in F i g . 5.3. v = Poisson r a t i o . 0.2 i n tension-tension and com-pression-compression and has a stress-dependent value i n tension-compression and u n i a x i a l compres-sion. 33 Sp=0.145S^ + 0.130Se o Fig. 5.3 Concrete s t r e s s - s t r a i n model adap ted from work by Karsan & J i r s a . CO CO Values of E^ and which are po s i t i v e and greater than a s p e c i f i e d minimum p o s i t i v e value are acceptable i n the program. Values less than t h i s minimum value are re-placed by the minimum p o s i t i v e value, even on the unload-ing curve of the s t r e s s - s t r a i n diagram of Fi g . 5.3, to ensure that the assembled s t i f f n e s s matrix i s always p o s i -t i v e - d e f i n i t e . Errors introduced by t h i s are corrected i n the compilation of the associated residual load vector which i s compiled from the stresses e x i s t i n g i n the element. Thus the square roots used i n equation (5.3) are always r e a l . Cracking i s modelled by reducing the associated e l a s t i c modulus to the minimum p o s i t i v e value and again any stresses across the crack are reduced to zero i n each i t e r a t i o n by the subsequent r e s i d u a l load vector. The following six crack configurations are included i n the program l o g i c : no crack, one crack, f i r s t crack closed, f i r s t crack closed and second crack open, both cracks closed, both cracks open. The b i a x i a l strength envelope of Kupfer and G e r s t l e 3 9 F i g . 5.4, i s imposed upon the p r i n c i p a l stresses by suitably s c a l i n g the stresses i n the concrete model of F i g . 5.3 up or down. 5. 3 Non-Linear Plane Frame Program This program was written to provide the approximate "beam on Winkler springs" model previously mentioned. Again the displacement method i s used with an i t e r a t i v e incremen-t a l solution of the Newton-Raphson37 type and eithe r nodal 1.4 / N o r m a l i s e d p r i n c i p a l s t r e s s R = 1/,/ 1 T c Fig. 5.4 B i a x i a l s t rength enve lope for c o n c r e t e a f te r Kupfer & G e r s t l e CO displacements or nodal loads may be applied. A l l elements in t h i s program are l i n e members; however, f i v e d i f f e r e n t types are included: 5.3.1 Linearly E l a s t i c Members: These members (type 1 i n Fi g . 5.5), are included for the simulation of faceplates and other members which always remain l i n e a r l y e l a s t i c . They may carry a x i a l loads, shear forces and bending moments. 5.3.2 Faceplate F r i c t i o n Simulator: This member i s l i n e a r -l y e l a s t i c , but ca r r i e s a x i a l compressive forces only, (type 2 i n Fig . 5.5). When such a member i s i n compression, a f r i c -t i o n load equal to the c o e f f i c i e n t of f r i c t i o n times the force i n the f r i c t i o n simulator i s applied at the j o i n t and i n the d i r e c t i o n s p e c i f i e d , as demonstrated i n F i g . 5.5. 5.3.3 Bond Spring: The bond spring i s l i n e a r l y e l a s t i c and ca r r i e s a x i a l forces only, but ceases to operate as soon as an a x i a l load of more than a s p e c i f i e d c r i t i c a l value i s im-posed, (type 3 i n Fig. 5.5). 5.3.4 Concrete Spring: This i s a non-linear member that c a r r i e s compressive a x i a l forces only and i s used to simulate faceplate end-bearing (type 4a), and studs.bearing on con-crete (type 4 i n Fig . 5.5). The load-deflection curve for th i s spring has the same algorithm as the concrete model shown i n Fig . 5.3. The scales along the axes are altered 86 © Deno te s f i x e d joint o Deno te s pinned joint F i g . 5.5 F ive types of l i n e members bu i l t into p l ane frame model. max o max. D e f l e c t i o n D Fig. 5.6 L o a d - d e f I ect i on m o d e l for c o n c r e t e spr ings. t o r e f l e c t l o a d v e r s u s d e f l e c t i o n , however, as shown i n F i g . 5.6. The p r o p e r t i e s D , D and P . w h i c h d e f i n e o max max' t h e l o a d - d e f l e c t i o n c u r v e , a r e o b t a i n e d f r o m a p l a n e s t r e s s f i n i t e e l e m e n t a n a l y s i s v i a t h e p r o g r a m a l r e a d y d e s c r i b e d . 5.3.5 N o n - L i n e a r S t u d s : T h e s e a r e n o n - l i n e a r l i n e e l e m e n t s w h i c h may have a n o n - l i n e a r s t r e s s - s t r a i n c u r v e and a s i m i l a r m o m e n t - c u r v a t u r e c u r v e . The s t r e s s - s t r a i n and moment-curva-t u r e d i a g r a m s b o t h have t h e same a l g o r i t h m w h i c h i s a t r i -l i n e a r c u r v e , as shown i n F i g . 5.7. The B a u s c h i n g e r e f f e c t was n o t i n c l u d e d i n t h e m o d e l . The d i s p l a c e m e n t f u n c t i o n o f t h i s e l e m e n t i s l i n e a r a l o n g i t s a x i s and c u b i c a t r i g h t a n g l e s t o i t . The e l e m e n t s t i f f n e s s m a t r i x i s n u m e r i c a l l y i n t e g r a t e d w i t h t h r e e Gauss q u a d r a t u r e i n t e g r a t i o n p o i n t s 3 7 w i t h i n e a c h e l e m e n t . A x i a l f o r c e s a r e c o n s t a n t a l o n g t h e l e n g t h o f t h e e l e m e n t and a r e c o r d o f a x i a l s t r e s s e s and s t r a i n s a t t h e c e n t r e i n t e g r a -t i o n p o i n t i s s t o r e d , w h e r e a s b e n d i n g moments v a r y l i n e a r l y a c r o s s t h e member and a r e c o r d o f moments and c u r v a t u r e s f o r e a c h i n t e g r a t i o n p o i n t i s s t o r e d . A x i a l f o r c e s and b e n d i n g moments a t any c r o s s - s e c t i o n o f an e l e m e n t a r e g o v e r n e d by a f a i l u r e e n v e l o p e w h i c h may be d e r i v e d a s f o l l o w s : W i t h r e f e r e n c e t o F i g . 5.8, t h e a r e a i n b e n d i n g com-p r e s s i o n o r t e n s i o n i s g i v e n by: F i g . 5.7 T r i l i n e a r h y s t e r e s i s l oops used for both s t r e s s - st ra in and moment - cu r va tu re r e l a t i o n s h i p s for s tuds. 00 co A = 2r' whe re r I " \ s i n 2 e o ~ e o (5.4) u0 2a radius of stud a r c s i n a r depth of a x i a l compression or tension zone The bending moment lever arm i s given by: 3 2y = 2r~ cos 0 0 (5.5) 3A The ultimate a x i a l force acting at a cross-section i P = (irr - 2A) f su (5.6) where f = ultimate s t e e l strength su ^ The ultimate bending moment acting at the cross-section i s : M = 2Ay f su (5.7) When no bending moment i s present, the ultimate axia force i s given by: P , = irr f p i su (5.8) S i m i l a r l y , when no a x i a l force i s present the u l t i -mate bending moment i s given by: M . = 4r f p i z r SU (5.9) F i g . 5.8 u l t i m a t e a x i a l B e n d i n g A x i a l Stud c r o s s - s e c t i o n under f o r ce and bend ing moment. 0-2 0.4 0.6 0.8 1.0 M M Pi Fig. 5.9 Stud i n t e r a c t i o n diagram From equations (5.6) to (5.9) 3fl (5.10) M P i sin2 8Q + 6Q P i (5.11) By taking d i f f e r e n t values of 8^  between 0 and T T / , an i n t e r a c t i o n diagram of the form shown i n F i g . 5.9 i s ob-tained. From inspection of equations (5.10) and (5.11) i t i s d i f f i c u l t to get a d i r e c t r e l a t i o n s h i p between p / P p ^ a n c ^ M/M ^ for computation purposes. The approximate parabolic r e l a t i o n s h i p for the i n t e r a c t i o n curve 2 1PP1J + M M P i = 1 (5.12) gives very good r e s u l t s , however, as can be seen i n Table V. With further reference to F i g . 5.9, any combination of M and' P r e s u l t i n g i n a point on the diagram below the s t r a i g h t l i n e P i + M M P i = 1 (5.13) i s admissible. For a combination of M and P which re s u l t s i n a point between the s t r a i g h t l i n e and the i n t e r a c t i o n curve, the s t r e s s - s t r a i n and moment-curvature diagrams are scaled down by l i n e a r i n t e r p o l a t i o n to conform to the inter-action diagram, as described i n Appendix C. TABLE V COMPARISON BETWEEN APPROXIMATE FORMULA AND RIGOROUS DERIVATION OF STUD INTERACTION DIAGRAM p M M . P l % Difference V Rigorous equations (5.10) & (5.11) Approximate equation (5.12) 0 1. 000 1. 000 0.00 0.1 0. 991 0. 990 -0.10 0.2 0. 963 0.960 -0.31 0.3 0. 916 0.910 -0.66 0.4 0. 851 0.840 -1.29 0.5 0.765 0. 750 -1. 96 0.6 0. 660 0.640 -3. 03 0.7 0.533 0.510 -4.32 0.8 0. 384 0.360 -6.25 0.9 0. 208 0.190 -8. 65 1.0 0.000 0.000 0.00 93 CHAPTER 6. COMPARISON OF COMPUTER ANALYSES WITH LABORATORY RESULTS 6.1 Properties of Steel Anchor Bars and Studs Coordinates of the t r i l i n e a r algorithm of Fi g . 5.7, which i s used for modelling both the s t r e s s - s t r a i n and mo-ment-curvature properties of s t e e l anchor bars and studs, are recorded i n Table VI. The coordinates of the t r i l i n e a r s t r e s s - s t r a i n curves were chosen to approximate curves mea-sured i n the laboratory as c l o s e l y as possible. A moment-curvature diagram was calculated for each diameter of bar or stud on the usual assumption that s t r a i n s vary l i n e a r l y with distance from the neutral axis of a cross-section. A set of progressively larger curvatures was chosen, and from the st r a i n s at each curvature, a stress diagram across the cross-section was constructed with the use of the corresponding laboratory s t r e s s - s t r a i n curve. The bending moment at a cross-section was calculated from t h i s stress diagram and t r i l i n e a r moment-curvature coordinates were chosen to rep-resent the r e s u l t i n g moment-curvature diagram as c l o s e l y as possible. TABLE VI TRILINEAR PROPERTIES OF STEEL ANCHOR BARS AND STUDS Diameter Coordinates A x i a l Stress Curvature Moment (in.) (See F i g . 5.7) Strain (P- s. i . ) ( i n . r 1 (lb. i n . ) (X! , y i ) 0. 00217 65 000 0.00986 880 1* 2 (x 2 /Y2 ) 0.00251 70 000 0.03500 1 350 (x 3 ,Ya) 0.20000 90 000 3.25000 1 600 (X! , y i ) 0.00184 55 000 0.00840 750 2 (x 2 , y 2 ) 0.00215 60 000 0.03500 1 170 (x 3 rYs) 0.20000 80 000 3.25000 1 400 ( X i rYl) 0.00217 65 000 0.00690 1 560 5* 8 ( X 2 ,Y2 ) 0.00251 70 000 0.02770 2 600 ( X 3 , y s ) 0.20000 90 000 2.75000 3 100 ( X i , y i ) 0.00217 65 000 0.00578 2 690 3* 4 (X2 ,Y2 ) 0.00251 70 000 0.02310 4 160 ( X 3 rY3) 0.20000 90 000 2.50000 4 960 Anchor bars and machined studs Commercial studs Relevant properties from Table VI were used for the comparison between measured and calculated load-deflection curves for a 1/2-in.diameter s t e e l cantilever bar with an end load, F i g . 6.1. The measured and calculated r e s u l t s show good agreement. 6 . 2 F r i c t i o n Specimen A computer model of the f r i c t i o n specimen with 6 i n . long anchor bars i s shown i n F i g . 6.2. Loads from the o r i -g i n a l computer c a l c u l a t i o n s bore no resemblance to the mea-sured load-deflection curve for t h i s specimen and never ex-ceeded 1 kip. The secondary e f f e c t of a x i a l shortening of the anchor bars due to bending was subsequently included i n the computer program and the calculated loads were thereafter con-siderably improved. The calculated curves shown i n F i g . 6.3 were a l l calculated at a constant c o e f f i c i e n t of f r i c t i o n of 0.3. These curves show some d i s t i n c t differences from the laboratory curves which are probably primarily due to a vary-ing c o e f f i c i e n t of f r i c t i o n i n the laboratory specimen. I t appears that the r e a l c o e f f i c i e n t of f r i c t i o n i s much higher than 0.3 at short d e f l e c t i o n s and during i n i t i a l load cycles, and that i t decreases to below 0.3 with increased d e f l e c t i o n and an increased number of cycles. 6 . 3 End-Bearing Specimens The non-linear plane stress f i n i t e element program described i n Chapter 5 was used for computer analysis of the end-bearing specimens. The program was f i r s t tested on a computer model of a c y c l i c a l l y - l o a d e d concrete t e s t speci-300+ -Q Q_ (0 o 0 0.1 0.2 0.3 0.U 0.5 End d e f l e c t i o n (in.) 0.6 0.7 Fig. 6.1 Bending test on ^2 in. dia. bars compared with computer ca lculat ions. C O Membe r w i t h h igh ax i a l s t i f f ne s s f o r d i s p l a c e m e n t c o n t r o l (T) 6 at \" = 6 / 7 .0.69 -© ©~ 4 IP CD 1/2 d i a . a n c h o r b a r s f i © Denotes f i xed joint o Denotes pinned joint F r i c t i o n s i m u l a t o r s (T) — F a c e p l a t e -o ©- ~®——9 1 " 1 1 /2. Fig. 6.2 Computer model of f r i c t i o n s pec imen w i th 6 in. long anchor bars. Q. o 5 U 3 2 0 -1 -2 -3 — 0 Fig. 6.3 D e f l e c t i o n (in.) Measured and computed l oad -de f l e c t i on curves for friction spec imen F2. CO CO 99 men t e s t e d by Karsan and J i r s a 3 8 which i s shown i n F i g . 6.4. The computer c a l c u l a t i o n s were done u s i n g displacement con-t r o l where a l l the nodes along the top edge of the model were d i s p l a c e d the same amount i n the y - d i r e c t i o n t o simulate the e f f e c t of a heavy t e s t i n g machine p l a t e n . The average com-p r e s s i v e s t r e s s at the h o r i z o n t a l c e n t r e - l i n e of the model ver s u s the average s t r a i n along the v e r t i c a l c e n t r e - l i n e (over a d i s t a n c e s i m i l a r t o t h a t used i n the a c t u a l experiment by Karsan and J i r s a ) i s compared w i t h the o r i g i n a l r e s u l t s i n F i g . 6.5. The maximum nor m a l i s e d s t r e s s value of the c o n c r e t e model of F i g . 5.3 had to be m u l t i p l i e d by 0.85 to ensure agreement between computed and l a b o r a t o r y r e s u l t s . F i n i t e element models of the 1/4, 3/8 and 1/2 i n . end-b e a r i n g specimens, s i m i l a r t o t h a t f o r the 3/8 i n . specimen i n F i g . 6.6, were used f o r the a n a l y s e s . T h i s type of a n a l y s i s i s v a l i d o n l y f o r long and narrow end-bearing areas along the edge of the c o n c r e t e , as d e p i c t e d i n F i g . 6.8. Work by Hawkins 5 1' 5 2' 5 3 may r e a d i l y be adapted to compact end-bearing areas where r e s t r a i n t e f f e c t s are imposed by the surrounding c o n c r e t e . Each a n a l y s i s was run under displacement c o n t r o l by s p e c i f y i n g the displacements i n the y - d i r e c t i o n of the c o n c r e t e step under the f a c e p l a t e . • A l l three f i n i t e element models proved to be about three times too s t i f f , as shown i n the t y p i -c a l l o a d - d e f l e c t i o n curve of F i g . 6.7, while the maximum loads were, on average, 11% too h i g h , as shown i n Fig.. 6.8. A s i n g l e run on a f i n e r f i n i t e element g r i d d i d not improve r e s u l t s appre-Nodes a long top edge a l l have same y - di s p l a c e m e n t . R e i n f o r cement s i m u l a t e d with l i n e e l e m e n t s . Th i cknes s = 3 in. fc' = 5 000 p.s.i. Fig. 6.4 F in i te element grid of one quadrant of 3ft conc re te test spec imen used by Ka r s an & J i r s a : 1 J 1 T e s t A C 3 - 1 0 , f c = 5 010 p.s.i. , Ref. 38 0. 0.5 1.0 1.5 2.0 No rma l i s ed s t r a i n S=e/r o Fig. 6.5 Comparison of computed stress-s t ra in curve wi th te s t by Karsan & J i r s a . 102 Fig. 6.6 3/gjp e n c j bearing s pec imen model led by f i n i t e elements. 103 H S~ 1 1 h •01 .02 -03 -OU -05 F a c e p l a t e de f I ect ion (in.) Fig.6.7 Load - def l e c t i o n curve for % in. end bearing specimens. 104 Q £ - o o £ E X CO £ -a o —* u fd u_ 0 0.1 0.2 0.3 0.4 0.5 0.6 F a c e p l a t e t h i c k n e s s (in.) Fig. 6.8 Comparison of calculated and measured m a x i m u m l o a d s for end bear ing s p e c i m e n s . c i a b l y , leading to the conclusion that the program did not model the shear-type f a i l u r e of the concrete under the face-plate too well. The computer models do appear to confirm, however, that the r e l a t i o n s h i p between faceplate thickness and factored maximum load i s l i n e a r , F i g . 6.8. A better i d e a l i s a t i o n of the end-bearing specimen was the plane frame computer model of F i g . 6.9. The pre-viously described f r i c t i o n model was combined with an end-bearing spring modelled by the load-deflection curves of Fig . 5.6. The maximum load of the spring was taken from the laboratory curve of F i g . 6.8, and the d e f l e c t i o n at maxi-mum load was taken to be the average of those from the labo-ratory curves of Figs. 4.5 to 4.7, namely 0.025 i n . Similar-l y the d e f l e c t i o n at f a i l u r e of t h i s spring was 0.09 i n . The load-deflection curve of t h i s model bears a reasonable resem-blance to the laboratory curves as shown i n F i g . 6.10. 6 . 4 Single Studs i n Concrete Each single stud i n concrete was modelled by a beam on Winkler springs, as described i n Chapter 5. The f i n i t e element g r i d of F i g . 6.11 was used to determine the load-d e f l e c t i o n c h a r a c t e r i s t i c s , F i g . 6.12, of a s l i c e of concrete of unit thickness. Note that the Y-load recorded i n F i g . 6.12 i s for only one half of the s l i c e . This analysis i n d i -cates that the maximum load capacity of concrete under a stud i s : 106 CO M e m b e r w i t h h igh a x i a l s t i f f ne s s f o r d i s p l a c e m e n t c o n t r o l (T^)-S -0.69* 6 at 1 y = -© ©-1 " r~ '/2 d i a . a n c h o r bars ( 5 to i © D e n o t e s f i x ed joint o Denotes pinned joint F r i c t i o n s i m u l a t o r ( ^ ) V F a c e p l a t e (T) E n d - b e a r i n g c o n c r e t e spr ing(4a) -Fig.6.9 Computer model of end bearing specimen. Def I e ct ion (in.) Fig. 6.10 Measured and computed l o a d -de f l ec t i on curves for end-bear ing s pec imen s B3 & B4. 108 A U support nodes f i x e d against rotat ion. |Y Th i cknes s =1 in. fc = 4 800 p.s.i. S tud rep re sen ted by l i n ea r — e l a s t i c e l ement s Typical support for nodes — on c e n t r e - l i n e . Fig.6.11 F i n i t e e lement grid of one half of concrete s l i c e containing s tud shank. V e r t i c a l c r a c k in c o n -c r e t e on c o m p r e s s i o n °\ s i d e of s ud shank © L S t ud shank c rack s away from c o n c r e t e on t e n s i o n s ide. + 1 j 0 -001 -002 -003 -004 -005 d e f l e c t i o n of s t ud c e n t r e r e l a t i v e to point A (in.) Fig. 6.12 L o a d - d e f l e c t ion curve for c o n c r e t e s l i c e containing s tud shank. where R = maximum reaction per unit length of stud d = stud diameter. f^ = concrete cylinder strength. and t h i s maximum reaction occurs at a de f l e c t i o n of 0.004 i n . These properties were used together with the load-deflection curves of F i g . 5.6 to represent the concrete springs used i n the plane frame computer model of a stud i n concrete, F i g . 6.13. I n i t i a l computer runs resulted i n c a l -culated load-deflection curves which were too s t i f f and which had maximum loads which were far too low i n comparison with the laboratory curves. By t r i a l and error, i t was deter-mined that the reaction per unit length of stud should be increased to R = 5.0 df^ (6.2) with a d e f l e c t i o n at maximum load of .05 i n . and the deflec-t i o n at complete f a i l u r e of the spring was set at .25 i n . The much higher values are not too surprising because i t must be remembered that the f i n i t e element analysis i s only true for a plane stress condition. The concrete immed-i a t e l y under a stud can be heavily confined by surrounding concrete and i s therefore probably under a t r i a x i a l compres-Member w i t h high a x i a l s t i f f n e s s f o r d i s p l a c e m e n t c o n t r o l (T) C o n c r e t e s p r i n g s S t ud s hank ( ? ) -F a c e p l a t e (T)-© Denot es f i x e d joint o D e n o t e s pinned joint Fig. 6.13 Computer model of stud in concrete. 112 sive stress condition which can r e s u l t i n a far higher load capacity than the plane stress condition would indicate, as mentioned i n Chapter 5. Equation (6.2) and the deflections of .05 in.and .25 in. were used i n a l l of the computations of the load-de-f l e c t i o n curves of Figs. 6.14, 6.16, 6.18 and 6.20. The measured and calculated load-deflection curves of the studs i n concrete compare very favourably, as do most of the de-fl e c t e d shapes of Figs. 6.15, 6.17, 6.19 and 6.21. This lends credence to the shear force, bending moment and reaction d i a -grams which were extracted from the calculations at a point in each load-deflection curve well af t e r f i r s t y i e l d . The maximum points i n each bending moment diagram indicate that two p l a s t i c hinges have formed i n the stud shank, one at the faceplate and one at some distance into the concrete. The shear force diagram passes through zero at the second p l a s t i c hinge and most of the concrete reaction to the stud occurs between the two p l a s t i c hinges. 6 . 5 Complete Headed Stud Connection For comparison of a plane frame computer model with a complete connection, the laboratory r e s u l t s from connection E2, investigated by Spencer^ were chosen because i n his recent experiments with connections, str a i n s i n the studs have been monitored by means of s t r a i n gauges. A pair of these gauges were situated near the head of the stud and 6 + CL T3 O 3 + 0 0.1 0.15 Def I ect ion (i n.) Fig. 6.14 Measured and computed l o a d - d e f l e c t i o n curves for stud S1. CO 114 D i s t a n c e f r o m c o n c r e t e f a c e (in.) 0 0-5 1-0 1.5 2-0 D i s t a n c e f r o m back of f a c e p l a t e (in.) Fig.6.15 D e f l e c t e d shape comparison and computed forces on stud S1 we l l a f te r f i r s t y ield. 10 + 0 0.05 0-1 0-15 0-2 0-25 D e f l e c t i o n (in.) Fig. 6.16 Measured and computed l o a d - d e f l e c t i o n curves for stud S2. M 116 D i s t a n c e f r o m c o n c r e t e f a c e (in.) D i s t ance f r om back of f a c e p l a t e (in.) F ig . 6.17 De f l ec ted shape comparison and computed forces on stud S2 w e l l after f i r s t y ie ld . L o a d (kip.) /LIT D i s t a n c e f r o m c o n c r e t e f a c e (in.) 0 0.5 1.0 1-5 2.0 D i s t a n c e f r om back of f a c e p l a t e (in.) Fig. 6.19 D e f l e c t e d shape compar ison and computed forces on stud S3 well a f ter f i r s t y ie ld . CL _£ O 0.1 Def I e c t i o n (in.) Fig. 6.20 Measured and computed l oad -de f l e c t i on cu rves for studs S4, S5 & S6. D i s t a n c e f r o m c o n c r e t e f a c e ( in.) D i s t a n c e f r o m c o n c r e t e f a c e (in.) Fig. 6.21 De f l ec ted shape comparison and computed forces on studs S4, S5 & S6 we l l a f te r f i r s t y i e l d . 121 another pair i n the region of the p l a s t i c hinge furthest away from the faceplate. One of each pair was cemented on the top of the stud shank and the other on the bottom. Details of specimen E2 and i t s associated computer model appear i n Fig . 6.22. Measured and computed load-deflection curves are shown i n Fig . 6.23, together with the calculated i n d i v i d u a l loads c a r r i e d by studs, faceplate f r i c t i o n and faceplate end-bearing . The laboratory test and the computer calculations indicate that both the upper and lower studs y i e l d s u b s t a n t i a l l y i n a x i a l tension, as indicated i n Fig . 6.24. In the laboratory t e s t t h i s t e n s i l e y i e l d i n g occurs simultaneously with the peak load on the connection, whereas the calculations indicate that a x i a l t e n s i l e y i e l d i n g occurs a f t e r the peak load i s reached and at a much greater d e f l e c t i o n of the faceplate. There are no laboratory measurements nor any re s u l t s from the computer calculations which explain t h i s difference. One p o s s i b i l i t y i s that in the actual specimen the concrete being compressed under each stud appears to carry very high compressive stresses because i t i s confined by surrounding concrete. Expansion of the concrete, i n a d i r e c t i o n p a r a l l e l to the studs, i s probably restrained by the faceplate, thus applying t e n s i l e forces to the studs, which may explain the high a x i a l s t r a i n s i n the studs at an early stage i n the te s t . On the other hand, the computer model does not contain t h i s e f f e c t and i t i s a x i a l shortening of the stud due to bending alone that produces the a x i a l t e n s i l e strains at a l a t e r stage. 1/2'n. dia. x 6 in . s tuds 2 x 2 x 7 ^ in. angle — CXJ oo CNJ f r = 6 050 p.s.i. De ta i l s of spec imen E2 E n d b e a r i n g s p r i n g (4a . I V/ F r i c t i o n s i m u l a t o r ^ ) — ^ Faceplate (T^ D i sp l a cement c o n t r o l member(T^) © D e n o t e s f i xed joint. o D e n o t e s pinned joint Computer mode l 45 Fig. 6.22 Stud connection E2, due to Spencer , s imu l a t ed by computer model . to to 30 25 20 15 10 0 0 0.05 0-1 De f l e c t i on (in.) 0-15 0.2 6.23 Measured and computed l o ad -de f l e c t i on curves for connect ion E2 _45 125 The important point to note i s that both studs are in a x i a l tension when the maximum shear load on the connec-t i o n i s reached. This indicates that the common practice of considering moment equilibrium about some point i n the connection, while ignoring the horizontal i n t e r a c t i o n forces between faceplate and concrete, i s meaningless. Such a c a l -c u l a t i o n would indicate that the top stud i n F i g . 6.22 i s i n tension and the bottom stud i n equal and opposite compression. While the studs are y i e l d i n g i n tension, i t must be remembered that p l a s t i c hinges have formed i n each stud shank as well. The formation of each p l a s t i c hinge furthest away, from the faceplate i s confirmed by the strains measured for connection E2. In the computer cal c u l a t i o n s the stud forces have reached the y i e l d surface of F i g . 5.9. The computer calcu l a t i o n s indicate that the t e n s i l e forces i n the studs are about 30% of t h e i r f u l l y i e l d value with no bending moment present and the moments at the p l a s t i c hinges are therefore about 90% of the f u l l p l a s t i c moment with" no a x i a l forces pre-sent, i n accordance with F i g . 5.9. 126 CHAPTER 7. SIMPLIFIED ANALYTICAL MODELS 7.1 Ultimate Shear Load of a Stud i n Concrete Common features of the computed shear force, bending moment and reaction diagrams for single studs i n concrete i n the previous chapter suggest the simple a n a l y t i c a l approxima-t i o n of the forces on a stud shown i n Fig . 7.1. The reac-t i o n diagrams of Figs. 6.15, 6.17, 6.19 and 6.21 can re a d i l y be approximated by a rectangular diagram, between the two p l a s t i c hinges, of height kf^, where k i s a constant.Measure-ments of the areas of the reaction diagrams of the previous chapter, between the two p l a s t i c hinges, yielded values for k between 4.3 0 and 4.50. Thus the t o t a l reaction applied by the concrete i s given by: C = 4.4 f'dL (7.1) c where f^ = concrete cylinder strength d = stud shank diameter L = distance between p l a s t i c hinges. 127 For v e r t i c a l equilibrium of the short length of stud between the p l a s t i c hinges V u = C (7.2) where V = ultimate shear load on stud, u For equilibrium of moments about the p l a s t i c hinge at A V L = + 2M , u 2 p l where M ^ = F u l l y p l a s t i c moment of stud ^ shank which may be calculated from equation (5.9). r 2 M i i . e . V = ^ + —-P-±. (7.3) u 2 L From equations (7.1), (7.2) and (7.3) M L = / r r i T ^ d <7-4> c Once L i s calculated, i s e a s i l y found from equa-tions (7.1) and (7.2). The design ultimate shear load, V^, would be given by V = d>V u u where cj) = customary capacity reduction factor. For studs subjected to combined shear and t e n s i l e loads the f u l l y p l a s t i c moment would be reduced according to the i n t e r a c t i o n diagram of Fig . 5.9. IV u M A p i c M P i i n n u u .7.1 A p p r o x i m a t i o n of forces acting a stud in c o n c r e t e at m a x i m u m load. ////// F ig . 7.2 App rox imate l o a d -d e f l e c t i o n model of a connect i on . where P = Tensile load on stud. P u s = Ultimate t e n s i l e capacity of stud with no bending moment present. M = Reduced p l a s t i c moment of stud shank. 7.2 Ultimate End-bearing Capacity of Faceplate For a faceplate with thickness much less than the faceplate width, the straight l i n e of Fig . 6.8 may be used. i . e . P . = 1.2A, f (7.7) ub f c where = Ultimate end-bearing capacity of faceplate. = Bearing area of faceplate. The design ultimate bearing capacity of the faceplate would be given by Pub = * Pub ( 7 ' 8 ) 7. 3 Comparison of Proposed Theory and Current Shear F r i c t i o n Theory with Laboratory Measurements. Comparisons are made i n Table VII with fourteen con-nections, a l l of which are of the type shown i n Fig . 1.2. The studs of the connections of series A were p a r a l l e l to the side faces of the concrete panels in which they were cast (Fig. 2.4) and the remainder were at 45 degrees to the panel sides. For comparison purposes, no capacity reduction was TABLE VII COMPARISON OF CALCULATED AND MEASURED ULTIMATE STRENGTHS OF CONNECTIONS 6 in. d i a . Concrete Faceplate Angle Stud Ultimate Strength Ultimate Strength Measured Maximum Load Specimen Cylinder Strength (p. s. i . ) Size (in. ) Diameter (in.) Shear F r i c t . Eqn. 2.3 (kip.) Proposed Theory (kip.) Up (kip.) Down (kip.) Al 4600 4x3x3/8 5/8 33.2 35. 6 35.4 * A2 4600 4x3x3/8 5/8 33.2 35. 6 31. 5 30.2 A3 4600 4x3x3/8 5/8 33.2 35.6 ' 29.3 27.0 Bl 4600 3x3x3/8 5/8 33.2 33.6 31. 8 28.0 B2 4600 3x2x3/8 5/8 33.2 31. 5 30.0 32. 0 B3 4600 3x2x3/8 5/8 33.2 31.5 23.4** 24.0** CI 6450 3x3x3/8 5/8 33.2 42. 0 48.5 44.0 C2 6450 3x3x3/8 5/8 33.2 42.0 44.4 39.8 C3 6450 3x3x3/8 5/8 33.2 42. 0 41.9 44.8 C4 6450 3x3x3/8 5/8 33.2 42.0 38.2 41.3 C5 6450 3x3x3/8 5/8 33.2 42. 0 * 43.8 C6 6450 3x3x3/8 5/8 33.2 42.0 40.3 42.5 E l 5850 2x2x1/4 1/2 21.2 23.4 24.4 24.9 E2 5850 2x2x1/4 1/2 21.2 23. 4 29.4 * A & B Series from Ref. 28. C & E Series due to Spencer, Ref. 45 * Monotonic Test ** Test Results Unreliable because of Connection Fabrication Error 131 considered i n the ultimate load c a l c u l a t i o n s ; that i s , the capacity reduction factor, c|>, was taken to have the value of 1.0 throughout. In addition, each calculated ultimate strength i s the maximum possible, because no reduction due to the presence of possible t e n s i l e forces i n i n d i v i d u a l studs i s considered. It appears that for values of between 4000 and 5000 p . s . i . , shear f r i c t i o n equation 2.3 gives a reasonable predi c t i o n of the maximum loads carried by the connections, while for values of f* above 5000 p . s . i . , the shear f r i c t i o n c values become conservative. I t i s also l i k e l y that, for values of f^ below 4000 p . s . i . , shear f r i c t i o n theory w i l l prove to be unconservative because equation 2.3 i s not linked to the concrete strength f^. In instances where the end of the faceplate does not bear on concrete, shear f r i c t i o n equa-tio n 2.3 may be dangerously unconservative. End-bearing capacities calculated from equation 7.7 proved to be more than 4*0% of the t o t a l ultimate load for each of the fourteen connections under consideration. Shear f r i c t i o n theory does not take faceplate end-bearing into account and could thus over-predict ultimate loads by up to 40%, while the loads c a l -culated by the theory proposed above would be suitably reduced. Conversely, i n cases where faceplate end-bearing areas are much increased over the areas of faceplate angles of the con-nections under consideration here, the shear f r i c t i o n theory may be overly conservative. This has been shown by Spencer1*5 who has measured maximum loads of up to 50 kip. for connec-132 tions with two 1/2-in. diameter studs and a faceplate end-bearing area of 2 i n . x 2 i n . in concrete with a cylinder strength of 5850 p . s . i . Equation 7.7 may not be v a l i d for end-bearing areas of such proportions, but current research w i l l r e c t i f y t h i s deficiency shortly. 7.4 F r i c t i o n between Faceplate and Concrete F r i c t i o n forces between faceplate and concrete have been omitted i n the equations proposed above because both the laboratory experiments and the computer calculations i n -dicated that these were small and disappeared r a p i d l y with c y c l i c loading. This i s confirmed by the values i n Table VII and also negates the v a l i d i t y of shear f r i c t i o n equation 2.3. Monotonic t e s t s , or c y c l i c t ests with a few cycles of large displacements may r e f l e c t some faceplate f r i c t i o n with higher than average maximum loads. 7. 5 Load-deflection Model for a Connection Continuing along the l i n e s of the approximation of Fig . 7.1, a complete connection can be approximated by the model of F i g . 7.2 which i s comprised of four concrete springs of the type used i n previous chapters and a cantilever beam with t r i l i n e a r load-deflection properties as i n Fig. 5.7. Two springs model the concrete around the studs and two model the concrete on which the faceplate bears. With reference to F i g . 7.2, the ultimate shear force on a connection i s given by S = V + R 2 (7.9) where V = Ultimate shear force from a l l studs. R2 = Ultimate end-bearing capacity from equation (7.7) . From equation (7.3) 2M V = n ( | + (7.10) where n = t o t a l number of studs i n connection'. i . e . V = R 1 + R3 where R^  = = 2.2f^dLn from equation (7.1) = contribution from concrete reaction. L = distance between p l a s t i c hinges, equation (7.4). R_ = ^ p l n =contribution from stud bending. L Substituting into equation (7.9) S = R± + R 2 + R3 (7.11) A t o t a l load-deflection curve (R^ versus A) over the distance between p l a s t i c hinges, L, can be calculated for the studs using a moment-curvature diagram and conventional Moment Area Theorems. This curve may be approximated by the t r i -l i n e a r algorithm of F i g . 5.7. A computer sub-routine repro-ducing the model of F i g . 7.2 has been written, which requires the values of R^  and R2 and the coordinates of the R^  versus A curve, and i s reproduced i n Appendix B. The p o s i t i o n of the 134 concrete reaction force,\C, s h i f t s from just behind the face-plate at small loads to L/2 away from the faceplate at maximum load and beyond. The subroutine includes de-gradation of the load-carrying capacity of the stud by making use of equation 4.1, and a record of cumulative p l a s t i c hinge r o t a t i o n i s stored for pred i c t i o n of stud fracture.. Several load-deflection curves were calculated using t h i s subroutine and are reproduced below. The load-deflec-t i o n curve for a single 5/8 i n . diameter stud i n concrete of F i g . 7.3 compares favourably with those of F i g . 6.20, and the curve for connection E2, F i g . 7.4, i s reasonably s i m i l a r to that of F i g . 6.23. The calculated degrading load-deflec-t i o n loops of stud bending specimen SB5 appear i n Fig. 7.5 and may be compared with the measured loops i n F i g . 4.26. The studs i n the actual specimen fractured towards the end of the tenth cycle, whereas a cumulative rotation of more than 3.5 indicated stud fracture for the calculated loops at the beginning of the eleventh cycle. Calculated load-deflection loops of connection A3 (Fig. 7.6) have the same.maximum d i s -placements i n each cycle as do the measured loops of F i g . 2.7. The studs i n t h i s connection must have had d i f f e r e n t proper-t i e s from those used i n the stud bending te s t s , because these studs withstood a maximum cumulative rotation of more than 11.0 before fracture. The measured and calculated loops of connection E l are shown i n Figs. 7.7 and 7.8 respectively, and both sets have the same maximum displacements i n each cycle. Although the measured and calculated loops have very s i m i l a r maximum loads, the enclosed area of each c a l -culated loop i s larger than i t s measured counterpart, givinc a higher energy d i s s i p a t i o n per cycle. The computer sub-routine indicated stud fracture i n the sixteenth cycle, whereas fracture actually occurred at the beginning of the f i f t e e n t h . D e f l e c t i o n (in.) Fig.7.3 Calculated load-def lect ion curve for studs S4, S5 & S6. O 0.2 0.15 0.1 0.05 0 0.05 D e f l e c t i o n (in.) 0-15 0.2 Fig.7.5 Ca lcu lated load-def lect ion loops for stud bending spec imen SB5. i—1 CO CO Fig.7.6 C a l c u l a t e d l o a d - d e f l e c t i o n loops - connect ion A3. CHAPTER 8. CONCLUSION 8.1 Confirmation of I n i t i a l Assumptions This i n v e s t i g a t i o n was i n i t i a t e d on the supposition that a connection transfers applied shear loads to the sur-rounding concrete by three d i s t i n c t mechanisms, namely: f r i c t i o n between faceplate and concrete; bearing of end of faceplate on concrete; and i n t e r a c t i o n between studs and concrete. Tests on laboratory models designed to i s o l a t e i n d i v i d u a l aspects of these mechanisms confirmed that a l l three did e x i s t . From the laboratory models i t was found that forces normal to the faceplate, necessary for f r i c t i o n forces be-tween faceplate and concrete to e x i s t , were caused mainly by ax i a l shortening of the studs during bending. Measurements of str a i n s of the studs of a regular connection indicated that some other mechanism,possibly expansion of the concrete around each stud i n a d i r e c t i o n p a r a l l e l to the stud axis, enhanced the development of forces normal to the faceplate. From labora tory measurements i t was concluded that f r i c t i o n forces decreased rapi d l y under cycled loading, and were n e g l i -gible compared to the remaining forces i n a connection. On the other hand, differences between calculated and ob-served loads from a few monotonic tests indicated that f r i c t i o n forces could increase the t o t a l ultimate load of a connection by a small amount. Bearing of the end of the faceplate on adjacent con-crete was shown to contribute s i g n i f i c a n t l y to the t o t a l u l -timate load of a connection and an equation for c a l c u l a t i n g t h i s contribution was derived empirically. S i m i l a r l y , i n -ter a c t i o n between studs and surrounding concrete, by bearing of the studs on concrete and bending of the studs, was shown to make a s i g n i f i c a n t contribution to the t o t a l load capacity of a connection. Stud and concrete i n t e r a c t i o n was success-f u l l y modelled on a computer by means of a beam on Winkler springs, leading to the development of a simple a n a l y t i c a l model for the p r e d i c t i o n of the ultimate shear load capacity of a stud i n concrete. The model was extended to a simple computer model which predicted the load-deflection behaviour of i n d i v i d u a l studs i n concrete or complete connections with reasonable success under both monotonic and c y c l i c conditions. Existence of the three mechanisms whereby a connec-t i o n transfers shear forces to the surrounding concrete con-t r a d i c t s the shear f r i c t i o n concept, currently employed i n 144 the design of connections, which assumes that a l l of the applied shear forces are transmitted to the concrete across some sort of shearing plane, the normal forces being sup-p l i e d by reinforcement normal to t h i s plane. A n a l y t i c a l equations developed from the i n v e s t i g a t i o n indicated that the strength of a connection i s d i r e c t l y dependent upon the strength of the concrete i n which i t i s cast, as opposed to the expression for shear f r i c t i o n , equation 2.3, which does not contain concrete strength as a variable. 8 . 2 Future Research The research described i n t h i s d i s s e r t a t i o n forms but a small part of the o v e r a l l objective of predicting the behaviour of precast concrete panel buildings under earth-quake loads, and research i n t h i s f i e l d i s only beginning. With the aid of the shear load versus d e f l e c t i o n algorithm developed above, i t may now be possible to analyse a set of panels assembled i n one plane with headed stud connections, under earthquake loading i n the same plane. Before analy-ses of a three-dimensional assembly of panels can be under-taken, the i n t e r a c t i o n between the shear load capacity of a connection and d i r e c t loads transmitted by the connection ' must be investigated. Connections similar to ones with headed studs, and i n p a r t i c u l a r , connections which have de-formed bars at 45° to the faceplate instead of studs, may possibly be investigated i n a s i m i l a r manner to the research car r i e d out here. The differences between the measured and calculated a x i a l t e n s i l e s t r a i n s of E2, discussed i n Chapter 6, warrant of the development of a x i a l strains nection. the studs i n connection further investigation in the studs of a con-146 BIBLIOGRAPHY 1. PCI Design Handbook. Precast and Prestressed Concrete, Prestressed Concrete I n s t i t u t e , Chicago, I l l i n o i s , 1971. 2. 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Borges, J.F. and Ravara, A., "Structural Behaviour of Panel Structures Under Earthquake Actions", Rilem Inter-national Symposium on E f f e c t s of Repeated Loading of Materials and Structures, Mexico, 1966. 148 22. Armer, G.S.T. and Kumar, S., "Tests on Assemblies of Large Precast Concrete Panels", Precast Concrete, London, Vol. 3, No. 9, Sept. 1972, pp.541-546. 23. International Recommendations for the Design and Con-stru c t i o n of Large Panel Structures, Cement and Concrete Association, London, Translation No. 137, 1967. 24. Zienkiewicz, O.C., Parekh, C.J. and Teply, B., "Three Dimensional Analysis of Buildings Composed of Floor and Wall Panels", Proceedings of the I n s t i t u t i o n of C i v i l  Engineers, London, Vol. 49, July 1971, pp.319-332. 25. Yuzugullu, 0. and Schnobrich, W.C., "A Numerical Pro-cedure for the Determination of the Behaviour of a Shear Wall Frame System", Journal of the American Con-crete I n s t i t u t e , Vol. 70, No. 7, July 1973, pp.474-479. 26. 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Balmer, G.G., "Shearing Strength of Concrete Under High T r i a x i a l Stress - Computation of Mohr'd Envelope as a Curve", Structural Research Laboratory. Report No. SP-23, U.S. Bureau of Reclamation, 1949. 32. Kupfer, H.B., H i l s d o r f , H.K-. and Rusch, H., "Behaviour of Concrete Under B i a x i a l Stresses", Proceedings of the  American Concrete I n s t i t u t e , Vol. 66, No. 8, Aug. 1969, pp. 656-666. 33. Darwin, D. and Pecknold, D.A., "Analysis of R.C. Shear Panels under C y c l i c Loading", Journal of the Structural D i v i s i o n , ASCE, No. ST2, Feb. 1976, pp.355-369. 149 34. Cervenka, V. and Gerstle, K.H., " I n e l a s t i c Analysis of Reinforced Concrete Panels: Theory", Publications, Inter-national Association for Bridge and Structural Engineer-ing, Zurich, Vol. 31 - I I , 1971, pp.31-45. 35. Cervenka, V. and Gerstle, K.H., " I n e l a s t i c Analysis of Reinforced Concrete Panels: Experimental V e r i f i c a t i o n and Application", Publications, International Association for Bridge and Structural Engineering, Zurich, Vol. 32 -II , 1972, pp.25-39. 36. P h i l l i p s , D.V. and Zienkiewicz, O.C., " F i n i t e Element non-li n e a r Analysis of Concrete Structures", Proceedings of  the I n s t i t u t i o n of C i v i l Engineers, London, Part 2, No. 61, March 1976, pp.59-88. 37. Zienkiewicz, O.C., The F i n i t e Element Method in Engineer-ing Science, McGraw-Hill, London, 1971. 38. Karsan, I.D. and J i r s a , J.O., "Behaviour of Concrete under Compressive Loadings", Journal of the-Structural D i v i s i o n , ASCE, No. ST12, Dec. 1969, pp.2543-3563. 39. Kupfer, H.B. and Gerstle, K.H., "Behaviour of Concrete Under B i a x i a l Stresses", Journal of the Engineering  Mechanics D i v i s i o n , ASCE, No. EM4, Aug. 1973, pp.853-866. 40. Tinawi, R.A., Behaviour of Orthotropic Bridge Decks, Ph.D. Thesis, McGill University, Montreal, Canada, May 1972. 41. Abramowitz and Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1972. 42. Cook, R.D., Concepts and Applications of F i n i t e Element  Analysis, John Wiley and Sons, New York, 1974. 43. Cowper, G.R., Lindberg, G.M. and Olson, M.D., "A Shallow Sh e l l F i n i t e Element of Triangular Shape", International  Journal of Solids and Structures, Vol. 6, 1970, pp.1133-1156. 44. Timoshenko, S.P. and Goodier, J.N., Theory of E l a s t i c i t y , McGraw-Hill Book Company, New York, 1970, pp.117-121. 45. Spencer, R.A., Research on Headed Stud Connections, Uni-v e r s i t y of B r i t i s h Columbia, Canada, 1977. To be pub-lis h e d . 46. V i e s t , I.M. et a l . , "Composite Steel-Concrete Construction," Journal of the Structural D i v i s i o n , ASCE, Vol. 100, No. ST5, May 1974, pp. 1085-1139. 14 S 47. Chadha, G.S., The Behaviour of Precast Concrete Connec- tions Incorporating Headed Steel Connectors. Master of Science Thesis, Dept. of C i v i l Engineering, University of Washington, 1976. 48. Al-Yousef, A.F., Composite Shear Connections Subjected  to Reversed C y c l i c Loading with and without Ribbed Metal  Deck. Master of Science Thesis, University of Washington, 1975. 49. Finney, E.A. et a l . , "Structural Design Considerations for Pavement J o i n t s " , Journal of the American Concrete  I n s t i t u t e , Vol. 28, No. 1, July 1956, pp.1-28. 50. Vi e s t , I.M. et a l . , " F u l l Scale Tests of Channel Shear Connectors and Composite T-beams", University of I l l i n o i s  B u l l e t i n , B u l l e t i n Series No. 405, University of I l l i n o i s Engineering Experiment Station, Urbana, Dec. 1952, pp.59-74. 51. Hawkins, N.M., "The Bearing Strength of Concrete Loaded Through Rigid Plates". Magazine of Concrete Research, Vol. 20, No. 62, March 1968, pp.31-40. 52. Hawkins, N.M., "The Bearing Strength of Concrete Loaded Through F l e x i b l e Plates". Magazine of Concrete Research, Vol. 20, No. 63, June 1968, pp.95-102. 53. Hawkins, N.M., "The Bearing Strength of Concrete for S t r i p Loadings". Magazine of Concrete Research, Vol. 22, No. 71, June 1970, pp.87-98. APPENDIX A TRIANGULAR AND QUADRILATERAL PLANE  STRESS/STRAIN FINITE ELEMENTS The displacement functions for these elements are obtained by applying constraints to e x i s t i n g displacement functions of a complete cubic t r i a n g l e and a complete cu-bic rectangle i n much the same way as proposed by Tinawi 4 0 for a t r i a n g l e . A.1 Plane Stress/Strain Triangle With reference to the complete cubic t r i a n g l e 3 7 shown i n F i g . A . l , the displacement function i n terms of area coordinates i s : ~(3£i-l) ( 3 ? 1 - 2 ) ? 1 / 2 " (3£2-l) ( 3 £ 2 - 2 K 2 / 2 (35 3-l) ( 3 £ 3 - 2 K 3 / 2 9 £ i ? 2 ( 3 ? i - l ) / 2 u = 9 ? i C 2 ( 3 ? 2 - l ) / 2 9 C 2 C 3 (352-D/2 9 ? 2 C 3 (3? 3-l)/2 95i? 3 ( 3 C 3 - D / 2 9 5 ^ 3 (3SX-U/2 2 7 ? i C 2 C 3 r! r 3 r 5 r 7 r 9 r 11 r 13 r 15 r 17 r 19 T i . e . u = [Nj [ r Q d d ] (A.l) T S i m i l a r l y v = T N I [r ~\ J L J [_evenj (A.2) If the centroidal nodal displacements are forced to be a quadratic function of the corner node displacements and displacements at the midsides of the t r i a n g l e , then L r i 8J (A.3) Applying equation (A.3) to equations (A.l) and (A.2) " ( 3 S i - 1) (3? i ~ 2 ) 5 i / 2 - 95 i 5 2 5 3/2~ T r 1 ( 3 £ 2 - 1) (35 2 - 2 ) 5 2 / 2 - 95 i 5 2 5 3/2 r 3 (3? 3- 1) (3? 3 - 2 ) ? 3 / 2 - 95 i 5 2 5 3/2 r 5 9 5 i 5 2 (3? i - l ) / 2 + 2 7 5 i 5 2 5 3/4 r 7 9 5 i 5 2 ( 3 5 2 - l ) / 2 + 2 7 5 i 5 2 5 3/4 r 9 ( 3 5 2 - l ) / 2 + 2 7 5 i 5 2 5 3/4 r u 9 5 2 5 3 (35 3 - l ) / 2 + 2 7 5 i 5 2 5 3/4 r 13 9 5 l 5 3 (35 3 - l ) / 2 + 2 7 5 i 5 2 5 3/4 r 15 9 5 i 5 s ( 3 ? i - l ) / 2 + 2 7 5 i 5 2 5 3/4 r 17 i . e . u r 1 r 3 r 17 (A.4) y.v 4&a» x , U Fig. A.1 Comp le te cubic t r i ang le . {j«ju*> X | U Fig.A.2 A t yp i ca l s i de of an e lement 153 S i m i l a r l y v = [M] r 2 | ( A . 5) r is and the dependence upon degrees of freedom r 1 9 and r 2o i s removed from the equations. Constraints are applied to side node displacements so that they may be expressed i n terms of the displacements of the corner nodes. With reference to F i g . A.2, enforce displacements to be l i n e a r along any t y p i c a l edge AD. i . e . d 7/ + e 2 n (A.6) solving for 0i and 8 2 d y - d j whence d 3 d 5 2d x/3 + d 7/3 di/3 +2d7/3 (A.7) Allow displacements at r i g h t angles to edge AD to remain cubic. 2 3 i . e . d J ^ = 0 3 + 9 l * r l + 05H + 9 6 n Solving for 0 3 to 0 G 154 d 4 = y y d 2 + y y d 8 + -5-731- y y 3 2 ' (A.8) _ 7 , . 20, , 2L D 4L„ Transforming r e s u l t s to def l e c t i o n s p a r a l l e l to x and y axes: cosa sina 0 u a d 2 •-• -sina cosa V a (A.9) d 7 0 cosa sina d 8 -sina cosa Vd 4> \ u c V c cosa -sina sina cosa 0 cosa -sina sina cosa d 3 d 4 d 5 d 6 (A.10) Substituting equations ( A . l ) , (A.8) and (A.9) into (A.10). 2 2 •j cos a 20 . 2 + 2 y s m a — ^ s i n a c o s a 4L . — 2 y S i n a 1 2 y cos a , 7 . 2 -f-jySin a 27 sinacosa 2L. 27 sina -^•sinacosa 2 . 2 j sin a ^2 0 2 + 2 7 C O S a 4L 27< :osa 27 sinacosa 1 . 2 j sin a . 7 2 + 2~ cos a 2L -27< :osa 1 2 -j cos a 7 . 2 + 2 y S i n a 27 •sinacosa 2L_ *27 sina 2 2 j cos a ,20 . 2 + — s i n a 27 sinacosa 4L^ 27 •sina 27 sinacosa 1 . 2 •j s i n a 7 2 -H^cos a 2L 27^ :osa —jysmacosa 2 . 2 j sin a , 20 2 + ^ ycos a 4L -27* :osa (A.11) 156 Using equations (A.11) for a l l three sides of the tr i a n g l e r 1 1 • • • • • • • • "s r r 2 • 1 • • • • • • • s 2 r 3 • • • 1 • • • • • s 3 r 4 • • • • 1 • • • • S4 r 5 • • • • • • 1 • • s 5 r 6 • • • • • • • • 1 • s 6 r 7 a 31 a 32 2a 33 a 34 - a 32 - a 33 • • • s 7 r 8 a 32 a 35 2a 3 6 32 a 37 - a 36 • • • s 8 r 9 a 34 -a 32 a 33 a 31 a 32 - 2 a 33 • • • Sg rio - a 32 a 37 a 36 a 32 a 35 -2 a 36 • • • r ii • • • a n a 12 2ai3 a 14 - a 12 - a 13 r i 2 • • • a 12 a is 2a is - a 12 a 17 - a i6 r i 3 • • • a 14 - a 12 a 13 a n a 12 - 2 a i 3 r i 4 • • - a 12 - a 17 a i6 a 12 a is -2a i 6 r is a 24 — a 22 - S 23 • • • a 21 a 22 2a23 ris - a 22 a 27 - 3 26 • • • a 22 a 25 2a 26 r 17 a 21 a 22 -2a2 3 • • • a 24 - a 22 a 23 r is a 22 a 25 - 2 a 2 6 • • • -a22 a 27 a 26 (A.12) where a. i a. x. a. 2c' i + 3L 2 1 2b.c. i i 2~ 27L I 20b 2 7 L 2b. I ~2~T 7b' + 31/ I 2 271/ 2bT ^ 20c 27L 3L 2c. I ~2T 2 2 bZ , 7c I + i 3L: 27L b . + c . x x 157 > ..f.:.}: 2:? < A.i3) and b. = y. - y, x j J k c. = x. - x. i k 2 > i , j,k are c y c l i c permutations of 1,2,3 ^ Degrees of freedom s^ i n equations (A.12) refer to the degrees of freedom of the constrained t r i a n g l e shown i n Fig. A.3. Strains i n the element are given by e X e y Y xy °/3x 0 0 ^ S y 9 / 3 y ^3x u v (A.15) The r e l a t i o n s h i p between derivatives with respect to x and y and the derivatives with respect to area coordi-nates i s given by 3/ 3/ 3x 3y 2A hi b 2 C i c 2 3/ 3/ 3?: 3?: (A.16) 2A = b i c 2 - b 2 c (A.17) Applying equations (A.15), (A.16) and (A.17) to equations (A.4) and (A.5) 159 V a l u e s f r o m r e f e r e n c e 41 d=0 .1012865073234563 A c c u r a t e up to quint ic t e r m s in in tegrand. F ig . A.4 Numer i c a l i n t e g r a t i o n points for t r i ang le . 160 y <f> 3 l - 4 > 3 0 2 0 4> 3 5 - 4> 3 6 4> 3 5-$ 3 6 4> 31-4> 3 2 T r i ~ r 2 <f>l i-<t> i 0 2 0 <f> 15-4> 1 6 4> 1 5-4> 1 6 4> 11~4> 12 r 3 r 4 4>2 l -<f>2 0 2 0 4>2 5~4>2 6 4>2 5-4>2 6 4>2 1~4>2 2 3 r 5 r 6 4> 3 3+ 3 4 ) 3 2/2 0 4> 3 7 + 3 4> 3 e / 2 r 7 0 (J) 3 7 + 3(J) 3 6 / 2 <P 3 3 + 3(J) 3 2/2 r 8 <J> 3 l t + 3c{) 3 2/2 0 4> 3 8 + 34> 3 S / 2 r 9 0 4> 3 8+34> 3 6 / 2 <j> 3 4 + 34> 3 2/2 r 10 4>1 3 + 3 4 ) 1 2/2 0 4> 17+34> 1 e / 2 r 11 0 4> 1 7 + 3 0 i e / 2 4> 13+34> 1 2/2 r 12 4>1 4 + 3<j) 1 2/2 0 <f> 18 + 34) 1 e / 2 r 13 0 4> 18+34> i e / 2 4) 11+ 3 4 ) 1 2/2 r 14 4>2 3 + 34) 2 2/2 0 4>2 7 + 34)2 6 / 2 r 15 0 4>2 7 + 34> 2 S / 2 4>2 3 + 34>2 2/2 r le (J)2 4 + 3 cj> 2 2/ 2 0 4>2 8 + 34)2 6 / 2 r i 7 0 (J)2 8 + 34) 2 6 / 2 4>2 i* + 34)2 2/2 r is (A.18) 161 where i i 1 2 1 3 I t 1 5 1 6 1 ? 1 8 b. (^d-95,+D/2A J ^ J J 2 ( ¥ j ? k + b k ? i S + V k ? i ) / 2 A  9 { b j ^ ( 3 S - i ) + b k ? j ( I S " ^ ) } / 2 A 9 { b j ? k { l ? k - i , + V j ( 3 ^ - i ) } / 2 A c. (^ 7-? 2-95 j+D/2A 2 ( c i 5 j ^ + C k ^ i ? j + C j V i ) / 2 A 9 { c . C k ( 3 C j - | ) + c k 5 . ( | 5 j - | ) } / 2 A 9 { c ^ k 4 V l ) + C k V 3 ^ ) } / 2 A > (A.19) i , j , k are c y c l i c permutations of 1,2,3 From equations (A.12) and (A.18) • I <I»3K ^ 3 7~ T s 1 ty 3 2 ^3 5 ^ 3 8 s 2 ty 3 3 ^ 3 6 ^3 9 s 3 e X ty 11 ^ 1 4 ty 1 7 e y ty 1 2 4* 1 5 <P 1 8 s 5 Y xy ^1 3 ty 1 6 ty 1 9 s s tyz 1 ^2 4 ty2 7 s 7 ^2 2 ^2 5 ^ 2 8 s 8 ^2 3 <J>2 6 ^2 9 s 9 (A.20) where ty = (ty -ty ) + a (cj> +h> ) + a . (<b. +|<J> ) + a (<j) +|<J) )+a (ty +h>, ) 11 l l 12 l l I3 Z 12 1 4 1 4 ^ 12 K t K3 Z K2 K j K t, Z K2 i K = a. (ty -ty. )+a, (<J>, -<J>. ) 12 1 2 I 3 l i t K 2 Kt, K 3 a. a ty. = -^(9<j) +ity +2ty )-*±(9ty + 2ty + 4ty ) 1 3 ^ 12 13 14 ^ K2 K3 K!i| ty. = a . ((j). -<J> )+a (<J>, -<J) ) l i | 12 17 18 K2 K-S K7 ty. = (ty- ~ty. ) + a . (4>- +4<|)- ) + a . (4>- +lty. )+a . (<J>. ) + a i (*, ) r X 5 15 16 15 17 2 T 1 6 17 18 2 T16 k7 k7 2 k6 k s k s 2 k6 a. a, ty. = -^(9ty +4<j) +2<j> )-JSs-(9<J> +2ty, + 4<j> ) 16 / 16 17 18 z Ke K 7 J<8 i i . - (ty. -ty. ) + a . ((J). +- 3r4>. )+a. (c|). + ^ 4 . . ) fa. (cL t 3 ^ . )+a . (<J>. + 3 7 4 , 1 ) . y x 7 15 l e 11 17 2 T i 6 i i , i s 2 Y i 6 kii k 7 2 Y k 6 k i Y k 8 2 Y k 5 12 = (ty. -ty. ) + a . ( 4 ) . + | 4 > . ) + a (<j>. + | 4 > . ) + a (ty +fcj> k )+a +~ty )+ty l g l i I2 15 13 z I2 17 1 t ^ ! 2 K7 K3 Z K-2 K5 Ki* Z K2 l i ) a. a. a a ty. = -^(9ty. + 4ty. +2ty. )+-^-(9ty. +44>. +2ty. )-J£3-(9<|>. + 2 4 > ^ + 4*. ) - - 4 M 9 < J > . + 2$. + 4*. ) r i 9 2 Y i 6 17 i s 2 r i 2 13 i t 2 T k 6 k7 k s 2 T k 2 k 3 Tki» i , j , k are c y c l i c permutations of 1 ,2 ,3 (A.21) c ro 163 From e q u a t i o n (A.20) (e) = [ f ] ( s ) and t h e s t i f f n e s s m a t r i x i s g i v e n by T [K] = t | [ Y ] [E] [ f ] d A = t I F ( 5 i , £ 2 , S 3 ) d A A I n t e g r a t i n g n u m e r i c a l l y n [K] = Cw F ( ? t , ? 2 , ? t ) A t (A.22) i = l where w_^  = w e i g h t i n g f u n c t i o n A = a r e a o f e l e m e n t t = t h i c k n e s s o f e l e m e n t n = number o f i n t e g r a t i n g p o i n t s The i n t e g r a n d c o n t a i n s t e r m s i n t h e £ 1 s up t o t h e f o u r t h power and f o r e x a c t n u m e r i c a l i n t e g r a t i o n t h e s e v e n p o i n t i n t e g r a t i o n scheme shown i n F i g . A.4 i s n e c e s s a r y . A t t e m p t s a t a l o w e r o r d e r i n t e g r a t i o n w i t h l e s s i n t e g r a t i o n p o i n t s y i e l d e d an e r r a t i c e l e m e n t and were abandoned. A. 2 P l a n e S t r e s s / S t r a i n Q u a d r i l a t e r a l The d i s p l a c e m e n t f u n c t i o n f o r t h e c o m p l e t e c u b i c q u a d r i l a t e r a l 3 7 shown i n F i g . A.5, i n l o c a l s , t c o o r d i n a t e s i s : 164 Fig. A.6 Con s t r a i ned cubic q u a d r i l a t e r a l . u (1-s) (1-t){ -10+9(s 2+t 2 ) }/32 r i (1+s) (1-t){ -10+9(s 2+t 2 ) >/32 r 3 (1+s) (1+t){ -10+9(s 2+t 2 ) 1/32 r 5 (1-s) d+t) { -10+9(s 2+t 2 ) }/32 r 7 9(1-t) , -> 2. (1-s ) (l-3s)/32 r 9 9(1-t) (1-s 2) (l+3s)/32 r ii 9(1+s) (1-t 2) (l-3t)/32 r i 3 9(1+s) (1-t 2) (l+3t)/32 r 15 9(1+t) (1-S ) (l+3s)/32 r i 7 9(1+t) (1-s 2) (l-3s)/32 r 19 9(1-s) (1-t 2) (l+3t)/32 r 2 i 9(1-s) (1-t 2) (l-3t)/32 r 23 i . e . u = [N] ( r Q d d ) T S i m i l a r l y v = TNI (r ) J even (A.23) (A.24) Applying equations (A.11) to a l l four sides of the q u a d r i l a t e r a l of F i g . A.5, a r e l a t i o n s h i p between the r degrees of freedom and the s degrees of freedom for the constrained cubic q u a d r i l a t e r a l of F i g . A.6 i s obtained. 166 ri ~ 1 • • • • S l r 2 • 1 • • - s 2 r 3 • • 1 • S 3 r<4 • • • 1 S4 r 5 • • • • 1 s 5 r 6 • • • • s 6 r 7 • • • • 1 s 7 r 8 • • • • 1 s 8 r 9 a n -a 22 -2a 1 3 a 14 a^ a 13 Sg rw a is 2ai6 a 12 a 17 -a i6 s io rn a 14 a i2 -a 13 an —a^  2a i 3 S l l r i2 ai2 a 17 a i6 —a 12 a is -2 a is Sl2 r i 3 • • • a 21 —a.22 -2a 2 3 a24 a22 a23 rn • • • —a.22 a25 2a26 5.22 a27 -a^ ris • • • a24 a22 -a23 a 2i —az2 2a23 ris • • • a.22 a27 a26 - a.22 a25 -2a 2 6 r X 7 • • • • a 31 -a 32 -2a 33 a34 a^ a 33 rie • • • • —a32 a35 2a36 a32 a 37 -a se r i 9 • • • • a34 a 3 2 -a 33 a 31 -a 32 2a33 • • • • a32 a 37 a36 -a32 a35 -2aa6 r 2 i a 1* a42 a43 • • • • a 4i -a42 -2a43 r22 a^ . a47 - a45 • • • • -a42 a 45 2a46 r 2 3 a 41 -a42 2a43 • • • • a44 a42 -a 43 r24 —a 42 a 45 -2a46 • • • • a 42 a 47 a46 (A.25) where a, x1 12 a. 1 3 i t i s a. 16 c. 1 L. l + 2_0 27 L. l _2_ 27 2b. l 27 2 3 2c. l 27 b.c. x 1 f > c. X z + 7 fb.l X L. 1 iJ 27 L. 1 i j X 2 + 20 c . l L. 1 i j 27 L i I i 1 X 2 + 7 c . X ct • — 17 3 L. 1 i J 27 L. 1 i j > i = 1,2,3,4 (A.26) The r e l a t i o n s h i p between derivatives with respect to x and y and the derivatives with respect to s and t i s given by the chain rule 9s 9 9x 9s 9x 9t 9y 9s 9y_ 9t 9x m i (A.27) Choose a l i n e a r transformation between the x,y sys-tem and the s,t system 3 7 160 x = i { ( l - s ) (l-t)xi+ (1+s) (l-t)x 2+ (1 + s) (l+t)x 3+ (1-s) (l+t)x„} y = i {(1-s) ( l - t ) Y l + ( l + s ) ( l - t ) y 2 + ( l + s ) (1+t)y 3+(1-s) ( l + t ) y 4 } (A.28) Equations (A.28) may be d i f f e r e n t i a t e d and s u b s t i -t u t e d i n (A.27) to g i v e 9 9s J n J 12 9 9x 9 J 21 J 22 9 LsyJ T h i s may be i n v e r t e d to g i v e 9 9x In I 12 9 9s 9 L 9 Y J I 21 I 22 3 _ 9 TL (A.30) Note a l s o t h a t dxdy = d e t [ j ] d s d t (A.31) 169 Applying equations (A.15) to the above equations T xy I 1 10 1 1+1 1 2 0 1 4 0 I 1 1 4> 2 1 + 1 12^21 0 I 1 1 0 3 1+1 1 2 0 3 4 0 0 I 1 1 <f> 1 2+1 1 2 <J> 1 5 0 I 1 1 0 1 3+1 1 2 0 1 6 0 II 1 0 2 2+1 1 2 0 2 5 0 I l 10 2 3 + 1 1 2 0 2 6 0 I l 1 0 3 2 + 1 1 2 0 3 5 0 I 1 1 0 3 3 + 1 1 2$ 3 6 0 II 1042+Il2045 0 I l 104 3+Il2046 0 0 I 2 10 1 1+ I 2 20 1 it 0 I 2 102 1+12202 4 0 I 2 1 0 3 1+ I 2 2 ty 3 4 0 I 2 1 0 4 1 + I 2 2 0 4 4 0 I 2 1 0 1 2+12 2 ty 1 5 0 I 2 10 1 3+1 2 2 4> 1 6 0 I 2 1 0 2 2 + 1 2 2 0 2 5 0 I 2 1 0 2 3 + 1 2 2 0 2 6 .0 12 10 3 2+12 2 0 3 5 0 1 2 1 0 3 3 +12 2 0 3 6 • 0 1 2 1 0 4 2 + 1 2 2 0^5 •2 1 0 4 3 + 1 2 2 0 4 6 I 2 II I 2 II I 2 II I 2 II I 2 II I 2 II I 2 II I 2 II I 2 II I 2 II I 2 II I 2 II 0 1 1 + 12 2 0 1 4 0 1 1+1 1 2 0 1 4 0 21+ 1 2 2 02 4 0 21+ 1 1 2 024 0 3 1 + 1 2 2 0 3 4 0 3 1+1 1 2 0 3 4 0 4 1+12 2 0 4 4 0 4 1 + 1 1 2 0 4 4 0 1 2 + 12 20 1 5 0 1 2+1 1 2 0 1 5 0 1 3+12 20 1 6 013+ 1 1 2 016 02 2 + 1 2 2 02 5 02 2 + 1 1 2 02 5 02 3 + 1 2 2 0 2 6 02 3+1 1 2 0 2 6 0 3 2 + I 2 2 0 3 5 0 3 2+1 1 2 0 3 5 0 3 3 + 12 2 0 3 6 0 3 3+1 1 2 0 3 6 0 4 2 + 1 2 2 0 4 5 0 4 2 + I l 2 0 4 5 0 4 3 + 12 2 0 4 6 0 4 3+1 1 2 0 4 6 r 1 r 2 r 3 r 4 r 5 r e r 7 r 8 r 9 r 10 r 11 r 12 r i 3 r 14 r 15 r is r 17 r is r 19 r 20 r 2 i r 22 r 23 r 24 (A.32) 170 where 0 11 = - (1-t) {-10+9(s 2+t 2)}/32+9s(1-s)(l-t)/16 0 12 = -9s (1-t) (l-3s)/16-27(1-t) (l-s 2)/32 0 13 = -9s (1-t) (l+3s)/16+27(1-t)(l-s 2)/32 4> it = - (1-s) [-10+9(s 2+t 2)}/32+9t(1-s)(1-t)/16 0 15 = -9 (1-s (l-3s)/32 0 16 = -9 M 2 (1-s , (l+3s)/32 021 = (1-t) 1 [-10+9 (s 2+t 2)}/32 + 9s(1+s) (l-t)/16 022 = 9 (1-t 2) (l-3t)/32 0 23 = 9 (1-t 2) (l+3t)/32 024 = - (1+s)^ [-10+9(s 2+t 2)}/32+9t(1+s)(1-t)/16 025 = -9t (1+s) (l-3t)/16-27(1+s) ( l - t 2 ) / 3 2 0 26 = -9t (1+s) (l+3t)/16+27(1+s)(l-t 2)/32 0 31 = (1+t)\ [-10+9(s 2+t 2)}/32+9s(1+s)(l+t)/16 0 32 = -9s (1+t) (l+3s)/16+27(1+t) (l-s 2)/32 0 33 = -9s (1+t) (l-3s)/16-27(1+t) (l-s 2)/32 0 34 = (1+s)J [-10 + 9 (s 2+t 2) }/32 + 9t(1+s) (l+t)/16 0 35 = 9 (1-s ) (l+3s)/32 0 36 = 9 (1-s ) (l-3s)/32 0 41 = -(1+t)\ [-10+9 (s 2+t 2)}/32 + 9s(1-s) (l+t)/16 0 42 = -9 d - t 2 ) (l+3t)/32 043 ~ -9 [1-t 2) (l-3t)/32 044 = (1-s)< '-10+9(s 2+t 2)}/32 + 9t(1-s) (l+t)/16 0 45 = -9t :i-s) (l+3t)/16+27(1-s) ( l - t 2 ) / 3 2 0 46 = -9t '1-s) (l-3t)/16-27(1-s) ( l - t 2 ) / 3 2 (A.33) 1 7 1 From equations (A.25) and (A.32) xy 4) 14 4) 1 7 _ T ~ S l 4> 12 4) 15 4)18 s 2 41 13 16 4)19 s 3 ^21 4)24 4)27 S 4 ^ 22 4)25 4)28 s 5 ^ 23 4)26 4)29 s 6 4> 31 4) 34 4) 37 s 7 ^ 32 4) 35 4) 38 s 8 IP 33 4)36 4)39 S g 4^1 4)44 4)47 S 10 4*42 4>45 4)48 S 11 4)43 4)46 4)49 S 12 (A.34) where \p. - I <j>. +1 <J>. +a. (I cj) . +1 cj). )+a. (I c|> . +1 4>. )+a0 (I <J>0 +1 cf>0 ) l l 11 l l 12 11 l l 11 12 12 15 l l 11 13 12 16 * 4 11 X 2 12 ^5 +ap (I 4>p +1 4>p ) JO 1 1 1 X , 3 12 *6 i K = a (I <j> +1 <j> -I (J> -I (J) )+a (I cj) +1 cj> -I cj) -I $ ) ! 2 12 11 13 12 16 11 12 12 15 * 2 11 X- 2 12 * 5 11 *> 3 12 ^6 rt. = -a (2(1 d> +1 cj) )+I cj> +1 d> )+a (2(1 cj) +1 cj> )+I cj) +1 <j)p ) 1 3 13 11 12 12 15 11 13 12 16 * 3 11 *• 3 12 X. 6 11 * 2 12 ^5 <J>i = a (I cj) +1 <j>. -I 0. -I <f>. )+a. (I d> +1 ch - i cj> -I cj) ) l l * 12 2 1 1 3 2 2 1 6 2 1 1 2 2 2 1 5 2 21 * 2 2 2 *• 5 21 X, 3 2 2 X< 6 Ui. = I cj>. +1 cj) . +a. (I cj) . +l c(>. )+a. (I cj> . +1 cj> . ) 15 21 l l 22 14 15 21 12 22 15 17 21 13 22 16 +a„ (I 0 +i cj) )+a. (I cj) +i cj) ) £ 7 21 X<2 2 2 * 5 *5 2 1 * 3 2 2 X. 6 rt. = a. (2(1 cj). +1 cj). )+i cj). +l cj). )-a (2(1 <j> +1 <f> » e) +12 1 <J>. +1 * ? ) r l 6 16 2 1 12 2 2 15 2 1 1 3 2 2 1 6 X, 6 2 1 Jt 3 2 2 l lz 22 X, 5 rt . = I cj). +1 cj) . +a. (I cj) . +1 cj) . )+a. (I cj) . +1 cj) . ) 17 21 l l 22 l i | l l 2 1 12 2 2 1 5 1 4 2 1 1 3 2 2 1 6 +ap (I 0 +1 <}>„ )+a (I cj) +1 cj) )+rt 21 X-2 2 2 * 5 *. 1 21 3 2 2 * 6 1 2 rt. = I cj). +1 0. +a. (I cj). +1 cj). )+a. (I cj) . +1 <J> . ) 18 11 l l 12 14 15 11 12 12 15 17 11 13 12 16 +a0 (I Cj) +1 Cj) )+a (I Cj) +1 Cj) )+rt X/7 11 *2 12 * 5 £ 5 11 >i 3 12 X, 6 1 •* rt. = -a. (2(1 cj) . +1 <j). )+I cj). +1 cj). )+a. (2(1 <J> . +1 cj> . )+1 0. +l 0. ) l g 13 21 12 22 15 21 1 3 22 16 16 11 12 12 15 11 13 12 16 +a0 (2(1 cj, +1 cj) )+i cj) +1 cj) )-a„ (2(1 cj) +1 cj) )+l cj) +1 cj> ) X, 3 21 ^ 3 2 2 X-6 2 1 & 2 2 2 * 5 * 6 11 *3 12 *< 6 11 * 2 12 X. 5 i,j,k,x\ are c y c l i c permutations of 1,2,3,4 (A.35) —J From equation (A.34) (e) = [y](s) and the s t i f f n e s s matrix i s given by: T [K] = t [V] [E] [y]dxdy t / F(s,t)det [j]dsdt from equation (A.31) Integrating numerically n n 37 [K] = t /__, /__, H H F(s , t )det[j] > j=l i = i 1 3 i i (A.36) where H.,H. i 3 n weighting functions thickness of element number of integrating points i n each d i r e c t i o n The integrand contains terms i n s and t up to the s i x t h power, and for exact numerical integration a set of 4 x 4 Gauss quadrature points would be required. 3 7 Lower order integration on a 3 x 3 g r i d of integration points yielded a successful element and was therefore retained. A. 3 Patch Tests 1' 2 on Elements The displacement functions for the triangular and 174 q u a d r i l a t e r a l e l e m e n t s a r e c o m p a t i b l e b u t i n c o m p l e t e . E i g e n -v a l u e t e s t s on s t i f f n e s s m a t r i c e s o f b o t h e l e m e n t s y i e l d e d t h r e e z e r o e i g e n v a l u e s f o r e a c h , w h i c h p r o v e d t o be i n d e p e n -d e n t o f e l e m e n t shape o r o r i e n t a t i o n , i n d i c a t i n g t h e e x i s t -e n c e o f t h r e e r i g i d body modes. However, i t was f o u n d t h a t t h e r o t a t i o n c o n s t r a i n t s a t e a c h c o r n e r o f an e l e m e n t p l a c e d a r e s t r i c t i o n on t h e s h e a r s t r a i n t h e r e . I n any e l e m e n t i n a mesh u n d e r a p u r e s h e a r c o n d i t i o n , t h e s h e a r s t r e s s i s z e r o i n a r i g h t - a n g l e d c o r n e r and m a r k e d l y low i n a c o r n e r o f any o t h e r a n g l e . F o r a p a t c h 4 2 o f a s s e m b l e d e l e m e n t s , w i t h a t l e a s t one i n t e r n a l u n l o a d e d and u n r e s t r a i n e d node, u n d e r l o a d s c o n s i s t e n t w i t h a p u r e s h e a r c o n d i t i o n , i t was f o u n d t h a t t h e s t r a i n e n e r g y o f a p a t c h was c o n s i s t e n t l y 9.1% t o o low f o r p a t c h e s c o n s i s t i n g o f q u a d r i l a t e r a l s and 9.3% t o o low f o r t h o s e o f t r i a n g l e s . T h e s e v a l u e s r e m a i n e d c o n s t a n t i n d e p e n d -e n t l y o f mesh r e f i n e m e n t . I n a d d i t i o n , t h e t r i a n g u l a r e l e -ments c o n s i s t e n t l y y i e l d e d s t r a i n e n e r g i e s 4.9% t o o low, i n -d e p e n d e n t l y o f mesh r e f i n e m e n t , u n d e r p u r e d i r e c t s t r e s s . I n any a n a l y s i s u s i n g t h e s e e l e m e n t s i t w o u l d n e v e r be p o s s i b l e f o r t h e s t r a i n e n e r g y t o c o n v e r g e t o t h e c o r r e c t v a l u e . The s t r a i n e n e r g y w o u l d a l w a y s be t o o low b e c a u s e o f t h e i n c o m p l e t e n e s s o f t h e d i s p l a c e m e n t f u n c t i o n s . I t i s s i g n i f i c a n t t h a t p a t c h t e s t s on t h e s e e l e m e n t s i n d i c a t e t h a t t h e y a r e t o o s t i f f by c o n s t a n t amounts, w h i c h a r e i n d e p e n -d e n t o f mesh r e f i n e m e n t . T h i s makes i t p o s s i b l e t o m u l t i p l y t h e r e l e v a n t columns o f t h e e l e m e n t s t r a i n m a t r i x by a c o n -175 stant i n order that the resultant s t i f f n e s s matrix may be correspondingly more f l e x i b l e i n the pure stress states. The constants that were applied were determined on a t r i a l -and-error basis and are c l e a r l y v i s i b l e i n the subroutines presented i n the next section. In addition, the shear s t r a i n s i n the trian g u l a r element were made constant through-out by using the centroidal value of shear s t r a i n for a l l integration points i n each element. S i m i l a r l y , the s t r a i n c a l c u l a t i o n subroutines for these elements have constants applied to the relevant columns of the s t r a i n matrix to give much-improved s t r a i n s . The shear st r a i n s of the q u a d r i l a t e r a l element are l i n e a r l y i n -terpolated from the strains calculated at the four points with coordinates s=t=±0.55735. These points were found to have shear s t r a i n s very close to the correct values under a l l three d i f f e r e n t constant stress states. Patch tests on the improved elements indicated, with some exceptions, that stresses and deflections were consis-tent with boundary conditions and applied loads. The excep-tions were d i r e c t stresses at the t r i a n g l e nodes which were too low and the Poisson e f f e c t of the t r i a n g l e s , which was consistently 85% of what i t should have been. The s t r a i n energies for a t y p i c a l set of patch tests are presented i n Table VIII. Coarser triangular meshes y i e l d low s t r a i n ener-gies i n a l l three constant s t r a i n nodes, but appear to improve rap i d l y with further mesh refinement. TABLE VIII STRAIN ENERGIES FROM PATCH TESTS ON ELEMENTS STRAIN ENERGY (Kip. in.) QUADRILATERAL FINITE ELEMENT Finite Element Mesh CALC. EXACT CALC. EXACT CALC. EXACT Pure Shear x=8k.s.i. 1.33333 1.33333 1.33463 1.33333 2.66666 2.66667 Pure Compression o=-8k.s.i. 0.53333 0.53333 0.53409 0.53333 1.06666 1.06667 Pure Bending ~ t oP ^bot , 0 1 a c=-o =10k.s.i. X X 0.27327 0.27778 0.27337 0.27778 0.54146 0.55556 TRIANGULAR FINITE ELEMENT Finite Element Mesh CALC. EXACT CALC. EXACT CALC. EXACT Pure Shear x=8k.s.i. 1.35422 1.33333 1.34652 1.33333 2.80364 2.66667 Pure Compression o =-8k.s.i. X 0.53284 0.53333 0.53064 0.53333 1.04648 1.06667 Pure Bending top hot , o ^--o =10k.s.x. X X 0.26986 0.27778 0.277037 0.27778 0.55191 0.55556 A. 4 Element S t i f f n e s s and Strain Calculation Subroutines A.4.1 L i s t of Variables i n Subroutine Input/Output L i s t s PLSTRI X,Y EXY T S PLSQUA X,Y EXY TH SM Coordinates of element nodes Lower t r i a n g l e of e l a s t i c i t y matrix stored as column vector Element thickness Lower t r i a n g l e of s t i f f n e s s matrix stored as column vector TRISTN X,Y DEF EPS II QUASTN X,Y DEF EPS IJ Coordinates of element nodes Vector of nodal displacements Matrix of str a i n s with centroidal s t r a i n s i n f i r s t row and nodal str a i n s i n succeed-ing rows If value = 1, centroidal s t r a i n s only are calculated, otherwise centroidal and nodal strains are calculated PLSTRI - PLANE STRESS/STRAIN TRIANGLE STIFFNESS MATRIX SUBROUTINE PLSTRI(X,Y,EXY,T,S) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3),S(45),A(3,7),B(3),C(3),RL(3),P(3,8),Q(3,9) ,SPA CD (3,9) ,SPAD1(45),Z(7,3),W(7) ,EXY (6) DATA Z/.3 333333333333333D0,.0597158717897698D0,.4701420641051151D0 C,.4701420641051151D0,.7974269853530 872D0,.1012865073234563D0,.1012 C8 65 07 3 23 45 6 3D0,.3333333333333333D0,.470142 0641051151D0,.05 9715 8 717 C8 976 98D0,.4701420641051151D0,.1012 865 07 32 34 563D0,.797 42 6985353 0872 CDO,.1012865073234563D0,.3333333333333333D0,.47 0142 06 41051151D0,.4 7 C0142 06 41051151D0,.0597158717897698D0,. 1012865073234563D0,.1012 865 0 C7 32 345 6 3D0,.7 97 42 6 985 353 0872D0/ DATA W/.225D0,3*.1323941527885062D0,3*.1259391805448271D0/ DO 1 1=1,45 S (I)=0.D0 DO 2 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3)*3 B(I)=Y(J)-Y(K) C(I)=X(K)-X(J) RL(I)=DSQRT(B (I)**2+C(I)**2) A(I,1)=2.D0*C(I)**2/(3.D0*RL(I)**2)+20.D0*B(I)**2/(2 7.D0*RL(I)**2) A(I,2)=2.D0*B(I)*C(I)/(27.D0*RL(I)**2) A(I,3)=2.D0*B(I)/27.D0 A(I,4)=C(I)**2/(3.D0*RL(I)**2)+7.DO*B(I)**2/(27.DO*RL(I)**2) A(I,5)=2.D0*B(I)**2/(3.D0*RL(I)**2)+2 0.D0*C(I)**2/(27.D0*RL(I)**2) A(I/6)=2.D0*C(I)/2,7.D0 A(I,7)=B(I)**2/(3.D0*RL(I)**2)+7.D0*C(I)**2/(27.DO*RL(I)**2) AREA=(B(1)*C(2)-B(2)*C(1))/2.D0 DO 3 L=l,7 DO 4 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3) *3 P(I,1)=B(J)*(13.5D0*Z(L,J)**2-9.D0*Z(L,J)+1.DO)/(2.D0*AREA) P(I,2)=4.5D0*(B(I)*Z(L,J)*Z(L,K)+B(K)*Z(L,I)*Z(L,J)+B(J)*Z(L,K)*Z ( CL,I))/(2.D0*AREA) P(I,3)=9.D0*(B(J)*Z(L,K)*(3.D0*Z(L fJ)-0.5D0)+B(K)*Z(L,J)*(1.5D0*Z ( CL,J)-0.5D0))/(2.DO*AREA) P (I,4)=9.D0*(B(J)+Z(L,K)*(1.5D0*Z(L,K)-0.5D0)+B(K)*Z(L,J)*(3.D0*Z( CL,K)-0.5D0))/(2.DO *AREA) P (I,5)=C (J)*(13.5D0*Z(L, J)**2-9.D0*Z(L,J)+1.DO)/(2.DO*AREA) P(I,6)=4.5D0*(C(I)*Z(L,J)*Z(L,K)+C(K)*Z(L,I)*Z(L,J)+C(J)*Z(L,K)*Z ( CL,I))/(2.DO*AREA) P(I,7)=9.DO*(C(J)*Z(L,K)*(3.D0*Z(L,J)-0.5D0)+C(K)*Z(L,J)*(1.5D0*Z ( CL,J)-0.5D0))/(2.D0*AREA) P (I,8)=9.D0*(C(J)*Z(L,K)*(1.5D0*Z(L,K) -0.5D0)+C(K)*Z(L,J)*(3.DO*Z( CL,K)-0.5D0))/(2.D0*AREA) CONTINUE DO 5 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3)*3 M1=1+(J-1)*3 M2=M1+1 M3^ =Ml + 2 Q(1,M1) = (P (I,1)-P(I,2) )+A(I,l) * (P(I,3)+1.5D0*P(I,2) )+A (1, 4 ) * (P (1, 4 C)+1.5D0*P(1,2))+A(K,4)*(P(K,3)+1.5D0*P(K,2))+A(K,1)*(P(K,4)+1.5D0* CP(K,2) ) Q(1,M2)=A(I,2) * (P(I,3)-P (1,4) )+A(K,2) * (P (K, 4 )-P (K, 3 ) ) Q(1,M3)=A(1,3)*(4.5D0*P(I,2)+2.DO*P(I,3)+P(I,4))-A(K,3)*(4.5D0*P(K C,2)+P(K,3)+2.D0*P(K,4) ) Q(2,M1)=A(I,2) *(P(I,7)-P(I,8) )+A (K, 2 ) * (P (K, 8 )-P (K, 7 ) ) Q(2,M2) = (P(I,5)-P(1,6))+A(I,5)*(P(1,7)+1.5D0 *P(1,6) )*A(1,7)*(P(1,8 C)+1.5D0*P(1,6))+A(K,7)*(P(K,7)+1.5D0 *P(K,6))+A(K,5)*(P(K,8)+1.5D0* CP(K,6) ) Q(2,M3)=A(I,6)*(4.5D0*P(1,6)*2.D0*P(1,7)+P(1,8))-A(k,6)*(4.5D0*P (K C,6)+P(K,7)+2.D0*P(K,8)) Q(1,M1)=Q(1,M1)/1.025D0 Q(1,M2)=Q(1,M2)/1.025D0 Q(1,M3)=Q(1,M3)/1.025D0 Q(2,M1)=Q(2,M1)/1.025D0 Q(2/M2)=Q(2,M2)/l.025D0 Q(2,M3)=Q(2,M3)/1.025D0 IF(l-L)5,7,7 Q(3,Ml) = (P(I,5)-P (1,6) )+A(I,l) * (P (I,7)+1.5D0*P (1, 6 ) )+A (1, 4 ) * (P (1, 8 C)+1.5D0*P(I,6))+A(K,4)*(P(K,7)+1.5D0 *P(K,6))+A(K,1)*(P(K,8)+l.5D0* CP(K,6))+Q(l,M2) Q(3 fM2)=(P(I,l)-P(1,2))+A(I,5)*(P(1,3)+1.5D0*P(I,2))+A(I,7)*(P(I,4 C)+1.5D0*P(1,2))+A(K,7)*(P(K,3)+l.5D0*P(K,2))+A(K,5)*(P(K,4)+1.5D0* CP (K,2) )+Q(2,Ml) Q(3,M3)=A(I,3) * (4.5D0*P(I,6)+2.D0*P(I,7)+P (I,8))+A(I,6)* (4.5D0*P (I C,2)+2.D0*P(I,3)+P(1,4))-A(K,3)*(4.5D0*P(K,6)+P(K,7)+2.DO*P (K,8))-A C (K,6)*(4.5D0*P(K,2)+P(K,3)+2.DO*P(K,4)) Q(3,M1)=Q(3,M1)/l.165D0 Q(3,M2)=Q(3,M2)/1.165D0 Q(3,M3)=Q(3,M3)/1.165D0 CONTINUE CALL DIMULT(EXY,Q,SPAD,3,9,3,3,0) CALL REMULT(Q,SPAD,SPAD1,9,3,3,3,1,0,NC) DO 6 1=1,45 S (I)=S (D+SPAD1 (I) *W(L) *AREA*T CONTINUE RETURN END C TRISTN - CALCULATION OF STRAINS IN TRIANGLE SUBROUTINE TRISTN(X,Y,DEF,EPS,II) -IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3) ,Y(3) ,DEF(9) ,EPS (5,3) ,B(3),C(3),P(3,8),Q(3,9),Z(4,3) CSPAD (3) ,RL(3) ,A(3,7) DATA Z/.3 33 3 3 33 33 333 3 333D0,1.D0,0.D0,0.D0,.3333333333333333D0,0.DO C,1.D0,0.D0,.33 333 333 3333 3333D0,0.D0,0.D0f1.D0/ DO 2 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3)*3 B(I)=Y(J)-Y(K) C (I)=X(K)-X(J) RL(I)=DSQRT(B(I)**2+C(I)**2) A (1,1)=2.DO*C(I)**2/(3.D0*RL(I)**2)+2 0.D0*B(I)**2/(27.DO*RL(I)**2) A(I,2)=2.D0*B(I)*C(I)/(27.D0*RL(I)**2) A(I,3)=2.D0*B(I)/27.D0 A(I,4)=C(I)**2/(3.D0*RL(I)**2)+7.D0*B(I)**2/(27.D0*RL(I)**2) A(I,5)=2.D0*B(I)**2/(3.D0*RL(I)**2)+2 0.D0*C(I)**2/(27.D0*RL(I)**2) A(I,6)=2.D0*C(I)/27.D0 2 A(I /7)=B(I)**2/(3.D0*RL(I)**2)+7.DO*C(I)**2/(27.D0*RL(I)**2) AREA= (B (1) *C(2)-B(2) *C(1) )/2.D0 IF(II-1)7,8,7 7 NN=4 GO TO 9 8 NN=1 9 DO 3 L=1,NN DO 4 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3)*3 P(I,1)=B(J)*(13.5D0*Z(L,J)**2-9.D0*Z(L,J)+1.DO)/(2.D0*AREA) P (I,2)=4.5D0*(B(I)*Z(L,J) *Z(L,K)+B(K)*Z(L,I)*Z(L #J)+B(J)*Z(L,K)*Z ( CL,I))/(2.D0*AREA) P (I,3)=9.DO*(B(J)*Z(L,K)*(3.D0*Z(L,J)-0.5D0)+B(K)*Z(L,J)*(1.5D0*Z( CL,J)-0.5D0))/(2.D0*AREA) P (I,4)=9.D0*(B (J)*Z(L,K)*(1.5D0*Z(L,K)-0.5D0)+B(K)*Z(L,J)*(3.D0*Z( CL,K)-0.5D0))/(2.DO *AREA) P(I,5)=C(J)*(13.5D0*Z(L,J)**2-9.D0*Z(L,J)+1.DO)/(2.DO*AREA) P (I,6)=4.5D0*(C (I)*Z(L,J)*Z(L,K)+C(K)*Z(L,I)*Z(L,J)+C(J)*Z(L,K)*Z ( CL,I) )/(2.D0*AREA) P(I,7)=9.DO*(C(J)*Z(L,K)*(3.D0*Z(L,J)-0.5D0)+C(K)+Z(L,J)*(1.5D0*Z ( CL,J)-0.5D0))/(2.D0*AREA) P (I,8)=9.D0*(C(J)*Z(L,K)*(1. 5D0*Z(L,K)-0.5D0)+C(K)*Z(L,J)* (3.DO*Z ( CL,K)-0.5D0))/(2.DO *AREA) 4 CONTINUE DO 5 J=l,3 I=J+2-(J/2)*3 K=J+1- (J/3)*3 Ml=l+ (J -D+3 M2=M1+1 M3=Ml+2 Q(1,M1) = (P(I,1)-P(I,2) )+A(I,l) *(P(I,3)+1.5D0*P (1, 2 ) )+A (1, 4 ) * (P (I f 4 C)+1.5D0*P(1,2) )+A(K,4) *(P(K,3)+1.5D0*P(K,2))+A(K,1)*(P(K,4)+1.5D0* CP (K,2)) Q(1,M2)=A(I,2) *(P(I,3)-P(I,4) )+A(K,2) * (P (K, 4)-P (K, 3) ) Q(1,M3)=A(I,3) *(4.5D0*P(1,2)+2.DO*P(1,3)+P(1,4))-A(K,3)*(4.5D0*P (K C,2)+P(K,3)+2.D0*P(K,4)) Q(2,M1)=A(I,2)*(P(I,7)-P(I,8) )+A(K,2)* (P (K, 8)-P (K , 7) ) Q(2,M2)=(P(I,5)-P(I,6) )+A(I,5)*(P(I,7)+1.5D0*P(I,6) )+A(I,7) *(P(I,8 C)+1.5D0*P (1,6) )+A(K,7) * (P (K, 7 )+1. 5D0* P (K, 6 ) )+A(K,5) * (P (K, 8 )+1. 5D0* CP (K,6)) Q(2 fM3)=A(I # 6)*(4.5D0*P(I,6)+2.D0*P(1,7)+P(1,8))-A(K,6)*(4.5D0*P (K C,6)+P(K,7)+2.D0*P(K,8)) Q(1,M1)=Q(1,M1)*.9316D0 Q(1,M2)=Q(1,M2)*.9316D0 Q(1,M3)=Q(1,M3) *.9316D0 Q(2,M1)=Q(2,M1) *.9316D0 Q(2,M2)=Q(2,M2)*.9316D0 Q(2,M3)=Q(2,M3) *. 9316D0 IF(1-L)5,10,10 10 Q(3 fMl) = (P(I,5)-P(I,6) )+A(I,l) * (P (I,7)+1.5D0*P (I,6))+A(I,4)*(P(I,8 C)+1.5D0*P(I,6))+A(K,4)*(P(K,7)+1.5D0*P(K,6))+A(K,1)*(P(K,8)+1.5D0* CP (K,6) )+Q(l,M2) M CO to Q(3,M2) = (P (I,1)-P(I,2) )+A(I, 5) * (P (1, 3)+1. 5D0*P (1,2) )+A(I,7) * (P (1,4 C)+1.5D0*P(1,2))+A(K,7)*(P(K,3)+1. 5D0*P(K,2))+A(K,5)*(P(K,4)+1.5D0* CP(K,2) )+Q(2,Ml) Q(3,M3)=A(I,3) * (4.5D0*P (1, 6)+2 . DO *P (1, 7 )+P (1,8) ) +A (I,6)*(4.5D0*P(I C,2)+2.D0*P(I,3)+P(1,4))-A(K,3)*(4.5D0*P(K,6)+P(K,7)+2.D0*P(K,8))-A C(K,6)*(4.5D0*P(K,2)+P(K,3)+2.D0*P(K,4)) Q(3,Ml)=Q(3,Ml)/1.131D0 Q(3,M2)=Q(3,M2)/1.131D0 Q(3,M3)=Q(3,M3)/1.131D0 CONTINUE CALL MAMULT(Q,DEF,SPAD,3,9,1,3,9,3,0,0) DO 6 1=1,3 EPS(L,I)=SPAD(I) CONTINUE RETURN END PLSQUA - PLANE STRESS/STRAIN QUADRILATERAL STIFFNESS MATRIX SUBROUTINE PLSQUA(X,Y,EXY,TH,SM) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(4),Y(4),SM(78) ,EXY(6) ,SS(3),W(3),RL(4),P(4,6),A(4,7),Q C(3,12) ,SPAD(3,12) , SPADl (78),B(4),C(4) DATA SS/-.774596669241483D0,0.DO,.774596669241483D0/ DATA W/.555555555555556D0,.888888888888889D0,.555555555555556D0/ DO 1 1=1,7 8 SM(I)=0.D0 DO 2 1=1,4 J=I+1-(1/4)*4 B(I)=Y(J)-Y(I) C (I)=X(J)-X(I) RL(I)=DSQRT(B(I) *B (I)+C (I) *C (I) ) A (1,1) = ( (C(I) /RL(i') ) **2) *2.D0/3.D0+( (B (I)/RL(I) ) **2)*20.DO/27.DO A(I,2) = (B(I) *C(I)/(RL(I) **2) ) *2. DO/27. DO A(I,3)=B(I)*2.DO/27.DO A (I, 4)= ( (C(I)/RL(I) ) **2)/3.D0+( (B(I)/RL(I) ) **2)*7.DO/27.DO A (1,5)= ( (B(I)/RL(I))**2)*2.D0/3.D0+((C(I)/RL(I) )**2)*20.DO/27.DO A(I,6)=C(I) *2.D0/27.D0 A (I,7) = ( (B(I)/RL(I))**2)/3.D0+((C(I)/RL(I)) **2)*7.DO/27.DO DO 3 K=l,3 DO 4 J=l,3 T=SS(K) S=SS (J) RJ11=( (T-1.D0) * (X(l)-X(2) ) + (T+l.D0) * (X (3)-X (4) ) ) / 4 . DO RJ12=( (T-1.D0)*(Y(1)-Y(2) ) + (T+l. DO) * (Y (3)-Y (4 ) ) )/4.D0 RJ21=((S-l.DO)*(X(1)-X(4))+(S+1.D0)*(X(3)-X(2)))/4.DO RJ22=((S-l.DO)*(Y(l)-Y(4))+(S+1.D0)*(Y(3)-Y(2)))/4.D0 DET=RJ11*RJ2 2-RJ12*RJ21 RI11=RJ22/DET RI12=-RJ12/DET RI21=-RJ21/DET RI22=RJ11/DET CON=-l0.DO+9.DO*(S*S+T*T) P (1,1) = ((T-1.D0)*CON+18.D0*S*(1.D0-S)*(1.D0-T))/32.D0 P (1,2) = (T-1. DO) * (18.D.0*S* (l.D0-3.D0*S)+2 7.D0* (l.DO-S*S) J/32.D0 P (1,3)= (T-1.DO)*(18.D0*S*(l.D0+3.D0*S)-2 7.D0*(l.DO-S*S) J/32.D0 P (1,4)=((S-l.DO)*CON+18.D0*T*(l.DO-S)*(l.DO-T)J/32.D0 P (1,5)=-9.DO*(l.DO-S*S)*(l.D0-3.D0*S)/32.D0 P (1,6)=-9.DO* (l.DO-S*S)*(l.D0+3.D0*S)/32.D0 P (2,1)= ( (l.DO-T)*CON+18.D0*S*(l.DO+S)*(l.DO-T)J/32.D0 P (2,2) = 9.DO*(l.DO-T*T)*(1.DO-3.DO*T)/32.DO P(2,3)=9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P (2,4)= (- (l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO-T)J/32.D0 P (2,5)=- (l.DO+S)*(18.D0*T*(1.DO-3.DO*T)+27.DO*(l.DO-T*T)J/32.D0 P (2,6)=-(l.DO+S)*(18.D0*T*(1.DO+3.DO*T)-27.DO*(l.DO-T*T)J/32.D0 P (3,1) = ( (l.DO+T)*CON+18.D0*S*(l.DO+S)*(l.DO+T)J/32.D0 P(3,2)=(l.DO+T)*(-18.D0*S*(1.DO+3.DO*S)+27.DO*(l.DO-S*S))/32.D0 P (3,3)= (l.DO+T)*(-18.D0*S*(1.DO-3.DO*S)-27.DO*(l.DO-S*S))/32.D0 P (3,4)= ( (l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO+T))/32.D0 P (3,5)=9.DO* (l.DO-S*S)*(1.DO+3.DO*S)/32.DO P(3,6)=9.D0* (l.DO-S*S)*(1.DO-3.DO*S)/32.DO P(4,1)=(-(l.DO+T)*CON+18.D0*S*(l.DO-S)*(l.DO+T))/32.D0 P(4,2)=-9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P(4,3)=-9.DO*(l.DO-T*T)*(1.DO-3.DO*T)/32.DO P (4,4)=((l.DO-S)*CON+18.D0*T*(l.DO-S) *(l.DO+T))/32.D0 P (4,5)=- (l.DO-S)*(18.D0*T*(1.DO+3.DO*T)-27.DO*(l.DO-T*T))/3 2.D0 P(4,6)=-(l.DO-S)*(18.D0*T*(1.DO-3.DO*T)+27.DO*(l.DO-T*T))/32.D0 DO 5 1=1,4 Ml=l+ (1-1)*3 M2=M1+1 M3=Ml+2 L=I-1+(1/1)*4 Q (1,M1)=RI11*P(I,1)+RI12*P(1,4)+A(1,1)*(RI11*P(I,2)+RI12*P(1,5))+A C(I,4)*(RI11*P(1,3)+RI12 *P(1,6))+A(L,4)*(RIll*P(L,2)+RI12*P(L,5))+A C(L,l)*(RI11*P(L,3)+RI12 *P(L,6)) Q(1,M2)=A(I,2)*(RI11*(P(I,3)-P(1,2))+RI12*(P(I,6)-P(I,5)))+A(L,2)* C(Rill*(P(L,2)-P(L,3))+RI12*(P(L,5)-P(L,6))) Q(1,M3)=-A(I,3)*(Rill*(2.D0*P(I,2)+P(1,3))+RI12*(2.DO*P(1,5)+P (1,6 C)))+A(L,3)*(Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.D0*P(L,6)+P(L,5))) Q(2,M1)=A(I,2)*(RI21*(P(1,3)-P(1,2))+RI22 *(P(1,6)-P(1,5)))+A(L,2)* C(RI21*(P(L,2)-P(L,3))+RI2 2*(P(L,5)-P(L,6))) Q(2,M2)=RI21*P(I,1)+RI22*P(1,4)+A(1,5)*(RI21*P(I,2)+RI22*P (1,5))+A 0(1,7)*(RI21*P(I,3)+RI22*P(I,6))+A(L,7)*(RI21*P(L,2)+RI2 2*P(L,5))+A C(L,5)*(RI21*P(L,3)+RI2 2*P(L,6)) Q(2,M3)=A(I,6)*(RI21*(2.D0*P(I,2)+P(I,3))+Rl22*(2.D0*P(I,5)+P(1,6) C))-A(L,6)*(RI21*(2.D0*P(L,3)+P(L,2))+RI22*(2.DO*P(L,6)+P(L,5))) Q(3,M1)=RI21*P(I,1)+RI22*P(I,4)+A(I,1)*(RI21*P(I,2)+RI2 2*P(1,5))+A C(1,4)*(RI21*P(I,3)+RI22*P(I,6))+A(L,4)*(RI21*P(L,2)+RI22*P(L,5))+A C(L,1)*(RI21*P(L,3)+RI22*P(L,6))+Q(1,M2) Q(3,M2)=RI11*P(I,1)+RI12*P(1,4)+A(1,5)*(RI11*P(I,2)+RI12*P (1,5))+A C(I,7)*(RI11*P(I,3)+RI12*P(1,6))+A(L,7)*(RIll*P(L,2)+RI12*P(L,5))+A C(L,5)*(RI11*P(L,3)+RI12*P(L,6))+Q(2,M1) Q(3,M3)=-A(I,3)*(RI21*(2.D0*P(I,2)+P(1,3))+RI22*(2.DO*P(1,5)+P(1,6 C)))+A(I,6)*(Rill*(2.D0*P(I,2)+P(1,3))+RI12*(2.DO*P(1,5)+P(1,6)))+A C(L,3)*(RI21*(2.D0*P(L,3)+P(L,2))+Rl22*(2.DO*P(L,6)+P(L,5)))-A(L,6) C*(Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.D0*P(L,6).+P(L,5))) Q(3,M1)=Q(3,M1)/0.1048 808D+01 Q(3,M2)=Q(3,M2)/0.10488 08D+01 Q(3,M3)=Q(3,M3)/0.104 88 0 8D+01 CONTINUE CALL DIMULT(EXY,Q,SPAD,3,12,3,3,0) CALL REMULT(Q,SPAD,SPADl,12,3,3,3,1,0,NC) DO 6 1=1,78 SM(I)=SM(I)+SPAD1 (I) *W(K) *W(J) *TH*DET CONTINUE CONTINUE RETURN END C QUASTN - CALCULATION OF STRAINS IN QUADRILATERAL SUBROUTINE QUASTN(X,Y,DEF,EPS,IJ) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(4) ,Y(4) ,DEF (12) ,EPS(5,3) ,B(4) ,C (4) ,SS (5) ,TT(5) ,P (4,6) , 00(3,12),SPAD(3),A(4,7),RL(4),R(4,12) DATA SS/0.D0,-1.D0,1.D0,1.D0,-1.D0/,TT/0.D0,-1,D0,-1.D0,1.D0,1.D0/ DATA CT/.57735D0/ IF(IJ-1)130,120,130 120 NN=1 GO TO 140 130 NN=5 140 DO 2 1=1,4 J=I+1- (1/4)*4 B(I)=Y(J)-Y(I) C(I)=X(J)-X(I) RL(I)=DSQRT(B(I) *B(I)+C(I) *C(I) ) A(I,1) = ( (C(I)/RL(I) ) **2) *2.D0/3.D0+( (B(I)/RL(I) )* *2 ) *2 0. DO/27 . DO A (I, 2) = (B.(I) *C(I)/(RL(I) **2) ) *2. DO/27. DO A(I,3)=B(I)*2.DO/27.DO A(I,4)=((C(I)/RL(I))**2)/3.D0+((B(I)/RL(I))**2)*7.DO/27.DO A (I,5)=((B(I)/RL(I))**2)*2.D0/3.D0+((C(I)/RL(I))**2)*20.DO/27.DO A(I,6)=C(I)*2.D0/27.D0 2 A(I,7)=((B(I)/RL(I))**2)/3.D0+( (C(I)/RL(I))**2)*7.DO/27.DO M=0 DO 15 0 K=1,NN IF (M)7,7,8 7 S=-CT T=S N=l GO TO 9 12 S=-T N=2 GO TO 9 13 T=S N=3 GO TO 9 h- 1 oo N=4 GO TO 9 M=l N=5 T=TT(K) S=SS(K) RJ11=((T-1.D0)*(X(l)-X(2))+(T+1.D0)*(X(3)-X(4)))/4.DO RJ12= ( (T-1.D0) *(Y(1)-Y(2) ) + (T+l.D0) * (Y ( 3)- Y (4 ) ) )/4 . DO RJ21=((S-l.DO)*(X(l)-X(4))+(S+l.DO)*(X(3)-X(2)))/4.D0 RJ22=((S-l.DO)*(Y(l)-Y(4))+(S+l.DO)*(Y(3)-Y(2)))/4.DO DET=RJ11*RJ22-RJ12*RJ21 RIll=RJ22/DET RI12=-RJ12/DET RI21=-RJ21/DET RI22=RJ11/DET CON=-10.D0+9.D0*(S*S+T*T) P (1,1)= ( (T-l.DO)*CON+18.D0*S*(l.DO-S)*(l.DO-T))/32.D0 P(1,2)=(T-l.DO)*(18.D0*S*(l.D0-3.D0*S)+27.D0*(l.DO-S*S))/32.D0 P(1,3)=(T-l.DO)* (18.D0*S*(l.D0+3.D0*S)-27.DO*(l.DO-S*S))/32.D0 P (1,4)=((S-l.DO)*CON+18.D0*T*(l.DO-S)*(l.DO-T)J/32.D0 P (1,5)=-9.DO*(l.DO-S*S)*(l.D0-3.D0*S)/32.D0 P (1,6)=-9.DO*(l.DO-S*S)*(l.D0+3.D0*S)/32.D0 P(2,1)=((l.DO-T)*CON+18.D0*S*(l.DO+S)*(l.DO-T))/32.D0 P(2,2)=9.D0*(l.DO-T*T)*(1.DO-3.DO*T)/32.DO P(2,3)=9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P(2,4)=(-(l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO-T))/32.D0 P(2,5)=-(l.DO+S)*(18.D0*T*(1.DO-3.DO*T) +27. ODO*(l.DO-T*T))/32.DO P (2 ,6)=- (l.DO+S)*(18.DO*T*(1.DO+3.DO*T)-27. DO*(l.DO-T*T) )/32.D0 P (3,1)=( (l.DO+T)*CON+18.D0*S*(l.DO+S)*(l.DO+T)J/32.D0 P(3,2)=(l.DO+T)*(-18.D0*S*(1.DO+3.DO*S)+27.DO*(l.DO-S*S)J/32.D0 P(3,3)=(l.DO+T)*(-18.D0*S*(1.D0-3.DO*S)-27.DO*(l.DO-S*S))/32.D0 P(3,4)=((l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO+T)J/32.D0 P (3, 5) = 9. DO* (l.DO-S*S) * (1. DO+3 . DO*S)/32 . DO' P (3,6) = 9.DO*(l.DO-S*S)*(1.DO-3.DO*S)/32.DO P(4,1)=(-(1.DO+T)*CON+18.D0*S*(l.DO-S)*(l.DO+T))/32.D0 P (4,2)=-9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P (4,3)=-9.DO* (l.DO-T*T)*(1.DO-3.DO*T)/32.DO P (4,4)=((l.DO-S)*CON+18.D0*T*(l.DO-S)*(l.DO+T))/32.D0 P (4,5)=- (l.DO-S)*(18.D0*T*(1.DO+3.DO*T)-27.DO*(l.DO-T*T))/32.D0 P (4,6)=- (l.DO-S)*(18.D0*T*(1.DO-3.DO*T)+27.DO*(l.DO-T*T))/32.D0 DO 5 1=1,4 Ml=l+(1-1)*3 M2=M1+1 M3=Ml+2 L=I-1+(1/1)*4 IF (M)10,10,11 11 Q(1,M1)=RI11*P(I,1)+RI12*P(1,4)+A(1,1)*(RI11*P(I,2)+RI12*P (1,5))+A C (1,4)*(RI11*P(I,3)+RI12*P(1,6))+A(L,4)*(RIll*P(L,2)+RI12*P(L,5))+A C(L,1)*(RI11*P(L,3)+RI12*P(L,6)) Q(1,M2)=A(I,2) * ( R i l l * (P(I,3)-P(I,2) )+RI12* (P (1, 6 )-P (1, 5 ) ) )+A (L, 2 ) * C(Rill*(P(L,2)-P(L,3))+RI12*(P(L,5)-P(L,6))) Q(1,M3)=-A(I,3)*(Rill*(2.DO*P(1,2)+P(1,3))+RI12*(2.DO*P(I,5)+P (1,6 C)))+A(L,3)*(Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.DO*P(L,6)+P(L,5))) Q(2,M1)=A(I,2) * (RI21* (P (1, 3 ) -P (1, 2 ) )+RI22* (P (1, 6 ) -P (1, 5 ) ) )+A (L, 2) * C(RI21*(P(L,2)-P(L,3))+Rl22*(P(L,5)-P(L,6))) Q (2,M2)=RI21*P(1,1)+RI22*P(1,4)+A(I,5)*(RI21*P(1,2)+RI22*P(1,5))+A C (1,7)*(RI21*P(I,3)+RI22*P(1,6))+A(L,7)*(RI21*P(L,2)+RI22*P(L,5))+A C(L,5)*(RI21*P(L,3)+RI22*P(L,6)) Q(2,M3)=A(I,6)*(RI21*(2.D0*P(1,2)+P(I,3) )+RI22 *(2.DO*P(1,5)+P (1,6) C))-A(L,6)*(RI21*(2.D0*P(L,3)+P(L,2))+RI2 2*(2.D0*P(L,6)+P(L,5))) DO 17 II=M1,M3 Q(3,II)=R(1,II)*(l.DO-(S+T-S*T/CT)/CT)/4.D0+R(2,II)*(1.D0+(S-T-S*T C/CT)/CT)/4.D0+R(3,II)*(1.D0+(S+T+S*T/CT)/CT)/4.DO+R(4,II)* (l.DO-(S C-T+S*T/CT)/CT)/4.DO 17 CONTINUE GO TO 5 10 Q(1,M2)=A(I,2)*(RIll*(P(1,3)-P(1,2))+RI12*(P(I,6)-P(I,5)))+A(L,2)* C(RI11*(P(L,2)-P(L,3))+RI12*(P(L,5)-P(L,6))) Q (2,M1)=A(I,2) * (RI21* (P (1, 3 )-P (1, 2 ) ) +RI22* (P (1, 6 )-P (1, 5) ) )+A (L, 2 ) * C(RI21*(P(L,2)-P(L,3))+Rl2 2*(P(L,5)-P(L,6))) R(N,M1)=RI21*P(I,1)+RI22*P(I,4)+A(I,1)*(RI21*P(I,2)+RI2 2*P(1,5))+A 00 C(1,4)*(RI21*P(I,3)+RI22*P(1,6))+A(L,4)*(RI21*P(L,2)+RI22*P(L,5))+A C(L,1)*(RI21*P(L,3)+RI22*P(L,6))+Q(1,M2) R(N,M2)=RI11*P(I,1)+RI12*P(1,4)+A(1,5)*(RI11*P(I,2)+RI12*P(1,5))+A C (1,7)*(RI11*P(1,3)+RI12*P(1,6))+A(L,7)*(RI11*P(L,2)+RI12*P(L,5))+A C(L,5)*(RI11*P(L,3)+RI12*P(L,6))+Q(2,Ml) R(N,M3)=-A(I,3)*(RI21*(2.DO*P(I,2)+P(I,3))+RI22*(2.DO*P(I,5)+P(I,6 C)))+A(I,6)*(Rill*(2.D0*P(I,2)+P(1,3))+RI12*(2.D0*P(1,5)+P(1,6)))+A C(L,3)*(RI21*(2.D0*P(L,3)+P(L,2))+RI22*(2.DO*P(L,6)+P(L,5)))-A(L,6) C* (Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.DO*P(L,6)+P(L,5))) 5 CONTINUE GO TO(12,13,14,15,16) ,N 16 CALL MAMULT(Q,DEF/SPAD,3,12,1,3,12,3,0,0) DO 110 1=1,3 110 EPS(K,I)=SPAD(I) 15 0 CONTINUE RETURN END O MAMULT - MATRIX MULTIPLICATION SUBROUTINE MAMULT(A,B,C,L,M,N,NDIMA,NDIMB,NDIMC,NAT,NBT) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(NDIMA,1),B(NDIMB),C(NDIMC,1) DO 1 1=1,L DO 2 J=1,N C (I,J)=0.DO DO 3 K=1,M IF(NAT)4,5,4 IF (NBT)6,7,6 C (I, J) =C (I, J) +A (I,K) *B (K, J) GO TO 3 C(I,J)=C(I,J)+A(I,K) *B(J,K) GO TO 3 IF (NBT)8,9,8 C (I, J)=C (I, J)+A(K,I) *B (K, J) GO TO 3 C (I, J)=C(I,J)+A(K,I) *B(J,K) CONTINUE CONTINUE CONTINUE RETURN END DIMULT - MULTIPLICATION OF SQUARE SYMMETRIC MATRIX BY A RECTANGULAR MATRIX WHERE UPPER TRIANGLE OF SYMMETRIC MATRIX IS STORED AS VECTOR SUBROUTINE DIMULT(A,B,C,N,M,NDIMB,NDIMC,NBT) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(l),B(NDIMB,1),C(NDIMC,1) DO 1 1=1,N DO 2 J=1,M C (I, J) =0. DO 11=1 DO 3 K=1,N IF (NBT)7,6,7 C(I,J)=C(I,J)+A(II) *B(K,J) GO TO 8 C(I,J)=C(I,J) + A (II) *B(J,K) IF (I-K)4,4,5 II=II+N-K GO TO 3 11=11+1 CONTINUE CONTINUE CONTINUE RETURN END C REMULT - MULTIPLICATION OF TWO RECTANGULAR MATRICES WHERE THE PRODUCT IS C KNOWN TO BE A SYMMETRIC SQUARE MATRIX STORED AS A COLUMN VECTOR SUBROUTINE REMULT(A,B,C,M,N,NDIMA,NDIMB,NAT,NBT,NC) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(NDIMA,1),B(NDIMB,1),C(1) 11=0 DO 1 1=1,M DO 2 J=I,M 11=11+1 C (II) = 0.DO-DO 3 K=1,N IF (NAT)4,5,4 5 IF (NBT)6,7,6 7 C(II)=C(II)+A(I,K)*B(K,J) GO TO 3 6 C(II)=C(II)+A(I,K)*B(J,K) GO TO 3 4 IF (NBT)8,9,8 9 C(II)=C(II)+A(K,I)*B(K,J) GO TO 3 8 C(II)=C(II)+A(K,I)*B(J,K) 3 CONTINUE 2 CONTINUE 1 CONTINUE NC=II RETURN END A.5 Performance of Elements i n Selected Problems Both the triangular and q u a d r i l a t e r a l elements are used i n three d i f f e r e n t problems. A.5.1 Cantilever with Parabolic End Load: The dimensions of the cant i l e v e r and the d i f f e r e n t grids used are shown i n F i g . A.7. In Fi g . A.8 the s t r a i n energy convergence toward the exact value with progressive grid refinement i s compared with the well-known constant and li n e a r s t r a i n t r i a n g l e s . Note the non-monotonic convergence of the q u a d r i l a t e r a l s , which i s at t r i b u t a b l e to the lower order integration pre-viously described. A higher order integration scheme ensuring exact integration of a l l terms i n the integrand matrix r e s u l -ted i n monotonic convergence for t h i s problem. Some of the calculated stresses i n the cantilever are compared with engineering beam theory i n F i g . A.9. A.5.2 P a r a b o l i c a l l y Loaded Square Plate: This problem has become a popular t e s t of the c a p a b i l i t i e s of a plane stress element. The dimensions of the plate and the d i f f e r e n t grids used are shown i n F i g . A.10, while s t r a i n energy convergence i s again compared with the constant and li n e a r s t r a i n t r i -angles i n F i g . A.11. A comparison of stresses and deflec -tions with exact values i s tabulated in^Table IX. A.5.3 Short Deep Beam: This problem was included because, unlike the previous two, a r e l a t i v e l y large proportion of the t o t a l s t r a i n energy comes from shear. Bearing i n mind the pro-195 blems that were encountered with the constrained cubic elements i n pure shear, i t was considered that the short deep beam would provide a rigorous test of these elements. The beam dimensions and f i n i t e element grids are shown i n Fi g . A.12. Unfortunately an exact solution to t h i s problem does not e x i s t , and although v a r i a t i o n of s t r a i n energy with g r i d size i s compared with the constant and l i n e a r s t r a i n t r i a n g l e s i n F i g . A.13, there i s no i n d i c a t i o n of how close-l y the s t r a i n energy approaches the exact value. Calculated stresses compare favourably with an approximate series solu-t i o n by von Kantian1*1* i n F i g . A. 14. 196 Y Thickness = 1 in. E = 30x10 k.s.i. V=0.25 c C M • 1 -ag 48in. Support nodes 40 k ip. x = y = f i x e d ro ta t ion = f r e e 1 x 4 gr id 2 x 8 g r id \ / / \ \ I / / 4 x 16 g r id Fig.A.7 C a n t i l e v e r with parabol ic end load. 110 + 80 70 + 60 50 A C o n s t r a i n e d cubic t r i a n g l e 0 L i nea r s t r a i n t r i a n g l e © C o n s t a n t s t r a i n t r i a n g l e • Cons t ra ined cubic quadr i l a te ra l j 1x4 2x8 4x16 8x32 G r i d s i z e . Fig.A.8 S t ra in energy convergence for c an t i l e ve r . CD C I CL> i _ c Q) U r -c CD CD > o cd <u (j c nd LO Q 6 4 3 + 0 A I n t e r po l a t ed f r o m t r i a n g l e c e n t r o i d s • A v e r a g e d at q u a d r i l a t e r a l n o d e s I n t e r p o l a t e d f r o m q u a d r i l a t e r a l c e n t r o i d s B e a m theory -A1 H 1-0 20 40 X- s t r e s s (k.s.i.) 60 Beam theory 4- + + -10 0 10 Y- s t r e s s (k.s.i.) Fig. A.9 Stresses in c an t i l e ve r at cross -sec t ion 12in. f rom support for 4 x 1 6 grid. -As l — h 0 1 2 3 4 5 6 Shea r s t r e s s (k.s.i.) F ig . A.10 One quadrant of pa rabo l i ca l l y loaded square plate. 100 90-88 + 1x1 j3 A Const ra ined cubic t r i ang le . 0 L i nea r s t r a i n t r i ang le . 0 C o n s t a n t s t r a i n t r i a n g l e . • C o n s t r a i n e d cub i c q u a d r i l a t e r a l 2x2 4x4 8x8 Gr id s i ze . Strain energy convergence for parabo l i ca l l y loaded square plat TABLE IX COMPARISON OF STRESSES AND DEFLECTIONS  FOR PARABOLICALLY LOADED SQUARE PLATE Refer to Fi g . A.10 for Key Element Grid in. u c in. V c in. V d in. 0 xa k.s.i. 0 ya k.s.i. xb k.s.i. yb k.s.i. a xc k.s.i. U kip. in. RECTANGLE l x l 2x2 4x4 8x8 -.004521 -.004382 -.004339 -.004328 .001373 .001154 .001152 .001162 .005077 .004452 .004233 .004162 .015849 .016273 .016381 .016407 -1.79 -4.81 -6.81 -7.34 47.10 47.36 46.23 45.92 -10.41 - 5.00 - 1.79 - 0.60 12.63 18.57 20.89 21.61 8.45 1.58 -1.01 -1.31 7.1088 7.2055 7.2239 7.2292 TRIANGLE Lxl 2x2 4x4 8x8 -.003272 -.004087 -.004087 -.004086 .000932 .001103 .001386 .001527 .006353 .005052 .004574 .004367 .014668 .016211 .016463 .016522 -6.41 -8.21 -8.73 43.19 44.88 45.45 - 1.41 -0.92 -0.63 28.19 24.83 23.35 0.84 0.61 0.20 6.9910 7.2655 7.2859 7.2982 EXACT -.005066 .000595 .004258 .016912 -7.518 45.816 0 21.902 0 7.4495 202 2 00 kip. Fig. A.12 One half of short deep beam. 6.0 5.0 2.0 1.0 ^ ^ ^ ^ ^ ^ \^^^ c c i A C o n s t r a i n e d c :ub ic t r i a n g l e ( 0 L i nea r s t r a i n t r i a n g l e © C o n s t a n t s t r a i n t r i ang le • Cons t ra ined cubic q u a d r i l a t e r a l l I 1 j. 1x1 2x2 UxU 8x8 Grid s i ze . Fig. A.13 Strain energy versus grid s i ze for short deep beam. o A I n t e r p o l a t e d f rom t r i a n g l e c e n t r o i d s . • Ave raged at q u a d r i l a t e r a l nodes. ® I n t e r p o l a t e d f rom q u a d r i l a t e r a l c e n t r o i d s . f - H 1 1 1 1 1 1 1— —! 1 1 — — I 1 1 {--30 -20 -10 0 10 20 30 40 -10 -5 0 0 5 10 15 X - s t r e s s (k.s.i.) Y - s t r e s s (k.s.i.) S h e a r s t r e s s (k.s.i.) F ig .A.K Stresses in short deep beam at cross-section 3 in. from midspan for 8x8 grid. Results are compared with an approximate series solut ion. APPENDIX B LOAD-DEFLECTION CURVE GENERATOR FOR HEADED STUD CONNECTIONS B.l L i s t of Variables i n Subroutine Input/Output L i s t DELTA PR(I,J) NDIMPR I RL R1MAX R2MAX Shear d e f l e c t i o n increment applied to connec-t i o n I Matrix which stores load-deflection and mater-i a l property history of connection I. I dimension = number of connections i n problem or greater. J dimension = 18 The f i r s t dimension of PR(I,J) Connection number. Stud bending length as calculated i n Chapter 7 Total ultimate force i n concrete under studs a calculated i n Chapter 7. Ultimate end-bearing resistance of faceplate a calculated i n Chapter 7. 206 XC(J),YC(J) Three X and Y coordinates of t r i l i n e a r t o t a l load-deflection curve, for a l l studs i n a con-nection, over bending length RL, defined i n Fi g . 5.7. P Shear load on connection aft e r a p p l i c a t i o n of de f l e c t i o n increment DELTA. D Shear d e f l e c t i o n of connection a f t e r applica-t i o n of d e f l e c t i o n increment DELTA. IFNEW =-ve,0, elements i n PR(I,J) are not reset, allowing for another t r i a l on the next i t e r a -t i o n . =+ve, elements i n PR(I,J) are reset. Rl That portion of P r e s i s t e d by concrete under studs. R2 That portion of P re s i s t e d by end-bearing of faceplate on concrete. R3 That portion of P r e s i s t e d by studs i n bending. B. 2 Notes B.2.1 The shear load Po on a connection I, before a p p l i -cation of d e f l e c t i o n increment DELTA, i s stored i n PR(I,D • The shear d e f l e c t i o n D0 of a connection I, before application of d e f l e c t i o n increment DELTA, i s stored in PR(I, 2) . B. 2. 2 The cumulative rotation of the studs i n a connec-t i o n , equation may be used as (4.2), i s stored i n PR(I,16) and an indicator of stud fracture. 207 B.2.3 A l l elements of PR(I,J) must be set to zero before subroutine STUDCO i s f i r s t c a l l e d . B.2.4 The subroutine makes use of the concrete degrada-t i o n model of F i g . 5.3 and the t r i l i n e a r hystere-s i s model of Fig . 5.7. B.2.5 A l l input loads may be i n any unit of measurement desired, provided that the same unit i s used con-s i s t e n t l y throughout. The unit for deflections i s the inch, but may be changed to any other unit by changing the DATA l i n e i n STUDCO. The variables D10, D20 and D2MAX i n t h i s l i n e are s p e c i f i e d i n inch units and should be m u l t i p l i e d by the appro-pri a t e factor for consistency with the new unit of length measurement. B. 3 Subroutine STUDCO A l i s t i n g of the subroutine appears on the pages that follow. STUDCO - LOAD-DEFLECTION CURVE GENERATOR FOR HEADED STUD CONNECTIONS SUBROUTINE STUDCO(DELTA,PR,NDIMPR,I,RL,RlMAX,R2MAX,XC,YC,P,D,IFNEW C,R1,R2,R3) IMPLICIT REAL*8(A-H,0-Z) DIMENSION PR(NDIMPR,1),XC(3),YC(3) DATA DlO/0.5D-01/,D20/0.25D-01/,D2MAX/0.9D-01/ DTHETA=DELTA/RL DD2=RL*DTHETA DD3=DD2 THETA=PR(I,2)/RL D2=RL*THETA D3 = D2 D=PR(I,2)+DELTA PR(I,16)=PR(1,16)+DABS(DELTA/RL) IF(.1D0-DABS(D))60,60,61 FACT=2.D0 GO TO 62 FACT=.1D+01+.3D+03*D**2-.2D+04*DABS(D**3)) D1=PR(I,15) DDl=D/FACT-Dl IF(R1MAX-.1D-05)90,90,91 R1=0.D0 GO TO 5 IF(D1+DD1)1,2,2 IF(PR(I,3) )3,4,4 F0=0.DO GO TO 7 0 F0=PR(1,3)/RlMAX S0=D1/D10 IF(DABS(S0)-.lD-05)80,8 0,81 S0=.1D-05 DELS=DD1/D10 SE=PR(I,4) SEHAT=PR(I,5) J=l CALL STRESN(F0,S0,DELS,0.DO,SE,SEHAT,J,F,S,DFDS) IF(IFNEW)8,8,9 9 PR(I,3)=F*R1MAX PR(I,4)=SE PR(I,5)=SEHAT PR(I,15)=S*D10 8 R1=F*R1MAX GO TO 5 1 IF(PR(I,3))6,6,7 7 F0=0.D0 GO TO 71 6 FO-PR (1, 3) /RlMAX 71 S0=-D1/D10 IF(DABS(SO)-.1D-05)82,82,83 82 S0=.lD-05 83 DELS=-DD1/D10 SE=PR(I,6) SEHAT=PR(1,7) J=l CALL STRESN(FO,S0,DELS,0.DO,SE,SEHAT,J,F,S,DFDS) IF(IFNEW)10,10,11 11 PR(I,3)=-F*R1MAX PR(I,6)=SE PR (1,7)=SEHAT PR(I,15)=-S*D10 10 Rl=-F*RlMAX 5 IF(R2MAX-.1D-05)92,92,93 92 R2=0.D0 GO TO 25 93 IF(D2+DD2)21,22,22 22 IF(PR(I,8))23,24,24 23 F0=0.D0 GO TO 72 2 4 F0=PR(I,8)/R2MAX 72 S0=D2/D20 IF(DABS(S0)-.lD-05)84,84,85 84 S0=.lD-05 o >^ 3 85 DELS=DD2/D2 0 SE=PR(If 9) SEHAT=PR(I,10) J=l CALL STRESN(FO,SO,DELS,O.DO, IF(S-D2MAX/D20)51,50,50 50 F=0.D0 SE=10.D0 51 IF (IFNEW)28,28,29 2 9 PR(I,8)=F*R2MAX PR (I, 9)=SE PR(I,10)=SEHAT 2 8 R2=F*R2MAX GO TO 25 21 IF(PR(I,8))26,26,27 27 F0=0.D0 GO TO 7 3 26 F0=-PR(I/8)/R2MAX 73 S0=-D2/D20 IF(DABS(SO)-.1D-05)86,86,87 86 S0=.lD-05 87 DELS=-DD2/D20 SE=PR(I,11) SEHAT=PR(I,12) J=l CALL STRESN(F0,S0,DELS,0.D0, IF (S-D2MAX/D20) 53,52,52 52 F=0.D0 SE=10.DO 53 IF(IFNEW)30,30,31 31 PR(I,8)=-F*R2MAX PR(I,11)=SE PR(I,12)=SEHAT 30 R2=-F*R2MAX 25 Y0=PR(I,13) XP=PR(I,14) SE,SEHAT,J,F,S,DFDS) SE,SEHAT,J,F,S,DFDS) CALL TRILIN(D3,Y0,DD3,XP,XC,YC,X,Y,DYDX) IF(IFNEW)41,41,42 42 PR(I,13)=Y PR(I,14)=XP 41 R3=Y DO=XC(2) CALL DEGRAD(PR,NDIMPR,R3,D,DO,I,IFNEW,FACT) P=R1+R2+R3*FACT IF(IFNEW)43,43,44 44 PR(I,1)=P PR(I,2)=D 4 3 RETURN END r—1 STRESN - STRESS-STRAIN DEGRADATION MODEL FOR CONCRETE SUBROUTINE STRESN(FO,SO,DELS,FTENS,SE,SEHAT,IFCRAK, F , S ,DFDS) IMPLICIT REAL*8(A-H,0-Z) S=S0+DELS IF(4.D0-S)l f 1,2 SE=5.D0 F=0.D0 DFDS=0.D0 GO TO 3 IF(4.D0-SE)4,4,5 IF(DELS)6,7, 7 ** LOADING CURVE **** IF(DABS(FE(SO)-F0)-l.D -6)8,8,9 F=FE(S) DFDS=DFDSl (S) SE=S GO TO 3 IF(SP(SE)-S)10,10,11 IF(F0)12,4,4 IF(SR(SE)-S)8,8,13 IF(SC(SE)-S)14,15,15 F=(SO-SP(SE))*DFDS2(SE) IF (FO-F)16,12,12 IF(SE-.250+00)12,17,17 SI= (FO+SP(SE)*DFDS2(SE)-S0*DFDS3(SE))/(DFDS2(SE)-DFDS3(SE)) IF(S-SI)18,12,12 F=F0+(S-SO)*DFDS3(SE) DFDS=DFDS3(SE) GO TO 3 F=(S-SP(SE))*DFDS2(SE) DFDS=DFDS2(SE) GO TO 3 IF(FO-FC(SC(SE)))35,35,36 F=F0+(S-SO)*DFDS4(SE) IF(F-FE(S)) 37 ,37,8 F=FC(SC(SE)) + (S-SC(SE) ) *DFDS4 (SE) 37 DFDS=DFDS4(SE) IF(SE-.25D0)3,3,19 19 SEHAT=(F-FE(SE)+SE*DFDS1(SE)-S*DFDS3(SE))/(DFDS1(SE)-DFDS3(SE)) GO TO 3 C **** UNLOADING CURVE **** 6 IF(SEHAT-SE)31,31,32 32 SE=SEHAT 31 IF(S-SP(SE))20,21,21 . 20 IF(IFCRAK)23,22,23 23 F=0.D0 DFDS=0.DO GO TO 3 22 F=(S-SP(SE))*DFDS2(SE) IF(F-FTENS)2 4,24,25 2 4 IFCRAK=1 GO TO 23 25 DFDS=DFDS2(SE) GO TO 3 21 F=FT(ST(SE))+(SO-ST(SE))*DFDS2(SE) IF (FO-F)26,26,27 26 F=FT(ST(SE))+(S-ST(SE))*DFDS2(SE) DFDS=DFDS2(SE) IF(F)28,28,3 28 F=0.D0 DFDS=0.DO GO TO 3 27 IF (SE-.2500)26,26,29 29 SI= (FO-FT (ST (SE) )+ST(SE) *DFDS2 (SE) -S.0*DFDS3 (SE) )/ (DFDS2 (SE) -DFDS3 ( CSE) ) IF (S-SI) 26,26, 30 30 F=F0+(S-SO)*DFDS3(SE) DFDS=DFDS3(SE) 3 RETURN END FUNCTION FE(A) IMPLICIT REAL*8(A-H,0-Z) FE=A*(2.7182 8182 85D+00**(l.DO-A)) RETURN END FUNCTION DFDS1(A) IMPLICIT REAL*8(A-H,0-Z) DFDS1=(l.DO-A)*(2.7182 818285D+00**(l.DO-A)) RETURN END FUNCTION SP(A) IMPLICIT REAL*8(A-H,0-Z) SP=(0.145D+00*A*A)+(0.13D+00*A) RETURN END FUNCTION SR(A) IMPLICIT REAL* 8(A-H,0-Z) SR=(-.091D+00+DSQRT(.8281D-02+(.372D+00*SP(A))))/.186D+00 RETURN END FUNCTION SC(A) IMPLICIT REAL*8(A-H,0-Z) SC=(-0.141D+00+DSQRT(.19881D-01+(.68D+00*SP(A))))/.3 4D.00 RETURN END FUNCTION DFDS2(A) IMPLICIT REAL*8(A-H,0-Z) IF (A-.1D-01)1,2,2 1 DFDS2=2.7182818285D+00 2 3 4 5 GO TO 5 IF(A-.25D+00)3,4,4 DFDS2=SLOPE(A) GO TO 5 DFDS2=FC(SC(A))/(SC(A)-SP(A)) RETURN END FUNCTION FC(A) IMPLICIT REAL*8(A-H,0-Z) i — ' FC=A*(2. 7182818285D+00**(1.DO-1.17D0*A)) RETURN END FUNCTION SLOPE(A) IMPLICIT REAL*8(A-H,0-Z) SLOPE=FE(A)/(A-SP(A)) RETURN END FUNCTION DFDS3(A) IMPLICIT REAL*8(A-H,0-Z) IF(A-.25D+00)1,2,2 DFDS3=DFDS2(A) GO TO 3 DFDS3=(FE(A)-FC(SC(A)))/(A-SC(A)) RETURN END FUNCTION DFDS4(A) IMPLICIT REAL*8(A-H,0-Z) IF(A-.05D+00)1,2,2 DFDS4=DFDS2(A) GO TO 3 DFDS4= (FE (SR (A) ) -FC (SC (A) ) ) / (SR (A) -SC (A) ) RETURN END FUNCTION ST(A) IMPLICIT REAL*8(A-H,0-Z) Sl= (-.218D+00+DSQRT(. 47 52 4D-01+(.64D0*SP(A))))/.32D+00 DFDS=.675D0*(1.DO-1.17D0*A)*2.7182818285D0**(1.DO-1.17D0*A) ST=(FT(SI)-FE(A)+A*DFDS3(A)-S1*DFDS)/(DFDS3(A)-DFDS) RETURN END FUNCTION FT(A) IMPLICIT REAL*8(A-H,0-Z) FT=.675D+00*A*(2.7182818285D+00**(1.DO-1.17D0*A)) RETURN END C TRILIN - TRILINEAR HYSTERESIS LOOP GENERATOR SUBROUTINE T R I L I N ( X O , Y O , D X , X P , X C , Y C , X , Y , D Y D X ) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X C ( 3 ) , Y C ( 3 ) X=X0+DX I F ( X C ( 3 ) - D A B S ( X P ) ) 9 , 9 , 1 0 10 I F ( X C ( 3 ) - D A B S ( X ) ) 9 , 9 , 1 1 11 I F ( X P - X ) 1 , 1 , 2 2 IFBEL=-1 XP=-XP X=-X DX=-DX X0=-X0 Y0=-Y0 GO TO 3 1 IFBEL=1 3 I F ( D X ) 4 , 5 , 5 5 IF ( X Q ( X C , Y C , X P ) - X ) 6 , 6 , 7 6 Y=Y3(XC,YC,X) DYDX=DFDX3(XC,YC) GO TO 8 7 IF ( X T ( X C , Y C , X P ) - X ) 1 2 , 1 3 , 1 3 12 Y=Y2(XC,YC,XP,X) DYDX=DFDX2(XC,YC) GO TO 8 4 XP=XPNEW(XC,YC,X0,Y0) 13 Y=Y1(XC,YC,XP,X) DYDX=DFDX1(XC,YC) .8 I F ( I F B E L ) 1 4 , 1 5 , 1 5 14 XP=-XP Y=-Y X=-X DX=-DX X0=-X0 Y0=-Y0 GO TO 15 NJ r-1 9 XP=2*XC (3) Y=O.DO DYDX=O.DO 15 RETURN END FUNCTION DFDX1(X,Y) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) DFDX1=Y(1)/X(1) RETURN END FUNCTION DFDX2(X,Y) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) DFDX2= (Y(l)-Y(2) )/(X(l)-X(2) ) RETURN END FUNCTION DFDX3(X,Y) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X (3) ,Y (3) DFDX3=(Y(2)-Y(3))/(X(2)-X(3)) RETURN END FUNCTION XT(X,Y,XP) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) XT=(Y(1)+DFDX1(X,Y)*XP-DFDX3(X,Y)*X(1))/(DFDX1(X,Y)-DFDX3(X,Y)) RETURN END FUNCTION XQ(X,Y,XP) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) XQ=(Y(2)-Yl(X,Y,XP,XT(X,Y,XP) )+DFDX2(X,Y)*XT(X,Y,XP)-DFDX3(X,Y)*X ( C2) ) / (DFDX2 (X, Y) -DFDX3 (X, Y) ) RETURN END NJ FUNCTION Yl(X,Y,XP,XX) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) Y1=DFDX1(X,Y)*(XX-XP) RETURN END FUNCTION Y2(X,Y,XP,XX) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) XOXT (X,Y,XP) Y2=Y1(X,Y,XP,X0)+DFDX2(X,Y)*(XX-XO) RETURN END FUNCTION Y3(X,Y,XX) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) Y3=Y(2)+DFDX3(X,Y)*(XX-X(2) ) RETURN END FUNCTION XPNEW(X,Y,XN,YN) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3) ,Y(3) XPNEW=XN-YN/DFDX1(X,Y) RETURN END OO C DEGRAD - CONTROLS DEGRADATION OF STUD LOAD-DEFLECTION LOOPS SUBROUTINE DEGRAD(PR,NDIMPR,R3,D,DO,I,IFNEW,FACT) IMPLICIT REAL*8(A-H,0-Z) DIMENSION PR(NDIMPR,1) IF(R3)1,2,3 1 IF(PR(I,1) )4,4,2 3 IF(PR(I,1))2,4,4 2 IF(DABS(R3-PR(I,1))-.1D-10)7,7,8 7 CROSS=D GO TO 9 8 CROSS=D+(R3/DABS(R3-PR(I,1)))*(PR(I,2)-D) 9 DIST=DABS(PR(1,17)-CROSS)/2.DO DEL=PR(I,18)+(.822D+01*(DIST/DO)+0.935D+02*(DIST/DO)**2)*0.5D-03 FACT=1.DO-DEL IF(IFNEW)5,5,6 6 PR(I,17)=CROSS PR(I,18)=DEL GO TO 5 4 FACT=1.D0-PR(I,18) 5 RETURN END APPENDIX C CONTROL OF AXIAL LOAD AND MOMENT INTERACTION FOR STUDS IN ITERATIVE CALCULATIONS Suppose that at some cross-section i n a stud the value of a x i a l load i s P', and the bending moment M', at a p a r t i c u l a r stage of a ca l c u l a t i o n . The next increment in the c a l c u l a t i o n , using tangent s t i f f n e s s e s from the previous increment, y i e l d s approximate increments of AP1 and AM' i n P' and M1 respectively. This r e s u l t s i n a point on the in t e r a c t i o n diagram of Fig . C.l with coordinates M1+AM1 P'+AP1 M , P l V > The slope, e, of the l i n e AB i n F i g . C.l i s P'+AP1 e -- p l (C.l) M1+AM1 M P l Any adjustments to AP' and AM' w i l l be carri e d out along l i n e AB. F i g . C . l i s the same in t e r a c t i o n diagram as Fig . 5.9 which i s described by equations (5.12) and (5.13). 221 By substituting e - J2i i n equation (5.12) M M P i i t can be shown that P i l+4e 2e 1 (C2) = A+4e 2 - 1 (C.3) V 2 e 2 The distance i n F i g . C .l i s given by From equations (C.2) and (C.3) r = e 'l+4e - 1 2e 2 r 2^ 1+e 2 I e J (C.4) Using a s i m i l a r s u b s t i t u t i o n of e i n equation (5.13) p l 1+e (C.5) M M P l 1+e (C.6) whence r -i 1+e' (1+e) (C.7) 222 The distance r in Fig. C.l i s given by r = P'+AP1 p i J + M'+AM1 (C.8) Let F p and F m be scaling factors with values between P l and 1.0 for F , and between M M p l and 1.0 for F . The m purpose of these scaling factors w i l l become clear shortly, (i) For any point which l i e s below the st r a i g h t i n t e r a c -t i o n l i n e , 0 £ r £ r , F = F = 1.0 p m ( i i ) For any point which l i e s outside of the curved i n t e r -action l i n e , r ^ r , e' F = P P l F = m M M P l ( i i i ) For any point which l i e s between the st r a i g h t and curved i n t e r a c t i o n l i n e s , r < r < r , i e r - r r - r e i 1.0-between 1.0 and 'Pl P P l which i s a l i n e a r i n t e r p o l a t i o n S i m i l a r l y F. m r - r r - r e i 1.0- M M P l The scaling factors F and F are applied to the ^ p m ^ s t r e s s - s t r a i n and moment-curvature diagrams respectively, for the cross section under consideration. This forces the diagrams to conform to the i n t e r a c t i o n curve as y i e l d -ing commences. Consider, for example, the moment-curvature diagram of F i g . C.2, which i s self-explanatory. 224 C u r v a t u r e Fig. C.2 Moment-curvature diagram. 

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